An Invitation to Critical Mathematics Education

An Invitation to Critical Mathematics Education

An Invitation to Critical Mathematics Education Ole Skovsmose Aalborg University, Denmark

SENSE PUBLISHERS ROTTERDAM/BOSTON/TAIPEI

A C.I.P. record for this book is available from the Library of Congress.

ISBN: 978-94-6091-440-9 (paperback) ISBN: 978-94-6091-441-6 (hardback) ISBN: 978-94-6091-442-3 (e-book)

Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands https://www.sensepublishers.com

Printed on acid-free paper

All Rights Reserved © 2011 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Table of Contents Acknowledgements

vii

Introduction: Preoccupations

1

Chapter 1: Mathematics education is undetermined 1.1 Mathematics education is disempowering 1.2 Mathematics education is empowering 1.3 Being undetermined

5 7 10 14

Chapter 2: Diversity of situations 2.1 A bias in mathematics education research? 2.2 Contrast through globalisation and ghettoising

17 18 19

Chapter 3: Students’ foregrounds 3.1 Foreground 3.2 Intentionality and learning 3.3 Meaning in mathematics education

21 21 24 27

Chapter 4: Landscapes of investigation 4.1 Entering a landscape of investigation 4.2 Milieus of learning 4.3 Leafing through a newspaper 4.4 Moving between different learning milieus 4.5 Zones of risks and possibilities

31 32 39 42 45 47

Intermezzo: The modern conception of mathematics Mathematics and natural science Mathematics and technology Mathematics and purity Modern mathematics education

49 49 52 53 56

Chapter 5: A critical conception of mathematics 5.1 Mathematics, discourse, and power 5.2 Dimensions of mathematics in action 5.3 Wonders, horrors, and reflections

59 60 62 68

v

TABLE OF CONTENTS

Chapter 6: Reflection 6.1 Reflections on mathematics 6.2 Reflections with mathematics 6.3 Reflections through mathematical inquiries

71 72 76 77

Chapter 7: Mathemacy in a globalised and ghettoised world 7.1 Mathematics education world-wide 7.2 Practices of marginalised 7.3 Practices of consumption 7.4 Practices of operation 7.5 Practices of construction

81 82 84 87 88 89

Chapter 8: Uncertainty

93

References

99

Name Index

109

Subject Index

111

vi

Acknowledgements Let me acknowledge immediately that this book did not turn out as I expected. I wanted to make an uncomplicated presentation of critical mathematics education and to draw only on material and examples with which I was already familiar and which I had written about. Sure, I did use such material and examples, but this did not ensure the straightforwardness and clarity I had hoped for. Maybe I also need to acknowledge that it is not possible, a least not possible for me, to provide a short and clear presentation of critical mathematics education. I certainly want to acknowledge that I have received much help in writing this book. I want to give thanks for the very many suggestions and improvements given me by Peter Gates, Aldo Parra and Miriam Godoy Penteado, my wife. I also want to thank Kristina Brun Madsen for her careful language revision and for organising the manuscript ready for print. Rio Claro, January 2011 Ole Skovsmose

vii

Introduction Preoccupations In many cases it does not make sense to distinguish between a phenomenon and the discourse about the phenomenon. The discourse constitutes the phenomenon and comes to make part of it. Different languages not only provide different world views but also different worlds. Our life-worlds are discursively constructed. However, there might be limitations to an extreme discourse relativism, which not only blurs the distinction between language and reality, but claims this very distinction to be meaningless. To me the interesting thing is not to repeat an extreme relativism as if it was a universal truth, but to address it critically. Does it make sense, in some situations, to talk about ‘reality’ and ‘discourse about reality’ as interacting, although this interaction does not annihilate the very distinction? Or do we in fact end up with one category: a discourse-reality unity? I find that there are shortcomings to any universal discourse relativism. To make a simple observation: According to statistics, different groups of people have different opportunities in life, including life expectancy. Such an observation is not easily changed through change of discourse. We cannot eliminate poverty and the implications of poverty through such a change. We have to make real changes. There is, however, something which is highly dependent on discourses, and that is our preoccupations. Through a discourse we can express specific preoccupations; and through a change of discourse we may come to change the preoccupations too. For example, the way one looks at poverty depends upon the discourse one uses to talk of it. One might, for instance, claim that the cause of poverty is lack of education, which in turn is due to lack of willingness to be educated. Accordingly one might develop a discourse within which poverty comes to be seen as self-inflicted. One might also develop a discourse according to which poverty is a consequence of exploitation exercised by a neoliberal economic system. However, such a change of discourse is no simple thing, as discourses are deeply ingrained in traditions, priorities, 1

INTRODUCTION

culture, ideologies and political systems. Changing a discourse means changing life-worlds, if not worlds. A variety of discourses are applied when one engages in talk about education. The teachers’ staffroom discourse concerns, for instance, the handling of difficult students. The administrative discourse addresses the organisation of staff and school. Political discourses may see schools as part of society’s production schemes and talk about education in input-output figures. A variety of theoretical discourses embrace different interpretations of learning, teaching, meaning, evaluation, etc. Radical discourse relativism would say that ‘education’ and ‘discourses about education’ melt together and that there is no educational reality as such. I prefer, however, to consider ‘education’ and ‘discourses about education’ as a blurred distinction, as an interacting relationship, but still as a relationship. I do not find that education can be subjected to absolute discourse relativism. Certainly there are very many different ways of expressing our preoccupations, and I will present some concepts through which I try to express some preoccupations with respect to mathematics education. It is through the formulation of these preoccupations that I want to elucidate critical mathematics education.1 I do not see critical mathematics education as a special branch of mathematics education, nor do I relate it to certain classroom pedagogy or particular curriculum content. Instead I see it as an expression of some preoccupations or concerns with respect to mathematics education. I will present a few terms by means of which one may be able to express some of these. To think of this fragile conceptual network as a discourse of critical mathematics education is a massive exaggeration. However, even though the network might be rudimentary and fragile, the preoccupations might be extensive and profound. I see mathematics education as being undetermined. It is without ‘essence’. It can be acted out in many different ways and come to serve a grand variety of social, political, and economic functions and interests. A rich exploration of issues of critical mathematics education is found in Alrø, Ravn, and Valero (Eds.) (2010); Appelbaum with Allan (2008); Ernest, Greer and Sriraman (Eds.) (2009); Greer, Mukhopadhyay, Powell, and NelsonBarber (Eds.) (2009); Mora (Ed.) (2005); and Sriraman (Ed.) (2008). See Skovsmose (in print) for a discussion of critical mathematics education in terms of concern. 1

2

PREOCCUPATIONS

I will use the notion of situation in order to emphasise the importance of discussing processes of teaching and learning with respect to social, political, cultural, and economic contexts. In our globalised and ghettoised world, there is a huge diversity of sites for teaching and learning mathematics, and this diversity has to be addressed. Through the notion of students’ foreground I want to address how students might experience possibilities. This further relates to notions like intentionality and meaning. How students construct meaning depends on how they may connect their learning activities to their foreground and to their situation in general. Through landscapes of investigation I want to explore educational possibilities, in particular those that reach beyond the school mathematics tradition. Such an exploration brings us to discuss notions like inquiry, comfort zone, and risk zone, which in turn represents a zone of possibilities. A critical conception of mathematics will be explored through mathematics in action and the variety of forms in which mathematics is brought into effect. This could be in technological, economic and business settings; it could be in all kinds of trade and everyday settings. Our life-world is deeply structured through mathematics in action. All form of actions need reflection, which also applies to mathematics in action. Through this observation, the very notion of reflection gets amplified, which in turn brings us to consider the notions of mathemacy and dialogue. Mathemacy can be interpreted in different ways, and I interpret it as also referring to social response-ability. This makes it possible to formulate some of the aspirations of critical mathematics education, including what it could mean to establish a mathematics education for critical citizenship. It should never be forgotten that as soon as one wants to operate with grand notions like ‘social responsibility’, one is on thin ice. The concerns of critical mathematics education cannot be formulated with reference to any well-defined framework of ideas and priorities. I see, instead, any critical activity as connected to a profound uncertainty. This has to be recognised as part of the formulations of preoccupations of critical mathematics education. What preoccupations, then, can be formulated though this fragile conceptual network established by: undetermined, situation, student’s 3

INTRODUCTION

foreground, landscapes of investigation, critical conception of mathematics, reflection, and mathemacy? I do not have any well-defined list mind, but throughout the rest of this book I will try to express some preoccupations, and let them be interwoven with uncertainties.

4

Chapter 1 Mathematics education is undetermined In order to understand what might be meant by ‘mathematics education is undetermined’, let me start by saying a few words about ‘mathematics’, ‘mathematics education’ and ‘undetermined’.2 I consider ‘mathematics’ an open concept, which, depending on the discourse one uses, might acquire many different possible meanings. In Philosophical Investigations, Ludwig Wittgenstein talks about the variety of language games, and ‘mathematics’ might well be operating as a variety of such games. While mathematics as a research field includes a vast domain of unsolved issues and conceptions in development, mathematics as a school subject refers to a well-defined body of knowledge parcelled out in bits and pieces to be taught and learned according to pre-formed criteria. Mathematics could, however, also refer to domains of knowledge and understanding that are not institutionalised through research or curriculum structures. Thus for example, we can locate mathematics in many work practices. Mathematics is part of technology and design; it is part of procedures for decision making; it is present in tables, diagrams, graphs. We can experience a lot of mathematics by just leafing through any daily newspaper. According to the language-game metaphor, this variety of mathematics need not just be different expressions of the same underlying ‘genuine mathematics’; rather, alternatively very different concepts of mathematics could be in use. We might only be dealing with the same word or phrase whose meaning and operationalisation could be different. As a consequence, perhaps we had better give up the assumption that A preliminary version of this chapter has been presented at the 32nd Encontro da Associação de Pós-Graduacao e Pesquisa em Educação (ANPED), Caxambu, Minas Gerais, Brazil, 4–7 October, 2009. 2

5

CHAPTER 1

it is possible to provide a neat and universal defining clarification of mathematics. Well-intended definitions, as for instance suggested by logicism (describing mathematics as a set of tautologies) or by formalism (describing mathematics as a formal game governed by explicitly stated rules), might just be shrouding the possibility that there are no unifying characteristics of mathematics. I shall try to keep these observations in mind when I talk about mathematics. I do accept this might well bring me to a difficult position, since I have already used the word ‘mathematics’ a fair number of times and will continue to use it. However, I will continue to let it be so. Similarly, ‘mathematics education’ refers to a variety of activities. We could think of both teaching and learning and the very many different contexts in which they both occur. Mathematics education takes place in schools, where the teaching is mainly taken care of by the teacher, and the learning mainly by the students. Yet ‘mathematics education’ could also refer to activities outside of the school. Mathematics could be taught and learnt in work places and in many daily activities; it could be taught and learnt even when the whole setting has little to do with mathematics, say, when someone is shopping, checking accounts, discussing the news, etc. I will keep all such examples of mathematics education in mind. And finally, addressing the meaning of ‘undetermined’ – just what could that mean? A social process could be undetermined in the sense that it could result in very different things. The situation is open, and so is its outcome. The use of ‘undetermined’ could remind us of a common use of the word ‘critical’ in medicine. One could find the situation of a patient to be critical. This means that his or her situation is not stable and could dramatically change for the worse at any moment; it could turn ‘both ways’ – and it certainly makes a dramatic difference which way it turns. In general, I consider something to be undetermined if it could develop in very different ways, depending on factors which might not be possible to comprehend. The development could be simply ‘out of control’ and proceed randomly. This gives the following reading of the headline of this chapter ‘mathematics education is being undetermined’: mathematics education – understood in a broad sense – can be acted out in very many different ways, and this could really make a difference, for the good or for the bad. 6

MATHEMATICS EDUCATION IS UNDETERMINED

1.1 Mathematics education is disempowering In the literature we can find many examples of mathematics education which looks ghastly, often personalised by a mathematics teacher who for example dominates the students and with devastatingly cold sarcasm, castigates those who do not grasp the elegance of a mathematical proof. Mathematics education may operate with socio-political naivety and blindness. The film Life is Beautiful, directed by Roberto Benigni, includes a scene which provides a grotesque illustration of this. The first, more humorous, part of the film takes place in a provincial Italian city before the Second World War, where the fascination with Nazi Germany was part of the fascistic outlook. In a short scene, we listen to an Italian educator who has visited Germany and was impressed by what she saw. There, 7 year old German children were able solve a problem such as the following: A lunatic costs the state 4 Marks per day. A cripple, 4.5 Marks per day. An epileptic, 3.5 Marks per day. The average is 4 Marks per day, and the number of patients is 300,000. How much would we save if these individuals were eliminated? The Italian educator could not believe that seven year olds had to solve a problem like this. These are after all difficult calculations! The children would need at least some notion of algebra. A man listening to the educator’s explanation emphasises that it is just a multiplication (apparently assuming an equal number of lunatics, cripples and epileptics): “300,000 times 4. Killing them all we will save 1.200.000 Marks a day. It is easy, right?” The educator agrees, but her point is that in Germany, 7 year old children can do it, while such a problem is far beyond the capacity of Italian children that age. Exercises play a crucial role within the school mathematics tradition. Thus, during their time in school, most children will be solving more that 10.000 exercises. However, not much mathematical creativity is cultivated through working on such exercises. Could it be that some deep socio-economic irrationality is maintained as part of mathematics education? Could it be that this part of the educational system the world over sustains a dysfunction? Or could it be that this is not dysfunction, but rather a kind of functionality which is actually much 7

CHAPTER 1

appreciated in today’s labour market, but which we as mathematics educators are not really prepared to acknowledge? Let us take a more careful look at a possible exercise: A shop is offering apples for 0.12 Euro apiece, and for 2.8 Euros for bags containing 3 kilos. There are 11 apples to each kilo. Calculate how much Peter will save if he buys 15 kilos of apples in bags of 3 kilos instead of buying them individually. As most other exercises from the school mathematics tradition, this exercise has just been invented at a desk. There is no need to do any empirical investigation in order to come up with similar exercises within this tradition. Furthermore, we can observe that concerning this exercise, all the information given can be considered to be exact. Thus, when doing the calculation, one can be sure there are 11 apples, and exactly 11 apples, to each kilo, just as we can be sure that the price is exactly 0.12 Euro for one apple. That we are dealing with two different kinds of truths is of no significance, and need not be addressed in any way as part of formulating the solution. Any information provided in the text of an exercise can be considered exact and sufficient. And furthermore, the information provided in the exercise is both sufficient and necessary for solving the problem. Based on the given information, it is possible (and legitimate within mathematics classrooms) to calculate the one and only correct answer. It is not necessary for the students to try to get more information. Certainly there is no need for them to leave the classroom in order to search for supplementary information about prices or to check if it makes sense to assume that 11 apples weigh one kilo. This could remind us of the principal step in industrialisation: controlling the workforce. A simple device of the industrial revolution was to bring the workers all together in factories, and there to provide them with all the necessary tools in one place and all possible reasons for the workers to leave the factory were eliminated. A similar logic of control also makes up part of the school mathematics tradition. All necessary information is provided and the students can solve the exercise while remaining seated at their desks. An exercise establishes a micro-world, where all measures are exact, and where the information given is both necessary and sufficient in order to calculate the one and only correct answer. 8

MATHEMATICS EDUCATION IS UNDETERMINED

Such exercises have to be solved correctly and the correctness of the answer depends on many things. If a student may have made a wrong calculation, it could be that he or she has chosen a wrong algorithm. A student might have copied the exercise wrongly from the textbook and, for instance, written 0.22 instead of 0.12 and such a mistake will result in a wrong answer. One can also have solved the wrong exercise: “Oh, Johnny, this exercise is not for today. Right now you have to do the exercises on page 34.” Michel Foucault has talked about a ‘regime of truths’. According to him every society endorses some categories, which come to designate what counts as truth. The establishment of ‘regimes of truth’ is a historical process, as all categorical frameworks are part of an epoch. All discourses are culture- and context-bound, and thus come to determine what to count as true: “Each society has its regime of truth, its ‘general politics’ of truth – that is, the types of discourse it accepts and makes function as truth; the mechanisms and instances that enable one to distinguish true and false statements; the means by which each is sanctioned; the techniques and procedures accorded value in the acquisition of truth; the status of those who are charged with saying what counts as true.” (Foucault, 2000: 131) Similarly, the school mathematics tradition also exercises its regime of truths. From the perspective of understanding mathematics many regulations and corrections, so characteristic for the school mathematics tradition, appear irrational. However, when students have been directed through the 10.000 exercises, they might have learnt something which need not have much to do with any mathematical understanding. Their learning might crystallise into a prescription readiness.3 Just take a look at the formulations of exercises: “Reduce the expression…!” “Solve the equation…!”, “Find x, when…!” “Calculate how much Peter will save…!” These exercises seem to take the form of a long sequence of instructions. Could it be that the school mathematics tradition cultivates a prescription readiness, which prepares the students for participating in work processes where a careful following of step by step instructions without any question is essential? Could it be that such a prescriptionreadiness is serviceable for very many job functions in our society and 3 For a discussion of ‘prescription readiness’, see Skovsmose (2008a). See also Christensen, Stentoft and Valero, P. (2007).

9

CHAPTER 1

that the school mathematics tradition serves society perfectly well in exercising this readiness? Could it be that a prescription-readiness, including submission to a regime of truths, cultivates a socio-political naivety and blindness that is appreciated at today’s labour market? Could it be that a prescription-readiness fits perfectly well the priorities of a neo-liberal market, where hectic and unquestioned production serves the economic demands?

1.2 Mathematics education is empowering The term ‘empowerment’ can be interpreted in different ways with reference to mathematics: we can consider a classic notion of intellectual empowerment; we can talk about empowerment in pragmatic (and individual) terms; and we can think of empowerment in socio-political terms. I am sure that there are many other interpretations of empowerment with reference to mathematics, but for the moment I restrict myself to considering just these three. The classic idea of intellectual empowerment through mathematics education draws from a long tradition in philosophy and epistemology. It has been pointed out that while many sorts of assumed knowledge and ways of thinking have deceived the human mind, mathematics has not. Instead mathematics represents a unique example of genuine knowledge. Since the advent of ancient Greek philosophy, the notions of knowledge and certainty have been related. Thus, Plato claimed that knowledge with certainty was within human reach. In fact the most splendid example was mathematics. According to Plato, intellectual capacity enabled human beings to discover properties of the world of ideas. Later, through the scientific revolution, the powers of mathematics reached a new format. It became recognised that the laws of nature had a mathematical format. Thus, through mathematics, and only through mathematics, it became possible to grasp basic features of God’s creation. Both lines of argumentation – concerning certainty and insight into nature – established mathematics as a sublime form of intellectual empowerment. The pragmatic (and individual) interpretation of empowerment developed along a different line of argumentation. It emphasises the power that mathematics brings to bear through its applications, and 10

MATHEMATICS EDUCATION IS UNDETERMINED

a range of such applications emerged along with the industrial revolution. There are many examples to highlight: spectacular applications in technology and applications that makes part of everyday routines. Furthermore, mathematics education can empower people by providing them with qualifications that are important for participating in a variety of practices. In particular, mathematics education could ensure many people obtain a good position in the labour market, which means (personal) empowerment. A socio-political interpretation of empowerment brings the discussion in a different direction. Here I can refer to very may different formulations of mathematics education for social justice. 4 The claim is that through mathematics education it is possible to develop an insight that has a broad social and political significance. This has been expressed through different conceptual frameworks drawing from more general formulations of critical education. Thus Paulo Freire has talked about an education that brings about a concientização; Theodor Adorno has talked about an education for Mündigkeit; others have talked about ‘emancipation’ as an educational notion; others still have talked about an education that could bring about critical citizenship.5 It should also be noted that many of these formulations belong to a first phase of critical education. There is a real need for renewed consideration. In order to illustrate an attempt to provide a socio-political interpretation of empowerment with reference to mathematics education, let me refer to the project ‘Energy’ which I have described elsewhere.6 The students that participated in the project were 14–15 years old, and the teacher Henning Bødtkjer conducted the project. The overall ‘empowering’ idea was that the project was to bring about an insight that made the students able to understand and address some socio-economic issues of general relevance, and which at the same time could be explored more specifically through mathematics. The project addresses input-output models for energy. The students were invited to have breakfast at the school where they carefully measured everything they drank and ate, and calculated how high an energy input the breakfast represented. The calculations were based 4 5 6

See, for instance, Gates (2006). See, for instance Freire (1972), and Adorno (1971). For a more detailed description for the project, see Skovsmose (1994).

11

CHAPTER 1

on all the information available about the energy content, measured in kJ, that any kind of food contains. The energy output was obtained through the activity of cycling. It was calculated how much energy each student used on a particular bicycle trip. The calculation was based on simplified formulas from sports research. The formulas expressed the use of energy as a function of different parameters like speed, length of the trip, type of bike, and the ‘frontal area’ of the cyclist. It was possible to measure the length of the trip, which was the same for all students, and the speed, which was individual. The ‘frontal area’ of each of the cyclists was a parameter more difficult to handle. However, a method was found and the students could complete their calculations of the consummation of energy. Bringing the two calculations together they could get a first experience of what input-output calculations with respect to energy could mean. After this introduction, the project turned to a different yet major issue, namely input-output figures for farming, in particular with respect to food production. The calculations were carried out with reference to a particular farm, not far from the school. The first step in the input calculation was to estimate how much energy, in terms of petrol, was used in order to cultivate a particular field in the space of a year. The field had to be gone over several times with different tools: the plough, the harvest, the sprayer, etc. The students took notes of all the procedures and measured the breadth of the different tools. They measured the size of the particular field to which all the calculations were related, and they calculated how many kilometres a year the farmer had to drive the tractor in preparation of the field. The students were notified of the tractor’s use of petrol per kilometre, and on this basis one part of the energy-input was estimated. The field was used for the growth of barley, and the energy content in the seeds used for sowing was also estimated. The next step was to estimate the energy output from the field. At this time, students found out how much barley could be produced on the particular field, and they looked up statistics on how much energy was contained in the produced amount of barley. From these calculations the first input-output factor was estimated. According to the students’ calculations, the harvested barley contained about 6 times the energy that had gone into the field. There seemed to be a good ‘energy growth’ in such and endeavour. Yet, as energy does not come 12

MATHEMATICS EDUCATION IS UNDETERMINED

from nothing, there must be some important supplier, and, sure enough, the sun is the most reliable one. The students’ calculation could be compared to the official statistics in Denmark revealing that the actual ‘energy growth’ factor is only about 3. Thus there are many more parameters to consider, for instance all the transports that are necessary in order to complete the field work. At any rate, the students got a fairly good idea about one example of input-output calculations with respect to farming. The next step in the input-output calculations was to investigate what happens in meat production. Barley can be used for feeding pigs, as was the case on the farm in question. The feeding process could be observed almost directly, as an automatic feeding machinery was geared in such a way that barley was transferred from the pile of barley in the barn in proper measures and at the proper times to each of the pigs’ feeding troughs. The transfer was made in accordance with an algorithm that considered the number of pigs and their size. This transfer also represented a transformation of food from barley to meat, and one could then look at this transformation in terms on input-output figures. The students calculated the energy contained in the barley that the pigs were eating and compared it to the energy contained in the meat from the pigs when slaughtered. The students collected the information about how much barley a pig would eat, depending on their weight, and what their weight was when sent to the bacon factory. The ratio between the weight of a pig and the amount of meat it produces when slaughtered was also clarified, together with the energy content of meat. On this basis the students estimated a new input-output figure, namely 0.2. From an energy point of view, meat production has a really bad ‘growth economy’. The statistics provided by agricultural research show that also in this case the students’ results were similar to the official results with respect to Danish farming. During the project the students became familiar with inputoutput calculations with respect to energy. The whole project was related to a particular farm, but the issue that was addressed was of a general format. In this sense, the project was ‘exemplary’: Through a study of a particular case the students got an insight into a problem of a general format.7 Naturally the students’ calculations were based on 7 For a discussion for ‘exemplarity’ as an educational concept, see Skovsmose (1994).

13

CHAPTER 1

some extreme simplifications; nevertheless the project illustrated some of the principal ideas in input-output calculations with respect to farming. In particular, the role of mathematics was important in order not only to conduct the calculations, but also to formulate the whole idea of input-output estimations. The ‘Energy’ project provided a basis for addressing many principal discussions with respect to farming, the uses of energy, and the supply of food on a global scale. One can compare input-output figures with respect to different types of production and for different countries. By looking through statistics, the students found that farming in the USA demonstrated the most problematic figures, as the highest amount of energy supply, not least in terms of petrol, was used in this type of farming. Through the project the students were able to address several issues of global relevance. In this sense one could think of the project as illustrating how mathematics education could empower students and thereby contribute to the development of a critical citizen-ship.

1.3 Being undetermined Here we might have run into some confusion. Some interpretations could bring us to see mathematics education as disempowering, others as empowering. Mathematics education being undetermined has to do with this confusion as well as with the open character of both ‘empowerment’ and ‘disempowerment’. Very different perspectives can be applied with respect to empowerment and disempowerment. It is, for instance, possible to extract from a conservative economic discourse categories of competencies, and claim that a main element in empowering students through mathematics education is to ensure that they obtain competencies to meet the demands of the labour market. These demands can be thought of both from the perspective of the individual (an empowered person will get an adequate salary) and from the perspective of the company (employing empowered persons will lead to a satisfactory level of profit). According to this perspective, empowered persons can be compared to wellfunctioning batteries; a mathematics education has to ensure that the batteries become charged in a proper way. However, the discourse of ‘empowerment’ and ‘disempowerment’ has also taken a completely 14

MATHEMATICS EDUCATION IS UNDETERMINED

different route. Thus, the discourse about mathematics education for social justice has outlined how, through mathematics education, students can develop a new self-esteem that makes it possible for them to ‘talk back to authority’ as might be illustrated by the ‘Energy’ project. ‘Empowerment and ‘disempowerment’ are contestable concepts: the meanings of both can go in almost any direction. Therefore, it might not be surprising that it is possible to claim that mathematics education is disempowering, and then follow that up with the claim that mathematics education is empowering. Both statements, contradictory as they might appear, seem possible to support with a wealth of observations. This brings us to see mathematics education as undetermined. This means that one cannot attach any essentialism to the functioning of mathematics education. The whole picture of possible disempowering functions of mathematics education could turn out true, but it need not be true. Nor do the possible empowering functions of mathematics education necessarily become fact. Not even in a carefully elaborated project, like ‘Energy’ for instance. Here the students might have greatly enjoyed having breakfast in the school. Some might have enjoyed the ride on the bike. Some might have feared that they would be riding more slowly than the others. Some might have liked walking around the field. Some might have disliked the smell of the pigs. Some might enjoyed doing the calculations. Some might have looked forward to returning to regular mathematics lessons. The students might have had very different experiences. There is no simple fact that demonstrates that the students became able to ‘talk back to authority’. There are no essentials in mathematics education. This, however, does not imply that mathematics education operates in a neutral way. In one context it could appear a disaster, while in another it could provide wonders. It should also be noted that the two-dimensional formulation, that mathematics education could be either empowering or disempowering, is highly problematic. Mathematics education could have very many different functions, which cannot simply be labelled ‘good’ or ‘bad’. A mathematics education could be empowering in different senses of empowerment. It could be empowering for some, and disempowering for others. It could be empowering for some as they could obtain competencies that are valued in the labour market. It could also be considered disempowering precisely because people may come to 15

CHAPTER 1

assume a prescription-readiness. So when I describe mathematics education as undetermined, I refer to a great uncertainty with respect to the possible functions mathematics education might have in a particular socio-political situation. This uncertainty reflects the openness of the situation as well as the openness of the conceptual framework through which we try to grasp the situation. The undetermined nature of mathematics education is important to acknowledge. If mathematics education was a closed process without social significance, there would not be much for a critical mathematics education to be concerned about. But there is.

16

Chapter 2 Diversity of situations Let me describe some photographs from the book The Cradle of Inequality with texts by Cristovam Buarque and photos by Sebastião Salgado.8 On the cover of the book a girl, sitting in a dark classroom, is concentrating on taking notes. She might be about 7 years of age. She is wearing a white dress; she has bare feet; she looks poor. Inside the book we see several scenes representing similarly poignant moments. We see a photograph of a group of young people sitting in the shadow of big trees. They are refugees from the Sudan who have arrived in Kenya. Together with a blackboard balancing on an easel, the shadows of the trees make up the classroom. A camel is passing by on the outskirts of the classroom. At another photo we see a group of children with bunches of branches in hand on their way to school. They are from Iraqi Kurdistan. It is winter, and they have to ensure the heating of their classroom. A photo shows students from Afghanistan, completely engrossed, following a lesson on how to recognise different types of bombs and landmines that, still unexploded, might be scattered around in the vicinity. We see photos of dark and sinister classrooms missing all kinds of educational equipment and facilities, but crowded with pupils. These photos show teaching and learning conditions very different from those normally assumed in mathematics education research. Statistics show that the number of children in what has been referred to as ‘the developed world’ makes up just 10% of the total population of children in the world. The number of children in what is referred to as ‘the developing countries’ makes up an astonishing 86%9 Considering the statistic, and considering the world as a whole, the photographs in The Cradle of Inequality seem to show teaching-learning conditions which are quite more common than not. 8 9

See Salgado and Buarque (2005). See UNESCO (2000). See also Skovsmose (2006c).

17

CHAPTER 2

2.1 A bias in mathematics education research? If we consider the classroom settings that are described in much research literature in mathematics education, we find a dominance of what I have referred to as the prototypical mathematics classroom. (I could just as well have talked about the stereotypical classroom.) Such a classroom always reflects good order and affluence. We do not find students that tend to ‘disrupt’ the process of education. We might find students who have difficulties with mathematics, but, according to the research literature, they struggle with the mathematical tasks and concepts presented to them. They might provide examples of conceptual misunderstandings, which can be analysed in greater detail. However, one does not find extensive transcriptions of overtly disruptive conversations, or presentations of students whose behaviour spoils the lesson.10 The prototypical classroom is cleansed of ‘noise’. Naturally part of the dialogue selected for transcription might include some side remarks made by the students that provide the transcription with a little agreeable amusement. In the mathematics education research literature, we very rarely find hungry students, or students suffering from illnesses or psychoses. We do not find mathematics classrooms in warlike areas. Nor do we find discussions of what poverty means in terms of obstructions to doing homework. Some very strong paradigmatic criteria might be operating within mathematics education research. Such criteria have shaped and constructed the prototypical mathematics classroom. This prototype dominates the research literature, but it might be far from representative of the variety of classrooms in this world. Naturally I could try to develop many statistics with respect to research in mathematics education as it has developed over time, but let me just present a simple conjecture: 90% of research in mathematics education concentrates on the 10% the most affluent classroom environments in the world, while 10% of the research addresses the remaining 90% of the classrooms.11 Let me make a few comments on this conjecture. The prototypical classroom is not to be associated simply with classrooms in the so-called ‘developed countries’. There are very many classrooms in ‘developed countries’ that do not demonstrate any prototypical features: the students One such example is found in Alrø and Skovsmose (2002, Chapter 5). Observations referred to in Skovsmose and Valero (2008) and Skovsmose (2006c), brought me to formulate this guess. 10 11

18

DIVERSITY OF SITUATIONS

might demonstrate a behaviour that is noisy and disruptive and far from stereotypical. And it should not be forgotten that in addition these countries include a solid share of the world’s poverty with devastating implications for the school systems. Furthermore there are many affluent educational environments in the ‘developing countries’; prototypical as well as non-prototypical classrooms can be found the world around. Research addressing non-prototypical educational settings does exist.12 In particular, the ethnomathematical research programme has had an important impact, establishing research in non-prototypical settings. The situation however is not straightforward. My 90%–10% conjecture is only a tentative guess, but I have not been able to find evidence that refutes it. I would, however, be more than happy if it could be documented that my guess was wrong. Until that happens, we have to consider the possibility that research in mathematics education involves a problematic bias with serious implications with respect to both theory and practice. The bias is established through the priorities adopted in the selection of empirical domains, and it becomes multiplied through to the theoretical frameworks that become developed on the basis of the empirical bias. It might be that the teaching and learning of mathematics is largely addressed in the literature as if everywhere we are dealing with ideal classrooms, ideal teachers and ideal students.13

2.2 Contrast through globalisation and ghettoising When we talk about teaching and learning, we unavoidably talk about teaching and learning in diverse situations. Yet, I have considered other variables than ‘the situation’. We could for example try to express similar phenomena through the notion of ‘culture’. It is common sense to talk about teaching and learning in different cultural settings, and also to emphasise that the interpretation of any form of teaching and learning has to be done with reference to the cultural setting in which it operates. 12 See, for instance, Valero (2004, 2007); Valero and Zevenbergen (Eds.) (2004); Vithal (2007, 2009); and Vithal and Valero (2003). 13 See Valero (2002) for at discussion of the idealised students and idealised teachers.

19

CHAPTER 2

I completely agree. Particularly in connection with the ethnomathematical research programme, the notion of culture has been elaborated carefully. Thus Ubiratan D’Ambrosio associates ‘ethno’ with culture.14 However, it is also important to talk about teaching and learning in different socio-economic contexts. We should remember that poverty and favelas as well as extreme wealthy neighbourhoods are distributed around the world: in São Paulo, Johannesburg, Bombay, New York, Madrid, etc. Rich neighbourhoods might be situated immediately adjacent to favelas and squatter settlements. To refer to such contrasts in terms of cultural differences makes some sense, but the very notion of ‘culture’ might also provide a false picture of the differences. Extreme contrasts are distributed around the world according to a violent logic of globalisation and ghettoising, and I see the distribution of teachinglearning conditions as a socio-economic structuring, and not only as a cultural structuring. We can also consider teaching and learning in different political contexts. Wars and violence may structure the whole meaning of ‘going to school’. One could address this structuring in terms of culture and talk about the culture of a war zone. Children who live in the vicinity of unexploded land-mines might have a different culture, but I find that in such circumstances the notion of culture is inadequate to fully describe the root differences. I prefer to talk about the diversity of political conditions. The photographs in The Cradle of Inequality show cultural differences, but above all they demonstrate socio-economic and political differences. This observation brings me to the notion of situation. By this term I refer to cultural, socio-economic and political contexts of the teaching-learning processes. I want to use an expression that draws our attention to this variety of contexts without including an overloaded set of assumptions about the nature of the contexts. A concern of critical mathematics education is to recognise the diversity of situations in which the teaching and learning of mathematics is taking place around the world. This would have an impact on the concepts and theories that become developed. In particular, it is a concern of critical mathematics education to prevent from repeating the bias that is established through discourses centred around the prototypal mathematics classroom. 14 See, for instance, D’Ambrosio (2006). See also D’Ambrosio (2010) for a broader exploration of a mathematics education for survival with dignity.

20

Chapter 3 Students’ foregrounds “Consider two South African children born on the same day in 2000. Nthabiseng is black, born in a poor family in a rural area in the Eastern Cape province, about 700 kilometres from Cape Town. Her mother had no formal schooling. Pieter is white, born in a wealthy family in Cape Town. His mother completed a college education at the nearby prestigious Stellenbosch University. One the day of their birth, Nthabiseng and Pieter could hardly be held responsible for their family circumstances: their race, their parents’ income and education, their urban or rural location, or indeed their sex. Yet statistics suggests that those predetermined background variables will make a major difference for the lives they lead. Nthabiseng has 7.2 percent change of dying in the first year of her life, more than twice Pieter’s 3 percent. Pieter can look forward to 68 years of life, Nthabiseng to 50. Pieter can expect to complete 12 years of formal schooling, Nthabiseng less than 1 year. Nthabiseng is likely to be considerably poorer than Pieter throughout her life. Growing up, she is less likely to have access to clean water and sanitations, or to good schools. So the opportunities these two children face to reach their full human potentials are vastly different from the outset, through no fault of their own.”15 This is how the World Bank Report of Equity and Development is introduced. The opportunities of life of these two children are different, and I am going to develop further considerations about opportunities.

3.1 Foreground By the foreground of an individual, I understand the opportunities which the social, political, economic and cultural situation provides See, Word Bank (2006: 1). Renuka Vithal referred to this formulation in her lecture at the Symposium Mathematics Education, Democracy and Development: Challenges for the 21st Century. Faculty of Education, University of KwazuluNatal, Durban, 4 April 2008. 15

21

CHAPTER 3

for the person.16 This formulation, however, needs a number of clarifications. Do we consider the description of Nthabiseng’s and Pieter’s opportunities, two radically different foregrounds are outlined: “the opportunities these two children face to reach their full human potentials are vastly different”. However, the foreground is not uniformly determined by the social, political, economic and cultural situation. It might be that Nthabiseng lives much longer than 50 years; it might be that Pieter dies in a traffic accident, still a promising university student. The statistics through which their opportunities in life are characterised only reveals tendencies, which set the foreground of the person. I do not see the foreground of the person as a ‘social fact’ or as a configuration of tendencies. A foreground does not exist in any ‘objective’ sense that can be specified through statistical investigations. I see the foreground of a person not only as composed of tendencies but also as being formed through interpretations of future possibilities. A foreground brings together expectations, hopes, frustrations, etc. One could relate the notion of foreground to the notion of life-world, which refers to life conditions as they are experienced. Edmund Husserl provided a rich interpretation of life-world which I find inspiring. However, he also buried the notion beneath a pile of phenomenological assumptions which I find problematic.17 Thus, the notion of life-world refers not only to given social facts or to situations as they might be described through statistics; it also refers to how facts and situations are experienced. I see foreground as referring to a particular province of the life-worlds. It is the province which is directed towards the future. The life-worlds of Nthabiseng and Pieter are different, and so are their foregrounds. The notion of foreground relates to the notion of background. One could see the background of a person as a determining factor for his or her foreground, and to some extent this makes sense. Thus, the backgrounds of Nthabiseng and Pieter frame their foregrounds. Some tendencies that are part of their foreground are constructed as part 16 For the introduction of the notion of foreground, see Skovsmose (1994). See also Skovsmose (2005, 2007b); and Alrø, Skovsmose and Valero (2009). See also Lindenskov (2010). 17 In particular I do not interpret life-world in terms of a ‘stream of consciousness’ as he suggested. See Husserl (1970), as well as Skovsmose (2009b).

22

STUDENTS’ FOREGROUNDS

of their background. Nevertheless, foreground and background are different kinds of matter. A person’s background comprises events that have taken place, while their foreground is composed of events that might take place. While the foreground of a person is an open situation, the background has somehow solidified into history. (This simple observation needs, however, a modification as a background also includes an interpretation of what has taken place.) I have talked about the foregrounds of Nthabiseng and Pieter in individual terms, but we could also talk about the foreground of a group of people. The people from the community to which Nthabiseng belongs share foregrounds. They are submitted to the same statistics. Their foreground is formed through shared statistic parameters. Still this does not rule out that opportunities in life can be experienced differently and acted out differently. Peiter shares foregrounds with many other young people in his neighbourhood: statistically speaking they have similar life-opportunities – being radically different from the opportunities of the group with whom Nthabiseng shares statistics. Framed by similarities in tendencies, individuals can act out possibilities in different ways. This means that the notion of foreground includes reference to both collective and individual features of life-worlds. Foregrounds include experiences and interpretations, which are elaborated through interaction and communication. Foregrounds include shared hopes and frustrations. Thus the foregrounds of both Nthabiseng and Peiter are elaborated through interactions – although interactions taking place in widely separate communities. The construction of foregrounds of young people takes place with reference to many different groups: their friends, their parents, their ‘stars’. A foreground may include contradictions. We should not expect a foreground to be an expression of an all-embracing rationality which ensures a wall-to-wall consistency. A foreground could be inconsistent and multi-layered. It is possible that a young person in one situation could establish a foreground in a most captivating format, while in a different situation only the most desolate possibilities appear to be available. Foreground could include impossible dreams, realism and frustration in an extensive and contradictory mix. A foreground can be bleak. When only desolate possibilities are experienced by the person, I talk about a ruined foreground. That a foreground is ruined does not mean that there is no foreground, only 23

CHAPTER 3

that it seems devoid of attractive possibilities. A ruined foreground does not support the development of aspirations, but more likely of frustrations. A foreground can be ruined through social, economic, political or cultural acts. In the most direct way the ruination of black people’s foreground was an integral part of the apartheid system of South Africa. Black people did not have the same opportunities as white people. Black people could only live in certain designated areas; they could not own property; they could not use the same facilities; not go to the same hospitals, etc. The different foregrounds of Nthabiseng and Peiter have to be interpreted in light of what was taking place during the apartheid era. There are many Nthabisengs and Peiters around the world. Differences between their foregrounds are recapitulated by the foregrounds of students from poor and rich neighbourhoods respectively in any metropolis of the world. They are recapitulated between the foregrounds of students coming from rural and from urban areas. They are recapitulated between foregrounds of immigrant students and other students.

3.2 Intentionality and learning The notion of intentionality was elaborated by Franz Brentano as part of his psychological and philosophical framework.18 His aspiration was to characterise human consciousness in a way that clearly separated it from any mechanical phenomenon. It does not make any sense to talk about a stone falling towards the earth due to its intentions of doing so. A stone has no intentions, no motives for falling. Its falling should be explained in terms of cause and effect, and not in terms of intentions and motives. According to the mechanical world view this observation not only applies to stones but to any natural phenomena, and also to human actions. As a conesquence, human action must be interpreted not with reference to any intentions on the part of the acting person but by being located as part of mechanical system. When explained adequately, any human action is identified as part of the operation of a mechanical structure. One has 18

See Brentano (1995a, 1995b).

24

STUDENTS’ FOREGROUNDS

to present the line of causalities that end up in the physical operations which characterise the act. Contrasting the mechanical world view, Brentano emphasised the importance of the directedness of the human consciousness, and he established the notion of intentionality as being crucial for understanding human activities. The notion of intentionality was elaborated further by one of Brentano’s students, namely Husserl.19 He did not see human consciousness as derived from some underlying mechanical operations, but as a phenomenon that had to be investigated for its own sake and within its own conceptual framework. According to this way of thinking the notion of intentionality becomes important. I want to assume a connection between human actions and intentionality; I want to interpret an action in terms of its directedness. The notion of intentionality can be related to the notion of foreground. If we look for motives for an action, it is important to consider the foreground of the actor. Naturally it makes sense to look at the background as well, as this is part of the framing of the foreground. However, I see intentionality as relating more directly to the foreground of the person, as it represents a directedness. The foreground represents the raw material for establishing motives.20 Actions can take many forms, and in particular I want to see learning as action, i.e. as performed by the person.21 Learning is an action which includes intentions and motives. When we want to investigate learning phenomena, we have to consider the intentionality of the learners. One could naturally ask if any form of leaning could be seen as action. Maybe this could only be done in case we propose an impressive concept streaching of the notion of action. One could think of the small child learning the mother tongue. Does this learning take the form of an action by the child? Well, one could claim that this learning is an expression of the intentionality of the child. One could also think of soldiers learning how to march. This learning seems to be an expression of direct orders, although one could claim that the soldiers See, for instance Husserl (1998). While the notion of motive refer to the foreground of the person, the notion of motivation, in particular as developed by the behaviourism, refers to the background of the person. 21 For a presentation of learning as action, see Skovsmose (1994). I also see teaching as action, but what this implies I am not going to explore here. 19 20

25

CHAPTER 3

somehow have the intention of following orders. Whatever comes from such considerations, I want to emphasise that learning, as we most often see it in school, can be interpreted in terms of actions. This draws our attentions to the students’ intentionality, their foreground, and their motives (or lack of motives) for learning. When learning is seen as action, we can interpret different learning phenomena – for instance the students’ engagement (or lack thereof ) and their achievements (or lack thereof) – with reference to their foreground. In particular a ruined foreground can obstruct bringing intentions into the learning process. Let us consider again Nthabiseng’s and Pieter’s situation. How could we interpret their activities in school and their learning achievements? Let us take a look at what has been referred to as white research in black education.22 In this research it seems to have been ‘documented’ that black childrens’ achievements are lower than white children’s. Such results can easily be enveloped in a discourse about how to organise a compensatory education for these children. If, however, we pay particular attention to the foreground of children, their achievements might be interpreted differently. In South Africa under the apartheid system, many career possibilities were simply not accessible to black students, such as becoming an engineer, a doctor, or a dentist, for instance. To put it directly: any black child’s foreground was ruined. The apartheid era belongs to the past, but economic differences are not simply eliminated. Nthabiseng’s and Pieter’s life conditions are different, and so are their foregrounds. Their school performances might be different as well. My suggestion is that we are careful when we try to formulate a discourse with respect to such differences. When learning is seen as action, the differences in learning achievements may be related to differences in acting conditions. Such differences have to do with the different possibilities society makes available to different groups. Weak performances in learning can be provoked by a ruined foreground, which in turn can be caused by socio-political acts of exclusion and suppression. Let us again take a look at the photos from The Cradle of Inequality. The refugees are sitting beneath a tree in Kenya. What life opportunities do they face? What would statistics show? What could they themselves 22 For a critical investigation of ‘white research in black education’ see Khuzwayo (2000). See also Skovsmose (2005).

26

STUDENTS’ FOREGROUNDS

imagine? What about the children who are bringing firewood to the school to ensure the heating? And the children from Afghanistan; how does their foreground look? This could inspire us to do some foreground investigation, and this is in fact an idea which has guided a whole project ‘Learning from Diversity’ in which I have participated. 23 We have considered the foreground of immigrant children in Denmark, of children from a favela, and of Indian students in Brazil. There are many different observations to be made. The more general one is that considering students’ foreground is part of interpreting the way they approach learning.

3.3 Meaning in mathematics education Different theories of meaning have had a strong impact on mathematics education. I will shortly outline the compositional theory of meaning as proposed by the so-called modern mathematics education and the background-theory of meaning, which has been particularly elaborated within the ethnomathematical approach. In place of these theories of meaning I will suggest that meaning become investigated with respect to action, intentionality and foreground, and I will illustrate what this could imply. The discussion of the meaning of concepts has long been a key concern in mathematics education; thus what can be referred to as modern mathematics education, emerging in the late 1950’s, paid particular attention to the notion of meaning. The principal assumption was that the meaning of a complex concept should be established by addressing the meaning of its constituting parts.24 Furthermore, it was assumed that the logical organisation of mathematical concepts (in particular as provided by Nicolas Bourbaki) also defines an educational ‘Learning for Diversity’ was directed by Helle Alrø, Paola Valero and myself. See, for instance, Alrø, Skovsmose and Valero (2009). 24 In analytic philosophy a particular concern was to investigate in what sense and to what degree the meaning of a complex or molecular concept could be seen as a composition of meaning of its constituting atomic elements. Frege made a distinction between sense (Sinn) and reference (Bedeutung), which was crucial to this clarification. See Skovsmose (2009b) for a discussion of Frege’s distinction. 23

27

CHAPTER 3

organisation of how complex concepts should be taught and learnt. For instance, the mathematical notion of group can be defined as a not empty set, G, organised with an operation, *. The operation, *, can be defined as a function from G X G to G, fulfilling certain conditions.25 A function can be defined as a set of ordered couples which fulfils a particular condition. And in order to get started on this sequence of definitions, one has to grasp the meaning of the basic set-theoretical concepts. As a consequence, one could think of the curriculum in terms of a sequence of concepts to be learnt, as follows: set, ordered couple, set of ordered couples, relation, function, operation, etc. According to the modern mathematics education, the logical structure of mathematics defines the educational structure of the curriculum. This implies that the meaning of molecular mathematical notions are going to be established through the meaning of their atomic parts. The Modern Mathematics Movement has been criticised, but its compositional theory of meaning has been broadly assumed. A different theory of meaning has also got a broad application in mathematics education. It has been assumed that meaning relates to the background of the person, which implies that in order to establish meaningful education one has to relate the educational content to the students’ background. The background-theory of meaning has been emphasised in ethnomathematical studies, which have scrutinised the cultural background of the students in order to establish a meaningful mathematics education for them. I find the compositional as well as the background-theory of meaning to contain limitations, and I suggest a different approach to discussing educational meaning. It is possible to talk about the meaning of activities or actions, and it is this kind of meaning on which I would like to concentrate. The meaning of an action I relate to the intentionality included in the action, which in turn relates to the foreground of the acting person. The meaning of a classroom activity is constructed by the students, and this construction depends on what the students may see as their possibilities; it depends on their foregrounds and intentions. Thus, I operate with a close relationship between meaning, intentionality and foreground; let me illustrate with a couple of examples. 25 The conditions are: the operation, *, is associative; there exists a neutral element e in G; and every element in G has a inverse elements also in G.

28

STUDENTS’ FOREGROUNDS

In a public school in Rio Claro, a city in the interior of the São Paulo State, the teacher wanted to introduce project work in mathematics. She asked what topics the students wanted to work with, and one suggestion was surfing and surf boards. The teacher did not find this to be a good possibility. The school was located in a poor neighbourhood. Most likely the students had never been at the beach and never seen the ocean. How could working with surfing make sense to them? If one relates meaning merely to already established experiences and to the background of the students, surfing does not appear meaningful to the students. However, one might miss some important aspects of what meaningfulness might include. It could very well be that surfing makes part of the students’ foreground, and, as a consequence, elaborating a project about surfing could be extremely meaningful.26 For a long period of time, I participated in a mathematics education project in South Africa. 27 We struggled with the challenge of what could make sense to children living in a village ‘beyond the mountains’? As mathematics educator one could try to investigate activities in the village through a mathematical archaeology and try to indentify how mathematics might be part of the way the field work is conducted, how the crops are divided, how the cooking is done, etc. One could identify mathematical activities as related to these everyday activities, and on this basis try to organise a meaningful mathematics education for the children. But we need not be surprised in case meaningful mathematics education for these children could be developed around what we might call ‘pilots mathematics’. The children from the village beyond the mountains may only have experienced the airplane in the form of a thin, white, downy line high up there in the sky. But piloting might nonetheless be part of their foreground. The intentionality of the children might seek beyond their actual situation; it might be pointing away from their background. As part of the project ‘Learning from Diversity’, we conducted different foreground investigations, for instance of young people from Brazil with an Indian background.28 One Indian student expressed a In a conversation this example about surfing has been presented to me by Miriam Godoy Penteado. 27 See Vithal (2010). 28 See Skovsmose, Alrø and Valero in collaboration with Silvério and Scandiuzzi (2008); and Skovsmose, Scandiuzzi, Valero and Alrø (2008). 26

29

CHAPTER 3

strong interest in continuing to work in the fields and to stay in the Indian community. This perspective provided one kind of meaning to the mathematical activities he experienced in school. Another student wanted to study medicine. He expressed the clear opinion that a major problem for Indian communities in Brazil has to do with health. To him it was important to come to study medicine and to be able to return to the Indian communities as a qualified doctor. This perspective provided a different set of meanings to classroom activities. Thus he felt sure that even though he did not know the particular relevance of the different mathematical notions and theories, mathematics was relevant for conducting further studies, not least within medicine. His foreground provided a set of meanings to the classroom activities that could not be identified if one concentrated solely on his background. I have participated in a project, conducted by Miriam Godoy Penteado, which concentrated on students whose conduct had been highly problematic, not only in the classroom but for the whole school environment. What could be thought of as meaningful education for such students? What could make these students who have resisted any form of learning engage in something? What could they experience as meaningful mathematics education? It turned out that the students liked to work with mathematical games and with puzzles. They liked to work out winning strategies. Dynamic geometry where one could work on a computer also turned out to captivate these students. How could that be? Let me just emphasise that the construction of meaning is a complex process. One experiences meaningfulness when one’s intentionality is part of one’s learning activities. But what might count as meaningful could include many surprises. There are no regulations and simple guidelines for establishing meaningful education and for anticipating students’ intentionality.29

29

See also Penteado and Skovsmose (2009).

30

Chapter 4 Landscapes of investigation In order to establish meaningful mathematics education and to make the students active learners, there are no simple principles to be applied. Meaningfulness is something that needs to be searched for. One suggestion is to search outside the school mathematics tradition, and I have been fascinated by the possibilities that project work may offer. However, project work cannot be presented as a universal recipe for providing meaningful activities for the students. In order to broaden the search for educational possibilities, I will explore landscapes of investigation.30 Such a landscape provides an environment for teaching-learning activities. While sequences of exercises, so characteristic for the school mathematics tradition, establish a one-way route through the curriculum, the possible routes through a landscape of investigation are not welldefined. A landscape can be explored in different manners and through different routes. Sometimes one must proceed slowly and carefully, sometimes one can jump around and make bold guesses. One important idea of establishing landscapes of investigation is to provide meaning to the activities in which the students are participating. Inquiry processes include possibilities for constructing meaning which I do not find along the one-way route defined by sequences of exercises. I see the notion of inquiry as closely related to the notion of intentionality, as an inquiry presupposes involvement. Thus the interpretation of learning as action brings us directly to the notions of inquiry and investigation. Such activities exemplify a particular form of learning as action. One can invite students into a landscape of investigation, but cannot force them to do inquiries. It might be that the students accept the invitation. They might be fascinated by the possibilities the landscape provides. But it might also be that they are not intrigued. They might decline the invitation. This depends on so many different 30 For a presentation for landscapes of investigation see Skovsmose (2001), and Alrø and Skovsmose (2002).

31

CHAPTER 4

factors. Some landscapes might be attractive to some students in some situations, while not to others.

4.1 Entering a landscape of investigation What could it mean to open an exercise and try to enter a landscape of investigation through this opening? Let us consider exercises which have to do with linear functions. Two functions f and g from R to R (R referring to the set of real numbers) are defined through the following equations: f(x) = 2x + 3 g(x) = -x + 5 With this information, one can formulate exercises of the following form: Find the equation that defines the function f –1. Find the equation that defines the functions fRg and gRf. Draw the graphs of f and f –1. And so on and so forth. There is no end to all the exercises that can be formulated following this pattern. However, it is also possible to turn this kind of exercises into a landscape of investigation. Maybe in the following way: Consider two functions, f and g, of the form f(x) = ax + b and g(x) = cx + d. (The parameters a, b, c and d could take any values from R, and f and g are functions from R to R.) Is it possible to provide some characteristics of the graphs of the functions: f, g, f –1, g –1, gRf, fRg, f –1Rg –1, etc.? One could introduce a new notion, //, where f//g signifies the intersection (if it exists) of the graphs of the functions f and g. One could then try to identify intersections, like: f//g, f –1//g –1, f//gRf, gRf//g –1 etc. There is no end to the intersections one could consider. One could try to identify some kind of pattern, at least among some of the intersections. One could try to express the intersections in terms of the parameters a, b, c, and d. In this way the initial set of exercises could provide an opening to a spacious landscape of investigation. 32

LANDSCAPES OF INVESTIGATION

I can still remember, many years ago, when some of my students were exploring exactly these properties of linear functions. I remember their smiles when they started making their first discoveries. They found an expression for calculating the intersection f//f–1. They discovered the relationship between the intersections f//g and f–1//g–1. There were so many properties to discover. They had initiated the investigations by looking at some particular functions. This way they got some ideas of the possible properties they might be able to discover, when they embarked on the more ambitious calculations using the parameters a, b, c and d. The students were familiar with functions of first degree as presented in their textbook, while matrix calculation was far beyond their curriculum. They expressed all their discoveries in a basic algebraic format. This, however, did not limit their investigations. In fact I find it important not to try to relate mathematical discoveries to more advanced mathematical notions. Discoveries can be made and expressed on all levels and in all formats. The students’ involvement in the investigation made me aware of the close relationship between inquiry, intentionality and experience of meaning. One could imagine the classroom turned into a research campus. The students are working in different groups. Each group is researching some properties. The research will go on for the mathematical lessons for a whole week, and only on the following friday will there be a seminar, where the research groups will present their discoveries. The research groups could have been working on related topics if not on the same topic. At the seminar they could see what the other research groups had discovered. The research seminar could be open, as students from other classes could be invited; maybe also other teachers. It is easy to open more advanced topics to investigation. The research projects could concern functions, F, from R to R defined in the following way: ( ax 2 + bc + c ) ( dx 2 + ex + f ) 2

F(x ) =

2

How does the graph of functions of this format depend on values of the parameters a, b, c, d, e and f ? (Naturally one also has to consider for which values the function might not be defined.) The students could initiate the investigations by making some simplifications. They could 33

CHAPTER 4

assign some specific values to the parameters and see what happens. Using mathematical software such as Winplot could be useful. One could assume specific values to all parameters except one, and then see what happens to the graph when this parameter changes. A different approach in doing simplification is to consider the function F to be defined in the following way: F(x ) =

( ax + b )( cx + d ) ( ex + f )( gx + h )

Here there are more parameters to consider, but nevertheless it might provide other ways of formulating the research results. It could be that a research group prefers to initiate the investigations by considering a function F defined in the following way: F(x ) =

( x – a )( x – b ) ( x – c )( x – d )

This formulation gives a simple formulation of the 0-points of both the nominator and denominator. One could discuss the principal format of the graph of F as related to the principal relationships between the values of a, b, c and d. Let us consider a different landscape. Let us talk about ‘small animals’ as being composed of small squares as illustrated in Figure 1. It is possible to make one and only one animal of size 1, consisting of one square. Apparently is also possible only to make one animal of sizes 2, consisting of two squares. But what about an animal of size 2 that

Figure 1. A few 2-dimensional animals. (This and the following photos by Mikael Skånstrøm.) 34

LANDSCAPES OF INVESTIGATION

is just connected by the corners of the squares? Could an animal be so thin that it has a waistline consisting of only one point? We could decide that animals need to be properly united by sides of the squares and not only by corners, and this leaves us with only one animal of size 2. Of size 3 there are two different animals. (Here we apply the notion of symmetry as normally applied in mathematics. However, we could instead decide to apply the stronger notion of identity as applied in language. Here one finds the letters d, b and p to be different, although one could claim that at the letter d is just a p that is standing up, and that a b is a d sleeping on the other side.) Let us look at animals of size 4. Figure 2 shows 5 different animals of size 4, but could there be more? The Figure 3 shows an animal of size 9. How many animals could there be of size 9? One could imagine that for each natural number n one could determine the number, A(n), of animals of size n. One could start off by determining A(n) for some small numbers n, and we have already noted that A(1) = 1, A(2) = 1, and A(3) = 2. Could one imagine a sort of induction to be brought into play? Could it be possible to say at least something about A(n+1), in case one knows A(n)? One could also consider 3-dimensional small animals. Different such animals of size 5 are shown in Figure 4. If we refer to the number of 3-dimensional animals of size n as B(n), one could try to determine B(n) for some natural numbers.

Figure 2. 2-dimensional animals of size 4. 35

CHAPTER 4

Figure 3. A 2-dimensional animal of size 9. In in-service courses for teachers, I have used the example with small animals many times. What continues to surprise me is that so many different suggestions for solutions crop up, as well as for procedures to reach solutions. Once I had about 120 teachers working briefly in small groups trying to determine the number A(6), i.e. the number of 2-dimentional animals of size 6. Hardly any of the many groups came up with the same proposal. I have had teachers drafting the animals on paper, and also working with centi-cubes. Sometimes the differences between dimensions have been addressed, sometimes it has not been acknowledged. We have talked about flat 2-dimentional animals and 3-dimentional animals. In mathematics one could also talk about, say, 4-dimentional cubes. Does it then make sense to talk about 4dimensional animals? They, too, could be of different sizes, depending on how many 4-dimentional cubes they are composed of. 31 31 As an introduction to this discussion one can consider how makes faces a 4 dimensional dice might have. For a clarification of this see Rønning (2010).

36

LANDSCAPES OF INVESTIGATION

Figure 4. Some 3-dimensional animals of size 5. One could consider the number C(n) referring to the number of the 4-dimentional animals of size n. One could go on considering D(n), referring to the number of 5-dimensional animals of size n, etc. One could consider the sequences of numbers A(n), B(n), C(n), … for a particular value of n. We started out considering animals of dimension 2, but we could have started with animals of dimension 1. They will look like worms that will be defined by their length. Let us call the number of 1-dimentional animals of size n O(n). We then have O(1) = O(2) = O(3) = … O(n) … = 1. Such worm-like animals are shown on the Figure 5. 37

CHAPTER 4

Figure 5. Some 1-dimensional animals. Some observations are indicated in Figure 6, but there is much more to explore. Dimension/Size 1 2 3 4 5

1 1 1 1

2 1 1 1

3 1 2 2

4 1 4

5 1

6 1

Figure 6. A few observations about small animals. Carrying out an investigation or making an inquiry is not restricted to any particular mathematical domain, and certainly not just to advanced mathematics. One might claim that the investigation of intersections of graphs of linear functions does not provide an insight that is new in any objective sense. But this is not relevant. The important activity for the students is their own researching, it is not to provide genuine research results. It is also interesting to note that the issues that could be researched by the youngest children, as for instance the world of small animals, may grow directly into the most difficult issues. Landscapes of investigation are not restricted to certain domains or to certain ‘levels’ of mathematics. 38

LANDSCAPES OF INVESTIGATION

As part of an inquiry what-if questions have a particular purpose. The teacher may have formulated some possibilities through what-if questions (having the investigations of the functions of the form F(x) = (ax2 + bc + c)/(dx2 + ex + f) in mind: What will happen if we assume that the parameter a is positive? What if we assume a to be 1? That the students then take charge of the inquiry process is demonstrated by the fact that they themselves start formulating what-if questions: Yes, what if we assume all parameters to be 1? Let us assume all of them to be 1, and then, using the graphic calculator, we can change one of the parameters to get a value higher than 1, and then smaller than 1. Then we get an impression of the significance of what each of the parameters might mean for the graph of the function. Well, this is an idea, but why take 1 as the initial value? Why not take 0? Then we see the significance of each of the parameters turning positive or negative. Yes, well, but if we start with the value of all parameters being 0, what do we then in fact start with? Following such a conversation, we may get an idea of how communication might be crucial for an inquiry process. Processes of interaction and communication play quite a larger role in inquiry processes compared to processes located in an exercise paradigm.32

4.2 Milieus of learning Classroom practices based on landscapes of investigation contrast with practices based on sequences of exercises. In this sense we can see landscapes of investigation and sequences of exercises as establishing different milieus of learning. But differences in learning milieus can also be established in other ways, and I want to consider here the references made when students are engaged in the activities. The references can be made to mathematical ideas and notions; thus solving a mathematical equation need not make reference to any non-mathematical entities or issues. On the other hand, reference can be made to reality-like entities. In this sense activities can be located in a semi-reality. Finally, references can be to real-life situations.

32 Consider, for instance, the Inquiry Co-operation Model as investigated in Alrø and Skovsmose (2002).

39

CHAPTER 4 Sequences of exercises References to pure mathematics References to a semi-reality Real-life references

Landscapes of investigation

(1)

(2)

(3)

(4)

(5)

(6)

Figure 7. Milieus of learning. If we combine the distinction of different types of references and the two paradigms of classroom actives, we get at matrix of learning milieus as shown in the Figure 7. Let me comment on each of the 6 milieus. The learning milieu of type (1) is positioned in a context of pure mathematics as well as in the tradition of exercises. This learning milieu is dominated by exercises, which can be of the form: (a) Reduce the expression… (b) Solve the equation… (c) Calculate… Mathematics textbooks are filled with exercises of this type with references only to mathematical entities. The learning milieus of type (2) are characterised as landscapes of investigation located in numbers and geometric figures. The examples, presented in the previous section, about investigations of functions and ‘small animals’ can serve as an example. The learning milieu of type (3) milieu is located in the paradigm of exercises with references to a semi-reality. What this could mean is illustrated by the exercise I referred to previously: A shop is offering apples for 0.12 Euro apiece, and for 2.8 Euros for bags containing 3 kilos. There are 11 apples to each kilo. Calculate how much Peter will save if he buys 15 kilos of apples in bags of 3 kilos instead of buying them individually. Certainly, there is talk about a shop, apples and prices. But we do not have to do with real apples, or real prices, or any real shop. The situation is artificial and the exercise is located in a semi-reality. It might be that the references to this semi-reality might help students contextualise the mathematics calculation. However, there are certain ways of operating with respect to a semi-reality. As already mentioned, this reality functions like a Platonic world, where all information is exactly true. 40

LANDSCAPES OF INVESTIGATION

The milieus of type 4 are also located in a semi-reality, which takes the form of a landscape of investigations. As an example, we can consider the simulation programme Simcity4 that was used as part of the project: ‘City Planning’.33 Simcity4 represents features of real city planning, but certainly we have to do with a semi-reality. Simcity4 is structured as a game, and the participants play the role of a mayor. As part of the game different aspects of city planning can be addressed, like the health system, the educational system, pollution, land value, transports, the locating of re-creative areas, law issues, supply of electricity, water supply, sanitary facilities. The project ‘City Planning’ took place in a Brazilian community, so several aspects of the game had to be discussed, as Simcity4 refers to a USA reality. Within the semi-reality of the game, the students could make suggestions for city planning, and be involved in different forms of calculations and decision making. Playing with Simcity4 gives many opportunities for conducting investigations and such a game illustrates what a learning milieu of type 4 could mean. Certainly, there are many such milieus to be constructed, both with and without the use of computers. Milieu 5 refers to real-life situations, and it is possible to provide many exercises with such references. Thus one could imagine that the whole idea of addressing input-output figures in farming become turned into sequences of exercises. Thus the project ‘Energy’, as referred to in Chapter 1, could have been elaborated into a sequence of exercises including very many real-life references. These exercises would be different from the exercises formulated in the milieus (1) and (3). In order to formulate exercises in the milieu (5) one has to conduct investigations of the situation to which one is referring. This is quite different from formulating exercises in milieu (3), where the whole construction of exercises could, as mentioned, be done at the desk. The milieu (6) is a landscape of investigation with real-life references. As an example one can think of the project ‘Energy’ as it in fact was organised. This project takes place in a school context, meaning that a landscape of investigation of type (6) does not constitute a reallife project as such, but an educational activity with real-life references. The whole project ‘City Planning’ is presented by Biotto Filho (2008), and I will return to it in Chapter 6, when I discuss the notion of reflection. The computer game Simcity4 was launched in 2003 by the publisher of electronic games Electronic Arts (EA Games). 33

41

CHAPTER 4

4.3 Leafing through a newspaper It is a huge challenge to develop learning milieus of type (5) and of type (6), but there is much inspiration to get when leafing through a newspaper.34 As an experiment I leafed through Folha de São Paulo of Monday, July 14, 2008. On the first page a headline imparts that the number of people killed in traffic accidents has been reduced by 57% due to a newly introduced Brazilian law which makes it illegal to drive with even a minimal level of alcohol in the blood. This number can be related the information that the number of guests to bars is reduced by 30%. One could compare this information to situations in other counties. One could also consider what the 57% reduction means in real numbers during, say, a 1-week period or a 1-year period. There are several articles in this Monday edition of the newspaper which address a case of economic crime. There is information about amounts of money that have been declared, or rather not declared. Money has been used for bribery. There are many exercises with reallife references to be made with respect to such numbers. For instance, one could calculate how much money the state might have missed in taxes, with reference to a specific case. The tax-perspective could be explored further: How does the general level of taxation in Brazil compare to other countries? What is the total amount of declared income? Could one make any estimation of the total amount of nondeclared income? Naturally, in order to formulate exercises or landscapes of investigations with such real-life references, some research would need to be carried out. Folha de São Paulo contains a section about Finance, which is overloaded with information expressed in numbers. As the stock market is closed during weekends, the Monday edition of Finance might contain few diagrams, but as soon as we get to the Tuesday edition, one could dive into diagrams and tables showing the actual exchange rate, the development of the exchange rates, and all kinds of numbers from the stock market. There are articles about inflation and deflation as well as other kinds of background information. Exercises could be constructed, and landscapes of investigation could be opened through all such references. 34 Once I was listening to a lecture by Philip Davis, where he conducted this exercise.

42

LANDSCAPES OF INVESTIGATION

What might be missing in numbers in the Monday section about Finance is compensated for by all the numbers that occur in the Sports Section. The reports on the sports events from the weekend, and let us just concentrate on soccer, take two different formats. One is the descriptions of what happened during the match: one team started very powerfully, but then the referee overlooked a penalty, and during the next 10 minutes… The other format is provided in terms of numbers and diagrams. We read about the new ranking of the teams. How many victories and how many losses? How many victories on home ground, and how many abroad? How many goals scored and how many suffered. Furthermore, when looking at the Sports Section the following days, one will find certain games to be subjected to a further mathematical X-raying. The number of passes performed by each team would be enumerated and compared. The number of successful passes also. The number of kilometres covered by each player is calculated. Naturally the number of free kicks, corners, yellow and red cards and so on make up part of this mathematical X-raying. The weather forecast includes a lot of information put into numbers. At first this information appears different from the mathematical X-raying of a soccer match. While the X-raying from the Sports Section is part of a description of what has taken place, the mathematical X-raying of the weather situation serves as part of a weather forecast. This prediction is based on complex mathematical models, of which we see only the surface in the newspapers. However, the whole set of sports information turns into a scheme of forecasting when gambling becomes the issue. One section of Folha de São Paulo opens with a photograph, covering almost half a page, showing two girls in front of a concrete wall painted dark green. The wall was painted long ago and now dirt and rain have turned the wall unsightly and dismal. Behind the wall appears a two-store favela building in red tiles. The cement between the tiles is protruding a bit here and there. Some of the red-brown tiles are damaged. Like most other houses in a favela, nothing has been done to the surface. No surface plastering, no paint. The rough red-brown tiles just turn more and more dark and ugly from season to season. The sky above the favela is dark and grey, as if ready to cry. It seems surprising that the two girls are smiling. But does the photo contain mathematics? 43

CHAPTER 4

First let us consider the geometry of the house. From the outside it is a box. Towards the street it has no windows, but a clothesline filled with laundry indicates life within the box. This could well be an overcrowded life. One could consider how the rooms of the house are distributed.35 How is the financing of house building in a favela organised? On what conditions? And why is it so difficult to complete the construction of the house? One could consider the geometry of the favela: How is the geometry of town planning? How is the network of streets organised? The article in Folio de São Paulo that accompanies the photo of the two-stored favela house discusses the growth of favelas. As there is no more space left, the favelas must grow upwards. But how could the feeble favela constructions be expanded upwards? They seem predestined to grow sideward. Well, the photo illustrates that at least two-storied favela-houses are possible. Leafing through a newspaper can provide inputs to creating learning milieus of both type 5 and type 6. When I try to develop a landscape of investigation, I always consider the possible mathematical depths of the landscape. For instance, when one considers developing a landscape with respect to an advertisement of a special offer, and there are many such advertisements in the Monday newspaper, then one could do so acknowledging the profundity of the mathematics of finance. Naturally, the different topics of the mathematics of finance need not be explored as part of investigating the landscape, but the landscape makes such explorations possible. An investigation of games, in particular gambling with respect to sports results, can be addressed through much elaborated mathematical theorising. Again such studies need not be completed as part of the students’ work with the particular landscape of investigation, but they are possible. Leafing through Folha de São Paulo was a search for meaningful examples. However, we have to remember that the students’ experience of what is meaningful is an expression of relationships. There is no simple logic of meaningfulness. There is no guarantee that the two girls standing in front of the greenish concrete wall will have any interest in As part of the project about ‘City Planning’, referred to previously, the students were asked to draw a map of their house. The teacher got surprised of the results. Had the students not understood what making a map would mean? Subsequently the teacher realised that they had, and he came to understand more about the poor living conditions provided by houses in a favela. 35

44

LANDSCAPES OF INVESTIGATION

working with the geometry of a favela house. Even the most careful investigations of actual issues from today’s newspaper need to appear meaningful to the students. The experience of meaningfulness depends on whether or not students establish their intentionality as part of their learning activities. Investigations and explorations are acts; they cannot take place as forced activities. They cannot be completed without the students actually doing the investigations; and this presupposes that the students’ intentionality is part of the inquiry process.

4.4 Moving between different learning milieus The matrix shown in Figure 7 represents a considerable oversimplification. The distinction between the tradition of exercises and landscapes of investigation is not clear-cut. In fact there is a huge terrain of possibilities stretching between these two alternatives. Exercises can be more or less narrow. A closed exercise can be opened a little, and this opening could make space for problem-solving activities. Problem solving could take the more open form of problem posing. Landscapes of investigations could be narrow and specific. They could be explored in the form of project works. Many different learning milieus could be located along the horizontal direction of Figure 7. We could also look at the distinction made with respect to the references of the activities. According to Figure 7, references could be to mathematics, to a semi-reality, or to real-life events. There could be many overlapping possibilities, while other possibilities might be simply ignored by the matrix, like the possibility of formulating landscapes with, say, historical references. The school mathematics tradition is safely located in the milieus (1) and (3). However, one should not be tempted to think of a solution to educational problems by moving rapidly to a learning milieu of type 6. One can think of the matrix in Figure 5 as suggesting a way of reflecting on what has taken place in the classroom. The teacher can consider in what milieu he or she has been operating. One can consider last school year: how was the movement between the different milieus? Where were most classroom activities located? One can think of the matrix as a planning devise: How to proceed between the different milieus in the coming year? 45

CHAPTER 4

It makes sense to think of an educational process in terms of a travel between different leaning milieus. There are no milieus that are ‘good’ as such, while others as such are ‘bad’, but there could be different forms of travels. I find it problematic to place all classroom activities in milieus of type (1) and (3), as we should not forget that the development of prescription readiness seems associated to milieus if type (1) and (3). However, it might make good sense after, say, investigating the intersections of functions of first degree to set up some consolidation work where the students work with exercises related to the such functions. Thus, after working in a milieu of type (2), one can return to a milieu of type (1), before one proceeds, say, to a milieu of type (4). Long ago, I was engaged in a mathematical project involving young children, about 7 years old.36 The main aim of the project was to plan and to construct a playground outside the windows of the classroom where there was a small piece of ground available. Certainly, this activity took place in a learning milieu of type (6), and, as a result of the project, a small playground was in fact set up with the active help of parents during a few weekends. Before that, however, much activity had taken place. First off all, the children visited other playing grounds in order to test which one was a ‘good’ one. Children of seven are experts in carrying out this kind of test. More difficult, however, was to specify the exact quality of the good playing ground. How tall are the swings? How much sand is needed? etc. Many things have to be measured, and in order not to forget such measures it becomes important to make notes about the observations. Not an easy task! As part of the project, which lasted a couple of months, there were periods of ‘office work’, which actually looked like an excursion into a learning milieu of type (1). The children were organised in small groups working in their ‘offices’. As in any public office, voices were low. The children had juice or lemonade in plastic cups put in front of them. They were sitting at their desks which, by some magic, now looked like real office desks. Sometimes the office workers nibbled at a cookie while they added up numbers. Sometimes the radio poured out low, soft music. Sometimes the teacher played the guitar. The papers scattered around the desks contained mostly exercises in adding and subtraction. 36 The following presentation for the playground project is taken from Skovsmose (2001) with only a few modifications.

46

LANDSCAPES OF INVESTIGATION

The point is that the children during the more intensive periods of project work had recognised the importance of being able to add numbers, and to add them correctly. During office hours, this kind of skill was consolidated, and reasons for doing such office work were found in the previous periods of the project work. The actual set-up of ‘office work’ broke with the pattern of the normal exercise paradigm, although the activity as such was of type (1). This illustrates that the route between the different milieus might help to imbue the students’ activities with new meaning. The office work did not take place in an atmosphere of the school mathematics tradition, although it took place within the exercise paradigm.

4.5 Zones of risks and possibilities The different learning milieus presented in the Figure 7 can also be referred to as different teaching-learning milieus. We are dealing not only with milieus for the students, but certainly also for the teacher. Let me say a few words about the milieus as they might be experienced by the teacher. From the teacher’s perspective moving away from the milieus of type (1) and type (3) might appear as moving from a comfort zone into a risk zone. This notion has been introduced by Miriam Godoy Penteado in her study of teachers’ experiences in a new learning environment where computers play a crucial role.37 Moving between different possible learning milieus, and paying special attention to landscapes of investigation will cause a great deal of uncertainty. This is illustrated in the Figure 8. Teachers could experience uncertainties with respect to how to solve a problem. If their students are working with small animals, soon issues will emerge which are far from straightforward to handle. Just imagine that a group of students presents an algorithm for how to proceed from animals of type 5 to animals of type 6. They start off from a specific animal of type 5 and consider in how many different ways one can add a square (if they are considering 2 dimensions) or a centicube 37 See Penteado (2001) for an exploration the notion of risk zone. See also Yasukawa (2010).

47

CHAPTER 4 Sequences of exercises References to pure mathematics References to a semi-reality Real-life references

1

Landscapes of investigation 2

3

4

5

6

Figure 8. Zones of comfort (indicated by light grey) and risks (indicated by dark grey) as related to the different milieus of learning. The figure has been elaborated by Biotto Filho. (if they are considering 3 dimensions). Would it be possible to refine such an approach to a more strict procedure? When students are exploring graphs, in particular by making use of mathematical software, all kinds of things might occur. Many more questions than any teacher could possibly answer will arise. The exercise paradigm establishes a way of keeping the students’ questions in a predictable format. When we have to do with preformulated exercises, a grid of right-wrong dichotomies can be applied to all activities in the classroom. Such a ‘regime of truths’ might provide a comfort zone for the teacher; in fact it might provide a comfort zone for the students as well. They know what to do, and when they have done things in the right way. Measures of performance appear transparent. However, as soon as students and teachers enter a landscape of investigation, the right-wrong grid turns obsolete. Uncertainties emerge. The comfort zone is left behind as risks always accompany landscapes of learning. However, a risk zone is a zone of possibilities. Dealing with risk also means creating new possibilities.

48

Intermezzo The modern conception of mathematics In this Intermezzo I want to outline what we might call the modern conception of mathematics, which I associate with the development of Modernity. It is a broad concept, and I will present it as including three different sets of ideas: that mathematics is essential for understanding nature; that mathematics is a powerful resource for technological invention; and that mathematics is a pure rationality which operates almost as an intellectual game, divorced from other human activities. These three sets of ideas – inconsistent as they might be – establish discursive elements of how to think of and how to address mathematics. They have been developed, integrated, and refined from the time of the emergence of the Scientific Revolution and until the conception of Post-Modernity was formulated. The modern conception of mathematics does not provide an adequate platform for formulating the concerns of critical mathematics education. I will return to this issue in the following Chapter 5, where I will present a critical conception of mathematics and discuss mathematics in action. In this way I try to move beyond the modern conception for mathematics. But let us consider more carefully what it is we are going to move beyond.

Mathematics and natural science The so-called Scientific Revolution proposes that science makes progress. Let me just recapitulate some elements of this revolution. Not least inspired by the ancient Greek philosophers, in particular the Pythagoreans and the neo-Platonists, Nicolaus Copernicus presented a heliocentric world picture. This was a radical alternative to the geocentric 49

INTERMEZZO

description provided by Ptolemy, authorised by the church as providing the proper picture of the universe. Copernicus assumed that the movements of the planets were circles, but through a careful investigation of the movement of the planet Mars, Johannes Kepler suggested that the planets were moving in ellipses, with the sun positioned in one of their loci. Through Kepler’s formulations, mathematics obtained a unique position. It seemed possible to describe exactly the orbits of the planets through mathematics. In this way it became acknowledged that mathematics captured the structures of nature. The Pythagorean idea that everything is numbers got a new, powerful interpretation. Mathematics could express the master plan for God’s creation of the world. And we have to remember that an unquestioned belief in God’s existence dominated the outlook of all the people who contributed to the Scientific Revolution. Atheism as an intellectual possibility only emerged later. Galileo Galilei distinguished between primary and secondary sense data. While the primary sense data refers to positions, movements, shapes, as well as to the number of entities, the secondary data refers to colour, smell, sound, taste and texture. And, as pointed out by Galilei, only the primary qualities have significance for understanding nature. Exactly these qualities can be depicted mathematically. While the secondary sense data signifies what we impose on nature, mathematics helps to delineate the primary sense data, which signifies what nature imposes on us. In short, mathematics brings forward the essence of nature. (An aside: If Galilei were to read the Sports Section of Folha de São Paulo, he might have associated the secondary sense data with the narratives about what took place during the soccer game, and the primary sense date with what is expressed in numbers and figures about the match. And he might have claimed that a soccer game, and nature as a whole, can most adequately be X-rayed by means of mathematics.) Such observations bring us closer to the formulation of a mechanical worldview, as suggested by René Descartes, who has often been referred to as the first modern philosopher. At that time, the spring driven clock had been invented, and the functioning of its subtle mechanics was seen as similar to the functioning of nature. All of nature was a mechanism with cogwheels and gears that were running according to laws laid down by God, when he designed the universal clockwork. Nature operated as a perpetual motion machine that 50

THE MODERN CONCEPTION OF MATHEMATICS

humankind could only dream of copying in minor form. However, humankind was capable of grasping the laws according to which God had created nature, and these laws had a mathematical format. In this way mathematics obtained a paramount role in the understanding of nature. Descartes found that bodies, including heavenly bodies, either stayed at rest or kept moving in straight lines as long as no external force turned them in new directions. In fact Descartes talked about the earth ‘falling’ towards the sun. Such an idea caused, however, new difficulties. If the natural movement of the earth is a straight line, while it is in fact moving around the sun, then there must be some tremendous force involved in crafting the elliptic movements. Here a strict mechanical world view was not of much help: where to find the cogwheels and gears that provided the ‘fall’ of the earth towards the sun? Isaac Newton provided the elegant completion of the Scientific Resolution by formulating the laws which govern all motion – on earth as well as in heaven. The essential point is that we have to do with the same laws for both earth and heaven. The same explanation applies to the trajectory of a stone being thrown and the movement of the earth around the sun. Furthermore, Newton brought the whole picture together with the idea of gravity. Any two units of mass, wherever they might be located in the universe, are attracting each other by a force, which is proportional to the product of the two masses and inverse to the square of their distance. Gravity operates across the whole universe. The nature of this universal force, however, is not specified in any mechanical way. This whole development provided mathematics with a crucial position. Mathematics ensures the basic insight into nature. It was well known that mathematics was used to achieve the beauty and the ideal proportions of any architectural construction. Saint Peter’s Cathedral in Rome was a magnificent example of detailed mathematically formulated architectural design. And now it became evident that God, as the architect of all nature, had used a mathematical blueprint as well. As a consequence, mathematical insight was important for establishing insight into God’s creations. In fact mathematics represented an overlap between human knowledge and God’s knowledge and wisdom. After the Scientific Revolution, mathematics became an integral part of the development of physics, and of any form of natural science, 51

INTERMEZZO

for that matter. No physical theory could be formulated without mathematics. One need only think of how Albert Einstein conceptualised the theory of relativity. Mathematics had become the language of natural science.

Mathematics and technology The notion of progress, as associated to Modernity, not only includes the idea that science makes progress, but that society as a whole makes progress. If we consider the general outlook of the Middle Ages, life before death was of no particular significance in itself; what mattered was life after death. The modern outlook, however, opened the way for a different perspective. The notion of progress comprises the idea that it is possible to improve the quality of life on earth. The principle task becomes to indentify recourses for progress, and one important idea is that scientific progress is the motor of progress on a grand social scale. Scientific insight may help to eliminate different forms of superstition which could obstruct progress, but more directly, insight into nature makes it possible to master nature. Many problems appeared to be related directly to nature. Humankind was surrounded by a hostile environment, and one could think of all the sufferings caused by storms, floods, draughts, etc. which could end in hunger and disease. Nature was powerful, but through the powerful natural sciences it seemed possible to master nature and to apply natural forces for the benefit of humans needs. The insight into nature initiated by the Scientific Revolution could be used for other purposes than understanding nature itself. It could be used for technological enterprises, and scientific insight made it possible to see nature as a resource for welfare and progress. Francis Bacon was somehow on the sideline of the Scientific Revolution. For instance, it is not clear to what extent he was aware of the Copernican revolution. Furthermore, he did not grasp the point of using mathematics for identifying the laws of nature. But he formulated what I refer to as the technological enterprise: Knowledge can turn powerful when insight into nature and technology are brought together. 52

THE MODERN CONCEPTION OF MATHEMATICS

Soon it became clear that technological enterprises had to be pursued via mathematics. The whole idea of mathematics-based technological invention was brought into effect. Important steps in this process were taken when mathematics was recognised as a crucial topic to be addressed at the polytechnic institutions that emerged after the founding in 1794 of École Polytechnique in Paris. Being indispensable to technological enterprise, mathematics came to signify the rationality of progress. With respect to natural sciences, mathematics had become a necessary descriptive tool. As part of the technological enterprise, mathematics became an indispensable constructive tool. It was not possible to realise any technological fabrication without the use of mathematics. The Industrial Revolution initiated the whole domain of mathematics-based construction. And as it was assumed that technology ensures progress, the whole technological endeavour was formulated in the most optimistic terms. In 1932, Charles A. Beard celebrated technology as the fundamental basis of modern civilization, and he emphasises how technology “supplies a dynamic force of inexorable drive, and indicates the methods by which the progressive conquest of nature can be effected” (Beards in Burry, 1932: page xx).

Mathematics and purity A very different notion of mathematics has developed focussing on intrinsic qualities of mathematics. This notion grew from developments in mathematics research during the 19th century and turned into an elaborated perspective at the beginning of the 20th century. It established an extreme purification of mathematics at a time when mathematics had turned indispensible in both science and technology. The notion of mathematics as a pure discipline has always been part of the conception of mathematics through Modernity, but let us look at its extreme formulation which included three elements: the development of an all- embracing axiomatic; an interpretation of mathematical concepts without assuming the existence of any metaphysical mathematical entities; and an interpretation of mathematical truth as a pure formal property. In Principia Mathematica I–III, written by Alfred N. Whitehead and Bertrand Russell and published 1910–1913, the building up of 53

INTERMEZZO

mathematics starts with axioms, five in total, and proofs are constructed by applying only two simple rules of deduction. In this way, Whitehead and Russell wanted to demonstrate that mathematics as a whole can be built on a foundation of logic. 38 Principia Mathematica follows the classic Euclidean paradigm by assuming that mathematics can be demonstrated to consist of absolutely true statements. In order to demonstrate the truth of all mathematics, the axiomatic organisation of mathematics is crucial. An axiomatic reduces the seemingly overwhelming task of assigning truth to all mathematical theorems to assigning truth to the axioms of the construction. How, then, to assign truth value to axioms? This assignment cannot be based on deduction. Instead, we have to rely on intuition. This was Aristotle’s approach, and this is how the Euclidean paradigm has been dealing with the problem ever since. But how can we rely on intuition in such important matters? The simplicity of the axioms seems to be a precondition, and the axioms of Principia Mathematica seem so simple that their truth can be grasped by the otherwise not too reliable human intuition. From then on, intuition has no role to play in mathematics. The theorems rely on logical deduction, and this deduction has the property that if ‘A implies B’, and ‘A’ is true, then ‘B’ is true also. So when intuition has provided the axioms with truth, deduction, like the most reliable postal system, will deliver truth to all theorems. So runs the ideal of the Euclidean paradigm, and Principia Mathematica represents this paradigm. It demonstrates what it could mean to include mathematics in an all-embracing axiomatic. However, Principia Mathematica was to be the last major work in mathematics to assume that mathematical truths could be established through an axiomatic. In Grundlagen der Geometrie, first published in 1899, David Hilbert suggests that Euclidean geometry does not presuppose references to any entities at all. This idea allows for a way of working with mathematical notions without having to subscribe to any particular ontology. The axiomatic geometry includes concepts like point, line and plane, but they have nothing to do with the empirical interpretation of point, line and plane, nor does a Platonic world provide referents for these terms. That the project of logicism, the reduction of mathematics to logic, was not completed by Principia Mathematica I–III (some elements of intuition an empirical observations seem impossible to eliminate) is a well known. And soon the whole project of logicism was given up. 38

54

THE MODERN CONCEPTION OF MATHEMATICS

According to Hilbert, the axioms of a geometric theory express relationships between undefined entities. These relationships are all we know about these entities. Any particular qualities they have are irrelevant to geometry. The development of geometry, then, consists in developing the implications of the axioms, and here only a logical deduction can be used. Any reference to intuitive properties of points, lines and planes would be illegitimate. Thus, a mathematics textbook in geometry would not be in need of any illustrations. Hilbert provides a surprising and provocative answer to the question of mathematical existence. He simply suggests that mathematical existence is equivalent to mathematical consistency. To prove that geometric entities such as points, lines and planes in fact exist does not entail any metaphysical considerations about the nature of such entities. The whole discussion turns into a question of proving that the geometric theory is consistent.39 Just as Hilbert dissolved the discussion of mathematical existence into a meta-mathematical discussion of consistency, the formalists transformed the discussion of mathematical truth into a discussion of provability. In Outlines of a Formalist Philosophy of Mathematics, first published in 1951, Haskell B. Curry presents the notion of mathematical truth as a purely formal property. A formalised mathematical theory presupposes a formal language, the basic units of which are symbols. Symbols can be organised in sequences and some of these sequences count as formulas. Some formulas are nominated as axioms. The rules of deduction state when a formula is a consequence of other formulas. A proof can then be described as a sequence of formulas with the Hilbert wanted to address both consistency and completeness. Consistency of a mathematical theory presupposes that it is not possible to prove a statement, say p, and at the same time (by using a different deductive route) to prove the statement non-p. Completeness of a theory means that of any two pairs of statements, the one being a negation of the other like p and non-p, one of these statements can be proved within the theory. So while consistency presupposes that not too much can be proved, completeness presupposes that not too little can be proved. Hilbert’s hope was to provide formalisations of mathematical theories that were proven to be both consistent and complete. Had this task been completed, Hilbert would have transformed the whole philosophy of mathematics into a logical endeavour. In the most dramatic way, however, Kurt Gödel proved that Hilbert’s hope must remain a dream: A formalism, rich enough to contain the theory of natural numbers, would be incomplete in case it was consistent. 39

55

INTERMEZZO

following property: Any formula in the sequence must either be an axiom or a consequence of some of the previous formulas in the sequence (according to stated rules of deduction). A theorem, then, is any formula which occurs as the last formula in a proof. Based on this clarification, mathematical truth can be described in the following way: A formula can be true in the sense that it can be a theorem in a formalised theory. Thus, the truth value of a mathematical formula becomes relative to a certain type of formalism. The truth of a mathematical statement (a formula) is identified as its provability within a certain formalised theory. The implication of this is that any higher conception of mathematical truth becomes abandoned. There is no need for a Platonic world of ideas. In fact there is no need for anything to refer to, as mathematics in not about anything. Mathematics is pure formalism.40

Modern mathematics education We have now seen three ways of looking at mathematics: mathematics as a sublime way of obtaining an understanding of nature; mathematics as an indispensable resource for technological development; and mathematics as pure rationality. As mentioned these three ways of looking at mathematics may be incompatible. Nevertheless, they supplement each other in presenting an attractive image of mathematics. They bring together the modern conception of mathematics which, in turn, frames what I refer to as modern mathematics education. This education emerged in a distinguishing format during the late 1950’s. The explicit formulation of modern mathematics education was initiated with reference to the importance of mathematics for developing sciences and technology. Thus in 1959 the mathematician Marshall H. Stone claimed that the teaching of mathematics would come to be recognised “as the true foundation of the technological Proponents of mathematical purity were not blind for the applicability of mathematics, but how to maintain the purity of mathematics when it in fact is applied? According to Curry, it makes sense also to discuss the empirical applicability of a mathematical theory, but this discussion is completely different from the investigation of the truth of mathematical statements. 40

56

THE MODERN CONCEPTION OF MATHEMATICS

society which it is the destiny of our time to create” (OEEC, 1961: 18). Thus it became pointed out that not only mathematics, but also mathematics education forms an integral part of technology and progress alike. Modern mathematics education also celebrated the logical architecture of mathematics for its own sake. Thus, it was claimed that a mathematical curriculum could replicate the logical structures of mathematics. The pure features of mathematics were presented as the principal educational elements. This is clearly reflected in the way modern mathematics education interprets educational meaning, as mentioned previously. The meaning of a complex concept could be seen as a composition of the meaning of its different elements, and in this way the presentation of mathematics, as in particular it has been presented in the works of Bourbaki, starting with the set-theoretical notions, could be recapitulated in any educational context. Modern mathematics education presents mathematics as an indispensable tool for insight into nature and for the completion of all forms of technological enterprises; and it celebrates mathematics in its pure format. As a consequence, there is no need to address mathematical rationality critically. According to modern mathematics education, mathematics teachers should serve as ambassadors of mathematics. The concern is how to provide learning environments, including textbooks and curriculum structures, which open a main road for students into mathematics and ensure that students come to appreciate mathematics. Modern mathematics education has a strong impact on the formulation of theories about teaching and learning. Let us look at an example: the genetic epistemology as formulated by Jean Piaget. He assumed that the logical structuring of mathematics, as developed by Bourbaki, is in fact anticipated by the natural learning potential of the child. Through this assumption, Piaget established a close connection between the formulations of theories of learning mathematics and the purified picture of mathematics. To Piaget the important notion in understanding the growth of mathematical knowledge is reflective abstraction.41 Such an abstraction represents the epistemic action of the child, for instance when he or she considers a certain set of operations with objects and then, by identifying some regularities in these operations, makes a further step by recognising some of the unifying principles in these 41

See Beth and Piaget (1968); Piaget (1970).

57

INTERMEZZO

operations. In this case reflective abstractions deal with properties of operations on objects (and not with properties of the objects). Reflective abstractions, as described by Piaget, represent individual faculties by means of which the child establishes the basis of certain operations. These abstractions become a mode of constructing more abstract mathematical notions, and in this way the child finds his or her way into mathematics. Reflective abstractions are presented as a modus of constructing mathematical knowledge in its structural format. Nowhere in Piaget’s works, I would argue, is the idea presented that mathematical rationality may be disputable. Piaget’ genetic epistemology assumes a (blind) trust in mathematical rationality. Such a trust characterises much of the theorising that is associated with the modern mathematics education.42

42 I see the so-called French tradition in mathematics education as an example of modern mathematics education, where a trust in mathematical rationality makes part of the notion of ‘didactical transposition’.

58

Chapter 5 A critical conception of mathematics Mathematical rationality can be presented in a rosy picture if we pay particular attention to the way mathematics facilitated the Scientific Revolution and helped provide an insight into nature. More rosy colours can be added when the mathematical resources for technological development are portrayed. And finally a grandiose picture of mathematics as the sovereign of science may emerge when we pay attention to the intrinsic qualities of mathematics. This celebration represents the modern conception of mathematics including an unquestioned trust in its rationality. The assumption of a close correlation between scientific development and progress in general is part of the modern outlook. We should not forget, however, that this outlook developed in connection with the so-called great discoveries which also included some of humankind’s most brutal colonisations. This was accompanied by the slave trade and an explicit formulation of racism, later to be turned into a scientifically based racism. Such events were part of the modern outlook, too. So, one should not be surprised if the relationship between science and progress is not quite so simple. The notion of risk has been associated to nature. Humankind has been surrounded by a hostile nature, and it has been our task to master this environment. This idea of conquering nature was pointed out by Francis Bacon, but today the idea of technology, by definition, as siding with humanity against nature becomes questionable. Technology also envelopes humankind in a techno-nature, which contains risk. The creation of atomic energy may serve as an illustration of this. An atomic power plant establishes an enormous resource of energy, but it also includes new risks. There might not be any catastrophe, but there could. In this sense, we can talk about a risk society, where the very production of risk is part of technological development. As a consequence, the supposed intrinsic connection between scientific development and progress in general appears dubious. 59

CHAPTER 5

Risks are far from distributed in a uniform way around the globe. We are not entering the risk society shoulder to shoulder. Here we do not find equality and brotherhood. Some become much more exposed to risks than others. The location of the English atomic power plant, Sellafield, can serve as an example. We can also think of the risks associated with different kinds of production. Thus, globalisation includes the relocation of particularly risky and polluting types of production to poorer parts of the world, where ‘bad’ jobs are preferred to no jobs. Mathematical rationality is an indispensable resource for all those forms of technological construction, initiatives and decision-makings which form our techno-nature. This indicates that a mathematical rationality can also be a doubtful rationality. This, however, does not mean that it is a rationality which brings about problematic conclusions by necessity. It is a rationality which can provide important innovations, but also bring about catastrophes. It is rationality without an essence. It is an undetermined rationality. It is a critical rationality. It can go ‘both ways’. I will try to outline an interpretation of mathematics different from the modern conception of mathematics. I will present a critical conception of mathematics by relating mathematics to discourse and power and then discussing different dimensions of ‘mathematics in action’. On this basis, I will try to formulate some preoccupations with respect to mathematics and mathematics education.

5.1 Mathematics, discourse, and power In breaking with the outlook of modernity, including the celebration of science-guided progress, Michel Foucault opened the way for an investigation of knowledge and power.43 He addressed the conceptions of madness, control, the birth of the clinic, etc., and he demonstrated that a scientific terminology might impress an order on the phenomena that it is supposed to describe. The scientific language in use might not represent a straightforward reflection of the reality which it is assumed to be describing. Instead the assumed reality might reflect categories incorporated in the language of description, which in this way turns into a powerful tool of prescription and formatting. Power can be acted 43

See, for instance, Foucault (1989, 1994). See also Valero (2009).

60

A CRITICAL CONCEPTION OF MATHEMATICS

out through the applied language. In his investigations, Foucault, however, did not address the natural sciences in any detailed way, and certainly he did not address mathematics. It might appear surprising, since the science-technology conglomerate represents a profound knowledgepower interaction; and mathematics provides an important site for studying this. Linguistic relativism, as formulated by Edward Sapir and Benjamin Lee Whorf, suggests that language not only describes but also shapes what is experienced. Language provides a grammar, not only for what to say and not to say, but also for what to see and not to see. Linguistic relativism relates to Immanuel Kant’s idea that what we experience is not things as such. Instead our experiences are structured by our categories, which are imposed on our experiences. According to Kant, such categories have an eternal permanence, but according to linguistic relativism, categories are historically and culturally developed and integrated in the basic grammar of language. The basic categories of our life-worlds are manufactured. This brings language into a crucial position for understanding what we refer to as our reality. Language provides a formatting of reality by imposing presumptions, categories, metaphysics, priorities, understandings, as well as misunderstandings. And, returning to Foucault, we can claim that also scientific discourse provides such a formatting trough their particular “regimes of truths”. Language also includes resources for acting. This aspect of language was suggested by John L. Austin and Ludwig Wittgenstein. Any utterance, statement, expression, formulation, question, etc. includes acts. Thus, to make a promise is more that saying something. Promising means doing something, and such an act can be discussed in terms of its content, force and effects. The locutionary dimension of a promise refers to the content of the statement; e.g. I could promise a friend to visit him tomorrow. The illocutionary force refers to the element of promising, which, in this example is the obligation entailed in me making the promise. The perlocutionary effect refers to the consequences which my promising might have: my friend might become a bit displeased by the prospect of having me disturbing him tomorrow. The overall point being that all these dimensions are involved in any form of speech acting. If we combine the two ideas, i.e. that language is part of a formatting of reality and that language includes actions, then the way is opened for a performative interpretation of language and of the 61

CHAPTER 5

power-language interaction – and in particular with respect to mathematics. In many cases, mathematics has been described as a language. For instance, a principal element of the purification of mathematics was to present it as formal language that operated without any references. It was presented as a transparent tool. Seeing mathematics as a language can, however, be developed in quite a different direction when we draw attention to the performative aspect of language. This aspect can in fact be associated to all different forms of mathematics: in engineering, economy, daily life, different cultural settings, research, etc. I will try to illustrate in what sense we can talk about mathematicsbased performances and in this way explore the critical conception of mathematics.44

5.2 Dimensions of mathematics in action I shall try to be more specific about the performative aspects of mathematics by exploring five aspects of mathematics in action: (1) Technological imagination, which refers to the possibility of exploring technical possibilities. (2) Hypothetical reasoning, which addresses consequences of not-yet-realised technological constructions and initiatives. (3) Legitimation or justification, which refers to possible validations of technological actions. (4) Realisation, which takes place when mathematics comes to make part of reality, for instance through processes of design and construction. (5) Dissolution of responsibility, which appears when ethical issues related to the implemented action become eliminated.45 Thus in the following I will refer to certain forms of applied mathematics. 44 For a discussion of linguistic relativism and speech acts as initiating a performative interpretation of mathematics, see also Skovsmose (2009b). 45 I have presented aspects of mathematics in actions in different ways (see, for instance, Skovsmose, 2005, 2009b), and I have not reached any conclusion about which way provides the most adequate overview. In collaboration with Ole Ravn (Christensen) and Keiko Yasukawa I have analysed examples of mathematics in action, and we have explored different conceptual frameworks for expression ‘agency’ related to mathematics. See, for instance, Christensen, Skovsmose and Yasukawa (2009). These shared efforts I am drawing on in the following presentation. See also Baber (2010); Jablonka (2010); and Ravn (2010).

62

A CRITICAL CONCEPTION OF MATHEMATICS

Technological imagination Technological development is based on imagination. This applies to any form of design (be it of machines, artefacts, tools, schemes for production, etc.) and decision-making (concerning management, promotion, economy, etc). In all such areas we find mathematics-based technological imagination. As a paradigmatic example of such imagination, one can think of the conceptualisation of the computer. The mathematical conception, in terms of the Turing machine, was investigated in every detail. Even the computational limits of the computer were clarified before the construction of the first computer. The information and communication technologies are deeply rooted in mathematics-based imagination. Thus, powerful possibilities for cryptography were identified through mathematical clarifications of number-theoretical properties. Such possibilities could not be anticipated through a commonsense conception of cryptography. The mathematics resources brought the technological imagination into a new landscape. My general point is that many innovations depend completely on mathematics. There is no commonsense-based imagination equivalent to a mathematics-based imagination. Let us consider an example from daily practices where a mathematics-based technological imagination is brought into effect: price fixing. Here we can take air-fares as an example. In this domain we see very different schemes for pricing. Thus, airlines deliberately overbook. However, the overbooking is carefully planned, and it is part of the whole computational scheme for fixing the prices. This scheme cannot be organised without bringing a mathematical model into effect. A lot of experimental pricing has to be carried out before a price policy can be decided upon. In fact the pricing becomes an ongoing process. The identification of the degree to which a flight can be overbooked can be based on the statistics of the numbers of no-shows for a particular departure. (A ‘no-show’ refers to a passenger with a valid ticket who does not show up for the departure.) The cost of bumping a passenger can be estimated. (‘Bumping’ a passenger means not allowing a passenger with a valid ticket to board the plane, as the plane is already fully booked.) The predictability of a passenger for a particular departure being a no-show is naturally an important parameter in designing the overbooking policy. This predictability can be improved when the types of tickets are grouped in different types associated with different 63

CHAPTER 5

conditions, for instance considering the possibility of changing the ticket. The whole overbooking policy can be experimented with mathematically, until one has identified how to maximise profit. Such experimentation and economic decision-making takes place in all kinds of business, in marketing, in production planning, in big companies, in small companies, in any economics sector of society. A clear impression of the mathematics-based approach to pricing is provided by leafing again through the Folha de São Paulo, for instance considering the special offers on cell-phones. In many cases an item, which can be thought of as a service, does not have a particular price defined in a clear and unambiguous way. Instead there are schemes of payment, which establish a set of economic transactions and obligations between the company and the buyer. This kind of pricing is an expression of a mathematics-based technological imagination, and the result of such imagination permeates our daily practices. Hypothetical reasoning Hypothetical reasoning is counterfactual. It is of the form, ‘if p then q, although p is not the case’. This form of reasoning is essential in any kind of technological enterprises as well as to our everyday decisions. If we do p, what would be the consequence? It is important to address this question before we in fact do p. In order to carry on with any such hypothetical reasoning, mathematics may be brought into effect. We can think of decisions like: Should we buy an energy-saving fridge? Should we buy the expensive one? Or should we carry on with the old one for another year? What decision to make? How to consider the implications of each decision? One can try to do some calculations of costs. The way of addressing such questions from daily life may take the same form as does complex decision-making, the difference being that in more complex cases, the hypothetical reasoning normally presupposes the use of elaborate mathematical modelling. The mathematics model comes to represent an imagined situation, p, which could refer to any form of technological design, construction or decision-making. The mathematical representation of the imagined situation, p, we can refer to as Mp. Through investigation of Mp, one tries to come to grips with the implications of realising p. However, the implications that are identified by investigating Mp are not real-life implications; they are just calculated implications. And it is far from 64

A CRITICAL CONCEPTION OF MATHEMATICS

obvious what the relationship might be between calculated implications and real-life consequences of completing the technological enterprise. This observation applies to any economic initiative, to any form of engineering construction. One carries out calculations based on a model in order to estimate consequences of not-yet performed actions. For instance, the stability of new aircraft design is carefully modelled and estimated long before any real construction takes flight. In many cases it appears that only through mathematics is it possible to investigate details of a not-yet-realised construction. Here we also find how risks can be produced. When we identify implications of completing a certain construction and the identification of implications is based on a mathematical model and not on any real construction, there is always a risk of something being overlooked. In fact very many aspects are by definition overlooked, as mathematics only represents particular features of a situation. There is no direct similarityrelationship between an imagined situation, p, and its mathematical representation, Mp. And certainly when realised, the technological construction may result in something which is very different from what was modelled. Mathematics-based hypothetical reasoning is formulated within a certain logical space provided by mathematics, implying that only a certain space of consequences can be grasped. The ‘blind spot’ of a mathematics-based hypothetical reasoning might be a tremendous blind region. The emerging of the risk society is related to this region. Most financial decision-making is based on a careful risk estimation, which in turn has developed into an advanced mathematical discipline. But exactly by being mathematical, risk estimations come to include rather extensive blind regions which might turn into fertile ground for economic crises. Legitimation or justification The notions of legitimation and justification are different. According to a classic perspective in philosophy, justification refers to a proper and genuine logical support of a statement, of a decision, or of an action. Naturally, what is proper and genuine and what is logical are not simple to define, but the notion of justification includes an assumption that some degree of logical honesty has been exercised. The notion of legitimation does not include such an assumption. One can try to legitimate an action by providing some argumentation, although without 65

CHAPTER 5

much logical significance. The point of providing a legitimation for an action is to make the action appears as if it is justified. In general, a legitimation is an as-if justification. However, it might only be within an idealised philosophical framework that it is possible to distinguish between legitimation and justification. Mathematics might blur such a distinction. When a mathematical model is brought into effect, it can serve as both a legitimation and a justification. It has been pointed out that in the case of huge engineering design processes, like bridge building, the mathematical modelling plays an active part in investigating implications of completing a particular form of the construction, for instance with respect to the impact on the environment. However, in such cases the decision-making is often based on one and only one model. In some cases we find that the mathematical modelling serves the purpose first and foremost of legitimating an already made decision. Through the mathematical model, one provides a mathematical description of the construction in terms of Mp, and one tries to identify implications of completing the bridge construction by investigating Mp. The consequences identified through the investigation of Mp need not, however, reflect any real consequences. For instance, the mathematical model can be formulated in such a way that the calculated environmental implications are seen to be within an acceptable range. The difference between the model-based calculated implications and the real-life ramifications, to be experienced when the construction is completed, might be tremendous. But the modelbased calculations have served their legitimating purposes, as the construction when completed cannot be moved or removed. In many cases detailed mathematics-based analyses cannot be substituted by any form of analyses. Mathematics simply provides a space for justification (as well as of legalisation) which is unique. Thus, a justification for a specific design of an aircraft might have no substitute in a commonsense-based argumentation about stability. As mathematics might bring about a unique space for technological imagination, a mathematics-based technological imagination might also bring us into a unique space for legitimation and justification. Realisation A mathematical model can become part of our environment. This is the most direct exemplification of the remark I made previously about 66

A CRITICAL CONCEPTION OF MATHEMATICS

speech acts and discourse. A language is not simply a descriptive tool: it also includes performances. Our life-world is formed through categories and discourses many of which emerge through mathematics in action. Technology is not something ‘additional’ which we can put aside, as if it were a simple tool, like a hammer. We live in a technologically structured environment, a techno-nature. Our life-world is situated in this techno-nature, and we cannot even imagine what it would mean to eliminate technology from our environment. Just try to do the subtraction piece by piece. We could start by removing the computer, the coffee machine, the fridge, the TV set, the phone. Then we continue by removing medicine, newspapers, houses, cars, bridges, streets, shoes. We have no idea about the kind of life-world into which such a continued subtraction would bring us. In this sense our life-world is submerged in techno-nature. Mathematics is an integral part of both techno-nature and lifeworld. Thus all the things referred to: coffee machine, fridge, TV set, phone, medicine, newspapers, houses, cars, bridges, streets, shoes are produced through processes packed with mathematics. But not only the objects which make part of our techno-nature are formatted through mathematics, so are many practices. Mathematics establishes routines. The travel business can again serve as an example. When I want to buy a ticket, the assistant at the travel agency can easily provide information about prices and schedules. The whole computational survey of information is part of the routines of the agency. Furthermore, much of the information is available on the Internet, which makes it possible to organise the booking from home. In all such cases the procedures are determined through computerised algorithms. Another domain where mathematics-based routines have been established is medicine. Here many routines of making diagnosis are established with reference to the numerical definition of what it is to be in normal health. The diagnosis and decisions about which treatment to consider adequate depends on the deviation from the norm according to certain parameters (concerning the level of cholesterol and blood pressure, for instance). Decision-making can be routinised, treatment can be routinised, and the extension of such routinisation can ensure efficiency. At the same time the procedures include new risks, as, being defined through a mathematics-based norm setting, they need not apply adequately in all situations. 67

CHAPTER 5

Dissolution of responsibility Mathematics-based action may include an dissolution of responsibility. Let us again consider the example with the travel agency. The assistant can tell the customer the price of the ticket and whether tickets are available on a certain day or not. The assistant cannot provide a ticket if they are sold out. Even if the costumer might be able to demonstrate that the travel is of extreme importance, the assistant cannot do anything. The assistant is in no way responsible for what the computer states. Nor is he or she responsible for the price of the ticket, the conditions of payment, or for anything that transpires on account of algorithmically defined procedures. One could ask who is responsible for the actions exercised through a computer? Somehow responsibility seems to evaporate. It cannot be the assistant using the model who is responsible. Nor can it be the model itself. Mathematics cannot be responsible, even when it is brought in action. But might we not say, at least, that a certain way of thinking is responsible? Could the people who constructed the model be responsible? Are the responsible ones those who have ordered the model? My point is that actions based on mathematics easily appear to be conducted in an ethical vacuum. Actions we normally associated with an acting subject. However, mathematics in action appears to be operating without such a subject. And when the acting subject disappears, the notion of responsibility seems to be blowing tin the wind. Mathematicsbased actions may appear as the only actions relevant in the situation. They might appear to be determined by some ‘objective’ authority as they represent the necessity provided by mathematics. In this way the elimination of responsibility might be part of mathematical performances, which in turn makes part of a knowledge-power dynamics.

5.3 Wonders, horrors, and reflections Mathematics in action can take many different forms. Wonders can be associated with mathematics in action, and medical research seems to provide many examples of this. In fact it is very difficult to think of any medical research without mathematics playing an integral part. It is also possible to provide examples of horrors facilitated by mathematics. 68

A CRITICAL CONCEPTION OF MATHEMATICS

Military enterprises cannot be carried out without mathematics. Economic restructurings resulting in the firing of workers become realised through mathematics. One could obviously retort that it is unfair simply to associate mathematics with horrors by referring to military application, as one could argue that a military is necessary for national security. One could also claim that firing people is part of the improvement of production efficiency, which is necessary for the general welfare. This brings us to the observation that the wonder-horror dichotomy with respect to mathematics in action may not actually be relevant to apply. It might be better to acknowledge that it is very difficult to establish any uniform scheme for evaluating mathematics in action. Like other forms of action, mathematics in action can have very many different implications, which could be judged in different ways depending on perceptive and context. This brings us to the critical conception of mathematics. Mathematics represents a rationality which could serve any purpose. Mathematics does not contain any essence, which provides mathematics-based actions with any particular qualities. Mathematics in action could come to serve any interests. As a consequence, mathematics in action is in need of reflections. Such reflections must be conducted with reference to all the particularities of the action including its context. If one applies a broad notion of ethics, one could also talk about an ethical demand associated to the critical conception of mathematics. Like those adhering to the modern conception of mathematics, so also advocates of the critical conception acknowledge that mathematics operates within a wide range of scientific disciplines. However, from a critical perspective, there is no automatic celebration of the role of mathematics within these disciplines. As any language, mathematics might bring along a range of metaphysical assumptions and, for instance, facilitate a mechanical outlook. In this way a mathematical discourses may provide an formatting of scientific discourses that need to be addressed critically. The modern conception of mathematics appears to dispense with carrying out reflections with respect to technology, due to the overall confidence that when mathematics is applied, progress will be ensured. But the discussion of mathematics in action brings about a different conclusion. Mathematics is an integral part of different ways of formatting our environment, our techno-nature, but this formatting does not 69

CHAPTER 5

ensure any automatic improvement. Technology has an impact on all spheres of daily life. Technology makes changes, but techno-nature does not develop according to any standards of progress, nor is there anything ‘natural’ about this growth. The modern conception of mathematics considers mathematics pure rationality. This means that mathematics is seen as a rationality which can be a resource of reflection. It can be seen as a sublime format of critical thinking. However, studies of mathematics in action call attention to the need for addressing mathematical rationality critically. This rationality cannot be related to purity. Mathematics in action means action, and as any other form of action, it requires reflection. Actions can be dangerous, courageous, risky, harmless, benevolent, praiseworthy, etc. And so can mathematics-based actions. Critical reflection is needed, and the ethical demand comes to operate as an overall challenge with respect to mathematics.

70

Chapter 6 Reflection Today’s societies include many processes, which establish a tremendous amount of feedback on society. As an illustration, one can think of the car industry. It has the explicit aim of producing cars, selling cars, and making this a profitable business. The car industry may experience prosperous periods; or it could go into recession. Such considerations refer to the explicit dimension of car production. However, there is also an implicit dimension of this production. It needs resources, and this brings about a competition in getting access to resources. Cars need petrol, and this may provoke international conflicts related to the problem of controlling access to oil. One can also think of pollution as an example of an implicit production conducted by the car industry, and of the entire network of motor roads, including the accidents that take place, as being part of the implicit production of the car industry. The point is that the implicit production makes up a part of the whole scheme of production. It might, however, be a simplification to try to separate between explicit and implicit production. We have to consider the full scope of intended and unintended aspects of the production, when we want to reflect on the car industry. This applies to any form of production, of any form of enterprise of economic, organisational, political, or technological format. We have to reflect on the full scope of any form of action including its feed-back on society. This brings us to a broad notion of reflection, which I also find important with respect to mathematics in action. In fact mathematics in action is part of these very many different processes including intended as well as unintended implications. The ethical demand, as referred to in the previous chapter, signifies the need for reflection with respect to all kinds of social processes, and not least those where mathematics in actions makes part. Reflections have to do with judgement of actions. (One can also reflect on, say, descriptions, statements, theories, etc., but here I concentrate on actions.) Reflections can be related to profound ethical considerations with respect to actions and be seen as a philosophical concept. However, I also see reflection as an everyday notion of giving 71

CHAPTER 6

thought to actions. Everyday life includes many decisions to be taken and actions to be carried out. It is filled with reflections. Reflection is an important educational notion. All kind of teachinglearning issues can be addressed through reflections. The students could consider what they are supposed to be doing in the classroom: Does it make sense what the teacher is talking about? What would happen if we did not do our homework? If the teacher asks us to form groups, will I get into the same group as Peter? Will I be bullied during the next break? The teacher will consider how things are working out: Are the students active? Will Michael again start making trouble? Will the next break be long enough for me to prepare the following lesson? I will try to operate with a notion of reflection that includes very many different aspects. (Thus I see the notion of reflection that grows out of Piaget’s contribution to modern mathematics education as extreme limited.) I want to establish a comprehensive social and political dimensions of reflection; dimensions that I tried to illustrate with reference to the car industry. I will view reflection as an expression of an ethical concern as well as being an everyday activity. With this in mind, I will discuss reflections with respect to the leaning of mathematics. I will be fully aware that we might have to do with a network of different notions and ideas. ‘Reflection’ might escape any attempts to be summed up by a clarifying definition, but I find the notion important in order to formulate some preoccupations of critical mathematics education. And better now to get to some examples. Through these I will try to illustrate what it could mean to reflect on mathematics, reflect with mathematics, and reflect through mathematical inquiries.

6.1 Reflections on mathematics Reflections can concern actions that are carried out with reference to mathematics. As an example I refer to the project ‘Terrible Small Numbers’, which addresses the notion of risk.46 There are naturally The example was developed in collaboration with Helle Alrø, Morten Blomhøj, Henning Bødtkjer and Mikael Skånstrøm and was described in Alrø and Skovsmose (2002). I have also summarised the example in Skovsmose (2006a) with a special emphasis on the notion of reflection. See also Skovsmose (2007b). 46

72

REFLECTION

very many different ways to contextualise risks, and in the project the topic was salmonella-infected eggs. This was due to the fact that in Denmark at that time, there had been much debate about salmonella infection. A number of people had become seriously ill, and one person had died. We have tried out the project with different groups of students (between 12 and 15 years of age). Here, I only give a general description of the project. One overall idea was to make it possible for the students to experience a situation where mathematics was brought into action and to reflect on such actions. We tried to create a situation where students came to face questions like: Can we trust information obtained from samples in order to draw conclusions about the whole population? What does it mean to make decisions based on figures and numbers? As part of the planning of the project we discussed how to illustrate eggs. The suggestion from Henning Bødtkjer, one of the participating teachers, was that eggs could take the form of empty film cases. Such black cases could easily be opened and checked. When the project started a whole population of eggs was brought into the classroom in a trolley. It had been easy to collect empty film cases from photo shops (but maybe not so easy any longer due to the advent of digital photography). The first population contained 500 eggs, of which 50 were infected by salmonella. This was known to every student in the classroom. The 450 eggs contained a healthy yolk, in the form of a yellow piece of plastic, while the remaining 50 contained a blue piece of plastic, indicating salmonella infection. The students worked in groups, and their first task was to select a sample of 10 eggs each from the trolley. This was exactly what an egg tray from the super market normally holds. The students then checked the eggs of the sample and made a note of the number of salmonella infected eggs. Then they collected a new sample, and in this way they collected a small amount of empirical material about how many salmonella infected eggs were included in a sample. Many students expected one egg out of the sample of 10 to be infected by salmonella. What they realised, however, was that this was far from always the case. They became aware of the fact that a sample does far from always reveal the ‘truth’ about the population from which it is drawn. When the information was put together, it revealed that less than half of the samples contained one and only one salmonella-infected egg. The students tried to find 73

CHAPTER 6

explanations for this observation. Could it be that the eggs in the trolley were not mixed up well enough? Would it be correct to imply that if one had a perfectly good mix, then a sample would contain one and only one salmonella-infected egg? Or are samples rather unreliable messengers regarding properties of the whole population from which they are drawn? Such considerations naturally point towards a more profound problem. In almost all real-life situations, we know nothing about the whole population except what is revealed through samples. This applies to any form of quality control of a product. In this way the project opened up to a broader discussion of the reliability of samples and of statistics and of information provided by numbers. At the same time it seems clear that information via samples cannot possibly be substituted by other more reliably sources of information. So we have to operate in the best possible ways with this kind of (more or less reliable) information. The discussion of reliability served as a first step in addressing mathematics in action. The next step was to bring the students into a situation where they had to make a decision based on numbers. In this part of the project, two trolleys were brought into the classroom. Each trolley contained a collection of eggs: Eggs from Greece; and eggs from Spain. Each group of students were asked to think of themselves as an egg-import company, and they had to make a decision as to which country, Greece or Spain, should provide the imported eggs. Eggs from both countries were infected by salmonella, but to different degrees. The degree of the infection was not known by the students, nor by the teacher as he had randomly added some infected eggs to each trolley. The economic conditions were explained to the students. The prices of the Greek and the Spanish eggs were the same, the equivalent of 0.50 Danish Kroner per egg. They could also expect to sell the two types of eggs for the same price, namely 1.00 DKr per egg. The salmonella control was not cheap. It cost 10.00 DKr to have one egg checked for salmonella. Furthermore eggs opened in the salmonella control were destroyed. They could not be sold later on, meaning that a careful check of each an every imported egg would not leave any eggs for selling. The students were asked to provide a procedure for decisionmaking and to make a budget of the whole business, including the 74

REFLECTION

amount of eggs they were ready to import and the amount they would inspect. In other words, they were asked to outline a procedure for decision-making, and in this way they came to experience mathematics in action. The students faced a dilemma. One the one hand, they could make a careful and elaborate statistical investigation in order to make sure that they did in fact import the eggs of the best quality, but the more elaborate they made their decision procedure, the less profitable business they were doing. On the other hand, they could try to reduce the cost of salmonella sampling, but that would make their decisions more tentative. This dilemma is fundamental to almost any kind of mathematics-based decision making. Any kind of quality control is a costly affair, so the more well-justified decisions one wants to make, the less profitable business one seems likely to be doing. This could bring about a more general discussion of what it could mean to make responsible decisions when facing such a dilemma. I find that the issues of reliability and responsibility are of general significance for reflecting on mathematics in action. They help to introduce an ethical perspective on mathematics in action. Reflections could concern all the mentioned aspects of mathematics in action. Thus, one could reflect on the nature of a mathematicsresourced technological imagination with respect to particular issues. As illustrated previously, such an imagination could bring about new business principles and new schemes for calculating prices and conditions for payment. It could bring about actions that could not be concepttualised by a commonsense-based imagination. But what is the strength and weakness of building an imagination on mathematical sources? One could reflect on the hypothetical reasoning that could be performed with respect to mathematical modelling. Some advantages might be associated with a mathematical X-raying of a situation, but there are also many hazards. One can reflect on the format of legitimisation or justification for certain actions and decisions that are made with reference to mathematics. One can consider what might be established or realised though mathematics. And one could consider to what extent an illusion of objectivity brings about a dissolution of responsibility. All aspects of mathematics in action are important to address through reflections. This applies to all the different forms of mathematics one may have in mind, including all ethnomathematical variations. 75

CHAPTER 6

6.2 Reflections with mathematics Although sometimes dubious, mathematical rationality is rationality. Even though there are many reasons to reflect on mathematics, reflecting with mathematics is still a crucial activity. Let us again consider the example of the mathematical X-raying of a football match. It might be that such an X-raying helps to identify some of the strong features of a team’s performance as well as things that could be improved. It is also clear that a mathematical X-raying leaves out very many aspects of what has taken place. The X-raying leaves out the body and the flesh of the match. This observation applies not only the mathematical representation of a soccer match, but to any form of mathematical X-raying. Still it can be extremely useful. As part of the project ‘City Planning’, as referred to previously, it was clarified that only 53% of the water that was delivered into the water supply system of the city Rio Claro, a city in the interior of the São Paulo State, was in fact registered by the customers.47 Information about the disappearance of water could be expressed verbally, but putting things in numbers makes it possible to reflect in a more systematic way on the efficiency of the water supply system. One could consider, first, if the information is correct. How is the amount of delivered as well as received water in fact calculated? Do some costumers not figure in the system of measuring? One can also start addressing the possibility of locating the problem. Are there some ways of tapping water from the system without being measured? Is the water-supply system leaking? Are there ways of estimating the water supply for specific neighbourhoods? Could the disappearance of the water be related to the amount of time the water supply system has been in service, which certainly might vary from neighbourhood to neighbourhood? One could also start considering the possible improvement of the water supply system? What could be the maximum percentage of delivered water that could possibly be registered by the customers (assuming that 100% would be a practical impossibility)? What is the situation in other cities? With an estimation of the optimal percentage, one could start gauging the yearly gain of having the 53%-system repaired. Such a gain could in turn be compared to the cost of repairing the system. 47

See Biotto Filho (2008).

76

REFLECTION

All such reflections can be carried out, and Denival Biotto Filho’s main conclusion based on the project ‘City Planning’ is that mathematics is an important resource for formulating, strengthening and specifying a broad variety of socio-political and economic reflections. It is possible to reflect with mathematics, and in many cases mathematics is a resource for strengthening reflections. A similar conclusion can be drawn from the Energy project referred to in the first chapter. Through this project it became possible to formulate some issues about the use of energy in a more specific way. The description of the transformation processes from barley to meat through input-output calculations made it possible to grasp in a more specific way the ‘energy costs’ connected to such transformations. This does not mean that the particular energy costs were estimated correctly, but the idea that it is possible to associate energy costs to different transformations is an import insight. Such a cost could be expressed verbally, but mathematics gives the formulations a different format. Naturally, this format need not represent any truth, and an important element of the energy project was to compare the transformation figures identified by the students with already established research results. Through such comparisons, the students came to consider what uncertainties could be connected to their procedures; furthermore they got the opportunity to consider what kind of uncertainties might be included in agricultural research in general.

6.3 Reflections through mathematical inquiries The different learning milieus, as presented in Figure 7, provide different possibilities for reflection. While the exercise paradigm imposes many prescriptions for what to do and how to do it, landscapes of investigation make spaces for inquiries. Naturally, one can make many reflections in relation to doing exercises: Are the result calculated correctly? Did I do the right exercise? Did I use the algorithm correctly? etc. Such reflections are conducted within the limited exercises-generated space. However, inquiries cannot be prescribed in detail. Instead inquiries presuppose a degree of analytical freedom, and this encourages reflection of very many forms. 77

CHAPTER 6

Let us consider an example, referred to previously, related to learning milieus of type 2. To approach a mathematical investigation of the graphs of functions of the format ( ax 2 + bc + c ) ( dx 2 + ex + f ) 2

F(x ) =

2

needs reflections. Could one try to clarify the features of the graphs in terms of the patterns of asymptotes? How to relate the different patterns to different combinations of values of the parameters? Would it be easier to express the asymptotic pattern if the functions are described in the following way: F(x ) =

( ax + b )( cx + d ) ( ex + f )( gx + h )

Or should one start with: F(x ) =

( x – a )( x – b ) ( x – c )( x – d )

In fact it seems impossible to separate mathematical inquiries and mathematical reflection. The project ‘Caramel Boxes’ will illustrate the kind of incentive for reflections that a learning milieu of type 4 could provide. Such a milieu includes references to non-mathematical issues, and this provides a broadening of the scope of reflections.48 The project concentrated on the design of boxes intended to contain different amounts of caramels. The project, therefore, called for exploring the relationship between the length factor, l, the area factor, a, and the volume factor, v, for different boxes of the same proportion. The principal properties of this relationship were not explicated to the students, but they could be discovered through working with the design of the boxes. The students were invited to design a caramel box that could contain twice the amount of caramels of an already given box. Would one need a double-sized The project ‘Caramel Boxes’ is described in Alrø and Skovsmose (2002). The short presentation here is based on the summary of the project in Skovsmose (2006a). 48

78

REFLECTION

sheet of paper in order to construct such a box? And what would a double-sized sheet in fact mean? What would happen if one constructed a new box where the length of all sides are twice the length of the sides of the original box? As part of the project the students had the possibility to reach the insight that if the length factor for two proportional boxes were l, then the area factor a would be l 2, and the volume factor v would be l 3. The project invited students to reflect on mathematical properties, and to address different aspects of proportional reasoning. In the project we find references to caramels and to boxes although in a semi-real setting, and such references provide other possibilities for reflection. Box design is part of a huge industry, and this could be addressed from an ecological perspective. Boxes could be different shapes, and one could consider the shape of the box in relation to the amount of material used and its volume. Based on such observations one could address questions of use of resources in a more profound way. Let us return to the “Energy” project, illustrating a learning milieu of type 6. This project provided a broad basis for reflecting on consumption of energy, but it also illustrates the intimate connection between inquiry and reflection. One of the issues of the project was to calculate the ‘front area’ of a cyclist. But how to do this? Each of the students had to fix a square made of cardboard the size of 1 dm2 with two safety pins on the pullover, before riding towards the camera. A picture was taken, and, as indicated by Figure 9, the square of 1 dm2 was easy to identify. The students then squared the whole picture, and from that they could make a good estimation of the front area. The estimation of this

Figure 9. Calculating the front area of a cyclist. 79

CHAPTER 6

area, a, could then be used for calculating the bike resistance, r, through the following formula: r = c1av2 + c2 where r refers to the ‘bike resistance’, a to the front area of the cyclist, v to the velocity, while c1 and c2 are two constants that depends of the type of the bike (being a normal bike, a sports bike or a racer). The bike resistance, r, makes part of other formulas through which the use of energy through the trip on the bike can be calculated.49 That the different learning milieus provide different possibilities or reflections has much to do with the patterns of communication relating to the different milieus. In Dialogue and Learning in Mathematics Education, Helle Alrø and I have discussed the relationship between communication and inquiry. We find that processes of inquiry are closely linked to dialogic process.50 In general we find that landscapes of investigation invite for dialogues, although there is certainly no guarantee that dialogue will in fact arise. In particular, we find that reflections need dialogue. Based on our observations with respect to the ‘Caramel Boxes’ project, we found that dialogue, including challenging questions, is important in order to facilitate and to provoke reflection. Reflection may be an expression of interaction more than an of individual processes. We do not claim that personal reflections do not exist; but in order to address profound questions concerning mathematical insight and mathematics in action, dialogue appears relevant.

For more detailed calculations, see Skovsmose (1994). The notions are brought together through an Inquiry Co-operation Model. For discussions of communication and dialogue see also Alrø and JohnsenHøines (2010); and Planas and Civil (2010). 49 50

80

Chapter 7 Mathemacy in a globalised and ghettoised world ‘Globalisation’ is a popular term, although globalisation is far from a popular phenomenon. Globalisation can refer to a new global order of domination and exploitation. It can refer to a network of production lines, running from poor locations with cheap labour force where products are fabricated onto affluent areas where the products are delivered and consumed. Processes of globalisation mean both inclusion (of some groups) and exclusion (of other groups). Therefore, I consider ghettoising as being part of globalisation. The notion of globalisation is sometimes interpreted in terms of a growing concern for each other based on new forms of communication. News is spread immediately, and we become aware of problems all over the world. It is possible, through the internet, to communicate observations and opinions in ways which make it impossible for governments with dogmatic or dictatorial aspirations to maintain control of what people know and do not know. The universal stream of information makes a variety of issues universal. This not only applies to global conflicts, but also to sports events and entertainment. A strong economic currency runs beneath all such events, and I let globalisation (always including ghettoising) refer to deeper socioeconomic and cultural trends, implying that it is not a simple question of ‘voting’ against globalisation. The processes of globalisation are not determined by those parliamentary and governmental forums where political decisions are taken. Processes of globalisation are powerful, but they are not governed by any political institutions. They operate with a different logic and represent an interplay between technological development and economic, political and military interests. Although I see globalisation as a determining process, meaning that other parameters of socio-political development easily become overruled by the dynamics of globalisation, I do not see it as a predetermined 81

CHAPTER 7

process. It does not operate like an engine put on rails. Instead globalisation includes a set of propensities, which could work out and be reworked in very different forms. There are many trends, some even contradictory, involved in processes of globalisation. The complexity might be such that it is impossible to grasp its dynamics through any available theoretical concepts. A complexity which extensively surpasses the conceptual constructs and theoretical insights of social theorising, I refer to as a happening.51 People experiencing a happening do not have the opportunity to grasp and to predict what is going to take place. The possible logic of what is taking place turns out to be far more complex than a logic established through theoretical constructs is able to grasp. In this sense I consider globalisation a world-wide happening.

7.1 Mathematics education world-wide The informational economy has been analysed as encompassing knowledge and information as particular resources. According to classic economic theory, productivity is a function of two variables: work and capital. But according to the basic assumptions of informational economy, knowledge in its very many forms has become a principal resource of value. The overall debate on informational economy, however, does not pay much attention to the different types of knowledge. Obviously there are differences between the productivity and the value that can be extracted from, say, knowledge of football and knowledge of mathematics. Different forms of knowledge may play different economic roles., and mathematics in action plays a significant role in the informational economy. Schooling may provide access to the funds of knowledge which is important for the further development and maintenance of the ‘machinery’ of globalisation and its associated economy. This observation brings us directly to mathematics education and to the notion of mathemacy seen as a competence in handling mathematical techniques.52 51 For a discussion of ‘happening’ with respect to social theorising, see Skovsmose (2005). 52 For a discussion of mathematics education and globalisation see Ernest (2009).

82

MATHEMACY IN A GLOBALISED AND GHETTOISED WORLD

Thus mathemacy can be discussed in terms of abilities in understanding and operating with mathematical notions, algorithms and procedures; it can be discussed in terms of abilities in applying all such notions, algorithms and procedures in a variety of situations; and it can be discussed in terms of abilities in reflecting on all then applications. Mathematics education can be interpreted as a universal preparation by which young people acquire a certain competencies, maybe including a prescription readiness, relevant for their further career opportunities and for the effectiveness of a huge variety of practices. Thus mathematics education can be seen as a universal form of socialising students into certain perspectives, discourses and techniques which are imperative for the present technological and economic framework. Thus mathematics education can develop the functional dimensions of a mathemacy. However, mathemacy may include other dimensions as well. I usually prefer to indicate the meaning of such a more radical interpretation of mathemacy by relating it to the notion of literacy as described by Paulo Freire.53 Literacy not only refers to reading and writing competencies in the regular sense of the words. It refers to much more, and in order to illustrate this one can take ‘text’ to be interpreted in a wide-screen hermeneutic format as referring to any form of life situation. This way a ‘text’ becomes a ‘life-world’. This provides a new set of meanings to the notions of reading and writing. Thus, one could interpret ‘reading’ as the actions through which one tries to grasp social, political, cultural, economic features of one’s life-world, and one could interpret ‘writing’ as the active way of changing this world. One could interpret mathemacy along the same lines. This way mathemacy can be seen as a way of reading the world in terms of numbers and figures, and of writing it as being open to change.54 Different groups of students in different contexts might experience the learning of mathematics in very different ways. There are Ntabesengs and Pieters all around the world, and we have to consider very many different situations when we try to understand what a mathemacy, including its radical dimensions, could mean. See Freire (1972, 1974). For an interpretation of mathematical literacy along these lines, see Gutstein (2006, 2008, 2009). See also Jablonka (2003) for a presentation of different notions of mathematical literacy, and Chronaki for an exploration of mathemacy. 53 54

83

CHAPTER 7

In order to be more specific I will discuss mathemacy with reference to different types of practices, and consider what mathemacy might mean with respect to these. I will consider practises of construction. Here I refer to the construction and further refinement of all forms of technologies which draw on mathematical resources. We can think of a practice of construction as a practice of expertise. I will consider mathemacy with reference to practices of operation, that is, mathematicsbased work procedures as experienced by laboratory assistants, bank accountants, assistants at a travel agency, etc. Naturally, mathematics need not be explicit in such practices. I consider mathemacy with reference to practices of consumption, meaning the buying or receiving of whatever kind of ‘goods’ we can think of in relation to shopping, watching TV, travelling, etc. I will also consider mathematics education with reference to the practises of the marginalised referring to situations of those very many people that are marginalised by the globalised economic order. It should be emphasised that in talking about these different practices, I do not have any classification in mind. I am merely talking about different types of practices, and a person could participate in different practices depending on the situation. Naturally, discussing these four different types of practices cannot substitute a more specific approach with respect to particular situations. Yet the following discussion might give some indication of what mathemacy might mean with respect to mathematics education. I start with the last group of practices.55

7.2 Practices of marginalised The notion of the ‘marginalised’ does not suggest that we are dealing with a minority group. Processes of ghettoising are so powerful that they set life conditions for huge groups of people the world around. These processes might emerge in many ways: from previous patterns of colonisation, from present patterns of exploitation, from a neoliberal capitalism, etc. Mathematics makes up part of the practices of the marginalised in many different ways. One could consider the mathematics of street 55

See also Skovsmose (2007c).

84

MATHEMACY IN A GLOBALISED AND GHETTOISED WORLD

sellers, as have been explored by Madalena Santos and João Filipe Matos; the mathematics of street children as discussed by Monica Mesquita; the mathematics of sugar cane farmers as investigated by Guida Abreu; and the mathematics of agriculture as presented by Paulus Gerdes.56 Many of the ethnomathematical studies have excavated the mathematics of practices of marginalised group``s. In my interpretation such practices are all examples of mathematics in action, which in turn can have all different kind of qualities. How to think of mathematics education for children from marginalised groups? One immediate concern could be for the education to be related to the background of the children. This implies, for instance, that the children of the families engaged in the farm work, as described by Gerdes, should be offered a mathematics education which relates to the mathematics of farming. The idea is that mathematics education should be based on the mathematics that makes up part of the cultural practices with which the children are familiar. There are many different examples that illustrate how mathematics education could be rooted in such cultural practices. I find it important to recognise that mathematics operates in very different cultural settings, and it is crucial that a mathematics education acknowledge this diversity. However, let me comment on some of the limitations of the idea that a mathematics education should relate first of all to the cultural background of the group of students in question. I became aware of such limitations when working in a South African context. Some of the apartheid rhetoric included an appreciation of cultural differences. Thus, one might encounter a rationale such as this: The Zulu culture is fascinating, just think of the cultural values expressed through dancing, rhythm, colours, house building, etc.; such cultural values have to be preserved. However, assuming that a mathematics education for the post-apartheid era had to ensure that the curriculum for Zulu-students be embedded in Zulu-cultural traditions appeared problematic. An important idea of the post-apartheid education was to eliminate limitations that had been imposed on black people. In particular it became important to provide equal opportunities for all. Instead of trying to organise a mathematics education with a particular See, Santos and Matos (2002); Mesquita (2004); Abreu (1993); Gerdes (2008); and Skovsmose and Penteado (in print). 56

85

CHAPTER 7

reference to the background of students, I found it important to consider their foregrounds. The same observation has presented itself to me in other situations. In Barcelona there are many immigrant groups, and some neighbourhoods take the form of immigrant favelas. What was referred to as a critical mathematics education programme was developed for children from such a neighbourhood. The content of this critical curriculum was formulated with a particular reference to the everyday-life situations known to the children. Every activity was carefully contextualised. There was sufficient time to spend on each topic. Considering the overall approach one might assume to be looking at an example of critical mathematics education. But a direct implication of this ‘critical mathematics education’ programme was that none of the children from this neighbourhood had the opportunity to get into further education. Instead, due to the educational programme, the children became stuck in their situation.57 For me there is no simple step to be taking from recognising the mathematics that may be ingrained in a particular cultural setting and the mathematics education that could appear meaningful to the students from that grouping. This observation particularly applies when we consider groups that, one way of another, can be characterised as marginalised. We have to consider mathematics education with reference to their foreground and certainly not only with reference to their background. I find it important to consider what would provide such students with more opportunities. One has to consider the empowerment that is established when marginalised students come to master competencies and techniques important for accessing further education. This brings us to more radical dimensions of mathemacy. One could think of mathemacy in terms of response-ability. This reading of responsibility has been suggested to me by Bill Atweh.58 This turns the discussion of mathemacy into a question of how to make students able to respond to different challenges in different situations. A crucial concern of critical mathematics educations is how to ensure social responseability for marginalised groups of students. Educational approaches The information about this educational programme in Barcelona has been provided to me by Núria Gorgorió and Núria Planas. 58 See also Atweh (2007, 2009). 57

86

MATHEMACY IN A GLOBALISED AND GHETTOISED WORLD

that make up part of the Movimento Sem Terra (the landless people’s movement in Brazil) exemplify what this could mean.59 However, there are no general guidelines to be expected for such educational approaches. Instead one has carefully to consider the particular situation of the students in question, when one try to explore mathemacy in terms of response-ability.

7.3 Practices of consumption Experts’ statements are expressed each and every day on television and in newspapers. Let us just recall some of the advertisements from Folha de São Paulo. At the front page Hyundai announces the possibility of buying a car interest-free. On the following pages we find advertisements for travel agencies in which the prices in huge print appear to be very small (only the amounts given are not single payments but must be paid as each of ten instalments). There are special offers from Dell; again the rate of interest is announced to be 0%, while the instalments have to be paid twelve times. And so it goes throughout the whole newspaper. Such advertisements are addressed to somebody, to whom I will refer as a consumer. Mathematics education also means preparation for consuming, and one can consider what social response-ability might mean in this situation. Consumers will face an overwhelming amount of ‘goods’ (which certainly also include an overwhelming amount of ‘bads’). We can think of any kind of product: TV set, tooth brush, coffee machine, holyday trip, or a special offer on a cell phone. It could also be ‘goods’ in an indirect sense, as when one as a citizen is met with numbers and figures in political advertisement, or when newspapers provide poll results regarding the candidates in an election. As citizens we are exposed to actions, initiatives, advertisings, designs, and decisions representing mathematics in action. 60 As citizens we are going to respond to all such forms of mathematics in action, and one possibility is that we do so through an almost blind acceptance. See, for instance, Knijnik (2009). A careful study of an example of such information is presented by Greer (2008), who investigates how the discounting of Iraqi deaths are addressed in public discourses. 59 60

87

CHAPTER 7

Thus a functional consumption, understood as a preparation for (blind) consumption, can be supported through the development of the functional aspects of mathemacy. This means, for instance, that people become able to manage all kinds of everyday economic transactions: with respect to buying and selling, receiving salaries, paying taxes, etc. If we consider consumption in broader terms as also referring to the reception and use of information expressed in numbers, then a mathemacy for consuming could be thought of in terms of a functional citizenship, meaning that one become able to receive information from a range of authorities and to act accordingly. However, mathemacy need not simply be functional; it could also include competencies in ‘talking back’ to authority by being able to critically evaluate all the ‘goods’ and ‘bads’ that become offered for consumption. This brings us to the interpretation of mathemacy as including a response-ability., which I consider crucial with respect to practises of consumption.

7.4 Practices of operation In many work situations people will operate with mathematics, although often in an implicit way. Mathematics might be available in ‘packages’, which it is important to be able to use. However, the details of how the packages function need not be grasped by the person in operation. Many times mathematics does not surface in the work situation of bank assistants, shop assistants, or accountants, who are all dealing with practices rich on mathematics compressed in packages. As already mentioned, one characteristic of the school mathematics tradition is the overwhelming number of exercises that students have to solve. It might be that a readiness to follow orders and to do so in a careful way is ‘functional’ for being an operator. That is, prescription readiness might be an important qualification for an operator. In order to have work processes or administrative procedures running according to schedule, it makes sense to engage people who have demonstrated a prescription readiness. A reliable workforce should be able to follow manuals in a careful and obedient way, and here I use ‘manuals’ in a broad interpretation, not only as related to computation. 88

MATHEMACY IN A GLOBALISED AND GHETTOISED WORLD

However, ‘blindly’ operating according to prescriptions might be problematic. Information and prescriptions expressed in numbers can be reliable or not, and this reliability has to be evaluated. Furthermore, an operator is not only ‘listening’ to numbers, he or she is also acting with reference to mathematics. This raises the question: What could it mean to make decisions and to act with reference to numbers and figures? Such a question calls for considerations of responsibility. If mathematics education is meant to prepare a person for a reflective practice of operation, we may consider how the issues of reliability and responsibility could be addressed. This provides one particular piece of input what a mathemacy might include. Naturally, it is difficult to simulate elements of the practice of an operator within a school practice. However, when the students involved in the project ‘Terrible Small Numbers’ were faced with the challenges of having to choose between the Greek and the Spanish eggs; to clarify their quality through a rather expensive test for salmonella; and to make a budget that could ensure the profitability of the egg business, they experienced aspects of a practice of operators. They had to calculate such figures, to evaluate their reliability, and to act on these figures being aware of the degree of their reliability.

7.5 Practices of construction Resources for technological innovation are continuously developed. This is part of what I refer to as the practices of construction. Mathematics is a crucial element of such practices, and I find it particularly important to emphasise that the preoccupations of critical mathematics education also concern the preparation of expertise. Many times critical mathematics education has been formulated as if it is primarily concerned with elementary mathematics education and with disempowered groups of people. Critical mathematics education is certainly thus concerned, but not exclusively so. For me it is important to emphasise that critical mathematics education also deals with the development of expertise. In particular: What does responsibility as well as response-ability mean with reference to the practices of construction? How do we interpret the not only functional but also radical features of mathemacy with respect to such practices? 89

CHAPTER 7

Almost any technological innovation today presupposes the activation of mathematics. It is the task of universities and other institutions of further education to prepare students for this, and any education of engineers, economists, computer scientists, pharmacists, etc. includes mathematics. How this education is organised is an important concern for critical mathematics education, as university education might ‘profile’ expertise in very different ways.61 An important phenomenon needs to be observed with respect to this education, namely the dissection of technological enterprises into particular sub-practices. Thus, the construction of, for instance, a destroyer is divided into a huge number of sub-tasks, as, for instance, the researching of so-called sandwich materials. Such material consists of different layers, and the hull of a destroyer is made of them. How to make a sandwich material particularly strong and adequate for the hull of a destroyer is a complex research question. However, the very construction of the sandwich material turns into a fascinating challenge in itself. The overall military issue can be broken down into a variety of issues, which from a scientific point of view are challenging in themselves. Different research groups can become absorbed in tricky problems, completely ignoring the fact that their solutions serve as part of an overall military research programme. Such forms of dissection are a general phenomenon within engineering education. Thus, the curriculum within any engineering education can be divided into a set of curriculum-specific activities, which in turn can be judged according to some internal criteria of quality. For instance, the relevant knowledge needed to pass a test in a course in mathematics can be formulated in mathematical terms. I see dissection of the curriculum as one basic element in letting ethical considerations with respect to engineering and technology be excluded from the education of expertise. I find it problematic that the modern conception of mathematics still dominates much university education which establishes expertise within technical domains. Here mathematical rationality is broadly celebrated. For me an important question is: How is it possible to include mathematical competence in broader technological competence without assuming or promoting the impression that mathematical techniques 61 See, for instance, Skovsmose, Valero and Christensen (Eds.) (2009); and Skovsmose (2006b, 2008b, 2009a).

90

MATHEMACY IN A GLOBALISED AND GHETTOISED WORLD

ensure neutrality and objectivity? A mathematical rationality should not be blindly celebrated, but questioned. An education for social responsibility with respect to practices of construction needs to acknowledge the critical conception of mathematics. This means that the different aspects of mathematics in action need to be reflected upon as part of a mathemacy for expertise.

91

Chapter 8 Uncertainty I have tried to characterise critical mathematics education in terms of a number of preoccupations. However, I do not see such preoccupations as taking up any systematic form. They cannot be enumerated. In fact I have only formulated preoccupations indirectly by referring to some more overall issues as follows. Mathematics education is undetermined. It has no essence. It can be elaborated in many different ways and come to serve very different socio-political, economic and cultural interests. One could see a mathematics education as submitting to a logic of domination and control. One could also imagine a mathematics education that could prepare for a critical citizenship. Furthermore, one could assume that any such dualistic interpretation might only be a gross simplification of the huge varieties of roles a mathematics education might play in society. I have talked about the diversity of situations for the teaching and learning of mathematics, and questioned the possibility of operating with prototypical, or stereotypical, assumptions about educational conditions. I find that the prototypical classroom has dominated much research in mathematics education, although this prototype in most cases appear far removed from experienced teaching-learning situations. I find it to be important not to embark on prototypical assumptions in the process of theorising. Through the notion of students’ foreground I try to identify important features of processes of learning and the construction of meaning. Experiences of meaning has to do with experiences of relationships. It could be relationships between what is taking place in the classroom and the students’ background as well as their daily-life experiences. However, I find that the experience of meaningfulness has much to do with experienced relationships between activities in the classroom and the students’ foreground. Furthermore, I find a foreground to be a dynamic entity. Foregrounds can be reconstructed, and meaningful mathematics education contributes to an ongoing construction and 93

CHAPTER 8

reconstruction of foregrounds. It helps to provide new opportunities in life. Through the notion for landscapes of investigation I try to expand the scope of educational possibilities beyond the exercise paradigm. I find that different teaching-learning milieus provides different opportunities, for instance with respect to meaning construction. While the exercise paradigm to some extent can be associated to a comfort zone, landscapes of investigation bring us into a risk zone. This zone, however, is also a zone of educational possibilities, and I find it important that such possibilities become explored. A critical conception of mathematics moves beyond the modern conception of mathematics, which has presented mathematics and mathematical rationality in a attractive format, and which has nominated mathematics teachers as ambassadors of mathematics. According to a critical conception of mathematics, mathematics makes part of a huge variety of actions within all spheres of life. Such actions could have all kind of qualities; they could serve many different interests. Thus mathematics does not preserve any sublime format. It makes part of daily-life processes as well as technological endeavours, some of which might be of dubious nature. This calls for the necessity of reflections. It is important to address any form of mathematical rationality through reflections. It is important to reflect on mathematics, including all the action in which they make part. Besides, one should not forget that it might be powerful also to reflect with mathematics, and that it is possible to reflect through inquiry processes. This brings us to the notion of mathemacy, which could play an important roles in formulating some of the aspirations of critical mathematics education. I have tried to formulate aspirations of critical mathematics education in terms of mathemacy in globalised an ghettoised world. I am inspired by the suggestion of reading responsibility as response-ability, and I read mathemacy as including a capacity of making responses and as reading the world as being open to change. I find it important that critical mathematics education explore what this could mean with respect different groups of people, from the groups of marginalised to the groups of experts. Then, what were the preoccupations? One might find that I have not been explicit, but only talked about issues that might relate to 94

UNCERTAINTY

preoccupations which characterise critical mathematics education. Time has come to become explicit. Nevertheless, I turn in a different direction. Instead of trying to be explicit I will emphasise that any critical approach, also critical mathematics education, is an expression of profound uncertainties, which also applies to the formulation of preoccupations. The notion of critique was explored through Critical Theory, which provided much inspiration for the initial formulation of critical education in general. It is important, however, to establish a departure from some of the assumptions that might be lingering in previous formulations of critical education in order to formulate a critical mathematics education for the future.62 In order to be more specific, I will outline some of the roots of critical thinking as they developed through Modernity. I want briefly to consider the conception of critique with reference to ideas of René Descartes, Immanuel Kant and Karl Marx. Descartes introduced a universal doubt as an epistemic device. He wanted to ground knowledge in a solid foundation, and to that end all forms of assumed knowledge had to undergo a critical revision. The idea was that everything it is possible to doubt should be doubted, and consequently eliminated (at least provisionally) from the established stock of knowledge. Left was only that which could not be doubted, and according to Descartes only one statement remained after the universal doubt had swept through all faculties of assumed human knowledge. This statement, cogito, ergo sum, represented not only a truth, but a truthwith-certainty. According to Descartes, knowledge should be composed of statements which are true-with-certainty, and only of such statements. This means that the purpose of a critical activity was to establish, through a universal doubt, a foundation for genuine knowledge, which in turn was characterised in terms of truth and certainty. This way a critical activity came to be part of establishing epistemic absolutism. Kant also wanted to address the whole stock of possible human knowledge: What could be known and what could not be known? Through his monumental work, Critique of Pure Reason, he tried to provide a study of the general conditions for obtaining knowledge. Critique was conducted as an a priori activity, which anticipated the formulation of particular forms of knowledge. Critique became a way 62 I have tried in different ways to formulate a critical mathematics education for the future. See, of instance, Skovsmose (2008c). See also Rasmussen (2010).

95

CHAPTER 8

of formulating general and a priori conditions for obtaining knowledge. But what epistemic resources could be available to Kant in conducting such an a priori investigation of knowledge? What epistemic layer comes before ‘knowledge’? As Kant wanted to address all forms of knowledge, he could not presuppose any particular piece of knowledge. Kant found, however, that it was possible to carry out a critical investigation of the general conditions for obtaining knowledge in the form of a transcendental philosophy. In this way critique became an expression of a transcendental certainty, and Kant found that his Critique of Pure Reason provided a clarification, once and for all, of human conditions for obtaining knowledge. In this sophisticated way, Kant established a connection between critique and epistemic absolutism. Marx’s critical approach was different. He wanted not only to establish a critical investigation of economic theories, but also to criticise the economic systems themselves. He formulated a critique, not only as an epistemological activity, as did Descartes and Kant, but also as a social and political activity. At the same time, he wanted to establish this broad scope of critical activities on a solid foundation, which was to take the form of a proper formulation of the logic which governs social development. This logic, in turn, was shaped through the laws that govern economic development, and it was Marx’ ambition to formulate these laws. When this was completed, any critical activity, including those that address the social-political an economic realities, could be given a solid foundation. Critical Theory provided an important step out of Marxist orthodoxy. I can think of two possible illustrations of this step. The Dialectics of Enlightenment by Max Horkheimer and Theodor Adorno anticipates much of the critique of the modern outlook that later became formulated my Michel Foucault and which turned into post-modernism and poststructuralism. It is difficult to locate any foundational assumptions in The Dialectics of Enlightenment. To me the work represents a step towards developing a notion of critique which does not incorporate foundational assumptions. The other book I will refer to is the Arcade Project, where Walter Benjamin tried to formulate a critical investigation of a whole period by taking as his point of departure the architectural innovations signified by the arcades in Paris, constructed during the first half of the 19th century. This critical investigation was not completed by Benjamin, but the posthumously published Arcade Project demonstrates Benjamin’s 96

UNCERTAINTY

anarchistic methodological approach. Through a highly elaborate patchwork of quotations, he provided an almost surrealistic presentation of his insight. The Arcade Project breaks with any assumptions of what could be considered a proper critical approach. Benjamin brings critique away from any predetermined methodological regulations. For me it is important to move beyond a conception of critique which includes any assumption of the possibility of building one’s approach on a solid foundation or through a well-defined methodology. This means to acknowledge that critique is a deeply uncertain activity. This is an important acknowledgement for any critical education that tries to move beyond the outlook of modernity. It is important for the formulation of a critical mathematics education for the future.63 This observation could bring us towards an opposite extreme: absolute relativism. According to absolute relativism there are no ‘constructive’ elements associated with a critical activity. It is not possible through such an activity to formulate suggestions about what could be done. With respect to education, absolute relativism cannot result in proposals for action. Such absolute relativism can be associated to some educational approaches formulated with reference to a post-modern or post-structural outlook, emphasising that any theory-based suggestions for ‘improvement’ is an expression of educational romanticism. There is no way of providing the notion of ‘improvement’ with meaning. I try not to be trapped by absolutism, nor by absolute relativism. But how to locate a critical activity somewhere between absolutism and absolute relativism? My proposal is to think of a critical position in terms of preoccupations. On the one hand, it is important to acknowledge that preoccupations are discursively constructed and depend on the formulated perspective. On the other hand, a discursive construction is not a completely free enterprise. Reconsidering the pictures from A Cradle of Inequality one could try to create a discourse, according to which a shadow beneath some trees is called a classroom. And one could try to formulate a discourse according to which children bringing wood to the school in order to ensure the heating of their classroom is part of a chemistry education. However, whatever discourse we create, the classroom in the format of a shadow has no electricity and there are no computers operating there. No change of discourse provides any 63 See also Ernest (2010); Knijnik and Bocasante (2010); Pais (2010); and Valero and Stentoft (2010).

97

CHAPTER 8

heating to any classrooms. A change of discourse makes changes, but not all kinds of changes can be established through a change of discourse. Different discourses establish different preoccupations. To me critical mathematics education is characterised through its preoccupations. I have tried to address some, although not in terms of any simple enumeration. I have tried to present some notions that provide a ‘grammar of preoccupations’. This whole approach, however, includes a profound uncertainty. This uncertainty applies to all the features I have suggested to become included in a grammar of preoccupations. Thus I do not believe a justification of a particular conceptual network for the expression of preoccupations. My uncertainty also applies to what I, in this Chapter 8, have said about uncertainty.

98

References Abreu, G. (1993). The relationship between home and school mathematics in a farming community in rural Brazil. Doctoral dissertation. Cambridge: Cambridge University. Adorno, T. W. (1971). Erziehung zur Mündigkeit. Frankfurt am Main: Suhrkamp. Alrø, H., & Johnsen-Høines, M. (2010). Critical dialogue in mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 11–21). Rotterdam: Sense Publisher. Alrø, H., Ravn, O., & Valero, P. (Eds.). (2010). Critical mathematics education: Past, present, and future. Rotterdam: Sense Publishers. Alrø, H., & Skovsmose, O. (2002). Dialogue and learning in mathematics education: Intention, reflection, critique. Dordrecht: Kluwer. Alrø, H., Skovsmose, O., & Valero, P. (2009). Inter-viewing foregrounds: Students’ motives for learning in a multicultural setting. In M. César & K. Kumpulainen (Eds.), Social interactions in multicultural settings (pp. 13–37). Rotterdam: Sense Publishers. Appelbaum, P., & Allan, D. S. (2008). Embracing mathematics: On becoming a teacher and changing with mathematics. New York: Routledge. Atweh, B. (2007). Pedagogy for socially response-able mathematics education. Paper presented at the annual conference of the Australian Association of Research in Education. Fremantle, West Australia. Retrieved from http://www.aaere.edu.au/07pap/atw07600.pdf Atweh. B. (2009). Ethical responsibility and the “What” and the “Why” of mathematics education in a global context. In P. Ernest, B. Greer, & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 7–17). Charlotte, NC: Information Age Publishing. Baber, S. A. (2010). Mathematics from the perspective of critical sociology. In H. Alrø, O. Ravn, & P. Valero, P. (Eds.), Critical mathematics education: Past, present and future (pp. 23–29). Rotterdam: Sense Publisher.

99

REFERENCES

Benjamin, W. (1999). The arcades project. Cambridge (USA) and London: The Belknap Press of Haward University Press. Beth, E. W., & Piaget, J. (1966). Mathematical epistemology and psychology. Dordrecht: D. Reidel Publishing Company. Biotto Filho, D. (2008). O desenvolvimento da matemacia no trabalho com projetos. Unpublished Master Thesis. Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas, Campus Rio Claro. Brentano, F. (1995a). Psychology from an empirical standpoint (A. C. Rancurello, D. B. Terrell, & L. L. McAlister, Trans.; with an introduction by Peter Simons). London: Routledge. (First German edition 1874.) Brentano, F. (1995b). Descriptive psychology (B. Müller, Trans. and Ed.). London: Routledge. (Lectures between 1887–1891: First German edition 1982.) Christensen, O. R., Skovsmose, O., & Yasukawa, K. (2009). The mathematical state of the world: Explorations into the characteristics of mathematical descriptions. In B. Sriraman & S. Goodchild (Eds.), Relatively and philosophically earnest: Festschrift in honor of Paul Ernest’s 65th birthday (pp. 83–96). Charlotte, NC: Information Age Publishing. (First published in Alexandria: Journal of Science and Technology Education, 1(1), 77–90.) Christensen, O. R., Stentoft, D., & Valero, P. (2007). Power distribution in the network of mathematics education practices. In E. de Freitas & K. Nolan (Eds.), Opening the research text: Critical insights and in(ter)ventions into mathematics education (pp. 131–146). New York: Springer. Chronaki, A. (2010). Revisiting mathemacy: A process-reading of critical mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 31–49). Rotterdam: Sense Publisher. Curry, H. B. (1951). Outlines of a formalist philosophy of mathematics. Amsterdam: North-Holland Publishing Company. D’Ambrosio, U. (2006). Ethnomathematics: Link between transitions and modernity. Rotterdam: Sense Publishers. D’Ambrosio, U. (2010). Mathematics education and survival with dignity. In H. Alrø, O. Ravn, & P. Valero(Eds.), Critical mathematics education: Past, present and future (pp. 51–63). Rotterdam: Sense Publisher. 100

REFERENCES

Ernest, P. (2009). Mathematics education ideologies and globalization. In P. Ernest, B. Greer, & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 67–110). Charlotte, NC: Information Age Publishing. Ernest, P. (2010). The scope and limits of critical mathematics education. In H. Alrø, O. Ravn, & P. Valero(Eds.), Critical mathematics education: Past, present and future (pp. 65–87). Rotterdam: Sense Publisher. Ernest, P., Greer, B., & Sriraman, B. (Eds.). (2009). Critical issues in mathematics education. Charlotte, NC: Information Age Publishing. Foucault, M. (1989). The archeology of knowledge. London: Routledge. (First French edition 1969.) Foucault, M. (1994). The order of things: An archaeology of the human sciences. New York: Vintage Books. (First French edition 1966.) Foucault, M. (2000). Power (J. D. Faubion, Ed.; R. Hurley and others, Trans.). New York: The New Press. Freire, P. (1972). Pedagogy of the oppressed. Harmondsworth: Penguin Books. Freire, P. (1974). Cultural action for freedom. Harmondsworth: Penguin Books. Gates, P. (2006). The place of equity and social justice in the history of PME. In A. Gutérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 367–402). Rotterdam: Sense Publishes. Gerdes, P. (2008). Exemplos de Aplicações da Matemática na Agricultura e na Veterinária (New ed.). Maputo: Lulu.com and TLANU. Greer, B. (2008). Discounting Iraqi deaths: A social and educational disgrace. In J. F. Matos, P. Valero, & K. Yasukawa (Eds.), Proceedings of the fifth international mathematics education and society conference. Lisbon: Centro de Investigação em Educação, Universidade de Lisboa – Department of Education, Learning and Philosophy, Aalborg University. Greer, B. Mukhopadhyay, S., Powell, A. B., & Nelson-Barber, S. (Eds.). (2009). Culturally responsive mathematics education. New York: Routledge. Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York and London: Routledge. Gutstein, E. (2008). Building political relationships with students. In E. de Freitas & K. Nolan (Eds.), Opening the research text: Critical 101

REFERENCES

insights and in(ter)ventions into mathematics education (pp. 189–204). New York: Springer. Gutstein, E. (2009). Possibilities and challenges in teaching mathematics for social justice. In P. Ernest, B. Greer, B., & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 351–373). Charlotte, NC: Information Age Publishing. Hilbert, D. (1968). Grundlagen der geometrie. Stuttgart: Teubner. (First published 1899.) Horkheimer, M., & Adorno. T. W. (2002). Dialectic of enlightenment (J. Cumming, Trans.). New York: Continuum. (First published 1947). Husserl, E. (1998). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy: First book. Dordrecht: Kluwer Academic Publishers. (First German version 1913.) Husserl, E. (1970). The crisis of European sciences and transcendental phenomenology (D. Car, Transl., with an introduction). Evanston, IL: Northwestern University Press. (First German version 1936.) Jablonka, E. (2003). Mathematical literacy. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 75–102). Dordrecht: Kluwer. Jablonka, E. (2010). Reflections on mathematical modelling. In H. Alrø, O. Ravn, & P. Valero, P. (Eds.), Critical mathematics education: Past, present and future (pp. 89–100). Rotterdam: Sense Publisher. Kant, I. (1933). Critique of pure reason (N. K. Smith, Trans.). London: MacMillan. (Original German edition 1791.) Knijnik, G. (2009). Mathematics education and the Brazilian landless movement: Three different mathematics in the context of struggle for social justice. In P. Ernest, B. Greer, & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 153–169). Charlotte, NC: Information Age Publishing. Knijnik, G., & Bocasante, D. M. (2010). Do ofício da pesquisa: Pílulas neoaforistica – About the craft of research: Neoaphoristic pills. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 101–120). Rotterdam: Sense Publisher. 102

REFERENCES

Lindenskov, L. (2010). Students’ curriculum in critical mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 121–131). Rotterdam: Sense Publisher. Mesquita, M. (2004). O conceito de espaco na cultural de criança em situatição de rua: Um estudo etnomatemático. In J. P. M. Ribeiro, M. do C. S. Domite, & R. Ferreira (Eds.), Etnomatemática: Papel, valor e significado (pp. 125–136). São Paolo: Zouk. Mora, D. (Ed.). (2005). Didáctica crítica, educación crítica de las matemáticas y etnomatemática: Perspectivas para la transformación de la educación matemática en América Latina. La Paz: Gidem. OEEC. (1961). New thinking in school mathematics. OEEC: Paris. Pais, A. (2010). Portrait of an influence. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 133–144). Rotterdam: Sense Publisher. Penteado, M. G. (2001). Computer-based learning environments: Risks and uncertainties for teachers. Ways of Knowing, 1(2), 23–35. Penteado, M. G., & Skovsmose, O. (2009). How to draw with a wornout mouse? Searching for social justice through collaboration. Journal for Mathematics Teacher Education, 12(3), 217–230. Piaget, J. (1970). Genetic epistemology. New York: Columbia University Press. Planas, N., & Civil, M. (2010). Discourse processes in critical mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 145–159). Rotterdam: Sense Publisher. Ravn, O. (2010). Step beyond mathematics: Investigations in the philosophy of mathematics. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 171–181). Rotterdam: Sense Publisher. Rasmussen, P. (2010). The critical perspective on education and on mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 161–169). Rotterdam: Sense Publisher. 103

REFERENCES

Rønning, F. (2010). Hvor mange kanter har en firedimensjonal terning? Tangenten, (2), 5–10, 17. Salgado, S., & Buarque, C. (2005). O Berço da desigualdade. São Paulo: Editora UNESCO no Brasil. Santos, M., & Matos, J. F. (2002). Thinking about mathematical learning with Cabo Verde Ardinas. In G. de Abreu, A. Bishop, & N. C. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 81–122). Dordrecht: Kluwer. Skovsmose, O. (1994). Towards a philosophy of critical mathematical education. Dordrecht: Kluwer. Skovsmose, O. (2001). Landscapes of investigation. ZDM The International Journal on Mathematics Education, 33(4), 123–132. Skovsmose, O. (2005). Travelling through education: Uncertainty, mathematics, responsibility. Rotterdam: Sense Publishers. Skovsmose, O. (2006a). Reflections as a challenge. ZDM The International Journal on Mathematics Education, 38(4), 323–332. Skovsmose, O. (2006b). Challenges for mathematics education research. In J. Maasz & W. Schloegelmann (Eds.), New mathematics education research and practice (pp. 33–50). Rotterdam: Sense Publishers. Skovsmose, O. (2006c). Research, practice, uncertainty and responsibility. Journal of Mathematical Behaviour, 25(4), 267–284. Skovsmose, O. (2007a). Foregrounds and politics of learning obstacles. In U. Gellert & E. Jablonka (Eds.), Mathematisation – Demathematisation: Social, philosophical, sociological and educational ramifications (pp. 81–94). Rotterdam: Sense Publishers. (First published in For the Learning of Mathematics, 25(1), 4–10.) Skovsmose, O. (2007b). Doubtful rationality. ZDM The International Journal on Mathematics Education, 39(3), 215–224. Skovsmose, O. (2007c). Mathematical literacy and globalisation. In B. Atweh, et al. (Eds.), Internationalisation and globalisation in mathematics and science education (pp. 3–18). New York: Springer. Skovsmose, O. (2008a). Mathematics education in a knowledge market. In E. de Freitas & K. Nolan (Eds.), Opening the research text: Critical

104

REFERENCES

insights and in(ter)ventions into mathematics education (pp. 159–174). New York: Springer. Skovsmose, O. (2008b). Critical professionalism in mathematics teacher education. Revista Pesquisa Qualitative, 3(1), 55–72. Skovsmose, O. (2008c). Critical mathematics education for the future. In M. Niss (Ed.), ICME-10 proceedings and regular lectures (on CD). Roskilde: IMFUFA, Department of Science, Systems and Models, Roskilde University, Denmark. Skovsmose, O. (2009a). Towards as critical professionalism in University science and mathematics education. In O. Skovsmose, P. Valero, & O. Ravn Christensen (Eds.), University sciences and mathematics education in transition (pp. 325–346). New York: Springer. Skovsmose, O. (2009b). In doubt – about language, mathematics, knowledge and life-world. Rotterdam: Sense Publishers. Skovsmose, O. (in print ). Critical mathematics education – in terms of concerns. In B. Sriraman, et al. (Eds.), The sourcebook on Nordic research in mathematics education. Charlotte, NC: Information Age Publishing, Inc. Skovsmose, O., Alrø, H., & Valero, P. in collaboration with Silvério, A. P. and Scandiuzzi, P. P. (2008). “Before you divide you have to add”: Inter-viewing Indian students’ foregrounds. In B. Sriraman (Ed.), International perspectives on social justice in mathematics education (pp. 209–230). The Montana Mathematics Enthusiast, Monograph 1. Charlotte, NC: Information Age Publishing, Inc. Skovsmose, O., & Penteado, M. G. (in print). Ghettoes in the classroom and the construction of possibilities. In B. Atweh, et al. (Eds.), Mapping equity and quality in mathematics education. New York: Springer. Skovsmose, O., Scandiuzzi, P. P., Valero, P., & Alrø, H. (2008). Learning mathematics in a Borderland position: Students’ foregrounds and intentionality in a Brazilian Favela. Journal of Urban Mathematics Education, 1(1), 35–59. Skovsmose, O., & Valero, P. (2008), Democratic access to powerful mathematical ideas. In L. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 415–438). New York and London: Routledge. 105

REFERENCES

Skovsmose, O., Valero, P., & Ravn Christensen, O. (Eds.). (2009). University sciences and mathematics education in transition. New York: Springer. Sriraman, B. (Ed.). (2008). International perspectives on social justice in mathematics education. The Montana Mathematics Enthusiast, Monograph 1. Charlotte, NC: Information Age Publishing, Inc. UNESCO. (2000). Education for all: Statistical assessment 2000. Paris: UNESCO. Retrieved from http://unesdoc.unesco.org/images/ 0012/001204/120472e.pdf Valero, P. (2002). Reform, democracy, and mathematics education: Towards a socio-political frame for understanding change in the organization of secondary school mathematics. Doctoral Dissertation. Copenhagen: Department of Curriculum Research, The Danish University of Education. Valero, P. (2004). Postmodernism as an attitude of critique to dominant mathematics education research. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 35–54). Greenwich, CT: Information Age Publishing. Valero, P. (2007). In between the global and the local: The politics of mathematics education reform in a globalized society. In B. Atweh, et al. (Eds.), Internationalisation and globalisation in mathematics and science education (pp. 421–439). New York: Springer. Valero, P. (2009). What has power to do with mathematics education? In P. Ernest, B. Greer, & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 237–254). Charlotte, NC: Information Age Publishing. Valero, P., & Stentoft, D. (2010). The ‘post’ move of critical mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 183–195). Rotterdam: Sense Publisher. Valero, P., & Zevenbergen, R. (Eds.). (2004). Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology. Dordrecht: Kluwer. Vithal, R. (2007). The ‘Uncivilised’ scientist. In U. Gellert & E. Jablonka (Eds.), Mathematisation – Demathematisation: Social, philosophical, sociological and educational ramifications (pp. 95–105). Rotterdam: Sense Publishers. 106

REFERENCES

Vithal, R. (2009). Researching, and learning mathematics at the margin: From ‘Shelter’ to school. In P. Ernest, B. Greer, B., & B. Sriraman (Eds.), Critical issues in mathematics education (pp. 475–484). Charlotte, NC: Information Age Publishing. Vithal, R. (2010). Democratising mathematics education doctoral research teaching and learning: Undoing the North-South divide. In H. Alrø, O. Ravn, & P. Valero, (Eds.), Critical mathematics education: Past, present and future (pp. 197–207). Rotterdam: Sense Publisher. Vithal, R., & Valero, P. (2003). Researching mathematics education in situations of social and political conflict. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 545–591). Dordrecht: Kluwer. Whitehead, A. N., & Russell, B. (1910–1913). Principia mathematica I-III. Cambridge: Cambridge University Press. Wittgenstein, L. (1958). Philosophical investigations (2nd ed.). (G. E. M. Anscombe, Trans.). Oxford: Blackwell. (First published 1953) World Bank. (2006). Equity and development: World development report 2006. Washington and New York: A copublication of The World Bank and Oxford University Press. Yasukawa, K. (2010). Educating critical mathematics educators: Challenges for teacher educators. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 209–224). Rotterdam: Sense Publisher.

107

Name Index A Abreu, G., 85 Adorno, T. W., 11, 96 Allan, D. S., 2n1 Alrø, H., 2n1, 18n10, 22n16, 27n23, 29n28, 31n30, 39n32, 72n46, 78n48, 80 Appelbaum, P., 2n1 Atweh, B., 86 Austin, J. L., 61

D D’Ambrosio, U., 20 Descartes, R., 50-51, 95, 96 E Ernest, P., 2n1, 82n52, 97n63 F Foucault, M., 9, 60-61, 96 Frege, G., 27n24 Freire, P., 11, 83 G Galilei, G., 50 Gates, P., 11n4 Gerdes, P., 85 Gödel, K., 55n39 Gorgorió, N., 86n57 Greer, B., 2n1, 87n60 Gutstein, E., 83n54

B Baber, S. A., 62n45 Bacon, F., 52, 59 Beard, C. A., 53 Benigni, R., 7 Benjamin, W., 96, 97 Beth, E. W., 57n41 Biotto Filho, D., 41n33, 48, 76n47, 77 Blomhøj, M., 72n46 Bocasante, D. M., 97n63 Bourbaki, N., 27, 57 Brentano, F., 24, 25 Buarque, C., 17 Bødtkjer, H., 11, 72n46, 73

H Hilbert, D., 54-55 Horkheimer, M., 96 Husserl, E., 22, 25 J Jablonka, E., 62n45, 83n54 Johnsen-Høines, M., 80n50

C Christensen, O. R., 9n3, 62n45, 90n61 Chronaki, A., 83n54 Civil, M., 80n50 Copernicus, N., 49, 50 Curry, H. B., 55, 56n40

K Kant, I., 61, 95, 96 Kepler, J., 50 Khuzwayo, H., 26n22 Knijnik, G., 87n59, 97 109

NAME INDEX

S Salgado, S., 17 Santos, M., 85 Sapir, E., 61 Scandiuzzi, P. P., 29n28 Silvério, A. P., 29n28 Skånstrøm, M., 34, 72n46 Sriraman, B., 2n1 Stentoft, D., 9n3, 97n63 Stone, M. H., 56

L Lindenskov, L., 22n16 M Matos, J. F., 85 Mesquita, M., 85 Mora, D., 2n1 Mukhopadhyay, S., 2n1 N Nelson-Barber, S., 2n1 Newton, I., 51

V Valero, P., 2n1, 9n3, 18n11, 19n12, 22n16, 27n23, 29n28, 60n43, 90n61, 97n63 Vithal, R., 19n12, 21n15, 29n27

P Pais, A., 97n63 Penteado, M. G., 29n26, 30, 47, 85n56 Piaget, J., 57–58 Planas, N., 80n50, 86n57 Powell, A. B., 2n1 Ptolemy, 50

W Whitehead, A. N., 53–54 Whorf, B. L., 61 Wittgenstein, L., 5, 61 Y Yasukawa, K., 47n37, 62n45

R Rasmussen, P., 95n62 Ravn, O., 2n1, 62n45 Rønning, F., 36n31 Russell, B., 53–54

Z Zevenbergen, R., 19n12

110

Subject Index A ambassador of mathematics, 57, 94 axiomatic organisation of mathematics, 54

ethnomathematics, 10, 27, 28, 75, 85 Euclidean geometry, 54 Euclidean paradigm, 54 exemplarity, 13n7

B background, 22–23, 25, 28–30, 42, 85, 86, 93

F foreground, 3, 21–30, 86, 93–94 foreground investigation, 27, 29 formalism, 6, 55n39, 56 formatting of reality, 61

C Caramel Boxes project, 78–79, 80 City Planning project, 41, 44n35, 76–77 comfort zone, 3, 47–48, 94 concientizaçao, 11 Copernican revolution, 52 critical conception of mathematics, 3, 4, 49, 59–70, 91, 94 critical rationality, 60

G genetic epistemology, 57–58 ghettoising, 19–20, 81, 84 globalisation, 19–20, 81, 82 H human consciousness, 24–25 hypothetical reasoning, 62, 64–65, 75 I illocutionary force, 61 input-output figures, 12–14, 41 inquiry, 3, 31, 33, 38, 39, 45, 72, 77–80, 94 intentionality, 3, 24–27, 28, 29, 30, 31, 33, 45

D dissolution of responsibility, 62, 68, 75 diversity of situations, 3, 17–20, 93 E educational possibilities, 3, 31, 94 empowerment, 10–11, 14, 15, 86 Energy project, 11, 14, 15, 41, 77, 79–80

J justification, 62, 65–66, 75, 98 K knowledge-power interaction, 61 111

SUBJECT INDEX

meaning in mathematics education, 27-30, 31, 93 mechanical world view, 24, 25, 50, 51 modern conception of mathematics, 49–58 Modernity, 49, 52, 53, 60, 95, 97 modern mathematics education, 27–28, 56–58, 72 Modern Mathematics Movement, 28 Movimento Sem Terra (landless people’s movement), 87 Mündigkeit, 11

L landscapes of investigation, 3, 4, 31–48, 77, 80, 94 language-game, 5 learning as action, 25, 31 Learning from Diversity project, 27, 29 legitimation, 62, 65–66 life-world, 1–3, 22, 23, 61, 67, 83 locutionary content, 61 logical structures of mathematics, 28, 57 logicism, 6, 54n38 M mathemacy, 3, 4, 81–91, 94 mathematical consistency, 55 mathematical existence, 55 mathematical rationality, 57–60, 70, 76, 90, 91, 94 mathematical truth, 53–56 mathematics, discourses and power, 60–62 mathematics and natural sciences, 49–52 mathematics and purity, 53–56 mathematics and technology, 52–53 mathematics in action dissolution of responsibility, 62, 68, 75 hypothetical reasoning, 62, 64–65, 75 legitimation or justification, 62, 65–66, 75 realisation, 62, 66–67 technological imagination, 62–64, 66, 75

N neo-Platonism, 49 P perlocutionary effect, 61 Post-Modernity, 49 power, 10, 60–62 Practices practices of consumption, 84, 87–88 practices of operation, 84, 88–89 practises of construction, 84, 89–91 practises of the marginalised, 84–87 preoccupations, 1–4, 60, 72, 89, 93–95, 97, 98 prescription readiness, 9–10, 16, 46, 83, 88 problem-solving, 45 112

SUBJECT INDEX

prototypical mathematics classroom, 18 Pythagorean, 49, 50

stereotypical classroom, 18 students’ foreground, 3–4, 21–30, 86, 93–94

R realisation, 62, 66–67 real-life reference, 41, 42 reflection, 68-70, 71–80 reflective abstraction, 57–58 risk society, 59, 60, 65 risk zones, 3, 47–48, 94

T teaching-learning milieus, 47–48, 94 technological imagination, 63–64, 66, 75 techno-nature, 59, 60, 67, 69, 70 tradition of exercises, 40, 45

S school mathematics tradition, 7–10, 31, 45, 47, 88 Scientific Revolution, 10, 49–52, 59 semi-reality, 39–41, 45, 79 situation, diversity of 3, 17–20, 93 speech act illocutionary force, 61 locutionary content, 61 perlocutionary effect, 61

U uncertainty, 3, 4, 93–98 undetermined, 2, 3, 5–16, 93 undetermined rationality, 60 W wonder-horror dichotomy, 69 Z zone of possibilities, 3, 47–48, 94

113