PERSPECTIVES ON SCHOOL ALGEBRA
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PERSPECTIVES ON SCHOOL ALGEBRA
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TABLE OF CONTENTS Introduction M)+"94#' ;4(,/*2"#'
vii
Approaches to Algebra M)942) E"9D)+ N%#+1 C/*/+" M)O"#)1 52"# 6/22 "#'@M)+"94#'@;4(,/*2"#'
1
The Historical Origins of Algebraic Thinking N4%+ 8: M"'7)*'
13
The Production of Meaning for 52$/3*"P a Perspective Based on a Theoretical Model of Semantic Fields M)942) E"9D)+ N%#+
37
A Model for Analysing Algebraic Processes of Thinking A/*'%#"#') 5*H"*/22)1 N4B%"#" 6"HH%#% "#' 8%"9D")2) E,%"DD%#%
61
The Structural Algebra Option Revisited L".%' <%*+,#/*
83
Transformation and Anticipation as Key Processes in Algebraic Problem Solving ?")2) 6)/*)
99
Historical-Epistemological Analysis in Mathematics Education: Two Works in Didactics of Algebra 54*)*" 8"22"*')
121
Curriculum Reform and Approaches to Algebra <"0/ ;("B/0 "#' !)22%/ !"B8*/$)*
141
v
vi
INDEX
Propositions Concerning the Resolution of Arithmetical-Algebraic Problems &4$/#%) A%22)01 C/*/+" M)O"#) "#' 84%22/*9) M43%)
155
Beyond Unknowns and Variables - Parameters and Dummy Variables in High School Algebra Q"." 62)/'0I R%##/*
177
From Arithmetic to Algebraic Thinking by Using a Spreadsheet 8%42%"#" L/(()*%1 M)++/22" 8"*4(% "#' &#*%B" N/94(
191
General Methods: a Way of Entering the World of Algebra ;)#%" -*+%#%
209
Reflections on the Role of the Computer in the Development of Algebraic Thinking N424 Q/"201 ;(/7"#) ?)HH% "#' M)+"94#' ;4(,/*2"#'
23 1
Symbolic Arithmetic vs Algebra the Core of a Didactical Dilemma. Postscript J%B)2"+ 6"2"B,/77
249
References
26 1
Index
273
INTRODUCTION
"A student who knows only arithmetic is quite right to say that 5 – 8 is impossible, but this impossibility is a gateway that leads to new knowledge, a stile that has to be got over, leading into a meadow in which fairer flowers grow than in the filed of arithmetic that is left behind, a passage through the looking-glass into a fairyland". Hudson, 1888, p.134 st
At the beginning of the 21 century school algebra remains a central concern of mathematics education around the world. Algebra is a powerful and succinct language for expressing and transforming mathematical ideas. Algebra is arguably an entry point into the virtual world of mathematics. This book results from a collective effort to understand and write about the complex issues which surround teaching and learning algebra, an effort that took several years to come to fruition. Throughout the process we became aware that school algebra is not an invariant across countries and this is reflected in the diverse viewpoints, which are represented in the book. This book provides new perspectives on school algebra at a time when little is known about how digital technologies will impact on education around the world. Thinking about the unknown is central to algebra and with this as a metaphor I hope that this book will enable you to move forward within the territory of algebra education. Hopefully as a reader you will be provoked to re-question your own perspectives on school algebra in order to re-think and re-energise your ongoing work. A working group of the International Group for the Psychology of Mathematics Education (PME) was the genesis of this book and PME continues to provide a forum for working on the issues which surround effective teaching and learning of algebra. Rosamund Sutherland University of Bristol May 2000
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ROMULO LINS1, TERESA ROJANO², ALAN BELL³ AND ROSAMUND SUTHERLAND4
APPROACHES TO ALGEBRA
1
In July 1991 the PME working group 'Algebraic Processes and Structure' met in Italy for the first time. Our aim was to characterise the shifts that appear to be involved in developing an algebraic mode of thinking, with a particular focus on the role of symbolising in this development. For the next four years we met in the United States, Portugal. Japan and Brazil and discussed papers which had been circulated before these meetings. The gestation period for this book has been a long and productive one, resulting from ongoing discussion by members of a group who. although often disagreeing. were willing to share and debate their ideas. In this way we believe that we have developed our own thinking about teaching and learning algebra, as well as about the nature of algebraic thinking, although the earlier algebra research formed an important background to much of the research represented here (see for example Kieran, 1990). In all chapters we find a concern with identifying the characteristics of what could be called an algebraic approach to solving a problem, together with a focus on the meanings the students construct/produce as they engage with mathematical problems, and not on the problems or the students' conceptualisations in isolation from the problem solving activity. Within this picture. however, we find two distinct trends. In one group of chapters the main concern is related to what has been previously called 'informal' or 'spontaneous' approaches/meanings, and how those take part in the development/acquisition of an algebraic approach/algebraic meanings, whereas in another group the main concern is related to what an algebraic approach is in itself. The former could be identified as a '%'"B(%B"2 trend, while the latter could be identified as " 7)4#'"(%)#"2%+( or (,/)*/(%B"2 trend. There is not, of course, a sharp divide here, as all the 'didactical' chapters deal, in one way or another, with some foundational sometimes in the form of implicit or explicit assumptions, and all ‘theoissues —
1
Mathematics Dept., UNESP, Rio Claro, Brazil. Centro de Investigación y de Estudios Avanzados-IPN, Mexico. 3 School of Education,University of Nottingham, Nottingham, U.K. 4 Graduate School of Education, University of Bristol, U.K. 2
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retical’ chapters deal, in one way or another, with issues about the teaching and learning of algebra. Nevertheless, these are useful categories which characterise the work presented here. The interplay between these two trends was always a very rich one within the group, as it represented a balanced effort to: (a) clarify and deepen our understanding of the processes involved in algebraic and non-algebraic thinking; and, (b) to make sure that this improved interest was always informed by and linked to our educational endeavour. Those more closely concerned with teaching and learning in the classroom were faced with questions beyond the effectiveness of teaching approaches, while those more closely concerned with theoretical issues were faced with relevant and informing input coming from work with students; we had the exciting opportunity to work within the environment of a ‘full table’ with each didactical approach being confronted with different theoretical approaches, and with each theoretical approach being confronted with different didactical ones. In any research concerned with algebraic education we should expect to find these two components, but our group had the opportunity to emulate them on a larger scale. In this sense we feel that this book should be read as a whole heterogeneous as it may appear to be -and not as a collection of individual contributions. Another useful way of characterising the chapters is by looking at the different horizons each one sets itself. In one chapter we may find, for instance, a concern which falls almost completely within the domain of mathematics and of mathematical meanings, while another chapter allows for social and cultural issues to become of central interest. Also, different chapters may deal with so-called local theories, while others propose a perspective which starts from a broader view of the issues at hand. This is not, of course, a simple matter of ‘wide vs, narrow approach’. The key issue here is that in each case we have, effectively, the proposition of a perspective for research in mathematics education. Although the same could be said about almost every edited book in the field. the difference here is that we actually 2%./' this diversity, and had to deal with it during our discussions. Each one of the controversial issues found its way to the spotlight, and that made us much more aware of them than is usually possible by reading the literature and attending conference sessions. Year after year, these differences would be revisited, and then either resolved, deepened or simply abandoned. At some point someone would say that history or epistemology was not relevant; some would say that our effort should be put into making available to teachers what had already been developed. We never reached a full consensus-as this book shows but this is only to say that we were always engaged in an effort to reconceptualise the field, and in such situations difference is a much needed fuel. —
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Until some ten years ago, research and development related to algebraic education was strongly dominated by a concern with #)("(%)#: There was a mostly implicit assumption that algebraic thinking could only happen in the presence of ‘letters’, and arithmetic was at best seen as part of ‘pre-algebra.’ Although acknowledging the power generated by the use of algebraic notation in many situations, and maintaining that one of the key objectives of school algebraic education is to get students to master its use, much of the research and development presented in this book takes into consideration an important issue: no matter how suggestively ‘algebraic’ a problem seems to be, it is not until the solver actually engages in its solution that the nature of the thinking involved comes to life. A related issue is that the signs of algebraic notation do not carry, in themselves, any meaning that can be apprehended by simply examining them. Taken at face value, this may seem all too obvious. But what the working sessions of our group slowly revealed was that it has subtle yet powerful implications. One of these is the crucial role played by teachers as opposed to a previous focus on some ‘natural’ development of algebraic thinking. All chapters in this book deal, more or less explicitly, with the need for intervention if the students are to make/survive the cuts/shifts/reconceptualisations (from arithmetic, mainly) required for the development of algebraic thinking. Sometimes it comes in the form of a statement of how students should think with respect to given aspects of algebra and, implicitly, that someone has to tell them this, otherwise they would not be making mistakes; sometimes it comes in the form of how other peoples thought in the together with a statement to the effect that this does not suggest any ‘natural’ past route. But there is another equally important side to this emphasis on the teacher’s role, and one which adds, we think, to previous approaches. Not only will the teacher have to have ways to %#(/*./#/-for instance, through sequences of well crafted problems and situations but s/he will also have to be able to read what the students are saying/doing, and the outcome of this will affect the course of teaching. Maybe this is a good point to remember the powerful notion, highlighted in the work of Vygotsky, that any process brought into play will necessarily cause its own transformation. Not that this has not been a tenet for good teachers for a long time, but it seems that finally it is more and more being %#I34%2( into development and in the research supporting it: teachers do not only need good material, they also need good ‘reading’ strategies. Again, the balance between ‘didactical’ and ‘theoretical’ issues seems important. Didactical approaches in this book will, as usual, focus on the intervention side. However, the theoretical approaches put forward here focus mostly on the reading side, rather then on prescription. A balance is achieved, but not one between ‘foun—
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dations of teaching’, in the sense of ‘how things are’ and ‘how teaching should proceed’, and ‘teaching’ the actual practice; there is, instead, a balance between teacher and student meanings, a balance between teacher and students %#(/*./#(%)#+: —
INTO THE BOOK Maybe we should now take a look at individual chapters. Chapter 2 . by Luis Radford ‘The Historical Origins of Algebraic Thinking’ opens the book with an insightful examination of the origins of the notion of the ‘unknown’, pointing towards proportional reasoning. Radford examines material from the mathematics of Babylonia, ancient Egypt, ancient Greece, the Middle Ages and Renaissance Europe. The objective is not to trace a direct route, but rather to analyse similarities and differences. In his way, he looks at aspects both internal and external to mathematics, something which will be reflected in his final remarks: '...each algebra (Mesopotamian and Greek) was conceived deeply rooted in and shaped by the corresponding sociocultural settings. This point raises the question of the explicitness and the controlling of the social meanings that we ineluctably convey in the classroom through our discursive practices.’ (page 34). In other words, there is an interplay between pragmatic needs and symbolic invention. and the pragmatic needs may well fall outside the internal needs of purely mathematical developments. Chapter 3, ‘The Production of Meaning of Algebra: a Perspective based on a theoretical model of Semantic Fields,’ was written by Romulo Lins. It provides a reconstruction of the notions of knowledge and meaning, and analyses the consequences of this reconceptualisation for algebraic education. The perspective is clearly epistemological, the main question being ‘what are those students saying?’. It is a ‘theoretical’ chapter, but the theoretical discussion leads to a classroom approach, based partly on a distinction between activities which are ‘problem-driven’ and those which are ‘solution-driven’. The main arguments are centred on a notion of (/K( which denies any intrinsic meaning to algebraic notation, looking instead at algebraic thinking as a way of producing meaning among many others and the implications for algebraic education deriving mainly from the need to establish the legitimacy of algebraic (mathematical) meanings and from the need to elicit the meanings students produce for a given (algebraic) text. The role of the teacher is crucial, but the broader notion of %#(/*2)B4()* is discussed. Chapter 4 is ‘A Model for analysing Algebraic Processes of Thinking’. Ferdinando Arzarello, Luciana Bazzini and Giampaolo Chiappini work within a horizon that includes both linguistics (Frege) and epistemology, providing an insightful look at what an algebraic expression represents S'/#)(/+T and that which it suggests or shows S+/#+/TY much analysis is directed towards the (negative) effect of collapsing —
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this distinction. It is certainly a theoretical chapter, but throwing light on classroom processes. The authors present a number of useful notions for analysing the process of meaning production/construction in the process of solving problems presented in algebraic notation, such as ‘evaporation’, ‘condensation’, ‘algebra as a game of interpretation’, and ‘conceptual frame.’ In all, there is as much a concern with mastering basic algebraic manipulation as with putting algebra to a more advanced, developed use. David Kirshner’s chapter 5 (’The structural Algebra Option Revisited’) takes a different approach. His main argument is neither for a greater attention to student’s approaches/meanings nor to what (formalised) mathematics prescribes. Instead, he argues that we will find a model for what is to be learned, in algebraic education, in what experts "B(4"220 ')Y rather than presenting syntactical rules, we should allow students to experience and match algebraic manipulation: ‘. ..learning (is) always grounded in perception and pattern matching as embedded in practices' (page 95). However, rationality of the sort found in syntactical rules, he argues, is part of a different, social, process, a process to be understood as social legitimisation, and that we should not expect it to come ‘...from engagement with inherently logical artefacts.’ Again, the role of the teacher is emphasised, though not directly. It is clearly a chapter dealing with epistemology and cognition, and the epistemology he proposes is one based on connectionism rather than on ‘(a) dualist philosophy (which) is the foundation of our culture’s common sense about mentality’ (page 88). This is an insightful chapter, which tackles old practices without simply discarding them as ‘wrong,’ choosing instead to discuss what is wrong with the environment in which they happen. Chapter 6 ‘Transformation and Anticipation as Key Processes in Algebraic Problem Solving,’ by Paolo Boero, extends in a certain sense and in a certain direction, the work of Arzarello, Bazzini and Chiappini (Chapter 4); the perspective is clearly epistemological. Overall. he proposes a way of reading what people are saying/doing when they engage in algebraic problem solving, a reading based on the notions of +/9 and 7)*91 while abandoning the traditional syntax/semantics distinction. Those two new notions allow one to apply the notion of +/#+/ (Chapter 4) both within (as in an example about the sum of four consecutive odd numbers) and outside the domain of mathematical meanings, in a powerful way; in many aspects it is also close to Chapter 3, by Lins. This is a very focused chapter which, through carefully selected examples, allows the reader to build an understanding of the implications of what the author proposes, Chapter 7, by Aurora Gallardo (‘Historical-epistemological Analysis in Matheas in Chapter 2 matics Education’), takes on history as a reference, but again not as a ready-made path for teaching. What Gallardo proposes is that history may provide a useful and important counterpoint to research in the classroom; the notions —
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of cycles and interplay. are crucial. Embracing Piagetian perspectives on cognition and learning, she sets out to investigate the similarities between what happens in the classroom and in history, acknowledging that ‘The body of mathematical knowledge (. ..) is something that cannot be fully apprehended through its formal dimension.. .’ (page 137). There are considerations both on how carefully chosen problems can further the process of acquisition of an algebraic competence and on student’s responses to those problems. Insightful both from a historical point of view and for those looking for insights into ideas for development work, this chapter manages to combine a mainly didactical perspective with historical and epistemological ones. Kaye Stacey and Mollie MacGregor wrote ‘Curriculum Reform and Approaches to Algebra,’ (Chapter 8) which in many ways seems to depart from the previous ones. Their particular concern has to do with discussing an assumption which has widespread acceptance in many countries: that generality as emerging from generalising patterns -beyond being a root of algebraic thinking. also makes. in itself, a better route to it than other approaches. The implications are many, but crucially the authors draw attention to the need for criteria for making curricular choices. a point of particular interest for those involved in or analysing situations like the socalled ‘maths war’ in the state of California (USA). but also a point of permanent and general interest; in particular. they approach the relationship between research and curriculum reform. They conclude that ‘... the greatest use of findings such as those reported in this chapter (is) not to advocate one teaching method over another but to highlight the ways in which students think about mathematical situations’ (page 152). If at first this chapter may seem a routine ‘method-testing’ exercise, it ends up at the other end: the students; starting from what seems to be a concern with teaching methods. they end up considering that ‘. ..curriculum designers are often concerned with how students ought to think instead of how they really think.’ This is a challenging chapter, as it proposes that concerns with good teaching methods should be accompanied by making hidden assumptions and intentions explicit. We then come to ‘Theoretical Theses on the Resolution of Arithmetic-Algebraic Problems,’ Chapter 9. by Eugenio Filloy, Teresa Rojano and Guillermo Rubio. Something outstandingly clear in this chapter is its %#(/#(%)#P the carefully chosen and examined solution methods for algebraic problems place this chapter clearly on the didactical side. And there it stands. but it is the willingness to make students overcome a ‘didactical cut’ that guides the whole expedition; in particular. the didactical cut refers here to manipulating the unknown, as in the authors’ previous work. Two classical solution methods the arithmetic and the Cartesian methods are described and examined. as well as two mi-conventional methods. The main point is not about basic manipulative skills in algebra, but about 9)'/22%#$1 although the focus remains throughout on examining how students deal with the relationships between the different pieces of data given in a problem. Epistemological and linguistic considerations are within the horizon of this chapter, which also draws —
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guistie considerations are within the horizon of this chapter, which also draws considerably from empirical work with students. In Chapter 10, ‘Beyond Unknowns and Variables: Parameters and Dummy Variables in High-School Algebra,’ Hava Bloedy-Vinner looks at the use of literal notation in algebra. considering that the traditional concern with unknowns and variables has to be extended to an analysis of the role of parameters and dummy variables. Although students’ work is used to illustrate the points being made, the line of inquiry falls clearly within the domain of mathematical meanings, that is, it is mostly concerned with what parameters and dummy variables are within mathematics itself, and with how students might interpret these new objects. She offers a number of questions which can serve to expose students’ understanding of the notions of parameter and dummy variable. Her didactic recommendations are to use these questions in classroom discussions, and also to take care to discuss the role of each letter in expressions when manipulative activity or problem solving is taking place. Giuliana Dettori. Rossela Garuti and Enrica Lemut wrote Chapter 11, ‘From Arithmetic to Algebraic Thinking by Using a Spreadsheet.’ In it they characterise algebra in intermediate school, and point out what determines the conceptual break from arithmetic, and analyse whether these features can be introduced by using a spreadsheet, focusing in particular on the elements to be taught and learned. In their approach they ‘.. . shift the focus from the potentialities of the software to the main characteristics of the subject to be taught, as (they) are convinced that. from an educational point of view. what matters is not to learn to find a numerical solution to algebraic problems but rather to understand the nature and the power of the theoretical solving scheme of algebra’ (page 191). Through the use of a carefully crafted set of problems, the authors examine the advantages and disadvantages of using a spreadsheet for their solution, particularly with respect to the possibility of expressing and manipulating relationships. They conclude that there are evident limitations if one is aiming at introducing algebra, but also that .. .using a spreadsheet (. . .) under the attentive guidance of a teacher ...’ ( 207) students can develop a number of important understandings related to the learning of algebra. This is a didactical chapter. placing great emphasis on the characteristics of the problems and also on teacher intervention. Chapter 12 was written by Sonia Ursini: ‘General Methods: a way of entering the world of algebra.’ As the title suggests, this chapter shares a concern with the previous one, that is methods in algebra. through working with the computer environment Logo. Ursini’s analysis, however, draws also from an examination of the history of mathematics, in particular the notion of $/#/*"2 #493/* as it appears in Vieta. Linked to this, Ursini proposes that the key difference between arithmetic and algebra is that, ‘while arithmetic deals with numbers and an important aspect of it is to perform computations obtaining numerical results, algebra deals with general mag-
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nitudes and arises historically from the interest in deducing general methods for solving sets of similar problems’. Her work with Logo is centred on the procedural characteristic of the software, not the geometric one as in other studies. Students use Logo to produce procedures to calculate, for instance. half of any number, and then Ursini proceeds to analyse what she calls 'the evolution of pupils' procedures.' to conclude that we may have here a powerful way of introducing students to the notion of 'general numbers,' one that she points out as essential for the development of algebraic thinking. Chapter 13, the last one, is ‘Reflections on the role of the computer in the development of Algebraic Thinking,’ by LuLu Healy, Stefano Pozzi and Rosamund Sutherland. In it they discuss school algebra in the UK, the influence of Piaget on algebra research, Vygotsky and an 'algebrising' culture and the role of the computer. A central point in this chapter is a move towards a more directive approach to teaching. after the realisation that students would not ‘naturally’ engage in algebraic activity, no matter how suggestively algebraic the proposed task seemed to be. The authors associate this move with a change in theoretical paradigm from Piaget to Vygotsky, put in a very broad way. The chapter ends with a statement of a current approach to algebraic education. The notions of 'setting' and 'tool' have become critically important, and an emphasis on the communicative function of symbol systems has emerged, leading to an approach to the learning and teaching of algebra which departs both from traditional symbol-manipulation and from an under-emphasis on the role of algebraic language in the development of algebraic thinking. —
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ACROSS THE BOOK
Many of the difficulties we encountered when working as a group related to views about what is and what is not algebra. and what is and what is not algebraic thinking. Whereas we may have disagreed on what algebra is, we did agree that it is important to analyse the different approaches which pupils bring to solving particular problems. A related issue which soon emerged from the work of the group is that school algebra differs in its emphasis from country to country. However, to understand why this is the case would entail an investigation of the socio cultural influences in each country, which is beyond the scope of this book. What we wish to draw attention to here is that in some countries students are likely to engage with carefully crafted word problems which have traditionally been designed to engage students in constructing algebraic methods (see for example Chapters 9 and 1 1), while in other countries these types of problem have almost disappeared from the curriculum. As we have pointed out before, the meanings which students construct for algebra will be related to the types of problem which are prioritised in the mathematics class-
APPROACHES TO ALGEBRA
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room, and we think that the cross-national picture offered in this book might offer a contribution in this direction. Also. it is interesting to compare this approach with the earlier research on student’s understanding of algebra (for example Booth 1984. Küchemann 198I ) , which first drew our attention to the multiple idiosyncratic meanings which students construct when they solve ‘algebra problems’, often explaining those meanings as being related to individual psychological development which would evolve almost spontaneously. whatever problems the students were engaging with. Sometimes starting from the point mentioned in the previous paragraph, but not always, many of the chapters in this book are influenced by a socio cultural perspective to learning. This is particularly the case in Luis Radford’s chapter (Chapter 1) which presents a detailed analysis of the historical origins of algebraic thinking, emphasising that the development of mathematics in ancient times was linked to the social, political and economic development of cities. This motivated a search for solutions to problems, such as how to calculate the area of a piece of land, how to solve inheritance problems and how to calculate the price of different commodities. In a similar way Aurora Gallardo (Chapter 7) discusses how the Chinese (ca. A.D. 250) were motivated to accept negative numbers well before other cultures, because of a need for these objects within problems involving ‘gains’ and ‘losses’. Gallardo’s iterative approach to analysing classical texts. linking the analysis to empirical work carried out with students, illuminates students’ developing conceptions, while Radford, on the other hand, stresses that the social and cultural aspects in the development of algebra cannot be reproduced in the classroom. In any case, a historical analysis highlights what the axiomatic presentation of mathematicians leaves out, which is a discussion of both the practical and theoretical motives that lead to the resolution of certain problems, and the obstacles that are inseparable counterparts of the development of fundamental ideas (Gallardo, Chapter 7 ). A socio cultural approach aims to understand how mental action is situated in cultural, historical and institutional settings (Wertsch,1991, p15). Whereas cultural differences between Western society and ancient or developing societies have often been interpreted as different stages in a normative development, both Radford and Gallardo show through their analysis that there is not a simple temporal progression in the development of mathematical language and thinking. What is crucial here is the dynamic interaction between the problems of the time and the technological. semiotic and epistemic tools available. This, however, is no simple relationship as the problems to be solved also drove the need for new technological and semiotic tools. In his chapter Radford makes conjectures about the meanings which the ancients constructed from interactions with a particular combination of problems. tools and language. From a present-day vantage point it is very difficult for us not to view the
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notion of ‘false position’ from the perspective of the algebraic idea of the ‘unknown’ because the idea of an algebraic ‘unknown’ i s now an established concept (together with a means of symbolisation). It i s just this sort of difficulty which occurs when we analyse pupils’ meaning construction as they solve a range of problems. It is difficult for the researcher not to ascribe meanings to pupils which are influenced by the researcher’s own knowledge perspective. Whether we are analysing students’ solutions to a range of problems or the solutions of an ancient historical group we will always be constrained by our own knowledge and perspective. This is why Lins (Chapter 3) stresses that any theoretical model has to be careful not t o depend completely on what ‘we’ are in order to say what ‘other people’ are, or should be. Still in the same direction, some authors advocate the introduction of computerbased symbolic languages as a type of intermediate language (tool) which can provide an entry point to the introduction of the algebra language. There is enough evidence that pupils who might find the algebra language alienating can more readily learn computer-based algebra language. However the nevi symbolic tool subtly changes the meanings which pupils construct and thus the associated learning. Moreover. computer-based approaches are often associated with mathematical modelling and problem solving and a focus on problem solving can work against the learning of algebra, because the focus becomes that of ‘solving the problem’ and not on the method for solving the problem. an awareness well developed within the group (for further discussion of this see the Royal Society/Joint Mathematical Council of the UK Report (1997) C/"B,%#$ "#' N/"*#%#$ 52$/3*"1 ?*/IXZT: Mathematics learning centres around solving problems, and historical analysis tells us that the nature of the problems makes a difference to what is learned. But this is not enough because a particular problem can be solved in many different ways, as illustrated by the work of Filloy, Rojano and Rubio (Chapter 9), Dettori, Garuti and Lemut (Chapter 11) and Lins (Chapter 3). Lins suggests that pupils have to become aware of the different approaches and the epistemological limits of each approach. Lins also stresses that the teacher should get pupils to make explicit justifications as a constitutive part of knowledge in order to allow shifts in meaning, that is construction of new knowledge. By establishing meaning through explicit statements it is then possible to provoke pupils to explore other meanings. Getting pupils to produce new meanings is what our school culture wants. and the role of the teacher comes in here to enable pupils to engage in producing meaning in a new way. The traditional approach of presenting students with so-called algebra problem may have worked for some students when the teacher more or less imposed an algebraic solving approach. The current trend however i s to encourage students to solve problems for themselves, without imposing a problem solving approach. We now know that students are likely to solve these problems in many non algebraic ways
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which can result in an unresolvable tension. We have to ask the question ‘can algebraic approaches develop in a seamless way from these non-algebraic approaches’. Some people believe that this is possible and others would suggest that the non algebraic approaches might even provide an obstacle to algebraic approaches. This is clearly an area where we need more research as curriculum reformers continue to fall into the trap of believing that ‘good’ problems alone will provoke algebra learning (as discussed by Stacey and MacGregor). The chapters in this book come together to show that ‘mathematics problems’ are only part of what would constitute an appropriate school algebra culture for students to learn algebra. The complexity for the teacher is that pupils do already know arithmetic and so are bound to be influenced by these arithmetic approaches. The teacher has to know and understand these approaches if he/she is to make sense of student’s constructions. The teacher has to understand the past history of learning which students bring to the classroom, which can emanate from both school and out of school experiences. As Wheeler (1996) has pointed out we have to take account of what students know but this does not then imply that we have to use an evolutionary model of learning. Instead he suggests that ‘learning appears to me to be very largely a discontinuous, non-linear, business’ (p 149). Reflecting on this issue would provide an appropriate backdrop to reading the chapters in this book.
NOTES 1
PME is the International Group for the Psychology of Mathematics Education.
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LUIS G. RADFORD*
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
INTRODUCTION Barely some fifteen years ago, a new and completely different interpretation of Mesopotamian mathematics arose due, on the one hand, to new philological analyses and, on the other, to the development of new methods in the historiography of mathematics; these have been increasingly incorporating anthropological and sociological categories in the understanding of the mathematics of the past. The new results have been changing the traditional view of Ancient Near East mathematics and their influence on Greek mathematics and have led us to review old questions and to raise new ones. One of these questions deals with the transition from arithmetic thinking to algebraic thinking. Another question is the role played by language and symbolism in algebraic thinking. In this paper we will deal with these two questions 1 . Concerning the first question. we shall attempt to show that (numeric) algebraic thinking emerged from proportional thinking as a short, direct and alternative way of solving ‘non-practical’ problems. In order to do so, in the first section of the chapter, we briefly describe some of the features of Proportional Mesopotamian thinking. The second section is devoted to the elaboration of our thesis. The historical technical evidence and its argumentation is presented in following sections. Since a certain knowledge of Diophantus’ algebraic methods is required, we give an overview of Diophantus’ monumental treatise 5*%(,9/(%B"1 and deal with Babylonian geometric algebra. The second question with which we deal in this paper that of the role of language and symbolism in algebraic thinking - is submitted to the same methodological approach that we used in the previous sections deriving from our perception of what mathematics is: we see mathematics as a semiotic manifestation of the culture in which mathematics is practised. Consequently, we suggest that the role of symbolism in algebraic thinking needs to be studied through the social meaning of alge–
* Ecole des sciences de1'éducation, Université Laurentienne, Ontario, Canada 13 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 X[\[]: U VWWX <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
14
L.G. RADFORD
bra and through the different forms of symbolisation that the culture under consideration uses to symbolise its objects (regardless of whether or not these are mathematical). This is why, in many instances, we shall refer to some sociocultural components of the world of Mesopotamian scribes. The final section of the chapter contains some pedagogical remarks. MESOPOTAMIAN PROPORTIONAL THINKING
Mathematics arose in ancient civilisations intimately linked to the social, political and economic development of the cities. For instance. ancient historical records suggest that the numerical cuneiform signs used in Mesopotamia were, in fact, the achievement of an important semiotic phenomenon preceded by the commercial counting activities first based upon incised bones and later on tokens (SchmandtBesserat, 1992; Jasmin and Oates, 1986; Le Brun et Vallat, 1978). The expansion of the cities required new and more elaborate forms of internal organisation which led to new and complex problems –e. g . how to calculate the area of a certain piece of land, how to calculate the interest of a loan, how to solve inheritance problems, eg. 2 how to calculate the price of different commodities . Nevertheless, we also find ‘non-practical’ problems, i.e. problems that, although being formulated in terms of the concrete semiotic experience of everyday life, have no direct relation to practical 3 needs . Practical and non-practical problems appear in two of the most important mathematical currents observed in ancient Egyptian and Babylonian civilisations: a geometrical one and a numerical one. In the first current, we find problems which deal with the calculations of area and perimeter of geometrical figures while, in the 4 second, we find problems about (contextual) numbers . In the two previous categories, problems are often solved using proportional thinking, which was, in fact, one of the most important areas developed in Mesopotamian mathematical thought. Not only is it attested to by the Babylonian tables of reciprocals, which were a byproduct, but also by the clever ancient methods based upon proportional tools that were developed to solve problems. One of the three oldest known problem texts SB": third millennium BC), which was found in 1975 when the Italian Archeological Mission discovered the Royal archives of the city of Ebla, is the text TM.75.G.1392 which contains a type of problem whose statement and problem-solving procedure reveal some features of early proportional thinking. The problem (concerning the assignment of cereals) is stated (according to Fribergs’ reconstruction) as follows: Given that you have to count with 1 $^I3"* for 33 persons, how much do you count with for 260,000 persons? (Friberg, 1986, p. 19). The problem obviously leads to division. However, the division is not easy to perform within the sexagesimal system. Therefore, the text shows a sequence of
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
15
calculations: 3 $^I34* for 99 persons, 30 $^I3#* for 990 persons. and so on S%3%'1p. 20). The sequence is based upon proportional tools which allows the scribe to obtain an (approximate) answer. The most sophisticated proportional problem–solving procedure deals with the methods of false position which appear as the greatest achievement of Mesopotamian arithmetical thinking. These methods are based upon the idea of assuming some false values for the sought-after quantities and then adjusting them through a ‘proportional adjusting factor’ which allows one to modify –in a proportional waythe false values in order to transform them into the true values. The following problem is an example of a non-practical problem solved by a false position method. It is included in a tablet from Susa. probably slightly after the 1st Babylonian Dynasty, i.e. at the end of the 17th century BC. (see Bruins and Ruta ten, 1961, pp. 101-103) . The problem is to find the sides of a rectangle whose width is equal to the length minus a quarter of the length. and the diagonal is 40‘. (Note that in a real situation, anybody who knows the stated relationship between the sides of the rectangle obviously already knows the actual sides of the rectangle. This problem is merely a riddle). To solve the problem. the scribe assumes a false quantity for the length. He says: ‘You, put one for the length’. He then calculates the width, by subtracting 1/4 (that is 15') from 1o, which gives 45_. He calculates the square of both false sides, which gives 1 = 1 and (45‘) = 2025` = 33_45`. The sum of the squares is 1o 33_45`. He then takes the square root of 1o 33'45` which is 1o 15_. This is the value of the 7"2+/ diagonal. The (*4/ diagonal, 40_, is less than the obtained value. He then calculates the inverse of 1o15_, which is 48_: he multiplies this number by 40_; the result is 32_. This is the ‘proportional adjusting factor’ by which he multiplies the false length (i.e. 10) and the false width (i.e. 45_): the scribe then obtains 32_ x 1o = 32_ and 32_ x 45_ = 24_. Therefore, the resulting numbers 32_ and 24' are the true length and width, respectively. In modern terms, the problem requires one to find the length, x, and the width, y, of a rectangle whose diagonal is a given number d. From the relation
²
d=
=
=
²
x >/ can deduce that x =
the scribe actually assumes a false length, 1 3 = = . He calculates the –
=
=
x0 =
5
(because
. d . In contrast,
which gives a false width false diagonal using
= I ) . The proportional argument under-
lying the procedure leads the scribe to calculate the inverse of arid to multiply this inverse by the given diagonal d=40. This problem clearly shows the functioning of the ancient false position method and will suffice to elaborate our historical reconstruction of the transition from arithmetic to algebra in Mesopotamian land. However, before going on to our next
16
L.G. RADFORD
stop in our historical journey, we need to make the following cultural and epistemological remark: the idea of using a 'false quantity' to start the false position method. leans on a deeper and more complex idea: at the beginning of the problem. the 'true quantity' (i.e. the exact solution of the problem) may be legitimately thought of (,*)4$, another quantity. 'False quantities' thus appear as 9/("D,)*+ of 'true quantities'. Furthermore, this is not a phenomenon restricted solely to mathematics: Mesopotamian thinking is full of metaphors. Odes. epic poems, literary and religious texts, for instance, show an intricate system of metaphorical expressions (see e.g. Wilson. 1901). Algebra, we shall suggest. was couched in such a system. ALGEBRAIC THINKING AS A METAPHOR OF THE FALSE POSITION METHOD As we shall see in later sections of this chapter, where we focus on some technical details, the influence of false position methods in the emergence of algebraic ideas can be discerned through some important structural similarities between false position reasoning and early algebraic thinking. One of the studies of the ancient connection between the Babylonian false position method and algebra was made by François Thurcau-Dangin (1938a). Following the trends of the old interpretation of Babylonian mathematics based on the possibility of translating the calculations shown in many of the tablets into modern algebraic symbolism, he noted a strong parallelism between the calculations done in some problems solved by false position methods and those of the modern algebraic methods6. He then claimed that, indeed, some Babylonian procedures were algebraic procedures. Thureau-Dangin’s main idea was supposedly supported by the fact that, in some problems, the scribe takes the number )#/ as the false solution (such an example could be the problem discussed at the end of the previous section) and when, according to Babylonian procedures. we replace the number )#/ by our modern unknown ‘x’, the problemsolving procedures look much like the modern algebraic procedures. He, (as well as others. e.g. Vogel, 1960), claimed that the number one was actually taken as a */D*/+/#("(%)# of the unknown and i f we cannot. straight out, see the unknown, it is simply because the scribes did not have a symbol with which they could represent it. However, the idea that Babylonians developed an _invisible algebraic language’ (i.e. a genuine algebraic language without symbols) has been abandoned (see Radford. 1996a. pp. 39-40). Effectively, there is no clear and safe argument supporting the statement that the Babylonian scribes actually (,)4$,( that the number one */D*/+/#(/' the unknown in an algebraic sense (see Høyrup, 1993b, p. 260). On the contrary. this peculiar numerical choice for the unknown seems to have allowed the scribes to +0+(/9"(%+/ the numerical problem-solving methods and hence to reach an important step in the conceptual development of ancient proportional thinking. In
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
17
fact, when ancient problem-solving procedures may be safely identified as «algebraic», which is the case of the problem mentioned in footnote 10, the unknowns are not */D*/+/#(/' by the number ‘1’ or anything else for that matter; instead, the unknowns bear their contextual name (e. g. the length and the width of a rectangle). Rather, the "2$/3*"%B %'/" )7 4#=#)># seems to have been thought of as a 9/("I D,)* of the ‘false quantities’ used in the ancient false position method. It happened, we suggest, when scribes stopped thinking in terms of 7"2+/ b4"#(%(%/+ upon which the false position method is based and. looking at the false quantity metaphorically, began to think in terms of the sought-after object %(+/271 accepting to consider this object as a #493/* (i.e. an operable, inanageable number) regardless that it was not c yet known . This could happen in solving non-practical problems previously solved by false position proportional methods like the following (Tablet YBC 4652, No. 7), where the method of solution is unfortunately omitted: ‘I found a stone, but I did not weigh it; after I added one-seventh and added one eleventh (of the weight and its one seventh). I weighed it: 1 ma-na. What was the original weight? The origin(a1 weight) of the stone was ma-na, 8 gin, (and) 22 1 še ‘ (Based on Neugebauer and Sachs'reconstruction; 1945, p. 101). 8
In modern notations, this problem reads as follows : x +
–
+
–
x+
–
=
1.
The algebraic way of thinking could have even been conceived when ancient scribes 9
faced an even simpler problem. For example, a problem of this type : x + 1 11
–
x = 1.
Let us suppose that this problem concerns the weight of a stone. The false position method is as follows: we assume (according to the usual line of thought in Babylonian mathematics) that the sought-after quantity is 11; then, the stone and one eleventh of its weight is 12. However, we should have 1. This means that we need to reduce the ‘false position’, that is the false value that we assumed at the beginning (i. e. 11). To reduce it, an elementary proportional argument shows that we need to take one 12th of our initial assumption, so the answer to the problem is 11/12 (or 55/60 = 55_ in the sexagesimal system). To see the metaphor that we are suggesting at work, let us, instead of beginning by assuming a 7"2+/ D)+%(%)# or false solution. start the problem-solving procedure by reasoning on the exact unknown sought-after quantity. In this case, the calculations unfold in a different way: first, we multiply both sides of the ‘equation’ by 11 thereby transforming the ‘equation’ into an ‘equation’ without fractional parts. This leads us to an equation that we would write as 12x = 11. Now, following a recurring Babylonian procedure, we just need to find the inverse of 12, which is 5 _ , and to multiply this inverse by 11, which gives us the answer 55_. (Note that the procedure
18
L.G. RADFORD
of multiplying both sides of the ‘equation’ by a number is attested to in many Baby10 lonian problem, e. g. C/K(/+ !"(,d9"(%b4/+ '/ Suse ). The type of problems that we have just discussed were frequent in ancient civilisations. For instance, one problem of this type is found in the Egyptian Rhind’s Papyrus; another is found in Babylonian tablets. This is the case of problem No. 3 of tablet YBC 4669 which could be translated into modern notations as follows11: 2 1.x=7. [ [@ @
The conceptual connection between false position ideas and algebraic ones can also be found in post-Greek mathematics. It can be retraced to some mediaeval mathematical treatises. It is particularly enlightening that, in the false position methods, mathematicians. at the beginning of the problem-solving procedure, used to refer to the action of choosing the false numbers as ‘making a position’. In the same way, when a problem is solved by algebra, the introduction of the unknown is sometimes referred to as ‘making a position’. For instance, in Filippo Calandri’s -#" *"B)2(" '% *"$%)#% (15th century). problem 18 deals with a problem that we may translate into modern notations as:
X
= x 1 x+1 (‘Trouva U numero, che partito per uno. più ne vengha un meno’. Santini (ed.), 1982, p. 19). Solving this problem through algebra and by calling the unknown (,/ (,%#$ (‘la cosa’, according to the Italian mediaeval tradition) Calandri says: ‘Farai posizione che que1 numero sia una co(sa)’ (I will make a position so that the (sought-after) number is a thing). An early example ( 14th century) is found in problem 6 of Mazzinghi’s C*"(("() '% A%)**((%: In this problem Mazzinghi says: ‘El primo (modo) e che si faccia positione che lla prima parte sia 5 et una chosa’ (Arrighi (ed.). 1967, p. 23). The connection between false position ideas and algebraic ones is more explicit in an anonymous abacus treatise of the 14th Century: F2 (*"(("() 'e"2$%3*": In this treatise, the unknown is defined just as a position: ‘...in prima noi diremo che sia questa chosa, onde dirò che non sia altro se non f una posizione che si fa in molte questioni...’ (first of all we shall say what this thing is, where I shall say that it is no more than a position that one makes in many questions; Franci and Pancanti, eds. 1988, p. 3, my translation). We can go one step further in our connections between algebraic and false position ideas by referring to a book written in 1522: Francesco Ghaligai’s ?*"((%B" '_5*%(,9/(%B": However. in this case. algebraic ideas have been developed enough to be taken as the explanatory substratum in which the false position ideas are set up. Ghaligai says: —
–
‘We can notice that the position is a concept assimilated to the thing that is chosen according to the knowledge of the intellectual realm Speaking in the case of a thing not known to you, right away the mind will think as if it already knew and say: position is a
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
19
quantity placed according to the case (the problem) and even though there are two positions, sometimes with only one position the case can be solved and one finds that which is necessary.’ (Ghaligai. 1538. p. 62: my translation).
Ghaligail’s ?*"((%B" 'e5*%(,9/(%B" shows then that the conceptual development of the unknown has completed a loop, now changing the roles of ideas: at that time, the metaphorical-analogical process was reverted and one explains the false position ideas in terms of algebraic ones. methods have to arise, thereby making it possible to handle the algebraic unknown. The metaphorical-analogical process underlying the passage from arithmetic to algebra will map or %#'4B/1 as is the case in inany metaphors (see Lakoff and Núñez, in To end this section let us stress the fact that the new algebraic object of unknown does not come to life alone: it emerges along with new methods. False position methods deal with numbers only. So, new print), important structural features of the first domain here the arithmetical one _ into the second domain here the algebraic one. This metaphorical induction is very clear in inany passages of Diophantus' 5*%(,9/(%B": Indeed, in any of Diophantus' algebraic methods arc hardly understandable without linking them to the ancient false position methods, as we will see in the next section. In order to understand this specific aspect of problem-solving methods. we now need to examine the place of Diophantus' 5*%(,9/(%B" in the development of algebraic ideas. –
–
ALGEBRAIC IDEAS IN DIOPHANTUS' ARITHMETICA
As we know, Diophantus' 5*%(,9/(%B" (written B%*B" 250), is made up of 13 books (3 of them are lost) dealing with the resolution of problems about numbers. Book 1 contains a short introduction in which a division of numbers into +D/B%/+ or B"(/$)*%/+ is presented: the squares, the cubes, the square-squares. the square-cubes. the cube-cubes. Each category contains the numbers that share a similar form or shape (the same /%')+T: Instead of being merely riddles like the Babylonian nonpractical problems, the problems of the 5*%(,9/(%B" were formulated in terms of the mentioned +D/B%/+: For instance, problem 10 from Book 2 reads as follows: ‘To find two square numbers having a given difference’ (Heath. 1910, p. 146). Undoubtedly, within the philosophical principles of Classical Greek thinking (where the search of origins and rational organisation was a starting point), Babylonian numerical word-problems and all the subsequent numerical activity surrounding similar problems in the post-classic Greek period. an activity particularly attested to by the 5#(,)2)$%" $*"/B" (Paton, (ed.), 1979), did not find a suitable niche to prosper 12. By transforming the Babylonian numerical word-problems into problems about abstract Greek species and other ancient well-known riddles that Diophantus supposedly disguised in abstract terms in his 5*%(,9/(%B" (e.g. Book I, problems 16-
20
L.G. RADFORD 13
21). Diophantus elevated this unscientific discipline to a scientific one . This was not the only important new aspect incorporated by Greek algebraists. There is another one related to the introduction of %#'/(/*9%#"(/ #493/*+ to the mathematical realm. This was done through a new use of the concept of "*%(,9)+ (ariqmoV), that is, the g#493/*e: ‘The arithmos’, says Diophantus, ‘is an indeterminate multitude of units’ (cf. Ver Eeck, 1926, p. 2) – although, in the solution of many problems, it can be an indeterminate multitude of fractional parts. The subtle, yet fundamental, step made by Diophantus in introducing %#'/(/*9%I #"(/ #493/*+ to the mathematical realm can be better appreciated if we see it within the heritage of the ancient philosophers. In this line of thought, it would be worthwhile to remember that. in one of the few extant fragments of the first Pythagoreans, Philolaus says: ‘Actually, everything that can be known has a Number; for it is im,14 possible to grasp anything with the mind or to recognise it without this SJ493/*T ; and here, Number means a determinate multitude. Thus, by introducing the "*%(,9)+ as an indeterminate multitude Diophantus extended the borders of what can be known. By the same token, the aforementioned concept of arithmos gave way to the creation of a completely new theoretical calculation on %#'/(/*9%#"(/ amount of units (e.g., in modern notations. rules dealing with calculations like x x x2 = x3 or x x 4 = x3) that proved to be very powerful in the resolution of problems. It is important to note that these mathematical accomplishments at the end of the Antiquity were linked to an increasing (albeit not complete) abandonment of Greek classical principles and the spreading of neo-Platonistic speculations that made it possible to think in new, different and promising ways (see Lizcano, 1993). In most of the problems of the 5*%(,9/(%B" the formal structure of their statement follows the same pattern: the problem is stated in terms of operations performed on some categories of numbers (the square numbers in the previous example). At the beginning of Book 1, Diophantus gives a description of the heuristics that one should follow in order to solve the problems. This is the very first explicit ancient description about how to solve problems on numbers that we know. For our discussion we will quote the following extract: ‘...if a problem leads to an equation in which certain terms are equal to terms of the same species (eidos) but with different coefficients, it will be necessary to subtract like from like on both sides, until one term is found equal to one term. If by chance there are on either side or on both sides negative terms, it will be necessary to add the negative terms on both sides, until the terms on both sides are positive, and then again to subtract 15 like from like until one term only is left on each side.’ (Heath, 1910, p. 131) .
In modern terms, this passage tells us that if in a problem we are led to an equation of this type: numbers
squares
cubes
square squares... = numbers
_squares
_square squares ±...
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
21
we have to add or subtract the terms sharing the same /%')+ in order to reduce the problem to the case in which a +D/B%/+ equals another +D/B%/+1 that is, an equation of the type axn =bxm. To solve the problems, Diophantus often chooses some quantities involved in the statement of the problem. For instance, in the previous problem, he chooses the difference equal to 60. Then he chooses the root of one of the sought-after numbers equal to the "*%(,9)+1 which plays the role of an algebraic unknown. The other sought-after number is chosen as the "*%(,9)+ plus 3. This leads him to the equation that could be translated into modem symbols as (x + 3)² – x² = 60 where he is able 1_ to obtain the equation 6x+9=60 and finally the solution x = 8 2 . So the sought-after 1 _ 1_ square numbers are the fractional numbers 72 4 and 132 4 . Of course, these are not all the solutions of the general problem. This is why Diophantus' solutions are often seen as incomplete. However, we have to be very cautious at this point. In fact, finding out how to express (in an explicitly verbal or symbolic way) "22 the couple of members of the +D/B%/+ )7 +b4"*/+ that verify the stated condition in the previous problem (or in another problem) is a rather modem problem and not an ancient one. The problems of the _Arithmetic Books’ of Euclid's &2/9/#(+ are not concerned with the task of '/+B*%3%#$1 in an explicit form, each of the members of a certain class of numbers (e.g. the perfect numbers; see Euclid's &2/9/#(+1 Book IX, proposition 36). Another example can be the gformula’ to produce polygonal numbers found by Diophantus himself. This 'formula’ does not '/+B*%3/ the elements of a class but D*)'4B/+ as many numbers as we want (triangular, square, pentagonal numbers and so on: see Radford, 1995a). By the same token, Diophantus' problem-solving methods do not aim to find nor '/+B*%3/ all the solutions of a given problem (except, of course in the cases where the problem has only one solution) but to D*)'4B/ as many solutions as we want. The alleged incompleteness of Diophantus' solutions are relative to our modern point of view. Without taking into consideration Diophantus' own conceptualisation, Diophantus' 5*%(,9/I (%B" becomes just a mere compendium of problems solved in a way that ‘dazzles rather than delights’ and Diophantus himself appears ‘unlike a speculative thinker who seeks general ideas’ but as someone looking only for ‘correct answers’ (e.g. this is the case of Kline's perception of Diophantus: see Kline, 1972, p.143, from whom the quotations were taken). Certainly, for a Babylonian scribe, Diophantus' 5*%(,9/I (%B" would be seen as the product of a genuine speculative thinker.
THE TRACE OF PROPORTIONAL MESOPOTAMIAN THINKING IN GREEK ALGEBRAIC THINKING We are now ready to technically tackle the first question raised in the introduction of this chapter. The essence of our manoeuvre consists in showing that the functioning
L.G. RADFORD
22
of the algebraic concept of unknown in Diophantus' 5*%(,9/(%B" is too closely related to the functioning of the concept of false quantities of the Babylonian false position method to be regarded as a mere accident. On the contrary, the structural coincidence in both concepts is fully understood on the basis of the idea that the algebraic unknown was conceived as a metaphor of its correlated arithmetical concept -the false quantities. Although there are many structural coincidences between the two concepts, here we shall refer to one of them: reasoning in terms of fractional parts. Let us refer to problem number 18 from a tablet conserved at the British Museum (BM 85 196) that goes back to the ancient Babylonian period. It concerns two rings of silver. 1/7 of the first ring and 1/11 of the second weigh 1 +%B2/: The first, diminished by its 1/7, weighs just as much as the second diminished by its 1/11 (See Thureau-Dangin, 1938b, p. 46). In modern notations, the problem can be stated as 1 1 _1 follows: _ x + _ y= 1; x- x= y- _1 y. 7
11
7
11
The scribe’s reconstruction of the solution given by Thureau-Dangin (1938a, pp. 74-75) suggests that «the first ring diminished by its 1/7» is transformed into «6 times the 1/7 of the first ring ». By the same token, «the second ring diminished its 1/11»is transformed into «10 times the 1/1 1 of the second ring». The reasoning is then carried out on the above-mentioned fractions (i.e. «1/7 of the first ring» and «1/11 of the second ring»). These fractional quantities are in a 10 to 6 ratio. Therefore, by employing the false position method the scribe assumes 10 for the 1/7 of the first ring and 6 for the 1/1 1 of the second. He then adds the false assumed values and gets 16. However, he was supposed to get 1. The canonical Babylonian proportional process leads to the question of finding a 'proportional adjusting factor' which, in this problem, corresponds to the inverse of 16. The scribe finds that the inverse of 16 is 3_ 45’. To find «1/11 of the second ring», he multiplies the false value (i.e. 6) by the ‘proportional adjusting factor', 3_ 45’, and finds 22_30’. Next, to find «1/7 of the first ring» he multiplies 3_ 45’ by 10 and gets 37_ 30’. He multiplies 22_ 30’ by 11 and gets 4 o 7_ 30’ the weight of the second ring. He multiplies 37_@30’ by 7 and gets o 4 22_ 30’; the weight of the first ring. Let us now examine the Greek counterpart. In problem 6 of Book 1. Diophantus tackles the problem to divide 100 into two numbers such that 1/4 of the first exceeds 16 1/6 of the other by 20 . This problem cannot be solved by the Babylonian false position method17. However, Diophantus' method of solving the problem begins by following the Babylo nian pattern seen above: the reasoning is based on the 7*"B(%)#+ of the sought-after numbers. Diophantus takes the 1/6 of the second part as the unknown (which he calls the "*%(,9)+1 that is, the #493/* and represents it by the letter V). Thus, the second number becomes 6 times the number. ‘Therefore, he says, the quarter of the first number will be 1 number plus 20 units; thus, that the first number will be 4
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
23
numbers plus 80 units. We want it so that the two numbers added together form 100 units. Therefore, these two numbers added together form 10 numbers plus 80 units which equal 100 units. We subtract the similar terms: 10 numbers equal to 20 units remain and the number becomes 2 units (...)’. (Ver Eecke, 1926, p. 12-13; my translation). Having found that the unknown is 2, Diophantus finds that the sought-after numbers are 88 and 12. Problem 5 of Book 1 of the 5*%(,9/(%B" shows also another example of reasoning performed on fractions of the sought-after numbers. The structural coincidence between the algebraic concept of unknown and the false quantity of arithmetical, proportional thinking can be traced to a time preceding Diophantus, as we will see in the next section.
THINKING IN TERMS OF FRACTIONAL PARTS: EARLY HISTORICAL EVIDENCE The preserved fragment of a Greco-Egyptian papyrus, dated B%*B" the first century and called Mich. 620, contains three mathematical problems with the following being one of them: ‘There are four numbers, the sum of which is 9900; let the second exceed the first by one-seventh of the first; let the third exceed the sum of the first two by 300, and let the fourth exceed the sum of the first three by 300; find the numbers’ (According to Robbins' reconstruction, 1929, p. 325). Our modern notations allow us to write the problem in question as shown in the rectangle :
The first part of the solution is not completely preserved, but it can be reconstructed from a kind of tabular arrangement or ‘matrix’ placed at the end of the solution. It is used to display the calculations and functions as an aid to help solve the problem. The ‘matrix’, which is comprised of 4 columns divided by a vertical line, suggests that the choice of the unknown, which the scribe represents as , like Diophantus did in his 5*%(,9/(%B" to designate 'the number', S"*%(,9)+T1 is 1 / 7 of the first number. It is, therefore, also the same pattern found in Babylonian mathematics.
L.G. RADFORD
24
The first sought-after quantity, which appears at the left of the first column of the table (that is, at the left of the first vertical line; see below), is equal to seven (which is an abbreviation of the whole expression «7 numbers»); the second number From that, the (found to the left of the second vertical line), is equal to eight scribe finds that the third number is 300 plus fifteen and that the fourth number is 600 plus thirty . The sum of the numbers then is 900 plus sixty which must equal 9,900. The scribe gives the answer 150, which corresponds to and then arrives at the sought-after quantities: the first one is 1,050, the second one is 1,200, the third one is 2,550 and the fourth one is 5,100.
(Table appearing in the Mich. 620 papyrus according to the reconstruction of Frank Egleston Robbins, 1929, p. 326). It is worthwhile to note, at this point, that the separation of numbers into columns, allows the scribe to divide each number into an unknown part (found to the left of the vertical line) and a known part (found to the right of the vertical line). This suggests an explicit and systematic way of dealing with the first literal symbolic algebraic language. Notice that this is basically the same pattern which is used to carry out calculations with symbolic expressions some fourteen or fifteen centuries later (e.g. Stifel, 1544). Nevertheless, we should note a difference: in the first case that of the Mich. 620 papyrus the algebraic language is seen as a ,/4*%+(%B tool (one calculates >%(, symbolic expressions); in the second case that of late mediaeval and early renaissance mathematics the algebraic language begins to be seen as a quasiautonomous object leading to a #/> (,/)*/(%B"2 )*$"#%+"(%)# (one calculates )# symbolic expressions). Many treatises will then begin to display rules on how to carry out calculations )# symbolic expressions (see the excerpt from Stifel’s 5*%(,I 9/(%B" F#(/$*"1 page 239). –
–
–
–
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
25
THE BABYLONIAN NAÏVE GEOMETRY
So far, our work has dealt with the #49/*%B"2 origins of algebraic ideas. We claimed that the Babylonian mathematical proportional thinking provided the conceptual basis for the emergence of elementary #49/*%B"2 algebraic thinking. There is, however, another Babylonian mathematical current which leads to another kind of 'algebra’. In fact, J. Høyrup, through an in-depth analysis of the linguistic sense of the terms occurring in Babylonian mathematical tablets, has suggested that a large part of problems was formulated and solved within a geometrical context. using what he calls a ‘cut-and-paste geometry’ or ‘naïve geometry’ (e.g. Høyrup, 1990, 1985, 1993a, 1994). In particular, this is the case of problems that have been traditionally seen as problems related to ‘second-degree equations’. We cannot discuss here, at length, the Babylonian Naïve Geometry. For our purposes, we shall just look at two examples of the new interpretation of the second-degree Babylonian algebra (see also Radford, 1996a).
Problem 1 of the tablet BM 1390 1 deals with a square whose surface and a side 18 equal 3/4. The problem is to find the side of the square .
26
L.G. RADFORD Then, the scribe cuts the width 1 into two parts and transfers the right side to the bottom of the original square.
Now, the scribe completes a big square by adding a small square whose side is ½ The total area is then ¾ (that is, the area of the first figure) plus ¼ (that is, the area of the added small square). It gives 1. The side of the big square can now be
The basic idea is that of bringing the original geometric configuration to a square-configuration. However, not all the problems can be solved by cut-and-paste methods alone. For instance, problem 3 of tablet BM 13901 deals with a square whose area less a third of its area plus a third of its side equals 20‘.
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
27
As Høyrup suggests, the procedure followed by the scribe is that of removing a third of the original square (fig. 1). After that, a rectangle of Width 1 is projected over the side, obtaining a configuration like fig. 2, A third of the projection is kept, which leaads to the next configuration (fig. 3). Finally, ‘in order to obtain a normalised situation (square with attached rectangle), the vertical scale is reduced with the same factor as the width of the square, i.e., with a factor 2/3 (...)’ (Høyrup, 1994, p. 13). Now the scribe can apply the procedure which solves problem 1 from Tablet 13901 discussed above. We have discussed this last example because it shows how proportional thinking also permeates the cut-and-paste geometrical thinking. Changing the scale is, in fact, a proportional idea. On the other hand, it is important to note that cut-and-paste procedures involve known and unknown quantities in a very particular way. Firstly, in Naïve Geometry, semantics plays a strong role throughout the problem-solving procedure. In Numerical Algebra, rooted in proportional thinking, the original Semantics is lost once the equation is reached (cf. Mich. 620 and Diophantus' 5*%(,9/(%B"T: Secondly, in Numerical Algebra the unknown is directly involved in the calculations. For instance, in problem 6, Book 1, mentioned above, (and translated here into modern notations in order to abbreviate our account), Diophantus performs the following calculations: 6x + 4x+80 = 100, and gets 10x+80=100. He then operates on the unknown >%(,%# a side of the equation. In other problems he performs calculations involving the unknown in 3)(, sides of the equation (e.g. Book 1, problems 7-12. In problem 7, for instance, Diophantus solves the equation 3x-300=x-20. See Ver Ecke, 1926, p. 13 ff.).
28
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In contrast, the algebraic concepts rooted in cut-and-paste geometry (i.e. Naïve Geometry), do not seriously involve the unknown quantities in direct calculations (see Radford, 1995b, footnote 6). For instance, in problem 1, tablet 13901 seen above. the projection, and not the unknown-side, is halved. The previous discussion suggests that very different conceptualisations underlie the Algebra embedded in Naïve Geometry and the Numeric Algebra. As seen above, their 9/(,)'+ and their B)#B/D(+ are essentially different. The difference can also be seen in terms )7@ D*)32/9+: Most of the problems in Naïve Geometry deal with problems beyond the tools of first-degree or homogeneous algebra. However, it is possible to detect some interactions between both kinds of algebras. In fact, problem 27, in book 1 of Diophantus' 5*%(9/(%B"1 is a classical problem stated in the realm of Naïve Geometry. Although stated in a numerical form, Problem 27 has the traces of its old geometrical fomulation (see Høyrup, 1985. p. 103) and appears then as a numerical */B)#B/D(4"2%+"(%)# of the old cut-and-paste technique (we shall return to this point in the section, below). However, some connections between the cut-and-paste technique and first degree algebra could happen even within Babylonian mathematics themselves: this is what the solving procedure of problem 8 of tablet 1390 1 suggests. In fact, using Høyrup’s notations. we can represent the squares by Q1 and Q2 and their side by s1 and s2, respectively. The problem can then be formulated as follows:
The solution begins by taking the ,"27 of the sum of the sides. Taking a new side which is equal to the half of the sum of the sides is a recurrent idea in many of the Babylonian geometrical problem-solving procedures. The ‘half of the sum’ idea also appears in Babylonian numerical problem-solving procedures (see our discussion of tablets VAT 8389 and 8391, in Radford. 1995b). This suggests an early link between geometrical and numerical algebraic ideas.
LANGUAGE AND SYMBOLISM IN THE DEVELOPMENT OF ALGEBRAIC THINKING In this section, we would like to make a first attempt at exploring the problem of the development of early algebraic thinking with regards to language and symbolism. First of all, it is important to stress the fact that it is completely misleading to pose the problem of the development of algebraic thinking in terms of a transcultural epistemological enterprise whose goal is to develop an abstract and elaborate symbolic language. Indeed, language and symbols play an important role in the way
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
29
that we communicate scientific experiences. Nevertheless. their use is couched in sociocultural practices that go beyond the scope of the restricted mathematical domain. A more suitable approach to the study of the relationships between symbols and language on the one hand, and the development of algebraic thinking on the other, might thus be to analyse language and symbolism in their own historical sociocultural semiotic context. The case of Mesopotamian scribes may help us to illustrate this point. In this order of ideas, it is worthwhile to bear in mind that the oldest tablets suggest that they were first of all seen as a complementary tool to record information. The signs that formed a 'text' in the proto-literate periods known as Uruk IV and III (3300-2900 BC) reflected the key words of the messages inscribed on the clay tablet without any «syntactic relations» (Nissen, 1986, p. 329). The meaning was often suggested by the pictographic form of the sign; this is the case, for instance, of the sign SAG, 'head'. where the sign shows the profile of a head with the eye, the nose and the chin19 . The archaic pictographic writing was later replaced by cuneiform writing in all likelihood related to the needs arising from the transactions of the Sumerian administrative bureaucracy and the emergence of a new technological artefact: an oblique new stylus leaving the impression of 'nails' SB4#/%Ton the clay. The cuneiform writing was increasingly used to reproduce the oral language and when the Semitic Akkadian language became the spoken language, Akkadian was written following the cuneiform syllabic tendency. These changes brought about two important modifications: first. the way to convey the meaning of the text changed radically; it no longer relied upon pictographic insights. Second, there was a radical diminution in the number of signs. While by 3200 BC a writing system had some 30 numerical signs and some 800 non-numerical signs (Ritter, 1993. p. 14), by the first half of the second millennium Akkadian could be written with some 200 cuneiform signs reproducing the spoken language with very little ambiguity (Larsen, 1986, pp. 5-6). This point connects us with two of the most salient particularities of ancient Mesopotamian mathematical texts which have very often puzzled historians and mathematicians who attempt to understand Babylonian mathematics from the perspective of modem mathematics: firstly, the texts do not show any ‘specialised’ or 'mathematical' symbols to designate the unknowns; secondly, the texts do not display grandiose eloquence concerning the /KD2"#"(%)# of the problem-solving methods followed to reach the answer of the problem. In fact, concerning this last point, with the very moderate exception of the C/K(/+ !"(,d9"(%b4/+ '/ ;4+/1 the scribe limits him/herself20 to indicate only the calculation to be carried out to solve the 21 problem (if not, as is the case of many problems, to simply mention the answer) . The reason of what >/ see as a «silence» is not the absence of mathematical ‘specialised’ signs like 'x', 'y' (signs which are but merely inconceivable and unnecessary within the realm of Babylonian thought). This «silence» is due neither to an –
30
L.G. RADFORD
incapacity to write a mathematical expression and eventually, in the Old Babylonian period (2000-1600 BC), any algebraic one but an accepted sociocultural way to transmit and record the information which demarcates the frontiers of >,"( has to 22 be written and ,)> . To clarify this last point, remember that the tablets bearing some «algebraic» content were, in all likelihood. not produced in professional activities (e.g "BB)4#(I ing, book-keeping. surveying) but in the Scribal Tablet-Houses, that is, in the institutions created to train the future scribes. The clay tablets were, without a doubt, privileged tools in the teaching practices. However, they do not merely reflect the B)#(/#( of the teaching but they also mirror the method of teaching and its symbolic forms as well. Concerning the method of teaching, some tablets show that instruction was heavily based on the mastering of cuneiform writing; another, no less important point. was to prepare oral recitations (that were under the supervision of the second person in the hierarchy of the Tablet-House, the +/+$"2 or «elder brother». the assistant of the «father» of the School). In the written component of the scribal training, students had to copy what the teacher said or did. In many cases, the clay tablet shows a sentence (or a passage of a literary work) on one side and, on the other side, with a less confident calligraphy. a copy (visibly the student's copy) of the given sentence (see Lucas, 1979. p. 3 11 ff.) . It is not difficult to imagine the enormous difficulties that young students had to face trying to master the stylus and the rules of cuneiform writing. In a tablet. known as 'In the Prise of the Scribal Art', we can read: 'The scribal art is not (easily) learned. (but) he who has learned it need no longer be anxious about it.' (Sjöberg, 1971-72, p. 127). Another very well known text. 'Examination Text A'. that dates back to the Old Babylonian period. deals with the examination of a scribe in the courtyard of the Tablet-House. Besides the precise idea that this text provides us with an examination scene, the test uncovers some accepted teacher-student relationships such as symbolic forms emerging in the dynamic of the scribal school. The examination covers topics such as the translation from Sumerian (by that time. a dead language, as mentioned in footnote 22) to Akkadian and vice-versa. different types of calligraphy, the explanation of the specialised language (or jargon) of several professions, the resolution of mathematical problems relating to the allocation of rations and the division of fields. When the teacher starts asking questions about the techniques employed in playing musical instruments. the candidate gives up the examination. He complains that he was not sufficiently taught. Then the teacher says: –
–
‘What have you done, what good came of your sitting here? You are already a ripe man and close to being aged! Like an old ass you are not teachable any more. Like withered grain you have passed the season. How long will you play around? But. it is still not late! If you study night and day and work all the time modestly and without arrogance, if you listen to your colleagues and teachers, you still can become a scribe! Then you can share the scribal craft which is good fortune for its owner, a good angel leading you
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
31
a bright eye, possessed by you, and it is what the palace needs.’ (Quoted by Lucas, 1979,p.314).
In the previous passage we find mentioned explicitly the intensive work that is expected to be done by the scribe. More importantly still, we also find in the previous quotation clear instances of a symbolic form that emerges as a D"*(%B42"* social student-teacher relationship. The symbolic form is that which forces the student to show some specific attitudes: s/he is supposed to be modest and without arrogance as well as a good listener. By the same token, the same symbolic form allows the teacher to say what he said in the text. More eloquent is a passage of another text called 'Schooldays' (Kramer, 1949) in which the scribe tells us that he is caned by his teachers for doing unsatisfactory work. This symbolic form was supported by the scribes' relatives, who encouraged their sons to follow the teachers' requirements. In a tablet. the father says to his son: «Be humble and show humility before your school monitor. When you make a show of modesty, the monitor will like you.» (Quoted in Lucas, 1979. p. 321 ). We do not need to go further into the detail. It suffices to say that the aforementioned scenes clearly suggest that the teaching model relied heavily upon an incontestable imitation model concentrated on the proving of, among other things. the replication of passages from literary texts and procedural and computational mathematical skills. In this sense, the formal, semiotical content of the mathematical texts are but the mirror of the sociocultural web of relations on which schooling and. in general. all Babylonian social activities were based. To expect that the scribe product a mathematical text containing an "#"20(%B "2$/3*"%B /KD2"#"(%)# of the procedure is to expect him/her to do something that was out of all the enculturation with which s/he was provided in the Tablet-House. Explanation is, in fact, a sociocultural 23 value, not a transcultural item . Of course, Mesopotamian explanations did exist. Nevertheless, they were riot largely based upon deductive analytical principles but on metaphorical ones. A survey of literary and mythological texts is very enlightening in this respect (see for instance Kramer, 1961a, 1961b). The case of Diophantus was completely different. From the 5th century BC, arguing and explaining were two important social activities that shaped Greek thought. On the other hand, Diophantus had, at his disposal, an alphabetical language and a very well-established socially accepted system of producing and transmitting infor24 mation out of which books attained an autonomous life . Concerning algebra, even though Diophantus could use letters to represent the unknown he did not. As it is well established, the use of letters in Diophantus' 5*I %(,9/(%B" stand for an abbreviation of the word hence, contrary to our modern use, as merely an economic writing device. Even though he was not probably the first to do this. as suggested by the papyrus Mich. 620, one of Diophantus' most important semiotic contributions is to be found –
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L.G. RADFORD
in the peculiar use of the expression «arithmos» We previously said that he accomplished a transcendental act by including indeterminate units into the realm of calculations. But this is not to what we are referring now. Here we refer to the use of this expression of the Greek language to designate the algebraic experience that the concept of unknown carries out with itself. In the 5*%(,9/(%B" and in the specific definition of "*%(,9)+ we found this experience «uttered». While Mesopotamian scribes used the semiotic experience of everyday life and used words like the length or the width of a rectangle to handle their unknowns (using expressions like 'as much as' or 'the contribution of the length' to refer to what we now call the coefficients of a polynomial), Diophantus transformed the word «arithmos» into a more general concept. Because of its generality, this concept could apply to a great variety 25 of situations. The «arithmos» thus became a genuine algebraic symbol . Through this symbol, a numerical assimilating process of (some) geometric algebraic techniques was undertaken, leading to a new formulation of old problems and the rising of new methods in solving new versions of old problems. In turn, the new methods also allowed the Greek calculators to tackle new problems: Diophantus' 5*%(,9/(%B" contains problems which do not have any corresponding version in the algebra embedded in the Naïve Geometry. as is the case of problems concerning square-squares and cubo-square categories. Nonetheless, it is important to note that the great generalising enterprise was supported by the socially committed Greek conception of mathematics (further details in Radford, 1996b). The «arithmos»-symbolism played a particular role in a different semantisation of problems in the problem-solving phase and led to a more systematic and global treatment of problems. In order to illustrate this ideas more precisely, let us consider problems 27-30 of Book 1 of Diophantus' 5*%(,9/(%B": There is no doubt that these problems belong to an early tradition. According to Høyrup (1994, pp. 5-9), these problems, formulated, of course, in a geometrical language, go back to the surveyors' Mesopotamian mathematical tradition of the late 3rd millennium BC. These problems constitute part of a stock of problems that played a role in the rise of the Old Babylonian mathematics. Using Høyrup’s notations (modified very little), these problems can be stated as follows:
THE HISTORICAL ORIGIN OF ALGEBRAIC THINKING
33
In Mesopotamia, problem 27 was solved through the cut-and-paste technique. Diophantus does not use this technique directly. However, as was mentioned above. there are traces of the geometrical ideas in the new numerical algebraic problemsolving procedure (see Høyrup, 1985, p. 103-105; Radford, 1996a). Problem 28 appears in tablet BM 13901, problem No. 8 and in YBC 4714, problem No. 1. There is no direct Mesopotamian evidence of problem 29. (However, according to Høyrup, it is possible that originally problem 9 could be contained in a missing part of Text V of the C/K(/+ !"(,d9"(%b4/+ '/ ;4+/T: While the cut-and-paste technique docs not apply to "22 those problems in the same way, the revolutionary concept of the numerical algebraic unknown provides a similar way of tackling "22 these problems. The idea is to take the half of the sum of the sides added (or subtracted) by a certain number. (As was mentioned above, this is a procedure at the very cross-roads of geometrical and numerical ideas: see Radford, 1995b). Thus, in problems 27, 28 and 29, Diophantus represents the soughtafter numbers as «10 + 1 ‘number’» and «10 - 1 ‘number’» (in modern notations, it means 10 + x and 10 – x). In problem 30, the sought-after numbers are chosen as «X@g#493/*e@ + 2» and «1 ‘number’- 2». On the other hand, symbolism shifts the thinking from the figures themselves and makes it possible to carry out operations that do not have any corresponding sense with the initial statement of the problem. This is not the case of the cut-andpaste-technique, where it is possible to distinguish the sequence of geometrical transformations and its link with the original configuration. The introduction of the «arithmean» symbolic language provides an "4()#)9)4+ >"0 )7 (,%#=%#$ -autonomous with regards to the context of the problem. In contrast, it requires a new semantisation that has its own difficulties. This is to what Diophantus probably refers when he says, at the beginning of Book 1: ‘Perhaps the subject will appear rather difficult, inasmuch as it is not yet familiar (beginners are, as a rule, too ready to despair of success)’ (Heath, 1910, p. 129). Indeed, many of Diophantus' 5*%(,9/(%B" Scholia or comments (cf. Allard (ed.), 1983) deal with detailed explanations about the elementary symbolic treatment of the problems. Some of them use a geometric context to give a sense to the solving procedure. This is the case of a scholium of problem 26 from book 1, which explains the solution of the equation 25x2 =200X in terms of two rectangles of the same width. (Allard (ed.), 1983, p. 727).
SOME REMARKS FOR TEACHING The historical itinerary that we have followed in this work provides some information about past trends in the historical construction of very early algebraic thinking. These trends can help us to better understand the deep and different sociocultural and cognitive meanings of algebraic thinking and provide teachers
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L.G. RADFORD
with new paths to teach algebra in the classroom. In particular, our historical epistemological itinerary can shed some light on the didactic problem of how to introduce algebra in school. Of course, as we have pointed out in previous works, we do not claim that we 94+( follow the historical path. History cannot be normative for teaching. There are social and cultural aspects in the development of algebra that we cannot reproduce in the classroom. Furthermore, these aspects may not be necessary for our purposes. There are other aspects which could be more interesting, such as the following: (1) The epistemological meaning of algebra, i.e., that of mathematical knowledge developed around a problem-solving oriented activity, can provide some insights about the way of introducing and structuring algebra in the school; and this us to re-think, within a new perspective, the role of D*)3I 2/9+ in teaching algebra. (2) However, our study of Mesopotamian and Greek algebra clearly suggests that the specific form in which each algebra was conceived was deeply rooted in and shaped by the corresponding sociocultural settings. This point raises the question of the explicitness and the controlling of the social meanings that we inevitably convey in the classroom through our discursive practices. (3) Our epistemological analysis suggests that algebraic language emerged as a tool or technique and later evolved socio-culturally to a level in which it was considered as a mathematical object. Usually, in the modern curriculum, algebraic language appears from the beginning as a mathematical object D/* +/: Taking into account this result, it is possible to make some changes in the way of introducing algebraic language in the classroom. Following some insights of our historical studies of the development of algebra (see also Radford, 1995c), we elaborated a teaching sequence whose goal was to introduce students to algebraic methods based on different semiotic levels which culminates with the progressive introduction of symbolletters (Radford and Grenier, 1996). (4) A fourth point that we can consider, from a teaching point of view, could be the historical movement of arithmetisation of geometric algebra (which occurs in a recursive way through the history of algebra until the pre-modern epoch). Until now, the algebra embedded in cut-and-paste Naïve Geometry does not belong to the modern curriculum of mathematics; inspired by this historical research we were able to successfully develop a teaching sequence in the classroom that, through cut-and-paste geometry, has been allowing High-School students to re-discover the formula for second degree equations (Radford and Guérette, 1996).
THE HISTORICAL ORIGINS OF ALGEBRAIC THINKING
35
( 5 ) Another aspect to consider could be the link between proportional thinking and algebraic thinking. Ratio and proportions are not presented, in modern school curricula, as being linked to algebraic thinking in the way that history suggests it happened. It seems to me that the historical metaphorical link between proportions and algebra is another interesting element to be explored in the teaching of mathematics.
Acknowledgements I would like to thank Professors Nadine Bednarz, Louis Charbonneau, Jens Høryup, Barnabas Hughes, Jacques Lefebvre, Romulo Lins, Teresa Rojano and Rosamund Sutherland for commenting on a previous version of this paper. NOTES 1
2
3 4
5
6
7
8
9
10
This research was supported by a grant from FCAR 95ER0716 (Québec) and a grant from the Research Funds of Laurentian University (Ontario). A collection of problems from Tell Harmal shows how to calculate the total price of some current items used in commercial business using silver as the ‘monetary’unit of goods. Such items included sesame, dates and lard; see Goetze, 1951, p. 153. Some such ‘non-practical’ problems will be discussed in this paper. One of them is found at the end of this section. As pointed out by Damerow, numbers in Mesopotamian mathematics are not abstract entities; they are attached to specific contexts (e.g. weight of objects, grain production). A relative detachment from the context is suggested, however, by the tables of reciprocals in the early Old Babylonian period (ca. 2000 BC) (see Damerow, 1996, particularly pp. 242-246). In what follows, we will represent the numbers in base 60; for instance a number represented by 2'' 3' 6_ 11 2 _ 2 x 60 + 3 x 60 + 5 + 2 60 60 5º 6’ 11’ means It is important to note this nuance: Thureau-Dangin was very careful with the philological aspects of the translations; in contrast, the interpretation of such translations very often had recourse to modern algebra (see Høyrup, 1996, 7-9). For instance, when discussing one of the problems of the tablet VAT 8389, Thureau-Dangin refers to the equation 40’x-30’y=8’20 and says: ‘Le scribe ne formule pas cette equation, mais i1 1’a certainement en vue’. (The scribe does not formulate this equation, however, he certainly has it in view). (Thureau-Dangin, 1938b, p. xx). As we suggested in the previous section, false quantities were generated as metaphors of true quantities. Here, a new metaphor would be used by the scribes to generate a new concept -that of algebraic unknown. We will use modern algebraic notations in some passages of our paper in order to have an idea of the problems and the methods of solution under consideration. Modern notations are not used as structural artefacts in our interpretation of ancient mathematics. Stated, of course, in a Babylonian ‘natural’ context (e.g. a stone and its weight). The Textes Mathématiques de Suse were translated by Bruins and Rutten (1961). In these Textes, there are two problems called problems A and B of text VII, related to the width and the length of a rectangle. In a recent re-interpretation made by Høyrup (1993b). Problem A of Text VII concerns the _1 equation that, translated into modern notations. reads as follows: 7 (x+4 y) - 10 = x + y , where x represents the length and y the width of a rectangle; nevertheless, our modern translation does not
36
11 12
l3
l4
15
16 17
18
19 20 21
22
23
24
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L.G. RADFORD
distinguish some of the different conceptualisations between ancient numerical operations and the modern ones. Keeping this in mind, some of the steps of the translating solution include the following calculations: 1/7[(4-1) x + y (x+y)]10 = 4 . (x+y), 3x 10+(x+y) . 10=28 . (x+y), 3x . 10 = 18 . (x+y), x . 10 = 6 . (x+y) Then, the scribe chooses x=6 and 10=x+y and he arrives at y=4. (For a complete translation see Høyrup, 1993b). One of the points to be stressed here is the fact that the calculations showed in the previous sequence are based on an (implicit) analytical procedure: the scribe's calculations comprise the unknown quantities x, y (as seen in their own mathematical conceptualisation); the unknown quantities are considered and handled as known numbers, even though their numerical values are not discovered until the end of the process. ‘J' ai mange les deux tiers du tiers de ma provende: le reste est 7. Qu’était la (quantité) originaire de ma provende?’ (Thureau-Dangin, 1938b, p . 209). The problem of the transmission of algebraic knowledge and the sources of Greek (numerical and geometrical) algebra has been studied by J. Høyrup in terms of sub-scientific mathematical traditions. (see, e.g., Høyrup, l990a). The problem of whether a conceptual organisation is scientific or not is evidently a cultural decision. In the case of the Alexandrian algebra of the 3rd century B. C., it is hardly possible to ascribe to Diophantus the whole merit ofbuilding such a theory (Klein, 1968, p. 147). Nevertheless, we can say that, in all likelihood, his contribution was conclusive to this enterprise. Freeman, 1956, fragment 4, p. 74. When reading this quotation we have to keep in mind that Heath's translation is tainted by a modern outlook. Diophantus never spoke about 'negative terms'. Diophantus spoke rather of leipsis, i.e. of deficiencies in the sense of missing objects; this is why we might remember that a leipsis does not have an existence per se but was always related to another bigger term of which it is the missing part. For a complete translation of the problem, see Heath, 1910, p. 132 or Ver Eecke, 1926, pp. 12-13. This problem can be solved by the method of two false positions. Given that this method was invented later, we will not discuss it here. Høyrup’s translation of the problem-solving procedure is the following: ‘1 the projection you put down. The half of 1 you break. 1/2 and 1/2 you make span [a rectangle, here a square], 1/4 to 3/4 you append: 1, makes 1 equilateral. 1/2 which you made span you tearout inside I : 1/2 the square line.' (Høyrup, 1986, p. 450). Although the sign could be written in a stylised format, only a few variants were allowed. See Green, 1981, p. 357. There were also female scribes, although, in all likelihood, they were not a legion! Such a scribe is the princess Ninshatapada (see Hallo, 1991 ). For an example, see the solution to the problem No. 1,tablet BM 13901, note 18. It is important to note that although Sumerian language was a dead language in the Old Babylonian period, in mathematical texts the scribes kept using some Sumerian logograms and. in repeated instances, they added phonetic Akkadian complements to some logograms as well. This rhetorical twist indeed shows a deep mastering of a very elaborated writing. Note, however, that it does not mean that the scribes did the calculations by rote. Certainly, an understanding of what they wrote was part of the task of learning (some tablets show, for instance, that a good scribe was supposed to understand what s/he wrote). It would be teleologically erroneous to think that the non-alphabetical cuneiform language of the Old Babylonian period was a delaying factor to the emergence of algebraic symbols in the Ancient Year East. The cuneiform language was a marvelous tool to crystallise the experiences, the meanings and conceptualisations of the people that spoke Sumerian and later Akkadian. Alphabetic languages correspond to new ways to see, describe and construct the word. One language is not stricto sensu better than the other: they are just different (For a critique of the alphabetical ethnocentric point of view, see Larsen 1986, pp. 7-9). In the light of this discussion, it is easy to realise that it is an anachronism to see the development of algebra in terms of Nesselmann’s three well-known stages: rhetorical, syncopated and symbolic (Nesselmann, 1842, pp. 301-306); further details in Radford, I997
ROMULO CAMPOS LINS*
THE PRODUCTION OF MEANING FOR 5N8&6M5P A PERSPECTIVE BASED ON A THEORETICAL MODEL OF SEMANTIC FIELDS
INTRODUCTION Various characterisations of algebra and of algebraic thinking have been offered by different authors (for example, Arzarello et al., in this volume; Biggs & Collis, 1982; Boero, in this volume; Lins, 1992; Mason et al., 1985). Also, many articles, books and research papers have dealt indirectly with this issue. Choices made about what algebra and algebraic thinking are have a strong impact on the development of classroom approaches and material (Lins & Gimenez, 1997), that is, the discussion of this more theoretical issue is directly related to mathematics education in the classroom. Each author makes epistemological assumptions implicitly or explicitly, the former being much more frequently the case. Almost all those sets of assumptions have at least one common feature, inherited from traditional epistemologies. The first part of this chapter deals with the analysis of that common feature, arguing that it does not allow a sufficiently fine understanding of the process of production of meaning for algebra. On the basis of this analysis, a new characterisation of knowledge is produced, leading to an epistemological model in relation to which algebra and the production of meaning for algebra are, then, characterised. The second part of this chapter consists of the examination. from the point of view of the theoretical framework developed in the first part, of two situations in which the production of meaning for algebra occurs actually or fictionally. The discussion proposed in this chapter is about ways of conceptualising cognitive activity, and the role of the little empirical evidence introduced is simply to provide a vehicle for this discussion. For the purpose of keeping a sharp focus, I will always use examples related to quite simple linear equations; I hope the reader can see in them ‘exemplary examples.’ The design model presented on page 5 1 is subject to the criticism that it —
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* Mathematics Dept., UNESP, Rio Claro, Brazil
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is artificial; I would certainly agree with this, and with the suggestion that realistic or real-life situations should be part of mathematical education in school. Nevertheless, we should not forget that many times what starts as artificial becomes quite real for pupils. even more so with the younger ones; also, many aspects of mathematical education in school are not ‘naturally’ found ‘in the streets,’ and I think algebraic thinking is one of these.
RE-THINKING EPISTEMOLOGY
h# 7"9%2%"* $*)4#' Saying what algebra ‘is’ is not a minor problem. nor one without significance within mathematics education. But we can avoid this discussion for a while, and start instead examining a much less controversial situation. I am quite sure that everyone in the mathematics education community will agree that solving an equation such as ‘3K+10=100’ is algebra. If for no other reason than because one has to deal with an ‘unknown’ expressed in literal notation, and dealing with literal expressions of this kind is algebra, even if it is not the whole of the subject. How can the task of solving ‘3x+10=100’ be fulfilled’? Certainly in a number of different ways: (1) Try different numbers, until you (hopefully) get it right.
(2) Think of 3 boxes which. together with a l0kg weight. balance a l00kg weight. (3) Think of a number, multiply it by 3, and add 10; the result is 100. Now undo it.
(4) 3 parts of a value as yet unknown, together with a part of value 10, compose a whole of value 100.
(5) Add or subtract the same number from both sides; multiply or divide by the same number. Aim at an expression of the form K =. .. . All these approaches are, in fact, so familiar that many of us tend to take them as being only different appearances of the same essence. with the likely exception of (1). I will however argue that this is riot the case. We start by characterising what the equation ‘is’ in cach case. In (1) it is a condition to be fulfilled.
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In (2) it is a scale-balance. In (3) it is the do-list of a function machine. In (4) it is a whole-part relationship. In (5) it is a relationship ivolving numbers (including K) , arithmetical operations and equality. What can be done with a condition to be fulfilled? Nothing, apart from substituting numbers and checking. If the condition is changed everything changes. What can be done with a scale-balance preserving the equilibrium? Well, a lot, for instance adding the same to both sides; or removing the same from both sides, given there is enough to be removed. Secondly, and as a consequence, one can double what is on each side, but this is not necessarily as visible as the two other operations. The equilibrium is also preserved if we have 2.7 of what was on each side, but that is even less immediately visible. And with a function machine? Undo. One can, of course. use it as a condition to be fulfilled. What does not make much sense if it makes any at all is to have a result, the number on the right side. expressed in terms of the number one is seeking: this would be the case with the equation ‘4x+10=x+100.’ (Fig. 1 ) Strictly speaking, although ‘3x+10=100’ is a natural for function machine, ‘ 100=3x+10’ is not: even more disturbingly, ‘ 10+3x=100’is not natural either. —
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With respect to whole-part relationships, many things could be done: separate the parts (decompose the whole), for instance. Or compare that whole with some other; or make a part into a whole. Finally, as to the object in (5). one can check any textbook on school algebra. For someone who is acquainted with these five possibilities for producing meaning for ‘3x+10=100,’ it is possible to speak of metaphors and of switching from
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one to another. But we can, instcad. think of a child who is presented for the first time with ‘3K +10=100,’ and told ‘it is a scale-balance.’ It is likely that the child will learn to deal with that situation, and to use the ‘add-remove-share’ approach to solve similar equations. The question now is this: what would s/he say of ‘3K +100=10’? My guess is: ‘that one can’t be,’ as a student wrote in a similar situation, in a study we conducted (Lins, 1992). and the reason is quite simple: there cannot be a balanced scale-balance with 3 things and 100kg on one side and only 10kg on the other. If, instead, the child is told that ‘it is a function machine.’ and supposing that this is a notion already familiar for the child, a completely different situation arises. Now it is perfectly possible to produce meaning for ‘3K+100=10,’ but not for ‘4K+10=K+100,’ particularly within an activity aimed at solving this equation. Three points arise. First, the text ‘3K+10=100’ can be constituted into objects in at 1 least five different ways . Second, depending on the objects constituted, there will be a certain logic of the operations, that is. peculiar ways of handling those objects, things which can be done with them. Third. and crucially important for mathematical education, there are other equations for which it is not always possible to produce meaning in ways similar to those possible for ‘3K+10=100’. The impossibility of producing meaning for a given statement is what I call an /D%+(/9)2)$%B"2 2%9%(: This is a useful notion; as it points to the fact that producing meaning in relation to, for instance, a scale-balance. is not always a metaphorical act. A remarkable instance is the impossibility in Greek mathematics of producing meaning for incommensurability as related to numbers. For some authors the separation between geometry and arithmetic is, in Greek classical mathematics, simply a trick to avoid technical problems: however, Jacob Klein (Klein, 1968) has conclusively shown that this is not the case, and that the separation is fully consistent with the ways in which meaning for number and for geometric objects was produced in Greek classical mathematics. (see also: Lins, 1992)
<#)>2/'$/ To approach the problem of knowledge, we consider two people who have produced meaning for ‘3K +10=100,’ one of them in relation to a scale-balance, the other in relation to "2$/3*"%B (,%#=%#$i: Both would plausibly state that ‘one can take the same (10) from both sides.‘ For the first subject, it would be so because ‘if the same is removed from both sides of a balanced scale-balance the equilibrium remains.’ while for the other it would be so because ‘one can add any number (-10, for instance) to both sides of an equality, and preserve it.’ The question here is not, of course, whether or not both of them will do the same thing, but why will they do so in each case. The key problem becomes, then, is it adequate to say that both subjects
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share a knowledge? To make things more dramatic, we might want to consider a five year-old child who says that g2+3=5 because if I put two fingers together with three fingers 1 get five fingers.’ and a mathematician who also says that ‘2+3=5,’ but with a justification built from Set Theory. I think in both cases the answer is #)1 it is not adequate, but there is a further difficulty with that inquiry: we did not say what w e mean by ‘knowledge.’ Such a key question has been many times overlooked, possibly because there is a strong traditional view about the subject. or possibly because people do not see this as a relevant question within mathematical education, but most likely a combination of the two. Although many reformulations of the original notion have been produced. traditional understanding of knowledge is still bound to a classical definition: 5 D/*+)# %+ +"%' () =#)> (,"( D %7P S%T (,"( D/*+)# 3/2%/./+ %# D Y S%%T D %+ " (*4/ D*)D)+%(%)#Y "#': S%%%T (,/ D/*+)# ,"+ "**%./' "( D*)D)+%(%)# D (,*)4$, "# @ "BB/D("32/ 9/(,)'1 (,"( %+1 (,/ 3/2%/7 %+ O4+(%7%/': S;//1 7)* %#+("#B/, L"#B0 SXZZ[1 B,: VTT The above definition is usually taken as saying that to know is to have a justified true belief. Let’s examine some consequences of this definition. First, there has to be some knowledge-independent criterion of truth: some alternatives offered in this direction are objectivism, Platonism and Cartesianism. Second, and this is a very interesting aspect of the classical definition, saying that a method i s acceptable means that someone judges it acceptable; for some people casting shells is not an acceptable method for forecasting the weather, but for others it might well be. The implication i s that ‘knowing’ is a socially constructed according to it is something of an absolute situation, although ‘knowledge’ nature. With respect to the classical definition. justifications have to do with the right of a person to say s/he knows, but not with the constitution of ‘knowledge.’ The classical definition is troublesome. as shown by what is called the Gettier rightly, from the Problem, a construction in which someone would be granted technical point of view the knowing of something s/he does not know; the argument leads to the fact that the three conditions for knowing are not sufficient (Gettier, 1963). There is also the criticism, of a non-technical nature, that the classical definition rules out %9D2%B%( knowledge; I will return to this later. From the classical definition I want to emphasise the fact that knowledge, according to this definition, has the status of a proposition, being ‘that which one knows’ (for a very good and accessible discussion on the traditional view of ‘knowledge,’ see, for instance. Ayer 1986, Chapter 1); in fact, this is true also for the practitioners of the ‘implicit knowledge’ idea. Saying what knowledge %+ is not, of course, a matter of finding out the truth, but rather a matter of conceptualising things in a way which produces useful insights which to some extent agree with our general experience on the subject. We must —
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then ask: is ‘knowledge is that which one knows to be the case’ a good definition? According to that definition, we must say that the child and the mathematician share a knowledge, namely that ‘2+3=5.’ A critical instance emerges if w e consider the case of a very young child who confidently says that ‘2+3=5,’ because his father — a fine mathematician told him. The child believes what he is saying is true it is actually true, and his father is a credited authority in the field; the fact is, however, that the child is not talking about numbers not even ‘finger numbers,’ but we would be led to say that the young child shares a knowledge with his father, the mathematician. Clearly, ‘knowledge is that which one knows‘ is not an adequate notion. That the notion of ‘implicit knowledge’ relies on such an inadequate notion. can be shown by observing that someone has to say that a person has this or that ‘implicit knowledge,’ that is, at some point it assumes a propositional form. and because the person one is talking about is not aware of having the said knowledge, all we have is that proposition. If that requirement is dropped, that is, if we do not require that someone say that a person has this or that ‘implicit knowledge,’ the difficulties are even greater. as we should come to the conclusion that we all know all the things as pet unimagined by any human being. Also, we must be aware that ordinary language may be very flexible with respect to the uses of the verb ‘to know.’ Not all we know is knowledge; for instance, I might say that ‘I know John,’ but that does not mean ‘John’ is knowledge. In a similar fashion. to say that someone know-how (to do or to make) something is different from saying that a person knows-that. We can now highlight what seems to be three key aspects of knowledge. First, the person must believe in something if that is to constitute part of a knowledge s/he produces, and that implies s/he is aware of holding that belief. Second, the only way we can be sure of that awareness is if the person states it, and here I am using the term ‘state’ freely, meaning some form of communication accepted by an interlocutor: it does not have to be linguistic in form. Third, it is not sufficient to consider what the person believes and states: as different justifications with the same statement-belief correspond to different knowledge. Moreover, justifications are related to what can be done with the objects a knowledge has to do with: in the case of the child saying that ‘2+3=5,’ for instance, ‘2+3’ is the same as ‘3+2,’ once the arrangement of the fingers does not make any difference. From the point of view of a set-theory based justification, spatial arrangements are not something having to do with ‘ 2 ’ and ‘3’ or with their addition. Justifications, then, play a double role in relation to knowledge. First, they are indeed related to the granting of the right to know, and this granting is always done by an interlocutor towards whom that knowledge is being enunciated. Second, they are related to the constitution of objects. —
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Within the view I propose, =#)>2/'$/ is understood as a pair, constituted by the stated proposition which one believes to be true (the +("(/9/#(I3/2%/7T1 together with ajustification the subject has for holding that belief knowledge = ( statement-belief, justification)
A justification has to fulfil the double role indicated above: it has to be acceptable (for some interlocutor), at the same time as it constitutes, for the subject of knowledge, objects, that is, s/he can say something about those ‘things.’ Moreover, the definition establishes that there is always a subject of knowledge, rather than simply a subject of knowing, as in the traditional views; it also establishes that knowledge is produced as it is enunciated. I am not saying, of course, that knowledge is all there is to human cognitive functioning; knowledge is part of it, a substantial and accessible part. I am saying, indeed, that knowledge is characteristically human, as sign-mediated activity is.
52$/3*"
As the main purpose of this chapter is to discuss the production of meaning for algebra, my next step will be to give a characterisation for algebra; given the previous discussion. it does not seem reasonable to say that algebra is knowledge, Let’s see why. We may start noticing that we would naturally say that ‘3K+10=100 3K=90’ is algebra. But we have also seen that it is possible to produce meaning for that statement in a number of different ways; if that is to be knowledge, there is at least the justification missing. Second, it is true that, generally speaking, we identify algebra with statements D)(/#(%"220 interpretable in terms of relationships (equality, and eventually inequality) involving numbers and arithmetical operations: in order to say that ‘3K+10=100 3K=90’ is algebra. we do not make reference to which meaning is being actually produced for it. Instead of saying that algebra is knowledge, we would do better to say that it is a set of statements with the characteristics described in the preceding paragraph. Notice, however, that we know nothing about the meanings which will be produced for them by a given person, in a given situation; they may well be related to a scalebalance or to a function machine. That characterisation of algebra is operational for the purposes of research; in a later section I will show that it is also operational for the purposes of development. Algebra being a set of statements, it is (for me) text (cf. Lins, 1996). Producing 3 meaning for a statement of algebra is producing meaning for a text , and producing meaning for a text is to constitute objects from that text and relationships between
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them. Justifications. as an integral part of knowledge, play a role in the constitution of objects from the text of the statement-belief. The characterisation of algebra I propose is , then. operational in two aspects. First, in pointing out that sharing statement-belief’s in algebra is not enough evidence of sharing knowledge. Second, in pointing out that we should examine justifications if we are to identify the objects being constituted from the statements of algebra.
;/9"#(%B A%/2'+ The notions of algebra and of knowledge I have presented so far enabled us to clarify two things. First, to distinguish algebra from knowledge in which algebra contributes statement-beliefs: this is an important distinction because it allows us to account for different meanings produced for algebra. Second, it draws attention to the fact that ‘people dealing with algebra’ is a demarcation heavily marked by our own system of categories, and that we should take this into consideration when 4 examining other people’s activity of producing meaning for algebra. But there are further consequences. For instance, if Seeger’s quite interesting question (Seeger, 1991, p. 138) ‘How do teachers convert content into forms of interaction and how do students convert those forms into content,’ is reasonably reframed to ‘How do teachers convert knowledge into forms of interaction and how do students convert those forms into knowledge.’ it becomes clear that it is the dual role of justifications which play the key role. On the one hand. justifications are interactional in nature. i.e., knowledge i s always produced (,*)4$, interaction be it physically or remotely established and "%9%#$ "( interaction (Lins, 1996); on the other hand, by establishing objects they produce ‘content’ by producing meaning. The question now becomes: ‘Is there noninteractional learning?’ I am not asking, of course, whether someone can learn alone in a room; I am asking whether such lonely learning i s or is not actually interactionfree in a wider cognitive sense. When new knowledge is produced, it can be new in two ways. It can be new in that the belief stated was not a belief before its constitution into knowledge. But it also can be new in that a new justification i s produced for a statement-belief which had already been part of another knowledge, with anotherjustification. First we consider the case of someone who has produced meaning for ‘3K+10=100’ as a balanced scale, and then enunciates that (K1=) ‘I can take 10 from both sides and preserve the equality, because it i s a balanced scale.’ S/he now enunciates, for some reason (related to the person’s present activity), that (K2=) ‘I can add 90 to both sides and preserve the equality, because it is a balanced scalebalance’; that might be new knowledge, in case the person did not hold, before its enunciation, that stated belief in relation to the object he has constituted from —
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‘3K +10=100.’ Notice that the two justifications are produced in relation to the same kernel, involving a scale-balance. Alternatively, we consider that after producing K1 s/he might enunciate that (K3=) ‘I can take 10 from both sides and preserve the equality, because it is like two equal piles of stones.’ K1 and K3 have the same statement-belief, but justifications which are not only different, being in fact produced in relation to different kernels (involving a scale-balance in the first case and piles of stones in the second). In order to provide a more flexible and complete account of such differences in knowledge production, we need another construct, which I call a ;/9"#(%B A%/2': We will say that a person is )D/*"(%#$ within " $%./# ;/9"#(%B A%/2' whenever s/he is producing knowledge (meaning) in relation to a given kernel: we will refer, for instance, to someone operating within a Semantic Field of wholes and parts. Alternatively, we may say that a Semantic Field is the activity of producing knowledge in relation to a given kernel. A kernel may involve a scale-balance or piles of stones, but also wholes and parts, function machines, a straight line, areas, money, a thermometer, algebraic thinking, all sorts of fantastic creatures, colours; indeed, it may be composed by anything conceivably existing. What is known about the kernel is not ‘justifiable’ within that Semantic Field; they are 2)B"2 +(%D42"(%)#+: I am just extending Nelson Goodman’s notion of a stipulation (Goodman, 1984; Bruner, 1986). Although 2)B"2 +(%D42"(%)#+ are ‘given’ within a certain meaning producing activity, they are not necessarily basic in the strong sense proposed by Goodman for his stipulations, that is, they might well be questioned, challenged or even provided with O4+(%7%B"(%)#+ within some other ;/9"#(%B A%/2': We could perhaps say that reality is a ;/9"#(%B A%/2' with a =/*#/2 constituted by stipulations in Goodman’s sense. The notion of Semantic Field allows us a dynamic view about meaning production. On the one hand, we are able to consider how and if new knowledge comes or not to be part of a transformed kernel. On the other hand, we and if knowledge is produced in relation to a given are able to consider how kernel relate to each other. Moreover, and this is a key aspect, we are able to make full operational use of the notion of epistemological limit, already mentioned. By an epistemological limit I mean the impossibility of producing meaning for a statement within a given Semantic Field; for instance, it is impossible to produce meaning for the text ‘3x+100=10’ as a balanced scale-balance. The operational importance of this notion is to establish that: (i) every time meaning is produced there is a restriction on the horizon for further incaning production, implying that, rightly, I think as learning to produce meaning, (ii) if learning is understood teaching must also aim at an explicit discussion of the limits created in that process. epistemological limit In a later section I discuss the relevance of this construct to development and the classroom. —
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The first of the two possibilities considered a few paragraphs above (the one involving K1 and K2) I call a ./*(%B"2@ '/./2)D9/#(, the constitution of new statement-beliefs into knowledge within the same Semantic Field. The second possibility (the one involving K1 and K3 is a ,)*%H)#("2 '/./2)D9/#(1 the production of knowledge from the same statement-belief, but within a different 5 Semantic Field. A horizontal development between Semantic Fields S1 and S 2 always implies a vertical development within S2. Also, a horizontal development characterises either the establishment of a metaphor (‘3K+10=100’ (an arithmetical relationship) is as if it were a balanced scale’) or of a reversed metaphor ('a balanced scale is as if it were 6 an equation (an arithmetical relationship)'). The constructs =#)>2/'$/ and ;/9"#(%B A%/2' form the core of the C,/)*/(%B"2 !)'/2 )7 ;/9"#(%B A%/2'+ (TMSF). On this basis we can speak of producing meaning for "2$/3*" (a (/K(T as the production of knowledge from algebra within Semantic Fields. Operating within different Semantic Fields means constituting )3O/B(+ to which particular 2)$%B+ )7 )D/*"(%)# apply. New knowledge can be produced through vertical and horizontal developments. F#(/*2)B4()*+ are the source of legitimacy for knowledge, and truth is relative, but not 'absolutely relative.' From the point of view of the TMSF truth is not a notion to be applied to the statement-belief, to the proposition which we know to be the case, but to knowledge, which implies that truth is a cognitive notion, and not objectively related to 'hard facts.' To be able to decide whether or not a statement is true, certainly one must make a decision on what is being talked about; but 'what is being talked about' is constituted precisely through knowledge enunciation, and truth is, thus, relative. Justifications have a role to play in the establishment of truth, and once justifications are always produced towards interlocutors, the 'individual' cannot any longer be taken as a source of truth, as assumed, for instance, by radical constructivism. What is produced is a relativism which has cultures, through the many practices which compose them, as the domains of relative validity of any given truths. Within the TMSF the distinction between algebra and algebraic thinking becomes natural. Moreover, thinking algebraically is to be seen primarily as a consequence of cultural immersion. GLANCING AT MEANING PRODUCTION FOR ALGEBRA In this second part, I will present and discuss some empirical material and a design model for classroom activity which has been tried with pupils; all the material presented is intended only as a vehicle for discussing the notions in the TMSF, giving the reader a chance to see how the notions proposed 'work in practice.'
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;%9D2/ >)*' D*)32/9+ As part of a wider research and development project7, we have separately interviewed two pupils, presenting them with word problems. Our main objective was to investigate two things: (i) the objects with which pupils were operating; and, (ii) the role of interlocutors in the process of producing knowledge related to the solution processes. The strategy adopted was to question pupils as to the justifications they had for making the statements they did; in the particular case of the problems we will discuss, the basic ‘solution’statement always had to be a choice of operation which solved the problem proposed. Secondarily, there were other statements related to the justifications offered by them. The two pupils interviewed studied in the same state school in Rio Claro, Brazil. FEE, a girl, 13years old, and FAB, a boy 12years 7months old. The interviews lasted about one hour, during which they solved three or four problems; only one of those problems is discussed here, the Oranges&Boxes problem: (1) To calculate how many oranges will fit into each box, we divide the total number of oranges by the number of boxes, i.e., number of oranges oranges per box = number of boxes (a) If I tell you the total number of oranges, and the number of oranges in each box, how would you calculate the number of boxes used? (b) If I tell you the number of oranges in each box, and the number of boxes, how would you calculate the total number of oranges? The reason for presenting the ‘algebraic’ formula was to ascertain whether the pupils would constitute it into an object, dealing with it in the process of solving the problem; neither of them made any reference whatsoever to this formula. At first FEE seems confused about what is given and what is not. After a somewhat long exchange, she says FEE Then ...for example, there are 40 oranges, I divide byyy...(softly) Wait a minute.. .(reads the problem again, very softly). . .Yeah.. .I divi...then I would divide the.. .the total num.. .you’ll, for example, you ... for example. I have 10 boxes, 4 go into each box, then I divide the total (number) of oranges by the total of.. .how many go into each box (writes down: ‘I would divide the total of oranges, by’; looks to the interviewer) By the oranges, isn’t it?
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R.C. LlNS I What is it you want to say? FEE That I would divide the total of oranges by the oranges that go into the box. (and writes down: ‘how many oranges go into each box.’) Can I move to b ? We asked her about justifications: I . ..why is it that calculation (a division) and not another? FEE Because I am, for example, I am, I am, I have to divide among the boxes, got it, to know (‘the number’; said together with the interviewer’s next question) (. . .) FEE So, but you told me how many go into each box, OK.. .I would divide them ...for example. you say that 4 go into, there are 40, I would divide them (the ‘I would divide them’ is accompanied by a gesture: open hand, palm down, touches the table as if indicating ‘lots’), got it.. .? I Divide means what? Sharing. you’re doing? FEE Yeah, for example I put 4 in a box, 4 in the other. ‘till it’s finished.. .then I would know how many.. . (. . .) I (...)...you’re saying you thought this way, but you did a division.. .How...why did it occur to you to make a division? FEE Well, because I had for me to, for example, for me to ...to know how many.. .how many will go, for example, you have 3.. .have.. .20 oranges to put each.. .to put 5 in each box, then I'll have to divide, got it.. . ? I can’t multiply it will increase the oranges, got it? Nor add, nor subtract... I Why not? FEE Because I can’t!! (laughs) How will I add? Look, there are 38 oranges, I will add to what? (One) has to divide with the boxes, I can’t add.
A number of things emerge. FEE used specific numbers, but they were always ‘number of something,’ boxes, oranges, oranges per box, and both elements are relevant. It seems that the role played by the chosen numbers is to check a reasoning in which the logic of the operations is primarily related to sharing oranges into boxes; there is an interesting interplay between using ‘divide’ to refer to an arithmetical operation and to sharing. Moreover, the numerical division and the sharing are so closely bound that she has difficulties in explaining why she did a division when she was ‘in fact’ doing a sharing; the explanation she gives reveals that both realistic constraints (‘I can’t multiply. it will increase the oranges!’) and
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dimensional constraints imposed on the quantities involved were part of the kernel in relation to which meaning was being produced. FAB, the other interviewee, was presented with the same problem. Prompted to ‘think aloud,’ he said,
FAB I thought ... I didn’t see this bit here ...I want to know how many oranges go into, then I.. .how many oranges would take.. .and would divide ...I would take all the oranges and divide ...by the number of boxes.. .(frowns) No.. .no...I wouldn’t have the quantity of boxes.. . I OK, then.. .but you want to know the quantity of boxes.. .what do you know’? You know how many oranges go into each box, how in any oranges altogether, you want to know how many boxes you need.. . FAB ...(lo oking at the problem) I would take the whole and put . . .I would divide by the number of bo.. .of oranges that go into.. . FAB (reads what he had written down) ‘I would take the quantity of oranges and divide by the quantity of oranges that go into each box.’ I Are you sure that you would get the number of boxes.. . FAB (nods, and moves to ( 1 b)) I What did you think that took you to this conclusion? FAB (smiles) The.. .(smiles). . .I.. . For a while he did not say anything, so we asked him to play as if he had to explain to a cousin why he did the problem that way; FAB said it couldn’t be because his cousin was too young to go to school. We decided to adopt 'a friend' instead of the cousin. Imagine your friend is there, at your side. and he asks ‘Listen, FAB, how do you know it? How did you think to.. .solve the problem?’ FAB . ...Well, I thought if I had the oranges with the box. I Hmm. Try to show me.. .How did you imagine it? Try.. .if you want to make a drawing, anything with the hands, to speak. whatever you want. FAB (drawing round shapes on the paper) I imagined I had a pile of oranges. I Hmm. FAB Then ...I took a box (draws a square 'u')...which would hold ...a certain amount (draws some round shapes inside the 'box') Then I thought ‘If I divide this amount of oranges (points to the shapes outside the box) by the amount which is inside (points to the box). . .which goes into here, 1’11find out.’ I
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FAB’s solution is substantially different from FEE’S in at least one key aspect: he #/./* mentions any specific numbers. The objects he is operating with are related to boxes and oranges. More precisely, he seems to constitute the following objects: unit oranges, a pile of oranges, and a box. Those objects have properties; for instance, the pile of oranges can be counted or separated into boxes. The logic of the operations engendered by his construction did not depend on specific numbers. It seems he thought of the operation ‘separate the oranges in the pile into smaller, equal, groups,' and only then indicated the arithmetical operation division which, as a tool, would be used in order to do an actual calculation. The use of an arithmetical operation is subordinated to the logic of the operations proper to the objects constituted. The boxes-oranges kernel is so solid that when first approaching the problem he says ‘I would take the whole and put...,’ and immediately changes to ‘I would divide by (. . .)’ (my emphasis); he is thinking around a kernel of boxes and oranges, but the problem says 'how would you B"2B42"(/:’ Another aspect of interest is that, although he had already hinted that he was operating with 'boxes and oranges' objects, it took a long exchange before FAB felt he could give the justification he apparently had in mind, and it is remarkable that this exchange involved precisely a proposed change in interlocutor, from the interviewer to a peer. The slip-of-the-tongue 'I put,' provides a strong indication that the justifications given later were not some 'rational reconstruction.' but rather a true enunciation, to a new interlocutor, of a faithful account of the 'actual' solving process. FAB also solved (1b) correctly, but this time it took him no time to produce a justification within a Semantic Field of oranges and boxes: I The same thing: how would you explain it to your friend? FAB I had ...as I imagined, that I had the boxes, some ten boxes. Then I would do the opposite to that (points to (la)). I would take as if I was going to count, but instead of counting I would find out how many oranges in each box and would do times. By the number of boxes used. ... I Then you did a multiplication...Why? FAB Because it's quicker, isn't it, than counting one by one. I But it is the same thing you're doing. just that to calculate... FAB Yeah.
A possibility for FAB’s need to specify 'some ten boxes,' is that in the ordinary experience of most people with boxes there are never too many and these can be precisely quantified without difficulty, while with respect to piles of things one is
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almost never expected to have a precise or near-precise idea of the quantity; put in other words, D%2/+ )7 )*"#$/+ "#' +/(+ )7 ,)+/+ 9"0 ,"./ ,"' '%77/*/#( D*)D/*(%/+ >%(, */+D/B( () b4"#(%7%B"(%)#: In solving (la) it is just natural that the number of boxes is not determined at first, and the other objects (a pile of oranges and lots of oranges to be put into boxes) do not suggest the need of a perceptual at-a-glance quantification. FAB clearly constituted a distinction between the operation actually carried out (counting of a number of equal lots) and the arithmetical operation used as a tool to evaluate the result of the counting: FEE, however, did not. Altogether, the interviews showed us different processes of meaning production for a text involving oranges and boxes. Numbers were constituted as different objects in each case, that is, they had different properties and played different roles. Distinct logics of operations were in place, but in neither of the two cases properties of the arithmetical operations played a part in these logics of operations, that is, arithmetical operations were not made into objects.
5 '/+%$# 7)* /KD2%B%( O4+(%7%B"(%)#+ In this section I will argue that justification, as a constitutive part of knowledge, has to become an explicit part of classroom environments. Presenting pupils with 'problems to solve' will focus the activity on producing a solution, and it is only natural that trying to get them to discuss their methods starting from a solutiondriven problem requires some considerable effort on the part of the teacher. Classroom common-sense, built both from tradition and from some scientific conmion-sense, suggests that the natural direction is from 'concrete' to 'abstract'; one possible interpretation here is that 'concrete' may refer, for instance, to problems with specific numbers, while 'abstract' would refer to problems with generic numbers. Freudenthal has already argued against such conception, pointing out that it is not true that generality is always achieved through generalisation (Freudenthal, 1974). More particularly, this comment was prompted by an analysis of Soviet work on the early introduction of generic, literal, expressions to pupils, and in the cases he examined, 'early' meant the first grades of primary school. If we take in particular the pioneer work of V.V. Davydov (e.g., Davydov, 1962), the most striking feature is a conceptual shift through which he departs from the accepted notion that 'numerical literals' (our terms) can only be made meaningful as 'variables', as generalisations of specific numbers. What Davydov proposes is to work from generic quantitative relationships, as one would find in a situation involving cars and trucks in a parking lot: 'In a parking lot there are two kinds of vehicles: some are trucks and some are cars. If all the cars leave, which vehicles are left?'
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A%$4*/ V As we have already indicated in Lins (1994), Davydov’s work is based on the notion that even if the original 'support' is provided by a trucks and cars situation, what one is in fact dealing with is the 'true' nature of simple algebraic relationships. i.e., pupils are given the chance to work with an embodied version of the essential whole-part relationship. From the point of view of the TMSF, however, what Davydov is proposing is that pupils produce meaning for literal expressions within a Semantic Field of cars and trucks, and then use these expressions as a departure point for beginning the development of Algebraic Thinking, as meaning for new statements gradually conies to be produced in relation to statements already made meaningful, rather than in relation to the original situation (kernel). This shift is not treated explicitly in Davydov’s activities, but it could have been. Based on Davydov’s original idea, I have developed an activity in which literal expressions are made meaningful within a given Semantic Field (of water tanks), and then a deliberate shift in the way meaning is produced for new expressions generated is proposed to pupils. This activity has been tried with sixth-graders in Brazil, and an overview of the results is presented in Lins (1994), will not be discussing here actual pupils’ work, but the overall shape of the approach proposed. The activity is introduced with the following text. What is being proposed is that the tanks situation will constitute a kernel in relation to which meaning will be produced for various statements. Objects or could be buckets and tanks. Implicitly, there is constituted in that kernel are also water, or some other liquid; another local stipulation, suggested by the drawing, is that the two tanks are of equal size. One possible first statement is. S1 ‘The tank on the right has more water than the tank of the left.’ There are, however, at least two different justifications for the enunciation of this statement-belief: ‘The line of water is higher on the tank on the right,’ or, J1A —
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‘9 buckets are needed to fill up the tank on the left, but only 5 are
needed on the right.’ Within the TMSF, the knowledge K1A= (S 1, J1A is different from K1B= (S1, J1B). The difference is not just a formal one; in K the drawing itself becomes an 1A object which has a place in cognition, while in K1Bthere is only reference to the 'verbal' part of the text. Notice that we are not claiming that the subject enunciating K1 B has never or will never constitute the drawing into an object: it is just that in the enunciation of K1 B that object does not seem to exist. While K1 A seems to have a more qualitative nature, K1 B seems to have a more quantitative one. To appreciate more fully the characteristic difference between and K1 B, we may consider whether similar justifications could be used in relation to the following question: ‘If I add one bucket of water to the tank on the left. will there still be less water in it than on the tank on the right?’ Operating with a quantitative relationship there would be no difficulty in providing an affirmative answer. ‘If one adds a bucket of water to the tank on the left, it will still have S2 less water than the tank on the right.’ ‘8 buckets will still be needed on the left, but only a on the right’ J2 Operating only with an object constituted from the drawing, however, it is impossible to produce an answer. It is possible, of course, to consider that the top 'white' space on the right corresponds to 5 buckets, and to estimate the 'slice' corresponding to a bucket, using that estimate to conclude visually that there would still be more water on the right than on the left. That operation, however, depends on also constituting 'a number of buckets' as an object. Returning to the activity, I would propose that the pupils produced valid statements together with justifications. It became more comfortable to assign singleletter names to the objects being referred to; thus. 'buckets' became 'b,' and equality was naturally indicated by '='. As to the tanks, we finally agreed on 'T'. The amount of water in the tank on the left was named 'X', and that in the tank on the right, named 'Y'. All these choices were made together with the pupils, and they seemed to have added no further difficulty to the activity. The pupils in question were Brazilian sixth-graders (12-13 years old), who would have had by then an introduction to simple equations using x’s or y’s. One might expect to get expressions like 'X+9b=Y+5b,' which are not to be understood as 'equations.' At first it is natural to get justifications which all refer back to the kernel, for instance: S3 ‘X+9b=Y+5b’ ‘If 9 buckets are added to X, we will get a full tank, and the same J3 happens if we add 5 buckets to Y.’ Or,
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S4 J4A
‘X+4b=Y’ ‘If 4 buckets are added to X, there will be 5 buckets missing on the left. and this is what is missing on the right.’
The enunciation of the knowledge (!4,"4A) produces meaning for S 4 within a Semantic Field of tanks, Meaning for a statement is produced by the constitution of objects froin that text. One could also produce the following justification for S4: J4B ‘If 9 buckets are missing on the left, and 5 buckets missing on the right, this means that Y has 4 buckets more than X.’ which is, of course, different froin J4A. The reason for eliciting justifications which refer back to the kernel is that the statements must first be made to correspond to objects already constituted, and the kernel, with its objects constituted through local stipulations plays the psychological role of reality: that is, of course, an approach radically different from objectivist theories of meaning, for which ‘hard core reality’ objects arc the things in which meaning is 'anchored'.8 Once meaning is established for a set of statements. within a Semantic Field of talks, it is possible to suggest that pupils explore another way of producing correct statements about the tanks situation, and this is done by examining possible relationships between already established statements: how could one ‘reach’ the already meaningful starting from the already statement ‘X+4b=Y’ statement ‘X+9b=Y+5b’? Our algebra-educated minds would meaningful certainly say, quite naturally, ‘Take 5b from each side.’ Before that can be taken as a natural step, however, we must consider that the very task proposed involves two crucial steps: (i) that the statements themselves become objects; and, (ii) that pupils’ thinking shifts quite strongly froin the tanks situation. Let’s examine the consequences of that. First, in order to constitute the statements into objects. we must be able to say something about the properties they have "+ >,)2/ +("(/9/#(+Y it is not enough to say what their constitutive elements "*/ nor what the statement says about those objects. But pupils are precisely being required to say something about what does not exist yet for them. +("(/9/#(+ 4+ )3O/B(+: There seems to be an epistemological paradox here. Second, any direct transformation froin ‘X+9b=Y+5b’ to ‘X+4b=Y’ will produce a new meaning for the latter. But meaning has already been produced for ‘X+4b=Y’; why would a pupil take aboard this new meaning instead of, or even in addition to, the original one, naturally produced by taking ‘X+4b=Y’ directly in relation to the kernel? There seems to be a didactical paradox here. —
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Before we set out to solve those paradoxes, a key question must be answered: why would >/ want to introduce a new way of producing meaning, when the previous one seems so promising? The answer to this question is as simple as it is crucial in solving the paradoxes: we do this because >/ >"#( pupils to be able to operate both ways: >/ >"#( them to be able to produce meaning linking a statement to a kernel and >/ >"#( them to be able to produce meaning by performing direct transformations of statements. But this is essentially a decision made on a cultural basis: that is what our culture expects from someone who is performing the function we are; there is no plausible reason to believe " D*%)*% that in any other given culture people are expected to be able to produce meaning through the direct transformation of statements. What makes this a key assumption in the solution of the paradoxes, is precisely that our function as interlocutors will provide the reference for the intention to constitute whole statements into objects, at the same time we, as interlocutors, are the agents of trying to get the pupils to engage in the activity of producing meaning in the new way we are proposing. The paradoxes are solved, then, by observing that there is an intermediate step in which the %#(/#(%)# to engage in a new activity is the key factor, and during which however brief it is authority plays a crucial role. It is never too much to observe that I am using the notion of authority just as to indicate a reliable point of reference. The statements are, then, first constituted into ‘objects whose properties I do not know (although I know they exist because my reliable interlocutor indicates so).’ This is not very different from pupils listening to a lesson about a distant country: ‘There is a place where people.. .’, and that is precisely what constitutes the country and the people the pupils engage in thinking about. It is certainly essential that they have already produced some meaning for 'people' and 'country.' Both epistemological and didactical paradoxes are solved at once: the first meaning for a statement as a whole is precisely 'statements can be treated as a whole', and the justification is the teacher’s authority, although probably nothing else is at first known about >,"( really can be made with them; on the other hand, pupils engage in that activity %7 (,/0 ') -because the teacher represents if s/he does the legitimacy of the newly proposed way of producing meaning, and because pupils want to belong to a social practice in which that way of producing meaning is legitimate and desireable. The paradoxes were rooted, in fact, in conceiving the possibility of a transition from the old to the new. But the formulation I present makes clear that it is not the case of a transition, but actually it is the case of a rupture, and that the rupture is promoted within a process of interlocution: ‘let’s do it differently,’ and someone has to have a reason for doing it differently.
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The next step would be the production of new correct statements in relation to the tanks situation, but with the requirement that two justifications are produced: one in relation to the kernel and another produced by a direct transformation of a previously accepted statement. The reason for this is precisely to make explicit the existence of two different ways of producing meaning for new statements. It is possible to produce meaning for the statement, S5 ‘X-1b=Y-5b’ both with J5A ‘If I take 1 bucket from X, 10 buckets will be missing; the same happens if I take 5 buckets from Y, because 5 buckets were already missing, and I am taking another 5.’ and with J5B ‘Take 5b from each side of X+4b=Y, which I had already established as correct.’ As soon as the property ‘one can take the same from both sides of a statement’ is established, it is possible to produce statements like ‘X-40b=Y-44b,’ which although correct according to the properties of the objects ‘statements,’ cannot be made meaningful within a Semantic Field of tanks, simply because it does not seem plausible that one can remove 40 buckets from X. It is now possible to produce a strong distinction between the two modes of producing meaning, on the basis of the fact that there are objects in one case which cannot be made into objects in the other. The crucial step is, I insist, to produce a rupture, not a transition.9 There are several possibilities to follow from here. One of them is to start working on producing ‘target-statements’ from statements already produced, for instance, from ’X - 1b=Y-5b’to produce a statement of the form ‘X=...’ or ‘Y=...’ or, later, of the form ‘b= ...’. In one of the groups, two solutions appeared to the ‘targetstatement’ ‘b=...’ from ‘X+4b=Y’: ‘b=Y-X-3b’ and ‘b = j \ k ' The latter came 4 from a student whose mother is a mathematician, who gave her ‘a hint.’ The activity described above is intended to provide a design model. On the basis of the rationale for that design is the fact that meaning can be produced for algebra in a number of different ways. It is meant to indicate that the so-called ‘concrete embodiments,’ if taken together with the assumption of the possibility of a transition the traditional didactical effort to produce a ‘silent transition’ is likely to hide from pupils the fact that they are being required to produce meaning within a new, and distinct, Semantic Field. Moreover, it indicates how it is possible to overcome such problems and still keep the possibility of starting from already constituted, familiar, kernels. Three axis were taken into consideration on the design proposed: (i) one which goes from +)24(%)#I'*%./# to 9/(,)'I'*%./# activities, the latter being characteristic
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of the Tanks activity; (ii) language, representations and notations; and, (iii) the logics of the operations, that is, kernels and Semantic Fields. SUMMARY AND CONCLUSION One key thing has been shown in Parts 1 and 2: that the characterisation of knowledge adopted by traditional epistemologies is inadequate, in particular for mathematics education. As an alternative, I have presented a characterisation of knowledge which incorporates as a constitutive element the justification a person has for believing that something is the case. The notion of knowledge as a pair (statement-belief, justification) is the basis for the construction of the Theoretical Model of Semantic Fields. Within that model, the production of meaning is an activity which happens around kernels constituted by local stipulations. That activity constitutes Semantic Fields. Justifications have the double role of constituting objects and of taking part in the process of a person being granted with the right of knowing that such and such is the case. Objects are, then, constituted within Semantic Fields. Whenever objects are constituted, there is a particular logic of operations which applies to them, i.e. what can be done with them. Knowledge is always enunciated to an interlocutor. Within the Theoretical Model of Semantic Fields, interlocutors are an essential part of cognition, as the production of meaning is always directed towards an interlocutor. When we produce meaning we are speaking to an interlocutor, either internal or external. Characterising algebra as a text, rather than as knowledge, allowed us to account positively for different meanings produced for it, without having to slide into a hierarchy in which 'official' ways of producing meaning arc at the top. There are two key consequences: (i) we are not forced any longer to treat children’s cultures nor any other culture as 'lacking';and, (ii) we are able to characterise the process by which meaning production might if not properly dealt with constitute limits for pupils’ learning. With respect to item (i), it is important to point out that any epistemology which characterises what is on the basis of the very culture within which it has been produced, is clearly unable to be of much use in helping us to move forward, to go beyond limits historically and materially produced. A number of educational consequences can be drawn. First, that instead of simply looking for ‘meaningful learning’, we must take into account the possibility of different meanings, and that we may be particularly interested in getting pupils to produce meaning in a specific way; in the case of algebra, the various ways of producing meaning are of interest, but we may be particularly aiming at getting them to think algebraically, although not to the exclusion of other possibilities. —
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Second, it is possible to organise classroom activity around different modes of thinking, rather than in terms of 'content' as given in mathematics itself. One key aspect of this new organisation, is that contents would be #"(4*"220 integrated. For instance, around 'thinking with wholes and parts' one would have fractions, some algebra, some geometry. Around ‘thinking with scale-balances’ one would have some physics, some algebra, some measurement. I am not mentioning the possibility that contents from outside mathematics are also taken aboard. In a sense, organising classroom activity around modes of thinking — as I characterise them — has some flavour of project-oriented approaches; I believe the two ideas can and should be thought of together. Third, that by getting pupils to make justifications explicit we may go far beyond the simple possibility of checking whether they ‘really’ know what they are saying. Through this process pupils and teacher will be able to produced +,"*/' meaning and +,"*/' knowledge. In terms of the teaching process, that enables the teacher to identify and approach, from inside, situations where learning is not occuring. Whenever the teacher has to deal with it only from the outside, there are two possibilities: (i) insist on the approaches already used, as if pupils needed a second chance to ‘see’ what they did not in the first try; or, (ii) leave it to the pupil, perhaps in the sense of assuming that the pupils was not yet ready to learn those ideas being proposed. In both cases the teacher is very much passive. But by entering into the pupils’ world of meanings, and by making explicit that at some points new ways of producing meaning are being proposed, both teacher and pupils become truly active in the constitution of a common, shared discourse. The sharing of statement-beliefs "#' justifications, on the other hand, are not seen any longer only as a politically correct attitude, in the solidarity sense of sharing; that is certainly important, but there is now also the fact that such a process is an essential, constitutive part of learning, as it is through this that the legitimacy of given modes of thinking is eventually established for those ‘listening’. I had already mentioned the role of the teacher as an interlocutor; the role played by pupils among themselves is quite similar to that of the teacher. Fourth, and last, there should be a shift away from the usual ‘concrete to abstract’ notion. The suggestion is that we stop thinking of scale-balances and function machines, for instance, only as intermediate steps in the road towards ‘what algebra really is.’ Instead of asking the question ‘how to bridge the gap,’ perhaps we would rather acknowledge that there is no possible ‘bridge,’ there is no transition. The idea of a transition is certainly rooted in the notion that there is a higher level of thinking to be reached from lower ones. Davydov’s work, and my own, shows that changing the perspective with respect to ‘concrete to abstract’ allows us to produce powerful classroom approaches.
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Although widely aimed as a model for epistemology, I think that the Theoretical Model of Semantic Fields provides a simple, yet powerful, tool for research and development in mathematics education, as well as for guiding classroom practices and for enabling teachers to produce a sufficiently fine, thus useful, reading of the process of meaning production in the classroom. Finally, I would like to emphasise that the Theoretical Model of the Semantic Fields is not a 'local theory' aimed only at the production of meaning for algebra; it is similarly applicable to all parts of Mathematics. In fact, it applies to any process where the production of meaning occurs, but this is certainly not the place for such a discussion.
Acknowledgement I would like to thank Alan Bell, Paolo Boero, Ole Skovsmose and Rosamund Sutherland for their insightful and sharp comments on many of the ideas presented here. I am also indebted to the other members of the PME Algebraic Processes and Structure Working Group.
577%2%"(%)# -#%./*+%(0 )7 ;") ?"42)1 6*"H%2 NOTES 1
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It is sufficient to say, at this point, that an object for a person is anything which that person can say something about. Strictly speaking, ‘3x+10=100’ might be constituted into objects even before any ‘mathematical’ meaning is produced. once the person may say that ‘3x+1 10=100’ can, for instance, be typewritten or handwritten, and in large, medium or small size. This is, however, a subtle aspect which will not be discussed any further here. I n Lins (1 992), I have characterised Algebraic Thinking as thinking arithmetically, internally, and analytically. Thinking arithmetically can be understood as ‘thinking in numbers'; thinking internally means not modelling back those numbers as some other objects (eg, measures or wooden sticks, currency or areas), ie. characterising them only as objects having given properties in relation to the operations and to equality and eventually inequality; and. thinking analytically means treating unknown numbers exactly as if they were known. Put together, these three conditions point to objects. numbers. which are known only as objects we operate on with the arithmetical operations. As suggested before, by a text, from here on, I will mean not only written text, but any residue o f an enunciation, sounds (residues of utterances), drawings and diagrams, gestures and all sorts of body signs. What makes a text what it is, is the reader’s belief it is indeed a residue o f an enunciation, that is, a text is framed by the reader: also, it is always framed as such in the context of a demand that meaning be produced for it. For instance: it may seem natural for us to place equations and functions close to each other, but this possibility depends on constituting them as objects with certain common features; such a constitution may not be, however, within the horizon of a given person’s ways of producing meaning for those objects. In the work ofthe Dutch group of Utrecht (see, for instance, van Recuwijk, 1995), we find the notions of vertical and horizontal mathematisation. Although similar, I would like to point out that vertical and horizontal devlopments within the TMSF are much more general notions than their Dutch counterparts, particularly as they are not aimed only at mathematical meaning In particular, the Dutch
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R.C. LlNS version characterises ‘mathematisation’ in terms of notation, but without clarifying whether or not this carries with it-within their model-‘meaning.’ This example is given assuming that meaning had first been produced within a Semantic Field of a scale-balance. A metaphor establishes the first Semantic Field as ontologically more primitive, while the reversed metaphor produces a restructuring of the ontological building: ‘now I know that in fact ... ’ Parts of this section have already been reported at PME XIX, Recife (Brasil), 1995. I am particularly indebted to Geraldo Garcia Duarte Jr., Rosamund Sutherland and Luciano Meira, for their insightful comments on the interviews discussed here. Producing meaning for algebra: a research and development project in teaching and learning, a cooperation project conducted under the direction of Rosamund Sutherland (University of Bristol, UK) and the author (Dept. of Mathematics, UNESP-Rio Claro. Brazil); the project is partially funded by CNPq (Brazil), grant 530230193-3, and by the British Council. An illuminating instance of the need of familiar kernels is found in the use of ‘examples.’ For instance, when I teach Group Theory, it is common that when presented with the definition of a Group my students are completely unable to say a word. After discussing a few examples, however, many of them become able to produce statements justified within a Semantic Field of the formal definition. Familiarity with the examples, here, allows them to say things. On a more technical note, it is worth indicating a general mechanism of production of new Semantic Fields, namely, the introduction of a new operation on objects previously constituted. In the history of mathematics we find a prime example of that mechanism in action, when Wessel introduces a multiplication of directed lines, objects which are first constituted as displacements (Wessel, 1959), and, as a consequence, produces new objects, which are distinct from the previous ones; in this particular case, Wessel then associates these new objects with complex numbers, by showing that the addition and multiplication of directed lines have the same properties as those of complex numbers. It is interesting that he is not seeking a foundational model for complex numbers; instead, he is trying to develop a ‘geometric calculus’-after all. he was a surveyor-, and what he shows is that complex numbers are helpful in dealing analytically with directions, that being his main objective.
FERDINANDO ARZARELLO¹, LUCIANA BAZZINI ² AND GIAMPAOLO CHIAPPINI ³
A MODEL FOR ANALYSING ALGEBRAIC PROCESSES OF THINKING
INTRODUCTION It is very well known that students show difficulties in learning the symbolic language of algebra; many authors have pointed out that a major problem consists of students’ incapability of relating symbolic expressions to their meaning: the roots of many misunderstandings lie in the inadequacy of such a relationship, leading to incorrect performance and blind manipulations with algebraic symbolism (see Sfard (1991), Kieran (1992)). Sometimes students do not only ignore the correct meaning of formulas and concepts, but even invent fresh meanings which surrogate the authentic ones. From a didactic point of view, it is very hard to convince students that they are wrong, in so far as the invented meaning often has its own justification, generally rooted in previously learned models, perhaps working appropriately in their own context. Hence, it may happen that the teacher and the student do use the same words, but with different meanings: a genuine comedy of errors is thus generated. A consequence of this is that many secondary school students do not master the sense of those symbols, which they have learnt to handle formally (for example, see the USA National Assessment of Educational Progress Report in Brown et al.( 1988), also quoted by Kieran (1994)). On the other side, some students, even if clever ‘algebraic calculators’, seem not to be able to see and use algebra as a means suitable to understand generalisations, to grasp structural connections and to argue in mathematics (see Laborde (1982)). Existing literature has shown the possibility of taking instant pictures of students’ difficulties, but it is not so easy to find the way to analyse the cognitive processes involved for longer periods of time and from a more global point of view and, consequently, to provide suitable suggestions for teaching. 1
Dept. of Mathematics, University of Turin, Italy.
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Dept. of Mathematics, University of Pavi, Italy. 3 Istituto per la Matematica Applicata, C N R , Genoa, Italy 61
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Algebraic thinking is recognised as being inseparable from the formalised language by means of which it expresses itself. However, it is reductive to believe that algebraic thinking lives only at this level; in this case everything would be reduced to a manipulative mechanism, which often does not work in the hands of students. In accordance with many people, particularly with Vygotsky’s theory about the nature of links between thought and language, we consider algebraic thought and language as two intertwined and mutually dependent aspects of the same process. In fact, analysing the word’s central role, Vygotsky states that it would be incorrect to consider thought and language as separate or independent, in spite of their separate origin. In particular, he stresses that a word’s meaning is a linguistic and intellectual phenomenon which evolves over time. This statement is of special interest when applied to the meaning of algebraic expressions; language is conceived of as a thinking tool progressing in time: in turn, the evolution of thinking inspires the use of more complex and sophisticated linguistic forms. It is our concern to expose here a theoretical model which is suitable for analysing algebraic thinking from such a point of view. This model has been developed over the last years by the authors in a n ongoing joint research project (see Arzarello et al. (1992), (1994)), and has as an empirical basis the behaviour of hundreds of students, from the 8th grade up to University, observed while solving (pre-algebraic and algebraic) problems. The chapter collects and exposes in a more systematic way the contributions given by the authors at PME Conferences, both in presentations and in the m)*=%#$ 8*)4D )# 52$/3*"%B ?*)B/++/+ "#' ;(*4B(4*/+: The emphasis is on the theoretical analysis, while didactic consequences, which arc the present object of investigations by the authors, will be discussed elsewhere.
;/#+/ "#' '/#)("(%)# )7 "# "2$/3*"%B 7)*942"P (,/ (*%"#$2/ )7 A*/$/ It is our aim to present a precise theoretic analysis of the meaning of symbolic expressions in algebra as a concrete tool to describe the dynamics of typical algebraic processes and misconceptions. Our starting point is the ideas of Frege on semantics (see Frege (1892a,b; 1918): in fact they seem suitable for looking at the interpretation of symbolic expressions of algebra, insofar as learning is concerned. In particular, we shall distinguish between Sinn (sense) and Bedeutung (reference, denotation, also meaning, but the English translations are ambiguous) of an expression (Zeichen): the Bedeutung of an expression is the object (Gegenstand) to which the expression refers, while the Sinn is the way in which the object is given to the mind (Figure 1), or in other words, it is the thought (Gedanke) expressed by the expression. Everyone knows the example of Frege concerning the two different senses of Venus, namely as Esperus, i.e. the
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Frege’s Semiotic t riangle A%$4*/ F
night’s star, and as Phosphorus, i.e. the morning’s star: the two expressions have the same denotation. that is the planet Venus, but different senses. Also in Mathematics there are expressions whose senses are different but which have the same denotation. For example, the expressions 4x + 2 and 2(2x+ 1 ) mean a different rule (sense) but denote the same function; likewise, the two equations (to be solved in R) (x+5)² = x and x²+x+1=0 denote the same object but have a different sense. So mathematics, as natural language, is full of expressions which have the same denotation but incorporate different senses. The most ‘evident’ sense of an algebraic expression represents concisely the very way by which the denoted object is obtained by means of the computational rules expressed in the formula itself; we call it the "2$/3*"%B +/#+/: For example, the formula n (n+ 1) in the universe of natural numbers expresses a computational rule, by which one gets the (denoted) set A = {0.,V1 6, 12, 20, ...}. But the same formula is able to incorporate additional senses. apart from the algebraic one. In fact, it can be used in different knowledge domains, mathematical or not, each generating (at least) a new sense, depending on the nature of the domain. For example, the expression n (n+1) in elementary number theory has the sense of ‘product of two consecutive numbers’. whilst in elementary geometry it may stand for the area of a rectangle of (integer) sides n, (n+1). We call the B)#(/K(4"2%+/' +/#+/ )7 "# /KD*/++%)# a sense which depends on the knowledge domain in which it lives (as such, it is different from the algebraic sense): in fact the above formulas express different thoughts, with respect to the different contexts where they are used (see the discussion in Frege ( 1918)).
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The variety of senses that an expression can have is shown in a very direct way by the formula itself, because of the %'/)$*"D,%B features of the algebraic language: which means that the formula mimics in its own form and shape the main relationships among the different objects involved. In the above example, the fact that the elements of A are obtained from the product of a number and its successor is shaped ideographically in the formula itself (compare it with the English sentence which expresses the same fact. where the representation i s purely symbolic, without any ideography). Ideography allows for changes of sense by suitable manipulations on the shape of the formula (for example, n (n+1) ---> n²+n). The power of algebra consists in the multiple senses which are incorporated by the same formula and/or which can be obtained by syntactic manipulations on it; whilst its didactic drama resides in the complete imbalance between senses. denotations and expressions, which make the status of algebraic signs very obscure for students. Let us make some examples. First, senses may change without a corresponding change either in the formula or in the denoted object (see the above example on successors and areas). Second, algebraic transformations can produce different expressions holding different algebraic senses, but with the same denotation. For example, transforming n (n+ 1) into n²+n does not change the denotation but does affect its algebraic sense, i.e. the computational rule. Third, it is not always true that two expressions having the same denotation can be mutually reduced by means of algebraic transformations (x+5)² = x and x²+x+1=0 in R. Fourth, algebraic transformations are not always invariant with respect to the denotation (for example, SQRT(x²) and x). Each of these imbalances creates uncertainty in students about the status of the formulas they are using and i s the source of possible misunderstandings. All possible senses of an expression constitute its so called %#(/#+%)#"2 aspects, while its denotation within a universe represents its so called /K(/#+%)#"2 aspect. It is worthwhile observing that the official semantics used in mathematics, and particularly in algebra, cuts off all intensional aspects, insofar as it is based on the assumption of the extensionality axiom (two sets are equal if they contain the same elements, independently from the way they are described or produced), which entails the %#."*%"#B/ of mathematical objects with respect to their intensional aspects. Of course, mathematicians do also use intensional aspects of the mathematical language. but these remain more or less implicit in their definitions and proofs, that is in the way they do mathematics, while their explicit sentences always assume extensionality; this game is very subtle and intriguing and is the cause of many misunderstandings in many pupils, at all ages. In fact, much research has pointed out (see Drouhard (1992). Kieran (1989)), that intensional aspects are very important because it may be very difficult for students to conceive the above invariance. Many algebraic difficulties can be described as
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deficiencies in the way pupils master the invariance of denotation with respect to the sense: there is a sort of rigidity that makes them act as if there were a one-one correspondence between sense, denotation and formal expression, so that identifying all three, they disappear and pupils remain with a trivial denotation: a symbolic expression denotes itself as a collection of signs. They own algebra only as a pure syntax: for them secondary processes, in the terminology of Sfard (1991), are purely syntactic rules and nothing morei, their real meaning, namely Frege’s triangle has collapsed into a trivial object, because of the identification of denotation with sign. As a concrete example of such an identification, let us consider the D+/4') +(*4B(4*"2 students described by Sfard (1992); the point is that they do not realise, in the case of equations, that 'the concept of a truth set the set of numbers which must not change under the ‘permitted’ operations is where the decision to call certain manipulations ‘permitted’ becomes clear' (Sfard (1992), p.2); since they do not see the invariance they become ‘formalist’, in the sense that they reveal a 'basic inability to link algebraic rules to the laws of arithmetic' and so 'formal manipulations ... (remain) as the only source of meaning' (ibid., p.8). Such difficulties in the semantics of algebra can be understood quite well if they are compared with the enormously easier semantics of arithmetic, where there is no imbalance between sense and denotation in a numeric expression; in fact a numeric expression denotes always a definite number and its sense is practically unique; hence, while in an arithmetic expression each time only one interpretation is available, on the contrary an algebraic expression can be generally interpreted in many different ways. —
—
E)#B/D(4"2 7*"9/+
To have a precise description of the dynamics of algebraic thinking, we still need an ingredient, that is the notion of conceptual frame. We shall introduce it by discussing a concrete example: it is a protocol of a typical 'average' student, whom we have called Ann, it is given here as a paradigmatic case. Ann is a 20 year old undergraduate student of mathematics and has to solve the following problem: ?*)./ (,"( (,/ #493/* SDIXTSbiIXTnl %+ "# /./# #493/*1 D*).%'/' D "#' b "*/ )'' D*%9/+: Protocol of Ann Episode 1. Ann develops the formula writing the words /./#1 )'' on the paper near the formulas: (p-l)(q²-1)/8 = (p-1)(q+1)(q-1)/8
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Ann points to the components of the formula and says: o/./#1@/./#1@/./# :::::::::::,999 :::::: (,/ */9"%#%#$ #493/* %+ #)( /./#@::::p Episode 2. Ann makes some algebraic transformations to the formula using the words even, odd: (p- 1)(q²-1)/8 = (pq²-q²-p+1)/8
Ann makes some oral calculations of the type «odd times odd is odd» then says: «...hmmm...it does not work!». Episode 3. As in the previous episode, but with calculations of the type «odd times odd is odd» referred to the factors (p-1), (q²-1); then Ann says «there must be some formula to use for primes! . Episode 4. Ann draws some scribbling on the formulas of the preceding episode and starts verifying the formula with some primes: data are collected into a table:
Ann comments: «So it is already q squared minus one that is a multiple of eight».
Episode 5. Ann changes the sheet of paper and writes down the following:
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Then Ann writes the following formulas:
(2h+1-1)((2k+1)²-1)/8=2h(4k²+4k+1-1/8 = 2h 4k(k+1)/8 and says: «...%7 = %+ /./#1 7)4* S(%9/+T = %+ " 942(%D2/ )7@l (Anne points to the 8 in the formula), so %( */9"%#+ " 942(%D2/ )7 V (Anne points to the 8 in the formula), "#' >/ "*/ h<:@F7 = %+ )'' :::::::::::: ( Ann reduces 8 with 4 in the usual written form, writing 2 near 8; then reduces the 2 with the 21 coefficient of h) F7@=@%+@ ) ''1 %( ')/+ #)( >)*= ...... Jhq %7@= %+ )'' (,/# = D24+ )#/ (Ann points at the k+1 in the formula) %+ /./# "#' >/ "*/@)./*qp:
Episode 6. Ann looks again at the text of the problem and says: o64(@D*%9/+@,"./ $)( #)(,%#$ () ') >%(, (,%+q h'' #493/*+ "*/@/#)4$,p: The example is typical of the way average attaining pupils reason with formulas. High attaining pupils instead, use at once the formula:
(2h+1-1)((2k+1)²-1)/8 = 2h(4k²+4k+ 1-1)/8 = 2h 4k(k+11)/8 and argue directly as follows: «if k is even then we are finished ‘cause four (times) k reduces with eight; if k is odd then (k+1) is even and the same argument applies». Some students also observe connections with triangular numbers. Low attaining pupils instead generally try the transformations of episodes 1-3 or similar ones; they seem to look (more or less in a random. purely syntactical fashion) for some more complicated formula which can solve the problem and seldom try to substitute numbers for letters to see ‘what happens’, so in the end get lost or solve partially the problem. If we look carefully at the dynamics of the protocol. we can observe that the crucial moment consists in the sudden change of strategy from episode 4 to episode 5:
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In fact, all actions and decisions of Ann in the first part (episodes 1,2,3) are ruled by her knowledge concerning even and odd numbers: such knowledge consists of an organised set of notions (i.e., mathematical objects, their properties, typical algorithms to use with them, usual arguing strategies in such a field of knowledge, etc.), which suggests to her how to reason, manipulate formulas, anticipate results while coping with her problem, that is how to switch on senses of formulas to be interpreted and to be manipulated in order to solve the problem. We call such an organised set of knowledge and possible behaviours a B)#B/D(4"2@7*"9/: We take the term 7*"9/ from artificial intelligence studies (for example, see Minsky (1975)); from this point of view, a frame is a structure of data that is able to produce a stereotyped representation of a piece of knowledge. Our notion of frame is wider than that of Minsky, in so far as it entails also specific conceptual aspects of knowledge as an organised set of conceptual notions and operational skills related to some precise pieces of mathematics: so we call it a conceptual frame. As such. it is related also to the notion of B"'*/ (setting), discussed in Douady (1986): its similarity with Douady’s notion rests on the fact that a conceptual frame has also a mathematical dimension (as well as socio-cultural and individual ones). In fact a sense whatsoever can be given to an algebraic expression, insofar as it is related to some piece of mathematics. The notion of B"'*/ in Douady entails a wider mathematical area and is based on mainly conceptual features; our conceptual frame is more specific and limited: a B"'*/ could be that of elementary number theory. a conceptual frame on the contrary must be a specific part of this (for example, evenodd, prime numbers, multiples) and contains not only organised notions and operative tools (as in Douady’s B"'*/T but also precise scripts (condensed stories), with which the subject can operate in an almost automatic way (as in Minsky’s 7*"9/+T: Each conceptual frame entails a certain piece of infomiation and is concerned with what one expects to happen as a consequence of the information and possibly also with what one must do if such expectations are not confirmed. In our example (episodes 1, 2), Ann looks at formulas in order to match a stereotyped sense of parts of) the formula within the conceptual frame (an even times an odd is an even, etc.) with the goal of her task (namely that the product of such parts simplifies by 8). A conceptual frame is a sort of a condensed story, which has its +B*%D(+ and where the subject is supposed to do something, because of the goal of the task. Conceptual frames are activated as virtual texts while interpreting a text, for example of a problem, according to context and circumstances; as such, it has socio-cultural and individual features (the activation of the conceptual frame /./#I)'' by Ann depends on her culture as a student in a certain socio-cultural context and as a specific student with her precise personal history).
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To indicate a wider space, within which a pupil can act and switch on her/his conceptual frames to solve a given task, we use the notion of +)2.%#$ >)*2': With respect to the knowledge domain, where a problem-situation is set, a solving world is featured both by systems of signs, which are used to mediate the subject's thought and action, and by interaction tools, which help the subject to produce meaningful and expressive objects to solve the given problem. For example, Ann does all her performances in a solving world which is featured by the possibility of using algebraic language, numerical tables to represent arithmetic data and natural language, while interacting with a test coacher, who records faithfully what she says. Within her solving world, Ann switches on at least three different conceptual frames: in fact, from the precise conceptual frame and scripts of /./#I)'' #493/*+ (episodes 1,2,3), she passes to the vague conceptual frame and groping scripts of D*%9/ #493/*+ (episodes 3,4), and arrives in the end at the definite conceptual frame and scripts of 942(%D2/+ (episodes 4,5). Of course, such adjectives as precise, vague, definite refer only to the way Ann owns such conceptual frames and stresses the individual dimension of the notion. But let us describe more carefully what happens in the changes from one conceptual frame to the other, in order to understand the dynamics of algebraic reasoning. In episodes 3,4, probably because of her failure to produce the expected result. Ann uses a second conceptual frame, that of D*%9/ #493/*+1 where she shows a less organised operative knowledge and a more groping strategy: this second conceptual frame is less ‘in focus’ than the previous one and remains in the background, while the former continues to rule her actions, even if in a less definite way, as long as her different attempts are unsuccessful. The first phase (up to episode 3), is marked by stereotyped syntactical transformations: the conceptual frame /./#I)'' #493/*+ guides them, but as it is clear from the involution of this phase, semantic control is very feeble; formulas are used as such and do not live really as objects to think with; they only incorporate the frame in a shallow way: the formula continues to be manipulated according to standard stereotypes which belong to Ann’s operative knowledge concerning evenodd numbers. The idea that some magic formula must be used culminates in the end of episode 3, which marks a change of the conceptual frame SD*%9/ #493/*+T and a deep change of approach: now the reasoning becomes arithmetic; semantics is monitored: the conceptual frame, which has strong numerical features, is really inhabited by numerical expressions, which are meaningful for her. As such, they are real (,%#=%#$ ())2+1 which in the long run activate hypothetical reasoning in a new precise conceptual frame S942(%D2/+T1 which overlap partially the first one. In episode 5 Ann can read the old formula with a new eye: it is written in such a way to incorporate the old conceptual frame even-odd and moreover is manipulated
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A%$4*/ V
according to an anticipated thinking related to the new conceptual frame S942(%D2/+T: It is not any longer the way that the formula is written to suggest standard manipulations to seewhat happens, but in order to prove a conjecture, the formula is written and manipulated in a certain way. In other words. in the new conceptual frame, the relationship sense-denotation of the formula does appear as a thinking tool. namely its double face is used dialectically to test a hypothesis: intensional aspects are guided and built by extensional ones and conversely (episode 5 , 1st part). This is the first aspect of what we B"22@7)*942"+@ "+ (,%#=%#$ ())2+: namely when formal manipulations are made in deep connection with denotative aspects, in order to produce a shape in the formula, that incorporates an expected sense, because of a supposed denotation. There is a second way in which formulas can be used as thinking tools. This is illustrated by the second part of episode 5. Here the formula presents some stiffness: the formula incorporates the sense that 4k(k+1) is a multiple of 8 in a transparent way only when k is even. Ann has some trouble with the case when k is odd; the formal aspects force her to simplify in a certain maimer and for a while the (conjectured) denotation of the formula (namely its being even after simplifications) does not cope with the right sense (k+1 is even if k is odd): to grasp this it is not required to manipulate the formula as before. but to activate a new part of the symbolic expression according to its denotation. It is the shift in the denotation
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A%$4*/ [
(even -- > odd) which makes this possible. With respect to the first aspect of formulas as thinking tools, the formal expression no longer changes; it is its sense which changes, insofar as we have looked at its denotation in a new way, shifting from the frame 942(%D2/+ to the /./#I)'' one. In other words, the first way of thinking with a formula is to transform (part of) its intension, manipulating it according to its (supposed) extension (Figure 2); the second way is to discover a new (supposed) intension, without doing formal manipulations (Figure 3), but looking at a new (supposed) extension (in a possibly new conceptual frame). In both cases such changes are activated because the intension and the extension of a formula are embedded in one or more conceptual frames. The second aspect is more evident in solutions given by secondary school students (17 years old) to the following problem: gE,/B= %7 "9)#$ */B("#$2/+ )7 " $%./# "*/" (,/*/ %+ )#/ >,)+/ D/*%9/(/* %+ " 9%#%9491 r4+(%70 0)4* "++/*(%)#+e: A typical average attaining student, whom we shall call Bob, has solved the problem as follows: first he has written the formula S=bh; then he has checked some numerical examples; afterwards, he has written the formula S=xy, has expressed the semiperimeter as x+S/x and then has used the standard machinery of calculus to find the minimum perimeter. (Students who have good marks in mathematics generally have written the last formula earlier and have used derivatives in a standard way.
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Low attaining students have got lost with numerical computations or at most have proved the formula for some specific instantiation of S). It is clear that the rewriting of the formula by Bob marks a change of the conceptual frame, namely from "*/" 7)*942"+ to 74#B(%)# 7)*942"+: The sense changes insofar as the extension of the formula changes because of the shifting of conceptual frames. In summary, our model for describing algebraic thinking is based on the following concepts: (i) Frege’s semiotic triangle; (ii) the notion of conceptual frame. Its dynamics develops with respect to: ( 1) the relationships among signs, senses and denotations of algebraic formulas; (2) the activation of conceptual frames. their mutual relationships and changes from one to the other (within a solving world); In other words, the real dynamic aspects of algebraic processes of thinking can be described properly by observing the way pupils modify their semiotic triangles within a frame or passing from one frame to another. For example, Ann’s thing can be described in the first part of her solution by looking at the way she activates senses for the given formula within the conceptual frame /./#I)''1 whilst in the second part one must look at the new conceptual frames: first, at the way the conceptual frame 942(%D2/+ activates suitable re-naming of variables and syntactic manipulations. so that the formula’s sense can match the goal of the task (that is an expected denotation); second. at the way the interplay between the previous conceptual frame /./*I)'' and the present one S942(%D2/+T suggests to Ann to read a new sense in the old formula in order to overcome the last difficulty (that is the case when k is odd). At this point, it is worthwhile observing that high attaining students seem to use the natural language code only as a framework, curtailing most of the conceptual frames used by low and average students and going directly to the last one. But, when interviewed, they spontaneously make explicit most of the curtailed frames with many details; this seems to confirm that also in their case there may be more than one frame (recall the definition of frame as a virtual text. given above). The presence of frames in solution processes may provide evidence of the fact that students are using a full semantics (namely with sense and denotation) which is generally marked linguistically in some way: for example. Ann writes in words and stresses in her explanations the three successive frames, where she develops her algebraic reasoning; Bob renames variables, when passing from one frame to the other.
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J"9%#$ Now our picture of algebraic thinking has the main ingredients and we can use them to describe what happens when a student solves an algebraic problem; as an exercise, one can look at the different behaviours of students who solve algebraic problems, for example those given in Arzarello (1992d): all of them can be focused clearly by means of our model. Also the detailed description given by Paulo Boero (this volume) of s a n e specific thinking processes, typical of algebraic manipulations, fits very well within our model. Now let us describe with some detail the typical way in which the solution process of an algebraic problem is coached by a pupil. First she/he interprets the text of the problem; in the interpretation activity her/his co-operation is the determinating factor which activates one or more conceptual frames. The way this interpretation activity is concocted is crucial. Subjective aspects determine the very way things are thought and the consequent course of thinking (i.e. anticipating thinking) and of writing (i.e. naming). The result of the interpretation will be another text (maybe written. but also written and spoken), into which the first text has been interpreted, and so on (possibly), interpreting this new text again into a further one. One can easily observe that this activity of repeated interpretations reveals remarkable differences from case to case, even in the same chain of interpretations. but the starting point is crucial for the subsequent development of the whole process. For example, suppose that the problem is standard, so that it reduces to the construction of a symbolic expression E (for example for modelling a situation) or to an interpretation of a symbolic expression (such as the problem of Ann). Generally, E contains some terms: unknowns. parameters or more complex expressions built up using variables and other symbols (specific constants, operations, etc.). The first construction-interpretation of letters is crucial insofar as it entails the very process of #"9%#$1 that is of putting ideas into formulas and this requires a good mastery of the ideographic features of the algebraic language. For example, to introduce or interpret variables (and terms in general) one must choose from among different virtual possibilities (within one or more frames): in fact, a variable (or a parameter or a constant) contains implicitly a form of D*/'%B"(%)# (for example, p may stand for a prime, in a generic or specific way). a term expresses a '/+%$#"(%)# (for example, the perimeter of a rectangle may be indicated with 2p or with 2(b+h)) - see Laborde (1 982) for a detailed discussionii. In the case of algebraic problem solving, the process of naming consists of assigning names to the elements of the problem and is aimed at constructing an algebraic expression able to make explicit the meaning of the problem. It implies the activation of a conceptual frame (at least) and of the dynamic relationship between
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sense and denotation of the algebraic expression within it (for more information, see Arzarello et a1. (1993)). in the naming process the role of constructing and interpreting letters (variables or parameters) is crucial. Letters can be used to give a name to the extra linguistic elements (mathematical or not) which are involved in the problem and to emphasise the relationship among the elements within an algebraic expression. The choice of names to designate objects is strictly linked with the control of the variables being introduced: in this process, one main difficulty, especially for novices, derives from the usual impossibility of sustaining the stream of thought by means of natural language: in fact, algebraic formulas are very seldom a linear record of corresponding sentences in natural language. Not only are relationships expressed in a different way, but as soon as formulas become complex, the algebraic language condenses in a concise and precise way relationships whose expression by means of natural language can be made only vaguely and with considerable difficulty. Let us consider a trivial example, to illustrate this. Problem: 60 9/"#+ )7 " $))' B,)%B/ )7 #"9/+ () '/+%$#"(/ (>) B)#+/B4(%./ )'' #493/*+1 +,)> (,"( (,/%* +49 %+ " 942(%D2/ )7@s:@ Experiments carried out with students, from junior secondary school (11-14 years old) up to University, have revealed similar typologies of errors, namely: 2h+ 1+2k+ 1; 2h+ 1+ 2k+ 1t2 x + y; instead of 2h+ 1+2h+3; moreover, a high percentage of students make only arithmetical checks. The example shows that some students, who seem to grasp the relationships among the elements of the problem using the natural language or the arithmetic code are unable to express them suitably by means of the algebraic code. Note that. already in this simple example, the use of natural language for expressing the problem is not possible (otherwise, one remains vague or must use the language in a very sophisticated and complex way), and this makes things already difficult for a significant proportion of students. More specifically, many students reveal that they are unable to use the algebraic code as a mediator between the identified goals of the problem and the relationships among its elements. Hence they are unable to express the algebraic meaning of the problem in adequate algebraic terms. By ‘algebraic meaning of a problem’ we intend the clarification of the relationships among its elements, when the text of the problem is interpreted according to the rules of the algebraic language. A good mastery of the algebraic code in the construction of an expression includes being able to incorporate the meaning of the problem: according to its goals. From the very beginning, good problem solvers usually have a glimpse of a possible path, and use this implicitly in their first trials at naming; usually they are able to incorporate the relationships among the elements and prefigure
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transformations suitable for reaching the solution. This is not the case for poor problem solvers, who proceed more randomly while choosing and naming variables: their naming is weaker and often superficial, in any case not linked with anticipatory aspects but rather influenced by rigid stereotypes. Sometimes the process of naming is impoverished or blocked when the subject constructs or interprets symbolic expressions in a rigid way, without understanding the complex and flexible relationship between sense and denotation in algebraic expressions. As a consequence, the subject usually does not grasp the potential of the algebraic code, i.e. the possibility of incorporating different properties within the naming process. Such an incapability may be an obstacle lo algebraic reasoning since it inhibits flexibility, which is a basic support of the functions underlying the use of algebraic languageiii.
52$/3*" "+ " $"9/ )7 %#(/*D*/("(%)# The activity of problem solving in elementary algebra can be pictured as playing a $"9/ )7 %#(/*D*/("(%)#P of a text in a semiotic system (for example, a problem in ordinary language) into a text in another system (for example, an equation), or from a text in a system (for example, an algebraic expression) into a text in the same system (for example, another algebraic expression). For example, consider a problem to solve (e.g., a word problem, a conjecture to prove, a phenomenon to model, etc.) and the request of an algebraic solution: the solution is nothing more than an %#(/*D*/(/*1 that is an %#(/*D*/("(%)# and the result of some (*"#+7)*9"(%)# (for example, an expansion or a condensation, see the chapter by Boero in this volume) of the text. with respect to the questions it poses. In fact. the interpreter is useful insofar as it makes it possible to know something more about what is interpreted: in the algebraic case the interpreter is a Frege triangle, produced by means of a suitable interpretation (possibly from another one); during such a process, the frame and even the solving world may change. To give a picture of algebraic thinking as a game of interpretations, let us consider our symbolic expression E (interpreter or to be interpreted): to such an expression is attached a set of Frege’s triangles with one vertex in common (the expression): they constitute all the possible interpretations of E. In this set there may be triangles with different denotations and to each of them there may be attached a cluster of triangles with the side expression-denotation in common, that is triangles with the same denotation (and expression): it is only their sense that changes. When the student starts her/his interpretative activity, by various reasons she/he activates one or more conceptual frames. in a solving world: her/his activity with Frege’s triangle(s) lives within such frame(s) and solving world(s). Once a conceptual frame is active, the student produces as a result of her/his interpretation a text (with some
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A%$4*/ s
sense and denotation among the possible interpretations living in the activated frame), and the process of solution of the problem consists in successive transformations of this text, possibly in the production of completely new texts. each with its own sense and denotation. according to the conceptual frames that are activated; also the transformations are made according to the conceptual frames that are active in that moment. The main goal of the game of interpretation is twofold. On the one hand, the chain of interpreters which can be built as a result of such a game is very fruitful for the learning of algebra, insofar as mental representations of pupils are generally made active by interpreters (think again of episodes 4 and 5 in Ann’s protocol). On the other hand, the game of interpretation helps students in producing metacognition about the processes by which they have produced their algebraic interpreters. The term ‘game’, which we have used for these activities, emphasises that, in a good didactic control of such processes, pupils’ acts of hypothetical thought may be verified and criticised by means of social practice coached by the teacher (i.e., social interaction in the class), or by other forms of interaction (i.e., interactivity with specific software, etc.). Here are some examples of interaction tools and activities
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which can help and sustain pupils to perform their game of interpretations. First, mediating tools which are meaningful and stimulating with respect to their culture (for example, computers, but also other machines: for a discussion in another context, see the Chapter 4 in Biehler et a1. (1994)): these can be useful, since they may facilitate connections between arithmetic practice and algebraic formulas. Second, activities of verbal interpretations of the algebraic expressions produced by pupils: the habit of discussing formulas they have produced and transformed, can stimulate reflections, social interactions and so on. The game of interpretation happens at least at two levels, namely among different frames, within a fixed solving world, or among different solving worlds; for example, among the similar frames activated by Ann in her solving world, or among the very different solving worlds activated by students who use a spreadsheet to construct algebraic formulas. Both may be very useful for algebraic learning. The former type of processes activates pupils’ mental work within the concrete constraints of the solving world and may be useful to overcome its own obstacles and difficulties, because of the shifting from one frame to the other. The latter may allow the subject to restructure totally the way under which she/he looks at the problem situation: concrete facts are always the same: not so for the sense and denotation, which changes radically, because of the possible imbalance between the two worlds (for example, think of the problem of Pythagorean triples solved within algebra or within analytic geometry: see for details Lang (1985)). The repeated construction of interpreters within different solving worlds is unified and ruled by the aim of solving the posed problem situation. or at least of grasping its meaning; of course, the latter docs not automatically ensure that the problem will be solved.
!/#("2 +D"B/+1 /."D)*"(%)# "#' B)#'/#+"(%)# A consequence of the preceding discussion is that in order to achieve a meaningful teaching of algebra, it is necessary to build up didactic situations which stimulate the developing of a fruitful game of interpretations, so as to overcome the typical blocks and difficulties of the algebraic learning, which are underlined in the literature (for example, the blind manipulation of formulas, or the incapability of producing a fruitful comparison of senses in formulas, etc.). To do this, micro-didactics is not enough, in so far as playing a game of interpretation means changing one’s habits and styles of thinking and hence entails long term interventions. made with a systematic new way of approaching algebra in the school. As we have already said, didactic interventions are beyond the aim of this chapter; however, it will be appropriate at this point to discuss some processes, which are crucial for the learning of algebra and can possibly be modified in the long term.
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F. ARZARELLO, L. BAZZINI AND G. CHIAPPINI
In fact, to understand something about the nature of blocks met by pupils in their game of interpretation. it is crucial to study these processes, deeply connected with their inner language, where pupils are requested to anticipate and to grasp globally (some portion of) a formula, in order to produce some insight of its different possible shapes and senses. Observations and interviews show that sometimes "#(%B%D"(%#$ (,)4$,( operates on the basis of rich numerical tables, constructed by pupils, possibly using the computer; sometimes it is a sort of hypothetical reasoning made in a mental space which has spatial, temporal and logical features (in interviews, pupils use words, metaphors etc, which mark clearly these features, not all necessarily present in the same subject). Its reality depends on the situation (namely, the problem, the pupils, the negotiation in the class, and so on), it gives real feedback to the pupils, who have built it, and allows them to activate their own conceptual models (see Arzarello (1989) for this notion) according to familiar scripts. The creation of such mental spaces usually helps students to avoid stumbling-blocks and to continue in the game of interpretation; in other words, it facilitates all those flowing processes, typical of problem-solving, by which data can be selected, processed according to one's conceptual models, integrated into new pieces of knowledge, and so on: perhaps it has some connections with the ideal environments in which scientists like Galileo, Faraday. Einstein made their well known mental experiments. In short, a mental space is a sort of fictitious real world which students themselves create, where they can make experiments and develop hypothetical reasoning. Hypothetical reasoning is essential both in general planning, when pupils elaborate possible solution strategies, activating different frames and senses and in controlling, contrasting, comparing them with each other. So it is deeply connected with the symbolic function of language (see the discussion of this notion in Cauty (1984)). As a typical metacognitive skill, it is a long-term process, which can be developed in the long term by means of a cognitive apprenticeship where the teacher encourages the thinking as a stream and not a single act of language. The most difficult point to analyse is that mental spaces are active if (and generally only if) problems are approached from the procedural side, that is doing calculations with concrete numbers, thought of as general objects. namely from a situation where there is a good semantic monitoring (see for example episode 4 in Ann's protocol). However, to get rid of arithmetic and its procedural aspects and to switch to algebra, with its relational and structural properties, pupils must work in mental spaces where objects lose their extra-mathematical and procedural tracks and must translate them into symbolic expressions, which are highly synthetic, ideographic and relational. To do this, a student is required to write concisely and expressively the amount of information of a term, whose complexity and generality cannot easily govern the language of arithmetic. In some sense, the stream of
ANALYSING ALGEBRAIC PROCESSES OF THINKING
79
thought which sustains her/his computations and arguments contracts and condenses its temporal, spatial and logical features into an act of thought, which grasps the global situation as a whole. Such an inner process happens in a dialectic back and forth movement with the formula (such processes are clearly visible when pupils interact positively with spreadsheets, symbolic manipulators, etc.): there is a sort of converging process, from the formula to the subject and conversely towards the final solution (this has been studied by Gallo in Arzarello & Gallo (1 994) and by Boero in this volume), which in the end culminates in a single act of thought in the pupil’s head and in a general formula on the paper (or on the screen). The former has some similarity with the process of %#(/*#"2%+"(%)# of speech studied both by Piaget and by Vygotsky. The latter has been called B)#'/#+"(%)#P the word is taken from semiology (see Eco (1984), p. 157) and from Freud (1905) (see his analysis of the wits); it has also some connection with the phenomenon of curtailing, described in Krutetski (1976), and with the property of shortening. typical of inner language (see Vygotsky (1934), chapt. 7). As an example, Arzarello calls condensation the process by which the same symbolic expression refers to two different meanings, possibly in two different conceptual frames, or when one shifts from a conceptual frame where she/he is interpreting an expression to another one, adapting the old Frege’s triangle to the new situation (for, example. when Bob renames variables in problem 2. and in doing so changes from an "*/" to a 74#B(%)# frame). Hence, condensation stresses semantic creativity insofar as it may be related to such things as analogies, metaphors, etc.; other examples are discussed in Arzarello (1992d). At the opposite side of condensation we find a typical process of poor algebraic performers, that we call /."D)*"(%)#: Whilst condensation entails a strong semantic control, on the contrary evaporation concerns the dramatic loss of the meaning of symbols met by most pupils when losing the semantic control in algebraic problem solving. Typically, this happens when they cannot any longer express the meaning of mathematical objects and relationships in ordinary language referring to a subject’s actions, to the very processes of their construction and generation and to any other extra mathematical information about them. In such cases they cannot any longer use algebraic language as a record of their processes of thinking, possibly mediating with natural language. This impossibility provokes the transition to the empty semantics of pseudo-structural students, discussed above, namely to a collapsed Frege’s triangle. As such, evaporation is one of the main obstacles in the developing of an algebraic way of thinking and to a non empty use of the algebraic code. It is interesting to observe that both condensation and evaporation obey a principle of economy: in fact both save space and time in the head and on the paper, compared with other more expensive ways of thinking and representing. But while condensation maximises the content of infomation of an expression preserving the
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right senses and denotations by means of condensing them suitably in the expression; evaporation. on the contrary, minimises senses and denotations of an expression trivialising them: the former produces expressions with a high content of information, the latter reduces this content to a minimum. So the two processes are in a duality with each other; both concern the symbolic functions of algebraic language, the former marking a good semantic control, the second its loss. In this delicate point, the role of language may be crucial, particularly as far as the dialectic between inner and outer language can bypass evaporation, provoking instead condensation. Condensation is apparently a sudden phenomenon, which happens on the spot, but it can be grasped properly if one does not look at the learning/teaching of algebra as a sequence of single acts )7 2"#$4"$/ but as a +(*/"9 )7 (,)4$,(1which breaks dramatically with the arithmetic way of thinking, which pupils are acquainted with from elementary school. Condensation marks deeply the passing from a procedural moment to a more abstract and relational one; it appears at once in the strategies of solutions of pupils and, like all processes which happen on the spot, it is very difficult to analyse, because attention is focused on the single acts of language, which reveal only a sudden change. For example, in all the cases we have studied (see Arzarello (1992b)), most pupils who work successfully at problems. even when explicitly asked, are not able to explain what has happened. The only positive observed fact (directly by some of the authors) is that interviews of pupils who ‘have condensed’, are marked by a massive use of complex expressions of ordinary language (for example, conditionals, subordinates etc.), whilst evaporation is at the opposite side (see examples in Arzarello (1992b)). Moreover, in all control classes, where the teaching style does not encourage the use of natural language in problem solving, only pupils with high verbal performances also get good achievements in algebraic problem solving. On the contrary, in classes where the teacher develops in her/his students the habit of using the verbal (spoken and written) code while solving algebraic problems, to discuss different strategies, solutions, etc., then also pupils of middle verbal and mathematical abilities get good scores. Our conjecture is that in such classes the students acquire the habit of working in different solving worlds, of switching from one conceptual frame to the other, of being flexible with respect to possible hypotheses, in other words of making the game of interpretations in a massive way. In fact, the habit of constructing different interpreters may develop in pupils those forms of flexibility of semantic control that can produce condensations, as a synthesis of different conceptual frames and activated solving worlds.
ANALYSING ALGEBRAIC PROCESSES OF THINKING
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NOTES 1
2
3
Secondary processes are the typical algebraic process by which computation rules generate abstract objects 'passing to the quotient' (for example: negative numbers from subtraction, by identifying
suitable pairs of numbers; an algebraic number like , from equation x2-2=0, by identifying polynomials modulo division by the irreducible polynomial x2-2). It is interesting to recall that, from a Frege analysis, a name designates an object (possibly an abstract one) and gives also criteria for its identification, that is gives a way for picking it out mentally as an object that has such and such features, which satisfy the given criterion. Designation and predication of algebraic language correspond to such an analysis. For example, let us look now at a typical case of poor algebraic performances, widely discussed in the literature, namely pseudo structural pupils (Sfard ( 1 992)). Roughly speaking, one can say that they act as i f there were a rigid one-one correspondence between expression, from the one side and sensedenotation, collapsed together, from the other; so for a psuedo structural student. transformations are not as in Figures 2. 3 but as in the following Figure:
PSEUDOSTRUCTURAL TRANSFORMATION
A%$4*/ )7 J)(/ [
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DAVID KIRSHNER*
THE STRUCTURAL ALGEBRA OPTION REVISITED
Some of these [school algebra] activities might be described by teachers or other adults as, say, 'expression simplifying,' 'equation solving' or 'problem solving.' Some others might describe them as 'function rewriting,' 'function comparisons,' or 'modeling,' respectively. Others might describe them as operations in and applications of rational or algebraic functions over the rationales or reals. 64( 9)+( +(4'/#(+ +// 2%((2/ 9)*/ (,"# 9"#0 '%77/*/#( (0D/+ )7 *42/+ "3)4( ,)> () >*%(/ "#' */>*%(/ +(*%#$+ )7 2/((/*+ "#' #49/*"2+1 *42/+ (,"( 94+( 3/ */9/93/*/' 7)* (,/ #/K( b4%H )* (/+(: (Kaput, 1995, pp. 71-72) My point of departure is a quote from Kaput's (1995) powerful and oft cited blueprint for algebra education research which relays common assumptions about rule memorising in school algebra that I take issue with in this chapter. My concern is with the possibility for a structural algebra option that is increasingly difficult to maintain in the face of the rule memorisation assumption. Kaput (1995) reaches an inevitable conclusion from the obvious failure of standard algebra curricula: Acts of generalisation and gradual formalisation of the constructed generality must precede work with formalisms -- otherwise the formalisms have no source in student experience. The current wholesale failure of school algebra has shown the inadequacy of attempts to tie the formalisms to students' experience after they have been introduced. It seems that, 'once meaningless, always meaningless.' (pp. 74-75, emphasis added) But algebra as generalisation is antithetical to structural algebra, which by its nature is formal and uninterpreted: The letters are mere 4#'/7%#/' 9"*=+ or 'elements' about which certain postulates are made .... The very point of elementary algebra is simply that it %+ abstract, that is, devoid of any meaning beyond the formal consequences of the postulates laid down for the marks. (Bell, 1936, p. 144)
*
Dept. of Curriculum & Instruction,Louisiana State University, U.S.A.
83 M: ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 l[\Zl: U VWWX <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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My argument in this chapter is that learning algebra in a traditional curriculum is not fundamentally a matter of explicitly memorising and applying rules, but of generating and consolidating +43B)$#%(%./ patterns (Hofstadter, 1985). Therefore, it is incorrect to read the failure of current curricula as indicating the bankruptcy of rule-based approaches to algebra. Rather: an explicit curriculum of rule forms has never been tried or tested. The chapter concludes with ideas about how to structure such a curriculum. Before launching into these arguments, let me acknowledge at the start some of the apparent difficulties and contradictions in the position I am advancing. First of all, the assumption that algebra learning (in traditional classrooms) involves rule mastery hardly seems to be open to question or dispute. Textbooks present rules; teachers expostulate on them; and students toil over endless exercises designed to consolidate them. The middle sections of this chapter examine the arguments of research and of common sense, respectively, for rule based learning. Secondly, what is the nature of these +43B)$#%(%./ D"((/*#+ that students are postulated to be appropriating? What kind of epistemological framework do they fit into, and how are they manifest in particular algebraic content? In the chapter I introduce a connectionist framework for pattern matching, and I provide empirical evidence and argument for pattern matching with respect to polynomial parsing. Finally, if, as argued here, fluency with the alphanumeric symbols system of algebra is a matter of subcognitive pattern matching, what can it mean to implement an explicit rule-following approach to algebraic symbol manipulation? Why would we want to? The final section of the chapter outlines a curricular approach to structural algebra, and takes up this apparent contradiction by returning to an epistemological meditation on the nature and function of rational discourse. STRUCTURAL AND REFERENTIAL APPROACHES TO ELEMENTARY ALGEBRA Broadly speaking, there are two approaches that can be taken to meaning making in elementary algebra. The structural approach builds meaning internally from the connections generated within a syntactically constructed system. Referential approaches import meaning into the symbol system from external domains of reference. This is similar to Peacock's (1 833) pure/applied distinction, except that his concern is with the D4*D)+/ of the investigation, whereas mine is with the 9/(,)'P The science of algebra may be considered under two points of view, the one having reference to its principles, and the other to its applications: the first regards its completeness as an independent
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science; the second its usefulness and power as an instrument of investigation and discovery. (p. 185, quoted in Menghini, 1994)
;(*4B(4*"2 "2$/3*": What is characteristic of structural approaches is the synthetic nature of the objects of study. Typically one starts with undefined terms and axioms, and explores the theorems that can be logically deduced from them. The structures developed often are experienced by mathematicians as having a life of their own (Sfard, 1994). Generally speaking, this is the method of pure mathematics. Structural algebra has been motivated by and applied to a wide range of systems, from numerical systems, to geometric symmetries, to matrix forms (Kaput, 1995). Importantly it is not the topic of the study that characterises structural algebra, but the method. For instance, one can study integers by developing and experimenting with patterns. Or one can logically explore a set of axioms that generates a system related to the integers. Only the latter is structural algebra. Whereas a motivating reference domain is common in pure mathematics, from the methodological point of view expressed here it is not essential. Any approach that builds from undefined symbols and explicit rules is structural according to my definition. This would include the traditional school algebra approach were the usual assumptions about rule memorisation correct. The structural approach has been a part of mathematics education since the time of Euclid. Indeed, it is only in the last fifty years in North America that the mantle of deductive mathematics was passed from geometry, which previously was pure Euclid, to algebra; One way to foster an emphasis upon understanding and meaning in the teaching of algebra is through the introduction of instruction in deductive reasoning. The Commission [on Mathematics] is firmly of the opinion that deductive reasoning should be taught in all courses in school mathematics and not in geometry alone. (College Entrance Examination Board, 1959, p. 23) Clearly this is an important legacy for algebra educators to consider. —
M/7/*/#(%"2 "2$/3*" Referential approaches to algebra all share the property that they import meaning to algebraic symbol systems from external domains. These can vary widely from real world situations (Fey, 1989; Nemirovsky & Rubin, 1991; Usiskin & Senk, 1990), to graphs and tables (Confrey, 199 1; Dugdale, 1990; Goldenberg, 1991;
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Kaput, 1987, 1989; Yerushalmy, 1991) , to arithmetic patterns (Linchevski, 1995: Schifter. 1996). As noted above, Kaput’s (1995) dictum that ‘acts of generalisation and gradual formalisation of the constructed generality must precede work with formalisms’ (p. 74) specifies a purely referential vision of algebra. But this also is the trend in almost all of the algebra education literature. Booth ( 1989) puts it this way: Without an understanding of the semantics of algebra, the mere manipulation of symbols becomes a fairly arbitrary exercise in symbol gymnastics, sometimes performed correctly and sometimes not, but in either case with little sense o f purpose. The essential feature of algebraic representation and symbol manipulation, then, is that it should D*)B//' 7*)9 an understanding of the semantics or referential meanings that underlie it. (p. 58) Similarly Thompson ( 1989) sees ‘the most damaging consequence of defining competence 9/*/20 as possession of correct rules is that we fail to look at incompetence as stemming from impoverished conceptualisations of inaterial from which correct rules should have been abstracted’ (p. 138). Part of the push towards referential algebra steins from the poverty of the referential component of usual curricula, which traditionally is restricted to standard word problems. As well, new technologies are pushing the possibilities for referential algebra in exciting new directions (Kaput. 1992; Thompson, 1989). But even taken together, these forces would motivate only calls for an enhanced and expanded referential component to the curriculum. The */D4'%"(%)# of structural algebra (as seen above) testifies to the fact that the failed curriculum of today is taken as proof that structural approaches are not viable. In arguing against this analysis, my objective i s not to develop and promote a new unitary structural curriculum. Rather I seek to enable a curriculum that honors both structural and referential possibilities for meaning making in algebra. INFORMATION PROCESSING ANALYSES OF ALGEBRAIC ERRORS Putting aside for the moment the common sense observation that learning school algebra is the acquisition of rules, there also is a research base consisting of twenty years of information processing analysis of students’ errors that supports this conclusion (Bundy & Welham, 1981; Carry, Lewis, & Bernard, 1980; Davis, 1979; Davis & McKnight, 1979; Matz, 1980; Wagner, Rachlin, & Jensen, 1984). For the most part, this research has concerned itself with identifying and explaining the 9"2 *42/+ (Sleeman, 1984) that characterise students' incomplete mastery of the symbol system (see Table 1 for a sampling). The usual explanation given for these error types is that they reflect a process of overgeneralisation of the
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correct rules (Matz, 1980). This analysis is consistent with constructivist interpretations of students as builders of the knowledge structures that enable their competence. The clear implication of this research program is that competence in symbol manipulation results when the student has developed an appropriate/viable set of rules. In other words. learning algebra is acquiring rules. C"32/ X: E)99)# &**)*+ "#' E)**/B( M42/+
Errors
Correct Rules
It should be noted, the conclusion that learning algebra is acquiring rules is not independent of the epistemological assumptions of the information processing paradigin itself. Information processing psychology understands the mind according to the model of the serial digital computer. The mind/brain, like the computer, is a D,0+%B"2 +093)2 +0+(/9 which can solve problems and manifest general intelligence through the physical manipulation of tokens (or instances) of symbols (Newell & Simon, 1985). The basic model for reasoning in information processing psychology is a formal one in which explicit rules dictate the processes of deriving new data structures from the givens. Indeed, most studies of algebraic skill have "++49/' a rule based explanation as part of the theoretical framework, prior to evaluation of empirical evidence. At the start of their report, Carry, Lewis, and Bernard (1980) assert that students need to have ‘the legal moves of the algebra game’ (p. 2) (such as the distributive law, difference of squares rule, etc.) explicitly represented in memory, Similarly, Matz’s (1980) framework begins with acquisition of a textbook curriculum:
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This proposal idealises an individual's problem-solving behavior as a process employing two components. The first component, the knowledge presumed to precede a new problem, usually takes the form of a rule a student has extracted from a prototype or gotten directly from a textbook. (p. 95) What can we conclude from the observation that research into algebraic symbol skills makes an a priori assumption of rule based learning? In the next section I argue that the agreement of cognitive research with common sense understandings of algebra as ru le based is not a coincidence. Both reflect philosophical assumptions about the human mind that are deeply entrenched in the modern world view of Western culture. EPISTEMOLOGICAL ASSUMPTIONS In the classical era, Plato advanced his philosophy of ideal forms. The sensory world was understood to be illusory; the true forms of things existing in an ideal realm approachable only through the rigors of logic and mathematics. At the start of the modern era, Descartes (1641/1979) confinned the separation of body (including sensory input) from mind through his meditations on existence and certainty. This S'4"2%+( philosophy is the foundation of our culture’s common sense about mentality. As Gardner (1 987) points out, cognitive science (which includes information processing psychology) is a philosophical descendant of Descartes’ dualism: René Descartes is perhaps the prototypical philosophical antecedent of cognitive science .... Mind, in Descartes’s view, is special, central to human existence, basically reliable. The mind stands apart from and operates independently of the human body, a totally different sort of entity. (pp. 50-5 1) How does the dualist position relate to rule based assumptions of cognition? Viewing the mind as a rule based mechanism provides a material explanation of how mind can be complete unto itself, yet still interact with the body to receive information and govern action. Descartes’ unlikely explanation placed the pineal gland as the interface between mind and body. Viewing the mind in analogy to the serial, rule based computer, resolves such speculations. Indeed as Haugland ( 1985) notes, computational (rule based) explanations of cognition solve a variety of perplexing problems that dualist philosophers have struggled with for centuries: (i) the metaphysical problem of mind interacting with matter; (ii) the theoretical problem of explaining the relevance of meanings, without appealing to a question-begging homunculus; and (iii) the methodological issue over the empirical testability ... of 'mentalistic'
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explanations. The computational idea can be seen as slicing through all three dilemmas at a stroke; and this is what gives it [cognitive science], I think, the bulk of its tremendous gut-level appeal. (p. 2) Mathematics has always held a privileged place within the metaphysics of dualism as the prototypical detached mental realm. Thus cognitive scientists and lay persons, alike, are led to understand algebra as a rule based competence, because of the prior assumption that mathematical cognition exists as a separate, self-consistent domain.
E)#(*"%#'%B"(%)#+ As appealing as the rule based gloss of school algebra is, several aspects of our experience with algebra education are discordant with it. First, and most prominent, is the demeanor of algebra students as regards the rule based curriculum. Students' interest is in seeing demonstrations of rule usage, not in hearing explanations of rules. Explanations usually are not welcome. The constant refrain is 'show me how to do it.' The popularity of John Saxon's (1991, 1992) 'incremental approach' is a case in point. Saxon's curriculum features several innovations including mixed practice sets, and reduced explanation and discussion to increase 'time on task.' The Saxon approach is so contrary to the usual espoused beliefs of mathematics educators that, astonishingly, for several years the National Council of Teachers of Mathematics of the United States and Canada refused him advertising space in NCTM periodicals (Hill, 1993). Nevertheless, his approach is well received by students, and relatively successful in comparison to other approaches to competence in standard tasks (Hill, 1993). The homogeneous grouping of exercises (which is the norm in standard textbooks) and the discussion of solutions (widely espoused by mathematics educators) both are vehicles for the mastery of explicit rules and principles. In contrast, Saxon offers students emersion in the domain of algebraic symbol tasks with minimal segmentation of that domain, and with minimal interference of classroom discourse. If the learning of algebra really is mastery of explicitly given rules, why are those explicit rules so unwelcome by students, and why is their absence so well received? A second indication that the rule based assumptions of cognitive psychology and of common sense may be suspect comes from reexamination of the sorts of errors that students typically make as they develop mastery of the symbol system. Examine, again, the error patterns displayed in Table 1. What is striking and so frustrating about these deviations is that they appear not to be based upon logical fallacy, but rather on their perceptual similarity to the correct rules. Thus there is a lurking suspicion that it is perception, not logic, that is operative in students' learning. Pat Thompson ( 1989) expresses the frustration of mathematics educators
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with the current curriculum: ‘students are prone to pushing symbols without engaging their brains’ (p. 138). I would argue that in a very real sense Thompson's observation is correct, if one takes ‘engaging their brains’ to refer to explicit, conscious cognition. CONNECTIONISM In psychology, B)##/B(%)#%+9 provides an alternative to the dualism of information processing theory (Cummins, 1991; St. Julien, 1997). In analogy to the neurology of the brain, connectionism asserts that cognition is parallel and distributed, rather than serial and digital. The primary cognitive functions are pattern matching and associative memory, not logic or rule following. Connectionism notices that the long chains of extended reasoning that serial digital computers do best, are hardest for humans. Things that humans do best, like recognising faces in different situations and from different angles, are the most difficult feats to simulate on serial computers, but the easiest to implement in connectionist architectures (parallel distributed processers, Salomon, 1993). Connectionist psychology posits dramatic redundancy and a superabundance of active elements, in contrast to the neat, linear processes of rule-based systems (Bereiter, 199 1). Information processing psychology and connectionism differ fundamentally with regard to the relationship between the mind and the environment. In the former, the environment needs to be pre-processed into discrete elements, facts, or attributes in order to be utilised by the inforination processor. This is what makes it so consistent with Descartes' dualist vision of mind separated from the body and from experience. In the latter, perception acts directly upon the cognitive system: Experience organises cognition, rather than vice versa. As Hofstadter (1985) has observed, it is those below the threshold of conscious attention that subcognitive processes organise the categories from which intelligence is manifest. The connectionist framework seems, in general terms, to afford the possibility of an alternative account of algebraic symbol skills that is more faithful to our observation as educators that students' work in algebra is non-reflective and patternbased. The purpose of the next section is to show how this framework can inform our understanding of specific cognitive functions in algebra manipulations. Following this we return to the educational dilemma newly formulated: How can we incorporate linear logical processes in a cognitive system that is not architecturally configured for them? —
—
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THE POLYNOMIAL PARSING PROBLEM The challenge for rule based theories of algebraic skill is to demonstrate the specific, definite rules that are used by skillful algebra symbol manipulators. Abstract algebra provides one model for an explicit rule system for elementary algebra. A tenet of abstract algebra is that the operations of addition and multiplication (and by extension of subtraction, division, exponentiation, and they map a pair of elements into a single radical) are binary operations element (Paley & Weichsel, 1966. p. 138). Thus addition maps (5.3) to 8, subtraction maps (7,9) to -2,: etc. From this perspective expressions like 5 + 3 + 1 are ill-defined, unless some convention dictates a binary parse: either (5 + 3) + 1 or 5 + (3 + 1). The fact that these two alternatives yield the same result, 9, is irrelevant to the mathematical principle, as is clear from the case of subtraction, 5 3 - 1, where, the two alternative parses, (5 - 3) - 1 and 5 - (3 - 1), are unequal. There is a standard convention that utilises a familiar hierarchy of operations for parsing expressions that are not fully determined by parentheses. The formulation of this convention presented in Table II comes from Schwartzman (1977). —
-
C"32/ FF: hD/*"(%)# Q%/*"*B,0
Level 1
addition
subtraction
Level 2
multiplication
division
Level 3
exponentiation
radical
;0#("B(%B E)#./#(%)# (a)
Higher level operations are precedent.
(b)
If adjacent operations are of equal level, the operation on the left is precedent.
J)(/: In this classification, Level 3 operations are said to be ,%$,/* than Level 2 operations which in turn are ,%$,/* than Level 1 operations. This version of the syntactic rule leads directly to the usual binary parse of expressions. For example 3x² is parsed as 3(x²) by rule (a), and 5 - 3 - 1 is parsed as ( 5 - 3) - 1 by rule (b). If the rule-based psychology of information processing is correct, then the expert algebraist must have come to internalise this nile (or some alternative), as the basis for competence in the domain of parsing. This claim is not the simple assertion that
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the competent algebraist 'knows' (i.e., can recite) the rule. Rather there must be some internalised representation of this rule incorporated into the cognitive structure that is activated in parsing situations. On the one hand, it is possible to 'know' the rule, but somehow never use it. On the other hand. one may have forgotten, or implicitly acquired, the rule but nonetheless have it as an active part of one's rule structures. Before presenting data that suggest that the basis for competence in parsing does not make use of a rule of the sort presented in Table II (or indeed, of any rule), it is necessary to map out the full range of implications associated with the hypothesis. The first observation is that a full binary parse implies extraordinary complexity, even for relatively simple manipulations. Consider the typical school task requiring collection of like terms for the polynomial 3x - 2y + 2x - 5y. The evidence of introspection suggests that the terms of the polynomial arc relatively autonomous. One can shunt terms around, with suitable care for the signs. But a fully parsed expression provides no such flexibility. In this case, one's mental representation is [(3x - 2y) + 2x] - 5y, and maintaining the full parse entails something like the following: [(3x - 2 y ) + 2x] - 5y = [(3x + -2y) + 2x] - 5y = [3x + (-2y + 2x)] - 5y = [3x + (2x + I2y)u - 5y = [(3x + 2x) + -2y] - 5y = [5x + -2y] - a 0 = [5x + -2y] + -5y = 5x + [-2y + -5y] = 5x + -7y = 5x - 7y. Whereas such a derivation is formally correct mathematics, it seems rather obtuse to presume that people actually process collection of like terms in this way.1 If we are willing to forego the purity of biliary parse, there are alternative logically consistent rule structures that preserve the information processing assumption of rule based intellection. For instance, one might start from the abstract algebraic definition that subtraction merely abbreviates addition of a negated term (Paley & Weichsel, 1966. p. 27), and move from there to dispense entirely with (b) of the syntactic convention presented in Table II. As Asimov (1959) advises, always adding a negative ‘gives us the opportunity to wipe out subtraction altogether’ (p. 32). This move resolves many of the problems associated with polynomial parsing. The parsing ambiguity of 5 - 3 - 1 is unproblematic, because the underlying mental representation. 5 + (-3) + (-1) is protected from inconsistency by the associative law for addition. Similarly the possibility of applying the commutative law for addition to x - y + z to get x - z + y is subverted because the representation of the former expression is x + (-y) + z. . Less obvious but still true is that dispensing with (b) of the syntactic rule in Table II does not lead to logical contradictions for the other operations2. But taking seriously the regimen of a consistent, rule-based conception of algebraic competence demands a certain discipline. If subtraction is to be taken to abbreviate addition of a negated term in this case, it must do so in all cases. Thus one would have to insist that, say, the 'difference of squares' transformation, x2 - x 2 = (x -y)(x + y) is cognitively represented as the 'sum of a square plus the negation of a
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square' transformation. This proposal has no more plausibility (as a cognitive hypothesis) than the complex polynomial simplification discussed above. There seems to be an inherent ambiguity in the representation of subtractio/negation that is not captured in the alternatives provided by consistent rule structures. EVIDENCE There is some evidence that novices do not operate from a logically consistent set of rules or axioms. Cauzinille-Marmeche, Mathieu, and Resnick (1984) have observed a fractured, ad hoc process in eleven-year-old students who were asked to judge the equivalence of expressions like 685 - 492 + 947, 947 + 492 - 685, 947 - 685 + 492, and 947 - 492 + 685. Kieran (1989) reports the findings: When students did not calculate, their most typical response was to make a judgment about equivalence on the basis of rules for transforming expressions. These rules were often incorrect. The most frequent incorrect rules could be interpreted, according to these researchers, as the dropping or adding of constraints on the principles of commutativity or associativity for addition. (pp. 38-39) In general it is eminently unsafe to generalise from novices to experts. However, there is further evidence that this same kind of ad hoc process underlies competence in algebra for the fully competent mathematician too. As part of a quite different study (Kirshner, 1989), a sample consisting of 137 fourth-year engineering students at the University of British Columbia was asked to evaluate 10 expressions for x = 2 (see Table 111) . C"32/ F 2 2 1 C/+( F(/9+1 7)* ;/#%)* &#$%#//*%#$ -#'/*$*"'4"(/+
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Students at this educational level are generally considered to be expert in elementary algebra, and, indeed, only 14 students in the sample did #)( score perfectly (a total of 14 incorrect responses and 1 omitted response). Clearly these errors are a marginal phenomena: however. they are not random. Twelve of the 15 errors (including the missing response) occurred with the trinomial expressions, #6, #8 , and #10 (seven of the errors occurred with #8). In each of these cases the response given (if any) was compatible with the incorrect parse of the expression (e.g., 19 - 4x + 2 = 19 - (4x + 2) = 9; for x = 2). What sorts of explanations are possible for these data? Perhaps these nearly-expert students construct a non-binary representation for polynomials, whereas their more expert cohorts employ a binary representations? Or perhaps for these students subtraction is just subtraction, while for their cohorts it is represented as addition of a negative? The problem with these rule-based analyses is that they propose radically different cognitive structures for students of very high accomplishment in elementary algebra. A third possibility does not require postulating such major anomaly in the cognitive structures of the erring students. Questions #6, #8, and #10 are nonstandard problems i n that usually no more than one of the terms in a polynomial is constant. It could be that the evaluation of polynomials. for competent performers, is instantiated as an ad hoc, perceptually based, left-to-right practice. (Similarly, perceptually based ad hoc practices would protect against other anomalies like applying the commutative law for addition to x - y + z to get x - z + y,) The need for initial focusing (for substitution purposes) on a middle term that is embedded between two constants may have been just sufficiently distracting to override this constraint for this small minority of students. This explanation has the advantage of leaving the syntactic structure of expressions and the representation of subtraction homogeneous for the entire sample, entailing only a slight modification of cognitive structure to explain the errant behavior. The implication, then, is that competence in algebraic skills is not a matter of knowing the rules, so much as of coordinating pattern-based perceptual cues (see also, Kirshner, 1989). TOWARDS A PEDAGOGY FOR STRUCTURAL ALGEBRA The key insight argued for in the preceding section is that competence, or rather fluency, in a domain like algebraic symbol skills does not imply access to the rules that motivate that domain. At some practical level, as educators, we may known this to be true. But our epistemological frame-of-reference does not permit us to honor and act upon that insight. Rather there is deep frustration and discouragement at the failure of our intensive investment in skill development to pay off with a rational, logical apprehension of the domain of algebra.
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The alternative (connectionist-based) epistemology alerts us to the fact that learning always is grounded in perception and pattern matching as embedded in practices, not in abstraction and rule following. Thus there is no reason to expect rationality to emerge from the solitary activity of working exercise sets. But what, then, should educators do to help students apprehend algebra as a logically structured domain? Indeed, what is rationality? Bereiter (1991) offers the following perspective: Harré’s theory about the social nature of rationality (1979, 1984) provides an illuminating way to think about this question and more generally about how the classical rule-based view relates to cognition. When people try to give a retrospective report of their mental processes, what they tend to do instead is provide a O4+(%7%B"(%)# of their actions (Nisbett & Ross, 1980). Rationality, according to Harré, originates in this essentially social process of justification. What we call logical reasoning, and attribute to the workings of the individual mind, is actually a public reconstruction meant to legitimate a conclusion by showing that it can be derived by procedures recognised as valid. (p. 14) It is too extreme to argue that rules play no role in competent performance, but it is an ancillary role informing cognition rather than constituting it. Bereiter continues: Explicit rules may play a part in learning to think, but (as suggested by the long history of failure of instruction in logic to improve thinking) a very limited one. Turning the social process of justification inward amounts to a kind of self-checking. In this process, one might B)#+42( logical rules in the same way that one might consult rules of algebra while solving a mathematical problem or consult an etiquette book when planning a formal dinner. Rules, thus, may play an important role as =#)..2/'$/ (,"( /#(/*+ %#() B)9D4("(%)#+1 but this is a fundamentally different role from the one traditionally conceived by philosophers and cognitive scientists, where *42/+ B)#+(%(4(/ (,/ B"9D4("(%)# "2 "2$)*%(,9+ (,/9+/2./+: (Bereiter 1991, p. 14) Rationality as social legitimisation suggests the outlines for a new pedagogy of structural algebra. If rationality is not inherent to algebraic manipulation skill, it needs to be fostered in the classroom through specialised activities and discourses. It is not enough to explain rules to students and have them practice their use (as in the usual North American curriculum). Students must be in the role of articulating and justifying their rule usage. Rationality accrues from rationalising. It is not inherent in symbol manipulation.
D. KIRSHNER
96
It is worthwhile, at this point, to revisit abstract algebra as a possible source for the rules that might motivate a rational accounting of algebra symbol skills. Indeed, abstract algebra is fundamentally concerned with justification and explanation. But its focus is on axioms and rules for defining extendible algebraic structures. Most of the preliminary theorems are logical niceties like the uniqueness of the additive identity, that have little to do with the derivations involved in school algebra. Furthermore, representational issues related to the parsing of expressions are (rightly) understood to be an independent notational concern outside of the purview of the theory itself. In these respects abstract algebra can be compared to D*/+B*%D(%./ $*"99"* that dictates how a language should be used, with little sensitivity to actual practices of speakers. (In basic linguistic usage prescriptive grammars attempt to impose one model of speech as the norm for a commnity. In contrast to prescriptive grammars which attempt to impose a standard on a community, descriptive grammar is a more scientific attempt to delve into speakers' (implicit) knowledge that enables their actual language competence (Crystal, 1987). The educational track that I recommend is based on a '/+B*%D(%./ $*"99"* (Kirshner, 1987) that attempts to model the actual practices of fluent users of algebraic language.3 This entails an elaboration of the parsing component as well as the usual transformational rules. Indeed, formalising the parsing rules turns out to play a crucial role in providing the discursive grounding for the transformations. An example can give a sense of this role, and of the pedagogical discourse I consider useful. Consider the rule for canceling common factors in a fractional "K expression ( ), so often overgeneralised to common terms = 3K 3 —
) . Now a discourse involving 'factors' and ‘terms’ clearly is 3+ K 3 relevant here. But the defining characterisation for these lexical items lies within the parsing component. To see this we need to refer to the notion of )D/*"(%)# D*/B/'/#B/ described in Table II. Consider the expression 1 + 3x². The hierarchy of operations determines the parse of the expression as { 1 + [3(x²)]}. The most precedent operation (in this case, exponentiation) is the one embedded within the most parentheses, in that if we were to evaluate the expression (given numerical values for all of the variables) it is the most nested operation that would be performed first. The least precedent operation is the least deeply embedded in the expression when fully parsed. The definitions for terms and factors now follow easily: If the least precedent operation is multiplication, the subexpressions it joins are factors; if addition. terms. Thus the above cancellation error for fractions cannot sensibly be discussed outside of an explicit and formal treatment of parsing rules. Trying to talk about terms and factors without this kind of grounded analysis results in the usual illusion of communication that characterises most school algebra discourse. –
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The irony of this approach to symbolic algebra is that although a rational discourse is sought, rationalisation is understood to be a secondary phenomenon, not always easily fitted to the perceptual base for fluent performance. In my judgment, creating rational discourses in mathematics is fundamental to acculturation of students into characteristically mathematical ways of thinking and participating. Great strides in this direction can be taken by attending to the structure of expressions, usually just glossed over in standard curricula. But even then, it is something of a balancing act to maintain a rationalist discourse in the face of a process of competence that essentially is pattern based and non-rational. Thus one probably would want to teach subtraction as addition of a negative rather than assign a binary parse to all expressions, though neither alternative exactly matches the perceptual basis for fluent performance.
SUMMARY The dualist legacy of Descartes gives us a perspective on mind as disembodied, logical, and complete unto itself. Following this legacy through modern information processing psychology to mathematics education gives us grounds to believe that competence in algebraic symbol manipulation entails a logical understanding of algebra. The problem is that the standard curriculum (in North America, at least) which attempts to instill such symbol skills through didactic presentation and individual practice demonstrably fails to result in students' appreciation of the logical, structural underpinnings of the discipline. Rather it leads, for most students, to the dead end of mal-rules that are resistant to logical explanation and that bear an uncanny perceptual resemblance to the correct rules. This failure of the standard curriculum is pushing mathematics education towards strictly empirical versions of algebra in which meanings stem from and are justified through their use in applicative domains like arithmetical pattern and 'real world' situations. The hope that students also can come to participate in mathematics as a formal, logical study is waning. An alternative epistemological framework which understands cognition as a massively parallel pattern-matching process rather than as linear and logical provides for a new understanding of past failures and future prospects. If the mind is not inherently logical, then simply engaging students in symbol manipulation will not necessarily produce a logical instantiation of those skills. Logic needs to be fostered through particular discursive practices. The irony is that abstract algebra doesn't provide a sound basis for such discursive practices. For instance it provides a most unwieldy model for parsing through binary representation of addition and multiplication. It can be likened to a prescriptive grammar which describes how a language +,)42' be spoken with little regard for the actual practices of speakers. A
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descriptive grammar can provide a closer fit to actual practices, although no rule set exactly matches the perceptual basis for fluent performance. But then, engaging student in rationalising their internal processes doesn't mean that those processes actually need to be rational. Logical discourse comes from the social function of rationalising, not from engagement with inherently logical artifacts.
NOTES 1
2
Some theorists (e.g., Ernest, 1987; Drouhard. 1988) have postulated n-ary operators that permit more spontaneous grouping procedures. The arguments here are somewhat technical. Multiplication is associative, so the parse of "3B as S"3TB or "S3BT is unproblematic. Other operations have their parse specified by the +4*7"B/ 7)*9 (Kirshner, 1987) of the operations. For instance " 3 B is ambiguous, but it is arithmetic, rather than "
–
3
"
algebraic, notation. Algebraically one would write either — or — with the parse being determined B 3 –
B
3
by the length of the vinculum. The parse of the other operations are similarly determined by surface form. This is a reinterpretation of Kirshner (1987), reflecting an epistemological turn-around for the author. In Kirshner (1987) the grammar was formulated as an explanatory model displaying the (unconscious) rule structures underlying competence (cf. Chomsky, 1957; 1965).
PAOLO BOERO*
TRANSFORMATION AND ANTICIPATION AS KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING
INTRODUCTION This chapter aims to deepen the idea that one of the crucial aspects of algebraic problem solving (which might be used to characterise it) is the transformation of the mathematical structure of the problem in order to be able to manage it better, and anticipation which allows the process of transformation to be directed towards simplifying and resolving the task. The process of transformation may happen without, before and/or after algebraic formalisation.When it happens >%(,)4( or 3/7)*/ "2$/3*"%B 7)*9"2%+"(%)#1 it is frequently based on the transformation of the problem situation through arithmetic or geometric or physical manipulation of variables (adding, subtracting, translating, equilibrating...). These problem solving strategies can be called ‘pre-algebraic’. When the transformation happens "7(/* "2$/3*"%B 7)*9"2%+"(%)#1 it is frequently based upon the ‘transformation function’ of the algebraic code. In this case, the manipulation of the algebraic expression extends enormously the range of possibilities of transformation. At least a partial ‘suspension of the original meaning’ of the transformed expression may happen during the transformation process (cf Bednarz & al., 1992). The process of transformation needs specific prerequisites and skills. In the case of transformation after formalisation, a crucial prerequisite is the mastery of standard patterns of transformation. A common ingredient of all the processes of transformation (without, before and/or after formalisation) is "#(%B%D"(%)#: In order to direct the transformation in an efficient way, the subject needs to foresee some aspects of the final shape of the object to be transformed related to the goal to be reached, and some possibilities of transformation. This ‘anticipation’ allows planning and continuous feed-back. In the case of transformations performed after formalisation, anticipation is based on some peculiar properties of the external algebraic representation.
* Dipartimentodi Matematica, Universitàdi Genova, Italy
99 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 ZZIXXZ: U VWWX <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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P. BOERO
A%$4*/ X: +/9n7)*9 '%"$*"9+
One focus of this chapter is to consider the educational strategies which could enhance the development of the ‘anticipation process’ and the chapter ends with an analysis of some traditional and innovative practices. Examples related to different school levels will be integrated into the presentation, in order to show different aspects of the same topics. AN HEURISTIC MODEL FOR THE TRANSFORMATION PROCESS For heuristic purposes, I will use the kind of diagrams as shown in Fig. 1. In these diagrams, form means any written expression based on the use of the algebraic language; this wide definition covers a great deal of mathematical expressions (eventually integrating special symbols used in different mathematical fields: mathematical analysis. linear algebra, probability....): from arithmetic expressions (such as 3 * [ 2+ 5 * (7+2 * 3)] ) to algebraic equations, from trigonometric equations (such as 1) to differential equations (such as y’(x)=ay(x)-by²(x)), from functional expressions (such as T(ax+by)=aT(x)+bT(y)) to matrix expressions. I will consider +/9 as a mathematical or non-mathematical cultural object (a mathematical statement, a relationship between physical or economical variables, and so on). +/9 consists of a mental representation "#' an external non-algebraic representation. These two expressions are suggested by the classification proposed by C. Janvier (1987. pp. 148-149): ‘the word ‘representation’ has roughly three different acceptations in the psychology literature: at first, ... material organisation of symbols,.which refers to other entities or ‘modelises’ various mental processes ....( see ‘external representation’ in the text above); the second meaning ... refer to a certain organisation of knowledge in the human mental ‘system’ or in the long-term memory... (see ‘mental representation’ in the text above); the third meaning refers to mental images. In fact, it is a special case of the second one’. See also Duval (l995, pp. 25-26) for similar definitions. I would like to point out the fact that the 7)*9 - +/9 distinction, as proposed in this article, does not follow the traditional +0#("K - +/9"#(%B+ distinction. Indeed,
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+/9 brings its own external representation (for instance. a geometric figure - and/or a sentence of natural language). This choice can be justified by the need of analysing some algebraic problem solving processes, especially the activities performed at the 7)*9 level and their relationships with the problem situation ‘represented’ at the +/9 level. For a discussion about possible ‘meanings’ attributed by students to 7)*91 see Demby, 1996; for an in-depth study of some, delicate questions related to the 7)*9 - +/9 distinction, see Arzarello, Bazzini & Chiappini, 1994 and Chapter 4 of this volume. In 7)*9n+/9 diagrams, upward arrows mean ‘formalisation’, downward arrows mean ‘interpretation’. Formalisation consists of a translation from +/9 into an expression of the algebraic language. Interpretation means generating a mental representation and an external lion-algebraic representation coherent with 7)*9: Continuous horizontal arrows mean ‘transformation’; or more precisely: horizontal arrows between 7)*9 X and 7)*9 V mean ‘transformation according to the rules of the algebraic language’, including not only standard algebraic transformations of a literal expression, but also resolution of differential equations, of systems of linear equations, etc.; also included are substitutions of numerical values to letters. In general, ‘transformation’ will mean any process, based on direct algebraic transformations or substitutions or general theorems proved through algebraic transformations, and expressed through formulas, which allow to get some new algebraic expressions from the original one. The following are some examples illustrating the above ideas: i)
transformation from: (a4-b4)/(a+b) to: a 3 - a 2 b + ab 2 - b 3 can be performed through decomposition: a4 - b4 = (a-b)(a3 - a2b + ab2 - b3) and simplification;
ii) transformation from: (sinx exp2x)' to: (cosx + 2sinx)exp2x can be performed according to the theorem concerning the derivative of a product, and application of the distributive property: iii) transformation from: y''(x)+4y(x)=1 to: y(x)=A sin2x + B cos2x + 1/4 can be performed through standard methods of resolution of linear differential equations. - horizontal continuous arrows between sem 1 and sem 2 mean ‘transformation of
mental and corresponding external representations’. The example concerning the evaluation of the area of a rectangular trapezium, illustrated on page 103, shows how this transformation can be performed in this case (through a change of the decomposition of the trapezium).
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I observe that 7)*9 X may be equivalent (through reversible algebraic transformations’) to 7)*9 V , and +/9 X may be equivalent to +/9 V (same example quoted above). - horizontal dotted arrows indicate a ‘guess’ (conjecture to be proved, etc.). I will consider now some examples of usage of the heuristic model I have introduced. These examples will show how some algebraic problem solving activities can be schematised through the model and prepare the analyses performed in the following sections. The complexity of mental operations involved in algebraic problem solving and revealed through the +/9 - 7)*9 diagrams suggests some, possible reasons for the difficulties met by students.
5DD20%#$ " 7)*942" () +)2./ " 9"(,/9"(%B"2 )* #)# 9"(,/9"(%B"2 +("#'"*' D*)32/9: In this case, we start from sem 1, we put the problem we must solve into a formula (form 1), we operate a standard algebraic transformation (for instance: solving a standard algebraic equation), and we produce a ‘result’ form 2; the interpretation of form 2 produces a new ‘meaning’, sem 2. In many cases, this process is a multistep process (with a chain of fundamental diagrams of the type considered before). #$%&'()* it is well known that the ‘stopping distance’ s of a car, from the point where the driver sees the danger, can be determined by adding a distance proportional to the square of the speed v to a distance proportional to the speed v (depending on the quickness of reflex). Let us consider the problem of determining the range of the speed which is compatible with the ‘stopping distance’ of 100 m; we may put the law stated before (sem 1 ) into a formula; we then get, for form 1 : s = Av2 + Bv = 100 ; then we may give values to A and B depending on the conditions of the road, on the condition of the braking system of the car, and on quickness of reflex (particularisation of the situation, bringing to sem 2 and: correspondingly, form 2 ). For a common situation (modern cars, normal conditions of the road, mean reflex speed) we may pose: A= 0.006; B= 0.08 (if v is expressed in km/h, and s in metres). So, we may solve the inequality : s = 0.006 v ² + 0.08 v = 100 (form 2). We get (through standard formulas): -136 = v = 123 (form 3). Interpreting this result, we may say that the speed must not exceed 123km/h(sem 3). The following diagram synthesises the whole process:
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We may observe that the root -136 (obtained through the resolution of the equation) is not relevant to our problem; this shows the importance of the ‘interpretation’ phase of the ‘algebraic result’ form 3. ?*)'4B%#$ #/> =#)>2/'$/ "3)4( "# )D/# D*)32/9 +%(4"(%)#: Suitable transformations at the sem level and/or at the form level can produce new knowledge. The new knowledge may concern: - a conjecture about the existence of a transformation between 7)*9 X and 7)*9 V, suggested by relationships existing between +/9 X and +/9 V This is illustrated by a simple example concerning the evaluation of the area of a rectangular trapezium:
7)*9 F = ",nV + 3,nV
7)*9 V = ", + S3I"T,nV
In this case, two different decompositions of the original figure into simpler figures generate two different formulas; but sem 1 is equivalent to sem 2, and this suggests that a transformation may exist between form 1 and form2 .
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the existence of an ‘object’ related to +/9 X1 whose existence is a consequence of the interpretation )7 7)*9 V, derived from form X according to more or less standard transformations This is illustrated by an )$%&'() suggested by Paolo Guidoni: it is not difficult to verify through measure that the equilibrium temperature T f reached by the mixture of two quantities of water, m 1 , and m2 is related to their respective temperatures T1 and T2 at the moment of the mixture according to the –
following formula (form 1 ):
This formula may be interpreted as: ‘Tf is the weighted mean of the temperatures T1 and T2’ (sem 1)
By a very easy algebraic transformation we may write :
This formula (form 2) may be interpreted as ‘conservation of the quantity of heat’ (sem 2). By a suitable algebraic transformation of this formula we may write the following formula:
This formula may be interpreted as ‘inverse proportionality between the quantities of water and the absolute variations of temperatures’(sem3). The following diagram synthesises the whole process:
KEY PROCESSES IN ALGEBRAIC PROBLEM SOLVING
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Some of the more spectacular applications of mathematics to physics concern this kind of usage of mathematics; physicists ‘put a physical situation S+/9 XT into a formula’ S7)*9XT (an algebraic formula, a differential equation, etc.); suitable (more or less standard) transformations of the formula may generate a new ‘formula’ S7)*9 2), the interpretation of which S+/9 VT may increase our knowledge of the physical world. Another remark: a kind of principle of ‘neutrality’(in relation with the real world) of algebraic transformations (already realised by Galilei) allows us to operate in such a way, that if an hypothesis +/9 X is appropriately put into a formula 7)*9 X 1 the interpretation of a transformedform V formula (obtained from form XT may be used as a tool to validate sem X: ?*).%#$ " B)#O/B(4*/ S%# 9"(,/9"(%B+1 D,0+%B+ "#' +) )#T: Algebraic formalism is a current tool in proving conjectures. In this case, frequently, sem 2 is known (the ‘content’ of the conjecture), sem 1 is known (information about data: physical situation, relationships between variables), form 1 and form 2 must be expressed in a convenient way in order to get form 2 starting from form 1 with suitable transformations.
In many situations the passage from 7)*9 X to form V (and, consequently, from +/9 X to +/9 V) needs intermediate steps, according to a chain which may be more
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or less complex. theorem:
As a very simple example, we may consider the following
‘The sum of two consecutive odd numbers is a multiple of 4’ Through a suitable formalisation, we may write the sum of two consecutive odd numbers as: (2K + 1) + ( 2K + 3); performing standard algebraic transformations we get: ( 2 K + 1 ) + ( 2 K + 3 ) = 2K+1 + 2 K + 3 = 4K+4= 4 ( K + 1 ) ; the interpretation of this formula allows the validation of the thesis.
TRANSFORMATION BEFORE ALGEBRAIC FORMALISATION Some transformations of the problem situation, having a counterpart at the form level, can be performed without any algebraic formalism. This subject was investigated in collaboration with Lora Shapiro. First of all we recall some essential contents of our research report (Boero & Shapiro, 1992). The purpose of this study was to understand better the mental processes ( i.e. planning activities, management of memory, etc.), underlying students’ problem solving strategies in a ‘complex’ situation. To this end the following problem was administered to students from grade IV to grade VIII: ‘With T liras for stamps one may mail a letter weighing no more than M grams. Maria has an envelope weighing E grams; how many drawing sheets, weighing S grams each, may she put in the envelop in order not to surmount (with the envelop) the weight of M grams ?’ Various numerical versions have been proposed to different groups of pupils:
The students’ resolution strategies have been analysed according to a classification scheme suggested by the data from a pilot study, and corresponding to the aim of exploring the mental processes underlying these strategies.
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Strategies were coded in the following manner: ‘Pre-algebraic’ strategies (PRE-ALG.). In this category the strategies involved taking the maximum admissible weight and subtracting the weight of the envelop from it. The number of sheets is then found by multiplying the weight of one sheet and comparing the product with the remaining weight, or dividing the remaining weight by the weight of a sheet of paper, or through mental estimates. If the problem would be represented in algebraic form, these strategies would correspond to transformations of the form : –
Sx + E
to :
up to :
x = (M - E)/ S
For the purposes of this research, we have adopted the denomination ‘prealgebraic’ in order to emphasise two important, strictly connected aspects of algebraic reasoning, namely the transformation of the mathematical structure of the problem (‘reducing’ it to a problem of division by performing a prior subtraction); and the discharge of information from memory in order to simplify mental work. ‘Envelope and sheets’ strategies (ENV&SH).This ‘situational’ denomination was chosen by us because it best represented students’ production of a solution where the weight of the envelope and the weight of the sheet are managed together. These strategies include ‘mental calculation strategies’, in which the result is reached by immediate, simultaneous intuition of the maximum admissible number of sheets with respect to the added weight of the envelope; ‘trial and error’ strategies in which the solution is reached by a succession of numerical trials, keeping into account the results of the prceeding trials (for instance, one works on the weight of some number of sheets and adds the weight of the envelope, checking for the compatibility with the maximum allowable weight); ‘hypothetical strategies’, in which one keeps into account the fact that the weight of one sheet is near to the weight of the envelope, and thus hypothesises that the maximum allowable weight is filled by sheets, and then decreases the number of sheets by one, etc. A preliminary review of the results (see Boero & Shapiro, 1992) showed that there is a clear evolution with respect to age and instruction from ENV&SH strategies towards PRE-ALG strategies within and between numerical versions (this is found in homogeneous groups of pupils: transition from IV grade to V grade; and from VI grade to VIII grade ). We see that the motivations and access to prealgebraic strategies may be different; but in all of them there is a form of reasoning that may derive from a wide experience involving production of ‘anticipatory thinking’. That is to say, with the aim of economising efforts, pupils plan operations which reduce the complexity of mental work. This interpretation provides a coherence amongst different results, concerning the evolution towards PRE-ALG. strategies with respect to age, as shown in the solutions produced in –
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grade IV to grade V and in grade VI to grade VIII, as well as with respect to the results involving more difficult numerical data (in the case (250, 14, 16), results show an higher percentage of PRE - ALG. strategies at every age level). Concerning research findings in the domain of pre-algebraic thinking, we may observe that there is some coherence between: our results, concerning the influence of numerical data on strategies in an applied mathematical word problem, proposed to students prior to any experience of representation of a word problem by an equation and prior to any instruction in the domain of equations; and Herscovics & Linchewski’s results ( 1991), concerning numerical equations proposed to seventh graders prior to any instruction in the domain of equations. For instance, they find that the equation 4n + 17 = 65 is solved by 41% of seventh graders by performing 4n= 65-17=48 and then n= 4814, while the equation 13n + 196 = 391 is solved in a similar way by 77% of seventh graders. Taking into account the Herscovics & Linchewski’s (199 1) and Filloy & Rojano’s (1989) findings, we have performed a further analysis of our data which gives evidence of two extreme opposite patterns, and many intermediate behaviours of pupils engaging in a PRE-ALG. strategy. Some students seem to transform the problem situation by thinking about the number of sheets and the weight of the envelope as physical variables; indeed they subtract the weight of the envelope and work with the remaining weight. Other students ‘put into a numerical equation’ the problem situation (even if they do not formally write the equation!) and transform the equation (they perform a subtraction. and then a division on pure numbers). The presence of these extreme patterns in the same problem situation in the same classes may explain a deeper relationship between our findings and other findings concerning purely numerical equations. It also allows us to understand better the degree to which different approaches to the 'transforming function' of the algebraic language are complementary. Our study gives some information about the B)$#%(%./ *))(+ of algebraic transformations. As we saw in the preceding paragraph, algebraic transformations (especially the more open and complex ones) require the student to integrate two or more of the following activities: transforming the nature of the problem (through horizontal and vertical arrows), in order to be able to manage the transformed problem in an easier way; anticipating (i.e. imagining the consequences of some choices operated on algebraic expressions and/or on the variables, and/or through the formalisation process ) making choices in order to obtain the solution in an economic way: suspending the original reference meaning (at the sem level) of algebraic expressions, and working at the level of algebraic transformations; –
–
–
–
–
–
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using the reference to the meaning (at the sem level) to plan further steps of transformation of form (cfr. Radford, 1994: ‘semantic deduction’), or to interpret the consequences of performed transformations. If we consider the ‘sheets and envelope’ problem and the resolutions achieved by students, we realise that (depending on age and instruction) many of them, while producing and managing PRE-ALG. strategies, were able to integrate some of these activities in an effective way. –
DEEPENING THE TRANSFORMATION FUNCTION OF THE ALGEBRAIC LANGUAGE
We observe that any algebraic expression may be transformed into different expressions, and any transformation may be achieved through different patterns, according to different aims and criteria. I will try to explore some aspects of this ‘transformation’ process related to its aims and components. To do this, I will start by analysing an example in some detail: #$%&'()* this is the case of trigonometric equations deriving from mechanics or geometry problems; the transformation process is performed in order to bring them to a well known, easy to process expression: 2 2 2 2 sin @x + cos2 2x = 3/2 becomes: sin x + (cos x- sin x ) = 3/2 , 2 and then: sin 2 x + (1 - 2sin2 x ) =3/2 ,.............., 4 2 and, finally, 4 sin x - 3 sin x-1/2 = 0, which is easy to solve through the substitution: sin 2x = y . As regards the next points i), ii), I observe that the (standard) transformation of 2 2 B)+ 2x in terms of +%# K is suggested by a guess concerning the possibility of 2 writing down an equation in the ‘unknown’ +%# K : in the case of a high school student familiar with trigonometric equations, the experience gained in similar situations and the initial shape of the equation allow a transformation suitable to facilitate the task of solving the (transformed) equation. The transformation process, guided by this intuition, is performed according to standard patterns. Taking into account the analysis performed in the preceding example, we may consider the following working hypotheses:
+, The transformation function is performed through a -+%()./+. 0)(%/+1234+' between 3/%2-%0- '%//)023 15 /0%23510&%/+12, deriving from instruction and practice, which produce the transformations, (for instance, considering the equality: 2
2
b -a = (b-a)(b+a);
or: a/b+c/d=(ad+bc)/bd; or: (fg)'=fg+fg') ) and anticipations which suggest a suitable ‘shape’ for the formula to be processed and the direction of transformations. Concerning the words utilised to express this working hypothesis, I observe that the word ‘anticipation’ means the mental process through which the subject foresees
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the final (and/or some intermediate) shape of an algebraic expression useful for solving the problem, and the general direction of the transformations needed to get it. Different elements may be concurrent in this process: the memory of past, successful transformations performed in similar situations (i.e., experience); the intuition of possible, final or intermediate shapes of the algebraic expression, suggested by its present shape; the capacity of relating the shape of a possible transformed expression to the aim of solving the problem. With the word ‘dialectic’ I want to emphasise the fact that if the subject has the necessary prerequisites and experience to attack a problem needing algebraic transformations, his success depends on a functional dynamic relationship between the two ‘poles’ (standard pattern of transformation and anticipation) whose characteristics are different and, in some senses, opposite. The continuous tension between ‘foreseeing’ and ‘applying’, ‘guessing’ and ‘testing the effectiveness’ allows the productive development of the process of algebraic transformation. In general, standard patterns of transformation without anticipation offer blind perspectives - with the exception, for expert people, of some easy school exercises on ‘simplification’ of algebraic expressions or standard resolutions of equations. Concerning the expression ‘blind perspectives’, here are two very simple examples: (example, grade VIII): the student is requested to generalise, in the case of the sum of four subsequent odd numbers, the property according to which the sum of two consectutive odd numbers is a multiple of 4. He immediately writes down: –
p + 1 + p + 3 + p + 5 + p + 7=4p + 16 (the choice of the letter p probably depends on the first letter of ‘pari’, which means ‘even’ in Italian ), then he stops: he does not anticipate the divisibility by 8; the probable, original meaning of p (‘even’) and the divisibility of p by 2 remain hidden; at first glance the student only finds the divisibility by 4; later on he writes:
4p + 16 = 4p + 8 + 8 = (4p + 8) + 8, then he stops again; (example, first university year): the student already knows the proof according to which if f and g are derivable functions, then fg is derivable and (fg)'= f 'g+fg';the student must find out what happens with 1/f (if f is a derivable positive function), The student writes down the ‘incremental ratio’ of 1/f at point x: –
(1/f(x+h)-1/f(x))/h=((f(x)-(x+h))/f(x+h)f(x))/h; then he stops: no relationship is recognised with known derivatives, no connection is made with a formula to be proved.
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On the contrary, anticipation may suggest the final expression of the transformed formula or intermediate steps, but - apart from very easy problems - such results cannot be obtained without a sufficient mastery of the standard patterns of transformation. Here are two simple examples: – (example, grade VIII): through examples, students have discovered that the sum of two subsequent odd numbers is divisible by 4; the teacher helped them to write down the initial and final step of the proof of this conjecture: 2k + 1+ 2k + 3 = C * 4 (where C is a suitable number depending on k)
A student writes: 2k + 1 + 2k + 3 = 2k + 2k + 4, then he stops and says: ‘I see, 4 seems to be there ... But I cannot figure it out in the formula’. The standard transformations 2k+2k = 4k and 4k+4=(K+1) * 4 seem to be out of reach of this student. Another student writes: 2k + 1 + 2k + 3 = 2k + 2k + 4 = 4k + 4; then he says: ‘how can it be proved that 4k + 4 is a multiple of 4? ‘.The standard transformation 4k+4=(K+1) * 4 seems to be out of reach of this student, although he knew the distributive property for numbers, as we realised from a previous interview. (example, first university year): students must prove that ‘if f is derivable and 2 positive, then l/f is derivable and (1/f)’ =- f/f ‘; a student writes down the same expression already considered in the previous example concerning derivation, then 2 he says: ‘Oh, yes, I see: f(x+h)f(x) approaches f (x) when h approaches 0.... oh, yes, f(x)-f(x+h) is like - (f(x+h)-f(x)) .... but how can I bring h under the difference (f(x+h)-f(x)) ? The place of h is occupied by f(x+h)f(x)!’ –
++, - such -+%()./+. 0)(%/+1234+' may have different characteristics and develop in different ways in different problems, as shown in the examples under i) and in the following two extreme cases: proving a conjecture (see 2.3.): frequently, the ‘shape’ of the final formula may be easily determined - or it is given; a convenient algebraic representation of the relationship between data must be constructed in order to facilitate the process of transformation towards the final formula, anticipating some aspects of this process, and standard patterns of transformation must be applied to achieve the transformations; constructing a conjecture (see 2.2., second example): the final formula is unknown; exploring, anticipating, transforming algebraic expressions must take place, based on generalising and synthesising numerical experiments and/or establishing algebraic relationships between the variables involved; –
–
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+++, - such -+%()./+. 0)(%/+1234+' needs support by algebraic, external representations (see Janvier, 1987) with suitable characteristics in order to manage ‘patterns of transformation’ and ‘anticipations’ (symmetry, references of literal signs to their meanings,.....). In particular, as we have seen in the preceding examples, and we will see later in more detail, sometimes the shape of the written algebraic expression may provide hints for the process of transformation (thus supporting anticipation), sometimes the shape of the written algebraic expression is suggested by the guess of a possible transfomation suitable for solving the problem. According to our experience, the shape of the algebraic expression, autonomously written by students (or suggested to them by the teacher) in order to solve a problem, has a very strong influence on their performance. As an )$%&'()6 few eighth graders are able to prove that the sum of two consecutive odd numbers is a multiple of 4 (see 2.3.) if the teacher suggests writing two consecutive odd numbers as d, d+2 or p+1, p+3. On the contrary if the teacher suggests taking into account that an odd number may be written as 2k +1, then proof becomes accessible to many students. Another )$%&'()* in order to prove that (p-1)(q 2 - 1) is divisible by 16 if p and q are odd numbers, frequently high school or university students write p=2m+1 and q=2n+1 and finally get (by standard transformations) the following expression: (2m-2)(2n+2)2n= 8(m-1)(n+1)n (cf Arzarello, Bazzini & Chiappini, 1994). At this point, if the teacher does not intervene, many students abandon this track because they do not ‘see’ that (n+1)n is the product of an even number and an odd number ! The presence of m-1 acts as a distracter, the shape (n+1)n hides the existence of an even number in this product! These working hypotheses, which offer a ‘way of viewing’ the process of transformation of algebraic expressions, have been used to plan some experiments with students. Collected data will allow some cognitive aspects of the transformation function of the algebraic language (see 5.), and some educational problems concerning it (see 6.) to be analysed. Thus, it will be possible to understand if the ‘way of viewing’ realised through the previous hypotheses provides some insight into the process of transformation; it will also be possible to deepen the meaning of these hypotheses. COGNITIVE ASPECTS OF ALGEBRAIC TRANSFORMATIONS: THE PROCESS OF ANTICIPATION AND THE ROLE OF EXTERNAL REPRESENTATION From the cognitive point of view, I will try to deepen the role of suitable written algebraic representations in enhancing the previously mentioned dialectic
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relationship or preventing it from taking place. Concerning this issue, two experiments were realised in 1994/95. The 5+03/ )$')0+&)2/ concerned university students with a wide, common university background in algebraic transformations (fourth year mathematics students). The aim of this experiment was to explore the dependence of the two poles of that dialectic relationship (standard patterns of transformation; anticipation) on the possibility of writing algebraic expressions. Do written algebraic expressions. when they are allowed, enhance standard patterns of transformation and/or anticipation? How do limitations in using written algebraic expression affect standard patterns of transformation and/or anticipation? This experiment consisted of proposing different tasks (proving a conjecture; constructing a conjecture) >%(,)4( "#0 */+(*%B(%)#+ for one group of students (group A), and >%(, (,/@7)22)>%#$@*/+(*%B(%)#+ for the other parallel group (group B): when ‘proving a conjecture’, only the ‘final’ and the ‘initial’ algebraic expressions may be written; the proof must be written verbally without using algebraic signs. when ‘constructing a conjecture’, all the explorations had to be expressed verbally; no algebraic sign was allowed neither in the explorations nor in the expression of the conjecture. Here are the conjectures to be proved: (C1) (following an idea by Arzarello): prove that the number (p-1)(q2 - 1)/8 is an even number, when p and q are odd numbers (C2) if K is a natural number, prove that the sum of 2K consecutive odd numbers is a multiple of 4K The conjecture to be constructed concerned possible generalisations, different from (C2), of the property: ‘The sum of two consecutive odd numbers is a multiple of 4’ The conclusion of the analysis of collected data may be summarised as follows: for a very simple initial expression (conjecture C1), anticipation and standard patterns seem to be inore easily developed by the students of group B in comparison with the students of group A (having no restriction in writing the intermediate steps): the possibility of intermediate steps seems to reduce the engagement in planning activities; in the case of a more complex task (conjecture C2), or more general conjectures constructed, both anticipation and recourse to standard patterns of transformation are substantially enhanced by the possibility of managing and transforming written expressions in a written form, even if the two processes seem of very different nature (see the later discussion of: ‘externalisation’ and ‘internalisation’). It is also interesting to see that students of group B produce only simple conjectures, compared to those produced by the students of group A ; again, in the case of the conjecture C2, the choices of the initial expressions (when produced) are more carefully made by the students of group B with suitable letters –
–
–
–
–
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and ‘shapes’ : the prevention from transforming (in written form) the algebraic expressions seems to enhance anticipation; this puts into evidence the nature of the planning process, which is inherent in writing down the starting expression. All this seems to be very strictly related to our observations concerning the role of external representation in problem solving (see Ferrari, 1992): good problem solvers orient the external representation of the problem situation towards its resolution; this means that, from its very beginning, external representation is ‘solving process oriented’, with a balanced relationship between ‘externalisation’ (external realisation
of shapes and steps anticipated in the mind) and ‘internalisation’ (taking and exploiting products of one’s own external actions, or products of other people). The following excerpt, concerning a group A student who tries to prove C2, shows mean in the case of algebraic what ‘externalisation’ and ‘internalisation’ expressions:
(after 2 minutes):
SJhC&PF "9 #)( +4*/ "3)4( (,/ 2"+( (/*9 )7 (,/ +49 Q)>/./*1 >/ >%22 +//T: (after 3 minutes):
(**) ?/*,"D+1 F B"# 3"2"#B/q
F( %+ #)( 0/( '%.%+%32/ 30 4K. N/( 4+ (*0 >%(, +)9/ ."24/+ )7 K K = 1 9/"#+P (>) )''1 B)#+/B4(%./ #493/*+ : 2m + 1 + 2m+1 F( ')/+ #)( >)*=q C,/0 "*/ #)( B)#+/B4(%./q Perhaps, the first is 2m +1 and the last is V9 t V<@t X for
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I consider only the second terms of the sums :
X1 [
OK, let us come back to V SV9 +
(***) now it works very well, I see s
We can see that: SwT %+ " 9/#("2 D*)'4B( S7*)9 g%#+%'/e () g)4(+%'/e), D*)3"320 */"2%+/' %# )*'/* () 7%K S"#' $"%# "BB/++ () 74*(,/* "#"20+%+ (,/ %#(4%(%)# )7 " g+,"D/e +4$$/+(/' 30 #49/*%B"2 /KD/*%9/#(+: C,/ g,/4*%+(%Be %#(/#(%)# )7 (,%+ +(/D %+ B)#7%*9/' 30 (,/ gJhC&e:
(**) (,/ g+,"D/’ D*)'4B/' +4$$/+(+ S7*)9 g)4(+%'/e () g%#+%'/eT (,/ D*)B/++ )7 3"2"#B%#$ "#'1 D*)3"3201 (,/ D)++%3%2%(0 )7 D4((%#$ %#() /.%'/#B/ 942(%D2/+ )f V9 "#' V<: F# )*'/* () 3/((/* */"2%+/ (,/ D*)B/++ )7 3"2"#B%#$1 "#)(,/* /KD*/++%)# %+ D*)'4B/' S7*)9 g%#+%'/e () g)4(+%'/ ’T .
(***) (,/ +(4'/#( 7)*/+//+ SD*)3"320 >%(, (,/ ,/2D )7 (,/ /KD*/++%)# >*%((/# "7(/* (**), >%(, " D*)B/++ 7*)9 g%#+%'/e () g)4(+%'/e "#' +43+/b4/#(20 7*)9 g)4(+%'/e () g%#+%'/e ) (,"( (,/ B,"#$/ '/B%'/' %# (,/ 7)*942%+"(%)#1 "#' %9"$%#/' %# (,/ /KD*/++%)#1 7%(+ %# >%(, (,/ #//' () +,)> (,/ '%.%+%3%2%(0 30 s: C,/# (,/ +(4'/#( D*)'4B/+ (,/ #/> /KD*/++%)#1 7)22)>%#$ (,/ )2' )#/ B"*/74220: The 3).12- )$')0+&)2/ was made with students of different ages and school experience in the domain of algebraic transformations (grade VIII, comparing groups of students having followed different curricula; first year university students coming from different kinds of high schools). The aim of the experiment was to explore the dependence of mental and written transformations on previous experience in this domain. Two kinds of tasks were proposed:
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transforming a given expression into a given expression without/with permission to write down intermediate steps of the algebraic transformation. Here is an example of a test used in this experiment for first year university students: Consider the expression: (b³-a³)(b²+a²)/(b4-a4); Can you transform it into : b + a ? Or into: b - a ? Or into: (a² + ab + b²)/(a+b) ? proving a conjecture, expressed in verbal terms, with permission to write down the initial and final steps, and without/with permission to write down intermediate steps of the algebraic transformation. The conjecture to be proved, for grade VIII students. concerned the fact that ‘the sum of four consecutive odd numbers is a multiple of eight’. –
–
For first year university students, the conjecture to be proved concerned the fact that ‘if K is a natural number, the sum of 2K consecutive odd numbers is a multiple of 4K’ In both situations, in the case of students who arc allowed to perform a completely written transformation process the previous school experience (quality and quantity) seems very relevant; in the case of students not allowed to write down the intermediate steps, only the quality of the previous school experience seems to be influential (especially as regards the ‘anticipation’ aspects of the transformation process). In the second situation, we also obtained confirmation of the fact (found in the other experiment) that prevention from writing intermediate steps forces students to find more suitable ‘shapes’ for their initial formulas (enhancing ‘anticipation’). EDUCATIONAL IMPLICATIONS From the educational point of view, I will consider some (more or less) current activities which may hinder or enhance the development of the -+%()./+. 0)(%/+1234+' considered previously in this chapter: - calculating standard arithmetic expressions; - transforming algebraic expressions in order to simplify them: - understanding and repeating algebraic proofs; - producing and proving conjectures expressed with algebraic formulas; - discussing the direction of transformations needed to obtain an algebraic expression with given characteristics. The content of this section largely agree with some findings obtained with different methods and from a different perspective by Y. Chevallard (1989).
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7%(.8(%/+29 3/%2-%0- %0+/4&)/+. )$'0)33+123* usually students are taught to manage algebraic notations according to very strict algorithmic rules SgB"2B42"(/ 942(%D2%B"(%)#+ "#' '%.%+%)#+1 (,/# "''%(%)#+ "#' +43(*"B(%)#+1 7*)9 (,/ %##/* D"*/#(,/+/+ )4(>"*'+eT ; the mechanism is blind, some small, formal changes modifying the usual ‘shape’ of the expression may cause serious problems, as happens in the calculation o f :
We observe that anticipation is not stimulated, nor is the application of algebraic properties of operations (distributive property...). For these types of problems. teaching might be considerably improved. but no effort is usually made in this direction.
:0%23510&+29 %(9);0%+. )$'0)33+123 +2 10-)0 /1 3+&'(+5< /4)&* in some cases, the final result is given (and it may stimulate some anticipation process, such as in an algebraic proof of a given formula); in other cases. anticipation is needed to perform transformations which (for some phases of the process) do not tend to ‘reduce the number of parentheses’. 3 3 2 2 4 4 For instance, the simplification of the expression: (b -a )(b +a )/ (b -a ) needs some 4 4 parentheses to be added temporarily, anticipating the fact that b -a may ‘liberate’ 3 3 b-a and b²+a², and that b -a may ‘liberate’ b-a. 3 3 2 2 We observe that executing the multiplication (b -a )(b +a ) results in a stalemate. Examining nine textbooks for Italian high schools, I found many examples of this kind only in two textbooks. On the contrary, most expressions proposed in the textbooks suggest moving towards progressive simplification, step after step, reducing the number of parentheses, as in this example:
Examples of this kind are very frequent in almost all the textbooks for Italian high school Usually, the exercise is followed by the final, ‘simple’ result (- V" ) which suggests this process of standard, progressive simplification.
=2-)03/%2-+29 %2- 0)')%/+29 %(9);0%+. '01153: this activity is very common from high school onwards, in algebra (algebraic equations, theory of groups, vector spaces, etc.) and in other domain when using the algebraic code. No anticipation is usually requested, while standard patterns of transformation are performed and may be better understood.
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Producing and proving conjectures expressed with algebraic formulas; discussing the direction of transformations needed to obtain an algebraic expression with given characteristics: these are uncommon activities in preuniversity mathematics education: they might be used to enhance anticipation and (under suitable guidance from the teacher) to stimulate awareness about the nature of processes of transformation (metacognitive aspect). It is not difficult to plan this kind of activity. For instance, the example illustrated on page 103. might be proposed to comprehensive school students as an introduction to conscious transformation of algebraic formulas. Unfortunately, no room is usually left for this kind of topics. Striking a balance between 'common' and 'uncommon' activities performed using the algebraic code, we find that the activities more suitable for ensuring development of ‘anticipation’ and a conscious management of the process of transformation are 'uncommon' in schools. At present, students are mainly forced to develop the ‘standard patterns of transformation’ component of the transformation process.
CONCLUSION The aim of this chapter was to show how the transformation of the mathematical structure of a problem is a crucial aspect in algebraic problem solving, and the role of anticipation in allowing the process of transformation to be directed towards simplifying and resolving the task. The point of view illustrated in this article brings some elements of novelty in the debate concerning the school approach to algebra and the relationships between arithmetic and algebra. Indeed. the transformation of the mathematical structure of a problem finds a natural tool in the algebraic language, but can be performed also before algebraic formalisation (as shown in the PRE-ALG. strategies of students, described earlier in the chapter). Suitable word problems, needing pre-algebraic strategies (like the ‘envelop and sheet problem’), could be largely introduced in early grades in order to develop anticipation. Other findings illustrated in this chapter concern the transformation of algebraic expressions in algebraic problem solving. Processes of anticipation integrated with the recourse to appropriate standard patterns of transformation play a major role in ensuring effective transformations. This point of view suggests that current classroom activities in the field of algebra are not equilibrated - most exercises are aimed at developing only standard patterns of transformation; other activities, which might be very useful in order to develop anticipation, are missing. This chapter introduces the +/9 I 7)*9 diagrams as an original, heuristic tool to schematise some crucial algebraic problem solving activities. In spite of the lack of ‘formal’ definitions of the elements of the +/9@I@7)*9 diagrams, their usage allowed the clarification of the complexity of mental operations involved in algebraic
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problem solving and might suggest possible interpretations of some difficulties met by students and some ways to overcome them. For instance, the example concerning the evaluation of the area of a rectangular trapezium, sometimes used to introduce students to algebraic transformations, might also be exploited to make them aware of the ‘sense’ of algebraic transformations in algebraic problem solving.
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AURORA GALLARDO*
HISTORICAL-EPISTEMOLOGICAL ANALYSIS IN MATHEMATICS EDUCATION: TWO WORKS IN DIDACTICS OF ALGEBRA
INTRODUCTION In this chapter, we will review some research in mathematical education whose salient feature is the use of historical-critical analysis as an important methodological component to investigate problems of learning and teaching elementary algebra. Our aim is on the one hand, to make explicit some of the common underlying assumptions and the methods used in these studies, and, on the other, to give a brief summary of some of the results obtained. The belief that a full understanding of mathematical concepts must take into account their historical development, is linked to a perspective about the nature of mathematical knowledge which, far from contemplating it as unshakable truths and eternal structures, looks at the results of a process of historical-cultural development. Already at the beginning of the century, mathematicians like Federigo Enriques (191 3), described the nature of mathematical knowledge as essentially historical, and pointed out that such a vision demands a critique of the foundation schools that try to reduce the whole theoretical content of matliematical knowledge to its formalsyntactical aspects. Even definitions, far from being arbitrary, are the result of a long process of construction and a persistent effort to understand. linked to a tradition of problems and to an order which, from Enriques' point of view, governs the development of the mathematical science. Several mathematicians and philosophers, like Brunschvieg, Cavaillès and Lautman in France and Cassirer, Becker and Weyl in Germany, formulated in the first half of this century, criticisms akin to Enriques' to the positions of the logicist and formalist schools. So we can affirm that there is a long genealogy of attempts to frame the nature of mathematical knowledge, and the practices of the mathematicians, within an historical-cultural perspective. Similar
*
Departamento de MatemáticaEducativa, Centro de Investigación y de Estudios AvanzadosIPN, Mexico.
121 M: ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 121-139. © 2001 <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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views can be seen. updated to the mathematics and epistemology of our times, in recent authors (Wilder 1981, Ernest 1991, Tiles 1991). Let us briefly review some ideas concerning historical-critical analysis advanced by Piaget, Bachelard and Brosseau, which are part of the antecedents for the theoretical frameworks of the works on didactics of algebra presented in this chapter. Historical-critical analysis is, for Piaget, one of the basic tools of epistemological inquiry. It is concerned with historical analysis, not as an account of a sequence of discoveries, but as the historical reconstruction of parts of knowledge, in order to carry out a ‘critical’ analysis, in a sense akin to the Kantian idea of criticism. Piaget resorts to history in order to search for the foundations of scientific knowledge, and to look for understanding of the processes of invention and discovery (Piaget, 1960). In Piaget’s work, the reconstruction of the history of science cannot be separated from the analysis of the formal structure of knowledge, with a psycho-genetical analysis based on empirical work of a clinical nature. Without endorsing the extreme thesis posited by Haeckel in his ‘biogenetic law’. that ontogenesis is a replica of phylogenesis, deep and far reaching analogies are recognised by Piaget within both courses of development. In his work, numerous cross references are found confronting data taken from the history of science, with empirical data concerning the ontogenesis of knowledge. Both on the historical and individual courses of development, advance is triggered by cognitive conflict. Hence the study of the development of the fundamental ideas of mathematics goes hand in hand with the study of fundamental errors and obstacles which are their inseparable counterpart; and the notions of epistemological and cognitive obstacle may constitute illuminative categories. The term /D%+(/9)2)$%B"2 )3+("B2/+ was introduced by Bachelard: ‘We must pose the problem of scientific knowledge in terms of obstacles. It is not just a question of considering external obstacles, like the complexity and the transience of scientific phenomena, nor to lament the feebleness of the human senses and spirit. It is in the act of gaining knowledge itself, to know, intimately, what appears, as an inevitable result of functional necessity, to retard the speed of learning and cause cognitive difficulties. It is here that we may find the causes of stagnation and even of regression. that we may perceive the reasons for the inertia, which we call epistemological obstacles.’ (Bachelard 1976, p15) He goes on to say: ‘We encounter new knowledge which contradicts previous knowledge, and in doing so must destroy ill formed previous ideas’,
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Bachelard pointed out that epistemological obstacles occur both in the historical development of scientific thought and can also be identified in educational practice. For him, epistemological obstacles have two essential characteristics: they are unavoidable and essential constituents of knowledge to be acquired; they are found, at least in part, in the historical development of the concept. Many authors have become interested in epistemological obstacles. Brousseau (1983) has pointed out that the notion of epistemological obstacle can be made fruitful only if obstacles are really identified in the history of mathematics, their analogues are recognised in the spontaneous models of the students, and pedagogical conditions for overcoming or avoiding them are carefully studied in order to design specific didactic projects whose outcome can be positive. In addition, Brousseau points out that this research requires a further effort of invention, because Bachelard’s idea does not fit well with the domain of mathematical education. Nonetheless, historical-critical analysis can provide valuable insights about fundamental ideas concerning different subject matters. even though it remains open to question whether these ideas arc necessarily best understood in terms of obstacles. Brousseau defines an epistemological obstacle as knowledge which functions well in a certain domain of activity and therefore becomes well-established, but then fails to work satisfactorily in another context where it malfunctions and leads to contradictions. It therefore becomes necessary to destroy the original insufficient, malformed knowledge, to replace it with new concepts which operate satisfactorily in the new domain. The rejection and clarifying of such an obstacle is an essential part of the knowledge itself; the transformation cannot be performed without destabilising the original ideas by placing them in a new context where they are clearly seen to fail. This therefore requires a great effort of cognitive re-construction. Once the assumption of the historical nature of mathematical knowledge is made, there emerges the question of the role historical analysis may play in research work in mathematical education. In the studies we review here the use of the historical-critical methodology is aimed at: Clarifying the conditions which make certain concepts possible and allow the formulation of certain problems. Clarifying the nature of some difficulties related to learning algebraic concepts. The study of the epistemological obstacles faced throughout the historical development of algebra may throw light on our understanding of the cognitive obstacles underlying didactic cuts observed in empirical work. On the other hand, knowledge of the results of empirical work allows a deeper reading of historical texts, which takes into account not only historical contingency and idiosyncrasy, but cognition as well. Improving our understanding of the evolution of languages and methods if solving problems, which precede the use of symbolic algebra.
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In the following section, we present two works concerned with different aspects of algebra, which use historical-critical analysis as an important methodological component. The approach to using historical-critical methodology in mathematical education in the studies presented, is characterised by a recurring back and forth motion from the analysis of classical texts in the history of mathematics, to empirical work carried out both in clinical situations, and inside educational systems. The back and forth motion between theoretical analysis of historical texts, and empirical work in clinical and teaching environments, situates these works in the realm of mathematical education, in contrast to pure historical or epistemological inquiry. Historical analysis becomes relevant for mathematical education when it throws light on understanding the emergence and evolution of concepts, when it helps in explaining the conditions which make a problem formulable, and when it allows the recognition, throughout the development of mathematics, of the persistence of certain problems inside the very solutions constructed for them over a period of time. THE ACQUISITION OF ALGEBRAIC LANGUAGE (FILLOY & ROJANO, 1984) This work is aimed at pointing out certain changes, both with respect to concepts and in the use of symbolisation, which mark a difference between arithmetic and algebraic thinking in the individual. From the analysis of the strategies and methods of solving problems found in the pre-symbolic algebra textbooks of the 13th, 14th and 15th centuries (Boncompagni, 1954; Arrighi, 1974; Hughes, 1981), a conjecture was made about the existence of a didactic cut between arithmetic and algebra. linked with a resistance to operating on the unknown. The existence of this cut was later confirmed and minutely studied through clinical research with 12 and 13 years old pupils undergoing the transition from arithmetic to algebra (Filloy and Rojano 1985a, 1985b, 1989 and Rojano 1985). These empirical studies, were concerned with a restricted class of problems leading to equations of the form Ax B = Cx D . The same phenomenon has been observed in broader classes of problems, and analysed theoretically in great detail without appealing to history, by Herscovies and Linchevski ( 1991). The analysis and comparison of L/ J49/*%+ L"(%+ by Jordanus de Nemore. and some of the so called ‘abacus books’, made it possible to conjecture the existence of a didactic cut in the process of acquisition of algebraic language. L/ J49/*%+ L"(%+ by Jordanus de Nemore, written in the XIIIth century, is considered by some historians of science to be the first textbook of advanced algebra. This book was intended to convey to students of Toulouse University skills for dealing with practical, non routine numerical problems. The so-called ‘abacus books’ are
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summaries of the practical arithmetic of medieval times and the early Italian Renaissance, which formulate numerical problems and describe their solutions in vernacular language (mainly in Venetian dialect), and sometimes made use of the methods of Oriental Mathematics imported by the Arabs (Eginond, 1980). As an example, we present a problem from Book I of De Numeris Datis. In the first column, there is a translation in English of the formulation of the problem, and of the steps for its solution, as written in the De Numeris Datis in plain language. The second column shows a possible translation of the steps of the solution into modem algebraic notation, and the third column displays some comments about the process.
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The first thing that comes out when analysing such problems and their solutions, is that they are formulated in a language which i s no longer spoken. The problems resemble typical word problems of a modern school text, but the strategies for their solution differ from present day thinking processes: we would not carry out the operations that appear to have been a necessary part of the process. Nowadays, from the point of view of the structured mathematical language of mathematicians, the languages of the Abacus books "#' )7 L/ J49/*%+@L"(%+@"*/@'/"'@2"#$4"$/+: Filloy and Rojano remark that different strategies and problem solving methods arising in history are linked to different languages with characteristic peculiarities. which, before the advent of symbolic algebra, are difficult to relate to each other. It is possible also to contrast analogous language strata by analysing diagnostic questionnaires and clinical interviews with 12-15 years old students. The following example shows part of a clinical interview illustrating the way in which one student makes two irreducible uses of language when interpreting the algebraic equation X[ξv[Z: S: gF( B)42' 3/P X[ 942(%D2%/' 30 +)9/ #493/* $%./+ [Z SX+( 4+/T1 )* X[ 2)(+ )7 +)9/ #493/* %+ [Ze SV#' 4+/T: I: gk/+e: S P gX[ 942(%D2%/' 30 " #493/*1 3/B"4+/ >,/# (,/0 "*/ ()$/(,/*1 %+ " >"0 () +%9D2%70 942(%D2%B"(%)# ’. I: g5#' (,/#1 m,"( >)42'@0)4@')xe S: g[Z '%.%'/' by X[ ‘. S-+%#$ B"2B42"()*T gF( >)42' 3/ [e: I: g5#' %7 0)4 >)42' 4#'/*+("#' %( %# (,/ +/B)#' >"0xe S: gX[ 2)(+ ) 7 K #493/* gives [Ze:
I: gQ)> ') 0)4 +)2./ %(:xe S: gF ')#e( =#)> ,)> () B"**0 %( ) 4 ( e @
I: gN/(e+ >*%(/ %(e: S : Sm*%(/+TP gC,%*(//# 2)(+ ) 7 K %+ $)%#$ () 3/ [Ze: gI 4#'/*+("#' (,%+ >"0P K %+ "# 4#=#)># #493/* >,%B, B"# 3/ 7)4#' >%(, "# /b4"(%)#: C,/ )(,/* 7)*9 F 4#'/*+("#' %( %+ S+,/ >*%(/+TP C,%*(//# 942(%D2%/' 30 K %+ $)%#$ () 3/ [Z ’. I: g5#' (,%+ gSC,%*(//# 30 K %+ $)%#$@()@,/ [Z eT >"+ +)2./' ')%#$ >,"(xe S : gL%.%+%)#e: I:
g5#' 0)4 9"'/ +4*/ %( >"+ B)**/B(1 ,)>x
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S : S-+%#$ B"2B42"()*Te2[ 30 [ /b4"2+ [Ze:
I: gC,"( %+1 0)4 =#)> ,)> () +)2./ (,%+ SgC,%*(//# 30 x %+ $)%#$ () 3/ [ZeT1 34( SgX[ 2)(+ )7 K #493/* $%./+ [Z ’), ')#e(@0)4@=#)> "(@(,%+ 9)9/#(xe S gJ) e: The above text illustrates that the student thinks that only the first use of language can be represented by the algebraic equation 13$=39. In the second use, she thinks of 13$ as a repeated addition. This fact impedes the inversion process, that is 39 divided by 13. This is the reason why she can not solve the equation In her second use of language. In the L/ J49/*%+ L"(%+1 transposition of unknown terms to the other side of the equation does not appear, due to the lack of symbolic language. This allows us to make plausible hypotheses, which can be tested within empirical studies. So, the analysis of texts that predate Viète’s 5#"20(%B 5*(1 together with the development of experimental teaching sequences, suggested to the authors the existence of a didactic cut in the child‘s evolutionary line of thought from arithmetic to algebra. This cut corresponds to the major changes that took place in the history of algebra in connection with the symbolic representation of the ‘unknown’ and the possibility of ‘operating on the unknown’. In terms of the curriculum, the cut is located at the transition between: (‘6/7)*/ ’) The students know how to solve arithmetical equations of the type A (Bx
C) = D
x/A=B x/ A = B/C
In order to solve these it is sufficient to invert or to ‘undo’ the indicated operations. F( %+ #)( #/B/++"*0 () )D/*"(/ )# )* >%(, (,/ 4#=#)>#: (‘57(/*’) Students have received no instruction on how to solve equations of the types. Ax
B = Cx D
To solve these it is not enough to invert the indicated operations. F( %+ #/B/++"*0 () )D/*"(/ )# >,"( %+ */D*/+/#(/':2 Whether ‘the resistance to operating on the unknown’ is or is not an epistemological obstacle in the sense of Bachelard may be the subject of discussion by epistemologists or historians of mathematics. But from the point of view of mathematical education, this analysis of historical texts suggests the possibility of
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studying empirically analogous obstacles within the problem solving approaches of beginning algebra pupils. We will not describe further details of this research. What we want to point out is that the theoretical results of the analysis of classical texts suggested a possible focus for empirical study as well as plausible categories for the theoretical framework of that study. On the other hand, once the results of the empirical work were known, the reading of the historical texts became deeper. Some things that in the first readings appeared as contingent and idiosyncratic, were later recognised as belonging to the way in which specific knowledge is constructed. THE STATUS OF NEGATIVE NUMBERS IN THE RESOLUTION OF EQUATIONS (GALLARDO & ROJANO, 1994).
As individuals acquire algebraic language, the extension of the numerical domain from natural numbers to integers becomes a crucial element for achieving algebraic competence in the solution of problems and equations. This statement led to the design and execution of this project. The didactic study of this research shows that negative numbers are interpreted by secondary school students in various ways: subtrahend, where the notion of number is subordinated to the magnitude, (for example. zero is considered less than negatives); signed number, when a plus or minus sign is associated with the number: relative number (or directed number), where the idea of opposite quantities in relation to a quality arises in the discrete domain and the idea of symmetry appears in the continuous domain; isolated number where there are two levels, that of the result of an operation or as the solution to a problem or equation. Finally, the formal mathematical concept of negative number is reached, within an enlarged concept of number embracing both positive and negative numbers. We intend to show that historical avoidances and recognitions of negative numbers, have a counterpart in different stages of conceptualisation of these numbers manifested by present day students. The first three stages mentioned do not follow a strict chronological order in the individual subject. The same student exhibits different levels of conceptualisation of negative numbers, depending on the task. For example, he may interpret 5-(-3) as 5-3, where the negative -3 is not ‘recognised’, while on the other hand, he ‘accepts’ the negative solution for some word problems, and rejects it in others. In the reification theory of Sfard and Linchevski (1994), the first four stages mentioned above could be described as operational, and the last as structural. The structural stage for negative numbers is usually not reached by 12-13 years old students. From the above work questions such as the following arose: with respect to equations and problems, what is the numerical domain that secondary school students ascribe to the constituent parts of an equation during the process of
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solution? Which numerical domain is accepted for the solution? What is the relationship between the numerical domain assigned to an equation and the type of language associated with the equation? Which methods or strategies obstruct or facilitate evolution towards the notion of number? These questions point to the need for research whose central concern is the study of the interrelationships between the processes of acquisition and use of algebraic language the methods for solving word problems and linear equations and the status of the negative number³ in word problems and linear equations. The general methodology of the project considers the interaction of these three components on two levels of analysis, the historical-critical level (evolution of meaning) and the didactic level (teaching-learning-cognition). With regard to the first of these, we found that the properties of subtraction, the laws of signs and certain elements necessary for operativity with negative numbers, appeared in remote historical times in the context of the solution of algebraic equations. The opposing concepts of gain and loss, property and debt, future and past, sale and purchase, are adequate interpretations for positive and negative. A crucial step towards acceptance of the negative number was to admit negative solutions to equations. The main difficulty facing medieval mathematicians in the solution of concrete problems was precisely the interpretation of negative solutions. The historical analysis carried out in this work allowed us to conclude that problems posed in old texts contributed to extend the numerical domain of natural numbers to that of the integers in the resolution of algebraic equations. In the following section we analyse three historical problems, one from the Chinese text Nine Chapters of Mathematical Art (ca. A.D. 250) one from the Hindu text Bijaganita (XIIth century) by Bhãskara and the third one from the French Triparty en la Science des Nombres (1484) by Nicolas Chuquet. Together with the historical analysis we also exhibit an analysis of the ways in which present-day 12-13 pupils approach the negative objects inherent in these problems. A%*+( &K"9D2/
The Fiu Zhang Suanshu (Nine Chapter of the Mathematical Art), is one of the earliest mathematical texts in China. Let us examine the eighth chapter entitled A"#$ E,/#$: Just like the other chapters in the text, the present version of the Fang Cheng chapter contains a number of problems together with their respective solutions. Firstly, we find the use of negative numbers, showing that the ancient Chinese had a clear concept of them and were able to apply negative numbers in mathematical problems as we would do nowadays. Secondly, the Fang Cheng chapter shows the formulations and solution of simultaneous linear equations of up to five unknowns. Thirdly, the Fang Cheng chapter introduces the methods of solving equations by tabulating the coefficients of unknowns and the absolute term in the forms of a
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matrix on the counting board, thereby facilitating the elimination of the unknowns, one by one. It must be emphasised that ancient civilisation had no ready made sets of notations. Conceptualisations were in a verbalised form, though the Chinese took a forward step when they used rod numerals to convert concepts onto the counting board. There are only two methods in this chapter. The first, called A"#$ E,/#$ or calculation by tabulation, is about solving a set of equations. The second method, called the positive-negative rules (zheng fu shu), comprises rules for the subtraction and addition of positive and negative numbers. The term Fang Cheng is defined as the arrangement of a series of things in columns for the purpose of mutual verification. The number of columns to be set up is determined by the number of things involved. In modem notation each column has two sections; the top section consists of the quantities aij (i, j = 1,2, ... n) of the various things while the bottom one shows the absolute terms bi (i = 1,2, ... n). Such an arrangement on the counting board can be shows as follows :
The whole process of operation is done on the counting board using the rod numerals to represent the various quantities.4 The unique place-value feature of this method of computation renders the use of symbols unnecessary. In each column of things on the counting board, the space between aij and bi, has the implicit function of an equal sign. The former matrix arrangement is transformed in such a way that all numbers in the upper side of the main diagonal are equal to zero (only columns are operated on). This transformed matrix corresponds to a diagonal set of equations, from which all the unknowns are successively determined. One can see that this method is essentially the usual method in present day algebra. Since the process of the Fang Cheng solution is the successive elimination of numbers through mutual subtraction of columns, there could be cases when a number to be subtracted from one column is smaller than the corresponding one in the other column. The opposite result obtained has to be indicated and certain rules have also to be established in order to continue the eliminating process. This gives
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rise to the creation of names: the term fu to indicate the resulting opposite amount to the term zheng for the normal difference. The concept of zheng and fu seems to have evolved from such ideas as ‘gain and ‘loss’ as clearly shown in Problem 8 which reads: ‘By selling 2 cows and 5 goats to buy 13 pigs, there is a surplus of 1000 cash. The money obtained from selling 3 cows and 3 pigs is just enough to buy 9 goats. By selling 6 goats and 8 pigs to buy 5 cows, there is a deficit of 600 cash. What is the price of a cow, a goat and a pig?’ The text considers the selling price zheng because of the money received and the buying price fu because of the money spent. The surplus amount is considered zheng and the deficit fu. These terms are merely names to indicate the nature of numbers. For the purpose of computation, numbers described by these terms have to be transcribed into a concrete form. There are two ways of doing this with rod numerals. If different coloured rods are used, then red ones represent zheng and black ones represent fu. Alternatively, if the rods are of one colour only, the fu numeral is indicated by an extra rod placed diagonally across its last non-zero digit. It is explained in the text that when a number is said to be negative, it does not necessarily mean that there is a deficit. Similarly, a positive number does not necessarily imply that there is gain. Therefore, even though there are red and black numerals in each column, a change in their colour resulting from the operations will not jeopardies the calculation. Negative numbers appeared as a c l a s s of numbers in the mathematical sense that is familiar to us today. The concept of positive and negative, which initially evolved from opposing entities such as ‘gain and loss’, ‘add and subtract’ and ‘sell and buy’, is now detached from linguistic associations. Its development has resulted in negative numbers being regarded as one group of numbers with properties which are connected with the other group of ‘normal’ or positive numbers. These properties are defined by positive-negative rules which correspond to the modern ones. Problem 8 involves selling and buying which equate to the concept of positive and negative respectively. The corresponding set of equations in tabulated form becomes: (3rd equation)
(2nd equation)
(1st equation)
(cows)
-5
3
2
(goats)
6
-9
5
(pigs)
8
3
-13
-600
0
1000
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When they had set up the equations in this form, they then operated the columns (in a way similar to the matrix reduction method for solving equations). They aim to make the numbers in the top right triangle zero. As can be seen, the Fiu Zhang has provided substantial evidence that, by the first century, the Chinese not only accepted the validity of negative numbers but understood their relationships with positive ones and were able to formulate rules and to compute with them. The historical analysis aforementioned suggested the use of what is called the ‘Chinese Model’ in the didactic realm. The 12-13 years students selected for this empirical study were familiar with the model of the number line and with the syntactical rules for operating on integers. These students showed conflict in the cases a-(-b) and -a-(-b) with a and b natural numbers. It was then decided to use a model which would give ‘concrete’ existence to negative numbers as the Chinese mathematicians did (they used black rods for negative numerals). The operativity employed in this teaching model corresponds to that of the Chinese mathematicians, that is, positive numbers are opposite to negative ones. The central concept arises, the sum of opposites is zero, which gives foundation to all the operations carried out within the model. The Chinese Model is based on 1) the counting of positive numbers extended to negative numbers; 2) an alternative representation of the minuend required in some cases in order to carry out the operation of taking away. Then, the addition of zeros is employed, as needed in each case. The students were presented with a diagrammatic version of the model. In the world of paper and pencils, positive numbers were white balls, and negative numbers, black balls. Zero was represented by the simultaneous presence of a white and a black ball. Operativity was carried out in the additive domain. For example, the addition 3 + (2) is presented; the numbers are described in the model
. They join together, provoking the
formation of zeros o . The result is a white ball which represents the number 1. Analogously, for the subtraction 2-3. The numbers are described in the model. Three cannot be subtracted from 2. A zero is added to the number 2. The representation of 2+ is obtained. Then the subtraction can be carried out, crossing out, A black ball is obtained, the representation of - 1. Some difficulties manifested by students when using the Chinese Model, are shown in what follows: The expression 2 - (-3) is represented as and zeros are formed . Instead of executing the action of taking away, the addition of opposites is effected.
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The diagrammatic representation of numbers in the model, is not adequately interpreted. For instance, a student says:
He does not identify the whole array as +1, because he fails to perceive opposites. On the other hand, some students identify opposites, but instead of cancelling, they try to give zero its positional value: is interpreted as 10. The following can be concluded from this analysis: competent users of the Chinese Model showed: differentiation of the actions of adding and subtracting negative numbers. recognition of the dual nature of zero, as a null element and formed by opposing elements (for example represents zero but also +l, - 1). distinction of negative numbers as relative numbers, and as isolated numbers (the students manifested five conceptual levels of negative numbers depending on the task, see page 128 of this chapter). It should be said that the main interest in modelling in this study is not its usefulness in teaching but as a resource which exhibits the different levels of acceptance of negative numbers by the students and allows them to develop different meanings for integer.5 ;/B)#' &K"9D2/
In the Hindu text Bijaganita (§140 in Colebrooke’s translation) Bhãskara treated equations of the second degree. He noted that, in some problems it may be that only one solution is acceptable even ifboth are positive. For example: ‘The fifth part of the troop less three, squared, had gone to a cave; and one monkey was in sight, having climbed on a branch. Say how many they were’. Bhãskara remarks: ‘Here a two-fold value is found, 50 and 5. But the second is in this case not to be taken: for it is incongruous. People do not approve a negative absolute number’. (The commentator of the text explains that the second value is five, its fifth part is one and can not be subtracted from three). The Hindu mathematicians used first letters of words for symbolising algebraic terms. They distinguished negative quantities by putting a point over the number and did not have a subtraction sign nor addition, multiplication or equal signs. The equation solving the problem above in both, Hindu’s notation and modern notation, is written as
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where ya (the first letters from the word yávat-távat) is x, ya v represents x², ru (from the word rúpa) is used for the independent term and Va
is
Va
This problem was presented to present day 12-13 year old students. The lack of equilibrium between the semantics and syntax of algebraic language during the process of the solution’s validity was observed. The following is a fragment of an interview after a student had solved correctly the Bhãskara’s problem. I: There are two solutions. What does it mean? Do these express how many monkeys exist?How do you interpret this? S: Could it be possible for 50 and 5 to exist? 1: But, could both answers be correct? S: Yes, they could be both. I Is there anything that might hinder it? S: Oh, sure! -2 monkeys can’t ever exist. I: -2 monkey can’t ever exist! Where did 0)4 get that answer of -2 monkeys? (Then the student explains how he obtained -2 by substituting 5 in the equation +1
=
x
S: Because, 5 divided by 5 is equal to 1, minus three, is equal to minus 2. Thus, there couldn’t be -2 monkeys in a troops.
The student attributes total validity to algebraic methods and the process of verification. However the context prevents the acceptance of one of the two positive solutions. This happens during the process of verification where the subject cannot abandon the meaning of the symbol (monkeys) and give sense to the process. C,%*' &K"9D2/
In the Appendix of the text Triparty en la Science des Nombres, Chuquet exhibits problems in which a negative solution is accepted and this is interpreted according
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to the context of the problem. Chuquet introduced a syncopated language with a general notation very similar to modern symbolisation. He writes the numbers with a 1 zero exponent. for example 12 as the linear termx as 1 ,the square term x²as 1² and so on. He abandoned any geometric referent that was associated with x1 x ² in ancient times. Also he used the symbols p for the addition sign and m for the subtraction sign. There is no equal sign in equations. The following is an example of one of Chuquet’s problems: 5 9/*B,"#( ,"+ 3)4$,( Xa D%/B/+ )7 B2)(, "( (,/ D*%B/ )7 X]W B*)>#+ SdB4+T1 )#/ =%#' )7 >,%B, B)+(+ XX B*)>#+ "D%/B/ "#' (,/ )(,/* X[ B*)>#+: m/ 94+( '/(/*9%#/ (,/ */+D/B(%./ b4"#(%(%/+ )7 B2)(, D4*B,"+/': Chuquet’s solution was very similar to the following in modern notation.
C"=/ x = K X "+ (,/ 4#=#)>#: C,4+ K V @v Xa I K1 "#' (,/ +/B)#' /b4"(%)# 3/B)9/+ XXK + X[ (XaIKTv X]W1 >,/#B/
Having obtained 17 1/2 for one unknown he said: gJ)> +43(*"B( Xc XnV 7*)9 XaY (,/*/ */9"%# IV XnV D%/B/+ 4( (,/ D*%B/ )7 X[ E*)>#+ " D%/B/e: After verifying that the second equation is satisfied, Chuquet remarks that such problems are considered impossible. In that case, the impossibility (i.e. the occurrence of the negative result) is due, he observes, to the fact that 0 (crowns) does not fall between 11 and 13 crowns, the given prices. Chuquet proposes the following interpretation. ‘The merchant bought 171 pieces at 11 crowns per piece with cash, thus paying 192% crown. He then took 2% pieces at 13 crown per piece on credit to, the amount of 32% crowns. Thus he has a debt of 32 ½ crowns, the subtraction(!) of which from 192 ½ gives 160. Following the same reasoning, Chuquet considers that the 2% pieces bought on credit must be subtracted from the 17 ½ pieces purchased, and that the merchant has only 15 pieces which are properly his’. This problem was posed to 20 students of the research study being discussed in this section. Following are the methods used by students when solving Chuquet’s problem.
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5*%(,9/(%B !/(,)' (used by 15 students). The students looks for multiples of 11 and 13 that add up to 160. When the students do not find the multiples needed to solve the problem. that is XX x XX t X[ x [ v X]W1 they use an additional interpretation to explain their results, for example, Student 1. He writes 66 + 91 = 157, and says: ‘lie bought 6 pieces costing 11 coins and he had 3 coins left over’. Student 2. He writes 154 + 0 = 167, and explains ‘he bought 14 pieces costing 11 coins each and none costing 13 coins’. Student 3. He writes 154 + 13 = 167. and says ‘he owned 7 coins’. 5''%(%./ !/(,)' (used by one student). The problem of the purchase of goods is modified such that the figures arc smaller in order to facilitate solution. The equations which model the problem in this case are: K t 0 v [ Y VK t [0 v sW: The student assumes that each one of the prices of cloth has a price different from that established in the statement of the problem in order to adjust the total price. He writes, X K V t X K [ v a: thus, sW I a v [a: He then says, g (,/ +"2/+94# 3)4$,( [ D%/B/+P X B)+(%#$ V B)%#+1 "#)(,/* B)+(%#$ [ B)%#+ "#' " (,%*' B)+(%#$ [a B)%#+e:
;,"*%#$ !/(,)': This is also found in the modified version ( xt y v [ Y Vxt[yvsW ) : A student divides the total price, sW1 by two. The result of the division, VW, is used with the other data of the problem 2, 3, and he formulates the sums: Xl t V v VWY F c t [ v VW: His answer %+ g,/ 3)4$,( Xl D%/B/+ >)*(,V B)%#+ "#' F c D%/B/+ >)*(, [ B)%#+ /"B,e: It is important to point out that. contrary to what might be expected, the modified version of the statement (with small numbers) renders the problem impossible for many students. The conflict is accentuated since the solution is sought in the positive domain and the lack of adjustment between the data of the problem is more notorious than in the previous version ( x t y v XaY XXxtX[yvX]W ) where the magnitude of the numbers tends to hide the conflict. This obstacle disappears when it is suggested to the student that he uses algebra to solve the problem. 52$/3*"%B !/(,)' (used by two students). Spontaneous formulation of a system of equations to solve the problem. Let us now look at the case of a student who, by using the process of substitution of the solution in a system of equations, managed to solve the problem which at first he had thought impossible. The student formulates the equations XXx t X[y v X]WY x t y v Xa: He obtains the solution; x v Xc:a1 The following dialogue then ensued: S: Totally impossible I: And now, how are you going to find
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1: Let's try anyway (student finds y = -V:a: Spontaneously he substitutes the values in the equations). S: It worked! I: What happened then? S: Instead of buying, he gave 2 and a half pieces of cloth to the person he was going to sell them to, and bought 17.5 of the other. That is, the buyer gave the seller 2 and a half pieces of cloth and the seller gave him 17.5 pieces. It's like barter. 1: Why did you say before it was impossible? S: Because it's impossible with positive numbers.
From the analysis of the different problems exhibited in the study6, the following was concluded. In the solution of different problems, student's conceptual level of negative numbers emerges. This level may vary from one problem to another, depending on the context. When faced with problems with negative solutions the student makes changes or adjustments to the data of the problem statement as well as constructing meaning which allows him to give plausible interpretations to the solution obtained. A problem which can appear impossible to solve with arithmetic methods, is thought of as possible using algebra, once the negative solutions is validated by being substituted in the corresponding equation or equations. The above shows the need to consider the mutual interrelationships between the processes of acquisition and use of algebraic language. the methods of solving word problems and linear equations, and the status o f the negative number in teaching situations. As we have said before, the study o f these interrelations is the central objective of the project.
SUMMARY
As pointed out in the introduction, the research discussed in this chapter, takes for granted that the study of the fundamental ideas underlying the historical progress of mathematics, is relevant for research in mathematical education. The body of mathematical knowledge is seen as something that cannot be fully apprehended through its formal dimension, but as the result of a long historic-cultural process of construction, formulation and clarification. Although the axiomatic skeletons of mathematics encapsulate and organise the results of this process of acquisition, they do not necessarily reflect, on the one hand. the practical or theoretical motives that lead to the formulation and resolution of certain problems, and on the other, the
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obstacles and errors that are not only inseparable counterparts of fundamental ideas, but also play a constitutive role in their development. In the first work, ‘The Acquisition of Algebraic Language’, an epistemological obstacle located in the historical development of algebra, helped to identify a didactic cut later studied in empirical work. From the analysis of methods and strategies for solving equations in pre-symbolic algebra, a conjecture was made of a ‘didactic cut’ appearing when it is necessary to operate with and upon ‘the represented’. Empirically. it was verified with students that, after having learned to solve linear equations with a single occurrence of the unknown (of the form Ax B = C ) , they fail to solve equations with an occurrence of the unknown on both sides of equality (of the form Ax B = Cx D), being unable to operate with or upon the unknown. The second work, ‘The Status of Negative Numbers in the Resolution of Equations’, is concerned with the conditions which make concepts like negative numbers possible. These numbers are subjected to a historical-critical analysis in the context of the resolution of equations in search of elements which might explain the facts observed in the process of teaching equations, regarding the incidence of negative numbers. Chapters of old books which dealt in some ways with negative numbers in the context of problems were studied. In the Chinese text Nine Chapters of Mathematical Art, a general method for solving systems of linear equations in a the negatives in a tabular setting. gives raise to a different class of numbers computational context free from the concrete meanings they used to have in ordinaryword problems. This suggested the use of a concrete model, in order to investigate the conceptual and operational difficulties of 12-13 year old students with negative numbers. The results disclosed different levels of acceptance of these numbers, which have a historical counterpart in the avoidance and recognition of negatives by different cultures. Although this teaching model is immersed within the arithmetic realm, historical-critical analysis, shows that the genesis of negative numbers belongs to algebra. There, sense is made of those numbers as solutions of problems and equations. In fact, the consolidation of algebraic language, is linked in a fundamental way, to the evolution towards a more advanced concept of the negative number (as general number, unknown, variable). From the methods used by the students in Bhaskara’s and Chuquet’s problems, it was concluded that the use of algebraic language becomes essential for the possible arrival at the negative solution, in fact, the extension of the numerical domain of natural numbers to that of integers becomes a crucial element for achieving algebraic competence in solving problem. —
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NOTES 1
2
3
4
The usual meaning of the word Abbaco was ‘The art of calculation, counting and arithmetic’. The term was used first by Leonardo Pissano (Fibonacci), who wrote a compendium about the practical mathematics known in his days (XIIIth. century). In this project, the didactic cut is analysed only in connection with very simple linear equations. However in history, we can observe for instance, that the solution strategies for equations such as x2 + c = 2bx and x2=2bx+c are completely different from each other [Hughes, 1981]. This difference would not exist if the authors had recourse to the rule of transposing terms from one side of an equations to the other for, at a syntactic level, the two equations would then be similar. But this facility would already imply an advanced ability to operate on the unknowns in the equations The term status of the negative number refers to the fact that the student has extended his/her numeric domain of natural numbers to integers in the posed problems. There are two types of numerals as shown below:
&
5
6
The A type numerals is for representing units, hundreds, ten thousands, etc., while the B type is for tens, thousands, etc. Since the 1970s, references to this model in its different forms have been made in the literature of the subject; its advantages and disadvantages for the teaching of integers has been analysed. See for example Janvier (1985), Rowland (1982), Kohn 1978), Bartolini (1976). The study analysed twelve problems of which two were selected for the purpose of this chapter.
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KAYE STACEY ¹ AND MOLLIE MACGREGOR ²
CURRICULUM REFORM AND APPROACHES TO ALGEBRA INTRODUCTION In a traditional curriculum, students are said to ‘begin algebra’ when they learn to write expressions using letters to stand for numbers and to interpret or transform these expressions. However educators and researchers are interested in identifying what else, other than use of a special notation, characterises algebra. One of the most obvious purposes of algebraic notation is to express general properties of numbers and operations on numbers. In algebra at higher levels, symbols continue to express general properties, although the referents of the symbols are extended to include functions, propositions, transformations, sets, etc. In their search for a definition and a shared understanding of what algebraic thinking might be, mathematics educators have come to regard working with generality as one of its characteristics. As a consequence, in recent curriculum documents for schools, the recognition and description of general rules for describing patterns are seen as part of ‘algebra’ or at least closely associated with it. Thus in 5 J"(%)#"2 ;("(/9/#( )# !"(,/9"(%B+ 7)* 54+(*"2%"# ;B,))2+ (Australian Education Council [AEC], 1991) work on number patterns is allocated to the Algebra Strand for children aged from 5 to 12, before any literal symbols are introduced. Similar activities for the early years are outlined in the Australian J"(%)#"2 !"(,/9"(%B+ ?*)7%2/ (AEC, 1993), !"(,/9"(%B+ %# (,/ J"(%)#"2 E4**%B4249 (Department of Education and Science [DES], 199 1) and E4**%B4249 "#' &."24"(%)# ;("#'"*'+ (National Council of Teachers of Mathematics [NCTM], 1989). It is clear that national curriculum documents are promoting the view that algebraic thinking begins to develop in the primary grades through experiences of generality and the recognition of general relationships in spatial patterns and number sequences. They go on to recommend that the first use of algebraic letters is for expressing the descriptions of these patterns and relationships. For example, a row of squares built from matches can be described by the rule ‘The number of matches you need is three times the number of squares plus one’, which can be abbreviated to 0 = 3x + 1. Introducing algebra and algebraic notation through this ‘pattern-based approach’ as a language for expressing relationships between two variables and not as a set of procedures for finding the ‘unknown number’ is a clear break with –
–
1 and 2
School of Science & Mathematics Education, Institute of Education, University of Melbourne, Australia 141 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 141–153. U 2001 <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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tradition. In contrast, traditional approaches to introducing algebraic notation are based on the use of letters to stand for specific but unknown numbers. In that approach, students’ first experiences are with the rules for using letters: simplifying and evaluating, writing simple expressions, and solving equations to find the unknown number. After learning how to transform expressions and solve equations, students come to deal with variables and functions and the broader notion of generality that they entail. The pattern-based approach deals with generality first, leading to an understanding of functional relationships and their algebraic description. Building on this understanding, students learn how to formulate and solve equations. The new approach introducing algebraic letters as pattern generalisers instead of specific unknown numbers is derived from a desire to identify early algebra in schools with algebraic thinking rather than only with its surface notational features. Since the premier curriculum statements of at least three countries recommend this change in approach, one might expect that it is supported by empirical research. We have been surprised to find, however, that there is no basis in the research literature for choosing the new approach over any other. Despite increases in government resources for educational research, recommended teaching methods continue to be based on conjectures, untested theories and the personal experiences of teachers rather than on outcomes of empirical studies. In published research studies, we have found little evidence that might support the change from the traditional ‘letter-as-specific-unknown’ approach to the patternbased ‘letter-as-variable’ approach. One important study (Küchemann, 1981) found that algebra test items which required a letter to be interpreted as a generalised number or variable were harder than most other items in which letters could be thought of as specific unknowns (see Küchemann, 1981, Table 8.12, p. 115). Küchemann’s results may be interpreted as advising against the shift to a patternbased approach, although Küchemann himself concluded that practical experiences with patterns may provide a firm basis for the recognition of general relationships and ‘ease the transition to formal operational thought’ (Küchemann, 1981, p. 118). Several studies (Herscovics, 1989; MacGregor & Stacey, 1992; MacGregor & Stacey, 1993b; Stacey, 1989) have documented students’ difficulties in generating algebraic rules from patterns and tables. On the other hand, the relative success of small samples of students in spreadsheet environments (Rojano & Sutherland, 1993; Sutherland, 1991) suggests that the interpretation of a letter as a variable is not of itself a major obstacle to learning. A similar conclusion was reached by Kieran, Boileau and Garancon (1996) who developed the CARAPACE program which presents unknowns within the larger context of variables. They found that students working with the program could switch easily between concepts of variable and unknown, using whichever interpretation was appropriate for solving a problem. –
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A. Look at the numbers in this table and answer the questions. y K 1 5 2 6 3 7
(i) (iv) (v)
4
8
5 6 7 8 ..
9 .. 11 .. ..
When K is 2, what is 0? .... (ii) When K is 8,what is 0? .... (iii) When K is 800, what is 0? .... Describe in words how you would find 0 if you were told what K is. ...................................... Use algebra to write a rule connecting K and 0: ................
B. The results of an experiment that measured two quantities N and y were: N
y
3 5 9 21
9 15 27 63
(i)
What would you expect y to be when N is 30? ......
(ii)
What would N be when y is 99? ......
(iii) Describe in words how you would find y if you were told what N is. ................................... (iv) Use algebra to write a rule connecting N and y: ............................ C. Look at the numbers in this table and answer the questions. K 0 1 2 3 4 5 6 0 2 5 8 11 14 17 When K is 6, what is 0 ? ... (ii) When K is 10, what is 0 ? ... (i) (iii) When K is 100, what is 0 ? .... (iv) Describe in words how you would find 0 if you were told what K is. .................................... Use algebra to write a rule connecting K and 0: .................... (v)
A%$4*/ X: C/+( %(/9+
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Another technology-based approach to beginning algebra E)9D4(/*IF#(/#+%./ 52$/3*" (see Heid, 1996) – was shown to help students develop the concept of variable as pattern generaliser. Our research with Australian students, described in the next section, shows that a pattern-based approach does not automatically lead to better understanding. Following a discussion of the results and the insights into students' thinking that they provide, we present some suggestions for improving a pattern-based approach. We point out, however, that no approach guarantees success, and we discuss the role of research in recommending changes to the curriculum. –
DOES A PATTERN-BASED APPROACH ENHANCE STUDENTS' UNDERSTANDING OF FUNCTIONAL RELATIONSHIPS? In Australian secondary schools, formal algebra is normally introduced in Year c (in certain schools delayed until Year 8) with a unit of about 20 lessons. Students study increasing amounts of algebra each year from Years 7 to 10. We discuss some results of research (MacGregor & Stacey, 1992, 1993b) carried out during 1991-3 with approximately 2000 students in the first four years of secondary school (Years 7 to 10, aged approximately 12 to 15). The study involved pencil-and-paper testing of whole classes in a variety of schools, and interviews with individual students. Some of the schools had used a pattern-based approach to algebra, whereas others had used a traditional approach. The test items used (examples are shown in Fig. I ) were designed to find out: (i) what patterns and relationships students perceive in function tables; (ii) whether they can extend the table (i.e., calculate further values of the variables) by thinking of a general rule; (iii) whether they can describe the pattern, the function or the rule clearly in ordinary language; (iv) whether they can describe it as an algebraic equation. Item A (see Fig. 1) was expected to be very easy for most students. Students should see the various recurrence relationships, unconnected (e.g. ‘xg keeps going up by 1, and so does 0 e T as well as linked (e.g., ‘when x goes up by 1, 0 goes up by 1’). They should also see the functional relationship linking K and 0 in any of several equivalent forms such as g 0 is greater than K by 4’ or ‘If you add 4 to K , you get 0e: The results showed that almost all students could interpret and extend the tables correctly for small values. Approximately half the students could describe the functional relationship between the two variables in words, although the majority
CURRICULUM REFORM AND APPROACHES TO ALGEBRA
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did not use clear or well-formed English. About 70% were able to calculate the high value of 0 S0 = 804), presumably using the functional relationship implicitly. In Item B, there are no regularities to be seen in the separate columns of the table. The only ‘pattern’ is the functional relationship linking L and Q. Only one third of the students wrote a correct equation. The omission rate was high; approximately one-quarter of the students omitted all parts of the item. Apparently they did not see the ‘multiply by 3’ relationship. The results for Item C suggested that the predominant perception of the pattern was of the recurrence relation (i.e., values o f y are increasing by 3). Although most students were able to find the missing values in parts (i) and (ii) by counting on, fewer than 20% were able to calculate 0 when counting on was impractical (part (iii)). Success in part (iii) requires the use of a functional rule linking the two variables multiply by 3 and add 2. Most of the small proportion of students who found a rule were able to express it verbally, and two-thirds of these wrote a correct equation. However many students wrote incorrect equations or expressions that were not equations at all, and a large proportion made no attempt to write any form of verbal description or algebraic rule. Performance varied greatly between schools. For example, on Item A, the success of Year 9 students in writing an algebraic rule ranged from 18% in one school to 73% at another school. Performance also varied between classes. For example. on Item C, in several classes at all year levels fewer than 10% wrote a correct rule whereas in other classes almost all students were correct. There was no evidence in our study to suggest that, on these pattern-oriented questions, students taught with a pattern-based approach were any more successful than those taught with a traditional approach. The performance of 10 classes taught with a pattern-based approach showed a lot of variation and was not better than the performance of 16 classes taught with a more traditional notational approach. –
DISCUSSION ;//%#$ " D"((/*# One of the most striking findings from both written testing and the interviews was the variety of patterns perceived and the many ways of describing them. The interview data confirmed that although students easily observed patterns in the tables only some of these patterns were useful for calculating higher values. For example, talking about Item A, Sarah (Year 10) said, ‘K@%(@+("*(+@"(@X1@V1@[1@s@"#' 34%2'+ 4D1 %(e+ $)%#$ %# )*'/*e1 and when asked for any more patterns she could see, gC,/ )'' #493/*+ "*/ >%(, )'' #493/*+ "#' (,/ /./# "*/ >%(, /./#e: Michael (Year 7) added the numbers in each row and noticed a recurrence pattern in the totals, saying, ‘m,/# 0)4e*/ "''%#$ 3)(, #493/*+ (,/0 "*/ "2>"0+ %#B*/"+%#$1 2%=/ X
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"#' a %+ ]1 V "#' ] %+ l1 [ "#' 7 is X W 1 /"B, #493/*+ $) 4D 30 2’. Students tended to search for a recurrence nile that would predict a number from the value of its predecessor rather than for a functional relationship linking pairs of numbers For example, in Item C (see Fig. 1) it is easy to see the recurrence relations, either separately (‘K $)/+ 4D by X "#' 0 $)/+ 4D 30 [eT1 or linked Sg"# %#B*/"+/ )7 X %# K B"4+/+ "# %#B*/"+/ )7 [ %# 0eT: Many students do not look for the functional relationship that links the two variables (0 = 3K + 2) unless they have confronted question such as C(iii). They may wonder why the functional relationship is important or useful, since the recurrence relation (‘0 =//D+ $)%#$ 4D 30 [e) is so obvious, so easy () comprehend and so easy to use for some questions. Many of the patterns that students saw, including the recurrence relationships, are valid but do not lead to an idea that can easily be expressed algebraically. It seems to us to be important that teachers recognise that their students see many patterns. Students need to be given time to discuss why some particular patterns and relationships are more helpful than others they have seen. They also have to be able to pick what is significant and what isn’t; for example, why are some tables vertical and some horizontal? why is the first entry 1 in some tables and not in others? why do some tables have sequential values whereas others do not? It is easy for teachers not to notice these extraneous details. A4#B(%)#"2 */2"(%)#+,%D+ Even when students do see the rule relating two variables, they are often reluctant or unable to write it as an equation. Some students want to describe a procedure for working out answers and not state a relationship. The rule that applies to Item C, for example, beginners sometimes want to write as ‘x 3 + 2e1 meaning, gC) >)*= %( )4(1 0)4 942(%D20 30 [ "#' (,/# "'' Ve (Stacey, 1987). It is hard for teachers to explain why rules written in this form are not adequate or useful for algebra, especially if they are working in a restricted textbook context. More than one-third of the students who did perceive the functional relationships described them in immature ways that did not support an algebraic translation. For Item A, for example, even the older students gave a range of responses based on counting rather than addition: g F# 3/(>//# K "#' 0 (,/*/ %+ 7)4*e1 gC,/*/e+ (,*// #493/*+ 9%++%#$eP gF(e+ 7)4* #493/*+ ,%$,/*e1 gC,/ 0 #493/* %+@7)4*@*)>+ 3/2)> (,/ K #493/*e1 gk)4 B)4#(@7)4*@D2"B/+e:@ Other verbal descriptions were not helpful as a basis for clear thinking about the function and its algebraic representation (e.g., g&./*0 #493/* %+ s (%9/+ 2"*$/*e1 gm%(, (,/ K1 7)* (,/@ 01@ 0)4 D4( s )# %(e1 "#' g!%#4+@s 7*)9 /"B,e T: Predictably, the syntax of algebra was a problem for some students. For example, one Year-8 student said g0 %+@7)4*@,%$,/*1F (,%#= %(e+ K /b4"2+ 0 7)4*e and wrote K = 04. There were some other occurrences of reversed equations, such as K@v
CURRICULUM REFORM AND APPROACHES TO ALGEBRA
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0 + 4, expressing this idea that g0 %+ s ,%$,/*e: (For a discussion of the reasoning
behind reversed equations to express additive comparison, see MacGregor & Stacey, 1993a). We were not surprised that students found it hard to write a rule for Item C, both in words and algebraically. However it was disturbing to see how many students could not express the simple rules ‘Add 4’ and ‘Multiply by 3’ in the other two items. 5++)B%"(%)# 3/(>//# "3%2%(0 () >)*= #49/*%B"220 >%(, " D"((/*# "#' ./*3"2 "#' "2$/3*"%B B)9D/(/#B/
Of special interest are the students who demonstrated that they understood the functional relationships in the tables by successfully calculating new values as required by A(iii), B(i) or C(iii), but who could not articulate them clearly and express them algebraically. Two examples of students who could calculate A(iii) but not express their procedures algebraically are discussed below. &K"9D2/ X: Sarah (Year 10) calculated correctly that if K = 100 then 0 = 104, and if K = 1000 then 0 = 1004. When the interviewer asked for an explanation of a general rule, Sarah offered the following five alternatives: gC,/ K ."24/ %+ 2)>/* (,"# (,/@0@ 30 se: gF( $)/+ 4D s 7)* /./*0 K (,"( (,/0 ,"./e: gm,"(/./* #493/* 0)4 ,"./ 7)* (,/@0@ 0)4 "'' s )#() %(e: [writes K = 0 4] ‘m%(, (,/ K, 7)* (,/@01@0)4@D4(@s@)#@%(e:@ gA)* (,/@01@%(@%+ @7)4* numbers ,%$,/* (,"# (,/ x’. After many prompts from the interviewer. such as ‘You know how to get y from x. Could you write something that starts with y?’, Sarah finally wrote y = K4. &K"9D2/ 2. Roberto (Year 10) used incorrect direct proportion rules for large number calculations but appeared able to use the ‘add 4’ rule for small numbers without being aware of what he was doing. In fact he was focusing on the ‘gap’ between K and 01 seeing the gap as a space for ‘three numbers’. In the interview he was constantly reminded to look for a simple way of saying his rule, but his only explanations were: gm%(, (,/ K "#' 0 #493/*+ (,/*/ %+ "2>"0+ 7)4* #493/*+ %# 3/(>//#e: g F 7 (,/*/ %+ K (,/*/@>%22@3/ (,*// #493/*+ "#' (,/# 0e: gm,"(/./* (,/ K (,/ 0 94+( 3/ (,*//1@#)1@7)4* '%$%(+e: [writes K = 30] g K /b4"2+ (,*// '%$%(+ 0 ’. The fact that many students doing the written test could use the functional relationships to calculate values but could not write explanations or algebraic equations indicates that for them, as for Sarah and Roberto, recognising and articulating the structure of the relationship was the stumbling block. Students who gave a correct verbal description were significantly more likely than other students to write a correct algebraic rule (MacGregor & Stacey, 1993b).
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Of the students who did not describe their patterns clearly, only one-quarter were able to write a correct algebraic rule. The difficulty experienced by many students in verbalising their understanding highlights the need for teachers to promote the development of language specifically related to mathematics. This need is often overlooked by mathematics teachers, who assume that their pupils come to class with language skills that they have already learned somewhere else. In the mathematics classroom, teachers should provide models of language structures that can be used for talking about patterns and relationships and for developing and refining ideas.
;(*4B(4*/ )7 " *42/ )* */2"(%)#+,%D There is currently an emphasis in mathematics education on students' writing and discussing mathematical ideas (AEC, 1991, 1993; DES, 1991; NCTM. 1989). As we observe how writing and discussion can enhance learning, it is easy to forget that much mathematical thinking is done without explicit verbalisation. Many students are able to calculate but cannot describe what they do; it is very possible they do not know consciously what they do. As Groen and Kieran (1983) have pointed out in their review of Piaget's contribution to our understanding of children's thinking, ‘the pupil will be far more capable of 'doing' and 'understanding in actions' than of expressing himself or herself verbally’ (p. 368). However, to learn algebra students need to be able to recognise and articulate the processes of arithmetic and the structures of relationships between numbers. One essential prerequisite for using algebra is that students can put their informal arithmetic knowledge into a formal arithmetic structure to know, for example, that doubling (‘k)4 D24+ %( 30 (,/ +"9/ #493/*’) is multiplication by 2, or that g(,*//@#493/*+@9%++%#$e between two numbers is expressed formally as subtraction (or addition) o f 4. A strong understanding of properties of numbers and operations is an essential foundation for beginning algebra (Stacey z MacGregor, 1997). –
m,"( %+ " *42/x The written responses and the oral responses in the interview sessions both showed that some students had used one rule for simple calculations and another for larger values of the variables. For example: the student Roberto described the rule for item A as gF7 (,/*/ %+ K (,/*/ >%22 3/ (,*// #493/*+ "#' (,/# 0e: He used this rule successfully for simple calculations. However to find y when K = 800 he decided to multiply 800 by 5. and explained that he had gone back to the first row of the table where K is 1 and 0 is 5. Another very common wrong answer for A(iii) was 1200, also based on a proportion rule. Students are tempted to believe that if 8 for K makes 0 equal to 12, then 800 for K must make 0 equal to 1200. The incorrect use of a simple direct proportion when large numbers were involved was common for Items A and C, and has been reported in other contexts (Stacey, 1989). There are several
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possible explanations for this, one being that some students are not aware of the difference between addition and proportion (see, for example, Hart. 198 1). They do not clearly understand that they are not applying the same rule, but think they are taking a valid shortcut to the answer. However, other students seem to be proposing different functions for different parts of the domain. For example, one student justified his answer of 1200 by writing g k)4 "'' s %7 %( %+ 4#'/* XWW "#' %7 %( %+ )./* XWW 0)4 "'' sWWe: In defining his function differently over different parts of the domain, he may have been drawing an analogy with a type of rule that frequently occurs in everyday contexts. such as the sale of raffle tickets at 50 cents each or 12 for $ 5 : In the school context, there is an unspoken convention that the rules lying behind the tables in items such as Items A and C are simple functions, defined in one way over all the integers. Teachers need to make assumptions such as these clear to students.i
?/*7)*9"#B/ */2"(/' () "DD*)"B, () "2$/3*" Our comparison of the success rates for classes that had been taught with patternbased and traditional approaches showed no evidence that one approach was any more effective than the other. The two best performing schools in the sample had both used a ‘concrete approach’ developed by Quinlan, Low, Sawyer and White (1989) in which algebraic letters stand for unknown numbers of hidden objects in containers. In this approach, students' first algebra experiences are to write expressions representing collections of objects. For example, a pile of six blocks together with two envelopes containing equal but unknown numbers of blocks might be represented as 6 + 2N or as 2(N + 3). Although the concrete approach has its critics, it is noteworthy that the two schools concerned outperformed the rest. none of whom had used a concrete approach. However it would be unwise to draw firm conclusions about the relative merits of different approaches from our data because of the method of selection of schools (teachers' interest in the research project) and uncontrolled factors. As well as differences in teaching style, amount of time in class spent on algebra, amount of homework, recent revision or practice before the test, and the general tone of the learning environment, there is the possibility that a teacher’s choice of approach or textbook is related to the level of academic skills of students at the school. These biases must be taken into account when drawing conclusions from the data. However there is no evidence in the results we obtained to indicate that a pattern-based approach to learning algebra was any better than a traditional approach in equipping students to identify relationships between variables and express them algebraically. Like many recommendations in the pedagogical literature, the quite definite positions adopted by national advisory councils (referred to earlier in this chapter) have no supporting research background.
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IMPLICATIONS FOR TEACHING M)4(/ 7*)9 D"((/*# () *42/ The route from perceiving a pattern or relationship in a table to writing an algebraic rule is not straightforward. Examining students' thinking has shown there are many critical steps along the way: knowing what features of the table are irrelevant and what are important (e.g. that K starts at 1 is irrelevant); looking beyond separate recurrence patterns and linking them to find a relationship between the two variables; checking that a pattern that has been identified holds for all values given in the table and also fits any other evidence available; being able to articulate the relationship in a useful way so that it can be used for calculating and generalising (e.g. ‘The sixth K is the same as the second 0e is not helpful); knowing what an algebraic rule looks like (e.g. ‘ K + 10 ’ does not have the structure of an algebraic rule); knowing what sort of rule the teacher will allow (Is a ‘raffle-ticket’ rule allowed?) knowing what can and cannot be expressed in elementary algebra (e.g. ‘Every time K goes up by 1, 0 goes up by 3’ cannot be easily translated to an equation). knowing the syntax of algebra (e.g. ‘ K + 04’ does not mean ‘Start with K, and add 3 to get 0’). C/"B,%#$ (,/ D"((/*#I3"+/' "DD*)"B,
A close look at the way in which different textbooks and teaching materials implement the pattern-based approach to beginning algebra reveals different routes from seeing a pattern to deriving an algebraic rule. These differences may be of crucial importance for learners. When the pattern-based approach was trialled by Pegg and Redden (1990), geometric features of a design were first described verbally and then suitable verbal descriptions were selected to be expressed algebraically. Our findings suggest that the verbal description phase is an important and perhaps necessary part of the process of recognising a function and expressing it algebraically. Unfortunately this part of the learning process is bypassed by some teachers and textbooks. In several textbooks, students are told to extend a geometric design, notice the recurrence patterns, make a table of values, determine the constant difference and insert this difference into a formula as the coefficient of K@: For example, one textbook in common use presents a table of ordered pairs (2,6), (3,10), (4,14), (5,18) and tells the reader to notice that
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as K increases by 1 (from 2 to 3, or 3 to 4, or 4 to 5), 0 increases by 4 (from 6 to 10,or 10 to 14, or 14 to 18). Therefore the rule could be of the form 0 = 4 x K . . . It may not be clear to students why a constant increase automatically becomes a multiplier, that is, why "''%#$ 4 is transformed to s (%9/+: Some students do not make the link between repeated addition and multiplication. We suggest that what many students remember about examples like this is merely a rote-learned procedure: ‘Look down the 0 column, find the constant difference and use it as the number in front of K ’. When it has no foundation of understanding, this procedure is easily forgotten. Moreover, some students believe that when they write 4K , they mean ‘add 4 to K’ or ‘4 more than K’, thus compounding their confusion. We suggest that geometric recurrence models such as the following matchstick pattern:
‘You need three more matches each time you add a square’ (AEC, 1991, p. 191) are probably not the best way to introduce algebra. The recurrence relation (keep on adding three) in this pattern is dominant over the functional relationship (multiply by 3 and add one). However, ‘keep on adding 3’ cannot be written using the algebra that students are learning. It would require suffix notation, which is not taught in the early years of algebra in Australian schooIs. We recommend the frequent use of items such as Item B, where there is no recurrence pattern in the x values. Such items can easily be set in real contexts, such as comparing taxi fares and distances travelled or the cost of petrol related to the number of litres bought. In making this recommendation, we do not deny the importance of having students develop the powerful problem-solving strategies of working systematically in order to find patterns. Rather we think that using questions with strong recurrence patterns is not appropriate when the main focus of a lesson is on recognising a functional relationship between two variables. The representation of recurrence patterns with suffix notation should be taught at some stage, as is done in the SMP scheme in UK, but the timing of this is a question for research. M/+/"*B, "#' B4**%B4249 */7)*9 Finally, we return to the observation that opened this chapter. When a new and different teaching approach is recommended in a national curriculum document, it is
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natural to assume that i n some way research has shown it to be better. However there has been little empirical research of relevance to the choice of ways of introducing algebra, although there are many ideas on what approaches ought to be effective. The difficulty that students experience in writing algebraic equations from tables of values or situations is well known. as is Küchemann’s conclusion that most 13-15 year old students could only deal with letters when they had a concrete interpretation, enabling them to be thought of as objects rather than as unknowns or variables. Possibly more importantly, there is the general observation. made in countless studies, that many students do not do well in algebra, which at least opens the door to change. Our data, referred to in this chapter, certainly showed that students taught with a pattern-based approach were not better at the algebra items in the test, even though these items closely matched the type they would have practised in lessons. But the difficulties of conducting any truly experimental design for long-term teaching confound the interpretation of the results. Is it the case that schools which choose textbooks with a new approach may be trying to cater for less able students, whilst schools where teaching is easier or comparatively successful are more content with tradition? Are the various approaches actually being taught well? Our analysis of the way the pattern-based method is presented, in at least some of the textbooks, shows that it is reduced to a routine, and is a far way from the rich lessons described by Pegg and Redden (1990). To what extent is it therefore sensible to think of all pattern-based approaches as the same? Our investigations have shown considerable variation in crucial features such as the amount, quality and purpose of verbal expression when students are recognising patterns and developing rules. Finally, when a national curriculum committee wishes to recommend particular approaches, which should it choose: those which theoretically have the best potential or those which are robust enough to withstand the impoverished treatment they will receive in some classrooms? What then can research contribute to curriculum recommendations if it cannot determine the choice between suggested teaching methods? We see the greatest use of findings such as those reported in this chapter as not to advocate one teaching method over another but to highlight the ways in which students think about mathematical situations. As we have discussed with Rob Davis, curriculum designers are often concerned with how students ought to think instead of how they really do think. Awareness of how students think about numerical relationships and of what they are likely to perceive in mathematical situations will help teachers to decide what aspects of presentation need special emphasis and what follow-up will be necessary, leading to better teaching and better outcomes regardless of the approach adopted.
CURRICULUM REFORM AND APPROACHES TO ALGEBRA
NOTES 1
There is nothing in the tables presented in the test items to indicate how they continue. A variety of quartics can be fitted through the data points given in Item B, for example. However, no response from any student indicated that complications of this nature had been considered.
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EUGENIO FILLOY 1, TERESA ROJANO² AND GUILLERMO RUBIO³
PROPOSITIONS CONCERNING THE RESOLUTION OF ARITHMETICAL-ALGEBRAIC PROBLEMS INTRODUCTION In recent years a large number of studies have drawn attention to a pragmatic point of view of meaning in the use of mathematical signs in preference to meaning in the abstract. We argue that the semiotics of Mathematics should not be focused on the study of signs but on the study of significance systems and significance producing processes. The mathematics involved includes not only the specific type of mathematical signs (the so called mathematical symbols) but their overlapping with the signs of natural language, geometric representations produced by individuals when learning mathematics, and with didactic tools (stone piles, scales, pieces of land, microworlds created in Logo; in general, any kind of software that can generate teaching sequences). That is, we must not talk about systems of mathematical signs but of Mathematical Sign Systems (from now on, MSS). There has been a change of direction away from competence in algebraic language towards the action of the user. As a consequence, in the study of the psychology of learning algebraic language, two essentially complementary domains are taken into account: on the one hand, grammar, that is the abstract, formal system of algebra and on the other, the pragmatics' constituted by the principles of the use of algebraic language. There exists a pragmatic component in the teaching field where learning is interconnected with an institutionalised social contract, so one must take into consideration not only the traditional use and ways in which MSS are produced in the educational field but also the historical evolution of such sign systems. What comes to light first is notation but this is not the only way in which, we now use the MSS and their applications in scientific, technological and social processes. There are various consequences of the above point of view. These have to do not only with the role and place which questions of formal grammar should occupy, but also the recognition that these should be fitted into a more general framework which 1
Departamento de Matemática Educativa, Centro de Investigación y de Estudios Avanzados-IPN, Mexico. 2 Centro de Investigación y de Estudios Avanzados-IPN, Mexico. 3 Universidad Nacional Autónoma de Mexico. 155 M : ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 Xaa\XXa: U VWWX <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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combines functional and formal explanations. Furthermore, in order to account for the full meaning of some mathematical messages which appear during the normal processes of teaching and learning mathematics, along with the strict meaning of the mathematical text, some other meanings of other messages have to be taken into account which are not expressed explicitly or intentionally by the transmitter or the receiver. Among these elements are: implicit assumptions, immediate consequences, conversational implications, and presuppositions which can carry a message. These aspects are found within the text itself or in the diverse contexts in which the acts of communication making up the processes of teaching and learning arc produced. On the basis of empirical observations on how mathematical systems of signs (MSS) are used during the exchange of messages within the process of mathematical teaching and learning, and the corresponding situations in which the subject uses these MSS in a problem-solving situation, it can be seen that the level of competence (which has to do with the formal system of algebraic language) and the pragmatic level are mixed in the cognitive processes involved, and this can be the result of several causes. One of these causes i s the presence of an element of pragmatics which is due to the cognitive structures of the individual subject and which appears at each stage of development and which gives preference to the mechanisms of different procedures, different ways of coding and decoding the mathematical messages belonging to the stage in question, strategies of problem solving, etc. Let us consider, for example, all the evidence that has accumulated on the tendency of subjects to maintain arithmetic interpretations in the majority of algebraic situations, even in quite advanced stages in the study of algebra (e.g. Bell and Galvin, 1977; Filloy and Rubio, 1993; Janvier and Bednarz, 1993; Bednarz, Radford, Janvier and Leparge, 1992). Given the stability of the phenomena and the replicability of the experimental designs that have been used in the studies of recent years, theoretical constructs have been proposed which take into account three important components in order to analyse empirical observations (Filloy and Rubio, 1991; 1993a; and 1993b). In this proposal, instead of emphasising one of the components (grammar, logic, mathematics, teaching models, cognitive and pragmatic models) emphasis is placed on adequate local theoretical models conceived only for the study of phenomena referring to specific D*)B/++/+ of teaching and learning. In the work we discuss in this chapter, the local model takes into account the following three theoretical components: a) 9)'/2+ 7)* (/"B,%#$ "2$/3*"P the different types of model for teaching algebra used in the processes of teaching and learning: b ) 9)'/2+ )7 (,/ %#.)2./' B)$#%(%./ D*)B/++/+1 both related to c) 9)'/2+ )7 D*)B/++/+ )7 7)*9"2 B)9D/(/#B/ which simulate the competent performance of an ideal user of the language of elementary algebra.
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In this chapter, we will present the elements of a local theoretical model based on experimental results concerning the types of competence necessary for use of four methods for solving word problems. The emphasis is on the need to be competent in the increasingly abstract and general uses of the mental representations required to achieve full competence in the algebraic method par excellence, which here we call the E"*(/+%"# 9/(,)': These uses are compared with the types of competence required for the other three methods which arc more rooted in arithmetic: (,/ 9/(,)' )7 +4BB/++%./ "#"20(%B %#7/*/#B/+1 (,/ 9/(,)' )7 +4BB/++%./ /KD2)*"(%)#+ and a +D*/"'+,//( "2$/3*"%B 9/(,)': In other words, we will describe two methods of solution for "2$/3*"%B >)*' D*)32/9+ which can be characterised as (,/ "*%(,9/(%B 9/(,)' and (,/ E"*(/+%"# 9/(,)': In general, educational systems intend that pupils get to learn the latter at the end of the secondary school. Nevertheless, it is well known that the majority of students tend to use non-algebraic (e.g. arithmetic) methods to solve problems. We analyse the characteristics of each solution method, showing the gap to be bridged and present (>) other ‘intermediate’ methods. In the sections L%+B4++%)# )7 (,/ /9D%*%B"2 */+42(+ and A%#"2 '%+B4++%)#1 we show how these ‘intermediate’ methods can help pupils to move towards a ‘more algebraic’ approach to solving problems, through two empirical studies carried out using such methods. An attempt is made to account both for why and for how the teaching approach proposed can work. TWO CLASSICAL SOLUTION METHODS
XT C,/ 9/(,)' >/ >%22 B"22 9/(,)' )7 +4BB/++%./ "#"20(%B %#7/*/#B/+ %+ O4+( (,/ B2"++%B"2 "#"20(%B"2 9/(,)' 7)* +)2.%#$ (,/+/ D*)32/9+ 4+%#$ )#20 "*%(,9/(%B: In this method the statements of the problems are conceived of as descriptions of ‘real situations’ or ‘possible states of the world’. These texts are transformed by means of analytical sentences, that is, using ‘facts’ valid in ‘every possible world’: logical inferences which act as descriptions of the transformations of ‘possible situations’ until one is reached which is recognised as the solution to the problem. We simply will refer to this method as (,/ "*%(,9/(%B 9/(,)': The resolution of the following problem illustrates this method. C,/ D*)32/9 )7 (,/ (/"B,/* A teacher has 120 chocolates and 192 caramels which she is going to share out among her pupils. If each pupil receives three more caramels than chocolates, how many children are there?
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We will now illustrate the two ways observed for solving the problem, using the "*%(,9/(%B 9/(,)' and the series of inferences that can be produced, the first being more usual than the second. ;)24(%)# J): F * Since in the statement of the problem it says that 120 chocolates and 192 caramels will be shared out among all the pupils, it can be concluded that 72 more caramels than chocolates were given to the pupils (192 - 120 = 72) and * since, furthermore, the text of the problem indicates that each pupil received three more caramels than chocolates, the number of pupils can be determined b y looking at how many times three caramels goes into 72 caramels; this can be easily calculated dividing 72 by 3. The result, 24, corresponds to the number of pupils. ;)24(%)# J): V * Solving the problem following the arithmetic method can produce the following logical inferences: * Since the statement of the problem says that 120 chocolates and 192 caramels are shared out among all the pupils, it can be concluded that 72 more caramels were shared out than chocolates to all the pupils (1 92 - 120 = 72) and * since furthermore, the text of the problem says that each pupil receives three more caramels than chocolates. the number of pupils can be determined as follows: * first of all, three caramels can be taken from the 72 extra caramels which were given to the pupils. Since each pupil was given t h e e more caramels than chocolates, this action begins a count of the number of pupils there are; that is. with this first sharing out it can be said that one pupil has been counted, * (,/#1 %7 "#)(,/* (,*// B"*"9/2+ "*/("=/# 7*)9 (,/ ]Z 2/7( ScV I [ v ]ZT1 "#' (,/+/ "*/ $%./# () "#)(,/* D4D%21 >%(, (,%+ "B( %( B"# 3/ +"%' (,"( (>) D4D%2+ ,"./ 3//# B)4#(/'1 2/".%#$ ]] B"*"9/2+ S]Z I [ v ]]T: * in the same way. if three more caramels arc taken from the ]] caramels. corresponding to another pupil. this action indicates that three pupils have now been counted and ] [ caramels are left S]] I [ = ][T: * With the same reasoning the caramels can continue to be shared out until the 72 more than the chocolates are used up, thus producing a series of successive inferences like the previous ones, finding out that at the end the number of pupils will coincide with the number of sharings-out, that is 24. 2) C,/ E"*(/+%"# 9/(,)': This approach to the solution of problems is the usual one found in current algebra textbooks. In this method, the process of solution occurs by means of the representation of some of the unknown elements in the statement of the problem by algebraic expressions, later translating the text of the problem into a series of relationships expressed in algebraic language which lead to one or various
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equations whose solution,via a return in the translation, leads to the solution of the problem. The (/"B,/*_+ D*)32/9 can be immediately solved by the E"*(/+%"# 9/(,)'1 although it does not seem very ‘natural’. * Take g x e to represent the number of pupils to whom sweets will be given, and * ‘y’ to represent the number of chocolates which are given to each pupil, then * x x y will be the total number of chocolates; but since we know there are 120, then * x x y = 120 &b4"(%)# X * Furthermore, since the statement says that there are a total of 192 caramels and we have already said that ‘x’ represents the number of pupils there are, then * 192/x will be the number of caramels each pupil will get, * furthermore, since ‘y’ represents the number of chocolates each pupil will get and in the problem it says that each child receives 3 more caramels than chocolates, then: * y + 3 will be the total number of caramels each pupil gets and thus: * y + 3 = 192/x &b4"(%)# V Since two equations with two unknowns were obtained to represent the problem: x x y = 120 &b4"(%)# X y + 3 = 192/x &b4"(%)# V then these should be reduced to one, to do this the ‘y’ is taken from the first equation and substituted in the second, thus: 120/x + 3 = 192/x This equation is solved using the rules of algebraic syntax and it is found that the number of pupils ‘x’ is equal to 24, the number of chocolates per pupil is 120/24=5, and the number of caramels per pupil is 5 + 3 = 8. TWO NON-CONVENTIONAL SOLUTION METHODS
3T 5#"20(%B"2 9/(,)' )7 +4BB/++%./ /KD2)*"(%)#+: This is a method of problem solving where in order to set the analysis of the problem in motion and thus to find the solution, the starting point is the identification of what has to be discovered in the problem, then, this is assigned a numerical value which is considered as a hypothetical solution of the problem. This, in turn, permits a reading of the problem where, instead of unknown elements, there are only numerical values with which one can establish numerical relationships between the data of the problem and the value assumed to be the solution. We will refer to this method as the successive explorations method. The teacher’s problem is solved by the method of successive explorations. ?,"+/ X * On reading the problem, the number of pupils is identified as ‘the unknown’. ?,"+/ V
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* If we assume, for example, that 12 pupils is the solution to the problem, and given that the statement says that there are 120 chocolates, then each pupil will receive 120/12 = 10 chocolates, * but since furthermore it says that each pupil receives 3 more caramels than chocolates, then each pupil will receive 10 + 3 = 13 caramels (13 caramels/pupil). ?,"+/ [P Two quantities that should represent the same are compared. * Since the problem also states that 192 caramels will be shared out among the pupils and it has been assumed as a solution that there are 12 pupils, each pupil will receive 192/13, 14.8 caramels (14.8 caramels/pupil); then, if we compare this value with the 13 caramels per pupil that has just been obtained in phase 2, it can be seen that the second quantity is larger than the first, which indicates that the numerical value assumed (12 pupils) is not the solution to the problem. The comparison is written thus: = 13? 14.8 caramels/pupil caramels/pupil (14.8 and 13 are two quantities which should represent the same but come from different relationships) ?,"+/ sP The recovery of operations * On the basis of the comparison between 14.8 and 13, the operations are recovered in a successive manner until a model or pattern of the problem is reached in which the relationships of the same are expressed, written in numerical form: 14.8 = 13? 192/12 = 1 0 + 3 ? 192/12 = 120/12 + 3? * Once this numerical ‘pattern’ is determined, the solution is varied, assuming another value for the latter; then, once again, all the phases of the method are followed until a numerical pattern similar to the one obtained is arrived at: 192/12 = 120/12 + 3. Here it is suggested that the student use between two and four hypothetical values as a solution in order that he find numerical patterns similar to the first one obtained. It is thought that this allows the user to gradually construct meanings for the operations between data and ‘the unknown’, preparing the way for the use of the algebraic relationships which conform the equation which represents the word problem. ?,"+/ s:F * In this phase the letter ‘x’ is used to represent ‘the unknown’ in the problem (the ‘exact’ solution); since this letter plays the same role as the numerical values assumed as solutions, operations can be carried out between it and the data, obtaining an equality instead of a comparison. Such an equality corresponds to the algebraic equation thus, the comparisons become an equation when a letter is assigned as the solution to the problem. ∗ In the application of this phase the fact that the relationships which are on both sides of the equality come from two different numbers should once again be underlined; they represent the same in the problem but they were obtained from
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different relationships. It is emphasised that the use of the letter ‘x’ to represent the solution of the problem no longer gives rise to a comparison, but to an equation which represents it algebraically, as we show here: 120/12 + 3? 192/12 = 192/x = 120/x + 3 caramels/pupil caramels/pupil x = exact number of pupils to whom sweets were given ?,"+/ s:V * The equation is solved algebraically in order to operate on ‘the unknown’ which has been assigned and the value obtained for the unknown is verified by the solution of the algebraic equation, substituting the numerical value obtained for the unknown in the equation.
4) ;D*/"'+,//( "2$/3*"%B 9/(,)': This method consists of representing one of the unknowns with a spreadsheet cell (this might be called x in the algebraic method); expressing with general formulae the relationships within the problem (between data and unknowns and between unknowns) in terms of this unknown; and varying the unknown value, a numerical solution is found. Varying the unknown can be done either by copying down the formulae or by changing the number in the cell representing the unknown. General formulae are typed with the spreadsheet symbolism. We will refer to this method as a spreadsheet method. Figure 1 illustrates the way The Theatre Problem can be solved by means of this method. We next describe the advantages and disadvantages of the use o f any of the four methods when solving ‘algebra’ word problems. We are interested in discovering the kinds of skills generated by the use of (,/ +4BB/++%./ /KD2)*"(%)#+ and +D*/"'+,//(+ 9/(,)'+ that can make the user competent in the use of (,/ E"*(/+%"# 9/(,)': And also, which skills generated by (,/ "*%(,9/(%B 9/(,)' are necessary for a competent use of (,/ E"*(/+%"# 9/(,)': EMPIRICAL STUDIES LEADING TO THESES ON THE COMPETENCE NECESSARY TO MASTER THE CARTESIAN METHOD In this section we discuss results obtained from two experimental studies with pupils aged between 10 and 16. Both studies focused on pupils' use of methods to solve arithmetic and algebraic word problems. One of the studies concerns the use of the arithmetic, the successive explorations, and the Cartesian method, whereas the other concerns the use of the spreadsheet method. The first of the two studies is part of the project ‘Didactic Models in the Solutions of Arithmetic and Algebra Word Problems’ii and aims to describe the advantages and disadvantages of the use of any of the three methods when solutions to word problems appearing in algebra textbooks are attempted. One of the main interests is in discovering the kinds of skills generated by the use of the +4BB/++%./
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A%$4*/@X:@C,/"(*/@C%B=/(+@52$/3*"@?*)32/9
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/KD2)*"(%)#+ 9/(,)' that can make the user competent in the use of the E"*(/+%"# 9/(,)': And also, which skills generated by the "*%(,9/(%B 9/(,)' are necessary for a competent use of (,/ E"*(/+%"# one. This research project contemplates the relationship between the solution of problems and a competent syntactic use of algebraic expressions, in such a way that we can: (a) detect and describe the knowledge skills needed to set the analysis and solution of certain types of problems in motion; (b) find out if the transference of algebraic operativity is promoted and/or reinforced, in the context of the problems, via the application of each one of the three methods for problem solution; (c) show whether a proposal for teaching which uses (,/ "*%(,9/(%B and (,/ +4BB/++%./ /KD2)*"(%)#+ 9/(,)' among its solution methods might allow the student to use language strata which arc close to algebra, yet more concrete than algebra itself, in the solution of problems; (d) analyse how the use of the elements of the Sign-System of Arithmetic, via the "*%(,9/(%B 9/(,)'1 could lead the user to give meaning to the symbols that are found in the algebraic expressions we call equations; (c) make it possible that algebraic expressions are given meanings external to mathematics; and that with the development of these skills, the foundations could be laid for the student to, (f) confer meanings, starting from the algebraic language itself, to the symbolic formations of algebra: and, (g) study the relationships between the ability to carry out the logical analysis of an arithmetical-algebraic word problem in the "*%(,9/(%B 9/(,)'1 and the ability to analyse and solve a problem via the other two methods, the +4BB/++%./ /KD2)*"(%)#+ and the E"*(/+%"# one. The following phases characterise this experimental project: Phase 1. Exploratory study. a) A theoretical analysis in order to formulate a local theoretical model (see Filloy, 1990) which would allow the empirical observations to be analysed in terms of three components: 1) Formal Competence, 2) Teaching Models, 3) Cognition. A brief description of these components can be found in Filloy (1991). b) An experimental process: the classroom activity of student (aged 15- 16) groups was monitored (1990-1992). The groups were classified according to three key areas of competence, and, essentially, as to the predominance of the following prerequisites for problem-solving: I) Arithmetic, II) Pre-algebra and syntax, and III) Semantics. In the teaching component of this phase, only (,/ "*%(,9/(%B 9/(,)' was used in some groups, and only (,/ +4BB/++%./ /KD2)*"(%)#+ 9/(,)' in others. Phase 2. (1991-1992). In this phase, five exploratory questionnaires were used to classify the population which was to be monitored in class. The results were used to select cases for clinical study, in order to observe in detail the theoretical theses of the project (Filloy and
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Rubio, 1991) as well as those which arose as the observation progressed throughout the school year. Essentially, the arithmetic method and successive explorations were used together in the teaching process so that, at the end, the students' performance with the Cartesian method could be analysed. For a detailed description of the study see Filloy and Rubio (1991, 1993a and 1993b) and Rubio (1994).
PROPOSITIONS In this section we enumerate some of the propositions which arose after the experimental work. The theses bring together ideas, both explicitly and implicitly, from different fields of knowledge such as history, epistemology, didactics, psychology and mathematics. However, they were obtained both from the interaction with the students solving problems in the classroom (and during the interviews), and from the interpretation of the results of their endeavours via the theoretical framework.
>01'13+/+12 ?@ h# (,/ %#(*)'4B(%)# )7 (,/ D,"+/+ )7 (,/ +4BB/++%./ /KD2)*"(%)#+ 9/(,)'1 %# (/*9+ )7 '%'"B(%B +/b4/#B/+: On the basis of the experimental results obtained, we can describe four important phases in the introduction of the successive explorations method in terms of didactic teaching sequences: Phase 1. Reading and making the unknowns explicit. The initial representation of the problem through reading is the consequence of a first analysis of the situation that the student is trying to understand. The ‘quantities’ that are to be determined (explicit unknown quantities) are separated out; the result of analysis, this ‘separating out’, helps some users to understand the problem and thus establish the relationships between the data and other unknown quantities in the problem (secondary unknowns). If one of the unknown quantities is not made explicit, the student cannot always carry the analysis through since, in PHASE 2, the unknown, for which a numerical value will be proposed as a solution, may or may not be known. Phase 2. A hypothetical situation is introduced, proposing numerical values for unknown quantities, thus assuming a possible solution to the problem and, from this, obtaining consequences. In this phase we have a first situation of analysis, especially in the creation of new unknowns. However, there is still not necessarily a notion of the relationships between the elements of the problem.
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If the problem situation is understood, that is, if the information is recognised, it is compared with some previously formed scheme which is stored in the long term memory. Cognitive mechanisms which anticipate the solution come into play, sketching a solution, showing themselves via a representation which uses language strata dominated by the user within a system of mathematical signs. However, if the problem situation is not understood, the user needs to bring other cognitive mechanisms into play, among which are those which lead to a deeper analysis. One possibility for a deeper analysis of the problem is to do so by means of the traditional method of translation. an adaptation of the E"*(/+%"# 9/(,)': However, the difficulty with this approach is that it requires the student to carry out the analysis with a strata of language which uses the mathematical signs system (MSS) of algebra, whose signs and rules have less semantic meaning for a problem solver of 15 or 16 years old, who only spontaneously uses the MSS of arithmetic. For this reason, the use of the strata of intermediate language provided by (,/ +4BB/++%./ /KD2)*"(%)#+ 9/(,)' allows the construction of a bridge between the two strata and a numerical approach can lead to the unfolding of the analysis of the problem situation and thus, its solution. Phase 3. In this phase a comparison must be established between two quantities which represent the same value in the problem, but at least one of them is the consequence of interrelating the data with the unknown or unknowns of the problem (in the E"*(/+%"# 9/(,)' this would correspond to formulating an equation representing the problem). Phase 4. On the basis of what is obtained in Phase 3 of the comparison, the operations that have been carried out are recovered and finally an equation is obtained. The construction of meanings for the representation of problem using equations has to pass through a stage of analysis of the situation where the relationships should come to have meaning to the student. Then, employing the intermediate language stratum used by the +4BB/++%./ /KD2)*"(%)#+ 9/(,)'1 it is hoped that the user can give meaning to the relationships in the problem and, finally and perhaps more importantly confer a meaning to the construction of the relation of equivalence represented by the equation, which points towards the solution of the problem. –
–
>01'13+/+12 ?? h# (,/ #"(4*"2 (/#'/#B0 () use #49/*%B"2 ."24/+ () /KD2)*/ D*)32/9+: When pupils make use of algebraic language, some experts and many beginners spontaneously employ numerical values (and arithmetic operations) to explore and thus resolve some word problems in algebra. This is because the use of numbers and arithmetic operations spontaneously confers meanings to the relationships found
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immersed in a problem and. in many cases, opens up more opportunities for the logical analysis of the problem to be set in motion. Since algebraic language is more abstract, it is more difficult to grasp the sense of the symbolic representations and, thus, to find strategies for solving the problem.
>01'13+/+12 ??? C,/ B)9D/(/#( 4+/ )7 (,/ E"*(/+%"# 9/(,)' %+ */2"(/' () "# /.)24(%)# %# (,/ 4+/ )7 +093)2%+"(%)#: The competent use of the Cartesian method to solve arithmetic-algebraic word problems implies an evolution of the use of symbolisation in which, finally, the competent user can give meaning to a symbolic representation of the problems that arises from the particular concrete examples given in the process of teaching, thus creating families of problems whose members are problems which are identified by the same scheme for solution. Use of the Cartesian method makes sense when the user is aware that by applying it, he can solve such families of problems. The Cartesian method is not conveyed by the unarticulated revision of examples (as is encouraged by conventional didactics, see Johnson, 1976). The integrated conception of the method needs the confidence of the user that the general application of its steps will necessarily lead to the solution of these families of problems. A)* " 4+/* () 3/ B)9D/(/#( %# " 9)*/ "3+(*"B( 9"(,/9"(%B"2 >01'13+/+12 A?? +0+(/9 )7 +%$#+1 2%/ +,)42' ,/ +) "2+) %# )(,/* 9)*/ B)#B*/(/ 9"(,/9"(%B"2 +%$#+ +0+(/9+: If a student is to become a competent user of the mathematical system of algebraic signs, which is the most abstract system used in this study, he has to be competent in other less abstract systems of signs, such as: the mathematical system of arithmetic signs, used in the "*%(,9/(%B 9/(,)'1 and be able to handle systems of signs between the two, such as in the +4BB/++%./ /KD2)*"(%)#+ 9/(,)': >01'13+/+12 A??? C,/ 9/"#%#$ )7 (,/ E"*(/+%"# 9/(,)' %+ */2"(/' 3)(, () (,/ "3%2%(0 () */(4*# () 9)*/ B)#B*/(/ 9"(,/9"(%B"2 +0+(/9+ )7 +%$#+1 "#' () (,/ B"D"B%(0 () */B)$#%+/ (,/ "2$/3*"%B /KD*/++%)#+ 4+/' () +)2./ (,/ D*)32/9 "+ /KD*/++%)#+ >,%B, %#.)2./ 4#=#)>#+: To give full meaning to the E"*(/+%"# 9/(,)'1 to solve word problems in algebra, the (competent) user has to have the ability to return to systems of signs with a greater semantic content (for instance, the system of signs associated with the +4BB/++%./ /KD2)*"(%)#+ 9/(,)'T: The acceptation of (,/ E"*(/+%"# 9/(,)' for solving problems requires the users to recognise the algebraic expressions used in solving the problem as expressions involving unknowns. We can say that there is competent use of expressions with unknowns when the operations carried out between the unknown and the data of the problem makes sense. In earlier stages, the pragmatics of these systems of signs leads to the use of letters as variables, passing through a stage in which the letters are only used as names and representations of
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generalised numbers and a later stage in which they are used only to represent the unknown in the problem. Both of these are quite distinct and precede the use of letters as algebraic unknowns and the use of algebraic expressions as mathematical relations of quantities and magnitudes, in particular as functional relations.
>01'13+/+12 ?B C,/ 9/(,)' )7 +4BB/++%./ /KD2)*"(%)#+ "+ " 3*%'$/ 7)* 4#%(%#$ +0#("B(%B >%(, +/9"#(%B '/./2)D9/#(: The didactic model based on the analytic method of successive explorations when used to solve verbal algebraic problems, serves as a bridge joining syntactic development with semantic development by means of the construction of meanings for arithmetic-algebraic operations, on the way from the use of the notion of variable to that of unknown. C,/ 9/(,)' )7 +4BB/++%./ /KD2)*"(%)#+ D*)9)(/+ '%77/*/#( >01'13+/+12 B? "2$/3*"%B %#(/*D*/("(%)#+ )7 " >)*' D*)32/9 C,/+/ %#(/*D*/("(%)#+ ') #)( "2>"0+ 7)22)> (,/ )*'/* )7 (,/ +("(/9/#(1 "+ 4+4"220 )BB4*+ %# (,/ (/"B,%#$ +/b4/#B/+ 4+/' () %224+(*"(/ (,/ E"*(/+%"# 9/(,)': >01'13+/+12 B??? C,/ +4BB/++%./ /KD2)*"(%)#+ 9/(,)' %# " B)9D4(%#$ /#.%*)#9/#( S7)* /K"9D2/1 +D*/"'+,//(+T ,"+ '%'"B(%B */2/."#B/ %# %(+/27 3/B"4+/ %( $%./+ 9/"#%#$ () #49/*%B"2 9/(,)'+ 7)* +)2.%#$ (,/ /b4"(%)#+ >,%B, "*%+/ >,/# D*)32/9+ "*/ '/7%#/': SOME PROPOSITIONS FROM A CLINICAL INTERVIEW (MARIBEL)
>01'13+/+12 A One of the cases in this research is that of Maribel, a pupil of 16 years of age and who at the moment of the interview could not work out the ?*)32/9 )7 (,/ F#./+()*+ No 1: ?/'*) "#' Q4$) %#./+(/' Vl1WWW "#' X]1WWW D/+)+ %# " 34+%#/++. C,/0 "$*//' () '%.%'/ (,/ D*)7%(+ D*)D)*(%)#"220 () (,/ 9)#/0 %#./+(/' 30 /"B, )#/: F7 Q4$) 9"=/+ " D*)7%( )7 c1WWW D/+)+1 ,)> 94B, D*)7%( ')/+ ?/'*) 9"=/x h# B)9D/(/#B/ %# (,/ 4+/+ )7 D*)D)*(%)#"2%(0 It would appear that Maribel is not competent in the uses of proportionality or multiplication. In order to locate the cut off point in the lack of competence, a ‘variation’ on the previous problem was presented, that is, the D*)32/9 )7 (,/ %#./+()*+ No 1.1 : ?/'*) "#' Q4$) %#./+(/' Vl1WWW "#' X]1WWW D/+)+ %# " 34+%#/++: C,/0 "$*//' () '%.%'/ (,/ D*)7%(+ D*)D)*(%)#"220 "BB)*'%#$ () (,/ 9)#/0 %#./+(/' 30 /"B, )#/: F 7 Q4$) 9"'/ " D*)7%( )7 s1WWW D/+)+ 1 ,)> 94B, D*)7%( '%' ?/'*) 9"=/x
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This is a problem of the same family as that of No 1 and Maribel solved it without difficulty, using the classical "*%(,9/(%B 9/(,)': Maribel reasoned as follows: F7@@Q4$)@9"'/@"@D*)7%(@)7@s1WWW@D/+)+@(,"(@%+@"@b4"*(/*@)7@>,"(@,/@%#./+(/'1@(,/#
?/'*) >%22 ,"./ () 9"=/ " D*)7%( >,%B, %+ (,/ b4"*(/* ) 7 >,"( ,/ %#./+(/'1 (,"( %+ Vl1 WWW K Xns = c1WWW: Once this problem was solved, Maribel was once again asked to try to solve the ?*)32/9 )7 (,/ F#./+()*+ No 1. For the second time she said she didn't understand the problem, although she was aware that it was very similar to the one she had just solved. Moreover, when she was asked about the difference between the two, she said ‘... in the quantities because ... in the other quantity 4,000 ...coincided ... that this ... is 1/4of 16,000 ... but here ... 7,000 and 16,000 don't coincide in anything ...’ The interviewer then showed her the ?*)32/9 )7 (,/ F#./+()*+ No 1.2 which has the same initial text as the previous two (No 1 and No 1.1) except for the part ‘.. F7 Q4$) 9"=/+ l1WWW D/+)+ D*)7%(1 ,)> 94B, D*)7%( ')/+ ?/'*) 9"=/x Maribel solved this problem correctly, incorporating a new unknown into her analysis, that is, the secondary unknown ‘the part of’, producing the following inferences on the basis of this: 8,000 (Hugo's profit) is 1/2 of 16,000 (Hugo's investment) then, 14,000 will be Pedro's profit since it is 1/2of 28.000 (Pedro's investment). At this point the interviewer tried to make Maribel aware of the scheme: ? x A = B, where ? represents the parts of a whole A. He said: ‘... here you know the part 8,000, you already know which part it is of ... ‘, and the pupil replied ‘... of 16,000 ... ‘ . Immediately after this short dialogue, Maribel was shown the D*)32/9 )7 (,/ F#./+()*+ No 1 again. As she began to solve the problem, it could be clearly observed that for Maribel, the use of the scheme X x A = B, where X represents the parts of the known whole A and B is the result of taking this part of the whole, had no meaning. Despite the fact that she herself had already created the secondary unknown, different from the one explicit in the problem, when she was asked ‘What do you have to ask yourself now?’ She replied ‘ ... which part of 16000 is 7000’, which does not correspond to the scheme X x A =B. In this case it can be observed that, as in other cases in the same study and in other studies, there are subjects >,) ') #)( "'9%(@ (,/@ D)++%3%2%(0@ )7 9"=%#$@ %#7/*/#B/+ "3)4(@ +)9/(,%#$@ (,/0@ ') #)( =#)>: In other cases, (,/@ +43O/B(@ +%9D20 ".)%'+ )D/*"(%#$ >%(, (,/ 4#=#)>#: That is, there is a resistance to bringing operations into play in the representation of something unknown, even when a use is already being made in the teaching of the +4BB/++%./ /KD2)*"(%)#+@ 9/(,)':@
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This is the case of Maribel who attempts to use the B2"++%B"2 "*%(,9/(%B 9/(,)' to solve the problem, avoiding operating with the unknown (Proposition V), despite being competent enough to solve various families of problems with the +4BB/++%./ /KD2)*"(%)#+ 9/(,)' and even with the E"*(/+%"# 9/(,)': It is only when she can give meaning to the arithmetic representation 7/16 x 16000 = 7000, that she can solve the problem writing: 7/16 x 28000 = 12,250.
Proposition III and IV In another part of the interview, Maribel is presented with the ?*)32/9 )7 (,/ ?/*749/ A"B()*0: F# " D/*749/ 7"B()*0 (,/*/ "*/ (>) */B%D%/#(+1 )#/ >%(, "# Xl{ B)#B/#(*"(%)# )7 E,"#/2 "#' (,/ )(,/* >%(, " B)#B/#(*"(%)# )7 s[{: Q)> 9"#0 92 )7 /"B, */B%D%/#( +,)42' 3/ 4+/' %7 " 3)((2/ )7 @V@92 B"D"B%(0 >%(, " B)#B/#(*"(%)# )7 []{ )7@D4*/ E,"#/2 %+ #//'/'x The difficulties that Maribel has with the percentage do not allow her to deal with this problem. and the interviewer goes on to present her with the following: C,/ )>#/* )7 " $*"%# +()*/ >"#(+ () B)9D/(/ >%(, (,/ D*%B/ )7 ,%+ 3/"#+: F7 ,/ ,"+ 3/"#+ B)+(%#$ aclWW D/* =%2) "#' ,/ >"#(+ () 9%K (,/9 >%(, 3/"#+ B)+(%#$ |XVWWW@D/* =%2)1 %# )*'/* () +/22 (,/ 9%K(4*/ "( |llaW D/* =%2)1 ,)> 9"#0 =%2)+ )7 (,/ 3/"#+ B)+(%#$ |clWW@D/* =%2) "#' ,)> 9"#0 )7 (,/ 3/"#+ B)+(%#$ |XVWWW@D/*@ =%2) ')/+ ,/ #//' () $/( VsW =%2)+ )7@"@9%K(4*/ B)+(%#$@|llaW D/* =%2)x At this time, Maribel already has a certain domination of the intermediate tactics mentioned in proposition III, which help her in the development of the ‘positive’ cognitive tendencies which are found in the processes for learning more abstract concepts such as: the return to more concrete situations, when a situation of analysis is presented, which is a necessary part of progress in the competence in the E"*(/+%"# 9/(,)': Maribel deals with the ?*)32/9 )7 (,/ 6/"#+ with the +4BB/++%./ /KD2)*"(%)#+ 9/(,)' in order to carry out the analysis and, finally, to understand it. This part of the interview with Maribel contains this episode:
M: ... ‘x’ the kilos of beans which are in ... in 7800 ... ‘y’ the kilos of beans costing 12000 pesos ... ‘x’ plus ‘y’ is equal to 240 kilos ... and ... I am missing a comparison .. I : Are you thinking at the same time of equations and comparisons? Maribel writes PHASE 1, which corresponds to her way of beginning the +4BB/++%./ /KD2)*"(%)#+ 9/(,)': M: ... when I find it difficult to understand things ... I start to use ... to use the method that I took ... thus I find out what is happening in the problem. I: ... Yes ... Maribel continues to write:
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PHASE 1 How many kilos of beans costing $7800 are needed How many kilos of beans costing $12000 are needed PHASE 2 kilos 7800 kilos 12000 M: ... 240 kilos ... then ... Maribel writes first 130 in front of 12000 (it appears to be an assumed value) and then obtains 110, and writes it in front of 7800, thus: PHASE 2 kilos 7800 110 kilos 12000 ... 130 Proposition VII At another moment during the interview, Maribel is once again presented with the ?*)32/9 )7 (,/ ?/*749/ A"B()*0 that she had not been able to solve. In this sequence of problems it can be observed that although Maribel is not yet competent in the use of the 9"(,/9"(%B"2 +0+(/9+ )7 +%$#+ of arithmetic and of that corresponding to the +4BB/++%./ /KD2)*"(%)#+ 9/(,)' (both more concrete than the mathematical system of signs of algebra), she already has developed a network of related competencies, through the use of these methods of problem solving, which are allowing her to advance in the uses of the concept of unknown and of equation. After Maribel reads the problem in silence, she says: M: .. shall I make an equation like this ..or not? I: .. you're going straight to the equation? M: .. because it seems similar to me ... I am comparing with the other statement ... I: .. why do you say it's similar? M: .. I say it's similar because ... here I am given ... in the other problem I am given the price of ... of a certain bean and here I am given a recipient with a 2 8% concentration of Chanel ... and in the other I am given ... beans costing 12000 pesos When Maribel recognises that the ?*)32/9 )7 (,/ ?/*749/ A"B()*0 is of the same family as the ?*)32/9 )7 (,/ 6/"#+ (which she solved correctly). she can make a logical sketch according to a stratum of language of the system of signs of algebra, establishing algebraic relationships between the data and the unknown ‘x’, and obtaining the algebraic equation: 18x+ .43(12-x) = .36 x 12 This is obtained through a reasoning similar to that used in the problem of the beans and, given insistence on the part of the interviewer, she manages to present the unknown.
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The above equation has the same structure as that obtained in the ?*)32/9 )7 (,/ 6/"#+P x x 1200 + (240 - x) 7800 = 240 x 8850. Thus we can observe how the need to represent the unknown is what generates new meanings which bring the possibility of making more abstract uses of the mathematical systems of signs used to make the representation of the problem, starting from a logical-semiotic sketch. Also, in the case of Maribel, it is clear that during the passage from the solution of a less complex problem (that of the beans) to a more complex one (that of the perfumes), processes of generalisation and abstraction are produced which operate on families of problems. It is also clear that an explanatory mechanism of why the problems of mixtures are more difficult than those of other families of problems, resides in observing the need to break with the use of only inferring on the basis of the representation of the unknown, in order to give a use to the representation in which these unknown parts vary. This happens, for example, in the passage from using a relation directly proportional to using various directly proportional relations together in the same problem.
A SPREADSHEET APPROACH FOR SOLVING WORD PROBLEMS As proposition XIII indicates, when the 9/(,)' )7 successive /KD2)*"(%)#+ comes alive in a computational context such as that of electronic spreadsheets, the search for the value of the unknown occurs at a completely numerical level. That is to say, unlike the 9/(,)' )7 +4BB/++%./ /KD2)*"(%)#+1 the +D*/"'+,//( 9/(,)' makes no explicit formulation of the equation and thus, neither is there an algebraic solution of the same. In this section we present the results obtained in an Anglo/Mexican project entitled 'Spreadsheets Algebra Project’iii in which the +D*/"'+,//( 9/(,)' is put to work with students between the ages of 10 and 15 years (Rojano z Sutherland, 1991, Sutherland z Rojano, 1993). The results are discussed in the light of the theoretical theses expressed above. Although this project was conceived of and developed independently of the work which led to the above-mentioned theses, its intention to close the gap between the non-algebraic methods of problem solving and the Cartesian method, by means of the use of intermediate methods, is a component shared by both projects. Furthermore, some of the facts observed in this project can be interpreted as an empirical counterpart of the theoretical work of Filloy and Rubio (apart from the empirical support which the proposition have in the experimental work of these researchers). The Anglo/Mexican project was constituted by a series of experimental studies in which pre-algebraic students (10-1 1 years of age) and students between the ages of 14 and 15 participated, all were reticent to use school mathematical methods. The
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studies were carried out in parallel, with two groups of students from each of the levels mentioned, one in Mexico and the other in England. the aim was to study the way in which the students used a spreadsheet to represent and solve algebraic word problems and how the use of this medium influenced their previous strategies for solving problems and vice-versaiv. In this chapter we have called this approach to problem solving (,/ +D*/"'+,//( 9/(,)' and it has been illustrated in the section Two Non-Conventional Solution Methods. Detailed reports of the teaching activities as well as how the results obtained in the research have been published at length in Rojano and Sutherland (1991, 1992 and 1993) and in Sutherland and Rojano (1993 and 1993). Perhaps one of the results which most surprised us at the beginning of the research was the ‘reasonableness’ of the non-formal strategies used in the solution of the problems posed in the test and interview previous to the experimentation. We can speak of two main categories of strategies used in some of the problems, both by the pre-algebraic children and the older ones: a) >,)2/nD"*(+ and b) (*%"2 "#' */7%#/9/#(: We will now illustrate these two approaches with the solution given by some children to the D/*%9/(/* )7 " 7%/2' D*)32/9 in the preliminary interview. ?/*%9/(/* )7 " A%/2' The perimeter of a field measures 102 metres. The length of the field is twice as much as the width of the field. How much does the length of the field measure? How much does the width of the field measure? An algebraic approach Let the width of the field = X metres. Let the length of the Geld = L metres. Then L = 2X–(1) 102 } (2) and 2L + 2X = So by substituting (1) in (2) 4X+2X= 102 6X = 102 X= 17 metres. A whole/parts strategy gF '%' XWV '%.%'/' 30 ] ::: F O4+( '%' (>) )7 (,/ 2/#$(,+ () 9"=/ %( +/#+%32/ :::: F O4+( (,)4$,( (,/*/ 94+( 3/ (>) )7 (,)+/ %# )#/ 2/#$(,::::::::e:@C,%+@+)24(%)# involves working with a known whole (the perimeter) and dividing this into parts to find the unknown lengths of the side of the field. This method can be seen in opposition to the "2$/3*"%B 9/(,)' which involves working with the unknown lengths.
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A systematic trial and error approach g>/22 F (*%/' sW1 %( was XVW.....+) F =#/> %( 94+( 3/ +9"22/* (,"# (,"(:::%# (,/ [W_+..."#' >,/# F (*%/' [] "#' %( >"+ XWl:::F =#/> %( B)42'#_( 3/ [a +) %( 94+( 3/ [s.......’. This strategy has also been identified by a number of researchers (for example, Bednarz et al, 1992) and involves working from the unknown to the known. Using (,/ +D/"'+,//( 9/(,)'1 the solution would look like this:
A%$4*/ V: 5 +D*/"'+,//( +)24(%)# () (,/ ?/*%9/(/* )7"A%/2' ?*)32/9
Within the study the pupils were explicitly taught this method for solving a family of algebraic word problems. Once the problem has been expressed in the spreadsheet symbolism pupils could then vary the unknown. In this way, through the +D*/"'+,//( 9/(,)'1 children are encouraged to deal with unknown quantities. Some relevant results from these studies are: 1) The majority of pupils do not spontaneously think in terms of a general algebraic object when first working in a spreadsheet environment. Their thinking is initially situated on the specific example with which they were working. 2) The spreadsheet environment supports pupils to move from focusing on a specific example to consideration of a general relationship. 3) The spreadsheet also supports pupils to accept the algebraic idea of working with an unknown. They use a spreadsheet cell to represent the unknown number and then, with the mouse or the arrow keys, express relationships in terms of this cell. 4) After the experimental work, there is an increased level of awareness of all the relationships between unknowns and between givens and unknowns. 5) Throughout the spreadsheet sessions it was observed that pupils evolved towards a ‘more general and algebraic method’ consisting of proceeding from the unknown to the known. 6) In the final interview, it was observed that some pupils integrated whole/parts and trial and refinement strategies with the +D*/"'+,//( 9/(,)':
DISCUSSION OF THE EMPIRICAL RESULTS The results of the Anglo/Mexican project can be easily identified with some of the theoretical theses put forward in earlier sections. For example, the spontaneous
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numerical approximation found in the >,)2/nD"*(+ and (*%"2 9%' */7%#/9/#( strategies corroborate proposition II on the need to use numerical values to explore the relations between the elements of a problem. Both the successive /KD2)*"(%)#+ 9/(,)' and the +D*/"'+,//( 9/(,)' maintain this numerical exploration as 9/(,)': In the case of spreadsheets, this medium allows the systematisation of this exploration since even with the relations of a problem expressed in terms of one of the unknowns and with the symbolism of the spreadsheet, the student continues a numerical search for a solution to the end. Results 3) and 4) described above can be an empirical support for our theses above since the exhaustiveness in the logical analysis of all the relations of the problem, in effect, is linked with the possibility of manipulating the unknown. In this computational environment, it is feasible that the student begins his work with the unknown without the reference being lost, since (as Figure 2 shows) the cells are labelled at the outset with the name of what they represent and this allows the student to incorporate the relationships of this unknown with the other elements of the problem, one by one, without the use of a more abstract system of signs (that of the spreadsheet) obstructing the analysis. Proposition V points in the direction of a need to evolve towards a use of algebraic symbolism so that the subject can identify families of problems with certain schemes of resolution previously employed. In the experimental phase of the Anglo/Mexican project, the step towards algebraic syntax was not reached. However, the case of an English girl, Jo (15 years old and with a history of failure in school maths) shows the feasibility of making this connection. since Jo, as well as becoming a competent user of the +D*/"'+,//( 9/(,)' to solve certain families of problem, in her final interview produced semi-algebraic expressions when the researcher asked her to write the expressions on the spreadsheet. naming the unknown with ‘x’. Jo proceeded to write the following:
On the other hand, the +D*/"'+,//( 9/(,)'1 together with the +4BB/++%./ /KD2)*"(%)# 9/(,)'1 can be considered as feasible ways of passage towards competence in the use of the E4*(/+%"# 9/(,)' That is to say, the experimental studies indicate that both methods constitute an antecedent of the elaboration of the meanings of algebraic relationships between the elements of a problem; and also they promote a flexibility in the treatment of these relationships (not necessarily following the textual order i n which these relations appear in the statement). FINAL DISCUSSION One of the principal aims of the application of solution methods which fall between the classical arithmetic method and the Cartesian method is that of facilitating the
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working out of the analysis of word problems; as well as using the pre-algebraic tendencies that pupils show when they approach these problems. It is on the basis of the theoretical analysis carried out and of the empirical observations obtained in the projects referred to in this chapter that aspects as important as that related to the different uses students make of ‘the unknown’ when they attempt to represent and solve arithmetical-algebraic problems appear. Another relevant aspect which arises from this research is that of the difficulties that these uses generate for the working out o f the analysis of different families of problems (Trujillo, 1987 and Rubio, 1990). For example, when ‘x’ is used to represent ‘the unknown’. The role played by intermediate methods in the passage from the classical arithmetic method to the Cartesian one has to do with the possibility of the user constructing meanings for the algebraic relationships between the elements of the problem. Although it should be pointed out that, on the other hand, the essential difference between the introduction o f algebra and all previous approaches lies in that in the latter, when solving problems, the unknown is represented, although it is not operated. Inferences are made with a reference to the representation of the unknown; but if operated, this is always done by means of the data: if a mention is made of unknowns, this is only in terms of the results of operations which are being done with the data. Finally, the employment in the research presented here of four solution methods (the arithmetic method, that of successive explorations, that o f spreadsheets and the algebraic method) to solve arithmetical-algebraic problems took into account the indisputable fact that to solve these problems, in particular with the algebraic method, it is necessary to consider the competencies, adjustments and limitations that other more concrete mathematical signs systems (in this case that of arithmetic and those of the intermediate methods) can impose on the more abstract system of signs that is to be taught (that of algebra) and, thus, the classical algebraic method which here we have identified as the Cartesian method. NOTES 1
The term is taken from linguistic pragmatics. This accounts for the uses of language produced in the exchange of messages which occurs in the act of speech (Levinson, 1983). That is to say, starting with a succession of statements. together with fundamental assumptions as to such use, very detailed inferences can be calculated regarding the type o f assumptions made by the participants and the objectives they seek through the use of the statements. In order to participate in the ordinary use of language, one has to be able to make such calculations both in production and in interpretation. This ability is independent of beliefs, feelings, and idiosyncratic usage (although it can refer to beliefs. feelings and idiosyncratic usage shared by the participants), and to a large extent is based on quite regular and relatively abstract principles. Pragmatics can be understood as the description of this ability, which acts both with concrete languages and language in general. There are authors who
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consider that the central aim of pragmatics is to study the relationship between language and context, and that this is the basis of dealing with the comprehension of language. Research project carried out from 1990-1994 in Centro de Investigación y Estudios Avanzados and the schools centres Hermanos Revueltas and Colegio de Ciencias y Humanidades, Plantel Sur in Mexico City. We worked with 10-1I year old pupils with no previous experience of algebra and 14-15 year old pupils who were all chosen because of their resistance to school mathematics. Case studies were carried out of eight Mexican and eight British pupils in the pre-algebra and algebra resistance group When working with spreadsheet some pupils seemed to be synthesising their informal strategies with a more formal algebraic approach.
HAVA BLOEDY-VINNER*
BEYOND UNKNOWNS AND VARIABLES — PARAMETERS AND DUMMY VARIABLES IN HIGH SCHOOL ALGEBRA THE NOTION OF PARAMETER
L%77/*/#( *)2/+ )7 2/((/*+ Although letters in algebraic expressions always denote numbers, there is a sense in which letters can be given meaning in different ways. These different meanings depend on the context and application of the expressions. Kuchemann (1981) identified several ways of interpreting and using letters three of which are related to our issue: (1) Specific unknowns — the letter has a particular but unknown value. Such usage is needed when required to solve equations, or to construct expressions which translate specific situations, e.g. the perimeter of a shape where ‘there are n sides altogether all of length 2.’ (2) Generalised numbers — the letter is able to take more then one value. This usage of letters is required in problems such as: ‘L+M+N=L+P+N is always true, sometimes true (when), or never true?’ Variables used when letters are seen as representing a range of (3) unspecified values, and a systematic relationship is seen to exist between two such sets of values. This usage is needed in problems such as: ‘Which is larger, 2n or n+2?’ According to Küchemann’s analysis, letters are used as variables when a problem requires the establishment of a (second order) relationship between relationships (in this case between the functions 2n and n+2). According to Küchemann these interpretations of letters represent different levels of understanding. In other words, the difficulty of problems increases with the level of meaning (in the above order) which is required to give to letters. For example, the success rate for the three problems quoted above, among 15 years old children, was 41%, 35%, 10% respectively. The different usage and meanings of letters arc also related to the issue of curriculum. According to Usiskin (1988), algebra curricula are determined by —
* Hebrew University. Centre for Preacademic Studies, Israel
177 M: ;4(,/*2"#' /( "2: S/'+:T1 ?/*+D/B(%./+ )# ;B,))2 52$/3*"1 177–189. U VWW X <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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different conceptions of algebra. He mentions three algebra conceptions which correlate with the different relative importance given to the various uses of letters mentioned above: (1) Algebra as a study of procedures for solving certain kinds of problems. These procedures include solving equations, where letters are used as unknowns. (2) Algebra as generalised arithmetic where letters are used as pattern generalisers, e.g. a+b=b+a which generalises 3+5=5+3. (3) Algebra as the study of relationships among (functions), where letters are used as variables (function arguments), as in the problem: ‘What happens to the value of 1/x as x gets larger and larger?’ Curricula which place emphasis on different conceptions of algebra may affect students' understanding of the different roles of letters. For example, Schwartz and Yerushalmy ( 1992) and Kieran (1994) designed curricula which use software to introduce the function concept at the beginning of algebra, basing all algebraic concepts and manipulations on functions. The designers of these curricula intend to explore the influence of this approach on students' understanding of these concepts. Regardless of the conception during the introduction of algebra, advanced high school algebra should contain all of the above applications, and students should eventually understand all these different meanings given to letters. C,/ *)2/ )7 D"*"9/(/* %# ,%$, +B,))2 "2$/3*" In more advanced high school algebra there is yet another usage of letters, which interacts with unknowns and variables, but should be distinguished from them. This is the usage of letters as parameters. As we shall see, this distinction too depends on the context and application of the algebraic expressions. In high school algebra parameter is encountered explicitly or implicitly when learning about families of equations, families of functions, and in some word problems and other mathematical problemsi. Here are some examples of problems and statements which involve equations or functions with parameters and with unknowns or variables: (a) In the following equation x is an unknown and m is a parameter: m(x–5)=m+2x. For what value of the parameter in will the equation have no solution? (b) The graph of (x–m) ² +(y–n) ² =R ² is a circle. (c) Find an equation for the line through ( 2 , 5 ) with slope 3. (d) Write the co-ordinates of the extremum of the graph of the quadratic function f(x)=ax ² +bx+c. (e) X is the price of one pencil. Find the price of one eraser if I paid $ K for 10 pencils and c erasers.
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A number A is given. Can we always find a number B so that (A–3)(B–2)=1? How do we know which letters are meant to be parameters and which are unknowns or variables? The answer is not built into the expression or the equation itself. The answer is to be found in the context, which consists of one of the following: (1) An explicit declaration about which letters arc used as parameters, and which are unknowns or variables, as in example (a). (2) Common knowledge about standard forms of equations or functions, including the common usage of x and y as names for axes in graphing, and the meaning of the notation f (x) . Thus, in example (b), this common knowledge indicates that m, n, and R are parameters, and x, y are variables. In example (d) the words 'quadratic equation' and the notation f(x) indicate that x is the function variable while a. b, c are parameters. ( 3 ) The formulation of the question, as in examples (e) and (f). In both, the letters which denote givens are parameters, and the letters to be found are unknowns. Moreover, the meaning of a letter as a parameter or as an unknown or variable. may change in the course of solving a problem. To see how it changes, let us look at some of the previous examples: In order to answer the question in example (a), one has to produce the equivalent equation (m–2)x=6m while considering x as an unknown and m as a parameter, as declared in the problem. Now one needs to find out when the coefficient m–2 equals 0. This is done by solving m–2=0, which indicates that m should temporarily be considered as an unknown. After that, x and m assume their original roles again. Solving example (c) starts with writing the equation y=ax+b. When out of context this equation could have many interpretations: is it an equation with 4 unknowns? is it a function with 3 variables? The problem context determines that we are looking at a line equation, therefore common knowledge determines that x and y are variables whereas a and b are parameters. The process continues by substituting 3 for a, to get y=3x+b, where the letters still hold the same roles. Now we look for a specific value of b. This indicates that b becomes an unknown whose value depends on parameters x, y. We substitute constants for x, y and solve the equation 5=3*2+b of unknown b. The process terminates by substituting the constants found for a and b, and by letting x and y be variables again in the line equation y=3x–1. The interaction of variables and parameters appears also in calculus. For example: The derivative f '(x) is defined for each x as the limit of [f(x+h)–f(x)]/h as h tends to 0. This definition contains two processes which involve varying values of letters: first, the limit process, where h is the variable while x is a parameter; this
(f)
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process yields the derivative at x (which is a number); when this process is done for every value of x, we get the derivative f '(x) which is a function with variable x.
C,/ B)**/B( 9)'/ )7 (,%#=%#$ %# (,/ B)#(/K( )7 D"*"9/(/* In order to analyse the difficulty students have in understanding the notion of parameter, and in distinguishing it from unknowns and variables, one should first make explicit these distinctions as understood by the mathematical community. Understanding this notion has two components: first, concluding from the context, which letters are meant to be parameters and which are unknowns or variables, as explained in the previous section; second, understanding the different roles of parameters and of unknowns or variables. When a group of students was asked what the role of parameter was, one student said: gF( %+ " B)#+("#(:e The whole group exclaimed: g64( %( ."*%/+qe Another student said: gF( %+ " ."*%"32/ >%(, " B)#+("#( ."24/:e In these answers there is a conflict between the parameter being a constant and being a variable. It feels like a constant, and yet it varies. A closer look at the mathematical meaning of parameter will tell us where this psychological conflict conics from. An equation or a function with a parameter stands for a family of equations or functions, where specific instances may be created by substituting numbers for the parameter, while the other letters still assume the role of unknowns or variables. A more formal definition: an equation or a function with a parameter is a (second order) function, the argument of which is the parameter, and the corresponding values of which arc equations or functions (with the other letters as unknowns or variables). This mathematical definition explains the psychological conflict, namely, why a parameter feels like a constant though it varies: on the one hand, the parameter is an argument of a (second order) function, and as it varies it determines Corresponding equations or functions; on the other hand, within each equation or function which corresponds to a specific parameter value, the parameter is a constant, while the other letters are unknowns or variables. We can see that distinguishing the role of parameter from that of unknown or variable involves a function of second order. This makes the distinction logically very complex. The distinction can be given another, more dynamic, formulation. Although not less complex, that formulation is, as we shall see, more closely related the mode of thinking needed with such to the cognitive aspect of these concepts equations or functions. That dynamic formulation involves some quantifiers. To specify the quantifier structures which correspond to equations or functions with parameters we shall use the equation m(x–5)=m+2x, from example (a), with parameter m and unknown x (a similar treatment can be given to equations or —
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functions with parameters and variables.) The roles of m and x in the equation can be formulated as: (al) A )* "22 m-substitution, there /K%+(+ an equation E, so that E is ‘m(x–5)=m+2x’ with constant m, unknown x. Note that after having quantified m-substitutions, in is a constant within E. Also note that the existential quantifier in this statement is of second order, namely, quantifying an equation, not a number. To statement (a 1) we add: (a2) and 7)* "22 x-substitution, the substitution either satisfies E or not. The order of quantifiers here is essential: first quantify in and get an equation; then, only after having an equation, quantify x. In specific parameter tasks, one is often asked to find values of the parameters for which the equations have special properties. To continue with example (a), we add the following statement: (a3) There /K%+(+ an m-substitution and a corresponding equation E, so that E has no solution. The latter part may be expressed as: (a4) 7)* "22 x-substitution, the substitution does not satisfy E. Here again, the order of quantifiers is essential: we quantify x only after having quantified m and E. The dynamic formulation given here is closely related to the cognitive aspect of these concepts. These ordered quantifier structures correspond to a potential of ordered possible substitutions, which is characteristic of equations or functions with parameters: first substitute for the parameter and get an equation or a function, then substitute for the unknowns or variables to check if the equality holds, or to obtain a function value. For the student who lacks a mathematical logic background and does not think in terms of explicit quantifiers, understanding the notions of parameter means being able to visualise the potential dynamic of ordered substitutions which captures the distinction between the roles of parameter vs. unknown and variable. (An exception to the above order should be noted: when we need to find a specific parameter value, for which the equation has a given solution, the dynamic of the problem is different from the dynamic of the equation itself, and one substitutes for the unknown before having a specific equation and a specific parameter value.) ;(4'/#(+_ '%77%B42(%/+ >%(, (,/ #)(%)# )7 D"*"9/(/* While there has been a lot of research about students' difficulties with more basic usage of letters, studies about difficulties with parameters have started only recently (Bloedy-Vinner 1994, Furinghetti and Paola 1994).
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There are two inherent difficulties with the notion of parameter. First, the fact that the roles of letters are context dependent and may even change in the course of an application. We have seen that even in a basic task like finding the equation of a given line, the roles change several times. As to the limit definition of the derivative. for example. one of the difficulties is understanding that our focusing on a variable changes: we start with f(x) where x is the variable, then we introduce a new process which focuses on h as a variable (and x as a parameter), only to go back to seeing x as a variable again, in f(x). Furinghetti and Paola (1994) asked students what the roles of x, y, and c in the expression x ² +y ² +c ² +2xy+2yc+2xc were. Despite the fact that there was no context to indicate the roles of the letters, most of the students answersd that x and y were variables and c was a parameter. This shows that the students attached roles to x, y and c according to some common convention that was irrelevant to any given meaning of the expression. The second difficulty with the notion of parameter is the great logical complexity needed to explain the roles and the difference between them. We have seen that no matter how one formalises it, we cannot get away from second order structures: either a second order function (with equations or functions as values), or second order quantifiers (quantifying equations or functions). In a study about parameters among students entering college (Bloedy-Vinner 1994) it was found that many of the students did not understand the distinction between parameters and unknowns or variables. They (implicitly) quantified unknowns or variables before quantifying the equations, which corresponds to a wrong order of substitution. Here are some examples of answers given by 48 science students and 34 social studies students: Problem 1. In his homework assignment Gadi was asked to explain why a line had infinitely many equations. Gadi answered: ‘ 5 +(*"%$,( 2%#/ /b4"(%)# %+ 5Kt60vE1 "#' (,/*/ "*/ %#7%#%(/20 9"#0 D)++%3%2%(%/+ 7)* K "#' 01 (,/*/7)*/ (,/*/ "*/ %#7%#%(/20 9"#0 /b4"(%)#+: ’ What do you think of Gadi’s answer? Since the straight line equation depend on the parameters A, B, and C, the correct analysis would be: A)* "22 A,B,C-substitution, there exists an equation E, so that there /K%+( infinitely many x,y-substitutions satisfying E. Gadi’s answer expresses almost explicitly the statement: A)* "22 x,y-substitutions, there /K%+(+ an equation E. In this statement x and y are quantified before the equation, which corresponds to a wrong order of substitutions. 29% of the science students and 63% of the social studies students claimed: ‘8"'%e+"#+>/* %+ B)**/B(: ’ These students thought that equations were obtained by specific values of x and y instead of A, B, C. Students gave explanations such as: ‘C*4/q /"B, #493/* >/ +43+(%(4(/ 7)* K1 0 $%./+ " '%77/*/#( */+42( 7)* (,/ /b4"(%)# SET: ’
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gQ%+ "#+>/* %+ B)**/B(1 (,/*/ "*/ %#7%#%(/20 9"#0 /b4"(%)#+ 3/B"4+/ )# (,/ 2%#/ (,/*/ "*/ %#7%#%(/20 9"#0 D)%#(+: ’ According to these quotes some students thought that different x, y substitutions make different equations because they make different C’s. Others thought that each substitution is in itself an equation. Problem 2. Is the straight line equation y=b a specific case of the straight line equation y=ax+b? &KD2"%# 0)4* "#+>/*q From the context we know that these are straight line equations, and we deduce that a and b are parameters and x and y are variables. In order to obtain a specific member of the family of equations, we need specific a or b substitutions, while x substitutions are irrelevant. The straight line equation y=b is obtained by a=0 only, not by x=0. 46% of the science students and 41% of social studies students answered one of the following: (1) ‘k/+1 >,/# KvW e1 SVT gk/+1>,/# "vW )* KvW:’ii These answers reflect, again, a wrong order of substitution: they substitute for x in order to get an equation, instead of substituting for x after having an equation. These students think that x values determine specific straight line equations, which demonstrates a lack of understanding of the idea of a family of equations and of specific cases which depend on the parameter values only. Problem 3. Students were asked when the following equation was not quadratic: mx²+(m+5)x+m–1=0 (m is a parameter.) Avi answered: ‘m,/# 9vW: ’ Michael answered: ‘m,/# KvW:e What do you think of each of the answers? (right, wrong, /KD2"%#q) Since x i s an unknown and not a parameter, substituting for x is not relevant for determining specific equations. Therefore only Avi’s answer is correct. 38% of the science students and 85% of the social studies students justified Michael (or both), giving explanations like: ‘6)(, "*/ *%$,( 3/B"4+/ Ki 3/B)9/+ H/*)’ )* ‘::: !%B,"/2 %+ *%$,( 3/B"4+/ (,/# ~>,/# KvWu 9\X\W1 "#' %( %+ #)( b4"'*"(%B:’ When interviewed, students who gave the latter answer said that they considered m–1=0 not to be quadratic because it didn’t have x ² (and not because it didn’t have m²). This shows that they were aware of x, and not in, being the unkown. Despite this, they used the wrong order of substitutions and substituted for x to get an equation. This is equivalent to the wrong order of quantification: A)* "22 x-substitution, there /K%+(+ an equation E, and for substitution x=0, E is not quadratic. These examples we saw show that these students' mode of thinking does not have the dynamics which is equivalent to the correct mathematical definition given above, namely, the correct order and role of substitutions. The students in this study were asked questions which were non-standard but did not require more than basic understanding of the concepts. These questions were specifically designed to reveal their mode of thinking. It may well be the case that standard parameter tasks would
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not reveal the same, and that students performance would seem skilful. This means that correct answers can sometimes be given without conceptual understanding and that non-standard questions are needed to reveal the mode of thinking about the concepts.
;)9/ '%'"B(%B */B)99/#'"(%)#+ As to the difficulty in determining which letters hold which roles, students should know that the roles are context dependent (an issue rarely discussed in the classroom). They should always be told what the role of each letter is, and where the information which determines it comes from. In problems where the roles change in the course of an applications, the temporary nature of these roles should be stressed. For example, in calculus, after having dealt most of the time with functions f(x) with x as a variable, the teacher should make it explicit when the focus changes unexpectedly to h as the variable of the limit process in the definition of the derivative. Of course, just telling students which letters hold which roles may not be enough if students do not understand the logical distinction between roles, specifying the names of the roles may be meaningless. We have seen that understanding the distinction between the roles requires the dynamic mode of thinking of visualising the ordered substitutions discussed above. One could help students understand this dynamic, and even the quantifier structures shown above, by using a concrete model. In order to do this, one might try the following idea: Let us use the equation 3mx²+(m+1)x–2m+3=0 with parameter m, for the demonstration. Now, let the students visualise the parametric coefficients as digital displays where numbers can go back and forth, including negative numbers, like the counter display of a VCR:
—
Students should imagine little calculators behind the display, so that when m varies, they calculate and display the values of the coefficients 3m, m+1, and –2m+3. (The display is shown here at a moment when m=2.) For equations with several parameters we should have several independent displays. Visualising this dynamic picture will help students see that as the parameter varies, it determines new
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equations, while x remains an unknown all the time. One can pose questions about properties of equations which are obtained at specific moments (i.e. for specific values of m). For example: What is the equation when m=10? When is the equation not quadratic and what does it look like in this case? etc. Students will see that only m determines equations, and that substituting for x is an irrelevant procedure unless one has a specific equation. (As we have noted before, the task of finding the value of the parameter for a given value of the unknown is an exception to this dynamic.) A computer could help make this concrete model even more concrete. It should not be too hard to write software which will simulate the digital display described here. Such a concrete dynamic model may help create the correct dynamic mode of thinking which cannot be created by telling students the corresponding mathematical definitions. THE NOTION OF DUMMY VARIABLE
m,"( /K"B(20 "*/ '4990 ."*%"32/+x The distinction between dummy variables and non-dummy iii variables is not related to the distinctions discussed in the previous sections. All uses of letters discussed there may appear as either dummy or non-dummy. For a letter, being a dummy variable, is not contrasted with its being an unknown, a generalised number, a variable, or a parameter. (In fact, according to the terminology of the previous sections, we should have used the term 'dummy letters' here, but we shall stick to the common term 'dummy variables'). As with the notion of parameter, the logic of dummy variables must be defined explicitly before analysing students' understanding of this notion. But first, let us look at some examples where x occurs as a dummy variable:
The difference between a dummy variable and a non-dummy variable may be defined as follows: A form containing a particular variable, say x, as a non-dummy variable, has values for various values of the variable; but a constant or a form which contains x as a dummy variable, has a meaning which is independent of x not in the sense of having the same value for every value of x, but in the sense that the assignment of particular values to x is not a relevant procedure. (The definition was adapted form Church (1956, pp. 40) by replacing the words free and bound with non-dummy and dummy respectively.) —
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Let us now look at the examples again. Notice that variables always become dummy after applying some operators (e.g. set notation, integral, limit, quantifiers) to forms where these variables were non-dummy. In (a) x becomes dummy by applying set notation to the form x>0 where x is non-dummy. While substituting for x determines the (truth) value of x>0. substitutions like {3:3>0} are an irrelevant procedure in determining the value of {x:x>0}. The integral (b) is a form whose value depends on the value of y only, not of x. The integral (c) and the limit (d) are constants despite the fact that they apparently contain a variable. Their constant value does not depend on values of x. It follows from the definition that the meaning of a constant or a form containing a dummy variable is not affected by replacing that variable by another letter: For with dummy x versus x>0 & x<5 with example, let us look at does non-dummy x. Replacing the dummy x in the first set, e.g. not affect meaning, while doing the same with the non-dummy variable, e.g. y>0 & x<5, changes the meaning. (This should not be confused with the fact that in algebra all letters are chosen arbitrarily, and replacing a letter never affects meaning, provided it is replaced in all of its occurrences.) Quantifiers (the universal quantifier ‘for all’ and the existential quantifier ‘there the quantified letters being dummy variables. Thus, exists’) are another example for the form ‘for all a and b, a+b=b+a’, substitutions like ‘for all 5 and 3, 5+3=3+5’ are an irrelevant procedure in determining its truth value. In the statement: ‘Solving equations of the types ‘x+3=5’ is easy for children’, x is used as a dummy variable. This means that substituting a number for x is irrelevant in determining the meaning of this statement. This is caused by the fact that in this context the value of ‘x+3=5’ is the equation as an object in itself (unlike the value of x+3=5 appearing without that context, which would be ‘true’ or ‘false’ depending on the value substituted for x). Similarly, when using a two variable equation to refer to a function as to an object in itself. the letters are dummy variables. e.g. x and y in the statement ‘The graph of the function y=x ² is a parabola.’ It follows from the definition of dummy variable, that any meaning given to a dummy variable has a restricted scope and is not related to any meaning given to the same letter occurring outside that scope. Referring to two functions, like y=x ² and y=3x–5, is an example where the occurrence of x and y i n one function is outside the scope of the other function. Thus, the meaning of y in the first function (i.e. being equal to the square of x) is not related to the meaning of y in the second function. It is as if we just happened to be using x and y to refer to each function. when we could have used any other letter or describe the functions verbally. —
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5*/ '4990 ."*%"32/+@'%77%B42(@7)*@+(4'/#(+x
As we have seen, dummy variables appear in a variety of contexts. In some of these contexts they do not cause students any problem, and in others they do. When introduced to algebra students are taught that different occurrences of the same letter should have the same value (and meaning). As was mentioned before, this rule does not apply to dummy variables: a dummy variable occurrence is not related (in value or in meaning) to previous occurrences of the same letter. For example, the set is a perfectly good answer to the question: ‘What is the codomain of the function y=x²?’ even though the codomain relates () y. This is a usually difficult to understand, and students and even teachers will find more appropriate answer. I myself witnessed a classroom situation in which a to the above question, and the teacher corrected her student answered answer and wrote instead. Like set notation, the following expressions also contain x as a dummy variable: ‘the truth set of x =9’ or ‘the solution of 5x+7=8x–3’. Wagner (1981) found that many junior high school and high school students did not conserve equations under transformation of variable, i.e. under replacement of the letter. This means that they did not think that ‘the solution of 5x+7=8x–3’ and ‘the solution of 5y+7=8y–3’ were the same thing. In the context of integrals dummy variables usually do not cause a problem. Most students know that the value of is a constant and does not depend on values of x. Quantifiers, though a hard concept in itself (as we have seen with the parameter notion), do not cause problems with regard to their dummy variables. Students understand that when a+b=b+a is stated as a universal rule (with an implicit universal quantifier), its truth value is a constant, and does not depend on specific substitutions. Many studies have shown that the concept of function is hard because students have to grasp the function as an object in itself. (The understanding of functions as objects is required when dealing with operations on functions like sums, derivatives, or graphs.) An additional difficulty with functions as objects is the fact that the letters used in their notations are dummy variables. A teacher may start a discussion by saying: ‘N/( 0v7SKT "#' 0v$SKT 3/ $%./# 74#B(%)#+: ’ Many students may infer that these functions are equal to each other because both are equal to y. In order to study this phenomenon I asked students entering college the following question: ‘Let y=f(x) and y=g(x) be given functions. Is the following statement true or false? (explain): The sum of these functions is the function 2y=f(x)+g(x).’ 27% of the science students and 67% of the social studies students answered that the statement was true. In their explanations, most of them treated the given functions as if they
²
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were simultaneous equations, and simply added both equations. Many students also explained that both functions were equal because y was the same number. This shows that these students did not know that for each of these functions y was a temporary name, which held only within the scope of that function, and that using the same letter did not mean that the values o f y were the same. This is a property of dummy variables which follows from the definition, and is part of the logical distinction between non-dummy variables in simultaneous equations and dummy variables in functions as objects. L%'"B(%B */9"*=+ The definition of dummy variables given above may be too complex to be stated for high school students. Studies have shown that students thinking is usually not guided by definitions, but rather by examples and their properties (Vinner 1991). Therefore, instead of the definition, the properties which are related to the definition should be explained: As for function notation, students should be taught the exception to the rule they have learned before, namely, the rule that a variable has the same value wherever it occurs. Using x and y to denote a particular function as an object i s the exception to this rule: x and y are temporary names, and hold their meaning only as long as we are referring to that particular function; when talking about another function, y has another meaning. The old rule applies again when we solve simultaneous equations of the two functions in order to find their point of intersection. As for set notation, notations like {x:x>0} and {y:y>0} should be discussed together with other notations of the same set, stressing that a set depends on its members only, and not on its notation.
CONCLUSION We have discussed two distinctions between uses of letters: the distinction between parameter versus unknown or variable, and the distinction between dummy versus non-dummy variables. We have seen that parameters appear everywhere in mathematics. and may cause a lot of difficulty which is very important to overcome. The difficulty can be explained by the fact that the notion of parameter includes implicit but unavoidable (second order) quantifier structures which are even more complex then advanced notions like, definitions in calculus, which teachers try to avoid. It seems that the complexity is inherent in the parameter concept itself, and is not merely a language difficulty. These difficulties are related to understanding the concepts of equation and family of equations themselves. Therefore the complexity cannot be
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avoided by using other notations, and the difficulties must be overcome by finding better ways of teaching this concept. As for the distinction between dummy and non-dummy variables, it seems that the misconceptions involved with it are not as deep, and that even when they exist they are less of an obstacle for students. Still, it is important that teachers be aware of the distinction, so that its logic can be explained when necessary.
NOTES I
2
3
The term parameter in mathematics refers to two quite distinct uses of letters. In this chapter we deal with one of these uses only, a letter whose value determines members of a family of equations or functions. The other use of the term parameter, which will not be discussed here, refers to letters in parametric representations of curves (or surfaces), e.g. x=t+1, y=t², where the value of the parameter determines points on the curve (or surface). The higher rate of science students here may be due to their higher technical skill, which led them to specify the substitutions g"vW )* KvWe1 while the social studies students gave unspecified answers like: gm,/# "KvWe or just ‘k/+e1 which were not included in the category reported here. In mathematical logic (Church 1956) the term for dummy variables is ‘bound variables’ and the term for non-dummy variables is ‘free variables’.
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GIULIANA DETTORI¹ , R O S SE LL A GARUTI² AND ENRICA LEMUT³
FROM ARITHMETIC TO ALGEBRAIC THINKING BY USING A SPREADSHEET INTRODUCTION Over the last few years we have worked on the impact of using a spreadsheet for introducing algebra. We developed this research by means of an "ID*%)*% analysis, preceded by some experiences of classroom use and followed by an experimentation which motivated and tested this analysis (Dettori et al, 1993, 1994). Both classroom experiences and experimentation were carried out with 12-14 years old pupils (intermediate school). Experiences performed before the a-priori analysis were of an exploratory nature, whereas the experimentation, planned to test the analysis. was structured in the form of a didactic sequence, as described below. This experience led us to develop opinions about some of the questions focused on within the PME Working Group, regarding in particular: the distinctive characteristics of algebraic reasoning; the algebraic notions that pupils can develop when working in a spreadsheet environment; the importance of incorporating methods of problem solving as a main component when analysing the nature of algebra. We start by sketching a characterisation of algebra for the intermediate school (12-14 years old), then analyse if the identified features can be suitably introduced by using a spreadsheet. In comparison with other papers in the literature on applying the spreadsheet to teaching a mathematical topic (Capponi & Balacheff, 1989; Malara et al, 1992; Rodgers & Thomlinson, 1993; Sutherland & Rojano, 1993). we shift the focus from the potentialities of the software to the main characteristics of the subject to be taught, since we are convinced that, from an educational point of view, what matters is not to learn to find a numerical solution to algebraic problems but rather to understand the nature and the power of the theoretical problem solving method of algebra. THE DISTINCTIVE CHARACTERISTICS OF ALGEBRAIC REASONING Learning algebra, or any other mathematical topic, means understanding and being able to apply its main concepts and formal tools. This entails defining what is algebra and what is relevant to teach and learn in intermediate school, when it is first approached by students. This step is made harder by the fact that there are various 1,2and3
Istituto per la Mathematica Applicata, C.N.R., Genova, Italy.
191 M: ;4(,/*2"#' /( "2: S/'+:T1?/*+D/B(%./+ )# ;B,))2 52$/3*"1 191–207. U VWW X<24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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views of what the content of algebra should be, and that different countries have slightly different school algebra traditions (for instance, see the Italian situation in the next paragraph, and the British situation in Healy et al’s chapter in this book). In order to base our analysis on a widely accepted and comprehensive definition, we examined both classical and modern views of algebra (Boero, 1989; Booth, 1984, 1989; Chevallard, 1989; Cortes et al, 1990; Kieran, 1989; Usiskin, 1988), trying to extract a common base of concepts and methods that can be considered as the core of algebra. This review led us to sketch out the following topics that should be tackled in intermediate school: 1) understanding the meaning of function (in one variable); 2) understanding the meaning of relation, in their various shades (equations, inequalities, formulas, identities, properties); 3) Understanding the differences between variables, unknowns and parameters; 4) learning to apply algebra, that is, to model problems or classes of problems by means of equations and inequalities; 5 ) learning to manipulate algebraic relations according to the rules of literal calculus; 6) learning to apply the algebraic calculus to the demonstration of simple theorems. This means that students should not be limited to performing algebraic activities, but, by reflecting on them, they should acquire the algebraic language as a working and expressing tool (Bell & Malone, 1993). This implies that learning algebra after arithmetic means developing a different way of thinking. Arithmetic is a prerequisite for algebra, in that algebraic manipulations are based on the four arithmetic operations and preserve their meaning. Both of them are suitable to solve a common corpus of problems, but their solving approaches are different, so that algebra can not be considered an extension of arithmetic. The break between arithmetic and algebra (Cortes et al, 1990) consists of starting a modelling process that B,"#$/+ (,/ #"(4*/ )7 D*)32/9 */+)24(%)# (from performing step by step computations to defining relations and transforming them by means of formal manipulations) "+ >/22 "+ (,/ #"(4*/ )7 (,/ 9"(,/9"(%B"2 )3O/B(+ 3/%#$@ ,"#'2/'@(not only numbers but also variables, unkowns, parameters). Unlike arithmetic, the main aim of algebra is not to perform numerical computations, but rather to provide an operative language to represent, analyse and manipulate relations containing both numbers and letters (Capponi & Balacheff, 1989). Since it is based on defining and manipulating relations, the style of algebra is essentially declarative, while that of arithmetic is essentially procedural (Chiappini & Lemut, 1991). From the above characterisation, it appears that the problem solving approach of algebra is also quite different from the solving approach of spreadsheets. Spreadsheets are tables with a labelling system for rows and columns. Their most salient characteristic is their facility to express formulas making reference to cell's addresses and using them to fill a table based on assigned or previously computed
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values, Other nice features include the possibility of inserting, deleting and moving rows, columns or sets of cells, hence adding a dynamic aspect to the produced tables. However, since spreadsheets are actually tables, problem solutions based on them are essentially numerical computations performed according to a trial-anderror strategy, rather than formal manipulations of relations as in the case of algebraic solutions. The spreadsheet’s approach is not fully algebraic, but only functional. SCHOOL ALGEBRA IN THE ITALIAN INTERMEDIATE SCHOOL The most recent official Italian programme for the intermediate school (1979) does not consider algebra as a specific topic to be taught. Algebraic elements are introduced within the topic ‘Problems and equations’, where it is recommended that ‘... the literal calculus without concrete references should not be the prevalent aspect in teaching and particularly not in assessment’. In practice, in many schools, the word ‘algebra’ is widely used both by text books, teachers and students to indicate mainly the introduction of variables in the context of functions, the resolution of equations and the first elements of the manipulation of symbols. This constitutes more than half of the classroom activities in the last year of intermediate school. A limited number of teachers put great emphasis on an early introduction of variables and unknowns, followed by the manipulation of symbolic expressions, applied to the proof of simple theorems on integer numbers. such as the following ones: ‘prove that the sum of two odd numbers is even’; ‘prove that the sum of two consecutive odd numbers is a multiple of 4’.
THE IMPORTANCE OF INCORPORATNG METHODS OF PROBLEM SOLVING AS A MAIN COMPONENT WHEN ANALYSING THE NATURE OF ALGEBRA Algebra shows best its power when applied to problem solving, especially at the school level which we are considering, where students are still too young to appreciate a mathematical formalism, just for its elegance and conceptual power. Classroom experience has shown that introducing basic concepts of algebra, such as those of variables or relations, outside of a problem solving context, would just make them meaningless for most students, and would later increase the learning effort when trying to make concrete these abstract concepts by applying them to the solution of problems (Arzarello et al, 1993). For this reason, we think that a good way to introduce algebra, either in a pen-and-paper environment or in a computerised one, is to introduce the basic concepts in a gradual way by solving problems of increasing complexity, starting with simple problems, which can be solved by one equation in one unknown, where the algebraic solution is similar to
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the arithmetic one, then passing to more complex ones, requiring the solution of more equations or inequalities in more unknowns, where it is easier to provoke students to remark on and appreciate the differences of various problem solving approaches. In this chapter we discuss the solution, of the following problems that we have found effective, in classroom experiences with 12-14 years old students. The problems are part of a didactic sequence to introduce or reinforce algebraic competencies both in a pen-and-paper and in a spreadsheet environment. Problem 1 (‘Book Problem’)
We want to distribute 100 books among three people so that the second person receives four times as many books as thefirst person, and the third person gets ten more than the second person. How many books does the first person receive?
This is a classical partition problem. Though problems of this kind are often used in the introductory phase of algebra, they are certainly not the most meaningful ones, since they can be easily and directly solved by using arithmetic, thus partially failing to motivate the introduction of algebra. An arithmetic solution can be developed according to the following reasoning: ‘I should subtract from the 100 books the 10 which are given extra to the third person; of what remains, 1 part is given to the first person, 4 parts to the second one and 4 to the third one. This makes 9 parts. I should divide 90 by 9. Hence, the first person receives 10 books’. This problem can be solved algebraically by writing and solving the equation X+(4X)+(4X+10)=100, where X is the number of books given to the first person. This equation describes the problem, and, at the same time, its solution in implicit form. By transforming it by means of the rules of algebra, we obtain X=10, which represents the solution explicitly. It is interesting to note that the operations made while manipulating this equation correspond to the steps of the arithmetic reasoning: moving 10 from the first to the second term of the equation actually corresponds to ‘I should subtract from the 100 books the 10 which are ...’; adding X+4X+4X=9X corresponds to ‘... 1 part goes to the first person, 4 parts to the second and 4 to the third one. It makes 9 parts’; finally dividing 90 by 9 is the same in the two solutions. This means that algebraic manipulations do not necessarily require, as students and teachers sometimes erroneously think, temporarily disregarding the meaning of the equations and blindly applying formal rules. It also means that what differentiates algebraic from arithmetic solutions is not the kind or the order of the performed operations, but rather the general organisation of the solution, since
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arithmetic proceeds step by step, while algebra focuses in every moment on a global, synthetic view of the problem. The simplicity of this problem makes it suitable as a starting point, to provoke pupils to compare different solution approaches. As we will see in the next paragraph, when solved in a spreadsheet environment, it eases the introduction of the concepts of formula and variable, which constitute the building blocks of algebraic equations a n d inequalities. Problem 2 (‘Theatre Problem’)
The theatre of a country town has 100 seats, divided into front section and rear section. The price of front seats is $8, that of rear seats is $6. When all seats are sold, the total income is $650. How many front seats and rear seats are there in the theatre?
This problem is, in a sense, more algebraic than the previous one, in that its algebraic solution is simpler and more direct than the arithmetic one. A possible arithmetic solution is as follows: ‘let us suppose that there are only rear seats; then all the 100 seats would cost $6 each, and the total income when the theatre is full would be 6*100=$600; but we know that the real income is $650: the difference $650-600=50 is due to the extra $2 paid for the front seats; since 50:2=25, there are 25 front seats and 100-25=75 rear seats’. Using algebra, the problem gives rise to a system of two equations {X+Y=100, 8X+6Y=650}, where X is the number of front seats and Y is the number of rear seats. The same considerations made for the previous problem can be applied to the solution of this one. Also in this case, the steps of the arithmetic solution correspond to the computations made in one of the possible reduction sequences of the algebraic system. However, the conceptualisation of the two solutions is different, since using algebra we simply convert the text of the problem into the two equations above, while the arithmetic solution passes through hypothesising a situation not present in the text. For this reason we said that this problem is more algebraic than the previous one. This problem allows the introduction of the concept of system of equations, and the distinction between variables and parameters (see Generalisation of problem on page 203).
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Problem 3 (‘False Generalisation Problem’)
The theatre of a country town has 100 seats, divided into front, middle and rear seats. The front seats cost $10, middle ones cost $7 and rear ones cost $5. When all seats are sold, the income is $700. Q)> many front, middle, and rear seats are there in the theatre?
This problem represents a false generalisation of the Theatre Problem, since its structure is apparently similar to that of the previous one, but actually it is different, in that it involves the same number of equations and one more unkown. Its main feature is that it has more than one meaningful solution, which is rather hard to find by using an arithmetic approach. In order to solve this problem with arithmetic, one could try to extend the solution of the previous problem, and suppose, for instance, that all seats are at the price of the rear section. This gives a total income of $500; the extra $200 to the real income is due to the extra cost of front and middle seats. At this point, however, it is not possible to proceed without any additional data, hence it is necessary to add some other condition not included in the given text, in order to reach a solution. In fact, it is possible to find more than one solution by imposing different extra conditions. However, it would be very cumbersome, if possible, to try all conditions. On the other hand, the algebraic approach, besides pointing out from the beginning that the given data are not sufficient to determine one result (since the problem gives rise to a system of two equations in three unknowns: {X+Y+Z=100, 10x+7Y+5Z=700}, where X,Y, Z are the number of front, middle and rear seats respectively), provides the tools to characterise the set of all possible solutions, hence leading in a short time to a better understanding of the problem. The above considerations suggest that this problem is yet more algebraic than the previous ones, in that only the algebraic approach allows an easy and complete solution, thus emphasising the descriptive and solving power of algebra. Moreover, this problem provokes students to face the fact that neither arithmetic nor empirical solution methods (like trial and error on a spreadsheet) can give the guarantee of the existence and unicity of solutions, hence justifying the need to learn to make simple formal proofs (see Proving results on page 202).
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Problem 4 (‘Needle Problem’)
An industry that produces needles, has $300.000 of fixed annual costs, not depending on the number of needlesproduced. The cost of materials for a single needle %s 2 cents. Each needle is sold for 17 cents. How many needles should be produced and sold in a year in order to have a positive balance?
Using arithmetic. we can find the number of needles for which the expenses equal the income, as: 17-2=15 cents; $300,000 15 c = 2,000,000 (number of needles that should be produced and sold in order to make the balance positive). Using algebra, this is a typical problem leading to the solution of an inequality involving two functions: 17 X 300.000 +2X, whereX is the number of needles. Also in this case, the same remarks made for the previous problems are valid. This problem is suitable for introducing inequalities and the concept of functions (see Introductory study of functions on page 20 1) Problem 5 (‘Fish problem’) We went fishing and caught a big fish; its tail weighs 4 Kilos, its body weights the same as the head p l u s the tail, and the head weighs one half of the body plus the tail. What is the weight of the whole fish? The arithmetic solution of this problem is rather difficult, since it requires subtle reasoning and great ability in manipulating the verbal language. A possible solution can be as follows: From the description of the mutual dependence of the weight of head and body, we can argue that the head’s weight is equal to ‘half head plus one half of 4 plus 4’. This means that the head is equal to ‘half head plus 6’, that is, its weight is 12; the body is 12+4=16; the whole fish is 32 kilos. Also when using algebra, this problem is definitely more difficult than the previous ones, since its solution requires a more complex reasoning because of the circular references present in the text. Indeed, the text gives rise to a system of four equations {T=4, B=H+T, H=B/2+T, W=H+B+T}, where T is the tail, B is the body, H is the head, W is the total weight. The system contains a circular reference, since B and T are defined in terms of each other. Its solution requires that students improve their ability with formal manipulation. It makes them appreciate the power of algebra in comparison with both arithmetic and the spreadsheet, since algebra allows the solution of circular references among variables without much effort. Moreover, once the system of algebraic equations is formalised, this gives the certitude that one solution exists and can be found by formal manipulation, that is,
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without much more mental effort. On the other hand, neither arithmetic nor the spreadsheet can give assurance of the existence of this solution, until the solution is reached, nor of its unicity (see Proving results page 202). ALGEBRAIC NOTIONS THAT PUPILS MAY DEVELOP WORKING IN A SPREADSHEET ENVIRONMENT
Using relations and synthesising equations Recognising which elements are involved in a problem and expressing part of the relations among them is the first fundamental step to perform when learning to apply algebra to problem solving. This phase can be smoothed, though not completely carried out, by using a spreadsheet. A first experience in this sense can be made, for instance, by solving the ‘Book Problem’ (see Figure 1). Column A contains the values that the child decides to try as possible solutions; columns B and C contain the formulas to compute the corresponding number of books given to the second and to the third person respectively; column D, computing the sum of the three previous values, allows the child to verify if the supposed value is actually the solution of the problem.
Figure I
A numerical result can be found by inserting different values in column A. either by copying down the first line several times or by successively substituting different values in cell A2, until the value in the corresponding cell in column D is 100. Columns B and C contain parts of the relations that model the problem; however, the equation X+(4X)+(4X+ 10)=100 is never explicitly expressed. The student solves it de facto by checking. by himself, the value of the sum in the last column, but he may never become aware of the equation since spreadsheets do not allow the handling of equations. However, learning algebra means learning to synthesise the
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partial relations of the spreadsheet table into one or more equations describing the given problem. At this point, a teacher’s intervention is necessary to guide the student towards developing this ability to synthesise. At the same time, the possibility of finding a solution simply by numerical attempts (which are easy to perform with the computer) can discourage the student from making an intellectual effort towards synthesis, hence missing a fundamental tool for the development of mature cognitive abilities. These observations lead us to point out a first major limitation of the spreadsheet for teaching/learning algebra: the sign of equality used in spreadsheets is actually the assignment of a computed value to a cell, while in algebra it states a relation. The inability to write relations in a spreadsheet means that it is not possible to use it to completely handle algebraic models.
Introducing variables and unknowns Variables and unknowns, strictly speaking, are absent from spreadsheet solutions. However, these two concepts can be introduced by reasoning on the spreadsheet operations. Based on our experience, students do not grasp immediately what is a variable or an unknown. In fact, we found that sometimes they feel unable to make reference in a formula to a cell which is still empty; or they feel that it is useless to refer, in a sum, to a cell containing zero, or, in a product, to a cell containing the number one. As in the case of synthesis, abstraction capabilities will hardly be achieved by students with the help of only a spreadsheet. In fact, if a student builds a table as in Figure1 (only one row). and tries several different values in cell A2, the teacher can guide him to the abstraction of the concept of unknown, letting him remark that he is looking for a value such that we obtain 100 in cell D2. On the other hand. if he builds a table as in Figure 2, the teacher can guide him to the abstraction of the concepts of variable and function, letting him remark the similarity of all rows. Moreover, becoming familiar with some other symbolic system, besides the verbal and arithmetic ones, can anyway play a role of mediator of mathematical thinking (Rojano & Sutherland, 1993), although this does not mean that the students will be able to use equally well any of them as descriptive tools (MacGregor & Stacey, 1993).
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Figure V
Finally, there is a point that deserves attention: letters inside formulas change their meaning when some formulas are composed into an equation, passing from generic values which vary in a given interval (variable) to well determined, though unknown, ones (unknown). Introductory study of functions
As pointed out in the above discussion, spreadsheets are function oriented, in that formulas involve the concepts of variable and function definition. Moreover, an interesting aspect of spreadsheet tables, as appears from our solution of the Theatre Problem, is that they are potentially dynamic, that is, columns and rows can be inserted, moved and deleted much more easily than in paper-and-pencil tables, hence making it easy to seek a more accurate solution of a problem (for instance, in Figure 3a the content of row 12 has been moved to its proper place in Figure 3b without rebuilding the table). The study of functions, besides being important in itself, is relevant also for the study of equations and inequalities, since these can be viewed as comparisons between two functions. In fact, the easiness of building and modifying tables, together with the graphic facilities to draw a diagram based on the tabulated values, makes the spreadsheet a suitable tool for formulating and testing conjectures about the function’s trends, as is the case in the Needle Problem. In this problem, students have to write two functions to describe expenses and income (see Figure 4), and to compare the numbers in the two correspondent columns, line by line, until the mutual relation bigger-smaller is inverted.
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Figure 5
Proving results The idea of proving the existence of solutions before completely solving a problem, or proving that all possible solutions have been found, can obviously be introduced with any problem, but it makes sense only with problems which are not completely straightforward. For instance, solving the Theatre Problem, if a spreadsheet solution is designed without implicitly solving the first equations with respect to one variable, we obtain a table which is clearly too cumbersome; however, bright students almost immediately notice that for each value of X it is necessary to try only one value of Y. However, without making a formal proof, it is not clear, before finding the result, that there is certainly one value, and, once the problem is solved. it is difficult to argue that the only solution has been found, while using algebra gives this certitude. When solving the False Generalisation Problem using a spreadsheet, it is evident that it is more difficult to choose suitable values by trials, but it is not evident where the difficulty comes from. It is clear that one of the three values is determined by the other two in order to make their sum 100, but this does not help to find suitable values to try in order to quickly reach a solution, and certainly does not make it clear that in this case there are many possible solutions (see Figure 5). An algebraic solution, on the contrary, shows that the two problem are structurally different. since in this case we have three unknowns and only two equations, hence the problem has an infinite number of solutions in R, all characterised by the relations X, and X, which allows us to choose a limited number of reasonable 2
2
solutions in N. This is certainly not easy to see if the problem is tackled only by spreadsheet, without any algebraic consideration.
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Reasoning on solution ranges Solving a problem by trial and error 1 , at first sight may appear a brute force approach. However, this can be a nice starting point to develop abilities to make ‘focused trials’. By this term, we mean a strategy aimed at pruning away all trials that are likely not to lead to a meaningful solution, based on constraints on the values the solution can take, depending on the problem context. Though not strictly related to algebraic competencies, learning to make focused trials is a nice side effect of using a spreadsheet for teaching the solving of algebraic problems. However, we wish to report here the observations about this which we made during our last experimentation, since we find them of some interest. When the teacher introduced the concept of focused trials, to our surprise students applied them only for the simplest problem (‘Book Problem’), but tried all numbers in a dense sequence when the presented problem was even slightly more complicated (from the ‘Theatre Problem’ up). This can not be attributed to the laziness induced by the use of the computer, since the students did this also when they were asked to simulate a spreadsheet with a pen-and-paper table so that they had to perform all computations by hand only with the help of a calculator. We attribute this behaviour to the fact that students seem to consider a table complete only if the independent variable is increased by regular values. Moreover, in more difficult problems, where children were not able to have a global view of the situation, they preferred to proceed step-by-step. Difficulties in implementing a focused trial strategy seems to suggest that this procedure requires the teacher not only to lead her pupils to a general view of the situation described in a problem, but also to reason about the meaning of the variation. For instance, in the Theatre problem, it is not meaningful to change the number of seats in a section by one, especially if the corresponding total income is still far from the required value. Generalisation of problems by means of parameters The distinction between parameters and variables can be introduced by generalising a problem, that is, by introducing the concept of a class of problems having the same structure but different numerical values. For instance, with the Theatre problem, let us suppose that we wish to answer the same question for a theatre of different size or with different ticket prices. It is clear that this new problem is basically the same as the original one and that also the structure of the solution will remain the same; only some numerical coefficients will be different. If we write these coefficients separately as in row 2 in Fig.6, it is possible to use their values in the formulas without explicitly writing them, hence obtaining a table that can be used to solve any problem with the same structure without any modification but for the new values inserted for the Coefficients. In the subsequent formulas, variables and parameters
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Figure 6
can be easily differentiated since the last ones must be referred to by using the absolute reference (containing two dollar signs), which leave unchanged the name of the referred cell throughout the copy-and-past operations. Thus the introduction of the absolute reference gives the teacher a tool to introduce an important conceptual distinction, which is not trivial to introduce in traditional environments. These coefficients play the role of parameters, which are, in a sense, second order variables, in that, given a class of problems (or functions), each value which they can take in a given range individuates a well determined problem (or function) in this class; in this phase, problems (or functions) are seen as atoms, and first order variables they contain do not play any role. Vice versa, once a problem in the class has been selected, only first order variables vary, while parameters behave just as constants. (For a deeper analysis of the difference between variables and parameters see Furinghetti & Paola (1994), Bloedy-Vinner ( 1994) and Bloedy-Vinner in this volume).
Solving circular references An interesting observation related to the above mentioned class experiments relates to the solution of the Fish Problem. The most straightforward implementation, based on a direct reading of the problem and then substitution in the written formulas, when run on the computer produced, as expected. the error message ‘can not solve circular references’. The pupils, however, did not give up, and produced another solution, where the values for ‘head’ were listed in two different columns (see
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Figure 7), namely ‘head’ (supposed values for the head) and ‘true head’(value for the head computed by means of the given formulas). The problem’s solution was found by comparing the values of the two ‘heads’. This smart trick can be adopted when solving algebraic problems involving circular references with a spreadsheet. In our opinion, this kind of solution, despite its empirical appearance, leads towards algebra, in that it requires the performance of a check on the coherence and consistency of the produced table.
Figure 7
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Learning the algebraic language From the above discussion of the proposed problem solving methods, it is clear that, when using a spreadsheet, formal manipulations are never performed. On the other hand, these are an essential part of learning the algebraic language. This includes both the ability to associate meanings with the symbols being used and the ability to manipulate symbols independently of their meaning. Only the first one of these abilities can be fostered using a spreadsheet. Another aspect of learning the algebraic language is the ability to read a relation in either direction. This is obviously impossible in spreadsheets, since the relations themselves are absent. If disregarded by the teacher, this risks reinforcing what is usually a difficulty of students at this school level. These considerations show that the spreadsheet can start the journey of learning algebra, but does not have the tools to complete it. Being able to write down parts of the relations among the considered objects, but not to synthesise and manipulate the complete relations, is like knowing words and phrases of a language, but being unable to compose them into complete sentences. JOINT INTRODUCTION OF ALGEBRA AND SPREADSHEET The didactic sequence described throughout this chapter has been implemented during an experimentation with 13-14 years old students in intermediate school. At the beginning of the experimentation, pupils had experience neither of algebra (in particular of equations) nor of spreadsheets. These two environments were introduced at the same time and were related to each other. Every problem was tackled in three different ways: spontaneous solutions suggested by the students, algebra and spreadsheet solutions. The different solutions were then compared, pointing out differences and analogies, and emphasising the algebraic aspects that could be detected in the spreadsheet solutions. For instance, for the Book Problem, the solution by trial and error produced by the students was close to that worked out by spreadsheet, but rather far from the algebraic one. The discrepancy between different solutions was still more evident in the Fish Problem, whose algebraic treatment is certainly not spontaneous for a beginner. The result of this activity was that the basic elements of algebraic problem solving were grasped by the class in less time and with less effort than we usually encountered in our previous teaching experience. CONCLUDING REMARKS The above discussion shows that the spreadsheet can be useful to introduce algebra, with some limitations: spreadsheets deal only with numbers, or addresses of cells containing numbers, and functions; variables, unknowns and relations can not be directly handled and formally manipulated, only assignments are made.
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However, using a spreadsheet, which by itself would lead students to solve problems by trial and error, under the attentive guidance of a teacher can lead them: 1) to become aware of activating a new modelling process for problem resolution; 2) to understand what solving an equation means, even before being able to handle equations; 3) to reason about the domain and constraints of a problem in order to decrease the number of trials necessary to reach a solution; 4) to introduce generalisation, abstraction and synthesis, which are fundamental cognitive abilities in mathematics (Tall 1991, p5) to become aware of the nature of algebra by comparing, as a metacognitive activity, different problem solving methodologies, such as arithmetic, algebra and spreadsheet. From a methodological point of view, we can question if it is still necessary to learn formal resolution methods such as algebra, since the more and more diffused application of computer-based tools allows us to find numerical solutions by other means. Our belief is that understanding theoretical resolution methods leads to a deeper comprehension of problems themselves and to a more precise awareness of what can be solved and what can not be solved. In fact, theoretical methods are the only ones that can prove properties such as correctness, completeness or existence of results. ACKNOWLEDGEMENTS Problem 6 was proposed during a lecture by F. Arzarello.
NOTES 1
For more discussion of this strategy see the chapter by Filloy, Rojano, and Rubio (eds).
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SONIA URSINI*
GENERAL METHODS: A WAY OF ENTERING THE WORLD OF ALGEBRA INTRODUCTION A large number of studies have highlighted the difficulties which pupils with an arithmetic background have when starting the study of elementary algebra. Focusing on some of the aspects characterising these two disciplines it has been emphasised that in order to work in algebra pupils need to develop a different understanding of the symbols that are used both in arithmetic and in algebra such as, for example, the equal sign and the operation signs (Kieran, 1988); as well as coping with completely new notions such as, for example, the idea of variable. Pupils’ tendency to use arithmetic methods for solving algebraic problems and their need to look for numeric results instead of accepting an algebraic expression as a result, is also highlighted (Booth, 1984). Research reports show that independently of which aspect of algebra is emphasised, namely, equation solving (e.g. Kieran, 1984, 1988; Filloy and Rojano, 1985, 1989; Herscovics and Linchevski, 199 1), generalisation and general expressions (Küchemann, 1980; Booth, 1984; Mason et.al, 1985; Lee and Wheeler, 1987; Ursini, 1990a), or the study of relationships (Dreyfus and Eisenberg, 1981; Markovits et al., 1986, 1988: Heid and Kunkle, 1988), beginners have difficulties in coping with algebra. Finally, it has also been suggested that pupils’ arithmetic background is an obstacle for approaching algebra (Filloy and Rojano, 1985; Kieran, 1988). A review of some texts (Bunt et al., 1976; Gillings, 1982; Klein, 1968; Neugebauer, 1969) discussing the historical developtnent of mathematics leads me to stress once more the differences between arithmetic and algebra. In particular, I want to underline one basic difference: while arithmetic deals with numbers and an important aspects of it is to perform computations obtaining numerical results, algebra deals with general magnitudes and it arises historically from the interest in deducing general methods for solving sets of similar problems. I consider that this difference deserves special attention when our concern is to help pupils with an arithmetic background approach elementary algebra. Vieta (1540-1603), who is considered by many historians to be the founder of modern algebra, focused his work on the development of a method, the ‘Analytic * Departamento de Matematica Educativa, CINVESTAV, Mexico.
209 R . Sutherland et al. (eds.),Perspectives on School Algebra, 209–229. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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Art’, which was intended to be an instrument for the solution of problems in general. The focus on deducing a method for solving problems, however, does not emerge for the first time with Vieta. We find this focus throughout the historical development of mathematics. From ancient mathematics texts, for example, there is evidence of the search for methods for solving problems involving unknown quantities and for making astronomical predictions. In Babylon and Egypt similar problems were grouped together and one of them was solved in detail as an example of how to solve the others (Neugebauer, 1969: Gillings, 1982). That is, the detailed numeric solution of one particular problem was intended to illustrate the general procedure to solve classes of problems (Bunt et al., 1976). Vieta’s work itself was strongly influenced by the method presented by Pappus in relation to geometric theorems and problems, and by the procedures of Diophantus’ Arithmetic. The study of these ancient texts led him to conjecture ‘... that a generalised procedure which is not confined to figures and numbers lies not only behind the geometric analysis of the ancient but also ... behind the Diophantine Arithmetic ...’ (Klein. 1968, p. 158). Other mathematicians and historians of mathematics have stressed that Greek mathematics did not lack general methods but appropriate means for expressing them (Mahoney, 1968). However, Vieta’s approach to general methods marks a turning point. The great innovation and crucial difference with respect to the ancient ‘general methods’ for solving problems is Vieta’s conception of the object handled by the general method. In ancient times methods were always applied to numbers, magnitudes, points. periods of time, etc. There was no conception of a ‘general object’ on which the method was applied. Vieta conceived this general object, calling it ‘species’, he proposed a symbolisin to handle it and treated it as an object he could operate on by well defined rules. This ‘general object’, while preserving its link with numbers, represented general magnitudes and became the object of a general mathematics discipline, algebra, not identifiable with an extension of arithmetic or geometry. This ‘general object’ represented a potential value. however, it was treated as an actual mathematics object capable of being manipulated and operated on (Klein, 1968). These considerations are intended to emphasise that the source of algebra was preceded by centuries of experience in working within the realm of numbers and in looking for general methods for solving problems of different mathematical nature (e.g. arithmetic problems, geometric problems. problems dealing with functional relationships). The methods were expressed by means of numbers, by particular examples and, Iater on, by using rhetoric and/or syncopated algebra. This tremendous experience constituted the mathematical basis for the appearance of algebra, its gradual development and acceptance. To conceive of the idea of ‘general object’ and to symbolise it using literals was the culmination of a long historical process which marked the starting point of the gradual development of symbolic
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algebra. Although Vieta’s work represents a turning point, it is worth emphasising that his symbolic system was not so concise and the manipulative techniques were not so fluid as ours. Vieta’s description of the ‘Analytic Art’, as well as of the operative rules to be followed when dealing with ‘species’, still approximate rhetoric or syncopated algebra as is suggested by Winfree Smith’s translation of the ‘Introduction to The Analytic Art’ (Klein, 1968). The fundamental rules of equations and proportions that appear in Chapter II of the Introduction, for example, are given in rhetoric form: 1. The whole is equal to its parts. 2. Things which are equal to the same thing are equal to each other. (Klein, 1968, p. 322). Although Vieta represented general magnitudes by literal symbols, he often verbally described the way symbols had to be operated on instead of performing the operations. So in Chapter IV, Precept IV, of the Introduction we read ‘... And thus, in the case of additions, let it be required to add Z to A plane/B. The sum will be (A plane)+(Zin B) B [i.e., a /b+z=(a +zb)/b]’, (Klein, 1968, p. 338). That is, the algebraic notation we use today for handling general magnitudes is also a consequence of a gradual evolution. When starting the study of elementary algebra pupils’ school mathematical experience is very often limited to learning arithmetic algorithms and geometric formulae, and to solving problems in which obtaining a numeric result is emphasised. There is often little work at school helping pupils to focus on the way in which an arithmetic or a geometric problem is solved. That is, there is little work helping pupils reflect on the methods and on the feasibility of generalising them to sets of similar problems. Moreover, there is little work to help pupils express general methods prior to the introduction of algebraic symbolism. A plausible explanation of the difficulties which pupils have in accepting and understanding elementary algebra might be an insufficient experience in focusing on methods and in expressing them. It is not my claim that in order to introduce pupils to algebra its historical development has to be followed. In fact, there are distinctive characteristics between the historical and the individual development of concepts. For instance, in the course of history completely new concepts are constructed and elaborated. On the contrary, at school pupils are expected to develop an understanding of already elaborated concepts. However, the historical source of algebra may suggest ways of approaching teaching and learning of algebra, that might facilitate pupils’
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understanding and acceptance of this discipline. For example, there is historical evidence that the interest in deducing methods for solving sets of similar problems evolved over centuries, and that this was a basis for the development of algebra. This suggests that to lead pupils to become aware of the methods used for solving sets of similar problems prior to the study of algebra, might help them approach some fundamental ideas underlying this discipline as, for example, the idea of general method and of the general object of a method. Initially a method can be expressed verbally or by giving a numeric example as a means of illustrating it, or by means of a symbolism that is not necessarily the same as algebraic symbolism, This kind of experience might provide a basis upon which, later on, our algebraic notation might be introduced and algebraic notions might be understood and accepted. CREATING PRE-ALGEBRAIC EXPERIENCE My conjecture is that prior to a formal introduction to algebra pupils can be helped to cope with situations which promote an approach to some fundamental algebraicideas. This kind of experience I shall call pre-algebraic experience. In the past years I have investigated the feasibility of helping pupils develop a pre-algebraic experience through Logo (Ursini, 1991. 1994). Pupils’ numeric background was used as support for approaching, for example. different uses of variable. In this chapter I discuss the results obtained when pupils worked in Logo with the ideas of general method and general number in the sense of Vieta, that is, the general object of a method. Throughout this discussion the term general number is used to refer to as the general object of a method. The word method is synonymous with the way in which sets of similar problems are solved. By focusing on the method I mean becoming awareof __ how __ sets of similar problems are solved as opposed to solving them. The subjects of the study were a group of 34 pupils aged 12-13 years old, attending the first year of a secondary school in Mexico City. They were middle class children coming from different primary schools without prior experience of formal elementary algebra. The study was developed during the pupils’ normal computer workshop. The researcher was the teacher of the group. Twelve out of the 34 pupils were case studied.
Why Logo? Educational research in the last fifteen years has devoted considerable efforts to creating and evaluating mathematical learning within Logo environments. In particular, it has been found that in Logo pupils have less difficulties in working
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with some algebraic ideas (e.g. Noss, 1985, 1986; Sutherland 1992; Hoyles and Noss, 1987; Hillel, 1992; Ursini, 199 1, 1994)). Working in specially designed Logo environments can help pupils understand that a variable represents a range of values (Noss, 1985). Constructing and using general Logo procedures can help pupils attach meaning to the symbols representing variables, to manipulate them and to see the result of doing this (Hoyles and Noss, 1987). When working in Logo pupils can accept, use and produce ‘unclosed’ algebraic expressions (Ursini, 1990b; Sutherland, 1991). Logo-based environments can support pupils’ work with different characterisations of variable, namely, general numbers. specific unknowns and variables in functional relationship (Ursini, 1994). Approaching general methods and general numbers through Logo A general Logo procedure is a formal way of expressing in Logo a general method for solving sets of similar problems. Symbolic variables are used to represent the general objects upon which the procedure is acting. Thus, the variables involved in a general Logo procedure may be viewed as representing general numbers in the sense of Vieta. Moreover, the interactive character of Logo suggests that in this environment a dialectic link may be established in a natural way favouring a gradual approach to the general, using the particular as support. Referring to pupils’ way of conceptualising procedures in Logo, Hillel (1992) points out that pupils tend to look at them as tools to be used or as a record of instructions that produce certain result. This suggests that they do not look at them as a way of expressing a method for solving problems. That is, although a Logo procedure is a way of expressing a method, being able to write it does not necessarily imply that a shift to focusing on the method has occurred. A fixed Logo procedure is often obtained from the record of instructions given in direct mode when solving a problem, without reflecting on how the problem was solved. Very often the idea of general Logo procedure is introduced by means of a fixed Logo procedure in which the elements that are expected to vary are substituted by a symbolic variable. This approach does not help focus on the method and its general object. It may rather foster pupils’ need to look for results and stress the idea of variable as place holder. However, a different approach may help pupils focus on the method and develop an understanding of variable as general number in the sense of Vieta, that is, the general object of a method, although, as will be discussed in this chapter, substantial teacher’s support and guidance are required.
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From looking for resuIts to focusing on the method The following activity was used to help pupils focus on the method. Pupils were invited to write three fixed Logo procedures to double a number, and to analyse these procedures in order to identify their similarities and differences. This activity requires the capability to regard Logo procedures as objects to be analysed and to be able to identify similarities, distinguishing them from differences. The development of these capabilities is not an easy process for the majority of pupils and it requires support. The main aim pupils usually have when writing a computer procedure is to solve a problem and obtain a result. Their attention is focused on the practical utility of the program. To consider the procedure as an object to be analysed it is necessary to shift the focus from this aspect to the characteristics that define the procedure itself. This implies shifting from situational thinking to abstract thinking. For analysing a procedure it is necessary to deconstruct it and internalise the construction process, that is to reconstruct the external activity internally highlighting each single step without loosing the whole coherence of the procedure. When working on this activity pupils spontaneously focused only on the arithmetic operations and on the results of applying them; they completely disregarded the other Logo commands and the common structure of the Logo procedures. In fact, they identified the operation used for doubling a number (addition or multiplication) as common to the three procedures, whilst the numbers involved and the results obtained were signalled as different. That is, pupils were able to regard arithmetic operations and the numbers involved as mathematical objects to be analysed but they were not able to shift to a level of abstraction in which procedures became objects to be analysed suggesting a tendency to look at Logo commands only from a practical point of view. In circumstances like this the teacher’s intervention to help pupils to focus on the structure of a procedure is crucial. The intervention was directed at analysing the three Logo fixed procedures row by row and command by command (deconstructing), stressing their similarities in order to help pupils become aware of the structural commonalties (internal reconstrcution). Subsequently through a guided group discussion, whose purpose was to make explicit the common structure of the three procedures, two common structures were extracted depending on the method used by pupils for doubling a number: one involving addition: the same number was added twice TO<procedure’s name> PRINT + END
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Figure 1. Procedures for doubling a number written by two pairs of pupils
the other involving multiplication: the chosen number was multiplied by 2 TO<procedure’s name> PRINT * 2 END
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Figure 3. Pupils_ general Logo procedures
In Figure 1 the fixed procedures analysed for identifying the two methods are exemplified. During the common discussion pupils spontaneously started using verbal expressions like ‘a number’ or ‘any number’ in order to signal the variable number involved in the operations. This suggests that focusing on the method leads to conceiving the general object involved in it. The verbal use of the general number might be considered a necessary background for introducing a symbolic representation of general numbers in a meaningful way. In fact, after this experience pupils accepted and were able to use symbolic variables that were introduced as a way of representing ‘a number’ or ‘any number’ upon which a method was acting. At the same time the syntax for writing a general Logo procedure to double a number was presented to pupils (Figure 2) emphasising that it is a medium for expressing a general method in Logo. An example of pupils’ production is shown in Figure 3. These procedures were written under the requirement to write general procedures _to add 10 to any number’,
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_to add 5 to the addition of any number with itself, ‘to divide any number by 10 and add 100 to it’. To conclude this section I want to stress that this way of introducing general Logo procedures helped pupils work with two fundamental algebraic ideas: general method and general number. In order to do this no prior formal algebra instruction was needed except for pupils’ arithmetic background and a capability for writing fixed Logo procedures. Fundamental, however, was the teacher’s guidance in helping pupils shift from a practical perception of Logo procedures to being able to look at them as objects that could be analysed. Moreover, Logo provided pupils with a language for expressing a general method and its object in a formal way, prior to the introduction of the algebraic language. The analysis of pupils_ work revealed the presence of different strategies for writing general Logo procedures and a spontaneous gradual evolution from one to the other was observed for all the pupils. An analysis of the pupils’ strategies highlights the reasoning processes pupils might undergo in order to deduce a general method. &.)24(%)# )7 D4D%2+ ’+(*"(/$%/+ Four different strategies were identified during pupils_ writing of general procedures (Figure 4). These were classified depending on the steps explicitly used by pupils when writing them and on the type of activity developed in each step.
;(*"(/$0 J[ S[ +(/D+T Following the approach used when they were introduced to general procedures, pupils started by verbalising several numeric examples. After this they wrote fixed procedures that executed the calculations just verbalised. Finally they wrote a general procedure.
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This strategy was used by some pairs when writing their first general procedure (m*%(/ " D*)B/'4*/ () B"2B42"(/ ,"27 )7 "#0 #493/*): After verbalising some numeric examples (e.g. g+%K '%.%'/' 30 (>) e1 g(/# '%.%'/' 30 (>) e1 g(>/#(0 '%.%'/' 30 (>)e) they expressed these in Logo, writing the corresponding fixed procedures. Subsequently, they wrote a general procedure in which they divided by 2 the symbol or word used to represent the variable number. Figure 5 shows the record of a general procedure written by a pair using this strategy. They did not record the fixed procedures on their work sheet. The use of this strategy suggests the need for both verbal numeric examples and their formal expression, by means of fixed Logo procedures, to support the search for a general method and its expression in Logo.
;(*"(/$0 JV3 SV +(/D+T After verbalising several numeric examples a general Logo procedure was written. In contrast with strategy N3, after the verbal numeric examples pupils did not write the corresponding fixed procedures. This strategy was used by some pupils when writing a general procedure ‘to add 10 to any number’. After some verbal numeric examples (e.g. ‘three plus ten’, ‘five plus ten’, ‘six plus ten’) a general procedure was written in which the number 10 was added to the symbol representing the variable number. Others used it ‘to calculate the half of any number’.
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The use of this strategy suggests that the step in which fixed procedures are explicitly written has been internalised. Pupils can analyse them at an internal level, extracting their common structure using the verbal examples as support for identifying the variable element. ;(*"(/$0 JV" SV +(/D+T
After verbalising one numeric examples a general Logo procedure was written. This contrasts with strategy N2b in which several numeric examples were verbalised. It was observed that some pupils used this strategy when writing a procedure ‘to subtract 10 from any number’. Others used it ‘to add 10 to any number’. In agreement with the viewpoint of Krutetskii (1976) about the development of the generalisation process this strategy may be considered more advanced than strategy N2b. In fact, it reflects an evolution from needing several particular examples to using only one particular example for devising a general method.
;(*"(/$0 JX SX +(/DT After reading loudly the given sentence pupils wrote a general Logo procedure. They did not verbalise numeric examples or write fixed procedures. Moreover, the reading of the given sentence was immediately followed by the writing of a general Logo procedure. The great majority of pupils were observed using this strategy for writing general procedures ‘to divide any number by 10 and add 100 to it’ and ‘to add 5 to the addition of any number with itself’, these being the last procedures which were required of them. It is worth observing that the complexity of the sentences (involving more than one operation and requiring the double appearance of the general number) was not an obstacle for using this strategy. The use of this strategy suggests a wholly internalised process of analysis, with the visible outcome being only the general procedure expressing in Logo a general method and a general number implicit in the given sentence. This suggests pupils_ capability for interpreting the given sentence as the description of a general method acting on an indeterminate number and shows their capability for expressing this in a formal language. Moreover, having internalised the analysis process seems to lead to a qualitative shift in the understanding of the idea of general number. ?4D%2+ +//9/' () ,"./ '/./2)D/' "# %'/" )7 $/#/*"2 #493/* "+ "# )3O/B( )# %(+ )># *%$,( "#' (,/0 >/*/ "32/ () >)*= '%*/B(20 >%(, %( >%(,)4( 9"=%#$ "#0 /KD2%B%( */7/*/#B/ () D"*(%B42"* #493/*+: The great majority of pupils did not use strategy N1 from the beginning. In general they started by writing general procedures using a more-then-one-step
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Strategy, namely, N3, N2b or N2a. But, a spontaneous tendency, that is, not promoted by the researcher, to evolve gradually to a strategy with less steps was noticed. There were pupils who started by using strategy N3 and after this, they used strategy N2b, N2a and finally N1 . There were others who started by strategy N3 and without using strategy N2b or N2a, they shifted to strategy N1. There were pupils who started by strategy N2b, then evolved to N2a and finally to N1; or who shifted from strategy N2b to N1. That is, different sequences of strategies were observed and not all the pupils went through using all of them. However, the general, spontaneous tendency was to evolve from using a more-then-one-step strategy to using strategy N1. Finally, all the pupils shifted successfully to strategy N1. Figure 6 shows the strategies used by the six case study pairs. From Figure 6 we see that for the great majority particular examples were initially crucial for helping them focus on the method, for identifying its general object and for expressing this formally through Logo. The spontaneous evolution to a one-step strategy suggests that to encourage pupils to rely on particular examples did not, in this environment, anchor them to the particular. The particular supported the formal expression of the general and it was used to validate the proposed general was established and this procedure. A dialectic connection particular helped pupils to break gradually the link with the particular, to make sense of the idea of general object of a method and to shift finally to work directly with it, without needing the explicit support of the particular any more. That is, in this environment pupils gradually developed an understanding of the idea of general number in the sense of Vieta, that is, as the general object of a method. This is illustrated by the record made by Valentin and Ernesto, a pair of case study pupils, of two general procedures they wrote when using strategy N I (Figure 7). This pair evolved from using strategy N3 to strategies N2b, N2a and finally they shifted to strategy N1.
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When individually interviewed these pupils explained that the sign used for the input represented any number and that any value could be substituted into it. They stressed that any sign, letter or word could be used but, to use signs was more amusing. It is worth noticing that in the first procedure the same sign is used for representing the double appearance of the same indeterminate value. This use of the symbol was common to all the pupils. This suggests that to work in this environment favoured the understanding that the same symbol represents the same indeterminate number. Moreover, experiences like this one might help avoid the development of the polisemia error’, namely: the belief that different values can be assigned to different appearances of the same symbol in an expression (Filloy and Rojano, 1989). When individually interviewed after this experience, the great majority of
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pupils did not present this error when asked. for example, for the possible values of X in the expression 5+X=X+5. The second procedure that appears in Figure 7, shows Valentin and Ernesto’s capability to operate on the variable. It has to be stressed that when writing this procedure they used strategy N1. This means that for operating on the variable they did not use the substitution strategy (Healy, Hoyles and Sutherland, 1990) suggesting that they were able to operate directly on the variable. This supports the conclusion that in this environment pupils could develop the idea of general number in the sense of Vieta, that is, as an actual mathematical object capable of being manipulated and operated on. Synthesising, the analysis of pupils ’ strategies suggests that in order to become aware of a general method and to construct the idea of general number it might be fundamental for the majority of pupils to rely initially on particular examples. But to work in an environment like Logo that facilitates the link might push them to break spontaneously the link with the particular and to shift to the general. It is just this possibility of going back to the particular that seems to allow pupils to break away from it.
&K(/#'%#$ (,/ B"D"3%2%(0 () 7)B4+ )# 9/(,)' "#' >)*= >%(, $/#/*"2 #493/*+ () (,/ (4*(2/ $*"D,%B+@/#.%*)#9/#(: The experience of focusing on the method and working with general numbers acquired in the Logo numeric setting awakened processes that helped pupils focus on the method and also work with general numbers in the turtle graphics environment. Repeating the approach used in the numeric environment it was suggested that pupils should first write three fixed procedures, analyse them in order to identify their similarities and differences, and use this analysis as support for writing general procedures in the turtle graphics environment. The following example analyses the way in which Etna, one case study pupil, wrote her first general procedure in the turtle graphics environment.
C,/ B"+/ "7 &(#" Working alone because her peer was absent Etna wrote three fixed Logo procedures for drawing three squares of different size. After this, by means of self directed speech out loud she analysed the three procedures making a first rough classification of what was invariant and what varied: gk/+1 "22 "*/ +b4"*/+ "#' >,"( %+ ."*0%#$ %+ (,/ +b4"*/e+ +%'/:@e After this she continued: g F >%22 >*%(/ */D/"( 7)4* "#' "7(/* (,%+ AL " #493/* "#' MC "#)(,/* #493/*e: While talking she was writing her procedure
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(Figure 8). Her comment shows her ability to look at fixed procedures as objects to be analysed and suggests that she was looking at them as representing a method for drawing squares in Logo. Moreover, her verbalisation g:::AL " #493/* "#' MC "#)(,/* #493/*... ’ and the way she symbolised ‘a number’ shows that during the process of deducing a general method for drawing squares Etna was not working with particular but with indeterminate numbers. This suggests that the experience acquired in the Logo-numeric environment created for Etna the capability to focus on the method for solving problems independently of the particular Logo setting (for example, drawing squares in the turtle graphics environment), and she could conceive the method acting on indeterminate numbers. Her capability for focusing on the method and for working directly with general numbers was confirmed when Etna was individually interviewed. Given a sketch of the letter F with particular dimensions (Figure 9) Etna was asked to write a general procedure for drawing the same shape in different sizes. After a verbal interchange with the researcher who misinterpreted her approach and tried to push her to use the given dimensions as support, Etna told: gA)*$/( %(1 F (,%#= %( %+ 3/((/* %# (,%+ >"0P F >%22 $) 7)*>"*' " #493/*Y (,"# F >%22 B)9/ ,"B= " ,"27 )7 (,"( #493/*Y F >%22 (4*# *%$,( ZWY (,/# AL " ,"27 )7 (,"( #493/* :::: e: Etna
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continued talking in this may and she was simultaneously writing her general procedure (Figure 9). Her approach to the problem clearly shows that she was not thinking with particular numbers but she was working directly with general numbers. She was expressing in Logo a general method for drawing letters F similar to the given one. The particular numbers helped her deduce the relation linking them and she could generalise this relationship. To write the general procedure she did not use a particular example as support by substituting a variable to a particular number but, she was working directly with a general number operating on it when necessary. This capability to shift to working directly with general numbers in the turtle environment was not, however, common to all the case study pupils. Although all of them were able to write general Logo procedures the great majority did this using the explicit support of a particular example expressed in Logo (a fixed procedure or the written record of the commands given in direct mode). However, spontaneous attempts to shift to a completely internalised analysis avoiding external supports
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were observed for all the case study pupils. Nonetheless the procedures obtained in this way usually failed and this pushed pupils to step back and to rely on explicit particular examples. During the working sessions as well as during the individual interview it was observed that the main obstacle to shifting to work directly with general numbers in this environment was the difficulty the great majority had in handling simultaneously several elements, namely, turtles turns, the invariant and the variable elements within a long chain of Logo commands. These results stress
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the crucial role of the environment chosen to support pupils’ approach to basic algebraic ideas.
&K(/#'%#$ (,/ 4#'/*+("#'%#$ )7 $/#/*"2 #493/*+ () D"D/* "#' D/#B%2 "2$/3*" ("+=+ The Logo-based activities were not designed with the aim of influencing pupils’ work with general numbers in paper and pencil algebra tasks. The purpose was to offer pupils a pre-algebraic experience that might be used later on as a basis for helping them understand the idea of general number in algebra. However, the Logobased activities in which pupils focused on methods and their general objects, influenced their capability for working with general numbers in more traditional paper and pencil algebra tasks. Evidence for this is provided by the difference between the number of correct answers given to a paper and pencil questionnaire (Ursini, 1994) before and after the Logo experience. After the Logo experience the great majority considered, for example, that in expressions like X+2=2+X, 4+S, X=X, the variable represents any value. The majority of pupils were able to operate on the variable to express the perimeter of a rectangle (Figure 10). The great majority of pupils could write an expression for adding 4 to n+5. Half of the pupils were able to write an open algebraic expression, although not always a correct one, when asked to write a formula for symbolising the sentence Add 3 to an unknown number and multiply the result by 8. At the beginning of the study the great majority of pupils did not answer these questions. It has to be stressed that when answering the questionnaire after the Logo experience, pupils had still very few or no experience at all of school algebra. This suggests that the observed progression was due mainly to the experience acquired during the Logo activities. That is, to help pupils focus on the method in Logo led them to construct in this environment the idea of general number and this experience enabled thein to cope with this idea of variable when it appeared in simple paper and pencil algebra tasks. CONCLUSION The analysis of the historical development of mathematics highlights the presence of qualitative changes characterised by the appearance of completely new ways of looking at mathematical objects. These qualitative changes are not determined only by previous mathematical knowledge and experience but also by the social and economic circumstances of the historical moment. Therefore, from an analysis of the historical development of mathematics only hints can be drawn that, reinterpreted
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from our historical moment, can become useful tools for helping improve pupils’ understanding of this discipline. In this chapter an analysis of the historical development of algebra was made based on some popular texts on the history of‘ the development of mathematics. The hint that emerged from this analysis was that Vieta’s conception of the general method and of the ‘general object’ of a method were fundamental for the development of algebra. This led to an exploration of the possibility of designing environments that might help pre-algebraic pupils approach and work with these ideas and so provide an experience that might facilitate later on the acceptance and understanding of a more formal approach to algebra. The results obtained show that pre-algebraic pupils could approach in Logo the idea of general method and of general number in the sense of Vieta, that is, as the general object of a method that can be treated as an actual mathematical object. Several elements contributed to the development of this capability. Pupils’ numeric background and their capability for working with particular cases were basic elements supporting this approach. Another important point was that working in an environment like Logo, favoured the development and the formal expression of the intermediate steps that initially supported pupils’ formal expression of a general method. The appearance of a spiral dialectic connection was observed. This dialectic process led pupils not only to the idea that a variable represents a range of values (Noss, 1985) but it supported a qualitative difference in pupils’ understanding of the idea of general number. Pupils gradually internalised the analysis of particular examples in order to work with the general. Finally they shifted to working directly with general numbers without needing the explicit support of particular examples to which, if necessary, they could drop back. But the central point for the development of pupils’ capability for working with general methods and general numbers was to provide the guidance that helped them shift from regarding Logo procedures as tools to do something to regarding them as objects that could be analysed. This was a crucial step that required shifting froin situational thinking to abstract thinking. I suggest that this experience favoured later on the spontaneous shift to regarding general numbers as actual mathematical objects that could be directly manipulated and operated on. Pupils developed the idea of general number in Logo settings and they worked with it in Logo-based activities. But, it was found that this experience substantially improved pupils’ capability of coping with general numbers when they appeared in simple paper and pencil algebra tasks. These results suggest that approaching the idea of general number by emphasising its character of general object of a method, and offering pupils a formal language for expressing themselves, helps them develop a firm enough starting point for this use of variable, allowing them to start coping
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with it in different contexts. Experiences like this might constitute a basis for a later more formal approach to algebraic concepts. The results discussed in this chapter suggest that teaching pupils to focus on the method for solving sets of similar problems and providing them with a medium for expressing their methods might be a promising way for entering the world of algebra. The results also show that from the analyses of the historical development of mathematics, promising approaches for helping pupils understand and accept mathematical ideas might be derived, provided that we take into account that there are fundamental differences between the historical development and construction of a concept and its understanding and acceptance by a pupil.
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LULU HEALY¹, STEFANO POZZI² AND ROSAMUND SUTHERLAND ³
REFLECTIONS ON THE ROLE OF THE COMPUTER IN THE DEVELOPMENT OF ALGEBRAIC THINKING BACKGROUND Throughout the 80’s and 90’s there have been numerous changes in both the content and teaching of algebra in the UK. Educators have endeavoured to create a more accessible mathematics curriculum and for some students this has actually resulted in the removal of much that could be called algebraic from the curriculum they follow. Influences on the school curriculum are also coming from the influx of new technology. Interaction with computer environments such as Logo and spreadsheets seems to provide students with alternative routes to algebra and algebraic ideas which put into question the validity of the current hierarchically organised curriculum. Further requisitioning of what should be taught in the classroom is being provoked by computer algebra systems, software which can perform algebraic manipulations. Within this chapter we shall discuss the results of our research on the teaching and learning of algebra mediated by the use of computer environments. All our research is classroom based, and involves long-term involvement with teachers and students. Inevitably, the activities we investigate are constrained and formed by curriculum demands. This chapter is concerned with the close inter-relationship between research and current practice. Firstly we discuss the changes in algebra teaching in the UK over the last 10 years and some of its Piagetian underpinnings which influenced the nature of our early research. We then describe the shifts in focus of our work, and how we changed our views on what students are able to do algebraically in their interactions with the computer. In parallel, we describe the theoretical frameworks which we explored, in order to provide better explanations for what we were consistently finding in our research. We describe a number of case studies from our research to illustrate these shifts of focus.
1
Institute of Education, University of London, U.K. Department for Education & Employment, London, U.K. 3 Graduate School of Education, University of Bristol, U.K. 2
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SCHOOL ALGEBRA IN THE UK Throughout the U.K., a great variation of teaching approaches have developed around how and when algebra is taught particularly in compulsory secondary education. One of the major influences in this development hac been the Cockcroft report (Cockcroft, 1982) which set out a reevaluation of the whole area of –
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mathematics education in schools. One of the main criticisms in the report was directed against the predominance of teacher exposition followed by individual skills practice. The authors recommended a wider range of teaching and learning approaches which place more emphasis on mathematical concepts. Partly as a result of this, a number of innovative curriculum schemes were developed to emphasise conceptual understanding as opposed to acquisition of skills particularly in algebra. Problem situations which exemplify this approach involve identifying and expressing general patterns (see for example Fig. 1). Alongside these changes, there have also been debates on the question of whether there has been an over-emphasis on the teaching and learning of algebra. Some advocate the removal of algebra teaching in pre-16 education on the grounds that most students leave school with little understanding of or use for it. A less extreme variant is that only some students should be taught algebra; that is those who wish to go on to study a subject in which algebra is required. The result of this is that some students leave school at 16 having experienced very little algebra teaching; especially those who take the lower tier version of mathematics GCSE. More recently one forum for these debates has been in the context of the Mathematics Curriculum (DES, 1991; SCAA, 1994), which has moved from specifying a core content of algebra to be taught from 7 years onwards. to its present form which only requires this from the ages of 1 1 to 16. The implication is that algebra is harder than. say, arithmetic and must be put off until a certain facility with number has been acquired. –
ALGEBRA RESEARCH: THE INFLUENCE OF PIAGET Research into the teaching and learning of algebra has also played a key role in the curriculum development of the last 15 years. The Cockcroft report drew on evidence from a number of research projects; in particular, the CSMS project in the mid-70s (Hart, 1981; Küchemann, 198 1). a large-scale project which investigated secondary school pupils’ understanding of a variety of mathematics (and science) concepts including algebraic understanding. For example, Küchemann found that students have considerable problems both in accepting and using the idea of a letter representing a variable in algebra and in understanding the systematic relationship between variables in algebraic expressions. Other research has also highlighted students’ difficulties in viewing a letter as representing a range of numbers (Collis, 1974; Booth, 1984; Matz, 1980) and conversely, in seeing that different letters can represent the same numbers (Wagner, 198 1). Research has also shown that students are reluctant to accept unclosed forms (e.g. p + 6) as objects to be operated on (see Jensen and Wagner, 1982; Thonipson and Dreyfus, 1985). Thus this research from the 70s and early 80s suggested that many students develop only a very rudimentary –
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grasp of basic algebraic concepts, indeed Küchemann concluded that ‘the majority of 13, 14 and 15 year olds were not able to cope consistently with items that can properly be called algebra at all.’ (Küchemann. 1981 p. 119). Most of the research into students’ conceptual difficulties in algebra including the CSMS project tends to be located within a Piagetian framework or at least draw from aspects of this framework. The developmental psychology associated with Piaget’s genetic epistemology (see Boden, 1979: Piaget and Inhelder, 1958) has had a major influence on educational practice particularly in primary education towards greater reliance on the spontaneous learning of the child. For Piaget, cognitive development proceeds through a series of successive psychological stages. in which each stage is dependent on previous stages of development. He describes the process of stage development as ‘reflective abstraction’; whereby the child transforms the (possibly mental) actions and processes at one stage, to objects of thought in the next stage of development. This reconstruction of processes in one stage into objects (or structures) in another stage underpins his idea that the invariance in the order of these stages is fundamental, and teaching can only accelerate pupils through these stages within limits. Indeed he believed that teaching a child a concept before they are ready could be counter-productive, resulting in miscomprehension covered up by superficial understanding. A number of mathematics educators (Sfard, 1991; Kieran, 1991; Sfard and Linchevski, 1994; Dubinsky, 1991; Tall, 1992) have taken these ideas and developed them into more elaborated theoretical descriptions of algebraic identifying different aspects of algebra as part of a hierarchy of development mathematical knowledge and explaining the different ways in which pupils ‘internalise’ (or not) the related structures which make up this hierarchy. Although differing in detail, the works cited above bring into play the idea of pupils moving from a process-oriented view of algebra to a structure-oriented view via some process of reflective abstraction (although Sfard and Kaput use the term ‘reification’, and Tall uses ‘encapsulation’). Within this context, the above research has often interpreted algebraic understanding as more sophisticated and more difficult than other forms of mathematising, suggesting that there are different stages of understanding which students need to move through in order to develop a clear grasp of algebra. For example Sfard (199 1) argues that structural conceptions should be preceded by operational conceptions. Different kinds of evidence are provided to support this view including making links between the historical development of mathematics and the mathematical development of individuals. One underlying implication of this approach is that algebra is more abstract, hence more difficult than arithmetic and that there is a developmental gap between the two. An over-emphasis on structural aspects which has not been grounded first in operational understandings –
–
–
–
–
–
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has been pinpointed as a source of major misunderstanding for students. These students may be able to ‘learn’ correct algebraic techniques only in superficial ways which break down when deeper levels of understanding are called for. This resonates with the commonly held view in mathematics education. As Resnick et al point out:
g"2$/3*" 2/"*%#$ %+ "''*/++/' 30 9)+( +B,))2IB,%2'*/# 4+ " D*)32/9 )7 2/"*#%#$ () 9"#%D42"(/@+093)F+@%# "BB)*' >%(, B/*("%# (*"#+7)*9"(%)# *42/+1@ >%(,)4( */7/*/#B/ () (,/ O4+(%7%B"(%)# )* 9/"#%#$ )7 (,/+/ (*"#+7)*9"(%)#+e SM/+#%B=1 E"4H#%22/I!"*"9/B,/ "#' !"(,%/41 XZlc1 D : XcsT It is easy to see why there was a backlash against the traditional teaching approach, which was seen to support only the acquisition of manipulation skill and not understanding. A great deal of innovation in the UK algebra curricula and the UK National Curriculum can therefore be seen as an attempt to guide pupils more carefully through a teaching (and hopefully learning) sequence which attempts to take account of pupils’ algebraic ‘stages’ of development. For instance, algebraic symbolism is usually introduced in the context of systems of reference, either reference to real world situations or other symbols systems (e.g. diagrams or tables, see for example Bell, 1992; Mason et al, 1982). The idea of expressing general relationships is often introduced in the context of patterns from numerical, spatial and practical situations, initially in words and then in algebraic expressions. Thus in the UK, research in mathematics education has led to the development of a more process orientated curriculum. Introducing algebra to pupils via an approach which places an emphasis on generalising from patterns (as shown in Figure 1 ) has become institutionalised within the UK mathematics curriculum. The intended emphasis on meaningful algebraic activity often becomes obscured because the structure of the situation from which numeric data is derived is often under-emphasised by the teacher (Hewitt, 1992). Research suggests that although students readily find and use patterns for calculating, they find it more difficult to describe these patterns verbally, let alone algebraically (for further discussion of this issue see Stacey & McGregor, in this volume). The generalisations pupils articulate are rarely justified except by reference to specific cases. Thus, while those students who do construct algebraic expressions may be more likely to experience letters as generalised number, they will not necessarily attach meaning to any expression they create by referring back to the pattern context. That is the pattern-spotting activity becomes divorced from the mathematical situation, often resulting in a de-emphasis on the structural aspects of the situation. The emphasis on process, producing a table of results, computing values, etc. in these new teaching approaches thus follows an essentially Piagetian
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framework, students are first introduced to algebraic notations through activities which place an emphasis on process before moving to the next stage which involves structure.
A%$4*/ V:@C,/@gN e ("+=
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THE ROLE OF THE COMPUTER Our early work with computers was also rather implicitly influenced by the work of Piaget and strongly influenced by the results of the CSMS study. This was the case with the Logo Maths Project (Hoyles and Sutherland, 1989) a three year longitudinal study investigating the way Logo could be used as an aid to pupils learning of mathematics. The theoretical rationale for the Logo Maths Project was that pupils learn mathematics through active construction of their own knowledge, and that this can be facilitated in a computer environment through an iterative process of conjecture and feedback. But whereas at the very beginning of this project we emphasised the importance of pupils constructing their own knowledge and deemphasised the role of the teacher, we very soon changed our position to recognise the role of the teacher in structuring the classroom situation particularly with respect to pupils’ learning of algebra-related ideas. ‘Towards the end of the first year of the Logo Maths Project we changed our strategy and intervened with teacher-directed tasks which introduced pupils to the idea of variable in Logo. This change in direction resulted from our ongoing analysis of the data. that is the finding that pupils were not choosing projects in which it was appropriate to introduce the idea of variable. We were now structuring the situation for the pupils’. (Sutherland, 1993, p 98) Within subsequent Logo projects (Healy, Hoyles and Sutherland, 1990; Sutherland. 1995) we presented pupils with the idea of variable within carefully structured activities (see for example Figure 2) and encouraged pupils to develop their own general Logo procedures (see for example Figure 3). Figure 3 shows how we also adopted what could be termed a process-orientated approach in which students were encouraged to move from specific (often seen as synonymous with arithmetic) representations to general (more algebraic) ones. The results deriving from our work with Logo provided evidence to support a growing conviction that some of the difficulties which students following a more traditional curriculum have, can be overcome through interaction with particular computational environments (see for example Sutherland, 1989; Noss, 1986; Thomas and Tall, 1986). We were finding that students could accept that a letter represented a general number, could accept and work with ‘unclosed’ algebraic expressions and could express general mathematical relationships expressed in a computer language. Moreover there was beginning evidence that pupils could make use of these computer-derived ideas in more traditional algebra settings (further evidence for this has emerged from the work of Ursini, this volume). Students were
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A%$4*/ [: R"*%"32/ ;="(/*+
found to be able to make use of the algebra-like notations underlying their interactions with computers when working on pencil and paper algebra tasks. In retrospect, we can see how with carefully structured activities in interactive computer environments, students necessarily used formal language to ‘converse’ with the computer giving them a more meaningful introduction to, for example, the notion of variables. We noted that in paper and pencil settings a similar set of structured activities often fails to lead students towards algebraic approaches as the ‘pre-algebraic’ activities do not engage many students in algebra related activity, but instead can be successfully negotiated through arithmetic problem-solving methods. Thus students are unlikely to spontaneously construct algebraic concepts and are unlikely to experience some of the power of the algebraic representation system. There needs to be a cognitive pay-off which motivates students to incorporate algebraic methods into their mathematical practices. In contrast, in the interactive computer environments, algebra-like notations are an essential part of the negotiation of the problem solution, and the feedback provided by the computer offers an additional resource for students to make sense of the mathematical ideas involved. This research with computer environments has challenged some of the ‘readiness’ hierarchies on which the UK curriculum is based. Students seemed to be ready to take on algebraic notions earlier than might have been expected if their development followed the stages suggested by previous algebra research. Our work with computers has been influenced by and has some similarities with the generalising-from-patterns approach discussed previously. As with the paperbased generalising activities we have to ask whether these computer activities provoke pupils to focus on mathematical structure. So for example, students might be happy in Logo to accept :X * 5 as an object but only because they see this as the
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process for obtaining a line 5 times the length of a geometrical object. Tall and Thomas (199 1) however, suggest that computer-based approaches can help pupils ‘cope with the process-product obstacle’. One explanation for this is that by giving the computer the responsibility for computations, students are more able through, reflecting on the computer feedback, to focus on the structure of the situation. The following example from our work with spreadsheets illustrates how computer-based activities could provoke a move from focusing on the computational processes associated with arithmetic to more structural aspects. Andrew and Graeme (aged 10 years) were using a spreadsheet to find expressions equivalent to 'multiply an unknown number by 4’. They entered the number 3 in cell A2 and then in another cell entered the formula A2*0.5*8 which produced a value of 12. In a new cell they then entered the expression A2*3 + 3 which they had worked out as being equivalent to ‘multiply an unknown number by 4’ because it also produced a value of 12. The second expression suggests that they were thinking with the specific number 3, and focusing on the process of calculation in order to construct the equivalent expression. When they varied the number in cell A2 they realised for themselves that the two formulae were not equivalent and changed the second expression without any need for intervention from the teacher. Through this interaction with the spreadsheet they began to construct equivalent expressions of the form A2*2*2, A2* 16/4,A2*32/8. In the above example we suggest that the pupils had moved from a focus on the process of calculating the value of the function X x 0.5 x 8 when the number 3 was input, to a consideration of the relationship between such functions as X x 0.5 xx 8 and X x 2 x 2 and X x 16/4 This involved analysing equivalent numerical expressions, for example 2 x 2 = 16/4, and thus involves a focus on structure. This type of activity can be extended to asking pupils to construct equivalent expressions K + __ K which involve manipulating an unknown (for example x+3 and __ 2 2 +3). These activities have been developed into an algebra course for vocational students (Sutherland, Howell and Wolf, 1996 ). Other work with spreadsheets (for example Sutherland & Rojano, 1993) has shown that pupils can use the spreadsheet to move from non-algebraic to more algebraic solutions of algebra word problems. Although a spreadsheet-algebraic approach is not the same as an algebraic approach (as discussed by Dettori et a1 in this volume) we maintain that work with spreadsheets can provoke a shift of emphasis from non-algebraic to algebraic approaches to solving problems.
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L. HEALY, S. POZZI AND R. SUTHERLAND VYGOTSKY AND DEVELOPING AN ‘ALGEBRISING’ CULTURE WITH COMPUTERS
As we analysed the results from our research we began to conjecture that work with
computers was doing more than allowing pupils to progress along stages. Although our results did not challenge the $/#/*"2 notion of a stage theory of learning, pupils seemed to transgress the particular learning paths set out in previous research and traditional curricula. We therefore sought out other paradigms for explaining the role of the computer, in particular we were attracted to Bruner’s notion of scaffolding (Wood, Bruner, and Ross. 1979). The notion of scaffolding derived from analysis of adult-child teaching interaction, whereby the activity of the child is supported (or scaffolded by the adult in a way which is contingent on what the child knows. As interaction progresses. the adult can pass more control of the activity over to the child until the point at which the child engages i n the activity independently of the adult. By exploring the idea that the computer may have a scaffolding functioni, we were interested in the ways in which the computer appeared to enable students to employ lines of development and ways of reasoning which could be understood but not constructed by individuals alone. And with software based on algebra-like notations, we were interested in whether the computer can scaffold the child’s "2$/3*"%B activity: especially the notions of constructing and interpreting generalisations, and accepting and manipulating unknowns. One area of focus was on how the tools available in Logo and spreadsheets are used to bridge the gap between specific and general (Hoyles, Healy and Sutherland, 1991). For example, the following extract shows that a mouse-driven interface on a spreadsheet such as Excel can provide invaluable support for pupils attempting to generalise algebraically where pupils tend to use gestures matching the actions on the spreadsheet as a means of scaffolding their ideas: –
–
Jamie was working on the problem of finding a rule to represent the square numbers in a spreadsheet. Having seen a relationship underlying the square numbers sequence he begins to explain this to Jake and together they used pointing strategies to negotiate the actions that when replicated represented a generalisation: –
Jamie first uses both pointing and reference to specific numbers to describe his rule: r"9%/P Um — times that’s add that plus 1, add that plus 2, add that plus 3 -how do we write that as a formula? Jake has not yet understood and Jamie attempts to explain again:
THE DEVELOPMENT OF ALGEBRAIC THINKING
24 1
A%$: s C,/ D)20$)# J493/*+ C"+=
So you — plus — so that's position, square number 1 plus position in the sequence 2, plus that (points at A4), equals, plus 1 ha, equals the next one and that square number 4 plus position 3 plus 2 equals that, that plus that plus 3 equals that. This time the combining of language and gesture provides the necessary scaffolding for the generalisation to be communicated. r"=/P So lets, let me see, so it is, is it that? r"9%/P That plus that plus 2. r"=/P Plus that. r"9%/P That plus that equals that. Yeah you got it. In the Logo environment the naming of the variable provides scaffolding in that it helps pupils move from viewing a fixed procedure as a specific case to viewing it as a generic example. Noss and Hoyles (1992) have discussed how the symbolic representations built up during students’ interactions with Logo provide a r"9%/P
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L. HEALY, S . POZZI AND R. SUTHERLAND
mediational means between actions and generalisations. For example. Alice and Joanna get started on a Logo task (Figure 4) by defining a fixed-head procedure; they then begin to discuss editing this procedure so that it will become a general procedure. r)"##"P (reads) I want one procedure that will draw heads of different sizes. you can choose which side to use as an input so which side is the input, which side shall we put? Alice does not initially understand Jessica’s suggestion and thinks she is referring to the image of a smaller head. 52%B/P We could try the bottom size couldn’t we, ha, ha r)"##"P ‘Side’ not ‘size’ 52%B/P Oh ‘side’, which ‘side’ do you use as the input. When Joanna explicitly names the variable this provides Alice with the necessary scaffolding to understand the next stage of the task. r)"##"P You know, like you do 45 is dot, dot S. 52%B/P Oh I know, I know, so shall I edit? Similar moves from the specific to the general can be bridged with environments such as Derive a computer algebra system. In the following example, the generalising is not in terms of algebraic variables but in terms of functions: Two pupils exploring different derivative quotients were able to see the functional pattern in the derivative but were only able to express it using particular examples of quotients, rather than as generic relationships between the two functions in the quotient (as it is usually expressed in textbooks). Although they were familiar with functional notation, this was invariably in the context of functions assigned to expressions (e.g. f(x)=sin(x)). By introducing them to unassigned functions in Derive, they were able to explore this notation and use it to express a part-general quotient derivative and verify it on the computer: —
–
–
–
From this, the pupils could go on to express a general quotient derivative on paper by themselves.
THE DEVELOPMENT OF ALGEBRAIC THINKING
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Still not completely convinced by their result, they could then go on and verify it
on the computer:
We also explored different aspects of Russian developmental psychology to see whether there could be a more radical reading of the role of the computer (or indeed any tool) in the types of algebraic activity pupils might engage in. Stemming from the work of Vygotsky (1962), Russian developmental psychology has offered a different mechanism for explaining algebraic development. In brief, this framework emphasises the importance of communication and enculturation in shaping pupils’ knowledge whereby knowledge is first developed between the individuals in a culture and then shifts to the cognition of the individual. This perspective provides a direct challenge to a Piagetian perspective, as mathematical progress becomes grounded in a culture and not located within a fixed developmental framework. Within the classroom, mathematical knowledge can be seen as shaped through individual and shared activities, as well as through the mathematical tools available. Thus, understanding different aspects of algebra need not be dependent on, say, understanding all aspects of arithmetic. This would depend on the activities of the school mathematical culture and the tools which were available to pupils. This further suggests that research which places pupils in different ‘stages’ (e.g. the CSMS study, 1981) should take into consideration the forms of algebraic activity the pupils engage in and the tools that are available to them. Thus, algebraic ‘stages’ may #)( be independent of the mathematical culture pupils are brought up in. For example, in a class of 13-14 year olds, where Logo work was an established part of their mathematics lessons, pupils analysed a proportionality task using a variety of algebraic methods a rare occurrence in paper and pencil activities: Jerome and Joe are working on a Logo task to design a number of proportional heads. They realise straight away that they need to write a general procedure in which the ratios between the lengths of the head are expressed, and that this relationship is invariant: –
—
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L. HEALY, S. POZZI AND R. SUTHERLAND
TO HEAD :L FD :L * 3 LT 60 FD:L*4 LT 30 FD :L * 4.5 LT 90 FD :L * 7 LT 90 FD : L*4.5 LT 30 FD :L * 4 LT 150 HAT :L/10 END
A%$4*/ a : C,/ Q/"' C"+=
They’re all same sort of degrees, and some sort of relationship to each other r/*)9/P What you want is Um something that will either make this size or any other r)/P okay, um so shall we call it lengths? r/*)9/P L for length After testing that their procedure produces heads of different sizes, they work out the commands to draw a hat using a HEAD procedure with an input of 10. They begin by defining a fixed procedure HAT. Both of them use a strategy whereby they visualise mentally the position and heading of the turtle after each command. As they are doing this Joe suddenly re-focuses on the aim of the task. He is worried that they ought to be writing a general procedure for HAT using variables:
r)/P
—
—
—
THE DEVELOPMENT OF ALGEBRAIC THINKING
r)/P
245
We’re just making an estimate, this isn’t variable though is it, this isn’t variable The pair continue to define the fixed HAT procedure, and try it out by calling HEAD 10. Jerome then suggests they try another size, he types HEAD 5. The screen feedback provides support for Joe’s argument that they had not written a variable procedure: r/*)9/P Shall I do a different size head? I told you it’s not variable r)/P Jerome, however, immediately solves this problem by ‘variabilising’ the fixed HAT procedure taking into account the fact that they had been working with a specific head of size 10. Seen from a Vygotskian perspective, our interpretation is that Jerome and Joe are using the Logo language to think with, which enables them to switch between the specific and the general in different ways sometimes focusing first on the general geometric relationships and, sometimes on specific details depending on the task requirements. We would argue that without the established use of Logo in the classroom, such strategies would be rare among the majority of pupils. In another example (Sutherland (1995)), Sue (aged 11) seems to have a structural algebraic in spite of the fact that she had great difficulties with approach to her Logo work arithmetic, and was therefore regarded as very weak by her teacher. From the beginning of her work with Logo Sue accepted the idea of a name representing any number in Logo: M;P You keep using the word variable — what’s a variable to you? ;4/P Well say you’ve got the two dots and then the n bit the n stands in for a number instead of putting in just 10 or something you can change the sizes by just putting a variable For the Heads task (Fig. 5), some of the pupils in Sue’s group thought of their general Logo procedure in terms of a specific shape, but Sue focused on the general aspects of the problem and successfully constructed a general Logo procedure which using commands of the form FD :C/3 and preserved proportional relationships BK 2*:C/3. In the interview situation, she indicated that she did this by thinking with the structure of the geometrical relationships in the problem, and the use of letters to represent a general length supported this focus on structure. So it seems that the Logo activity, with its focus on using letters to construct general relationships was allowing Sue to avoid her difficulties with arithmetic. She said of her computer work g0)4 ')#e( ,"./ () >*%(/ "22 )7 (,/9 ')># "#' 0)4 ')#e( ,"./ () 3/ D4HH2%#$ %( )4( %# 0)4* ,/"'e: This ‘puzzling it out in your head’ almost certainly refers to her difficulties with evaluating arithmetic expressions. Pupils like Sue might benefit from a more structural approach to arithmetic as an introduction to algebra, and through this work they might overcome their resistance –
—
—
—
—
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L. HEALY, S. POZZI AND R. SUTHERLAND
to number work. The difference between Sue and pupils who are recognised as being good at algebra is that ‘successful pupils’ are often very articulate and very good at arithmetic. Sue in contrast expresses herself poorly in English, can hardly write on paper and has been classified by the mathematics teacher as weak mathematically. Pupils like Sue are normally expected to overcome their difficulties with arithmetic before being introduced to algebra. An alternative approach would be to work with Sue on the structural aspects of arithmetic and through this work she may overcome some of her resistance to calculations with numbers. In fact, work in the Vygotskian perspective already exists to support the claim that algebra does not necessarily have to be grafted on to a sound arithmetic base. Davydov (Davydov, 1962; Freudenthal, 1974), working within a Vygotskian framework, argues that very young students can develop a strong conceptual framework for algebra >%(,)4( expressing quantitative relationships through numbers. Instead 1st grade students engage in activities involving the comparison of things (sticks, cubes weights) in terms of different parameters (length, weight etc.). They are introduced to the symbols for equalities and inequalities to describe the relationship between the objects. Davydov describes how the students learn to operate with quantitative characteristics directly in a generalised from which may be recorded in letter symbols. Lins has replicated some of this work by Davydov and he points out that : gC,/ 4+/ )7 2%(/*"2@ #)("(%)#@ >"+ %#(*)'4B/'@ *"D%'20@ "#'@ >%(,)4(@ (*)432/1 3/B)9%#$ " ."24"32/ "#' "'/b4"(/ ())2 %# (,"( B)#(/K(: m/ 4#'/*+("#' (,"( (,/ %9D)*("#B/ )7 (,)+/ +(4'%/+ %+ (>)7)2': A%*+(1 (,/0 D)%#( )4( () (,/ >"0+ %# >,%B, (,/ 4+/ )7 +093)2+ 9"0 D*)'4B/ " +,%7( 7*)9 (,/ +)24(%)# )7 D*)32/9+ () (,/ %#./+(%$"(%)# )7 9/(,)'+ )7 +)24(%)#: ;/B)#'1 30 +("*(%#$ 7*)9 (,/ 9"#%D42"(%)# )7 >,)2/ID"*( */2"(%)#+,%D+1 4+ " +4DD)*(1 "#' (,/# 9).%#$ () (,/ 9"#%D42"(%)# )7 (,/ /KD*/++%)#+ (,/9+/2./+1 (,/ >)*= )7 L".0'). +4$$/+(+ " 7*4%(742 34( #)( 74220 /KD2)*/' ./%# e SN%#+1 XZZsT: Whereas Davydov introduced young pupils to algebra without passing through arithmetic, we suggest that work with computers can support pupils to make the break from arithmetic, whilst at the same time making use of the arithmetic ideas which are a basis for algebra. OUR CURRENT APPROACH
The importance of the setting and tools available to pupils now plays a critical role in our thinking about pupils’ development (Balacheff & Sutherland, 1994; Hoyles, Healy and Pozzi, 1994). Recognising that the curriculum culture and sequence of learning will strongly influence pupils’ development of algebraic thinking is not dissimilar to the
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recognition that historical development is strongly related to the tools available in a culture and the problems being solved (as discussed by Radford in this volume). This type of analysis is rather different from the historical analysis presented by Sfard (199 1) which suggests a more hierarchical historical development from process to structure. We have found that Vygotsky’s emphasis on the communicative function of symbol systems provides some explanation for the ways in which pupils make sense of computer-based symbols. Work with computers suggests new approaches to teaching algebra which would place more of an emphasis on the communicative function of algebraic symbolism. This is different from the ways in which pupils have traditionally been introduced to algebraic symbols in school mathematics, and different also from the moves in the UK to under-emphasise the crucial role which the algebraic language plays in the development of algebraic thinking. In our work we have not yet taken on board the implications of the new computer algebra environments which are now available on hand-held calculators. In order to do this, we need to be clearer about the purpose of school algebra and we need to understand more about the differences between what a computer algebra system can do and what a human algebra system does. History teaches us that the tools available radically change approaches to problem solving and that new methods develop in response to new problems.
NOTES 1
Hoyles and Noss (1993) have coined the phrase computational scaffolding to describe this process.
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POSTSCRIPT BY NICOLAS BALACHEFF*
SYMBOLIC ARITHMETIC vs ALGEBRA THE CORE OF A DIDACTICAL DILEMMA INTRODUCTION I accepted the invitation to write this postscript, despite the fact that I could not write anything other than a view from an outsider to the field of research on teaching and learning algebra (even though about ten years ago I considered this question in the context of the use of spreadsheets). But this position may not be a bad one in order to try to extract from the book just a little more than the substantial results and ideas the authors already provide for their readers. Twenty years ago it was possible to pretend to comment on almost any research in mathematics education, but since then too much progress has been made, too many works must be known, so I will just take the content of this book as a frame of reference, augmented by some synthesis to confront it with the state of the art (Kieran 1990, Filloy and Sutherland 1996). I see this postscript as a way of opening doors on possible other perspectives on a domain well and extensively studied, this may be only possible because take a candid and straightforward position, possibly bypassing some difficulties or subtleties that the informed reader may identify to address some of the questions. So, let us first look for the front door to the algebraic Eden...
LOOKING FOR THE FRONT DOOR All the papers gathered in this book share a common preoccupation : to find the front door to the world of algebra for people coming either froin the world of arithmetic or that of everyday applications (the world of the manipulation of quantities). The investigation goes along different lines: historical analysis of the ancient times when algebra was not available with its efficient symbolic technology, analysis of situations in which algebra may appear as being the more efficient problem-solving tool, analysis of the possible contribution of modern computerbased technologies. This diversity of routes followed in order to make sense of this old and difficult problem speaks to the growing awareness of the research community about its complexity. What do we learn from the research efforts
* CNRS, Laboratoire Leibniz-IMAG, Grenoble, France. 249 M . ;4(,/*2"#' /( "2: S/'+:T1 ?/*+D/B(%./+ )# ;B,))2 52$/3*"1 249–260. U VWW X <24>/* 5B"'/9%B ?432%+,/*+: ?*%#(/' %# (,/ J/(,/*2"#'+:
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N. BALACHEFF
presented in the papers gathered here? and is there any progress towards a solution ? I will survey the content of this book bearing in mind these two questions. But, let u s start from the very beginning : why is the problem of teaching algebra so difficult? As Stacey reminds us: ‘in traditional curriculum, students are said to ‘begin algebra’ when they learn to write expressions using letters to stand for numbers and to interpret or transform these expression’ (page 141). This classical view expresses what is seen as an unavoidable link between arithmetic and algebra, one may even add for better or for worse. And, it seems that if it is for better when initiating the teaching of algebra, it turns out to be for worse in the long run for its learning. Stable and replicable research results provide nowadays plenty of evidence ‘on the tendency of subjects to 9"%#("%# arithmetical interpretations in the majority of algebraic situations, even in quite advanced stages in the study of algebra’ (Filloy. page 156, my italics). The construction of this link between algebra and arithmetic is seen as an answer to the problem of teaching a formalism which otherwise would have no source in students' experiences. Kirshner (page 83) identifies here a central difficulty, expressing that ‘Algebra as generalisation is antithetical to structural algebra which by its nature is formal and uninterpreted.’ Expressed in an other way, Dettori joins this author, when claiming that ‘what matters is not to Iearn to find a numerical solution to algebraic problems but rather to understand the nature and the power of the theoretical solving scheme of algebra’ (Dettori et.al. page 191). There is a common agreement that Algebra is marked by ‘the recognition and description of general rules for describing patterns’ (Stacey, page 141, Kirshner: page 83) and that Algebra consists of the study of structures and relationships (Bloedy-Vinner, page 178), but there i s also a large agreement-more or less explicit-that Algebra is about properties of numbers and operations on numbers. being in this sense a ‘generalised arithmetic’ (ibid.) If what matters is the ‘theoretical solving scheme of algebra’ as Dettori proposes it, then the pragmatic justification of algebra as a powerful tool to solve problems which are not seen at first as being algebraic may not be possible to avoid. Here arises the fundamental teaching problem clearly formulated by Kirshner as the opposition between a ‘structural approach’, and a ‘referential approach’: ‘the structural approach builds meaning internally from the connections generated within a syntactically constructed system. Referential approaches import meaning into the symbol system from external domains of reference.’ (Kirshner, page 84). But is there here more than a classical tension between form and content, or between syntax and semantic (a tension that Bocro in his chapter investigates though a 7*)9n+/9 diagram)1. The strong resistance of the effects of this tension,
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which is often described as the obstacle of arithmetic to the learning o f algebra, suggests that it could be so. All the papers acknowledge that the central issue is that of meaning, or more precisely the meaning for the students of their manipulation of symbolic writings. It is generally accepted that this meaning cannot be only a syntactical meaning, which would mean just knowing the rules of symbolic manipulation and conforming strictly to them-to some extent independently of any sense making. Because of the current development of student knowledge, this meaning can only be provided by the field of experience and the competencies they have previously constructed by manipulating quantities (word problems) or numbers (arithmetic): ‘to learn algebra students need to be able to recognise and articulate the processes o f arithmetic and the structures of relationships between numbers’ (Stacey, page 148) although ‘ it is very possible they do not know consciously what they do’ (ibid.) A way to understand the links between arithmetic and algebra is to look at the historical genesis of algebra for discovering the best conditions for its teaching and learning. In the first chapter of this book, Radford explores the historical origins of algebraic thinking. This author traces in Mesopotamian and Babylonian arithmetical thinking, the method of the false position and in Diaphantus’ ‘arithmos’ the anticipatory milestones of algebra. Such a journey into the ancient past to search for the first evidence of algebraic modes of thinking is rather classical. Less classical would have been the dual search for evidence of the resistance of arithmetic to the development of algebra. For example, a look at some not-so-old books shows the method of false position alive in the eighteenth century (see for example the g N_@5*%(,9d(%b4/ /# +4 D/*7/B(%)# 9%+/ /# +" D*"(%b4/ +/2)# 2g4+"$/ '/+ 7%#"#B%/*+1 $/#+ '/ D*"(%b4/1 3"#b4%/*/+ /( 9"*B,"#'+e by F. Legendre — ‘arithméticien’, says the text in 1774), the remaining traces of its existence in the nineteenth century (for example in thegC*"%(d 'g"*%(,9d(%b4/ • 2g4+"$/ '/ 2" 9"*%#/ /( '/ 2g"*(%22/*%/e by Bezout in 1832) and the manifestation of still strong oppositions to algebra considered as a too abstract approach like that of the g5*%(,9)(/B,#%/ )4 2g"*( '/ *d+)4'*/ 2/+ D*)32f9/+ +"#+ 2/ */B)4*+ '/ 2g"2$f3*/1 D"* 2gdb4"(%)# #d$"(%./e by Vaucher, 1866)² The reader may realise that all the above mentioned books target an audience of practical persons and not of mathematicians, Indeed, I would suggest here that one possible view we could adopt is that algebra and arithmetic do not address the same communities of practice. Whereas the former is valued by mathematicians and engineers, the latter concerns common practice and business at large. In his analysis: Radford emphasises that the role of problems in algebra is surely a crucial point, and that even the nature of problems could make the difference (Radford, page 13). This attention to the role of problems may be essential for the understanding of difficulties in learning algebra, in particular it seems that algebra —
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and arithmetic are in competition over the same corpus of problems (Dettori et.al., page 191). Paradoxically this suggests that the commonly shared idea that ‘a strong understanding of properties of numbers and operation is an essential foundation for beginning algebra’ (Stacey & MacGregor, page 148) could be pointing also to the source of difficulties. My coiijecture is that the more students are confident in their arithmetical competencies, the more they will be reluctant to accept the use of algebra on problems they recognise as pertaining to the field of arithmetic (or computation). Let us come back to a case developed by Lins in his chapter. The students observed, in the situation Lins reports, manipulate numbers of things (that is, quantities) and not numbers, and one may well claim that algebra is completely absent. A strong support to this interpretation of students’ "B(%.%(0 is given by the strength of their control of this activity by the quantities manipulated: the control is left in the reference world of the modelling process (page 37). Would it be possible to escape this difficulty? Maybe yes, but only if we were able to find situations in which students must ‘work in mental spaces where objects lose their extramathematical and procedural tracks and must translate them into symbolic expressions, which are highly synthetic, ideographic and relational’ (Arzarello, page 78). Only in such situations may students realise that ‘symbolism made it possible to carry out operations that do not have any corresponding sense with the initial statement of the problem’ (Radford, page 33). Thus, the relation between algebra and arithmetic may not be of the same nature as the relation between form and content. Arzarello, Bazzini and Chiappini in this volume suggest that the passage from one to the other could be negotiated in a game of interpretation (Arzarello et.al., page 75), managing the double obstacle of condensation (referring ‘to two different meanings [for the same symbolic expression]’ ibid. page 79) and evaporation (‘dramatic loss of the meaning of symbols’, ibid. page 79). Lins (page 10), recognising with other words these two central difficulties (he speaks of ‘epistemological limit’ in order to designate the impossibility of producing meaning for a given statement), insists on the idea that the crucial point is to produce a clear rupture not a transition. Both positions in fact acknowledge that arithmetic and algebra have their own form (or semiotic system) and their own meaning; if there is a rupture it is witnessed by a move in the means of control. The crucial question for students is that which way they could follow to ensure theniselves that the solution they propose to a problem is correct, or that the operations they have carried out are relevant and well used. As Arzarello, Bazzini and Chiappini point out in the introduction of their chapter: ‘it is very hard to convince students that they are wrong, insofar as the invented meaning often has its
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own justification, generally rooted in previously learned models, perhaps >)*=%#$ "DD*)D*%"(/20 %# (,/%* )># B)#(/+(e (Arzarello et al., page 61,my emphasis).
CONTROL, A DEADLOCK TO BREAK FOR A CONCEPTUAL MOVE Judiciously. Lins (page 10) reminds us that ‘justification is a constitutive part of knowledge’; I would have expressed this in a more general way by claiming that B)#(*)2 %+ D"*( )7 =#)>2/'$/: More precisely, to be in good command of any piece of knowledge, one must master a */D*/+/#("(%)# +0+(/91 a set of )D/"()*+ and a B)#(*)2 +(*4B(4*/€ in order to monitor one’s problem-solving activity and to take decisions (among which the decision that the problem at hand is solved). I would even claim that these three dimensions together with a characterisation of the ')9"%# )7 D*)32/9+ on which one is efficient is the best way to characterise the state of a personal conception associated with a piece of knowledge (Balacheff. 1999). The students Lins reports about in his chapter manipulate oranges and boxes of oranges; they manipulate quantities and not numbers, counting facts and not arithmetical operations. The control on their reasoning and other mental operation lies in the concrete world of reference with which they have been provided. In the next example Lins discusses the use of literals, but they are the means to speak about actions on objects of a referent world of buckets and tanks. Even if I recognise with the author that this is another level of abstraction. I believe that it is still not algebra because the control on the meaning of the symbolic expressions as well as the control on their manipulation depends on an interpretation in this referent world. I do take the question Lins asks: why would a pupil take on board the new meaning resulting from algebraic manipulation instead of, or even in addition to. the meaning directly constructed from the reference world (page 54)? But I would not see here a ‘didactical paradox’ (55), instead I would see a sign of a certain sense of an economy of practice. This same sense may be the origin of a contrast which Boero reports on in his chapter between students who, like those observed by Lins, ‘seem to transform the problem situation by thinking about the number of sheets and the weight of the envelope as physical variables’ (page 108) whereas others ‘put into a numerical equation the problem situation and transform the equation’ (ibid.). The dilemma of controlling the problem-solving process in most situations involving algebra in the school context is, taking Dettori words, the need for students to be able ‘to associate meanings with the symbol being used, and to manipulate symbols independently of their meaning’ (page 206). Bocro suggests the overcoming of this dilemma by the ability of ‘a partial ‘suspension of the original meaning’ of the transformed expression’ during the transformation process (Boero, page 99). But all authors insist on the tendency of students to come back to a
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reference world outside algebra in the case when they want to check either the relevance of the actions they performed or the validity of their solution. The anticipatory value of algebra, which Dettori, Garuti and Lemot (page 202) link to its proving value, is presented as one of the possible keys to open the way to algebra (as long as it is not used for trivial problems). How to prove the existence of a solution, or its completeness ? In the context of the use of a spreadsheet this author notices that ‘without making a formal proof, it is not clear, before finding the result, that there is certainly one, and, once the problem is solved, it is difficult to argue that the only solution has been found: while using algebra gives this certitude’ (ibid.) I think that this remark can be extended to all problem situations normally used in the context of teaching elementary algebra (again, provided that they have some complexity). The role of anticipation is especially emphasised by Boero in his chapter, because it is necessary for the control of symbolic manipulations which otherwise would develop in ‘blind perspectives’ (Boero, page 110). The role of anticipation is to allow ‘the process of transformation to be directed towards simplifying and resolving the task.’ (ibid. page 118). What seems to be dominating then is a view of algebra as a modelling tool. If students succeed in suspending the original meaning while they perform the symbolic computation, then the issue of meaning is left to the process of construction of the symbolic representation of the problem and then to the interpretation of the results obtained. We recognise here the schema proposed by Boero (page 99) where instead of modelling the author speaks of a formalisation as consisting of a ‘translation from +/9 into an expression of the algebraic language’ (ibid.) But in my opinion the word (*"#+2"(%)# hides the difficult task of identifying the relevant objects and their relations in the problem statement: a task which could involve an important effort of conceptualisation outside mathematics as such (even though one very often thinks that in word problems just common sense is involved). The main issue I would like to raise is that in a modelling process the validation of the reasoning as well as of the result is not internal to mathematics (here to algebra) but must take into account external constraints. For example, obtaining a negative number as a solution of an equation may destabilise the problem-solver. Gallardo (page 129) reminds us that interpreting negative solutions was the main difficulty facing medieval mathematicians, a difficulty which could be overcome either by constructing meaning for negative numbers or by discussing the so-called formalisation of the problem considered (the equation may not have been well chosen, or the problem was ill-posed). In the case of modelling, the problem solver will always be tempted to keep an eye on ‘the contextual sense of an expression’ (Arzarello et.al., page 63). and the ‘ideographic features of the algebraic language’ (ibid. page 63) facilitate this attempt to keep control on the computation from the related but external world of reference.
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The students’ solving world (ibid. page 68) will contain symbolic representations (we may call them algebraic) as well as means to manipulate them, but the control structures __ all through the solving process __will still refer to the external world of reference attached to the situation by the problem statement. Algebra is not there, but instead we see the functioning of what I would call +093)2%B "*%(,9/(%B which has it own rules and domain of validity. In some way I would say that symbolic arithmetic is to algebra what quantities are to numbers. Both may use the same representation system, and even common tools, but they are not submitted to the same control system (this phenomena is well illustrated by studies such as that of the carpenter and the apprentice by Nuñes 1993). Without making this explicit, teaching practices tend to turn symbolic algebra into '/ 7"B() content to be taught instead of algebra __a choice often made by ‘realistic mathematics educators’.
SYMBOLIC ARITHMETIC, A POSSIBLE WAY OUT Numeric algebra (Radford, page 28) or symbolic arithmetic? My suggestion is that the former is well adapted to describe the ancient mathematical practice of scientists who, like Diophantus, treated what should be Considered as genuine algebraic problems before the availability of the modern symbolic technology. The latter term is better adapted to describe the mathematical practices and tools of those who are confronted by arithmetical problems and who make use for this purpose of some symbolic technology 34( under the control of the initial context of the problem. Symbolic arithmetic uses a symbolic language like algebra, and it relies on operations close to those of algebra. Symbolic arithmetic and algebra seem to be competing for certain domains of problems, at least when one considers those problems which are proposed in the curricula for beginners. The idea that one particular problem could illustrate the general procedure to solve classes of problems (mentioned by Ursini) is often seen as a way to stimulate a focus on method. but it does not guarantee a shift towards algebra since on the contrary it shows the possibility of remaining within the frame of arithmetic. More crucial is the idea that whereas for some problems it is possible to trace the meaning of the steps of the symbolic manipulation, on the contrary for others the tracing is inore difficult, which suggests that they are ‘more algebraic’ (Dettori. page 195). But even in these cases, if it is more difficult, it is just ‘more difficult’ and not impossible; this is not enough to prove the necessity to leave the framework of arithmetic. On the contrary control may be what makes the difference between the two. In symbolic arithmetic one keeps as far as possible (except for exceptional eases) a parallel relationship between the computation and the referent world at all steps, the focus being on the meaning of the numerical evaluation of the symbolic expression.
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As Ursini writes in her chapter: ‘To conceive of the idea of ‘general object’ and to symbolise it using literals was the culmination of a long historical process’ (page 210). But back in the classroom this is not enough to achieve the breakthrough needed to enter the algebra world. and especially not for students (in the case of history, the study of the opposition to algebra has not been sufficiently carried out). In the case of students, the ability to manipulate a symbolic representation system (like the Logo language to which Ursini refers. or spreadsheets which we will consider later) does not mean that a shift of focus in the method has occurred (ibid, page 213). The absence of this shift may not in the first place be due to a lack of experience, as Ursini suggests (ibid. page 214) but simply to a lack of the need to do so _ I think that this is the case for most of the students observed and whose strategies have been analysed in this book. The case of spreadsheets is indicative of a possible confusion between algebra and symbolic arithmetic. I would even claim that the teaching of spreadsheets has represented the pinnacle of the constitution of symbolic arithmetic into an object of teaching. Indeed it is said that spreadsheets ‘can start the journey of learning algebra, but do not have the tools to complete it’ (Dettori et.al., page 206), but the missing tools are in fact precisely those which could provide students with the possibility of having algebraic control of the symbolic manipulation—on the contrary their controls are fundamentally based on numerical evaluations. All this is well illustrated by Filloy’s proposal to close the gap between the nonalgebraic methods of problem-solving and the Cartesian method (that is. the algebraic method) by a method of ‘successive exploration’ which ‘comes alive in a computational context such as that of electronic spreadsheet.’ (ibid. page 171). This attempt sounds like a confirmation of the role played by symbolic arithmetic in the teaching of algebra, it contains also the recognition of the impossible task for teachers since to really adopt the Cartesian method the student must accept an evolution of the use of symbolisation. But Filloy’s claim that ‘the method needs the confidence of the user’ (ibid, page 166) refutes the idea of a possible transition without rupture from arithmetic to algebra. If the rupture is not completed, then it is symbolic algebra which installs itself as the dominant student conception. Considering more deeply the issue of control, one may realise that an essential difference between symbolic arithmetic and algebra is a shift of emphasis from a pragmatic control to a theoretical control on the solution of the problems considered. This shift corresponds to the shift of confidence Filloy mentioned. It corresponds also to Dettori’s statement that ‘theoretical methods are the only ones that can prove properties such as correctness, completeness or existence of results’ (ibid. page 207), problems of proof are outside the scope of symbolic arithmetic. This is in my opinion far more important than even becoming aware of the differences in problemsolving methodologies between arithmetic, algebra and spreadsheets.
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Indeed. algebra may well be the student’s first occasion of an experience of the practice of a theoretical discourse, a practice which is characterised by the use of a specific language, but which cannot be reduced to this language, as perhaps Kirshner suggests.
CAS: AN ELECTRONIC HIGHWAY TO ALGEBRA Computer-based environments, because they required the manipulation of a symbolic language of communication seem to mathematics educators to provide a natural setting for the learning of algebra. They appear to be a good solution to the search for situations justifying the introduction of algebra while allowing the learner to keep enough control through the examination of a graphical display in Logo or of numerical results with spreadsheets. Logo was seen as a possible precursor of algebra. as Ursini reports: ‘Logo provided pupils with a language for expressing a general method and its object in a formal way prior to the introduction of the algebraic language’ (ibid. page 218, see also Healy et.al.. page 23 1). The generally shared expectation is that students would be able to transfer to paper and pencil algebra tasks the competencies they have developed by using algebra-like notations in the software environment (ibid. page 236). In my opinion designating the expression ‘algebra-like’ the kind of representation students use with spreadsheets is misleading, to speak of symbolic notation or language is a better choice for it does not imply a judgement on the nature of the conceptions involved. One may notice that Healy, et.al. recognises that ‘the generalisations pupils articulate are rarely justified except by reference to specific cases’ (ibid. page 235), a tendency that a spreadsheet environment does not discourage. More generally, ‘pupils do not spontaneously think in terms of a general algebraic object when first working in a spreadsheet environment. Their thinking is initially situated on the specific example with which they were working’ (Filloy et. al., page 173). Indeed spreadsheets keep a focus on numerical results and do not facilitate the validation o f the result by an examination of the relationships more or less hidden by the topography of the cells and the possibility of direct designation instead of naming of the variables. To use Dettori. Garuti and Lemut’s words 1 would say that spreadsheets provide students with an interesting experience of a symbolic technology, but they are ‘not fully algebraic’ (Dettori et.al., page 193). The dilemma mathematics educators and most researchers are faced with is due to the conviction that ‘introducing basic concepts of algebra [.. .] outside a problem-solving context would just make them meaningless for most students’ (ibid.). Spreadsheets seems to offer a way to overcome this dilemma, even though in the spreadsheet context ‘the equation ... is never explicitly expressed’ (ibid, page 198).
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Equations, standard algebraic manipulation of symbolic expression are now possible in Computer Algebra Systems (CAS), like DERIVE or MAPLE or MATHEMATICA .The form ofthe algebraic activity in these environments is rather different from that in the other ones mentioned above. If they share a symbolic system of representation, the nature of control the user can have on his or her action is necessarily different (if there is a reference to a numerical or a familiar problem situation, the environment does not encourage making a link to them). This suggests that the students’ conceptions emerging from the use of such environment would be different from these resulting from the paper and pencil environments or other software environments (Logo or spreadsheets). The use of CAS for initiating the learning of elementary algebra seems not yet to have been explored in a substantial manner, although these environments could be the place to explore Healy’s evocation of a ‘Vygotskian perspective [which] already exists to support the claim that algebra does not necessarily have to be crafted on to a sound arithmetic base’. (page 246). Continuing on this line I would suggest that the availability of fully algebraic manipulations on pocket calculators may open the access to situations of communication between students and the device, in order to obtain the transformation or evaluation of an expression in the framework of a genuine algebraic representation system. The expectation is simply that the computer will have a scaffolding function (ibid. page 240), the involvement of the students in algebra depending on their understanding and acceptance of the teacher intentions. Finally, the consequence of the often stated recognition. that the students’ world of reference is such that algebra is hardly needed, is that to enter a different D*)32d9"(%b4/ they have to be confident in the teacher’s promise that it is worth attempting the passage.
THE TEACHER: MASTER OF THE RUPTURE Let us pick up some quotations here and there in this book : Ursini recognises the need for a ‘substantial teacher’s support and guidance’ (ibid. page 213), in particular to struggle against ‘a tendency to look at Logo commands only from a practical point of view’ (ibid. page 214); Stacey and MacGregor suggest that ‘the difficulty experienced by many students in verbalising their Understanding highlights the need for teachers to promote the development of language specifically related to mathematics’ (ibid. page 147). In my opinion the interview presented by Stacey and MacGregor on page 145 illustrates this importance of the teacher. The didactical contract is what can permit learners to understand ‘what sort of rules the teacher will allow’ (ibid. page 150). In other words, the teacher plays a central role in allowing students to locate themselves in the relevant paradigm and to play the game of
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algebra, especially when it would be possible and even more economical for them not to do so. This emphasis on the role of the teacher is a sign, coming even from those who do not pose the question in these terms, that the ‘crucial step is to produce a rupture, not a transition’ froin arithmetic to algebra (Lins, page 56). Thus if it is not in terms of a transition, but in terms of a rupture, that the didactical problem must be described and solved, then the input of the teacher is to be recognised as the key clement of the learning to occur. The teacher has a role to play in allowing students to recognise this rupture, which should be lived as such. otherwise they may be involved in a silent transition (Lins, page 56) that some of them may not hear, and for that reason they may fail in this learning. What should be the behaviour of the teacher? The form of his or her involvement in the teaching is a difficult question left open beyond the recognition of this necessity.
CONCLUSION This book, rich with a great variety of analysis and of proposed solutions, leaves me with a question: what is algebra? Or what should it be from a didactical point of view? I agree with Lins that the choice of an answer to this question has strong effects on the research one may carry out, and on teaching. To define algebra by contrast with and with reference to arithmetic is an option defended in many places in this book, and it has consequences which several chapters examine in detail. The more severe consequences may be the risk of a massive Jourdain effect we may speak of (,/ +(4'/#(e+ "2$/3*" when students or the risk of perform just sophisticated activity involving symbolic manipulation teaching symbolic arithmetic without turning it into an object of teaching in an explicit way. Teaching symbolic arithmetic or teaching algebra? This appears to me to be the basic dilemma emerging from this book. Could symbolic arithmetic be taught as such? My answer would be ‘yes’, with the idea that it will imply strategies explicitly dedicated to the demonstration of the benefit of using and mastering a symbolic technology to treat problems of arithmetic. What about algebra? The passage from symbolic arithmetic to algebra implies the negotiation and the acceptation of a rupture with arithmetic. As in the case of linear algebra (Dorier 1997) or in the case of mathematical proof, there is no possible entrance to the world of algebra without a strong push and guidance from the teacher because there is no natural passage froin the D*)32d9"(%b4/ accessible from the child's world to the mathematical D*)32d9"(%b4/: —
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As suggested here and there in this book, algebra may be the first theoretical experience mathematics teaching offers to the student... NOTES 1
2 3
A)*9 means any written expression based on the use of algebraic language’ (Boero, Ch. 6), g;/9 consists of a mental representation and an external non-algebraic representation.’ (ibid.) All these examples come from a French library, but I have every expectation that analogous examples would be found in any library around the world. Control structure as a set of meta level rules rather than an operational level.
REFERENCES Allard, A. (1983) Les scolies aux Arithmetiques de Diophante d’Alexandrie dans le Matritensis Bibl. Nat. 4678 et Ies Vaticani Gr. 191 et 304, 60H"#(%)# Vol. 53, 2, 664-760. Arcavi, A. (1994) Symbol Sense: The Informal Sense-making in Formal Mathematics, A)* (,/ 2/"*#%#$ )7 !"(,/9"(%B+1 14, 3, pp. 21- 35. Arrighi, G. (1967) (ed.) !•5#()#%) '/_@ !"HH%#$,(P@ C*"(("()@ '%@ A%)*/*(%1@ ?%+"P@ L)94+@ 8"2%2/"#":@ Arighi, G. ( 1 974) C*"B("%) Lg533"B,): Pisa: Domus Galileana. Arzarello. F. (1989) The role of conceptual models in the activity of problem solving, 5B(/+ '/@ ?!&@ jFFF: Paris, 93 - 100. Arzarello, F. (I991) Procedural and relational aspects of algebraic thinking. ?*)B: )7@ ?!&@ jR1 Assisi. 8087. Arzarello, F. (1992a) La Ricerca in Didattica della Matematica, NgF#+/$#"9/#()@ '/22"@ !"(,/9"(%B" / '/22/ ;B%/#H/ F#(/$*"(/ I 5 (4) , 345-356 Arzarello, F. (1992b) Algebraic problem solving. in J.P. Mendes da Ponte et al. (eds ), !"(,/9"(%B"2 ?*)32/9 ;)2.%#$ M/+/"*B,1 NATO ASI SERIES VOLUMES, Springer Verlag, Berlin Arzarello, E. and Gallo, E. (eds.) (1994) Proceeding of the Workshop on the learning of Algebra Turin, 8 -10 October I992 (WALT), in M/#'%B)#(% '/22g-#%./*+%(• e '/2 ?)2%/B#%B) '% C)*%#)1 voll. 52/1. Arzarello, F., Razzini, L.. & Chiappini, G. (1993) Cognitive Processes in Algebraic Thinking: Towards a Theoretical Framework, ?*)B: jR ?!& E)#7/*/#B/: Vol. 11. Tsukuba, Japan, Vol. 1,138-145. Azarello, F., Bazzini, L. and Chiappini. 8: (1994a) 'The process of naming in algebraic thinking, ?*)B//'%#$+ ?!&IjRFFF1 Lisboa, vol. 11,40-47. Arzarello, F., Bazzini, L. and Chiappini, G. (1995) The construction or algebraic knowledge. Res. Forum Pres. in ?*)B: )7 ?!&IjFj1 Recitè,119-134. Arzarello, F., Chiappini. G.P., Lemut., E.. Malara, N. and Pellerey, M. (1992b) Learning to program a cognitive apprenticeship, in Lemut et al. (eds. ), E)$#%(%./ !)'/2+ "#' F#(/22%$/#(@ &#.%*)#9/#(@ 7)* N/"*#%#$ ?*)$*"99%#$1 NATO AS1 Series, vol. F I I I, Springer Verlag, Berlin, 284 -298. Asimov, 1. (1 959) M/"29 )7 #493/*+ Cambridge, MA: The Riverside Press. Australian Education Council (I991) 5@ #"(%)#"2@ +("(/9/#(@ )#@ 9"(,/9"(%B+@ 7)*@ 54+(*"2%"#@ +B,))2+1@ Curriculum Corporation (Australia). Carlton, Vie. Australian Education Council (1993) J"(%)#"2 9"(,/9"(%B+@ D*)7%2/1 Curriculum Corporation (Australia), Carlton. Vie. Ayer, A. J, (1986)C,/ D*)32/9 )7 =#)>2/'$/: New York: Viking Penguin Inc.. Bachelard, G. (1 976) N" A)*9"B%G# '/2 &+D%*%(4 E%/#(‚B): México: Siglo XXI Balacheff, N. (1999) !/"#%#$1 "@ D*)D/*(0@ )7@ (,/@ 2/"*#/*I9%2%/4@ +0+(/9@ (unpublished paper. 28 pages). Grenoble, Laboratoire Leibniz-IMAG. Balacheff, N., & Sutherland, R. (1994) Epistemological Domain of Validity of Microworlds: The Case of Logo and Cabri-géomètre. In R. Lewis, & P. Mendelson (Eds.). N/++)#+@ 7*)9@ N/"*#%#$@ S5Is]T (pp. 137-1 50). North Holland: Elsevier Science BV. Bartolini, P. (1976) Addition and subtraction of directed numbers. !"(,/9"(%B+ C/"B,%#$: Vol. 74 (March). pp. 34-41. Bednarz, N., Radford,L., Janvier,ß. & Lepage, A. (1992) Arithmetical and Algebraic Thinking in Problem Solving, ?*)B//'%#$+ )7 ?!&IjRF1 Durham, vol.1. 65 - 72 Bell, A. (1992) ?4*D)+/ %# ;B,))2 52$/3*"1 Plenary Lecture to the Algebra Working Group, ICME 7, Quebec, 1992. Bell, A. W. and Galvin W. P. (1977) g5+D/B(+ )7@ L%77%B42(%/+@ %# (,/@ ;)24(%)#@ )7@ ?*)32/9+@ F#.)2.%#$@ (,/ A)*9"(%)# &b4"(%)#+e1 Shell Centre for Mathematical Education. University of Nottingham. Bell, A., & Malone, J. (1993) Learning the Language of Algebra, ?*)B: jR FF ?!& E)#7/*#B/1 Tsukuba, Japan, Vol.1, 130-137. Bell, E. T. (1936) The meaning of mathematics. In W. D. Reeve (Ed.), C,/@ D2"B/@ )7@ 9"(,/9"(%B+@ %#@ 9)'/*#@ /'4B"(%)# (pp. 136-181). NCTM Eleventh Yearbook. New York: Bureau of Publications. Teachers College, Columbia University. Bereiter, E : (I991)Implications of connectionism for thinking about rules. &'4B"(%)#"2 M/+/"*B,/*1 VW1 10- I6 Biehler, R.. Scholz R.W., Strässer R. and Winkelmann, B. (I ( 994) L%'"B(%B+ )7 !"(,/9"(%B+ "+@ "@ ;B%/#(%7%B L%+%D2%#/1 Kluwer, Dordrecht.
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Stacey. K. (1981) 'Finding and using patterns in linear generalising problems', &'4B"(%)#"2 ;(4'%/+ %# !"(,/9"(%B+1@ VW1 147-164 Stifel, M. (1544) 5*%(,9/(%B" F#(/$*"1 Norimbergæ apud Iohan Petreium. Sutherland, R. (1980) Providing a Computer-based Framework for Algebraic Thinking, &'4B"(%)#"2 ;(4'%/+ %# !"(,/9"(%B+1 Vol. 20, No. 3, pp. 3 17-344. Sutherland, R. Howell, D. & Wolf, A. (1 995) 5 ;D*/"'+,//( 5DD*)"B, () !"(,+@ 7)*@ 8JRy &#$%#//*%#$1 Hodder Headline, London. Sutherland, R. (1993) Connecting Theory and Pi-actice: Results from the Teaching of Logo. &'4B"(%)#"2 ;(4'%/+ %# !"(,9"(%B+1 Vs1 pp 95-113 Sutherland, R. (1995) The Influence of Teaching on the Practice of School Algebra: What can we Learn from Work with Computers?, in M/#'%B)#(% '/2@ ;/9%#"*@ !"(/9"(%B) '/2 -#%./+(%"@ /@ '/2@ ?)2%(/B#%)@ '%@ C)*%#): Sutherland, R. (1991 ) 'Some Unanswered Research Questions on the Teaching and Learning of Algebra', A)* (,/ N/"*#%#$ )7 !"(,/9"(%B+ 11, 3, 40- 46. Sutherland, R. (1992) 'What is Algebraic about Programming in LOGO?', in Hoyles, C. and Noss, R. (Eds.), N/"*#%#$ !"(,/9"(%B+ "#' Nh8h1 M.I.T. Press, 37-54. Sutherland, R. and Rojano, T. (1993) 'Bridging the gap between Non-Algebraic and Algebraic Approaches to problem Solving in Mathematics', ?*)B//'%#$+ )7 (,/ R F#(/*#"(%)#"2 ;09D)+%49 )7 M/+/"*B, %# !"(,/9"(%B+ &'4B"(%)#1 Merida. Yuc. Mexico (in press). Sutherland, R. and Rojano, T. (1993) 'Spreadsheet Approach to Solv ing Algebra Problems'. C,/ r)4*#"2 )7 !"(,/9"(%B"@ 6/,".%)*:@ New Jersey, USA, Vol. 12,4, 1993, p. 353-383. Sutherland, K., & Pozzi, S. (1994) C,/ E,"#$%#$@ !"(,/9"(%B"2@ 6"B=$*)4#' )7 -#'/*$"'4"(/ &#$%#//*+P 5 M/.%/>)7 (,/ F++4/+1 The Engineering Council, UK. Sutherland. R.. & Rojano. T. (1993) A Spreadsheet Approach to Solving Algebra Problems. r)4*#"2 )7 !"(,/9"(%B"2 6/,".%)4*1 New Jersey, USA, XV (4), 1993, p. 353-383. Tall, D. (1992) L4"2%(01 593%$4%(0 "#' A2/K%3%2%(0P@5@?*)B/D(4"2@R%/>@)7@;%9D2/@5*%(,/9/(%B:@ Occasional paper, Mathematics Education Research Centre, University of Warwick. Tall, D. (1991) The psychology of Advanced Mathematical thinking. in: 5'."#B/' !"(,/9"(%B"2 C,%#=%#$1 D.Tall (ed.), Kluwer, Dodrecht. pp. 3-21. Tall. D, & Thomas, M., (1991) Encouraging Versatile Thinking in Algebra using the Computer, &'4B"(%)#"2 ;(4'%/+ %# !"(,/9"(%B+ VV1 125-147. Thomas, T., & Tall, D. (1986) The Value of the Computer in Learning Algebra Concepts. ?*)B//'%#$+ )7 (,/ C/#(, F#(/*#"(%)#"2 E)#7/*/#B/ 7)* (,/ N/"*#%#$ )7 !"(,/9"(%B+: London. Thompson, P. W. (1989) Artificial intelligence, advanced technology, and learning and teaching algebra. In Kieran, C. & Wagner, S. (Eds.), M/+/"*B, "$/#'" 7)* 9"(,/9"(%B+ /'4B"(%)#P M/+/"*B, %++4/+ %# (,/ 2/"*#%#$ "#' */"B,%#$ )7 "2$/3*" (pp. 135-16 1). Reston. VA: National Council of Teachers of Mathematics. Hillsdale, NJ: Lawrence Erlbaum Publishers. Thureau-Dangin, F. (1938a) La méthode de fausse position et l‘origine de l‘algebre, M/.4/ 'g5++0*%)2)$%/ /( 'g"*B,/)2)$%/ )*%/#("2/1 35, 71 -77. Thuruau-Dangin. F. (1938b) C/K(/+ !"(,f9"(%b4/+ 6"302)#%/+1 L eyde, Brill. Ex Oriente L UX ,1. Tiles. M. (1991) !"(,/9"(%B+ "#' (,/ 29"$/ )7 M/"+)#: London Routledge. Trujillo, M. (1987) -+) '/2@N/#$4"O/@ 52$/3*"%B)@/#@ 2" M/+)24B%G# '/ ?*)32/9"+ '/ 5D2%B"B%G# Tests de Maestria, Sección de Matematica Educativa del CINVESTAV del IPN. México, 1987. Ursini, S. (1996) General Methods, A Way of Entering Algebra World, this volume Ursini, S. (1990a) 'Generalisation Process in Elementary Algebra: Interpretation and Symbolisation', ?*)B//'%#$+ )7 (,/ A)4*(//#(, ? ! & E)#7/*/#B/1 México, 149-156. Ursini, S. (I 990b) 'El Lenguaje Arithmético-Algebráico en un Ambiente Computacional’. Cuadernos de Investigación, 15, IV, Julio. PNFAPM. México, 53-105. Ursini, S. (1991) 'First Steps in Generalisation Processes in Algebra’, ?*)B//'%#$+ )7 (,/@ A%7(//#(,@ ?!&@ E)#7/*/#B/1 Assisi, Italia. 3 16-323. Ursini, S. (1994) Pupils' Approaches to Different Characterisations of Variable in Logo. PhD Thesis, University o f London Institute of Education. Usiskin, Z (1988) Conceptions of School Algebra and Uses of Variables, in C,/@ F'/"+@ )7 52$/3*"1
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INDEX abstraction 94, 170, 199, 207, 2 14, 234, 253 additive method 136 algebraic formalisation 99, 118 algebraic manipulations 73, 192, 194, 231,258 algebraic method 156, 16 1, 172, 175, 256 algebraic reasoning 69, 72, 75, 107, 191 algebraic sense 16, 63, 64 algebraic symbol 32, 84, 85, 88, 89, 94,97 algebraic thinking 1, 16, 25, 28, 33, 40, 45, 46, 61, 65, 72, 75, 108, 124, 142, 261 algebraic understanding 233 anticipation 99, 109, 1 13, 116, 254, arithmetic Method 136 arithmetica 13, 18, 21, 31, 265, 271 arithmos 20, 22, 32, 25 1 axioms 85,93, 95
classroom 3, 4, 5, 34, 37, 45, 5 1, 58, 148, 163, 184, 187, 191, 193, 231, 237, 243,255 Cockcroft report 230, 233 cognition 5,53, 57, 88,95, 123 cognitive abilities 199, 207 cognitive processes 26 1, 270 cognitive psychology 89 comunication 42, 96, 156, 243 competent performance 95, 156 computer algebra systems 231 computer-based activities 339 conceptual frame 4, 65, 67, 73, 75, 79 condensation 252, 4, 75, 77, 79 conjecture 101, 105, 111, 116, 124, 212, 239, 252, 70, 75, 80 connectionism 90 content 30, 31, 44, 58, 79, 84, 105, 116, 121, 152, 166, 191, 201, 231, 233, 249, 252, 255, 264 culture 10, 243, 246, 5, 55, 57, 68, 77, 88 curriculum 141
Babylonian mathematics 16, 23, 28, 29, 32,265 Bachelard 121, 122, 127, 261 Balacheff 191, 192, 246, 249, 253, 262 Bell 59, 83, 156, 192, 235, 261 Booth 8, 85, 192, 209, 233, 262, Brousseau 123,262 Bruner 45, 240, 262
Davydov 51, 246, 263 De Numeris Datis 124, 266, 269 denotation 62, 70, 74, 75, 81 diagram 102, 104, 201, 250 didactic cut 124, 127, 138, 265 didactic sequence 191, 194, 206 didactical approach 2 didactical contract 258 didactical paradox 55, 253 didactics of algebra 121 Diophantus’ algebraic method, 13, 19
calculators , 32, 6 1, 184, 247, 258 calculus 60, 71, 179, 184, 188, 192, Chevallard 116, 263 Chinese mathematicians 132 Chomsky 98, 263
Eco 79, 264 educational Strategies 99
273
274
lNDEX
empirical solution methods 196 enculturation 3 1, 243 epistemological limit 40,45, 252 epistemological obstacles 122 epistemology 262, 269 Euclid’s Elements 21 evaporation 4, 77, 80, 252 external representation 100, 114
information 29, 33, 68, 74, 78, 86, 97, 105, 164, 184, 266 instruction and practice 109 interlocutor 4, 42, 43, 50, 55, 58 internalisation 79, 114, 1 13 interpretation 4, 13, 16, 25, 35, 51, 65, 73, 101, 107, 129, 152, 164, 175, 245,252, 135, 142
false position method 15, 22 formulae 161, 211, 239 frame 68, 75, 79, 94, 255, 121, 249 Frege 4, 62, 72, 264 function 8, 38, 43, 5 5 , 72, 78, 97, 130, 146, 178, 179, 199, 258, 58, 63, 108, 143, 149, 270 function machine 39,45
Janvier 100, 112, 139, 156, 361, 266 justification 41, 42, 54, 55, 57, 61, 95, 235, 250, 268 Kaput 83, 234, 267 Kieran 1, 61, 93, 142, 148, 178, 192, 209, 234, 249, 267 Klein 40, 209, 267 knowledge 4, 5, 10, 18, 34, 40, 51, 53, 54, 57, 63, 67, 78, 86, 95, 100, 121, 137, 148, 161, 179, 227, 234, 243, 251, 261, 265, 268, 270 Krutetski 79, 267 language 8, 16, 24, 28, 34, 42, 57, 61, 68, 72, 96, 108, 112, 118, 124, 134, 141, 155, 164, 188, 197, 206, 218, 228, 238, 245, 256, 263, 272 learning 1, 5, 44, 57, 76, 80, 86, 94, 121, 129, 142, 148, 169, 178, 192, 198, 211, 231, 233, 239, 246, 256, 264, 269, 272 linguistics 4 literal notation 6, 38, 246, logical rules 95 logo 7, 155, 212, 240, 266, 271
general number 7, 138,213, 221 general object 213, 255, generalised arithmetic 178, 250 generalising 6, 32, 11 1, 150, 21 1, 203, 235,242,271 genetic epistemology 234 geometric problems 2 10 geometry 25, 77, 34, 58, 85, 109, 210, 262 Greek mathematics 13, 18,40, 210 Hart 148, 233, 265 Herscovics 108, 124, 142, 209, 264, 265 Heuristics 20 history 34, 247, 265 Hofstadter 83, 90, 266 hypothetical thought 76 identities 192, 264 implicit knowledge 41 inequalities 193, 246, 271 informal 1, 148, 271
manipulation 5, 8, 77, 84, 95, 193, 197, 235, 246, 249, 257 mathematical objects 64, 67, 79, 192, 214,227 mathematical problems 1, 23, 30, 129, 178
INDEX mathematical structure 99, 107, 118, 238 meaning 3, 9, 29, 34, 49, 73. 99, 108, 129, 155, 163, 177, 185, 192, 201, 213, 235, 250 metacognitive 78, 118, 207
natural language 68,74,79 negative numbers128 notation 2, 98, 125, 130, 141, 151. 179, 185, 211, 242,257 obstacle 10, 75, 122, 127, 136, 189, 209, 220, 238, 250 open problem situation 103
275
semantic fields 4, 44, 46, 57, 268 semiotics 155 sense 2, 11, 25, 44, 58, 61, 96, 119, 138, 165, 195, 221, 247 Sfard 61, 85, 128, 246, 271 social 2., 9, 31, 55, 76, 95, 155, 182, 227, 269 socio-cultural 68, 269 spreadsheet 171, 191, 198, 206 structural algebra 83, 250 structural approach 84, 245, 250 symbol 8, 16, 32, 34, 84, 89, 97, 134, 219, 247, 250, 267 symbol system 87 syncopated 36, 134, 210 syntax of algebra 146, 150, 267 synthesis 80, 199, 207, 249 system 84, 89, 250
parameters 7, 73, 178, 188, 195, 203, 246 parentheses 9 1, 96, 1 17 parsing 84, 9 1, 96 pattern-based approach 141, 149 Tall 207, 234, 271 Piaget 8, 79, 121, 148, 233, 262, 269 teacher 262, 266, 269, 270, 272 teaching 1, 8,30, 34, 45, 58, 77, 85, 117, pre-algebra 2, 175 problématque 258, 259 129, 137, 149, 163, 165, 189, 199, process 3, 19, 37, 45, 56, 62, 75, 79, 92, 211, 229,240,254,259,269 104, 179, 225, 234, 246, 261 text 4, 14, 29, 35, 40, 51, 57, 67, 72, 100, proportional thinking 13, 16, 23, 34 125, 131, 138, 155, 168, 193, 197, 251 proving 105, 196, 290 textbooks 89, 117, 124, 150, 158, 161, pseudo structural 65, 8 1 242, 266 ratio and proportion 265 theorem 101, 106 rationalisation 96 theoretical approaches 2 recurrence relationships 143, 146 tool 8, 10, 24, 29, 34, 50, 59, 70, 105, referential approach 250 118, 192, 199,243,250,263 relationships 6, 29, 39, 43, 51, 64, 72, 79, transformation 3, 54, 75, 92, 99, 113, 105, 131, 141, 170, 209, 235, 238, 123, 187, 235, 253, 258 250, 257 turtle graphics 223, 224 representation system 238, 253, 258 unknown 4 scaffolding 240, 247, 258 second-degree equations 25
276
variable 7, 73, 138, 167, 179, 184, 199, 209, 219, 228, 233, 237, 241, 266, 272 Vygotsky 3, 62, 79, 239, 243, 272
INDEX
Wertsch 9, 272 Wittgenstein 272
Mathematics Education Library !"#"$%#$ &'%()*P A.J. Bishop, Melbourne, Australia 1.
H. Freudenthal: L%'"B(%B"2 ?,/#)9/#)2)$0 )7 !"(,/9"(%B"2 ;(*4B(4*/+: 1983 ISBN 90-277-1535-1; Pb 90-277-2261-7
2.
B. Christiansen, A. G. Howson and M. Otte (eds.): ?/*+D/B(%./+ )# !"(,/9"(%B+ &'4B"(%)#: Papers submitted by Members of the Bacomet Group. 1986. ISBN 90-277-1929-2; Pb 90-277-21 18-1
3.
A. Treffers: C,*// L%9/#+%)#+: A Model of Goal and Theory Description in Mathematics Instruction The Wiskobas Project. 1987 ISBN 90-277-2165-3
4.
S. Mellin-Olsen: C,/ ?)2%(%B+ )7 !"(,/9"(%B+ &'4B"(%)#: 1987 ISBN 90-277-2350-8
5.
E. Fischbein: F#(4%(%)# %# ;B%/#B/ "#' !"(,/9"(%B+: An Educational Approach. 1987 ISBN 90-277-2506-3
6.
A.J. Bishop: !"(,/9"(%B"2 &#B42(4*"(%)#: A Cultural Perspective on Mathematics Education. 1988 ISBN 90-277-2646-9; Pb (1991) 0-7923-1270-8
7.
E. von Glasersfeld (ed.): M"'%B"2@E)#+(*4B(%.%+9@%# !"(,/9"(%B+@&'4B"(%)#:@ 1991 ISBN 0-7923-1257-0
8.
L. Streefland: A*"B(%)#+ %# M/"2%+(%B !"(,/9"(%B+ &'4B"(%)#: A Paradigm of Developmental Research. 1991 ISBN 0-7923-1282-1
9.
H. Freudenthal: M/.%+%(%#$ !"(,/9"(%B+ &'4B"(%)#: China Lectures. 1991 ISBN 0-7923-1299-6
10.
A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.): !"(,/9"(%B"2@<#)>2/'$/P@ F(+ 8*)>(,@C,*)4$,@C/"B,%#$: 1991 ISBN 0-7923-1344-5
11.
D. Tall (ed.): 5'."#B/' !"(,/9"(%B"2 C,%#=%#$: 199 1
12.
R. Kapadia and M. Borovenik (eds.): E,"#B/ &#B)4#(/*+P ?*)3"3%2%(0 %# &'4B"(%)#: 1991 ISBN 0-7923-1474-3
13.
R. Biehler, R.W. Scholz, R. Sträßer and B. Winkelmann (eds.): L%'"B(%B+ )7!"(,I /9"(%B+ "+ " ;B%/#(%7%B L%+B%D2%#/: 1994 ISBN 0-7923-2613-X
14.
S. Lerman (ed.): E42(4*"2 ?/*+D/B(%./+ )# (,/ !"(,/9"(%B+@E2"++*))9:@1994 ISBN 0-7923-2931-7
15.
0. Skovsmose: C)>"*'+ " ?,%2)+)D,0 )7 E*%(%B"2 !"(,/9"(%B+ &'4B"(%)#: 1994 ISBN 0-7923-2932-5
16.
H. Mansfield, N.A. Pateman and N. Bednarz (eds.): !"(,/9"(%B+ 7)* C)9)**)>e+ k)4#$ E,%2'*/#: International Perspectives on Curriculum. 1996 ISBN 0-7923-3998-3
17.
R. Noss and C. Hoyles: m%#')>+ )# !"(,/9"(%B"2@!/"#%#$+:@Learning Cultures and Computers. 1996 ISBN 0-7923-4073-6; Pb 0-7923-4074-4
ISBN 0-7923-1456-5
18.
N. Bednarz, C. Kieran and L. Lee (eds.): 5DD*)"B,/+ () 52$/3*": Perspectives for Research and Teaching. 1996 ISBN 0-7923-4145-7; Pb 0-7923-4168-6
19.
G. Brousseau: C,/)*0 )7 L%'"B(%B"2 ;%(4"(%)#+ %# !"(,/9"(%B+: Didactique des Mathématiques 19701990. Edited and translated by N. Balacheff, M. Cooper, R. ISBN 0-7923-4526-6 Sutherland and V. Warfield. 1997
T. Brown: !"(,/9"(%B+ &'4B"(%)# "#' N"#$4"$/: Interpreting Hermeneutics and ISBN 0-7923-4554-1 Post-Structuralism. 1997 21. D. Coben, J. O’Donoghue and G.E. FitzSimons (eds.): ?/*+D/B(%./+ )# 5'42(+ N/"*#I %#$ !"(,/9"(%B+: Research and Practice. 2000 ISBN 0-7923-6415-5 20.
22.
R. Sutherland, T. Rojano, A. Bell and R. Lins (eds.): ?/*+D/B(%./+ )# ;B,))252$/3*": 2000 ISBN 0-7923-6462-7
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