Mathematics Education and Culture (Educational Studies in Mathematics, Vol 19, No 2, 1988)

EDUCATIONAL

STUDIES

Volume 19 No. 2

IN MATHEMATICS

May 1988

MATHEMATICS EDUCATION AND CULTURE Editedby

Alan J. Bishop Publisher's Announcement

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Editorial

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BETH GRAHAM / Mathematical Education and Aboriginal Children

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PAULUS GERDES / On Culture, Geometrical Thinking and Mathematics Education

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NORMA C. PRESMEG / School Mathematics in Culture-Conflict Situations

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ALAN J. BISHOP / Mathematics Education in Its Cultural Context

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MARC SWADENER AND R. SOEDJADI / Values, Mathematics Education, and the Task of Developing Pupils' Personalities: An Indonesian Perspective

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K. C. CHEUNG / Outcomes of Schooling: Mathematics Achievement and Attitudes Towards Mathematics Learning in Hong Kong

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THOMAS S. POPKEWITZ / Institutional Issues in the Study of School Mathematics: Curriculum Research

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RICHARD NOSS / The Computer as a Cultural Influence in Mathematical Learning

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Book Reviews Erich Ch. Wittmann, Elementargeometrie und Wirklichkeit (T. J. FLETCHER)

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C. C McKnight, F. J. Crosswhite, J. A. Dossey, E. Kifer, J. O. Swafford, K. J. Travers, and T. J. Cooney, The Underachieving Curriculum - Assessing US SchoolMathematics from an International Perspective (K. M. HART)

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Louise Lafortune (ed.), Women and Mathematics (M. ARTIGUE and M. F. COSTE ROY)

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J. Dhombres, A. Dahan-Dalmedico, R. Bkouche, C. Houzel, and M. Guillemot, Mathdmatiques au fil des dges (DETLEF D. SPALT)

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Educational Studies in Mathematics presents new ideas and developments which are considered to be of major importance to those working in the field of mathematical education. It seeks to reflect both the variety of research concerns within this field and the range of methods used to study them. It deals with didactical, methodological and pedagogical subjects rather than with specific programmes for teaching mathematics. All papers are strictly refereed and the emphasis is on high-level articles which are of more than local or national interest.

This publication is available in microform from University i Microfilms International. Calltoll-free800-521-3044.Ormailinquiryto: 300 North UniversityMicrofilmsInternational, ZeebRoad,AnnArbor,MI48106.

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PUBLISHER'SANNOUNCEMENT MartinusNijhoffPublisherswishto bringthefollowingto yourattention. MartinusNijhoff/ Dr W. JunkPublishers,D. ReidelPublishingCompany, and MTP Press have mergedto form a single companyto be known as KluwerAcademicPublishers. The abovecompanies,ownedby the Wolters-Kluwer concern,were already within the Kluwer Academic Publishers cooperatingclosely Group,and for the and distribution of books and manyyears marketing journalspublishedby thesecompanieswascarriedoutby centralservicedepartments of thisgroup. To face the challengesof the comingyears, it was felt necessaryto combine the publishingprogrammesof the three companies, and to restructure themintofourdivisions: - Humanities& SocialSciencesDivision - Science and Technology Division - Life Sciences Division

- HumanRights& International LawDivision. madeby the formercompanieswill Existingagreementsand arrangements notbe affectedby thischangeandwill be honouredto theirfull extent New will be madein thenameof KluwerAcademicPublishers. agreements As soon as practicable,all books and journalspreviouslybearingthe imprintsof MartinusNijhoff/Dr W. Junk,D. Reidel,andMTPwill appear underthe imprintof KluwerAcademicPublishers.

EDITORIAL Socio-culturalStudies in MathematicsEducation

The studies in this Special Issue all focus on the socio-cultural nature of mathematicseducation. This area has not received much attentionin the distant past, but in the last ten years there has been an increasing interest in it. Indeed, there was a time when the only disciplined enquiry into mathematicseducation was through the medium of philosophical analysis. Following that phase the psychologist's methods, constructs and rationales were seized on in the search for more relevant ways to challenge the problems and issues of mathematics teaching. Nowadays the influence of the social sciences is being increasingly felt. For example, ideas like 'understanding'and 'attitudes' which had respectability as goals in mathematicseducation throughtheir psychological pedigree are now being scrutinised through social 'lenses' and are being found to be social as well as psychological products- defined and negotiated by different groups and at differentlevels in the societal structure. For some people this influence representsa thoroughlyundesirabledevelopment. Mathematical knowledge has for them the attributesof clarity, universality and truth; values which imply certain specific educational goals, and therefore certain specific research tasks. The social perspective is for them at best an unnecessary diversion from the real tasks, and at worst an undesirable confounding of an alreadycomplicatedfield. For others this influence is felt to be thoroughlydesirable, enabling mathematics education to be recognised as the social process they feel it undeniablyis. They are happy to face the complex issues raised by this conception, because they welcome the potential it offers for the development of significant knowledge in our field. Whatever one's position on this matter it is clear that there is plenty of currentresearch activity on the socio-culturalfront, and in this Special Issue of Educational Studies in Mathematicswe present some examples of the different studies being undertakenby colleagues in different parts of the world. As with the other Special Issues of Educational Studies in Mathematics,the aim has not been to present a systematic overview of the field, but ratherto illustrate some of the research avenues currentlybeing examined. In the first three papers we see where some of the impetus for this research stems from. Beth Grahampresents us with some of the complex issues concerning Australian Aboriginal education, an area where the role of culture is fundamentalto any real understanding.Paulus Gerdes, from the perspective of Mozambique, is engaged in 'defreezing' the 'frozen' indigenous mathematicsof Mozambican culture in order to help in the educational process of cultural conscientialization, so critical for formerly colonised peoples. The next paper, 19 (1988)117-118. EducationalStudiesin Mathematics

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by Norma Presmeg, tackles the problems of overt culture conflict in mathematics education in South Africa - a situation similar in kind, if not in degree, experienced in many other countrieswith multi-culturalsocieties. In my own paper I present my attempts to conceptualise mathematics as a socio-cultural phenomenon. This structuringoffers two sets of ideas: firstly, the point of view that mathematics is a universal symbolic technology; and secondly, thatmathematicsis not a value-freephenomenon.The next two papers are both concerned with values: the first, by Swadenerand Soedjadi, deals with the potential contributionwhich mathematics,through education, can make to the development of Indonesian societal values; the second, by K. C. Cheung, locates a more traditionalkind of study of achievement and attitudes in the socio-culturalcontext of Hong Kong. The final two papers take us into relatively unchartedterritory.The first, by Tom Popkewitz, examines the role of the 'educational institution' in mediating socio-cultural influences in the USA. Institutionalstructureshave not figured prominentlyin researchin mathematicseducation,but Popkewitz shows us why they should, particularly in this context. Richard Noss begins the task of analysing the culturalrole of the computerin relation to our field. The computer is already shaping mathematicaldevelopments as well as educational thinking, and its influence needs interpreting. Perhaps indeed the recognition of this influence will demonstrate to sceptics that mathematics, like any human knowledge, continues to be, as it has always been, subject to societal and culturalpressures, and that mathematicseducation ignores those pressuresat its peril. A. J. B.

BETH GRAHAM

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ABSTRACT. This paper is concerned with the issues surrounding the mathematical education of traditionally oriented Aboriginal children. A wide-ranging review of the relevant literature is presented and discussed with a view to developing more effective educational procedures. The discussion will be found to be relevant to other culture-conflict educational situations.

1. INTRODUCTION

Aboriginal children from traditionally oriented communities in the Northern Territory of Australia are failing to learn mathematics effectively in school. The degree of this failure has been documented by Bourke and Parkin (1977) amongst others. However, research findings based on testing procedures do not fully reveal the nature of the difficulty experienced by Aboriginal children when faced with the task of learning Western mathematics in the school environment. Nor do such findings reveal the degree to which present approaches to mathematics education result in Aboriginal children perceiving school mathematics more in terms of a meaningless ritual than as a purposeful pursuit (Christie, 1985, pp. 48-49). The result is that after years of schooling many children have only learned the answers to 'sums' that have little relationship with the world as they know it. Much of this unusable mathematical knowledge is soon forgotten, both between periods of school attendance, and once school days are left behind. Even when some number facts are retained, many Aboriginal adults experience difficulty in knowing whether to add, subtract, multiply or divide when faced with the mathematical realities of the wider world that are inevitably encoded in the language of life rather than in the language of the classroom (Graham, 1982, 1984). If this situation is to be remedied, Aboriginal people and educators must together negotiate about the what, how and why of mathematics education for traditionally oriented Aboriginal children. 2. LEARNING

THE MATHEMATICO-TECHNOLOGICAL

CULTURE

In a discussion with teachers and others involved in mathematics education for Aboriginal children, Bishop (1985b) pointed out that the body of knowledge that is referred to as school mathematics is in fact one component of Educational Studies in Mathematics 19 (1988) 119-135. ? 1988 by Kluwer Academic Publishers.

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a pervasiveand almost world-wideMathematico-technological' (MT) culThis ture. knowledgehas been classifiedby those in the culturalgroup into number,measurementand geometry(space) strands.In addition,the mathematical and logical meaningsthat encode these understandingsand relationshipsare realizedin the lexico-grammatical systemsthat operatewithin the languagesof those societieswhich are part of the MT culturalgroup. Thus, mathematicalmeaningsare readilyavailableto childrenwho speak suchlanguagesbut may not be recognized,or will be difficultto constructin languagesthat are outside the MT culture(Bishop, 1985b, p. 2; Halliday, 1974).Hence, for childrenfrom traditionalsocieties,acquiringa mathematical educationinvolveslearninga secondlanguagein whichthesemathematical meanings and relationshipscan be realised,or adaptingtheir mother tongue so much meaningscan be conveyed. In addition,one can describethis culturalcomponentas being'technological',meaningthatmathematicsis a kindof technology- a symbolictechnology. Using White's (1959) terminology then, one can see that if this technologicalcomponentof MT cultureis to be fully understoodand mastered,the ideological(beliefs,etc.), sociological(institutions,etc.) and sentimental (attitudes, etc.) componentsof the MT culture,which provide the 'context of situation'(Malinowski,1923) for the technologicalcomponent must be acknowledged,if not accepted,by the learner(Bishop, 1985b,p. 3). Bishop concluded that, when these socio-cultural components of the MT culture are ignored by teachers,children from societies outside that culturegroupseemto findlittle or no sensein the curriculumofferingsof the school. Many teachersof Aboriginalchildrenwhen faced with the continualfailure of theirstudentshavesuggestedthat only enoughmathematicsshouldbe taught to enablechildrento functioneffectivelyin their home communities while the moredemandingmathematicalcontactswith the outsideworldare managed by others. This solution has certain appeal in the face of the realitiesof teachingWesternmathematicsin remote traditionallyoriented Aboriginalcommunities.However,such policies, while appearingto meet the particularneedsof Aboriginalchildren,would,in Cummins'view (1985, p. 4) continue to disempowerthem in any conitactwith the dominantnonAboriginal society. For without knowledge of Western mathematics, Aboriginalchildrenaredeniedaccessto furthereducationand to the knowledge and power inherentin the social institutionswhich, even today, influence the way Aboriginalpeople live their lives. Thus, if Aboriginalparents want theirchildrento achievethe kind of academicsuccessthat is perceived to be the normaloutcomeof a school education,teacherswill have to find

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ways of making the sociocultural as well as the technological components of the MT culture available to their students. Teachers and others see problems with this approach and express concern that if Aboriginal children are schooled in this way they will lose their Aboriginal identity. Teaching facts from the other culture is one thing, it seems, but teaching the value system quite another. However, as Bishop has pointed out, the so-called 'facts' make no sense without the other aspects of the MT culture that provide meaning for what is being learned. Wolcott (1967, p. 130) faced a similar situation when teaching Kwakiutl Indian children. He appreciated that these children needed to understand the values that gave meaning to school learning but believed that such values could be taught to minority children as skills. In that way they could be used when learning or working in the institutions of the dominant society but did not have to become values that dictated how children should live at home or in their community. Such a strategy, if used in schools, would enable Aboriginal children to establish domains, or separate areas, in their lives and so live confidently as bicultural people who are able to think, speak and act in a manner appropriate to the situation in which they find themselves. 3. THE OTHER SIDE OF THE COIN - WHAT DO THE CHILDREN BRING TO SCHOOL?

A brief examination of recent research into the teaching of mathematics in mainstream society reveals that while this subject has been taught in our schools for many years, few teachers fully comprehend the breadth and complexity of the subject matter involved. In addition, it would appear that we are only now beginning to appreciate the significance of what children know when they come to school and how they learn and think about what they know. In particular, any discussion related to providing mathematics education for traditionally oriented Aboriginal children must recognize the mathematical understandings (ethnomathematics) that Aboriginal children bring with them to the educational encounter. These understandings are encoded into the language they speak and they express a particular view of reality. However, Aboriginal children do not need to come to school just to learn their ethnomathematics. Indeed, Aboriginal people have demonstrated, for many centuries, that they can learn the mathematical relationships inherent in their own culture quite effectively without schooling. For example, research reveals that counting behaviour in traditional societies has not always been recognized by Western researchers because, for

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example, gestures rather than words may have been used (see Gelman and Gallistel, 1978, pp. 73-78, also Harris, J., 1982 in relation to counting systems among Aboriginal groups). However, while Aboriginal people traditionally make use of some numbers words, in Sayer's (1982, p. 187) experience, they place little value on precise counting and in traditional communities have little understanding of the concepts involved. Thus, while Aboriginal children may recognize and name groups to three when they come to school - a skill that is possibly reinforced by the need to recognize both dual (two) and plural (three or more) when using personal pronouns they are usually not counting. Hence, counting is something that has to be learned rather than refined in school. On the other hand, very young Aboriginal children who grow up in Central Australia are able to indicate cardinal directions as they move about the community (Laughren, 1978). Moreover Kearins (1976) has demonstrated that Aboriginal children from the Western desert have strong visual spatial memories. Davidson (1979), who studied the way that Aboriginal children at Bamyili played cards also noted this ability. He found that card players did not use the numbers on the cards to identify them, nor did they add up to find total scores: Ratherthey used complexsystemsof patternrecognitionand groupingin whichall combinationsof cardsfor all possiblescoreswerealreadyknownbeforethe gamebegan(Davidsonand Klich, 1984,p. 144).

Davidson noted (1979, pp. 277, 287-8) that these children were using simultaneous or synchronous rather than successive or serial methods of synthesis of perceptual information. In Davidson's view one implication is that such children are at a disadvantage in classrooms where language is the dominant medium of the teaching/learning process, and learning is based on successive or serial analyses and syntheses of ideas and facts typical of both literacy and Western scientific thinking. This type of research suggests that there is an imbalance in our approach to teaching mathematics which needs to be addressed particularly when discussing Aboriginal education. For example, Bishop (1986) argues that we need to enable children who experience difficulty in learning through language alone to have another avenue of attack and we need to allow children with a strong spatial orientation to make better use of it in learning. To achieve these goals, Bishop believes that children need to be encouraged to reflect on their particular spatial view of the world and through discussion be helped to focus on features of space that are of significance to mathematics. He also believes that children should be encouraged to represent their spatial

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understandings through modelling, drawing and language. In this way, spatial strengths can be used to provide a foundation for verbal learning. These insights are of particular importance if more effective mathematics programmes are to be provided for Aboriginal children who have a strong spatial orientation and yet need to develop the language skills through which much school learning is mediated. 4. THINKING, LEARNING AND TALKING - THE CRITICAL ROLE OF LANGUAGE

It is appropriate at this stage to reflect briefly on some of the literature about cognition. The influence of Piaget is still felt in this area and many writers refer to his findings (Liebeck, 1984; Dickson et al., 1984; Lewis, 1979, 1980, 1983a,b). Although the results of his particularstudies are not in doubt, many writers appear to question the interpretation of his results and have, for example, demonstrated that, while conservation indicates a certain level of logico-mathematical development, children can use a different kind of logic to solve a range of mathematical problems (Lewis, 1983b; Gelman and Gallistel, 1978; Donaldson, 1978). Piagetian tests when carried out in Aboriginal communities have indicated some developmental lag (Hunting and Whitely, 1983). The typical explanation for these results has been to see them as a function of environmental factors or different cognitive strategies (Seagrim and Lendon, 1980). Recent procedures developed by Halford (1984, 1985) have aimed to account for cognitive capabilities through measuring children's ability to process information. Boulton-Lewis and Halford (1985) believed that assessment of the information-processing capacity of Aboriginal children would provide a more definitive indication of underlying cognitive capacity than previous approaches. They found that a group of Aboriginal children at Cherbourg were able to process information as well as European children of the same age. Such a finding is encouraging for all those working with Aboriginal children who felt that this was the case but could not produce evidence to support their intuitions. However, the environmental factor that was revealed in earlier Piagetian and intelligence testing procedures simply cannot be ignored by teachers who must work in these communities. It would seem, then, that if Aboriginal children are to learn mathematics successfully in school, classrooms will need to resemble mathematical 'homes'. An environment will thus be created in which the cognitive processes which underlie the technological component of the MT culture can be revealed to children as they are involved in both living and learning mathematics in school.

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This aspect is importantfor anotherreason.The 'good practice'of using rods, blocks and otherconcretematerialin schoolswill not of itselfcreatea significantreality.Whilerecognizingthat using blocksor countersfor operationsmay makethe processmore'real'it does not makeit 'reallife',and the social meaning,and thus the purposeof the activity,is frequentlynot made availableto the children. Bishop(1985a) sees approachesthat enablechildrento be involvedin the 'social constructionof meaning'as a significantdevelopmentin our understandingof the processof mathematicseducation.He believesthat teachers need to move awayfrom 'thinking'too muchabout content,knowledgeand topics. Rather they need to think more about the kind of experiencesthat childrencan be involvedin that will enablethem to constructmathematical meanings for themselves. Key features of this approach are activities or experiences, communication,which is to do with sharing meanings, and negotiation, which is to do with developing meanings (Bishop, 1985a, p. 26).

Bauersfeld (1980) has also carried out research in this area. After analysing classroom texts in relation to mathematicslessons, he demonstratedthat, while teachersand childrenare using languageto interact,they are all behavingaccordingto their own actual subjectiverealities.Hence, teachersand studentsarefrequentlyat cross-purposeseventhoughtheyboth believethat they understandwhatthe otherpersonis saying.He, like Bishop, also favoursapproachesthat enablestudentsto be involvedin socialnegotiation of mathematicalmeaningsbut points out that in mathematicssuch negotiationsneed to continue until studentsbecome aware of the performanceof meaningthat gets the teacher'ssanction(Bauersfeld,1980,p. 35). For Aboriginalchildreninvolvedin learningWesternmathematicsin school suchnegotiationsneedto be lengthyso theycan, if necessary,recognizetheir particularAboriginalview of realityand also come to perceivethe meaning inherentin the MT culture. Approacheslike these, while essentialif Aboriginalchildrenare to gain mathematicalmeaningsand not just skill in respondingcorrectlyin some narrowlydefinedmathematicalsituation,presentother problemsfor their classroom teachers.Malcolm (1980) demonstratedthat many Aboriginal childrendo not findit easy to take partin the participantstructuresthrough which meanings are made and shared in the classroom. Kearins (1985), Harris (1984) and Christieand Harris (1985) have also documentedthe difficultiesAboriginalchildrenexperiencein classroomsand the many factors that lead to commuicationbreakdownin the context. The strategiesdevelopedby Gray(1983) for enablingAboriginalchildren to becomemore effectivelanguagelearnersin the school contextneed to be

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examined in this regard. Through an approach that has come to be known as Concentrated Language Encounters, he has enabled children to encounter language in a context that provides meaning onto which language can be mapped. Negotiation is a key feature of the approach and observers are impressed not only with the improved language abilities of the children but with the knowledge that children have gained through meaningful interactions in 'real-life' contexts. If Aboriginal children are to find meaning in their school mathematics programmes, Concentrated 'Mathematics' Encounters may provide an approach through which this may be achieved. 5. TWO LANGUAGES IN MATHEMATICS EDUCATION

While many children who learn mathematics in their mother tongue experience difficulty in acquiring the register associated with mathematics, these difficulties are exacerbated for children who must learn in a second language. Many of these children experience difficulties which can clearly be related to their inability to comprehend English mathematical terms and the patterns of discourse found in oral interactions and written texts (Newman, 1981). In Halliday's view (1975), learning language involves 'learning how to mean' and hence learning the language of mathematics involves learning how to make and share mathematical meanings using language appropriate to the context, which is more than recognizing and responding to words in isolation. Unless teachers of mathematics become more aware of this difference it seems that many second language learners will continue to be disadvantaged in school. However, for some children who are involved in second language education the situation is very different. For example, children in the St Lambert bilingual programme performed at significantly higher levels than controls, on measures of divergent thinking. Examination of the results achieved by individuals within the groups that were studied led to the development of the so-called 'threshold' hypothesis (Cummins, 1977, 1981). The form of the hypothesis that is most consistent with available data suggests there are two thresholds. Children who know neither language well may experience negative cognitive effects, while those who know both languages extremely well will experience positive cognitive effects. In between these two thresholds neither positive or negative cognitive effects have been noted. Cummins (1981) refers to several studies which have reported findings that are consistent with the general tenets of this threshold hypothesis. Hence, while it has been demonstrated that bilingual education, per se, is not necessarily detrimental and for some can be decidedly advantageous, there are groups of

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children,whose home languageis not being adequatelydevelopedand who are not becoming effectivespeakersof a second language in school, who must be consideredat risk. Evidencesuggeststhat such childrencan easily sufferlinguistic,intellectualand academicretardationand may ceaseto have identitywith theirculturalgroupwhilefailingto establishsuchlinkswith the contact group (Cummins, 1977;Cumminsand Gulutson, 1974). Cummins believesthat these conditionscan be createdwhen educatorsendeavourto replacea child'slanguageand culturewith that of the dominantgroup. He describessuch an educationalprogramas 'subtractive'while the bilingual education that results in educationaladvantagehe describesas 'additive' (Cummins, 1981, 1985). Dawe (1983) carriedout researchto discoverif there was any evidence with respectto the abilityof bilingualchildrento reasonin mathematicsin Englishas a secondlanguagethat would supportCummins'hypothesis.He foundthat mathematicalreasoningin the deductivesenseis closelyrelatedto the ability to use languageas a tool for thought, and that the ability of a child to make effectiveuse of the cognitivefunctionsof his firstlanguageis a good predictorof the abilityto reasondeductivelyin Englishas a second language. He also found that there was a complex relationshipbetween visuo-spatialand verbal-logicalreasoningand that bilingualchildrenoften switchedfrom one mode to the otherduringthe reasoningprocess,and also that this switchwas often accompaniedby a languageswitchas well (Dawe, 1983,pp. 349-350). Cathcart(1980) also exploredthe matterof cognitiveflexibilitywith bilingual and monolingualchildren.The numberof second rationalizationsa child could give for conservationwas consideredto be indicativeof this quality, for it demandedthat the child look at the phenomenain different ways. The study found in favourof the bilingualchildrenand in Cathcart's view providedfurtherevidencein supportof the thresholdhypothesisthat had been formulatedby Cummins(Cathcart,1980,p. 8). Both Cathcartand Dawe concludedthat firstlanguagemaintenancefor minoritylanguagestudents was an importantfactor in predictingsuccessin the area of mathematics education, a finding that has implications for the mathematical educationof remotetraditionallyorientedAboriginalchildren. Thus one educationalimplicationis that we need to considerAboriginal childrencarefully,for if, as it would seem,becominga mathematicalperson involvesconstructingmathematicalmeaningsand communicatingand negotiating about those meanings,it is in the child's first languagethat such interactionscould most easily occur. However,many languagesspoken by childrenin developingcountriesthat are outside the MT culturelack the

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register - both the vocabulary and the logical connectives - necessary to encode understandings inherent in that culture. As Halliday (1974) points out, this does not necessarily mean that people do not perceive some or many of the classifications inherent in the MT culture, it is just that they do not attend to them. All languages have evolved to meet particular needs of their speakers and given time and the need, languages that reflect simple technologies can evolve further and absorb some, at least, of the understandings inherent in the MT culture. Even while this development is taking place Halliday (1974) believes that such languages can be used as a point of departure for helping children learn Western mathematics in school. While these languages may never develop a full register of mathematical terms, concepts can be 'talked around' in the everyday language of life. Hence, while there may be no word for 'plus' in a language, children can be involved in and talk about experiences that enable them to 'bring together', 'add together', 'put with' and so on. In doing this it is inevitable that the meaning of some words will change. For example, Christie (1980) notes that in Gupapuyngu, a language spoken in Northeast Arnhem Land, 'bulu' the word for 'more' is used to denote 'extra' as in, 'I want more soup', but it does not mean 'relatively greater' as in, 'There is more sugar in this bowl than in that one'. However, it can have its meaning extended to carry that understanding if that is what people want to talk and think about. Extending meanings of words, borrowing words from the other language, and combining two or three words to create a new term or locution as in 'right-angled triangle', in a planned way, is referred to as language engineering (Morris, 1978). Leeding (1976) in North Australia, Gnerre (1984) from Brazil, Mwombogela (1979) from Tanzania along with a wide range of speakers from third world countries who attended a CASME workshop in Ghana in 1975 can all provide examples of just how this language planning may occur in countries where, frequently, there are political as well as pedagogical reasons for educating children in their mother tongue and thus incentive is provided to extend the local language to carry out the task. In very small languages of say 1000 speakers or less, there are probably practical rather than theoretical considerations as to why this development may not extend as far as is technically possible and educators need to ensure that children's mathematical development is not stunted by a lack of appropriate terms. Nevertheless a structured bilingual approach to Aboriginal education does find strong support in the research literature.

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6. WHAT ARE THE IMPLICATIONS FOR MATHEMATICS EDUCATION OF AN ABORIGINAL VIEW OF REALITY?

Aboriginal children grow up in a society in which the system that controls the economic realities of life are based on relationships between people rather than relationships between quantities of money, time, goods and other services, as is the case in MT culture. Bain, in Christie (1985, p. 9), has described it as 'interactional' rather than 'transactional'. Thus, Aboriginal children are much better at talking to establish personal relationships with their teachers than they are at talking to transact knowledge inside the classroom. The environment in which people live is also grounded in such interactional relationships which extend back to the Dreamtime and relate Aboriginal people to the land and to the dominant features of the land. Hence, questions like, 'How much land?' are immaterial. Instead, people focus on the relationship between a particular group of people who are 'owned' by the land.

6.1. A Concernfor Quality In such a society the emphasis is not on the quantity of the relationship but on the quality. Rudder (1983) examined the classificatory systems, the evaluative systems and cognitive structures of the Yolnu people of Northeast Arnhem Land. He used the term 'qualitative thinking' to describe the way Yolnu people reflect on their world. For example, Aboriginal people, when talking about what English speakers would think of as length - which signifies a quantitive approach to the attibute - focus instead on quality. Thus, the quality 'shortness' (gurriri) may be noted. Unlike English, that sees 'shortness' as part of a continuum that extends from 'short' to 'long', Yolnu see the quality 'shortness' as being discontinuous and so objects are either 'short' or they are 'not-short' or a non-expression of the quality 'short'. Once the initial assessment of the quality has been made the second choice is to do with describing the quality of that quality. Thus, something that is perceived as being short can be further qualified as very short or moderately short (Rudder, 1983, p. 36). Such knowledge has a place in a bilingual school for it enables children to recognize and reflect on their own particular view of reality. It may provide a bridge to English as children can add the English word to the concept of shortness already developed - though care should be taken to ensure that the situation is one that speakers of both languages perceive in terms of shortness. However, it may be inappropriate to continue to use the vernacular

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terms when the comparison between 'short' and 'long' is being stressed. That is a particularly Western way of looking at reality and reflects our concern with quantifying. Aboriginal people may, of course, be happy for their language to be used in this way, but such decisions need to be made after lengthy negotiation when all the participants concerned are conscious of the subtle but significant differences in the way language is used. 6.2. Many Languages: Many Systems of Knowledge However, what also needs to be appreciated is that there cannot be just one solution for resolving all these matters. The work of Stokes (1982), Sayers (1982), Harris J. (1979) and the collections of findings, based very largely on linguistic research, that has been gathered together by Harris P. (1980, 1984a,b, 1988 - in press) highlight both the differences and similarities that exist between language groups. For example, Laughren (1978) has noted that Aboriginal children who grow up in Central Australia demonstrate, at a very early age, an ability to use and respond to cardinal directions. While this knowledge can be capitalized on in both ethnomathematics and in learning Western mathematics it cannot be assumed that all Aboriginal children possess such precise knowledge. In my experience, while coastal children always knew where they were going they did not indicate direction in the way of children who live in or near the desert, hence such knowledge is localized. Other spatial knowledge inherent in the Aboriginal view of reality is quite widespread. When exploring the difficulties that many Aboriginal children experienced when working on number lines, it was found in one language that a word was being used for 'after' (e.g. What comes after 3?) that was related to the speaker's point of view. Thus, the word could be translated back into English as 'before', 'after', 'previously', 'following', etc., depending on the context. This confusion has been found to be quite widespread, but interestingly, it is only through extensive use of two languages in education that many of these confusions have been revealed. Previously, children in English-only programmes, who have been intelligently applying their Aboriginal view of the world to their newly acquired English terminology have simply appeared 'stupid' to teachers and others who frequently had little understanding of the difficulties children were experiencing were in making sense of what was being taught (Harris, J., 1979). 6.3. Ethnomathematicsand Formal Schooling It would seem, then, that there is a place in our schools for ethnomathematics and for the teaching of aspects of Western mathematics in children's own

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language.HarrisJ. (1979, p. 149), Halliday(1974) and others supportthis approach.For they claim, quite strongly,that unless schools recognizethe conceptualview of the world that childrenbringwith them to school, children will find difficultyin acquiringthe mathematicalunderstandingsthat are inherent in the MT culture. In contrast, Davidson and Klich (1984) expressconcernthat manyof the activities,like card playing,that appearto be mathematical,in the MT senseof the word,involvechildrenin processing informationin waysthatarequitedifferentto that of the school.The insights gained by HarrisP. (1984b) on the ways Aboriginalpeople handlemoney (e.g., papermoney is often simplyreferredto by colour of the note and the numericalvalueis not recognized)providesanotherexampleof how some of the artifactsof Westernsociety,have been 'Aboriginalized'since theirintroductioninto that culture.Davidsonand Klich (1984, p. 144)actuallyquery whethersuch 'street'activitiescan havea placein classroomswherethe goals of the programmeare not just rememberingbut are more too with understanding,generalizingand applying. The issues go deeperthan this of course. Aboriginalpeople have been happy to have their childrenbegin to be mathematicalpeople in the MT cultural sense of the word through,for example,encouragingchildrento recognize and representthrough drawing and language, the people that belong to a certainkin group. However,they may not be so happy if the kinshipsystemis dealtwithin schoolin sucha way that it becomesan 'open' systemin Horton's(1971, p. 230) senseof the word. For Aboriginalpeople, the kinshipsystemis given.To use it to encouragechildrento infer,predict, generalizeand so forth may be consideredinappropriate.By the time adults realizedwhat was happeningit could alreadybe too late. Thus, although Gay and Cole (1967) recommendedthat the "teacher shoulduse the Western,scientificmethodfor comprehending,clarifyingand organizingcontent drawn directlyfrom the child's familiar daily experiences",aftermanyyearsworkingin the Aboriginalcontext,I now say "Take care".Aboriginalmathematicalknowledgehas an importantplacein school education.It can:providea bridgebetweenhome and school;be part of the Aboriginal studies strand that is an integral part of any biculturalprogramme;and providea foundationfor some learningof the technological componentof the MT culture.Such activitiesshould make this knowledge availableto childrenat a consciouslevel so they can recognizeit as part of theirAboriginalityand realizewhen it is overlappingwith the Westernview of realitythat they talk about, at times in their own languageand at other times in English.However,whethertraditionalAboriginalknowledgeis to be exposedto the kind of so-calledhighordercognitiveprocessesthat arean

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inherent part of the MT culture is something that only Aboriginal people can decide. Until they are aware of the options involved we must move slowly. Otherwise in our eagerness to open doors to the knowledge and understandings that Aboriginal people may need if they are to survive as an identifiable independent people in a modern world we may destroy the very culture that provides that identity and that gives meaning to life. 7. CONCLUSION

It is clear that there are no easy solutions to the problems Aboriginal children experience in learning Western mathematics in school. No doubt, more children will learn more effectively when they and their families perceive a reason for doing so. In the meantime, this review of research has highlighted several features that should be inherent in any approach to the teaching of the MT culture in Aboriginal schools. These are: Aboriginality. Aboriginal children are first and foremost Aboriginal and it is what they and their parents want them to remain. Their knowledge, language and learning styles should be used and respected while they gain other knowledge, language and skills that are added to what is already there. Where there is conflict between ways of perceiving, talking and thinking about reality they must be presented as alternatives and children should be encouraged to see themselves as learning to be bilingual and bicultural people who will be able to act appropriately in the situation in which they find themselves. Time. Aboriginal children need more time, particularly in the early years, if they are to gain the level of understanding necessary to provide a solid foundation on which further mathematical studies can be built. Spatial awareness. More use should be made of the visual/spatial orientation that Aboriginal children bring with them to school. This skill should not just be used to assist remembering but should be used to assist children to learn to talk about what they perceive and so help them develop some of the skills essential for effective school learning. Experiences. Aboriginal children need to be involved in mathematical experiences that are not only 'real' but are 'real-life' and which enable them to adopt the roles of participants in such contexts and so begin to understand the purpose of the transactions involved.

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Language. Mathematics in an Aboriginal context must be viewed as a language activity and must provide contexts that will enable children to 'learn how to mean mathematically'. This highlights again the need for appropriate experiences that will provide the meaning onto which the language can be mapped. Two languages. Wherever possible, particularly in the early years, children should be encouraged to talk themselves into understanding the new concepts that they are meeting, through the use of the language over which they have most control - their mother tongue. However, care needs to be exercised to ensure that teachers are developing the idea concerned and not just the nearest approximation. Ethnomathematics.This knowledge which is encoded into the language spoken by the children must have an important role in their education. It should be seen as an end in itself and, if and when possible, be drawn on to provide a bridge into the mathematics of the wider world. However, care needs to be taken to ensure that such knowledge is not trivialized or in other ways harmed through its contact with the Mathematico-Technological culture which is inherent in the institution of Western schooling. Negotiation. Key factors in the development of more effective mathematics programmes for Aboriginal children are the interactions that must take place between teachers, students, parents and others about the role of Western mathematics in their children's education. Parents must be helped to become aware of the language and cognitive processes that are essential for success in school and work alongside teachers in ensuring that children are grounded in their own heritage while being given access to another if that is what they desire. In particular, Aboriginal children must be involved in negotiation both about their own learning and as a means of achieving that learning. Finally, Aboriginal people must decide what they want from school. An old Aboriginal man told Seagrim and Lendon: We wantthemto learnEnglish.Not the kindof Englishyou teachthemin classbut yoursecret English.We don't understandthat Englishbut you do (Christie,1985,p. 50).

Some Aboriginal people may soon be saying such things about our mathematics programmes. Aboriginal children are being taught mathematics in our schools, but they are not learning the things that matter. Such knowledge is not just to do with getting sums right, though that is part of it. Rather, it is to do with the way people talk and think about what they know. The MT

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culturewith its open view of realitypresentsmanyproblemsto traditionally orientedAboriginalAustralians.With it they may lose some aspectsof their Aboriginality,without it they will continue to be dependenton others to teach their childrenand managetheirprojects.However,that choice is one that Aboriginalpeople must make. The question that educatorsmust addressis simplythis, "If Aboriginalpeoplereallywanta mathematicaleducation for their childrenhave we the knowledgeand flexibilityto work with them to achievethat goal?" REFERENCES Bauersfeld,H.: 1980,'Hiddendimensionsin the so-calledrealityof a mathematicsclassroom', Educational Studies in Mathematics 11, 23-41.

Bishop,A. J.: 1985a,'Thesocialconstructionof meaning- a significantdevelopmentfor mathematics education', For the Learning of Mathematics 5(1), 24-28.

Bishop,A. J.: 1985b,'Mathematicsin Aboriginalschoolsfor Aboriginalchildren',paperpresentedat the National Seminaron MathematicsEducationfor AboriginalChildren,Alice Springs,27-31 August. Bishop,A. J.: 1986,'Whataresomeobstaclesto learninggeometry',in R. Morris(ed.), Studies in Mathematics Education, Vol. 5, UNESCO, Paris.

Boulton-Lewis,G. and G. Halford:1985,'Levelsof informationprocessingcapacityand culturalknowledge,in a groupof AboriginalAustralianchildren',paperpresentedat the National Seminaron MathematicsEducationfor AboriginalChildren,Alice Springs,27-31 August. Bourke,S. F. and B. Parkin:1977,'Theperformanceof Aboriginalstudents',in S. F. Bourke and J. P. Keeves (eds.), Australian Studies in School Performance, Vol 3: The Mastery of Literacy and Numeracy, Final Report, Australian GovernmentPublishing Service, Canberra,

1977. CASME:1975,'Languagesand the teachingof scienceand mathematicswithspecialreference to Africa',A CASMERegionalWorkshop,Ghana. Anothervariableinfluencing,conceptualdevelopCathcart,G. W.: 1980,'BilingualInstruction: ment in young children', Research in Mathematics Education in Australia, MERGA.

Christie,M.: 1980,'Gupapuynguequivalentsof Englishclassroomterms',Unpublishedpaper, Brisbane,1980. Christie, M.: 1985, Aboriginal Perspectives on Experience and Learning: The Role of Language in Aboriginal Education, Deakin University Press, Victoria.

breakdownin the Aboriginalclassroom',in Christie,M. and S. Harris:1985,'Communication J. Pride (ed.), Cross Cultural Encounters: Communication and mis-Communication, River

Seine,Melborne,pp. 81-90. Cummins,J.: 1977,'Cognitivefactorsassociatedwith the attainmentof intermediatelevelsof bilingual skills', The Modern Language Journal 61, 3-12. Cummins, J.: 1981, Schooling and Language Minority Students:A TheoreticalFramework,Eval-

uation,Disseminationand AssessmentCentre,CalifornianState University,Los Angeles. Cummins,J.: 1985, 'Disablingminoritystudents:Power, Programsand Pedagogy'.Unpublishedpaper,OntarioInstitutefor Studiesin Education. Cummins,J. and M. Gulutson:1974,'Bilingualeducationand cognition',TheAlbertaJournal of Educational Research 20(3), 959-969.

Davidson,G. R.: 1979,'An ethnographicpsychologyof Aboriginalcognitiveability',Oceania 49(4), 270-294.

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Davidson,G. and L. Z. Klich:1984,'Ethnography,cognitiveprocessesand instructionalprocedures', in J. R. Kirby (ed.), Cognitive Strategies and Educational Performance, pp. 137-153.

AcademicPress,Sydney. Dawe, L.: 1983, 'Bilingualismand mathematicalreasoningin Englishas a second language', Educational Studies in Mathematics 14, 325-353. Dickson, L., M. Brown, and 0. Gibson: 1984, Children Learning Mathematics: A Teacher's Guide to Recent Research, Holt, Rinehart and Winston, New York.

Donaldson,M.: 1978, Children'sMinds,Fontana/Collins,Glasgow. Gay, J. and M. Cole: 1967, The New Mathematics in an Old Culture:A Study of Learning among

the Kpelleof Liberia,Hold, Rinehartand Winston,New York. Gelman, R. and C. R. Gallistel: 1978, The Child's Understandingof Number, Harvard Univer-

sity Press,Cambridge,Mass. Gnerre,M.: 1980,'Nativelanguagevs. secondlanguagein teachingelementarymathematics.A case from the Amazon',in Proceedingsof ICME V, pp. 582-586, Berkeley. Graham,B.: 1982,'Can we count on maths?',TheAboriginalChildat School10(2), 4-10. Graham, B.: 1984, 'Finding meaning in maths: An introductoryprogramfor Aboriginal children', The Aboriginal Child at School 12(4), 24-39.

Gray, B.: 1983, 'Helping children to become languagelearnersin the classroom',paper presentedat the AnnualConferenceof the MeanjinReadingCouncil,Brisbane. Halford, G.: 1984, 'Cognitivedevelopmentalstages based on informationprocessinglimitations', paper presented at 23rd InternationalCongress of Psychology, Acapulco., Mexico. Halford, G.: 1985, 'A hierarchyof conceptsin cognitivedevelopment',paper presentedat Conferenceof the Society for Researchin Child Development,Toronto, Canada, 25-28 April. Halliday,M. A. K.: 1974,'Someaspectsof sociolinguistics',in InteractionsBetwenLinguistics and MathematicalEducation,pp. 64-73. Symposiumsponsoredby UNESCO, Nairobi, Kenya. Halliday, M. A. K.: 1975, Learning How to Mean -Explorations in the Development of

Language,EdwardArnold,London. Harris,J.: 1979,'Ethnoscienceandits relevanceforeducationin traditionalAboriginalcommunities',UnpublishedM.Ed. thesis,Universityof Queensland. Harris,J.: 1982,'Factsand fallaciesof Aboriginalnumbersystems',in S. Hargrove(ed.), Work Papers of SIL-AAB, Series B, Vol. 8, pp. 153-177, Language and Culture, Darwin. Harris, P.: 1980, Measurement in Tribal Aboriginal Communities, N.T. Dept of Education,

Darwin. Harris, P.: 1984a, Teaching About Time in Tribal Aboriginal Communities, Mathematics in

AboriginalSchoolsProject:2, C.D.C. and N.T. Dept. of Education,Darwin. Harris, P.: 1984b, Teaching About Money in Tribal Aboriginal Communities, Mathematics in

AboriginalSchoolsProject:3, C.D.C. and N.T. Dept. of Education. Harris, P.: 1988 (in press), Teaching the Space Strand in TribalAboriginal Schools, Mathematics

in AboriginalSchoolsProject:4, C.D.C. and N.T. Dept. of Education. Harris,S.: 1984,'Aboriginallearningstylesand formallearning',TheAboriginalChildat School 12(4), 3-23. Horton, R.: 1971,'Africantraditionthoughtand Westernscience',in M. F. D. Young (ed.), Knowledge and Control, Collier McMillan, London.

Hunting, R. and H. Whitely: 1983, 'Mathematics,prior knowledge and the Australian Aborigine', in Research in Mathematics Education in Australia, MERGA, pp. 13-24.

Kearins,J.: 1976,'Skillsof desertAboriginalchildren',in G. E. Kearneyand D. W. McElwain (eds.), Aboriginal Cognition: Retrospect and Prospect, pp. 199-212, Humanities Press, New

Jersey.

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Kearins, J.: 1985, 'Cross-cultural misunderstandings in education', in J. Pride (ed.), Cross Cultural Encounters: Communication and Mis-Communication, River Seine, Melbourne, pp. 65-80. Laughren, M.: 1978, 'Directional terminology in Walpiri', Working Papers in Language and Linguistics, No. 8, Tasmanian C.A.E., Launceston. Leeding, V.: 1976, 'Contrastive semantic units in the teaching of concepts in a bilingual education program', Unpublished paper, Darwin. Lewis, G.: 1979, 'Learning mathematical concepts in a second language: Problems for the young child', Australian Journal of Early Childhood4(1), 33-37. Lewis, G.: 1980, 'Premathematical knowledge in preschool children', in Research in Mathematics Education in Australia, MERGA, pp. 89-104. Lewis, G.: 1983a, 'Mathematics - how much do they know', in Links: For Teachers of Young Children, Dept. of Education, Queensland. Lewis, G.: 1983b, 'Rethinking young children's thinking: Neo-Piagetian perspectives', paper presented at SPATE, Brisbane. Liebeck, P.: 1984, How ChildrenLearn Mathematics:A Guidefor Parents and Teachers, Pelican Books, Middlesex. Malcolm, I.: 1980, 'Preparing teachers for interaction in Aboriginal classrooms', in F. Christie and J. Rothery (eds.), Language in Teacher Education in a Multi-cultural Society, Applied Linguistics Association of Australia Monograph. Malinowski, B.: 1923, 'The problem of meaning in primitive languages', Supplement 1, in Ogden, C. K. and Richards, I. A. (eds.), The Meaning of Meaning, The International Library of Philosophy, Psychology and Scientific Method, Kegan Paul, London. Morris, R.: 1978, 'The role of language in learning mathematics', in Prospects, UNESCO, Vol. VIII, No. 1, pp. 73-81. Mwambogela, A.: 1979, 'Language problems in teaching mathematics in Tanzania', paper presented to the Seminar on Developmentof TeachingMaterials For School Mathematics, held at Mbabne, Swaziland, 12-16 March. Newman, M. A.: 1981, 'Comprehension of the language of mathematics', in Research in Mathematics Education, MERGA. Rudder, J.: 1983, 'Qualitative thinking: An examination of the classificatory systems, evaluative systems and cognitive structures of the Yolnu people of North-east Arnhem Land', Unpublished M.A. Thesis, National University, Canberra. Sayers, B.: 1982, 'Aboriginal mathematics concepts: A cultural and linguistic explanation for some of the problems', in S. Hargrave (ed.), WorkPapers of SIL-AAB, Series B, Vol. 8, pp. 183-199, Language and Culture, Darwin. Seagrim, G. and R. Lendon: 1980, Furnishing the Mind: A Comparative Study of Cognitive Development in Central Australian Aborigines, Academic Press, Sydney. Stokes, J.: 1982, 'A description of the mathematical concepts of Groote Eylandt Aborigines', in S. Hargrave (ed.), Work Papers of SIL-AAB, Series B, Vol. 8, pp. 33-152, Language and Culture, Darwin. White, L. A.: 1959, The Evolution of Culture, McGraw-Hill, New York. Wolcott, H.: 1967, A Kwakiutl Village and School, Holt, Rinehart and Winston, New York.

130 Gordon Street, Balwyn, Vic 3103, Australia.

PAULUS GERDES

ON CULTURE,

GEOMETRICAL

MATHEMATICS

THINKING

AND

EDUCATION*

ABSTRACT.This articleconfrontsa widespreadprejudiceabout mathematicalknowledge, that mathematicsis 'culture-free', alternativeconstructionsof euclideangeoby demonstrating metricalideas developedfromthe traditionalcultureof Mozambique.As well as establishing the educationalpowerof theseconstructions,the articleillustratesthe methodologyof 'cultural in the contextof teachertraining. conscientialization'

1. SOME SOCIAL AND CULTURAL ASPECTS OF MATHEMATICS EDUCATION IN THIRD WORLD COUNTRIES

In most formerly colonized countries, post-independence education did not succeed in appeasing the hunger for knowledge of its people's masses. Although there had occurred a dramatic explosion in the student population in many African nations over the last twenty five years, the mean illiteracy rate for Africa was still 66% in 1980. Overcrowded classrooms, shortage of qualified teachers and lack of teaching materials, contributed toward low levels of attainment. In the case of mathematics education, this tendency has been reinforced by a hasty curriculumtransplantationfrom the highly industrialized capitalist nations to Third World countries.' With the transplantation of curricula their perspective was also copied: "(primary) mathematics is seen only as a stepping stone towards secondary mathematics, which in turn is seen as a preparation for university education".2 Mathematics education is therefore structured in the interests of a social elite. To the majority of children, mathematics looks rather useless. Maths anxiety is widespread; especially for sons and daughters of peasants and laborers, mathematics enjoys little popularity. Mathematics education serves the selection of elites: "Mathematics is universally recognized as the most effective education filter", as El Tom underlines.3 Ubiratan D'Ambrosio, president of the Interamerican Committee on Mathematics Education agrees: "... mathematics has been used as a barrier to social access, reinforcing the power structure which prevails in the societies (of the Third World). No other subject in school serves so well this purpose of reinforcement of power structure as does mathematics. And the main tool for this negative aspect of mathematics education is evaluation".4 In their study of the mathematics learning difficulties of the Kpelle (Liberia), Gay and Cole concluded, that there do not exist any inherent Educational Studies in Mathematics 19 (1988) 137-162. ? 1988 by Kluwer Academic Publishers.

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difficulties: what happened in the classroom, was that the contents did not make any sense from the point of view of Kpelle-culture; moreover the methods used were primarily based on rote memory and harsh discipline.5 Experiments showed that Kpelle illiterate adults performed better than North American adults, when solving problems, like the estimation of number of cups of rice in a container, that belong to their 'indigenousmathematics'.6 Serious doubts about the effectiveness of school mathematics teaching are also raised by Latin American researchers. Eduardo Luna (Dominican Republic) posed the question if it is possible, that the practical mathematical knowledge that children acquired outside the school is 'repressed' and 'confused' in the school.7 Not only possible, but this happens frequently, as shown by the Brasilians Carraher and Schliemann: children, who knew before they went to school, how to solve creatively arithmetical problems which they encountered in daily life, e.g. at the marketplace, could, later in the school, not solve the same problem, i.e. not solve them with the methods taught in the arithmetic class.8 D'Ambrosio concludes that "'learned' matheracy eliminates the so-called 'spontaneous' matheracy",9 i.e. "An individual who manages perfectly well numbers, operations, geometric forms and notions, when facing a completely new and formal approach to the same facts and needs creates a psychological blockade which grows as a barrier between the different modes of numerical and geometrical thought".10What happens in the school, is that "The former, let us say, spontaneous, abilities (are) downgraded, repressed and forgotten, while the learned ones (are not being) assimilated, either as a consequence of a learning blockage, or of an early dropout.. .".' For this reason, "the early stages of mathematics education (offer) a very efficient way of instilling the sense offailure, of dependency in the children".'2 How can this psychological blockade be avoided? How can this "totally inappropriate education, leading to misunderstanding and sociocultural and psychological alienation"13 be avoided? How can this 'pushing aside' and 'wiping out' of spontaneous, natural, informal, indigenous, folk, implicit, non-standard and/or hidden (ethno)mathematics be avoided?'4 Gay and Cole became convinced that it is necessary to investigate first the 'indigenous mathematics', in order to be able to build effective bridges from this 'indigenous mathematics' to the new mathematics to be introduced in the school: "... the teacher should begin with materials of the indigenous culture, leading the child to use them in a creative way",'5 and from there advance to the new school mathematics. The Tanzanian curriculum specialist Mmari stresses, that: "... there are traditional mathematics methods still A good teacher can utilize this situation to being used in Tanzania....

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underline the universal truths of the mathematical concepts".16 And how could the good teacher achieve this? Jacobsen answers: "The (African) people that are building the houses are not using mathematics; they're doing it traditionally ... if we can bring out the scientific structure of why it's done, then you can teach science that way".17For D'Ambrosio, it becomes necessary ". .. to generate ways of understanding, and methods for the incorporation and compatibilization of known and current popular practices into the curriculum. In other words, in the case of mathematics, recognition and incorporation of ethnomathematics into the curriculum".'8 ". .. this ... requires the development of quite difficult anthropological research methods relating to mathematics; ... anthropological mathematics ... constitutes an essential research theme in Third World countries... as the underlying ground upon which we can develop curriculum in a relevant , 19 way". 2. TOWARDS A CULTURAL-MATHEMATICAL REAFFIRMATION

D'Ambrosio stressed the need for incorporation of ethnomathematics into the curriculum in order to avoid a psychological blockade. In former colonized countries, there exists also a related cultural blockade to be eliminated. "Colonization - in the words of President Samora Machel - is the greatest destroyer of culture that humanity has ever known. African society and its culture were crushed, and when they survived they were co-opted so that they could be more easily emptied of their content. This was done in two distinct ways. One was the utilization of institutions in order to support The other was the 'folklorizing' of culture, its colonial exploitation.... reduction to more or less picturesque habits and customs, to impose in their place the values of colonialism". "Colonial education appears in this context as a process of denying the national character, alienating the Mozambican from his country and his origin and, in exacerbating his dependence on abroad, forcing him to be ashamed of his people and his culture".20 In the specific case of mathematics, this science was presented as an exclusively white men's creation and ability; the mathematical capacities of the colonized peoples were negated or reduced to rote memorization; the African and American-Indian mathematical traditions became ignored or despised. A cultural rebirth is indispensable, as President Samora Machel underlines: ". . . long-suppressed manifestations of culture (have to) regain their place".21In this cultural rebirth, in this combat of racial and colonial prejudice, a cultural-mathematical-reaffirmationplays a part: it is necessary

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to encouragean understandingthat our peopleshave been capableof developing mathematicsin the past, and therefore- regainingculturalconfidence22- will be able to assimilateand develop the mathematicswe need; mathematicsdoes not come fromoutsideour African,Asian and AmericanIndiancultures. We may concludethat the incorporationof mathematicaltraditionsinto the curriculumwill contributenot only to the eliminationof individualand social psychologicalblockade, but also of the related cultural blockade. Now, this raises an important question: which mathematical traditions? In

orderto be able to incorporatepopular(mathematical)practices,it is firstof all necessary to recognize their mathematical character. In this sense,

D'Ambrosiospeaks about the need to broadenour understandingof what mathematicsis.23Ascherand Ascherremarkin this connection"Becauseof the provincialview of the professionalmathematicians,most definitionsof mathematicsexcludeor minimizethe implicitand informal;... involvement with conceptsof number,spatialconfiguration,and logic, that is, implicitor explicit mathematics, is panhuman".24

Broadeningour understandingof what mathematicsis, is necessary,but not sufficient.A relatedproblemis how to reconstructmathematicaltraditions, whenprobablymanyof themhave been- as a consequenceof slavery, of colonialism...-wiped out. Few or almost none (as in the case of Mozambique)writtensourcescan be consulted.Maybefor numbersystems and some aspects of geometricalthinking,oral hstory may constitutean alternative.What other sourcescan be used? Whatmethodology? We developeda complementarymethodologythat enablesone to uncover in traditional,materialculturesome hiddenmomentsof geometricalthinking. It can be characterizedas follows. We looked to the geometricalforms and patternsof traditionalobjectslike baskets,mats,pots, houses,fishtraps, etc. and posed the question:whydo thesematerialproductspossessthe form they have?In orderto answerthis question,we learnedthe usualproduction techniquesand tried to vary the forms. It came out that the form of these objects is almost never arbitrary,but generallyrepresentsmany practical advantagesand is, quite a lot of times,the only possibleor optimalsolution of a productionproblem.The traditionalform reflectsaccumulatedexperience and wisdom.It constitutesnot only biologicaland physicalknowledge about the materialsthat are used, but also mathematicalknowledge,knowledgeabout the propertiesand relationsof circles,angles,rectangles,squares, regularpentagonsand hexagons,cones, pyramids,cylinders,etc. Applyingthis method,we discovereda lot of 'hidden'or 'frozen'mathematics.25The artisan, who imitates a known production technique, is,

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THINKING

generally, not doing mathematics. But the artisan(s) who discovered the technique, did mathematics, was/were thinking mathematically. When pupils are stimulated to reinvent such a production technique, they are doing and learning mathematics. Hereto they can be stimulated only if the teachers themselves are conscious of hidden mathematics, are convinced of the cultural, educational and scientific value of rediscovering and exploring hidden mathematics, are aware of the potential of 'unfreezing' this 'frozen mathematics'. Now we shall present some of our experiences in this necessary 'cultural conscientialization' of future mathematics teachers. 3. EXAMPLES

OF 'CULTURAL

CONSCIENTIALIZATION'

MATHEMATICS

OF FUTURE

TEACHERS

3.1. Study of Alternate Axiomatic Constructionsof Euclidean Geometry in Teacher Training Many alternate axiomatic constructions for euclidean geometry have been devised. In Alexandrov's construction,26 Euclid's famous fifth postulate is substituted by the 'rectangle axiom': D

C

if _

A

D

C

A

B

,then

B

i.e., if AD = BC and A and B are right angles, then AB = DC and C and D are also right angles. In one of the classroom sessions of an introductory geometry course, we posed the following provocative question to our future mathematics teachers - most of whom are sons and daughters of peasants -: " Which 'rectangle axiom' do our Mozambican peasants use in their daily life?". The students' first reactions were rather sceptical in the sense of "Oh, they don't know anything about geometry...". Counterquestions followed: "Do our peasants use rectangles in their daily life?". "Do they construct rectangles?". Students from different parts of the country were asked to explain to their colleagues how their parents construct e.g. the rectangular bases of their houses. Essentially, two construction techniques are common: (a) In the first case, one starts by laying down on the floor two long bamboo sticks of equal length.

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Then these first two sticks are combined with two other sticks also of equal length, but normally shorter than the first ones.

/ Now the sticks are moved to form a closure of a quadrilateral.

One further adjusts the figure until the diagonals - measured with a rope become equally long. Then, where the sticks are now lying on the floor, lines are drawn and the building of the house can start.

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(b) In the second case, one starts with two ropes of equal length, that are tied together at their midpoints.

A bamboo stick, whose length is equal to that of the desired breadth of the house, is laid down on the floor and at its endpoints pins are hit into the ground. An endpoint of each of the ropes is tied to one of the pins.

Then the ropes are stretched and at the remaining two endpoints of the ropes, new pins are hit into the ground. These four pins determine the four vertices of the house to be built.

"Is it possible to formulate the geometrical knowledge, implicit in these construction techniques, into terms of an axiom?". "Which 'rectangle axiom' do they suggest?". Now the students arrive at the following two alternate 'rectangle axioms':

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GERDES

(a) D

D

C

C

L

{J< , then

if

A

A

B

B

i.e. if AD = BC, AB = DC and AC = BD, then A, B, C and D are right

angles. In other words, an equidiagonalparallelogramis a rectangle. (b) D

C

if A

D

C

, then B

A

7 u

1--1---f

B

i.e. if M = AC r BD and AM = BM = CM = DM, then A, B, C and D are

rightangles,AD = BC and AB = DC. In other words,an equisemidiagonal quadrilateralis a rectangle. "After all, our peasants know something about geometry", remarks a student. Another, more doubtful:"But these axioms are theorems,aren't they?".... This classroomsession leads to a more profoundunderstandingby the studentof the relationshipsbetweenexperience,the possiblechoices of axioms, betweenaxiomsand theoremsat the firststagesof alternateaxiomatic constructions.It preparesthe futureteachersfor discussionslater in their studyon whichmethodsof teachinggeometryseemto be the most appropriate in our cultural context. It contributes to cultural-mathematical confidence. 3.2. An Alternate Construction of Regular Polygons

Artisansin the north of Mozambiqueweave a funnelin the followingway. One starts by makinga squaremat ABCD, but does not finish it: with the

CULTURE AND GEOMETRICAL THINKING

D

G G. .

F E

..r._

.

-H *

145

C

.HH

T

A

B

strands in one direction (horizontal in our figure), the artisan advances only until the middle. Then, instead of introducing more horizontal strips, he interweaves the vertical strands on the right (between C and E) with those on the left (between F and D). In this way, the mat does not remain flat, but is transformed into a 'basket'. The center T goes downwards and becomes C=D

F=H

A

B

the vertex of the funnel. In order to guarantee a stable rim, its edges AB, BC, and AC are rectified with little branches. As a final result, the funnel has the form of a triangular pyramid. So far about this traditional production technique.27 We posed our students the following question: "What can we learn from this production technique?" "The square ABCD has been transformed into a triangular pyramid ABC.T, whose base ABC is an equilateral triangle. Maybe a method to construct an equilateral triangle?".... Some reacted sceptically: "A very clumsy method to do so .. .". Counterquestions: "Avoid overhasty conclusions! What was the objective of the artisan? What is our

PAULUS GERDES

146

C

A

C

B

A

B

objective?" "Can we simplify the artisans' method if we only want to construct an equilateral triangle?" "How to construct such a triangle out of a square of cardbord paper?" An answer to these questions is given in the following diagrams: C

D

D

C

/

K /

T

A

/T B

B

A

foldingthe diagonals

F

D

C /

I

\

\ /

/

I

/

T \

/

A

B

foldingFT

CULTURE AND GEOMETRICAL THINKING D'-~

147

C

)-9

B

B

A

join the trianglesDFT and CFT until C and D coincide, F goes up, T goes down D=C

A

A

B

B

fix the 'doubletirangle'DFT to the face ATC, e.g. with a paperclip

"Can this method be generalized?" "Starting with a regular octagon, how to transform it into a regular heptagonal pyramid?" "How to fold a regular octagon?" A8

F

^"^A^7~~~~~~~~~~~~

A^^^

A2

A16

A4

foldingthe diagonalsand FT

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PAULUS

GERDES

A8

1\

i

A6

A2

A5

A3

A4

F goes up, T goes down and A7 and A8 aproximate until they coincide

"How to transform the regular heptagonal pyramid into a regular hexagonal pyramid?" As 2"-gons are easy to fold (by doubling the central diagonals when one starts with a square) and each' time that the simplified 'funnel-method' is applied, the number of sides of a regular polygon (or of the regular polygonal base of a pyramid) decreases by 1, it can be concluded that all regular polygons can be constructed in this way.28 Once arrived at this point, it is possible to look back and ask: "Did we learn something from the artisans who weave funnels?" "Is it possible to construct a regular heptagon using only a ruler and a compass?" "Why not?". "And with our method?"

CULTURE AND GEOMETRICAL THINKING

149

"What are the advantages of our general method in relation to the standard Euclidean ruler and compass constructions?". "What are its disadvantages?". "Which method has to be preferred for our primary schools?" "Why?" 3.3. From Woven Buttons to the 'Theorem of Pythagoras'29 By pulling a little lassoo around a square-woven button, it is possible to fasten the top of a basket, as is commonly done in southern parts of Mozambique (see photo 1). The square button, woven out of two strips, hides some remarkable geometrical and physical considerations. By making them explicit, the interest in this old technique is already revived. But much more can be made out of it, as will now be shown. When one considers the square-woven button from above, one observes the following pattern:

or after rectifying the slightly curved lines and by making the hidden lines visible:

In its middle there appears a second square. Which other squares can be observed, when one joins some of these square-woven buttons together? Do there appear other figures with the same area as (the top of) a square-woven

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PAULUS GERDES

button? Yes, if you like, you may extend some of the line segments or rub out some others.

What do you observe? Equality in areas?

CULTURE AND GEOMETRICAL THINKING

151

A

Hence A = B + C:

i.e. one arrivesat the so-called'Theoremof Pythagoras'. The teacher-studentsrediscoverthemselvesthis importanttheoremand succeedin proving it. One of them remarks:"Had Pythagoras- or somebody else beforehim - not discoveredthis theorem,we would have discovered it".... Exactly!By not only makingexplicitthe geometricalthinking 'culturallyfrozen' in the square-wovenbuttons, but by exploiting it, by

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PAULUS GERDES

revealingits full potential, one stimulatesthe developmentof the abovementionednecessarycultural-mathematical (self)confidence. "HadPythagorasnot... wewouldhavediscoveredit". The debatestarts. "Couldour ancestorshave discoveredthe 'Theoremof Pythagoras'?""Did they?"... "Why don't we know it?"....

"Slavery, colonialism...".

By

'defrostingfrozenmathematicalthinking'one stimulatesa reflectionon the impact of colonialism,on the historicaland politicaldimensionsof mathematics (education). 3.4. From TraditionalFishtraps to Alternate CircularFunctions, Football and the Generation of (Semi)regular Polyhedra

Mozambicanpeasantsweavetheirlighttransportationbaskets'Litenga'and fishermentheir traps 'Lema'(see photo 2) with a patternof regularhexagonal holes. One way to discoverthis patternis the following. How can one fasten a borderto the walls of a basket,when both border and wall aremadeout of the samematerial?How to wrapa wallstriparound the borderstrip?

What happenswhen one presses(horizontally)the wallstrip?What is the best initial angle betweenthe border-and wallstrip?In the case that both stripshave the same width,one findsthat the optimalinitialanglemeasures 60?. By joining more wallstripsin the sameway and then introducingmore horizontalstrips,one gets the 'Litenga'patternof regularhexagonalholes.

CULTURE AND GEOMETRICAL THINKING

153

By this process of rediscovering the mathematical thinking hidden in these baskets and fishtraps - and in other traditional production techniques - our future teachers feel themselves stimulated to reconsider the value of our cultural heritage: in fact, geometrical thinking was not and is not alien to our culture. But more than that. This "unfreezingof culturallyfrozen mathematics" can serve, in many ways, as a startingpoint and source of inspiration for doing and elaborating other interesting mathematics. In the concrete case of this hexagonal-weaving-pattern, for example, the following sets of geometrical ideas can be developed. a. Tilingpatternsand theformulation of conjectures.Regular hexagonal and other related tiling patterns can be discovered by the students.

\

/,x

^

&

hexagonal

8^~~

triangular

rhombic

pentagonal

154

PAULUS GERDES

With the so-found equilateral triangle, many other polygons can be built. By considering these figures, general conjectures can be formulated, e.g. the sum of the measures of the internal angles of a n -gon is equal to 3(n - 2) 60?.

*

/ \

A

\J

/

/ 'I

areas of similar figures are proportional to the squares of their sides.

*

r\ 5

A=1

s=l,

1=1 C*

s= 2,

A =4

1+3=4

s=3,

A =9

1+3+5=9

the sum of the first n odd numbers is n2.

Once these general theorems are conjectured, there arises the question of justifying, how to prove them.

CULTURE AND GEOMETRICAL THINKING

155

b. An alternate circular function. Let us return to the weaving of these 'Litenga' baskets. What happens when the 'horizontal' and 'standing' strips are of different width, e.g. 1 (unity of measurement) and a?

One finds a semi-regular hexagonal pattern. How does the optimal angle a depend on a? a = hex(a)

How does a vary? Both a and a can be measured. One finds

I I I I I I I I I I I i

2

1

0?

30?

60?

90?

We have here a culturally integrated way to introduce a circular function. After the study of the 'normal' trigonometric functions, their relationships can be easily established, e.g. a = hex-'(a)

=

2

2 cos a

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PAULUS

GERDES

c. Footballsandpolyhedra.The faces and edges of the 'Lema'fishtrapdisAt its verticesthe situationis differplay the regular-hexagonal-hole-pattern. ent. The artisansdiscoveredthat in orderto be able to constructthe trap, 'curving'the faces at its vertices,it is necessary,e.g. at the verticesA, B and C to reducethe numberof strips.At thesepoints, the six stripsthat 'circumscribe'a hexagon,have to be reducedto five. That is why one encountersat these verticeslittle pentagonalholes.

600

120

can be woven, that display at all their vertices pentagonal holes?

It comes out that the smallestpossible'basket',made out of six strips,is similar to the wellknown modern football made out of pentagonal and hexagonalpieces of leather.

woven ball

CULTURE AND GEOMETRICAL THINKING

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football

When one 'planes' this ball, one gets a truncated icosahedron, bounded by 20 regular hexagons and 12 regular pentagons. By extending these 20 hexagons, one generates the regular icosahedron. On the other hand, when one extends the 12 pentagons, the regular dodecahedronis produced. What type of 'baskets' can be woven, if one augments their 'curvature'? Instead of pentagonally woven 'vertices', there arise square-hole-vertices. By planing the smallest possible 'ball', one gets a truncated octahedron, bounded by 6 squares and 8 regular hexagons. Once again, by extension of

woventruncatedicosahedron

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PAULUS

GERDES

truncated icosahedron

its faces, new regular polyhedra are discovered, this time, the cube and the regular octahedron.When one augments still more the curvature of the 'ball', there appear triangular-hole-vertices and by 'planing' the 'ball', one gets a truncated tetrahedron, bounded by 4 regular hexagons and 4 equilateral triangles. By extension of its hexagonal or triangular faces one obtains a regular tetrahedron.

Photo 1

basket with square-woven button

CULTURE

AND GEOMETRICAL

THINKING

159

iw

0. e0

E i-i r4 0 10

&u

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PAULUS GERDES

Many interestingquestionscan be posed to futureteachers,e.g. - is it possibleto 'weave'other semi-regularpolyhedra?Semi-regular,in what sense? - did we generateall regularpolyhedra?Why? - what happensif one, insteadof reducingthe materialat a vertexof the basket, augmentsit? 4. CONCLUDING REMARKS

and the combatof Of the struggleagainst'mathematicalunderdevelopment' racial and (neo)colonial prejudice,a cultural-mathematical reaffirmation makesa part.A 'culturalconscientialization'of futuremathematicsteachers, e.g. in the way we described,seems indispensable. Some other conditions and strategies for mathematicseducation to become emancipatory in former colonized and (therefore) underdeveloped

countrieshave been suggestedelsewhere.30 ACKNOWLEDGMENTS

The authoris gratefulto Dr. A. J. Bishop(Cambridge)for his invitationto write this articleand to Dr. W. Humbane(Maputo) for proofreadingthis paper. NOTES * Thisarticleis dedicatedto SamoraMachel,the belovedPresidentof the People'sRepublicof Mozambique,eternalsourceof inspiration,who died on the 19th October1986,the day of conclusionof our article. "Colonizationis the greatestdestroyerof culturethat humanityhas ever known.... . .. long-suppressed manifestationsof culturehave to regaintheirplace. ." (SamoraMachel, 1978). "Educationmust give us a Mozambicanpersonalitywhich,withoutsubservienceof any kind and steepedin our own realities,will be able, in contactwith the outsideworld,to assimilate criticallythe ideas and experiencesof otherpeoples,also passingon to themthe fruitsof our thoughtand practice"(SamoraMachel, 1970). Cf. e.g. Eshiwani(1979), Nebres(1983) and El Tom (1984). 2 Broomesand Kuperes(1983, p. 709). 3 El Tom, 1984,p. 3. 4 D'Ambrosio(1983, p. 363). 5 Gay and Cole (1967, p. 6). 6 Gay and Cole (1967, p. 66). 7 Luna (1983, p. 4).

8 Carrahera.o. (1982). 9 D'Ambrosio(1984, p. 6). Cf. D'Ambrosio(1985b). 10 D'Ambrosio (1984, p. 6), italicsP. G.

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1 D'Ambrosio(1984, p. 8), italicsP. G.

12 D'Ambrosio (1984, p. 7). 13 Pinxten (1983, p. 173). 14

D'Ambrosio:spontaneous;Carrahera.o.: natural;Posner,Aschera.o.: informal;Gay and Cole: indigenous;Mellin-Olsen:folk-: Ascher and Ascher: implicit; Carrahera.o.: nonstandard;Gerdes:hidden,'frozen';D'Ambrosioa.o.: ethno-. I5 Gay and Cole (1967, p. 94). 16

7 18

Mmari (1978, p. 313).

Quotedby Nebres(1984, p. 4). D'Ambrosio (1984, p. 10).

19 D'Ambrosio (1985a, p. 47).

20

21 22 23

Machel (1978, p. 401). Machel (1978, p. 402). Cf. Gerdes (1982, 1985a). D'Ambrosio (1985, p. 45).

Ascherand Ascher(1981, p. 159),italicsP. G.; cf. Gerdes(1985b,?2). The firstresultsaresummarizedin Gerdes(1985b).Cf. Gerdes(1986a,f).By bringingto the surfacegeometricalthinkingthat was hiddenin very old productiontechniques,like that of basketry,we succeededin formulatingnew hypotheseson how the ancient Egyptiansand Mesopotamianscould have discoveredtheir formulasfor the area of a circle [cf. Gerdes (1985b,c, 1986d)]and for the volumeof a truncatedpyramid[cf. Gerdes(1985b)].It proved possibleto formulatenewhypotheseson how the so-called'Theoremof Pythagoras'couldhave been discovered[cf. Gerdes(1985b, 1986c,e)]. 26 Experimental coursedevelopedfor secondaryschoolsin the USSR (1981)by a teamdirected by the academicianA. Alexandrov. 27 The implicitgeometricalknowledgethat it reveals,is analyzedin Gerdes(1985b). 28 For more details,see Gerdes(1986b). 29 Another'culturallyintegrated'introduction to the 'Theoremof Pythagoras'is presentedin Gerdes(1986c, g). 30 Cf. e.g. Gerdes(1985a, 1986a),D'Ambrosio(1985b)and Mellin-Olsen(1986). 24

25

REFERENCES curriculain thepasttwo decades: D'Ambrosio,U.: 1983,'Successesandfailuresof mathematics A developingsocietyviewpointin a holisticframework',in Proceedingsof theFourthInternational Congress of Mathematical Education, Boston, pp. 362-364.

transmissionof mathematicalknowledge:Effectson D'Ambrosio,U.: 1984,'The intercultural mathematicaleducation',UNICAMP,Campinas. and its place in the history and pedagogy of D'Ambrosio,U.: 1985a, 'Ethnomathematics mathematics',in For the Learningof Mathematics,Montreal,Vol. p. 5, no. 1, pp. 44-48. D'Ambrosio, U.: 1985b, Socio-cultural Bases for Mathematics Education, UNICAMP, Camp-

inas. Ascher, M. and R. Ascher: 1981, Code of the Quipu. A Study in Media, Mathematics, and

Culture,Universityof MichiganPress,Ann Arbor. Broomes,D. and P. Kuperes:1983,'Problemsof definingthe mathematicscurriculumin rural communities', in Proceedings of the Fourth InternationalCongress of Mathematical Educa-

tion,Boston,pp. 708-711. Carraher,T., D. Carraher,andA. Schliemann:1982,'Na vida,dez,na escola,zero:os contextos de matematica',in Cadernosdepesquisa,Sao Paulo,Vol. 42, pp. culturaisda aprendizagem 79-86.

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El Tom, M.: 1984,'Theroleof ThirdWorldUniversityMathematicsInstitutionsin promoting mathematics',Adelaide. Eshiwani,G.: 1979,'Thegoals of mathematicsteachingin Africa:A need for re-examination', in Prospects,Paris,Vol. IX, no. 3, pp. 346-352. Gay, J. and M. Cole: 1967, The New Mathematics and An Old Culture: A Study of Learning Among the Kpelle of Liberia, Holt, Rinehart and Winston, New York.

Gerdes,P.: 1982,'Mathematicsfor the benefitof the people',CARIMATHS,Paramaribo. mathematicseducationin underGerdes,P.: 1985a,'Conditionsandstrategiesfor emancipatory developedcountries',in For theLearningof Mathematics,Montreal,Vol. 5, no. 1, pp. 15-20. Gerdes, P.: 1985b, Zum erwachendengeometrischen Denken, Eduardo Mondlane University,

Maputo. Gerdes,P.: 1985c,'Threealternatemethodsof obtainingthe ancientEgyptianformulafor the area of a circle',in HistoriaMathematica,New York, Vol. 12, pp. 261-268. Gerdes,P.: 1986a,'Onculture,mathematicsand curriculumdevelopmentin Mozambique',in Mellin-Olsenand JohnsenHoines,pp. 15-42. Gerdes,P.: 1986b,'Ummetodogeralparaconstruirpoligonosregulares,inspiradonumatecnica mo9ambicanade entrela9amento', TLANU-booklet,Maputo,no. 4. Gerdes, P.: 1986c, 'A widespreaddecorativemotif and the Pythagoreantheorem',For the Learning of Mathematics, Montreal (in press).

Gerdes, P.: 1986d, 'Hypothesenzur Entdeckungdes altmesopotamischen Naherungswertes pi = 38',TLANU-preprint,Maputo,no. 1986-4. Gerdes,P.: 1986e,'Did ancientEgyptianartisansknow how to finda squareequalin areato two given squares?',TLANU-preprint,Maputo,no. 1986-5 Gerdes,P.: 1986f,'How to recognizehiddengeometricalthinking?A contributionto the develmathematics',in For theLearningof Mathematics, opmentof anthropological Montreal,Vol. 6, no. 2, pp. 10-12, 17. Gerdes,P. and H. Meyer:1986g,'Pythagoras,einmalanders',Alpha,Berlin(in press). Luna, E.: 1983,'Analisiscurriculary contextosociocultural',Santiago. Machel,S. 1970,'Educateman to win the war,createa new societyand developour country', in Mozambique, Sowing the Seeds of Revolution, Zimbabwe Publishing House, Harare, 1981,

pp. 33-41. Machel,S.: 1978,'Knowledgeandscienceshouldbe for the totalliberationof man',in Raceana Class,Vol. XIX, no. 4, pp. 399-404. Mellin-Olsen, S. and M. J. H0ines: 1986, Mathematics and Culture. A Seminar Report, Caspar

Forlag, Radal. Mmari,G. 1978,'TheUnitedRepublicof Tanzania:Mathematicsfor socialtransformation', in F. Swetz (ed.), Socialist Mathematics Education, Burgundy Press, Southampton.

Nebres,B. 1983,'Problemsof mathematicaleducationin and for changingsocieties:problems in SoutheastAsian countries',Tokyo. Nebres, B.: 1984,'The problemof universalmathematicseducationin developingcountries', Adelaide. Pinxten, P. I. van Dooren and F. Harvey: 1983, The Anthropology of Space. Explorations into the Natural Philosophy and Semantics of the Navajo, University of Pennsylvania Press,

Philadelphia.

Faculty of Mathematics, Eduardo Mondlane University, C.P. 257, Maputo, Mozambique.

NORMA C. PRESMEG

SCHOOL MATHEMATICS

IN CULTURE-CONFLICT

SITUATIONS Towards a Mathematics Curriculum for Mutual Understanding when Diverse Cultures Come Together in the Same Classroom' ABSTRACT. In times of cultural change, education plays an especially important role. The writer suggests that even mathematics curricula, which have traditionally been considered culture-free, have a role to play in fostering mutual understanding amongst members of different cultures, after a period of cultural upheaval. Anthropological and educational sources are used to suggest points of relevance when a mathematical curriculum is designed for multi-cultural classrooms.

As indicated by the word "towards" in the subtitle, the writer does not regard this paper as providing ultimate solutions to problems which are extremely complex. Writing, as she is, about South African cultures and sub-cultures all of which are in a state of intense ferment at this time, she is aware of complexities and contradictions which are implicit in some of the issues (and some of which are not confined to South African society). The analysis of elements which may contribute to a mathematics curriculum for mutual understanding is offered as a first approximation to an answer to the question of how pupils from diverse cultures can best learn mathematics together after a period of cultural change.

A PERIOD OF INTENSE

CULTURAL

FERMENT

IN SOUTH AFRICA

University of Durban-Westville (U D-W) is an institution which, in line with the trend at all South African universities, is increasingly opening its doors to students of all races. At present there are approximately 6000 students, the majority (86%) of whom still are Indians. The remaining 14% are predominantly black students. 40% of the academic staff of 425 are Indians, almost all of the remainder being White. Of the total staff complement of more than 1000, about 70% are Indians. "Dr Presmeg, what must I do?" Yusuf, a fourth-year mathematics education student, stood in the doorway of the writer's office at U D-W with a piece of paper in each hand. One piece of paper was a police notice prohibiting unlawful gatherings on the campus of U D-W in terms of the state of emergency regulations; the other was a notice from the Students' Representative Council calling all students Educational Studies in Mathematics 19 (1988) 163-177. ? 1988 by Kluwer Academic Publishers.

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NORMA C. PRESMEG

to a mass protestmeeting.The studentswerenewlyreturnedto campusafter previous boycotts and disruptionshad resultedin the early closing of the universityfor the Julyholiday.(The academicyearendsin December.)Some studentleaderswere still in detention. Yusufdid not reallyhavea choice.He attendedthe meeting,whichin spite of a heavy securityforce presencedid not end in violence this time. As another studentinformedthe writer,at a previousmass meetingto decide whetherthe studentswould boycott lectures,one studentspoke againstthe boycott. A studentleadersaid, "Don't listen to him. We'll deal with him. Let'sget on with it!"No studentspokeagainstthe boycott afterthat. Voting was by a show of hands. Studentswho attended lectures that afternoon (some of whom were part-timerswho did not know of the decision)were forciblyremovedfrom some lecturerooms by intimidators. Yusuf's dilemmaillustratesthe conflict situation in which not only students but black pupils in many of the townshipsfind themselves.Burning and boycott of schools which are seen as symbolsof white dominationare actions which are encapsulatedin the slogan, "Liberationfirst, education later!"But the vocabularyis changingas fast as it emerges(Gangat, 1986), and this slogan, popularin 1985,has alreadybeen supercededby "People's educationfor people'spower!"The term"people'seducation"also seemsto have supercededthe term "alternativeeducation",which has been used in the last few years to describeforms of oppositionalor counter-hegemonic educationalprogrammesor approaches(Walters, 1986). It is beyond the scope of this paperto documentall the social, political, economicand ideologicalforcesat work in SouthAfricansocietytoday, but the foregoingservesto illustratethreepoints, as follows. (1) The social, culturaland ideologicalchangeswhich are taking place amount to an ongoing revolution. (2) Education is seen as "a prime catalyst in and towards change" (Gandat, 1986). (3) The emergentculturesas well as those they are replacing,are fraught with contradictions. The events on the campus of U D-W illustratea phenomenonwhich is widespreadin counter-hegemonicstrategiesin South Africa at present.In the protest againstlack of true democracy,methodsare employedwhichin fact perpetuatethis lack of democracy.It appearsthat culturalchangeis not a rational,logicalprocess.Indeed,some of its elementsmay not be conscious at all. Apple (1982, p. 24) wroteof the "contradictions,conflicts,mediations and especiallyresistances"whichhe foundin Americanschool sub-cultures. Negotiationand conflictare aspectsof culturalhegamony.Ironically,in the

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165

rejectionof mentallabourby workingclassboys,Applefoundthat"Theseeds of reproductionlie in this very rejection"(ibid., p. 99). It is a danger,then, in allculturalchange,thatunwantedelementsof a formerhegemonicideology may be reproducedunwittingly.One mastermay simply be exchangedfor another.However,the formationof ideologiesis not a simpleact of imposition. "It is producedby concreteactorsand embeddedin lived experiences that may resist,alter or mediatethese social messages"(ibid., p. 159). It may be askedwhat role mathematicseducationcan, or should, play in the reproductionof ideologies.The political,social and economic environmentin SouthAfricatodayis highlysensitive(Natal Teachers'Society,1986). It seemsto the writerthatmathematicseducationin SouthAfricain the times aheadcan play a positive role towardshealingrifts and bitternessesand in promotingunderstandingand toleranceof culturaldifferences. Writingin andabout Brazilduringa timeof socialtransition,Freirewrote: The time of transitioninvolvesa rapidmovementin searchof new themesand tasks. In such a phase,man needsmorethanever to be integratedwithhis reality.If he lacksthe capacityto perceivethe 'mystery'of thechanges,he willbe a merepawnat theirmercy(Walters,1986,p. 2).

The need to be "integratedwith his reality",particularlyduring cultural dislocationsor periods of intense social change, is discussedin a general anthropologicalcontext in the next section, while ways of achieving this integrationin the contextof mathematicseducationare suggestedin the final section. With Bishop (1985e) and Berry (1985), the writer is convinced that a mathematicscurriculumwhich is experiencedas real by a pupil can be developedonly by adultswho belongto the sameculturalgroupas the pupil. Thisaspectis complicatedby the multiplicityof culturesin SouthAfrica,but the writerbelievesthat thisis a goal whichis capableof realisation.However, the necessaryunderstandingand tolerancecannot be learnedin separate educationalsystems,or indeed in separateclassrooms.To gain this understandingand tolerance,childrenfrom all culturalgroupswill need to come togetherin the same classroomsin the future. Under these conditions, a mathematicscurriculumdesignedby a groupof people representativeof all culturesinvolvedwouldhavea positiveroleto playin promotingunderstanding and tolerance. SOME PROBLEMS OF ACCULTURATION: LIVING IN TWO WORLDS

In this sectionare discussedsome relevantaspectsof the dislocationwhich may occur when a "Western"school culturedoes not resonatewith the home cultureof pupils. This problemis well documented(Spindler,1974),

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and has severalimplicationsfor a mathematicscurriculumfor mutualunderstanding,which are listed at the end of the section. As Singleton(1974, p. 28) pointed out, cultureencompasses"patternsof meaning,reality,values,actionsand decision-makingthat aresharedby and withinsocial collectivities".All these patternsare relevantin the learningof mathematics(Bishop, 1985a,c,d).Culturaltransmissionincludesboth the transmissionof traditionfrom one generationto the next and the transmission of new knowledgeand culturalpatternsfrom anyonewho "knows"to anyonewho does not. This distinctionunderliesthat betweenenculturation, "the processof generationalcontinuity"and acculturation,"the processof individualand group change, caused by contact with variousculturalsystems" (Singleton, op. cit., p. 28). It is the dynamicaspectsof acculturation which are relevantin multi-culturalclassrooms. In the Englishpublicschool tradition,HiltonCollegein Natal is a private, boys' residential school which usually provides finalists in the national MathematicsOlympiads.Hilton admitsboys of all races.One of the mathematics teachersat Hilton told the writerthat black pupilsleave by train at the start of the school holidays wearingtheir school uniforms,but change theirattirebeforearrivingat theirdestinations.Thisactionis symbolicof the "livingin two worlds"which may be experiencedby such pupilsas cultural dislocation- but not necessarily.In the following case studies of school acculturation,elements are identifiedwhich shed light on the question of why, and underwhat conditions,dislocationis not inevitable. The establishmentof Westernschools,especiallyboardingschools,and curriculain non-Western societiesis likelyto constitutean extremetype of culturaldiscontinuityand may do much to force 'either-or'choiceson theirlearners(DuBois in Sindell,1974,p. 333).

This extremetype of culturaldiscontinuitywas experiencedby Mistassini Creechildrenwho left theirhomes at age fiveor six to attendthe residential school at La Tuquein Canada.Sindell(1974)describedmany aspectsof the culturaldislocationof these children.In the field of interpersonalrelations, they learnedthat dependentbehavioursuchas cryingwas effectivein gaining an adult's attention.At home, self-relianceand independencewere valued and thereforecryingwas ignoredfrom an earlyage and childrenlearnednot to cry. In school the childrenalso learnedthat the smallestscratchwould elicit concern, thus contradictingtheir early trainingin silently enduring pain.

In the case of the MistassiniCree,preschoolchildrenlearnedbehavioural patternsand values whichwere highlyfunctionalfor participatingas adults in the traditionalhunting-trappinglife of theirparents.

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Because they must go to school, their development into trappers or wives of trappers is arrested. Prolonged residential school experience makes it difficult if not impossible for children to participate effectively in the hunting-trapping life of their parents. Not only do they fail to learn the necessary technical skills, but they acquire new needs and aspirations which cannot be satisfied on the trapline. Yet most Mistassini parents want their children to return to the bush. It remains to be seen how the students will resolve their dilemma (ibid., pp. 340-1).

The extreme cultural dislocation and "either-or" choices facing Mistassini Cree children have also been documented for children in roughly similar circumstances in Papua New Guinea (Lancy, 1983), and for Hopi children during certain periods of their history (Eggan, 1974). However, discontinuity is experienced in a less extreme form, if at all, if the schooling is seen as relevant in the pupils' future without either-or decisions having to be made. This point is illustrated in the self-perceptions of Sisala pupils in Northern Ghana (Grindal, 1974). The distinction between the "traditional" and "modern" sectors of African life need not be perceived by the actors as a dichotomy of two worlds if they can simultaneously embody the continuity and values of the traditional society and the changes brought by colonization and modernization. This conception is illustrated in the apparent ease with which African tribal leaders such as Chief Mangosuthu Buthelezi of the Zulus switch back and forth from traditional roles amongst their people, to various roles at national and international levels as representatives of their people. One significant aspect of the problem of "two worlds" is implied in the following exchange between two new young teachers in charge of a village school among the Ngoni of Malawi, and a senior chief: The teachers bent one knee as they gave him the customary greeting, waiting in silence until he spoke. 'How is your school?' 'The classes are full and the children are learning well, Inkosi.' 'How do they behave?' 'Like Ingoni children, Inkosi.' 'What do they learn?' 'They learn reading, writing, arithmetic, scripture, geography and drill, Inkosi.' 'Is that education?' 'It is education, Inkosi.' 'No! No! No! Education is very broad, very deep. It is not only in books, it is learning how to live. I am an old man now. When I was a boy I went with the Ngoni army against the Bemba. Then the mission came and I went to school. I became a teacher. Then I was chief. Then the government came. I have seen our country change, and now there are many schools and many young men go away to work to find money. I tell you that Ngoni children must learn how to live and how to build up our land, not only to work and earn money. Do you hear?' 'Yebo, Inkosi' (Yes, 0 Chief) (Spindler, 1974, p. 308).

In this conversation the chief also pointed to the general shape of the solution to the problem of "two worlds", in his referenceto education as learning how to live.

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Margaret Mead made an important point when she wrote that children need the stability of the cultural heritage, especially when their society is marked by rapid change (Nash, 1974). At such a time, adults may also learn from their children (and not only children from adults) in what Mead called prefigurative enculturation. These points are illustrated in one more case study, set in Sch6nhausen, an urbanising German village in the Remstal. Incidentally, in its ethos, the Grundschule in Sch6nhausen, in the Federal Republic of Germany, bears a striking resemblance to Izotsha Primary School in Natal as it was a few decades ago, even to the nature excursions which took place ("Wanderungen"), disciplinary use of the "Ohrfeige" (cuff), and use of the German language (now an alternative medium of instruction only in the first two years of schooling at Izotsha). The Izotsha school was established more than a century ago by German settlers in Natal, the descendants of whom are predominantly sugar cane farmers in the area. In the years between 1945 and 1968, Sch6nhausen almost doubled its population (from 1300 to 2500) as migrants arrived from what was the east zone or from the outlying prewar German minorities, or from other parts of Western Germany. From being a region almost entirely devoted to the cultivation of wine grapes and subsistence farming, the whole area moved rapidly towards urbanisation and industrialisation. The importance of the case study of the four-year Grundschule in Sch6nhausen for this paper lies in the fact that although the divergencies in backgrounds between natives and newcomers set the scene for potentially explosive confrontations, these confrontations did not occur. Spindler (1974, p. 233) wrote as follows: Particularlyintriguingis the fact that on the whole this greatinfluxof diverse(thoughGermanic)populationwas assimilatedwithout any apparentdisturbance.The low incidenceof crime,suicideandjuveniledelinquencysuggeststhat therehas been no substantialincreasein the socialandpsychologicalills thatoftenaccompanyrapidurbanisation.Schonhausenandthe areaaroundit give everyappearanceof socialand economichealth.

According to Spindler's ethnography, Heimatkunde (learning about the homeland) and Naturkunde (learning about the land and nature) were important components of the Grundschule curriculum. In an atmosphere of freedom and exploration, during a six-and-a-half-hour Wanderung the pupils covered a distance of nearly eighteen kilometres and discussed elements of the forests, meadows and waterways as well as the new apartment buildings and industries plainly identifiable in the valley. On another occasion, the children were taught about the history of the four great bells in the tower of the local Protestant church, after which they walked across to the church to experience the bells first-hand. The Wanderung and the bells illustrate the role that knowledge of local culture and

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history can play in fostering stability and an understanding of change. As the Schonhausen children grew older they increasingly chose to follow the "easier" life of modern apartments and fixed salaries; but one gains the impression that the changes did not usually involve rejection, but ongoing appreciation of the old culture. From the foregoing, several relevant points emerge. For a mathematics curriculum for mutual understanding when diverse cultures come together, the following points appear to be of importance: (1) Children need the stability of their cultural heritage, especially during periods of rapid social change. (2) The mathematics curriculum should incorporate elements of the cultural histories of all the people of the region. (3) The mathematics curriculum should be experienced as "real" by all children, and should resonate, as far as possible, with diverse home cultures. (4) The mathematics curriculum should be seen by pupils as relevant to their future lives. CHANGING VIEWS OF THE MATHEMATICS CURRICULUM

As Howson, Keitel and Kilpatrick (1982) point out, curriculum means more than the syllabus: it must encompass aims, content, methods and assessment procedures - and curriculum reform is never completed. Amongst the pressures that serve to initiate curriculum development they list societal and political pressures, mathematical and educational pressures. In common with most Western countries, pressures in South Africa for curricularreform in mathematics resulted in the introduction of "modern mathematics" in the 1960s and more recent curriculum reforms which are still in the process of being implemented. In the future, however, the curriculum changes will be far more fundamental and far-reaching as first and third worlds attempt to come to terms with mathematics in the same classrooms. The magnitude of the issues is realised by some mathematics educators in South Africa at present. At a workshop open to all to discuss the subject, "People's Education and the Role of Mathematics", it was stated that "underlying all [the] debates are questions concerning the kind of society envisaged in the medium and long term future, and the role of mathematics education in achieving these goals. These are enormous issues, and the most that a workshop of this nature can achieve is to raise some of the main questions" (Taylor et al., 1986). The workshop was seen as an early step in a long process. The political aspects of the "democratisation of knowledge"

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are focal issues at present, but the writer is looking beyond these issues (important as these are) to what she sees as an essential stage, inevitable if growth and stability are goals, in which healing of prejudices, rifts and bitternesses must take place. It has been pointed out (Howson et al., 1982) that it is possible for teachers to teach a new curriculum as they taught the old, thereby undermining the intentions of the curriculum developers. The actions of mathematics teachers in the classroom are predicated on their beliefs about the nature of mathematics and the teaching and learning of mathematics (Cooney, 1984; Cooney et al., 1985). In-service and pre-service courses for mathematics teachers would therefore be important components in the implementation of any radically new mathematics curriculum. Furthermore, in all societies prejudiced beliefs about different cultures may be deep-seated or unconscious (Reynolds and Reynolds, 1974). However, studies of anti-semitic prejudice following World War II provided research evidence that if people can be induced to change their actions, then changes in their belief-systems in line with these actions are likely to follow (Selltiz et al., 1963). In order to reify the values of tolerance and mutual understanding, these values would necessarily be conscious goals on the part of people of all cultures in the classroom. It is in the philosophical assumptions underlying perceptions by mathematics educators of the nature of mathematics, that changes in world thinking have been taking place. It might be asked to what extent it is meaningful to speak of cultural mathematics. Several writers have argued convincingly that mathematics is not culture-free, but culture-bound (Bishop, 1985b,e; Breen, 1986; D'Ambrosio, 1984; Fasheh, 1982; Gerdes, 1985). The traditional view is that the propositions of mathematics are absolute and transcend questions of culture. After all, six plus six must equal twelve in any culture! But even this statement may be called into question. In Papua New Guinea there are more than 700 different languages (not dialects) and the counting systems may be classified into four main types only one of which includes our familiar base ten system (Lancy, 1977). If one's counting system goes 1, 2, 3, 4, many, or numerals are attached to various parts of the body, or the system changes according to what one is counting, then 6 + 6 = 12 as an absolute truth, becomes meaningless. However, the view that mathematics is culture-bound is probably too limited if stated baldly, like this, without qualification. Mathematicians from diverse cultural backgrounds have no difficulty in understanding each other. Indeed, this point is illustrated strikingly in the account of G. H. Hardy's discovery of the mathematical genius of Ramanujan. Ramanujan was a poor

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clerk from Madras who had not been able to enter Madras University because "he could not matriculate in English" (Hardy, 1979, p. 35). He was brought to Cambridge by Hardy, who recognised from his manuscripts that Ramanujan was, in terms of natural mathematical genius, in the class of Gauss and Euler. In his 1933 Spencer lecture at Oxford, Einstein said, Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics (Einstein, 1973, p. 274).

There is a universal element in the principle to which Einstein refers here. This element is present, too, in the curtailment, generalisation and logical economy evident in the mathematical thinking of Krutetskii's (1976) "capable" pupils. The universal or absolute aspect of mathematics must be available to those pupils of all cultures who can master it. In plural South African society it must be possible for future Ramanujans to learn the mathematics which will enable them to make their contributions. Perhaps the schools of the future will teach such mathematics in special groups to those pupils who desire it or who require it for their chosen vocations. The rigour of mathematical concepts needed in what Bishop (1985e) has called mathematico-technological (MT) culture could be developed gradually from the concepts of a cultural mathematics curriculum. For all pupils, cultural mathematics as a basic groundwork could provide a core curriculum which is meaningful in their reality. Some of the problems and possibilities of such a core curriculum are addressed in the following section. ELEMENTS OF A CURRICULUM FOR MUTUAL UNDERSTANDING

By virtue of the fact that they would be sharing a common school experience (first section), not all aspects of the realities of pupils from diverse cultural backgrounds would be different. Lawton (1975, p. 5) wrote appositely in this regard: One view is that a common curriculum must be derived from a common culture. But this in turn raises other difficult issues. What is meant by a common culture? Is it meaningful to talk of a common culture in a pluralistic society?

Lawton came to the conclusion that there were no convincing arguments against the existence of some elements of a common culture in English society - sufficient at least as a basis for a common curriculum.

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In the much more diverse South African context, it might be useful to ask, further, to what extent it is possible for cultures to combine. An anecdote is appropriate in this regard. The writer's children, who live on a small farm in Natal, frequently play with the black farmworker's children who live on the property. One day when the children were playing together, they found in the veld (grassland) a stray domestic cat, which had obviously been living wild. The black children indicated that this find was "ukudla, nyama" (food, meat); the white children indicated, no, and demonstrated that the cat was to be petted and stroked. The cat, tough from the ways of the wild, was named "Tiger" and became a family pet. It is suggested that individual elements of cultures cannot combine: one either pets the cat or one eats it. But the writer also suggests that social interchange permits and promotes understanding and tolerance of ways that are different. If these differences can be aired naturally in mathematics classrooms, in a non-evaluative atmosphere, mutual understanding may be facilitated especially among children, whose patterns of thought are more pliable than those of their parents (Spindler, 1974). It is inevitable that new cultural forms will evolve from the old in this "melting pot" experience, and in this sense cultures will grow closer to one another. It is possible to observe this growing together of cultures in language changes in South African townships. Nundkumar (1985) described graphically the "changing face of English" in the Indian township of Phoenix, near Durban. He quoted a poem written in "Township lingo", a verse of which is as follows: This is poetry,brazo,and if you think I'm 'g'-ingyou, readit, Twaaiit lucker,I come fromthe townshipmei brew, An' I can't skryf or choon anythingexceptin the townshipstyle. Don't worryabout the lingo bhai, the poem is more important.

English, Afrikaans, Hindi and American terms appear in the extract. Vital influenceson 'Townshiplingo' are televisionand films,especiallythose depictingBlack Americans.At the momentthereis a crazefor imitatingBlackAmericans- theirstyleof dress, the way they walk and talk. The 'Townshiplingo' is not restrictedto the IndianTownship. Thereis a constantdiffusionof wordsand phrasesfrom the othercommunities,especiallythe Blackcommunities.WhenBlackstalk of a friendtheycan him 'bra'(short for brother).In the Indiantownshippeopletalk of a 'bra'but use the term 'brazo'morefrequently(ibid., p. 69).

It is likely that the future lingua franca of South Africa will be English, because this is the only South African language which has international currency. The issue of language is important in the learning of mathematics because it is possible that the structure of the learner's mother tongue has a strong influence on mathematical cognitive processes such as classification and recognition of equivalences and relationships (Gay and Cole, 1967;

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Glick, 1974; Bishop, 1983, Lancy, 1983; Berry, 1985). In his illuminating article, Berry (1985) classified language-related mathematical learning problems into two types. Type A problems occur typically when the language of instruction is not the student's mother tongue. "Remedial strategies are linguistic, not mathematical-the treatment is to improve the learner's fluency in the instructional language" (ibid., p. 20). Type B problems result from the "distance" between the cognitive structures natural to the learner and implicit in his mother tongue and culture, and those assumed by the teacher or curriculum designer. Type B problems may occur among unilinguals being taught in their mother tongue, as Berry illustrated in a Botswana context in which the mathematics textbooks were English books simply translated into Setswana. The remedy for type B problems is to modify the curriculum and methodology to build on the learner's natural modes of cognition. Two points emerge. Once again the importance of a multicultured development team is stressed, so that each member of this team is au fait with the cognitive structures of at least one of the cultural groups involved, and each group is represented on the team. The second point is that analysis of the cognitive structures implicit in the home languages of the pupils for whom the curriculum is intended might provide a useful groundwork for the development of this curriculum. With regard to this second point, the work of Pinxten et al. in developing a Navajo geometry, is excitingly relevant (Pinxten, 1984; Pinxten et al., 1983). The fundamental principle used in the development of a Navajo geometry for the primary school was that the mathematical ideas must be based upon, and develop from, Navajo pupils' frame of reference. The model must derive from the Navajo child's own world. Analysis of the Navajo language revealed a preponderance of verbs and verb forms: their thinking was characteristically dynamic. Objects in their culture were typically viewed in terms of actions which could be performed. It was found that intuitive, action-based notions formed a basis for many standard geometrical concepts, e.g. parallelism may be understood in terms of two people running side by side, whose paths do not diverge or approach each other. However, no attempt was made to make the curriculum conform to preconceived ideas of Euclidean geometry. Rather, major elements in the Navajo child's world were studied in the classroom from a mathematical, but action-based, point of view, in a way which represented the authentic thinking of the Navajo child. Thus projects which were modelled and studied included the rodeo, the hooghan (traditional Navajo house), herding sheep, weaving and the school compound. Evolving concepts of continuity,

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circularity, size, projections of a three-dimensional model in two-dimensional representations, and many others, were developed naturally from the models. The emphasis was on action by the children in the making and studying of the models. Although it would of necessity differ from the development of mathematics for Navajo children (because the cultures are different), the Zulu input, for instance, in the development of a common core curriculum for the primary school in Natal, might adopt the same principles as those followed in the Navajo project. Thorough study of the language to identify characteristic elements of thinking, constructs and world views, would need to be followed by identification of mathematical concepts which are already embedded in Zulu culture. Projects based on the world of Zulu children (which would be done by all children in the common core curriculum), could then be used to develop and broaden these intuitive mathematical concepts. For instance, the traditional Zulu village is based on a pattern of concentric circles moving out from the circular cattle kraal in the centre, through the circle of huts arranged in a particular way, to the enclosing outer fence. Krige (1965, p. 39) wrote that "The village everywhere is built on the same plan with few variations, and even these are very slight, never disturbing the customary arrangements of the huts, cattle-kraal, etc." The arrangement of the huts is based on the status of different wives, the indlunkulu or hut occupied by the chief wife being situated at the top end of the kraal exactly opposite the main entrance. Although a decreasing number of Zulu children actually live in traditional villages, as discussed earlier it is important that all children be made aware of their cultural heritage, especially when social change is rapid. Furthermore it is likely that elements of cultural life are reflected in their language and therefore in their cognitive structures. Children from non-Zulu cultures would grow in understanding, through exposure to elements of Zulu tribal life in their learning of mathematics. Each culture would have its turn as activities or elements from that culture came to the fore in aspects of the mathematics curriculum. Interestingly, Bishop (1985a) suggested that a focus on activities rather than on lessons entailing the learning of a fixed piece of mathematical knowledge, would promote the negotiation of mathematical meaning in the classroom. Excellent work has been done by Gerdes (1985 and 1986) in analysing mathematical elements in traditional Mozambiquan life and culture, e.g. in weaving, fishing and building. Some of his writings are openly propagandist, however (only the hegemonic ideology being represented), and in a mathematical curriculum designed for mutual understanding of all cultures involved in a society, a more balanced representation would be essential.

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Inevitably, the success of a curriculum such as the one envisaged in this paper (and indeed whether its development could be realised at all), must depend at least partly on the kind of society which emerges in South Africa. Professor A. J. Thembela, Vice-Rector of the University of Zululand, expressed the view that political change must precede the finding of solutions to some of the problems experienced in black education in South Africa today (Thembela, 1986). In its consideration of some aspects of a common-core mathematics curriculum for diverse cultures, this paper has done little more than scratch the surface of a problem which may require a great deal of time, and which will certainly require the dedication of a large number of South African mathematics educators. Even some mathematical features which are common to all cultures, such as the need to measure, may admit of cultural interpretations and outcomes. This point was illustrated by Bishop (1985d, p. 2) in the following conversation which he had with a university student in Papua New Guinea. I asked him how he would find the area of a rectangular piece of paper. He replied: 'Multiply the length by the width.' 'You have gardens in your village. How do your people judge the area of their gardens?' 'By adding the length and the width.' 'Is that difficult to understand?' 'No, at home I add, at school I multiply.' 'But they both refer to area.' 'Yes, but one is about the area of a piece of paper and the other is about a garden.' So I drew two (rectangular) gardens on the paper, one bigger than the other. 'If these two were gardens, which would you rather have?'

'It dependson many things,I cannotsay. The soil, the shade...' I was then about to ask the next question, 'Yes, but if they had the same soil, shade. . .', when I realised how silly that would sound in that context."

What we are not aiming for in the schools of the future is the cognitive compartmentalisation epitomised in the comment, "At home I add, at school I multiply", even if a pupil is capable of embodying both worlds in an integrated personality structure (as discussed earlier). Cultures are not static, and if the various South African cultures are to move towards each other in the future, a sound understanding of each other's cultures will be a precondition. This understanding might be promoted, inter alia, by a common core mathematics curriculum, along the lines suggested in this paper.

NOTE The substance of this paper is the same as that in an article written for the journal, Cultural Dynamics.

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REFERENCES Apple, M. W.: 1982, Education and Power, Routledge & Kegan Paul, Boston. Berry, J. W.: 1985, 'Learning mathematics in a second language: Some cross-cultural issues'. For the Learning of Mathematics 5(2), 18-23. Bishop, A. J.: 1983, 'Space and geometry', in Lesh, R. and Landau, M. (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York. Bishop, A. J.: 1985a, 'The social construction of meaning- a significant development for mathematics education?', For the Learning of Mathematics 5(1), 24-28. Bishop, A. J.: 1985b, The Social Dimension of Research into Mathematics Education, Plenary paper presented at the research pre-session, N.C.T.M. Annual Conference, San Antonio, 16-17 April, 1985. Bishop, A. J.: 1985c, The Social Dynamics of the Mathematics Classroom, Talk to Austrian teacher educators in Klagenfurt, Austria, 22 May, 1985. Bishop, A. J.: 1985d, The Social Psychology of Mathematics Education, Plenary paper presented at the Ninth P.M.E. conference, Noordwijkerhout, Holland, July, 1985. Bishop, A. J.: 1985e, A Cultural Perspective on Mathematics Education, Paper presented at a conference of mathematics educators, Alice Springs, Australia, September, 1985. Breen, C. J.: 1986, Alternative Mathematics Programmes, Proceedings of the Eighth National Congress of Mathematical Association of South Africa, Stellenbosch, July, 1986. Cooney, T. J.: 1984, Investigating Mathematics Teachers' Beliefs: The Pursuit of Perceptions, Paper prepared for 'short communications' at the Fifth International Congress on Mathematical Education, Adelaide, August, 1984. Cooney, T., F. Goffree, M. Stephens and M. Nickson: 1985, 'The Professional Life of Teachers', For the Learning of Mathematics 5(2), 24-30. D'Ambrosio, U.: 1984, Socio-Cultural Bases for Mathematical Education, Plenary paper presented at ICME V, Adelaide, August, 1984. Eggan, D.: 1974, 'Instruction and affect in Hopi cultural continuity', in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Einstein, A.: 1973, Ideas and Opinions, Souvenir Press, London. Fasheh, M.: 1982, 'Mathematics, culture and authority', For the Learning of Mathematics 3(2), 2-8. Gangat, F.: 1986, Education: A Prime Catalyst in and towards Change, Open lecture, Faculty of Education, University of Durban-Westville, Oct., 1986. Gay, J. and M. Cole: 1967, The New Mathematics and an Old Culture: A Study of Learning among the Kpelle of Liberia, Holt, Rinehart and Winston, London. Gerdes, P.: 1985, 'Conditions and strategies for emancipatory mathematics education in underdeveloped countries', For the Learning of Mathematics 5(1), 15-20. Gerdes, P.: 1986, 'How to recognize hidden geometrical thinking: A contribution to the development of anthropological mathematics', For the Learning of Mathematics 6(2), 10-12. Glick, J.: 1974, 'Culture and cognition: Some theoretical and methodological concerns', in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Grindal, B. T.: 1974, 'Students' self-perceptions among the Sisala of northern Ghana: A study in continuity and change', in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Hardy, G. H.: 1979, A Mathematician's Apology, Cambridge University Press, Cambridge. Howson, G., C. Keitel and J. Kilpatrick: 1982, Curriculum Development in Mathematics, Cambridge University Press, Cambridge. Krige, E. J.: 1965, The Social System of the Zulus, Shuter & Shooter, Pietermaritzburg. Krutetskii, V. A.: 1976, The Psychology of Mathematical Abilities in Schoolchildren,University of Chicago Press, Chicago.

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Lancy, D. F.: 1977, The Indigenous Mathematics Project: A Progress Report, Address to the

LinguisticSocietyof PapuaNew Guinea,Lae, September,1977.

Lancy, D. F.: 1983, Cross-Cultural Studies in Cognition and Mathematics, Academic Press, New

York. Lawton, D.: 1975, Class, Culture and the Curriculum,Routledge and Kegan Paul, London.

Nash, R.: 1974,'The convergenceof anthropologyand education',in Spindler,G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Natal Teachers' Society (NTS): 1986, NTS Strategyfor Maintaining Relevanceand Impact in the

ChangingSouthAfricanContext,Documentpresentedat the 71stAnnualConferenceof the NTS (theme"Educationfor Liberation"),June, 1986. Nickson, M.: 1984, Perceptions of Mathematics and the Social Context of the Mathematics

Classroom,Paperpresentedat the FifthInternationalCongresson MathematicalEducation, Adelaide,August, 1984. Nundkumar,R.: 1985,'Thechangingfaceof Englishin the townshipof Phoenix',Universityof Durban-Westville: Faculty of Education Journal 2(4), 65-72. Pinxten, R., I. Van Dooren and F. Harvey, 1983, Anthropologyof Space: Explorations into the Natural Philosophy and Semantics of Navajo Indians, University of Pennsylvania Press,

Philadelphia. and Navajogeometry',Communication Pinxten,R.: 1984,'Navajospatialrepresentation InformationVI(23),266-289. Reynolds,D. A. T. andN. T. Reynolds:1974,'Therootsof prejudice:CaliforniaIndianhistory in schooltextbooks',in Spindler,G. D. (ed.), EducationandCulturalProcess,Holt, Rinehart and Winston,New York. Selltiz,C., M. Jahoda,M. DeutschandS. W. Cook:1963,ResearchMethodsin SocialRelations, Holt, Rinehartand Winston,New York. Sindell,P. S.: 1974,'Somediscontinuitiesin the enculturationof MistassiniCreechildren',in Spindler,G. D. (ed.), Educationand CulturalProcess,Holt, Rinehartand Winston,New York. in Spindler,G. D. (ed.), Singleton,J.: 1974,'Implicationsof educationas culturaltransmission', Education and Cultural Process, Holt, Rinehart and Winston, New York.

A studyin culturaltransmissionand instruSpindler,G. D.: 1974,'Schoolingin Sch6nhausen: mentaladaptationin an urbanizingGermanvillage',in Spindler,G. D. (ed.), Educationand CulturalProcess,Holt, Rinehartand Winston,New York. Spindler,G. D. (ed.): 1974,EducationandCulturalProcess,Holt, RinehartandWinston,New York. Taylor,N., J. Adler,F. Mazibukoand L. Magadla:1986,People'sEducationand the Roleof Mathematics, Workshopat a conferenceheldby the Departmentof Education,Universityof the Witwatersrand, 31 Oct.-3 Nov., 1986. Kenton-on-the-Jukskei, Thembela,A. J.: 1986,BlackEducation,Paperpresentedat the 71stAnnualConferenceof the Natal Teachers'Society,Durban,June, 1986. Walters, S.: 1986, People's Education: A ConceptualFrameworkfor Analysis, Paper presented at

a conferenceat the Centrefor Adult and ContinuingEducation,Universityof the Western Cape,July, 1986.

Faculty of Education, University of Durban-Westville, Private Bag X54001, Durban, 4000 Rep. of South Africa.

ALAN J. BISHOP

MATHEMATICS

EDUCATION

IN ITS CULTURAL

CONTEXT

ABSTRACT.This paperpresentsthe resultsof a seriesof analysesof educationalsituations aretheideasthatall culturalgroupsgenerate involvingculturalissues.Of particularsignificance mathematicalideas, and that 'Western'mathematicsmay be only one mathematicsamong many.The valuesassociatedwith Westernmathematicsare also discussed,and variousissues raisedby theseanalysesare then presented.

In this article I shall summarise the results of the analyses and investigations which have engaged me over the past fifteen years and which relate specifically to this Special Issue of Educational Studies in Mathematics. There have been two major and related areas of concern in that time, and both seem to have important implications for research, for theory development and for classroom practice.

Cultural interfaces in Mathematics Education The first concern is with what I think of as 'cultural interfaces'. In some countries like the U.K. pressure has mounted to reflect in the school curriculum the multi-cultural nature of their societies, and there has been widespread recognition of the need to re-evaluate the total school experience in the face of the education failure of many children from ethnic minority communities. In other countries, like Papua New Guinea, Mozambique and Iran, there is criticism of the 'colonial' or 'Western' educational experience, and a desire to create instead an education which is in tune with the 'home' culture of the society. The same concern emerges in other debates about the formal education of Aborigines, of Amerindians, of the Lapps and of Eskimos. In all of these cases, a culture-conflict situation is recognised and curricula are being re-examined. One particular version of this problem relates to the mathematics curriculum and its relationship with the home culture of the child. Mathematics curricula though, have been slow to change, due primarily to a popular and widespread misconception. Up to five or so years ago, the conventional wisdom was that mathematics was 'culture-free' knowledge. After all, the argument went, "a negative times a negative gives a positive" wherever you Educational Studies in Mathematics 19 (1988) 179-191. ? 1988 by Kluwer Academic Publishers.

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are, and triangles the world over have angles which add up to 180 degrees. This view though, confuses the 'universality of truth' of mathematical ideas with the cultural basis of that knowledge. The ideas are decontextualised and abstracted in such a way that 'obviously' they can apply everywhere. In that sense they are clearly universal. But as soon as one begins to focus on the particulars of these statements, one's belief in that universality tends to feel challenged. Why is it 180 degrees and not, say, 100 or 150?Where does the idea of negative number come from? Authoritative writers on mathematical history have given answers to these kinds of questions, of course, and they demonstrate quite clearly that mathematics has a cultural history. But whose cultural history are we referring to? Recently, research evidence from anthropological and cross-cultural studies has emerged which not only supports the idea that mathematics has a cultural history, but also that from different cultural histories have come what can only be described as different mathematics. One can cite the work of Zaslavsky (1973), who has shown in her book Africa Counts, the range of mathematical ideas existing in indigenous African cultures. Van Sertima's Blacks in Science (1986), is another African source as is Gerdes (1985). On other continents, the researchof Lancy (1983), Lean (1986) and Bishop (1979) in Papua New Guinea, Harris (1980) and Lewis (1976) in Aboriginal Australia, and Pinxten (1983) and Closs (1986) with the Amerindians, has also added fuel to this debate. The term 'ethnomathematics' has been revived (d'Ambrosio, 1985) to describe some of these ideas, and even if the term itself is still not yet well defined, there is no doubting the sentiment that the ideas are indeed mathematical ideas. The thesis is therefore developing that mathematics must now be understood as a kind of cultural knowledge, which all cultures generate but which need not necessarily 'look' the same from one cultural group to another. Just as all human cultures generate language, religious beliefs, rituals, food-producing techniques, etc., so it seems do all human cultures generate mathematics. Mathematics is a pan-human phenomenon. Moreover, just as each cultural group generates its own language, religious belief, etc., so it seems that each cultural group is capable of generating its own mathematics. Clearly this kind of thinking will necessitate some fundamental re-examinations of many of our traditional beliefs about the theory and practice of mathematics education, and I will outline some of these issues below. Values in Mathematics Education The second area of concern has been our ignorance about 'values' in Mathematics Education. In the same way that Mathematics has been considered

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for many years to be culture-free,so it has also been consideredto be value-free.How could it be concernedwith values,the argumentgoes, when it is about indisputablefacts concerningtriangles,fractionsor multiplication?Onceagainanthropologically-oriented researcherslike Pinxten(1983), Horton(1971), Lewis(1976)and Leach(1973)havepresentedus withplenty of evidence with which to challengethat traditionalview. Moreoverany mathematicseducatorwho works in cultural-interfacesituations, such as most of the otherwritersin this SpecialIssue,soon becomeacutelyawareof the influenceof value-conflictson the mathematicallearningexperienceof the childrenthey are responsiblefor.' Moreoverone can arguethat a mathematicaleducationis no educationat all if it does not haveanythingto contributeto valuesdevelopment.Perhaps that is a crucialdifferencebetweena mathematicaltrainingand a mathematical education? Indeedit wouldseemto me to be thoroughlyappropriateto conceptualise muchcurrentmathematicsteachingas merelymathematicaltraining,in that generallythere is no explicitattentionpaid to values. I am not saying that valuesare not learnt- clearlythey are- but implicitly,covertlyand without much awarenessor consciouschoice. Surelya mathematicaleducation,on the other hand, should make the values explicit and overt, in order to develop the learner'sawarenessand capacityfor choosing? Thereis even moreof a pressingneedtoday to considervaluesbecauseof the increasingpresenceof the computerand the calculatorin our societies. These devicescan performmanymathematicaltechniquesfor us, even now, and the argumentsin favourof a purelymathematicaltrainingfor our future citizensare surelyweakened.Societywill only be able to harnessthe mathematicalpowerof these devicesfor appropriateuse if its citizenshave been made to consider values as part of their education. For some pessimists however,like Ellul(1980) the situationis far too out of controlin any case for educationto be able to do anythingconstructiveat this stage.Nevertheless the ideas of other analystssuch as Skovsmose(1985) do offer, in my view, the potentialfor developingstrategiesfor change. My own perspectiveon this area of values has been stimulatedby the culture-conflictresearchmentionedearlierand it is this perspectivewhich I propose to enlargeon here.The fundamentaltask for my work was to find a rich way to conceptualisemathematicsas a culturalphenomenon. Mathematics as a Cultural Phenomenon

The most productivestartingpoint was providedby White (1959) in his book TheEvolutionof Culturein whichhe argues,as othershave done, that

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"the functions of cultureare to relateman to his environmenton the one hand, and to relateman to man, on the other"(p. 8). White, though, went further,and dividedthe componentsof cultureinto four categories: - ideological:composedof beliefs,dependenton symbols,philosophies; - sociological:the customs, institutions,rules and patterns of interpersonal behaviour; - sentimental:attitudes,feelingsconcerningpeople, behaviour; - technological:manufactureand use of tools and implements. Moreoverwhilstshowingthat thesefour componentsare interrelatedWhite arguesstronglythat "the technologicalfactoris the basic one; all othersare dependentupon it. Furthermore,the technologicalfactor determines,in a generalway at least, the form and content of the social, philosophicand sentimentalfactors"(p. 19). Writerssuch as Bruner(1964) and Vygotsky(1978) have also shown us the significanceof written language,and one of its particularconceptual 'tools', mathematicalsymbolism.Mathematics,as an exampleof a cultural phenomenon,has an important'technological'component,to use White's terminology.But White'sschemaalso offeredan opportunityto explorethe ideology, sentimentand sociology drivenby this symbolictechnology,and thereforeto attendto values as well. Mathematicsin this contextis thereforeconceivedof as a culturalproduct, whichhas developedas a resultof variousactivities.TheseI have described in otherwritings(Bishop, 1986;Bishop,1988)so I willjust brieflysummarise them here. Thereare, from my analyses,six fundamentalactivitieswhich I argue are both universal,in that they appear to be carriedout by every culturalgroupeverstudied,and also necessaryand sufficientfor the development of mathematicalknowledge. They are as follows: Counting.The use of a systematicway to compareand orderdiscretephenomena. It may involve tallying, or using objects or string to record, or specialnumberwords or names. (See for example,Lean, 1986;Menninger, 1969; Ascher and Ascher, 1981; Closs, 1986; Ronan, 1981; Zaslavsky, 1973.)

Locating.Exploringone's spatialenvironmentand conceptualisingand symbolisingthat environment,with models,diagrams,drawings,wordsor other means. (See for example,Pinxten, 1983;Lewis, 1976;Harris,1980;Ronan, 1986.) Measuring.Quantifyingqualitiesfor the purposesof comparisonand ordering, using objects or tokens as measuringdevices with associatedunits or

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'measure-words'.(See for example,Menninger,1969;Gay and Cole, 1967; Jones, 1974;Harris, 1980;Zaslavsky,1973.) Designing.Creatinga shape or designfor an objector for any part of one's spatial environment. It may involve making the object, as a 'mental tem-

plate', or symbolisingit in some conventionalisedway. (See for example, Gerdes, 1986;Temple, 1986; Ronan, 1981;Bourgoin, 1973; Faegre, 1979; Oswalt, 1976.) Playing. Devising, and engaging in, games and pastimes, with more or less formalised rules that all players must abide by. (See for example, Huizinga, 1949; Lancy, 1983; Jayne, 1962; Roth, 1902; Falkener, 1961; Zaslavsky,

1973.) Explaining.Findingways to accountfor the existenceof phenomena,be they religious, animistic or scientific. (See for example, Lancy, 1983; Horton, 1971; Pinxten, 1983; Ronan, 1981; Gay and Cole, 1967.)

Mathematics,as cultural knowledge,derivesfrom humans engaging in thesesix universalactivitiesin a sustained,andconsciousmanner.The activities can eitherbe performedin a mutuallyexclusiveway or, perhapsmore significantly,by interactingtogether,as in 'playingwith numbers'which is likely to have developed numberpatternsand magic squares,and which arguablycontributedto the developmentof algebra. I would argue that, in the mathematicswhich I and many others have learnt,these activitieshave contributedat least the followinghighly significant ideas: Counting: Numbers. Number patterns.Number relationships.Developments of numbersystems.Algebraicrepresentation.Infinitely large and small. Events, probabilities,frequencies.Numerical methods.Iteration.Combinatorics.Limits. Locating: Position. Orientation.Developmentof coordinates- rectangular, polar, spherical. Latitude/longitude. Bearings. Angles. Lines.

Networks. Journey. Change of position. Loci (circle, ellipse, polygon.. .). Changeof orientation.Rotation. Reflection. Measuring:Comparing.Ordering.Length.Area.Volume.Time. Temperature. Weight. Developmentof units-conventional, standard, metricsystem.Measuringinstruments.Estimation.Approximation. Error.

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Designing: Propertiesof objects.Shape.Pattern.Design.Geometricshapes (figuresand solids). Propertiesof shapes. Similarity.Congruence. Ratios (internaland external). Playing: Puzzles.Paradoxes.Models. Games.Rules. Procedures.Strategies. Prediction. Guessing. Chance. Hypothetical reasoning. Games analysis. Explaining:Classifications.Conventions.Generalisations.Linguisticexplanations- arguments, logical connections, proof. Symbolic explanations- equations,formulae,algorithms,functions.Figuralexplanations- diagrams,graphs,charts,matrices.(Mathematical structure-axioms, theorems, analysis, consistency.) (Mathematicalmodel- assumptions,analogies, generalisability, prediction.) From these basic notions, the rest of 'Western'mathematicalknowledge can be derived,2whilein this structurecan also be locatedthe evidenceof the 'othermathematics'developedby other cultures.Indeedwe ought to re-examinelabelssuchas 'WesternMathematics'sincewe know that manydifferent culturescontributedto the knowledgeencapsulatedby that particular label.3 However,I mustnow admitto what mightbe seen as a conceptualweakness. Thereis no real prospectof my being able to test whetheror not this 'universal'structurewill be adequatefor describingthe mathematicalideas of other culturalgroups.On the contrary,I would maintainthat it must be for othersfromthose culturalgroupsto determinethis. Far frommy inability beinginterpretedas a weakness,I believeit is importantto recognisethat in this kind of analysisone must be constantlyaware of'the dangersof culturo-centrism.It may well be the case that my analysiswill not hold up under cross-culturalscrutiny- it is my hope that it may in fact stimulate some other analytic developments,which again could be tested crossculturally. This kind of culturo-centrism is well explainedby Lancy(1983) who has a 'universal' proposed stage theoryof cognitivedevelopment.Lancyshows that his Stage I correspondsto Piaget'ssensory-motorand pre-operational stages "the accomplishmentsof this stage are sharedby all humanbeings" (p. 203). Stage2 is whereenculturationbegins:"Whathappensto cognition duringStage2, then, has much to do with cultureand environmentand less to do with genetics"(p. 205). This, for me, is the stage where different culturesdevelopdifferentmathematics. However Lancy also has a Stage 3 in his theory, which concerns the metacognitivelevel: "In addition to developing cognitive and linguistic

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strategies,individualsacquire'theories'of languageand cognition"(p. 208). For Lancy, therefore,the 'formaloperation'stage of Piaget'stheory represents the particulartheoryof knowledgewhichthe 'Western'culturalgroup emphasises.Otherculturalgroupscan, and do, emphasiseother theoriesof knowledge. This idea gives a usefulculturalentreeinto the area of values, linkingas it does with White's idea that the technologyof a culture(in our case the symbolictechnologyof mathematics)not only relateshumansto theirenvironmentin a particularway, but also 'drives'the otherculturalcomponentsthe sentimental,the ideologicaland the sociological.It is these that are the heartof the values associatedwith mathematicsas a culturalphenomenon. Beforeturningto examinethesein moredetailit is necessaryto point out that my own culturalpredispositionmakes it very difficultto attemptany moreat this stagethanmerelyoutliningthe valueswhichI feel are associated with the 'WesternMathematics'with which I am familiar.I do know that enoughevidenceexists to suggestthat White'sschemadoes have some credibilityin 'Western'culture.I am in no positionhoweverto arguethat for any otherculture.Once again that verificationmust be left to those in the other culturalgroups. The three value componentsof culture- White'ssentimental,ideological and sociologicalcomponents- appearto me to havepairsof complementary valuesassociatedwith mathematics,whichgive rise to certainbalancesand tensions. If we considerfirst the 'sentimental'componentwe can see that so muchof the powerof mathematicsin our societycomes fromthe feelings of security and control that it offers. Mathematics,through science and technologyhas givenWesternculturethe senseof securityin knowledge- so much so that people can become very frustratedat naturalor man-made disasterswhich they feel shouldn'thave happened!The inconsistencyof a Mathematicalargumentis a strong motive for uncoveringthe error and getting the answer 'right'. The mathematicalvaluing of 'right' answers informs society which also looks (in vain of course) for right answersto its societal problems. Westernculture is fast becoming a MathematicoTechnologicalculture. Wherecontrol and securityare sentimentsabout things remainingpredictable,the complementaryvaluerelatesto progress.A methodof solution for one mathematicalproblemis able,by the abstractnatureof mathematics, to be generalisedto other problems.The unknown can become known. Knowledgecan develop.Progress,though,can becomeits own rewardand changebecomesinevitable.Alternativismis stronglyupheldin Westerncultureand as with all the valuesdescribedhere,containswithinitself the seeds

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of destruction.It is thereforeimportantto recognisethat it is the interactions and tensions between those values of control and progress which allow culturesto surviveand to grow. If those are the twin sentimentsdriven by the mathematicalsymbolic technologythen the principleideologyassociatedwith WesternMathematics must be rationalism.If one were searchingfor only one identifiablevalue,it would be this one. It is logic, rationalismand reasonwhich has guaranteed the pre-eminenceof mathematicswithinWesternculture.It is not tradition, not status, not experience,not seniority,but logic which offers the major criterion of mathematicalknowledge.With the advent of computersthe ideology is extendingeven further,if that is possible. The Indo-Europeanlanguagesappearto have richvocabulariesfor logicGardner(1977) in his (English)tests used over 800 differentlogical connectives. The rise of physicaltechnologyhas also helped this development,in that 'causation',one of the roots of rationalargument,seems to be developed much easier throughphysicaltechnology than through nature- the time-scalesof naturalprocessesare often too fast or too slow. It was simple physical technologicaldevices which enabled humans to experimentwith process,and to developthe formidableconcept of 'directcausation'. However there is also a complementaryideology which is clearly identifiablein Westernculture,and that is objectism.Westernculture'sworldview appearsto be dominatedby materialobjectsand physicaltechnology. Whererationalismis concernedwith the relationshipbetweenideas, objectism is about the genesisof those ideas. One of the ways mathematicshas gainedits poweris throughthe activityof objectivisingthe abstractionsfrom reality. Through its symbols (letters, numerals,figures)mathematicshas taught people how to deal with abstractentities,as if they wereobjects. The final two complementaryvaluesconcernWhite'ssociologicalcomponent, the relationshipsbetweenpeople and mathematicalknowledge.The firstI call opennessand concernsthe fact that mathematicaltruthsare open to examinationby all, providedof coursethat one has the necessaryknowledge to do the examining.Proof grew from the desirefor articulationand demonstration,so well practisedby the early Greeks, and although the criteriafor the acceptabilityof proof have changed,the value of 'opening' the knowledgehas remainedas strongas ever. Howeverthereis a complementary sociologicalvaluewhichI call mystery. Despite that openness, there is a mysteriousquality about mathematical ideas.Certainlyeveryonewho has learntmathematicsknowsthis intuitively, whetherit is throughthe meaninglesssymbol-pushingwhich many children still unfortunatelyexperience,or whetherit is in the surprisingdiscoveryof

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an unexpected connection. The basis of the mystery again lies in the abstract nature of mathematics - abstractions take one away from a context, and decontextualised knowledge is literally meaningless. Of course mathematical ideas offer their own kind of context so it is very possible to develop meanings within mathematics. These then are the three pairs of values relating to Western mathematics which are shaped by, and also have helped to shape, a particular set of symbolic conceptual structures. Together with those structures they constitute the cultural phenomenon which is often labelled as 'Western Mathematics'. We certainly know that different symbolisations have been developed in different cultures and it is very likely that there are differences in values also, although detailed evidence on this is not readily available at the present time. How unique these values are, or how separable a technology is from its values must also remain open questions. Some Issues Arisingfrom This Analysis White's (1959) view of culture has enabled us to create a conception of Mathematics different from that normally drawn. It is a conception which enables mathematics to be understood as a pan-cultural phenomenon. It seems that what I have been referring to a 'Western' mathematics must be recognised as being similar to, yet different from, the mathematics developed by other cultural groups. There appear to be differencesin symbolisation and also differences in values. Just how great those differences are will have to be revealed by further analysis of the available anthropological and cross-cultural evidence. Hopefully the analysis here will help to structure the search for that evidence. But what educational issues has this analysis revealed to us? From an anthropological perspective mathematical education is a process of inducting the young into part of their culture, and there appear initially to be two distinct kinds of process. On the one hand there is enculturation, which concerns the induction of the young child into the home or local culture, while on the other there is acculturation,which is to do with the induction of the person into a culture which is in some sense alien, and different from that of their home background. Appealing as this simple dichotomy is, the real educational situation is rather more complex. Consider inducting a child into 'Western' mathematics - for which children is enculturation the appropriate model? Is it really part of anyone's home and local culture? It is certainly not the product of any one culture and therefore no-one could claim it as theirs exclusively.

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Moreover there are plenty of practising Mathematiciansin Universities all over the world who would object fundamentally(and rightly so in my opinion) to the suggestionthat they were engagedin developing Western culturalknowledge. So what, culturally,is the mathematicswhich until a few years ago was generallythought to representthe only mathematics?Is it betterto think of mathematicswhichall can learnto speakand it as a kind of internationalised understand?Or is it morea sort of Esperantoof the mathematicalworld,an artificial,pragmaticsolutionto a multi-culturalsituation?That doesn'tseem to be a good analogybecauseof the strongculturalvaluesassociatedwith it, and the fact that it was not deliberatelycreatedin the way Esperantowas. Perhapsit is more appropriateeducationallyto recognisethat different societies are influencedto differentdegrees by this internationalmathematico-technologicalcultureand that the greaterthe degreeof influencethe more appropriatewould be the idea of enculturation? What of acculturation?That clearlydoes raise other educationalissues. Whilstacculturationis a naturalkindof culturaldevelopmentwhencultures meet, thereis somethingverycontentiousto me aboutan educationwhichis intentionallyacculturative.Thereis a clearintentionimpliedin that notion to induct the child into an alien culturewithout any concernfor the ultimate preservationof that child'shomeculture.4Thosewhosechildrenare beingso acculturatedhave a perfectlyunderstandableright to be concerned. MoreoverI would stressthat only those peoplefor whom Westernmathematicsis an alienculturalproductshould decidewhat to do in the cultureconflict situation so created.It might be possible to develop a bi-cultural strategy,but that shouldnot be for 'aliens'like me to decide.In my view the same questionsariseover choice of teacher,and choice of educationalenvironment.I would argue that in generalin a culture-conflictsituationit is betterin the long run for the teacherto be from the 'home'culture,and for that teacherto be closelylinkedwith the local community.If culture-conflict is to be handledsensitively,thenschooling,and the teacher,shouldstay close to the people affected, in my view.5

Another set of issues relate to the mathematicscurriculumin schools, particularlyin those societieswherethereare severalethnicminoritygroups. Whatideas shouldwe be introducingthe childrento? To whatextentshould any of the mathematicalideas from other culturesbe used?And how is it possibleto structurethe mathematicscurriculumto allow this to happen? It certainly would seem valuable to use mathematicalideas from the child'shome culturewithin the overallmathematicalexperience,if only to enablethe child to makegood contactwith the constructof mathematicsper

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se. We know only too well some of the negative effects of insisting on children only experiencing alien cultural products - the meaninglessness, the rote-learning syndrome, the general attitude of irrelevance and purposelessness. So how can we overcome this? One possible way is to use as a structural framework the six activities which I described earlier. If those activities are universal, and if they are both necessary and sufficient for mathematical development, then a curriculum which is structured around those activities would allow the mathematical ideas from different cultural groups to be introduced sensibly. Is it indeed possible by this means to create a culturally-fair mathematics curriculum - a curriculum which would allow all cultural groups to involve their own mathematical ideas whilst also permitting the 'international' mathematical ideas to be developed? Finally, what about the education of values? One implication of the values analysis earlier could be a consideration of the emphasis given in present mathematics education to certain values. I do not think it would be too cynical to suggest that a great deal of current mathematics teaching leans more towards control than to progress, to objectism rather than to rationalism, and to mystery rather than to openness. Perhaps a greater use of such teaching activities as group work, discussion, project work and investigations could help to redress the balances in each of the complementary pairs. We may then move our mathematical education more towards 'progress', 'rationalism' and 'openness', a goal with which several recent writers appear to agree. Certainly I believe that we should educate our children about values and not just train them into adopting certain values, although I realise that different societies may desire different approaches. (Nevertheless I can't imagine how, or why, one would train a child to adopt a value like openness!) Again it seems to me to depend on the extent to which the particular society is influenced by these mathematico-technological cultural values, and relates once again to the enculturation/acculturation issue I described earlier. In Conclusion Perhaps the most significant implication for Mathematics education of this whole area lies in teacher education. It is clear that teacher educators can no longer ignore these kinds of issues. Mathematics education in practice is, and always should be, mediated by human teachers. Inducting a young child into part of its culture is necessarily an inter-personal affair, and therefore teachers must be made fully aware of this aspect of their role. More than that, they

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need to know about the values inherent in the subject they are responsible for, they need to know about the cultural history of their subject, they need to reflect on their relationship with those values, and they need to be aware of how their teaching contributes not just to the mathematical development of their pupils, but also to the development of mathematics in their culture. Teacher education is the key to cultural preservation and development.

NOTES Fasheh (1982) and Kothari (1978) are educators who also express the values-conflict clearly. 2 As mathematical ideas develop, of course, they become part of the environment also, ready to be acted on as with any other part of the environment. 3 Kline (1962) and Wilder (1981) are two authors who have explored the cultural history of 'Western' mathematics. 4 The papers by Graham, Gerdes and Presmeg in this issue are concerned with this area. 5 Taft (1977) in a wide-ranging article describes many of the complex issues surrounding people in culture-conflict situations, and also indicates just how widespread a phenomenon it is.

REFERENCES d'Ambrosio, U.: 1985, 'Ethnomathematics and its place in the history and pedagogy of mathematics', For the Learning of Mathematics 5(1), 44-48. Ascher, M. and R. Ascher: 1981, Code of the Quipu, University of Michigan Press, Chicago. Bishop, A. J.: 1979, 'Visualising and mathematics in a pre-technological culture', Educational Studies in Mathematics 10(2), 135-146. Bishop, A. J.: 1986, 'Mathematics education as cultural induction', Nieuwe Wiskrant, October, 27-32. Bishop, A. J.: 1988, Mathematical Enculturation: A Cultural Perspective on Mathematics Education, Reidel, Dordrecht. Bourgoin, J.: 1973, Arabic Geometrical Pattern and Design, Dover, New York. Bruner, J. S.: 1964, 'The course of cognitive growth', American Psychologist 19, 1-15. Closs, M. P. (ed.): 1986, Native American Mathematics, University of Texas Press, Austin, Texas. Ellul, J.: 1980, The Technological System, Continuum Publishing, New York. Faegre, T.: 1979, Tents: Architecture of the Nomads, John Murray, London. Falkener, E.: 1961, Games Ancient and Oriental - How to Play Them, Dover, New York. Fasheh, M.: 1982, 'Mathematics, culture and authority', For the Learning of Mathematics 3(2), 2-8. Gardner, P. L.: 1977, Logical Connectivesin Science, Monash University, Faculty of Education, Melbourne, Australia. Gay, J. and M. Cole: 1967, The New Mathematics in an Old Culture, Holt, Rinehart and Winston, New York. Gerdes, P.: 1985, 'Conditions and strategies for emancipatory mathematics education in underdeveloped countries', For the Learning of Mathematics 5(1), 15-20. Gerdes, P.: 1986, 'How to recognise hidden geometrical thinking: A contribution to the development of anthropological mathematics', For the Learning of Mathematics 6(2), 10-17. Harris, P.: 1980, Measurement in Tribal Aboriginal Communities, Northern Territory Department of Education, Australia.

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Horton, R.: 1971, 'African traditional thought and Western Science', Africa, Vol. XXXVII, also in M. F. D. Young (ed.), Knowledgeand Control, pp. 208-266, Collier-MacMillan, London. Huizinga, J.: 1949, Homo Ludens, Routledge and Kegal Paul, London. Jayne, C. F.: 1974, String Figures and How to Make Them, Dover, New York, (first published as 'String Figures' by Scribner in 1906). Jones, J.: 1974, Cognitive Studies with Students in Papua New Guinea (Working Paper, No. 10), University of Papua New Guinea, Education Research Unit. Kline, M.: 1962, Mathematics: A Cultural Approach, Addison Wesley, Mass. Kothari, D. S.: 1978, Keynote address in Proceedings of Asian Regional Seminar of the Commonwealth Association of Science and Mathematics Educators (New Delhi), British Council, London. Lancy, D. F.: 1983, Cross-cultural Studies in Cognition and Mathematics, Academic Press, New York. Leach, E.: 1973, 'Some anthropological observations on number, time and common-sense', in A. G. Howson (ed.), Developments in Mathematical Education, Cambridge University Press. Lean, G. A.: 1986, Counting Systems of Papua New Guinea, Research Bibliography, 3rd edn., Department of Mathematics, Papua New Guinea University of Technology, Lae, Papua New Guinea. Lewis, D.: 1972, We the Navigators, University Press of Hawaii, Hawaii. Lewis. D.: 1976, 'Observations on route-finding and spatial orientation among the Aboriginal peoples of the Western desert region of central Australia', Oceania XLVI (4), 249-282. Menninger, K.: 1969, Number Words and Number Symbols - A Cultural History of Numbers, MIT Press, Cambridge, Mass. Oswalt, W. H.: 1976, An Anthropological Analysis of Food-getting Technology, Wiley, New York. Pinxten, R., I. van Dooren and F. Harvey: 1983, The Anthropology of Space, University of Pennsylvania Press, Philadelphia. Ronan, C. A.: 1981, The Shorter Science and Civilisation in China: Vol. 2, Cambridge University Press. Ronan, C. A.: 1983, The Cambridge Illustrated History of the World's Science, Cambridge University Press. Roth, W. E.: 1902, 'Games, sports and amusements', North QueenslandEthnographic Bulletin 4, 7-24. van Sertima, I.: 1986, Black in Science, Transaction Books, New Brunswick. Skovsmose, O.: 1985, 'Mathematical education versus critical education', Educational Studies in Mathematics 16(4), 337-354. Taft, R.: 1977, 'Coping with unfamiliar cultures', in N. Warren (ed.), Studies in Cross-Cultural Psychology, Vol. 1, Academic Press, London. Temple, R. K. G.: 1986, China, Land of Discovery and Invention, Stephens, Wellingborough, U.K. Vygotsky, L. S.: 1978, Mind in Society, MIT Press, Cambridge, Mass. White, L. A.: 1959, The Evolution of Culture, McGraw-Hill, New York. Wilder, R. L.: 1981, Mathematics as a Cultural System, Pergamon Press, Oxford. Zaslavsky, C.: 1973, Africa Counts, Prindle, Weber and Schmidt, Inc., Boston, Mass.

Department of Education, University of Cambridge, 17 TrumpingtonStreet, Cambridge CB2 IQA.

MARC SWADENERAND R. SOEDJADI

VALUES, MATHEMATICSEDUCATION,AND THE TASK OF DEVELOPINGPUPILS' PERSONALITIES: AN INDONESIANPERSPECTIVE

ABSTRACT. This paper reviews the role andpsychological bases of values in education and their effect on teaching and learning.Contrastingviews of the role of values in educationin Indonesiaand the United States of America are given. Includedis extensive discussion of the role of social values in educationin Indonesian society, and of the resultingresponsibilityteachers (specifically mathematicsteachers)have in this context. This is translatedinto practiceby identifying value laden topics within mathematicsand examining social values inherentin these topics. Discussion includes an analysis of how the study of each of these topics may be used to promote social values which are consistent with national goals, and the promotionof nationaldevelopment

INTRODUCTION

Some issues in American education are often discussed and then pragmatically resolved either throughlegal means or by an evolving tradition.The teaching of values and the role of education in the furtheranceof the values of society are two of these issues. The overt teaching of values is controversialin American public education. The controversycenters aroundthreepoints: 1. Whose values are to be taught? 2. What are the qualificationsof the teachersinvolved in teaching the values; and 3. Providing equal treatmentof opposing values. What usually results in the classroom is one or a combination of two conditions. Either (a) there is minimal discussion of values in classrooms, or (b) values are discussed "objectively"with the hope that the students involved will not be unreasonablyswayed by the teacherto either agreementor disagreement with the values considered. The apparent goal is to let students decide for themselves what is to be valued in theirlife. American private schools (includingbut not limited to parochial schools) do not face the "teachingvalues" dilemma to the same degree as do public schools. Most American private schools are associated with a religious, or other group founded on a well defined set of values. Private education therefore is, by its nature,concerned with the values espoused by the organizationwhich sponsors the school. The values taught in private schools are the values of the sponsoring 19 (i988) 193-208. Educational Studiesin Mathematics ? 1988 by KluwerAcademicPublishers.

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organization.Parents of children attendingprivate schools know this in advance and desire this for their children. Many parents send their children to such schools for this reason. They can be more certain that the values their children are exposed to are ones which they "value"themselves. A second recurringissue is: Should public education be designed to produce students who 'fit' in society? Is one of the roles of public education the continuance of currentsocietal attitudes,behaviors, values, a "nationalplan," etc.? Should teachers actively teach and encourage the attitudes,behaviors, etc. of the society in which the studentswill function?There is some question whetherthis can be prevented, but the issue is whether such an orientation should be an active function of public education. Should public school programsbe designed to "shape" the student's mind, attitudes,behaviors, and values to be consistent with the society in which they reside? Whatever an individual's feeling on this issue, an immediate question is "Whatare the basic attitudes,behaviors, values, etc. of society?" There would be significant agreement on a limited listing of such characteristics within American society. However, it would be next to impossible to obtain widespreadagreementon an extensive list of such characteristics. America was founded on documents (the Declaration of Independence and the Constitution) which mention certain values. Inculcationof everyday values, by design, has not been the purpose of contemporaryAmerican public education. This is different from the situation in many (if not most) countries, especially in highly ideological, many developing, and third world countries. Many such countries are struggling for existence and the resulting role of education is what could be called "educationfor nationbuilding." In 1985 the writers of this paper had the opportunityto work together in Indonesia. Indonesia is clearly a developing country. It declared its independence in 1945 after 300 years of domination by the Dutch (the Dutch East Indies) and JapaneseoccupationduringWorld War II. The road to independence was difficult. Indonesia is the fifth most populous country after the America. It is made up of more than 13,600 islands (6,000 populated) in the south Pacific, and is considered to be part of Southeast Asia. Several hundredethnic groups and many native languages exist in Indonesia. The rich cultural tradition in Indonesia has developed over many centuries and is highly varied in some regions. Because of the rich, varied, and highly developed culture and the 300 year experience of being a colony of the Dutch, there are strong and widespread feelings of nationalpride and commitmentto nationaldevelopment. The principles, values and moralitywhich are the foundationfor the founding of Indonesia are specified in a document called the Panca Sila (or the "five principles"). Panca Sila is THE (emphasis intended) foundation of national values and morality. The Panca Sila specifies five fundamental principles.

(IndonesiaHandbook,p. 89)

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1. Belief in One SupremeGod; 2. A Just and Civilized Humanity; 3. Unity of Indonesia; 4. Democracy wisely led by the wisdom of Liberationamong representatives; and 5. Social Justice for the whole of the people of Indonesia. Indonesian history and culture are significant factors in education and the education system plays an active role in promoting nationalism, values consistent with Indonesiannationalgoals, and societal morality. Indonesian schools are organized within a national system. In each school there are three types of teachers; regular teachers of content (mathematics, language, science, etc.), teachersof "PancaSila ethics;"and teachersof religion. All teachers in all schools, including teachersof Panca Sila ethics and religion, are government employees. All schools (public and private) are subject to government regulation, and all students receive instruction in the areas representedby the three kinds of teachers. The purpose of the discussion in this paper is to bring to the attentionof the reader a perspective on the issues involved in "values education"which differs from American education practice. There is no intention to present the views expressed as better or worse than opposing views. All teachers(in all countries) should have a broad general knowledge of educationand the issues which affect them and their students.This paperis an attemptto broadenthe reader's view on the role of values in education. It is one treatmentof one set of issues. Hopefully the reader will obtain a "feel" for educational thinking in Indonesia, how it differs from thinking in America, and how such thinking affects all teachers, specifically teachers of mathematics. A discussion of the relationship between values and teaching mathematics with some discussion of the situation in Indonesiafollows. The stated goal of Indonesiannationalconstructionis "... to realize a just and prosperoussociety with an equitabledistributionof materialand spiritualwealth based on the Panca Sila principles within a free, sovereign, united, and democraticRepublic of Indonesia."(Outlines of State Policy, 1983) This is to take place "... in a secure, peaceful, orderly and dynamic atmosphereamidst a free, friendly,orderly and peaceful world community."(Outlinesof State Policy, 1983) One basis for supportof efforts to fulfill the goals of national constructionis in the statementof goals for Indonesianeducation. Nationaleducationis to be basedon the PancaSila principlesin orderto enhancethe devotionto the One God, improvingintelligenceandskills, strengthenreality,develop

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personality, and heighten the spirit of patriotism and love to the homeland. This is to develop citizens imbued with the ideal of construction,able to develop collective and self reliance, and possess a sense of collective responsibility in building the nation of Indonesia. (Outlines of State Policy, 1983)

Within this statement is the implied duty of all educational institutions to develop students' intelligence. Equally importantis the task of shaping students' attitudes to enable them to constructively participatein national construction. This implies that during the pupils' formal education, learning experiences should be directed towards the fulfillment of this task. A consequence of these goals is instruction oriented to the simultaneous development of pupils' intelligence and personality. The goals of Indonesian education can only be achieved through the collective effort of all persons involved in the educationalprocess. The nurturingand shaping pupils to become individuals with high ideals demands concentrated efforts driven by devotion and dedication to the nation of Indonesia. Teachers, while not solely responsible the students' education, must strive to develop positive values so that all students will be able to make constructive contributions to Indonesian society. All teachers share this responsibility. It is not acceptable to leave the task of nurturing pupils' personalities solely to teachers of Panca Sila ethics and teachers of religion. It is incorrectto assume that teachersin academic subjects, such as mathematics,are not needed and cannot actively participatein fulfilling these goals of education. Values inherentin academic subjects which can make a positive contributionto a student's developmentare best developed within that academic subject This is the most appropriateenvironmentto understandvalues and facilitate learningpositive societal values. Because a student's personalityis complex its developmentrequiresmultiple efforts. Single efforts will not succeed. Neither will a diversity of uncoordinated efforts. Therefore, this paper is devoted to shedding light on the following questions. 1. Does the development of students' personality and values apply to mathematics teaching, and if so, how? 2. Are there values inherent in teaching mathematics which are capable of contributingdirectly to the achievementof broadergoals of education? 3. Are there values inherentin studying (and teaching)mathematicswhich can constructively contribute to the development of students' personality, values, and intelligence? 4. Is studying (and teaching) mathematicscapable of adding to a students' skills and knowledge, at the exclusion of "valueseducation?" The views implied on these questions are prevalent in the design of In-

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donesianeducation.Becausepersonalityandvaluesarecloselyrelated,a general discussion of values and personalitywill be presentedfirst. This will be followedby discussionspecificto thestudyandteachingof mathematics. Lastly there will be discussionof specific topics in mathematicsand how teaching thesetopicscancontributeto "valueseducation."

I. VALUES

Values cannotbe removedfromthe environmentwithinwhich the valuesare held. Whatis valuedin one society will not necessarilybe valuedin another society.An individualin one communityis unlikelyto attachthe samevalueto certainbehavioras anotherindividualin thatsamecommunity.Valuesconflict occursamongindividualsas well as withinindividuals.This does not implythat it is impossiblefor a communityas a wholeto holdcertainvalues.Mostnations, eitherformallyor informally,adhereto a definitesystemof values.Indonesiais a good example.Indonesiawas foundedon the five principlesof PancaSila (see above). It is difficultto definewhata valueis. Someconceptof "good"and"bad"is necessary.The followingdefinitionsof valueshavebeenposed. "A value is an idea - a concept - about what someone thinks is important in life." (Fraenkel,1977) "Values are ideas about the worth of thinking,they are concepts, abstractions."(Fraenkel, 1977)

Valuerefersto an ideaor conceptabouttheworthof something.Valuesmay be dividedinto two categories,estheticand ethicalvalues.Estheticvaluesare relatedto objectsof beauty,whereasethicalvaluesareconcernedwith objects which can be valuedas good or bad, specificallygood or bad with respectto behavior.Educationand valuesare inseparable. Educationaccompaniespupils in theirdevelopmentto maturity.It enablesthemto constructively join society. In a moregeneralsense, consideration of the worthof a somethingimplies philosophicalvalues. The statement"Manproposes,God disposes."implies philosophicalvalue.It denotesthat"manexistswithinlimits?"andappliesto all societies.However,therearephilosophical valueswhicharelimitedto a specific community.Alternatively,values can be differentiatedinto culturalvalues, practical values, educational values, and historical values. It is in this connec-

tionthatlaterdiscussionattemptsto addressthequestionspresentedabove.

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There are many theories of personality.The following discussion of personality is not meant to exhaust these theories. Neither is the purpose to present a definitive theory of personality. The mentioning here of a several topics about personality is intended merely to present a base for later discussion of how teaching mathematicscan contributeto "values education." In this paper the terms personality and character are considered equivalent. The basic assumption throughoutthe discussion is that personality (character)is amenable to education. Freud's psychoanalytic theory of personality is one of many theories of personality. Gleitman (1981, p. 463) states that "Freud's emphasis was on the development of motives and emotions, and concentrates upon the distinction between desire and attainment,on the child's growing awareness that he or she has to do something to make a wish come true." According this theory, a child's basic personality structureis fixed by age six. Furtherdevelopment refines the basic structure.Efforts to shape a student's personality after age six should be directed to strengtheningalready existing positive personality traits,and weakeningnegative traits. According to Freud,personalityhas threeparts: 1. The Id, consisting of basic biological urges (immediate satisfaction, physical pleasure); 2. The Ego, which confronts reality, satisfaction and pleasure within the constraintsof the real world; and 3. The Super ego, which internalizes rules of good and bad as dictated by societal morality.

The id acts for the returnof pleasure to reduce tension arising from stimulation. The ego acts within reality in repressing the demands of the id. The super ego is the vehicle for morality and seeks perfection. The basic functions of the super ego are: A. B. C.

To block influences of the id which are not in accord with the norms of society or the environment; To encourage the ego to adheremore to morality thanreality;and The pursuitof perfection.

Freud (1933, p. 66) said "The past, the traditionof the race and of the people, lives on in the ideology of the super ego; and ... plays an importantpart in human life!" Although there have been criticisms of Freud's theory of personality,a large

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part of contemporary psychology is based on this theory, including psychoanalysis. To be consistent with these factors, the nurturingof a pupils personality (especially super ego) should be directed toward achieving harmony within oneself and society. For Indonesian education this means being consistent with the principles of Panca Sila. Students should become citizens with a positive personality,be self-directed,loyal, and preparedto carryout activities consistent with the principlesvalued by society. Efforts directed toward shaping such a person proceed throughthe development of philosophical values which start with the forming of habits. These habits are; A. realizing the need for and respecting the existence of norms, B. consideration of the consequences of decisions before the decisions are made, and C. reducing tension caused by contradictions within oneself or between oneself and the environment. The following discussion of the nurturing of a student's personality will contributeto answeringquestions mentionedin the introduction.

m. MATHEMATICS ANDEDUCATION Mathematics can be considered in at least two ways, as "science" and as teaching material. The quotation marks about science are to remind the reader that 'mathematicsas science' is an opinion based on a given set of criteria on what "science" is, and that mathematics is not necessarily a "science" by all criteria. A. Mathematicsas Science Mathematicshas as its main concern abstractobjects - facts, concepts, operations, and principles - created by man himself (Begle, 1979). It has been said that since mathematicsis createdby man, mathematicsshould not be difficult to comprehend;and because of its abstractnessit can be comprehendedonly with difficulty. BertrandRussell said "... mathematicsmay be defined as the subject in which we never know what we are talking about, or what we are saying is true."(Newman, 1956, p. 4) Moder mathematics is a deductive-axiomatic structure.Some of mathematics is structuredhierarchically, so that without understandingprerequisite parts it is difficult to comprehendlater parts. The deductive axiomatic structure of mathematicsis based on agreements about undefinedterms, definitions, and logical rules. Once an agreementabout a set of "objects"is reached,deductions about those objects must be capable of being explained or proven by applying

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those agreementsin a logically rigorousmanner.Departurefrom accepted whichareunacceptable andrenderthe structure agreementscausecontradictions invalid. Patterns in mathematicalthought exist which facilitate understanding Thesepatternshelp in describingotherpartsof differentpartsof mathematics. mathematics. The abstractnessof mathematical objects,the precisenessof agreementsin mathematics,andthe rigornecessaryin applyingthe deductivemethodenables the creation of abstractstructuresin mathematics.The existence of these structures,aided by generalizations,enhances the broad applicabilityof mathematicsas a tool for all branchesof scienceandotherfields of knowledge, includinglanguageandcommunication. B. Mathematicsas TeachingMaterial Many abstractobjects of mathematicsas a "science"do not have concrete Yet an understanding of abstractconceptscan be facilitatedby reperesentations. real objects.Thereare mathematical modelsof reallife situationsand thereare A commonpedagogicaltechnique structures. physicalmodelsof mathematical of dealingwith the concretebeforeabstraction applieshere.The characteristics of real objectscan be used as a startingpoint in comprehending the abstract of This mathematics. is in objects approach appropriate educationbecausethe recipientsof mathematicsas teachingmaterialare studentsin the processof development.Since mathematicsas teachingmaterialis an externalstimulusit creates tension in the learner.This tension motivatesstudentto learn new methodsto reducetension,theyadapt.Teacherscan contributeto the reduction of tensionby using a combinationof modelsof teaching,such as "advance organizers"(Ausubel, 1963) "inductivethinking"(Taba, 1966) and "stress reduction"(JoyceandWeil, 1980,p. 388). By using calculatedexternal motivation,adjustedto the stages of the student'sdevelopment,internalmotivationis encouraged.Internalmotivationis moreeffectivethanexternalmotivation(Skemp,1975).Effortsto encourageand maintaininternalmotivationresultin studentswho will morelikelyachievethe cognitive,affective,andpsychomotorgoals of education.If internalmotivation is achieved,mathematicsas teachingmaterialcan contributeto achievingthe goalsof education. Though efforts are made to "simplify mathematics,"the fundamental of mathematicsare not abandonedin the process.If they were characteristics wouldbe lost. Abstractmatheabandoned,importantfeaturesof mathematics matical objects (undefined terms, definitions, generalizations,structure, of contradiction, conclusions,etc.) are still necesprocedures,unacceptability sary.

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IV. VALUESIN MATHEMATICS EDUCATION The following are example topics from mathematics in which values are inherent.In each example a brief discussion of the interrelationsof values with the topic is given. In the examples and discussion the educationalfunction of the mathematics topic in improving the intelligence of the pupil is not the purpose of giving the example. It is assumed that such a function is accepted and thereforedoes not constitute the main topic of this paper. A. Universe In every mathematicalproblem it is importantto understandthe existence of a universe for the problem. The solution to a given mathematical problem will likely be different within differentuniverses. EXAMPLE IV.A.1 Give the solution to the equation, (X + 1)(X - 2)(X - 3) = 0, if X is a member of each of the following universes. Universe (1) real numbers (2) even numbers (3) naturalnumbers (4) properfractions

Solution (1) {-1,2,3) (2) (2) (3) (2, 3) (4) void

EXAMPLE IV.A.2 1 + 1 = ? in the following universes? Universe (1) Real numbers (2) Electrical circuits (3) Binary numbers

Solution (1)1+1=2 (2) 1 + 1 = 1 (3) 1 + 1 = 10

EXAMPLE IV.A.3 In metric space (R2,d) constructa figure of neighborhoodN( r) if (1) where

d(p,q)

= (P - ql)2 + (P2- q2)2

(P1 P2) in R2 (2) where

d(p,q)

= I(p - ql)l + I(P2- q2)

(q, q2) in R2

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Solutions: (1)

(2)

!A A

R

X

~~4

t

Clearly,these examplesshow thatperceivedtruthis "relativetruth."Thatis, somethingis trueonly withinthe universein whichit is considered.Whethera solution applies to a given mathematical problem depends on the universe considered. Not considering the universe of problemcan lead to misunderstanding, misleading routes to solution, an unconvincing or incorrect solution. The latterresult can have serious consequences. In studying mathematics,students should always consider the question of the universe of a problem, both in its narrowand its broad forms. Teacher's should encourage students to consider the universes that apply to any problem and the resulting alternative solutions. This promotes an awareness of the environment or community in which one is located, found, or applies. The solution for a specific problem is valid if it is in accord with the environmentor community (universe) in which the problem is considered. Furthermore, examining the concept of "universe" further develops the student's consciousness of limitations, i.e. the limitationsof the environmentin which the problem is considered, and ultimately the society in which the student exists. Such consciousness can reduce tension. This implies an educationaland philosophical value thatreducing tension is desirable. B. Agreement/Conventions Mathematics is replete with agreements about concepts, facts, and operations. Agreements in mathematics depend on the freedom to define mathematical "objects"as necessary to accomplish desired goals.

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EXAMPLE IV.B.1 The most common numericalsymbols, 0, 1,2,3,4,... are a nearly word-wide convention used from pre-elementaryschool throughout life. Also, the symbols for numericaloperationslike addition(+), subtraction(-), multiplication(x or .), etc., are by convention widely used. The notationused in mathematics, although not completely uniform, is one of the most widely used notation systems. Realizing that these symbols are an agreement, students can consider questions such as; Why is the numbersymbolized as '1'? Is anothersymbol permitted? The answer to the latter question is clearly YES. But asking and answering this latter question should result in considering other questions, such as; "Is there a need for using another symbol?" and "What are the consequences of using anothersymbol?" EXAMPLE IV.B.2 The priority of operationswithin a mathematicalexpression, when order is not otherwise indicated (which caused consternationin the past), is also an agreement (convention) among users of mathematics.Questionsdo arise however. Must multiplicationand division be given priority to addition and subtraction? Can it be otherwise? Is a change in the prioritypermitted? The answer to the latter two questions is YES. The importantconsiderationis the intended meaning of the expression, and avoiding misunderstanding. Avoiding misunderstandingis a socially desirablevalue, and is a consequence of the use of conventions. EXAMPLE IV.B.3 What is the value of 3 * 5, if * means the sum of two times the first numberand five times the second number? A problem of this kind is frequently encountered in secondary school mathematics.The result of the operation(*) dependson the meaning (definition)

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of thesymbol*. For themeaningmentionedabove; 3 * 5 = 2(3) + 5(5) = 6+ 25 = 31.

EXAMPLEIV.B.4 If f: x -x2 and g:x --3x, whatis the resultof f g (' *' is the compositionof functions)?For a properresponse,the meaningof the symbol ( *) must be determined.If the symbolis given the meaningof "mapping" g followedbyf, thentheresultwill be; f .g(x) =92. But if the symbolis given the meaningof firstmappingf thenfollowedby g, thentheresultwill be; f

g(x) = 3x2.

Fromthe above fourexamples,it can be seen thatin mathematicsone has the freedomto define"things"as desired,butonce an agreementhas beenreached, adherenceto the definitionis requiredto avoidmisunderstanding andmiscommunication.Latermathematical workmustbe guidedby this agreement.This to progressrapidly. freedomof definitionhasallowedmodemmathematics Discussing definitionaland notationalfreedomwithin mathematics,and discussingthe need for conventionsnurturesthe habitof responsiblefreedom. Respectfor existingagreementsand normsof society is relatedto the definitional freedomone has in developingmathematics.Consideringdefinitional freedomand resultingconsequencesin mathematics can encouragerecognition andunderstanding of socialnormsandencourageresponsiblesocialdiscipline. In Indonesiain particular, the mathematics teachershouldrelatemathematical agreementsto the highestagreementamongtheIndonesian people,thePanca Sila, which is the foundationof the nation. Building commitmentto and of agreements(conventions)in mathematicsfacilitatescommitunderstanding ment to a social structure.Valuesin mathematicsthus help to developsound overallpersonalitiesandresponsiblesocialcontribution. C. Contradiction Contradictoryresults are not permittedin mathematics.If contradictionis structure,the structurecollapsesand is therefore permittedin a mathematical The law of "dichotomy" from logic can be appliedto illustrate unproductive. thisstatement.

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EXAMPLE IV.C.1 The logical statement"p ->(not-p --q)" as an implicationis true irrespective of the truthvalue of p or q. By applying "modusponens" the following derivation can be carriedout. a) p ->(not-p -4q) b) p c) not-p ->q d) not-p e) q

always true if true true by modus ponens if true trueby modus ponens

If it is assumed that p and not-p are true, the result is that statement q is true, irrespectiveof the meaning and truthvalue of statementsp and q. If meaning is given to each symbol of the statementa contradictionis seen. Substitutefor p and q as follows. Substitutefor p the statement: "Corruptionis bad." Then not-p will have the meaning: "Corruptionis not bad." If these two statements are considered true (assumed in the process) then we may substitutefor q the statement: "Slanderoustalk is good." which may be valued as true. By the derivation above, q can also have the meaning: "Slanderoustalk is not good." which is also valued as true. What does this imply for the humancommunity?Will this cause confusion of values in social relations? Obviously if contradictionis accepted in a community, then that community will be deprivedof its guiding principles or values. The sensible applicationof values containedin the above mathematicallogic (with assumptionsin conformity with its base in the law of dichotomy) will help develop qualities within currentnorms,and avoid contradiction. D. Transformation In mathematicsthere are many formulaswhich translateone set of conditions in one universe into anotherset of conditions in anotheruniverse. Several transformation formulas may be used to change one quantityinto anotheror to translate a relation in one universe into anotherform in anotheruniverse.

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EXAMPLEIV.D.1 Whatis the valueof 2356expressedin denary(base10) numbers? formula By usingthetransformation abcp = a(p2) + b(pl) + c(pO)

we obtain; 2356 = 2(62)+ 3(61)+ 5(6) = 72 + 18 + 5 = 9510

EXAMPLEIV.D.2 Thefollowingtransformation formula x = r. cos(A) y = r.sin(A)

can be used to translatepolar coordinates(r, A) into equivalentCartesian coordinates(x, y). Likewise the formulasbelow can be used to transform Cartesiancoordinates(x,y) intoequivalentpolarcoordinates(r . A). r = (2 + y2)/2

A = ArcTan(x/y)

There are many other examplesof mathematicaltransformations. By using transformationformulas teachers and students can consider mathematical situationsfromone universein another.Presentingandapplyingtransformation formulascanawakencuriosityin transferring somethingfromone "community" into another.Suchtreatmentcan createan awarenessof the need for meansof communication amongindividualsanddifferingsocialgroups. formulasin mathematics, studentscan derive Using differenttransformation seemingly differentforms or images, when in fact they are mathematically identical. Indeed, the applicationof the concept of transformation contains estheticvaluesespeciallywithinthestudyof topicssuchas geometry. E. Analogy

In mathematicstheteachercanpayspecialattentionto analogies,i.e. similarities betweensets of circumstances withrespectto bothformandprocedure. EXAMPLEIV.E.1 In the"algebraof statements" (Booleanalgebra)we havethelaw not(p v q) = not-p

A

not-q

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In the "algebraof sets" we have the law (P u Q)C=C

) QC.

EXAMPLE IV.E.2 Analogies can be useful in proofs of theorems. For example, the proof of "de Morgan's law" for an infinite numberof open or closed groups. Dealing with analogies in mathematicsfacilitates the pupil's comprehension of mathematicalconcepts and principles. The teacher's active incorporationof extra-mathematicaldiscussion of mathematical the concepts of analogy and deductive reasoning can contributeto an attitudeof acceptance of societal rules as applied to individuals, including oneself. Pupils may become more familiar and comfortable, and exhibit greater mutual respect. This is implied in the Golden Rule: "Do unto others as you would have others do unto you." which is valued in education and society.

CONCLUSION In the discussion above a variety of topics in mathematicsare mentioned which in part answered the questions suggested in the introduction. 1. Does the development of students' personalityand values apply to mathematics teaching, and if so, how? 2. Are there values inherent in teaching mathematics which are capable of contributingdirectly to the achievementbroaderof goals education? 3. Are there values inherentin studying (and teaching)mathematicswhich can constructively contribute to the development of students' personality, values, and intelligence? 4. Is studying (and teaching) mathematicscapable of adding to a students' skills and knowledge, at the exclusion of "valueseducation?" Values can and must be investigated more intensively and extensively through systematic investigation than is possible in this paper. It seems however, that teaching mathematics does not solely enhance students' intelligence, the cognitive function. Mathematicscan contribute to the development of pupil's individuality. The question which presents itself is: How can a mathematics teacher actively encourage positive values while maintaining the cognitive function? Developing students' habits oriented towardpositive values, as discussed in part IV, encourages reductionof students' internaltension. This helps to develop

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the pupil's personality. However, transfer of learning, whatever the content (including values education) does not take place unless the teacher actively teaches so that the learning will transferto other circumstances.Transfer does not happen automatically. Mathematics teachers can actively contribute to a students' "values education" within the context of mathematics.Discussion of value laden mathematicaltopics in light of the values inherentin their use and discussing in addition their applicabilityin the more general universe of societal and individual interaction can be a positive step in a students' personal and cognitive growth.

REFERENCES Beeby, C.E.: 1979, Assessmentof IndonesianEducation:A Guide in Planning,New ZealandCouncilforEducational Research,OxfordUniversityPress,Wellington,New Zealand. Education:Findingsfroma Surveyof Begle, E.: 1979,CriticalVariablesin Mathematics Council of EmpiricalResearch,MathematicalAssociationof America/National D.C. Teachersof Mathematics, Washington, Wm. C. BrownandCompany, Bell, F. H.: 1981, TeachingandLearningMathematics, Dubuque,Iowa. Fraenkel,J. R.: 1977,Howto TeachAboutValues:AnAnalyticApproach,Prentice-Hall, EnglewoodCliffs,New Jersey. Lectureson Psycholanalysis, Translated Freud,S.: 1933,New Introductory by J. Stracky, W. W.: 1965,Norton& Company,New York. Gleitman,H.: 1981,Psychology,W. W.Norton& Company,New York. Indonesia. Ministryof Education:1976,IndonesiaHandbook,Jakarta, Inc., Joyce, B. and Weil, M.: 1980, Modelsof Teaching(2nd Edition),Prentice-Hall, EnglewoodCliffs,New Jersey. MPR-RI:1981,Garis-GarisBesarHaluanNegara(Outlinesof StatePolicy),Percetakan Indonesia. NegaraRepublikIndonesia,Jl. Percetakan Negara21, Jakarta, Metcalf,L. E.: 1971, ValuesEducation:Rationale,Strategiesand Procedures,41st D.C. Yearbook,NationalCouncilof SocialStudies,Washington, SimonandSchuster,New York. Newman,J. R.: 1956,TheWorldof Mathematics, Skemp, R. R.: 1975, The Psychologyof TeachingMathematics,Penguin Books, Middlesex,England. Taba,H.: 1966, TeachingStrategiesand CognitiveFunctioningin ElementarySchool Children,CooperativeResearchProject2404, San FranciscoState College, San Francisco,California. MARC SWADENER and School of Education Universityof Colorado Boulder, Colorado 80309-0249 U.SA.

R. SOEDJADI Fakultas Pendidikan Mathematikadan Ilmu PengetahuanAlam InstitutKeguruandan Ilmu Pendidikan Surabaya,Indonesia

K. C. CHEUNG

OUTCOMES OF SCHOOLING: MATHEMATICS ACHIEVEMENT AND ATTITUDES TOWARDS MATHEMATICS LEARNING IN HONG KONG ABSTRACT.High achievementof cognitiveskillsand the formationof favourableattitudes towardslearningareuniversallyacclaimedoutcomesof schooling.Thepresentstudyutilisedthe SecondIEA MathematicsStudydata(N = 5644)to examinewhichof the ten measuredattitude dimensionsarepertinentin explainingmathematicsachievementvarianceof FormOnestudents in Hong Kong. Correlationaland commonalityanalysisrevealedthat students'perceptionof theirabilityto do mathematics,the importanceof mathematicsto societyand the conceptof mathematicsbeinga creativesubjectare the most pertinentattitudedimensions.In particular, theperceptionof students'estimatedabilitiesto do mathematics madea substantial,uniqueand commoncontributionto the explainedmathematicsachievementvariance. INTRODUCTION

In past decades,researchworkershave triedwith some successto explain, evaluateand recommendvariouspracticesof schooling,based on hypothesisedconceptuallearningmodels.Thesemodelsrepresentan importantline of developmentin the mainstreamof researchon school learningin the two decadesfrom 1960to 1980(Noonan and Wold, 1983).Implicitor explicitin thesemodelsare some preferredoutcomesof schoolingwhicheducatorsand citizens highly value and endorse. Among those which have receivedthe greatest attention are the attainmentof academic skills and attitudinal outcomes. Cheslerand Caves (1981) devoted a chapterin their book to elaborate variousoutcomesof schooling.They assertedthat attitudinaloutcomesrelate to the socialisingfunctionsof the schoolsin compassingthe diversityof humanpersonality.School-relatedand society-relatedattitudesare the two majorgroupsof attitudesthat nearlyeveryschoolfocuseson. Typicalexamples of these attitudesare the views that studentshold about themselves, theirpeers,theirfamily,the educationalprocesses,the social issues and the broadersociety. They pointed out that attitudinaloutcomes consist of a numberof dimensions.For the purposeof this articleattitudewill referto affectivelytonedperceptionsof situationsin whichmathematicsis learnedas well as to views of mathematicsas a subject. In a reviewarticleon attitudestowardsmathematics,Aiken (1985) concludedthat attitudestowardsmathematicsbegindevelopingas soon as childrenare exposedto the subject,but the junioryears(age 11-13) appearto be particularlyimportant.He assertedthat this is the time when negative Educational Studies in Mathematics 19 (1988) 209-219. ? 1988 by Kluwer Academic Publishers.

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attitudes towards mathematics become especially noticeable. Whether the increase in negative attitudes at this stage of development is due to greater abstractions of the mathematical material to be learned, to social/sex preoccupations, or to some other factor is not clear. The investigation of these attitudinal outcomes may result in the improvement of mathematics learning. The present study sought to use the Second IEA Mathematics Study data (1980) to examine the relationship between mathematics achievement and attitudes towards mathematics in junior secondary schools in Hong Kong. The specific research questions were: (1) What attitudes towards mathematics learning are most intimately related to mathematics achievement? (2) To what degree does the measurement of such attitudes explain variance in mathematics achievement, uniquely and in common? SOURCE OF DATA

The sample of students studied in this article was drawn in the Second IEA Mathematics Study (SIMS) which was conducted in Hong Kong in 1980 (Brimer and Griffin, 1985). Data were obtained from 130 Grade 7 classes (Modal age = 13). Within each class all students were tested and achievement test and attitudinal data were obtained for 5644 students. The sampling design permitted analyses at the between-student level where there was a response rate of 94%. THE OUTCOME MEASURES

Mathematics achievement within the internationally defined curriculum was measured using specifically designed tests by the SIMS International Project Committee. After scaling the tests, a total score which represented the students' mathematics achievement in their formal curriculum was calculated. The descriptive summary statistics are shown in Table I. Ten, 5-point Likert scales, with 5 indicating the most positive views towards mathematics were used to elicit student's attitudes and perceptions on (1) Perceived Home-Support (HSSUP), (2) Perceived Home-Process (HSPRO), (3) Mathematics-Importance (IMPT), (4) Mathematics-Easy (EASY), (5) Mathematics-Like (LIKE), (6) Mathematics-Create (CREATE), (7) Mathematics-Rules (RULES), (8) Mathematics and Myself (MYSELF), (9) Mathematics and Society (SOC) and (10) Sex-Stereotyping (SEX). The scale directions of RULES and SEX have been reversed because

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TABLE I Descriptive statistics (N = 5644) of mathematics achievement of form one students in Hong Kong (1980-81) Maximum Possible Score = 74 Mean = 37.6 Standard Deviation = 14.2 Skewness = 0.07 Kurtois = -0.93

viewing mathematics, not as a set of rules, nor a male-domain subject, were construed as poles of their attitude dimensions (Lau, 1986). The rationale for measuring these attitude components have been reported in the published National Report (Brimer and Griffin, 1985). The description of the scales and sample items are included in Table II. Scale statistics such as mean, standard deviation, valid N, Alpha-reliability and discriminant validity are also tabled. Factor analytical procedures have also been performed (Lau, 1986) to assess the construct validity of the scales. All these analyses revealed that the ten attitude measures exhibited satisfactory reliability (all except RULES) and high construct and discriminant validity.

RESULTS

In order to investigate the nature and the degree of relationship between mathematics achievement and various dimensions of attitude towards mathematics, scatter diagrams were plotted and Pearson correlations (Table III) were calculated. The results indicated that the correlation between the attitude dimensions and mathematics achievement were positive, showing that the more positive the students' attitudes towards mathematics, the higher their achievement in mathematics. The greatest correlation was associated with SELF, a measure of the students' own estimation of their abilities in doing mathematics; it attained a value of 0.42. Another two larger correlations were associated with SOC and CREATE (0.37 and 0.31), which measured the students' perception of the usefulness of mathematics in society and of mathematics as a creative subject. The other dimensions of attitude towards mathematics were associated with lower correlation coefficients. Of particular interest was the moderate correlation (0.15) of SEX with mathematics achievement. Further analyses indicated that there was significant gender difference (alpha level of significance = 0.05) between

TABLE II

Summaryof attitudemeasuresand theirscale statisticsof form one students(N = 564 Scale

1. Perceived Home-Support (HSSUP) 2. Perceived Home-Process (HSPRO) 3. MathematicsImportance (IMPT) 4. MathematicsEasy (EASY)

5. MathematicsLike (LIKE)

Description

It measures the parents' ability to help in doing mathematics homework (4 items) It measures the parents' interest and value placed on mathematics (4 items) It measures the students' perception of the importance of mathematics content and process components (15 items) It measures the students' perception of how easy the mathematics content and process components are (15 items) It measures the students' perception of how much they likes the mathematics content and process components (15 items)

Sample Item

Scale statisti Mean

S.D

2.97

0.8

4.13

0.8

3.71

0.4

Using charts and graphs is easy

3.21

0.5

I like solving word problems

3.29

0.5

My father would usually be able to do my mathematics homework problems if I asked him to help My father thinks that learning mathematics is very important Memorising rules and formulae is important

6. Mathematics- It measuresthe students' Create perceptionof the creativity in mathematics(7 items) (CREATE) 7. Mathematics- It measuresthe students' Rules perceptionof mathematicsnot as a set of rules(5 items) (RULES)e 8. Mathematics It measuresthe students'own and Myself estimationof theirability in doing mathematics (SELF) (18 items) 9. Mathematics It measuresthe students' and perceptionof the usefulness of mathematicsin occupation Society and everydaylife (7 items) (SOC) 10. Mathematics It measuresthe students' and Sexperceptionof mathematics as a non-maledomainsubject Stereotyping (SEX)e (4 items) a

Mathematicsis a good field for creativepeople

3.61

0.50

Mathematicsis a set of rules

2.94

0.49

Mathematicsis harderfor me than for most people

3.39

0.46

3.61

0.58

2.45

0.7

Mathematicsis usefulin solvingeverydayproblems Boys have more natural abilityin mathematics than girls

Scalemeasuresrun from 1 to 5, with 5 indicatingthe most positiveattitudetowardsmathematics. Valid N refersto the numberof studentswho have non-missingdata at least on 80% of the itemsof w c CoefficientAlpha reliability. d Discriminant validityis the meancorrelationof a scale with the other nine attitudescales. ' Directionof scale has been reversed. b

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TABLEIII Pearson correlations of mathematics achievement and attitudes towards mathematics (N = 5644)

r

Attitudescale 1. PerceivedHome-Support 2. PerceivedHome-Process 3. Mathematics-Importance 4. Mathematics-Easy 5. Mathematics-Like 6. Mathematics-Create 7. Mathematics-Rules 8. Mathematicsand Myself 9. Mathematicsand Society 10. Mathematicsand Sex-Stereotyping

(HSSUP) (HSPRO) (IMPT) (EASY) (LIKE) (CREATE) (RULESa) (SELF) (SOC) (SEXa)

0.007 0.166 0.193 0.185 0.098 0.306 0.164 0.424 0.374 0.150

a Directionof scalehas been reversed.

the mean SEX scores (Boys = 2.11; Girls = 2.79). Boys generally viewed mathematics as a male domain, and they viewed themselves as haivng more natural ability than girls. The Pearson correlations between SEX and mathematics achievement were 0.107 and 0.250 respectively for boys and girls. These figures gave a moderate indication that high-achieving students tended to regard mathematics as a non-male domain and this tendency was stronger amongst girls than boys (see Table III). In order to understand better the relationships of the attitude variables CREATE, SELF and SOC to mathematics achievement, commonality analysis (Mayeske, 1970) and correlations amongst the outcome measures were calculated and the results are shown in Table IV. The Pearson correlations among these attitude variables and mathematics achievement indicated that about 18% of the variance in mathematics achievement was accounted for by SELF, also SOC and CREATE each contributed 14% and 9% to the variance of mathematics achievement. However, these variables were intercorrelated, with the correlations ranging from 0.39 to 0.48, resulting in commonly shared achievement variance. In total 22.6% of the achievement variance was associated with CREATE, SOC and SELF taken together. Reading down each column in Table IVb, it is possible to note how the proportion of variance accounted for by a given variable was partitioned into various components. The proportion of mathematics achievement variance accounted for by CREATE, was partitioned as follows: 0.008 unique to CREATE, 0.022 common to SELF and CREATE, 0.010 common to CREATE and SOC, and 0.054 common to CREATE, SELF and SOC. It

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TABLE IV Commonality analysis results of mathematics achievements - intercorrelations and variance components (a) Pearson correlations (N = 5644) Attitudes

X1

X2

X3

Create (X1) Self (X2) Society (X3) Achievement (Y)

1.000 .441 .391 .306

1.000 .481 .424

1.000 .374

(b) Percentage of explained mathematics achievement variance ATTITUDE MEASURES 1 CREATE Unique to 1 Unique to 2 Unique to 3 Common to 1 and 2 Common to 1 and 3 Common to 2 and 3 Common to 1, 2, and 3 Total explained variance

2 SELF

3 SOC

.008 .056 .028 .022 .010

.022

.054

.048 .054

.010 .048 .054

.094

.180

.140

was evident that CREATE made very little unique contribution; most of the variance accounted for by CREATE (0.094) was due to its commonalities with the other attitude variables. Similar results were obtained for the proportion of achievement variance accounted by SOC: Only about 0.028 of the total variance was unique to SOC; a greater proportion (0.112) of the total variance accounted for by SOC was due to its commonalities with the other atttitude variables; with 0.048 common to SELF and SOC, 0.010 common to CREATE and SOC, and 0.054 common to CREATE, SELF and SOC. In contrast, SELF showed relatively a large unique contribution of about 5.6%, with the remaining 12.4% in common with the other attitude variables. The reciprocal nature of the relationship amongst the attitude variables CREATE, SELF, and SOC with mathematics achievement (ACH) were examined by three other commonality analyses, taking CREATE, SELF, and SOC in turn and the results were shown in Table V. Similar interpretations, as those in the preceding paragraphs, could be made to note how the proportion of the variances of the three criterion attitude variables

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TABLEV Commonalityanalysisresultsof attitudestowardsmathematicslearning- explained variancecomponents (a) CREATE Explanatoryvariables 1 ACH Unique to 1 Unique to 2 Unique to 3 Commonto 1 and 2 Commonto 1 and 3 Commonto 2 and 3 Commonto 1, 2, and 3 Total explainedvariance

2 SELF

3 SOC

0.008 0.061 0.033 0.022 0.009

0.022

0.055

0.057 0.055

0.009 0.057 0.055

0.094

0.195

0.154

(b) SELF Explanatoryvariables 1 ACH Unique to 1 Unique to 2 Unique to 3 Commonto 1 and 2 Commonto 1 and 3 Commonto 2 and 3 Commonto 1, 2 and 3 Total explainedvariance

2 CREATE

3 SOC

0.047 0.053 0.067 0.023 0.046

0.023

0.065

0.054 0.065

0.046 0.054 0.065

0.181

0.195

0.232

(c) SOC Explanatoryvariables 1 ACH Unique to 1 Unique to 2 Unique to 3 Commonto 1 and 2 Commonto 1 and 3 Commonto 2 and 3 Commonto 1, 2 and 3 Total explainedvariance

2 CREATE

3 SELF

0.026 0.031 0.072 0.010 0.046

0.010

0.059

0.054 0.059

0.046 0.054 0.059

0.141

0.154

0.231

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accounted for by their explanatory variables were partitioned into variance components. The key results were summarised as follows: (1) The commonalities of both attitude variables in explaining their respective criterion attitude variables were consistently high in proportion (CREATE: 0.057; SELF: 0.054; SOC: 0.054). (2) The unique contributions of SELF were substantial in explaining the total variances of CREATE and SOC (0.061 and 0.072). The unique contributions of SOC and CREATE were substantial in accounting for the total variance of SELF (0.067 and 0.053), but with moderate unique contributions between themselves (0.031 and 0.033). (3) The unique contributions of ACH on the three criterion attitude variables were greatest for SELF, and weak for SOC and CREATE (0.047, 0.026, and 0.008 respectively). (4) The commonalities of ACH and the two attitude variables in explaining their respective criterion attitude variables were consistently substantial in proportion (CREATE: 0.055; SELF: 0.065; SOC: 0.059). (5) The commonalities of ACH and SOC, ACH and SELF were high in proportion in explaining SELF and SOC respectively (0.046, and 0.046), but were low in explaining CREATE (0.022, and 0.009). The commonalities of ACH and CREATE were small in proportion in explaining both SELF and SOC respectively (0.023, and 0.010). (6) 35.3%, 29.7%, and 24.5% of the total variances of SELF, SOC, and CREATE were explained by mathematics achievement and the other two attitude variables taken together. INTERPRETATION OF FINDINGS

If the students find mathematics useful in their daily lives, and through the activity approach that some teachers employ, then the students are more likely to consider mathematics to be a creative subject. Thus, the results of this study, which indicated that both CREATE and SOC were pertinent attitudes towards mathematics, tie in well with present views of mathematics educators in Hong Kong. As for the fact that SELF was found to be another pertinent dimension of attitude towards mathematics, it is plausible to consider that the more confident a person is, the better his performance, especially in academic subjects like mathematics. Individual help and supervision would assist students to restore confidence in learning mathematics and eventually to increase achievement in mathematics. The commonality analyses results also made clear that the relationship of mathematics achievement and attitudes towards mathematics learning is

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reciprocal in nature. Hence it is not appropriate to depict only a direct casual link from attitude to achievement and not indirectly or the other way round. The interrelated nature of the various outcomes of schooling, and the greater proportion of commonalities than uniquenesses in explained outcome variances reflect that these outcomes of schooling are mutually facilitating and inhibiting as in a cybernetic system. Teachers in Hong Kong tend to place a premium on academic achievement and less on attitude cultivation because of its visibility in the achieved curriculum. Thus, the findings that SELF, SOC, and CREATE were most intimately related to mathematics achievement signalled some areas for classroom teachers and curriculum planners to think about current practices so as to redress their emphases more on fostering favourable attitudes as a means to enhance mathematics achievement especially for the repeaters and under-achievers. CONCLUSION

The results of the present study indicated that the three attitude dimensions SELF, SOC and CREATE were the most pertinent dimensions in explaining the variance of mathematics achievement of Grade 7 students in Hong Kong. In particular, SELF showed a relatively large unique contribution of about 5.6%, with the remaining 12.4% being in common with other attitude variables. The influence, if the variables are malleable and responsive to enhancement, is that promoting the students' attitudes in these dimensions is likely to result in an increase in their achievement in mathematics in subsequent years of schooling. ACKNOWLEDGEMENTS

Acknowledgement is due to the IEA Hong Kong Centre for the permission to use the data for this study. Special thanks must be also be paid to Prof. M. A. Brimer, Prof. A. Lewy, Dr. L. Dawe and Mr. Crawford for commenting on earlier drafts of this paper. REFERENCES Aiken, L. R.: 1986,'Attitudestowardsmathematics',in Husen, T. and Postlethwaite,T. N. (editors-in-chief), The International Encyclopedia of Education, pp. 4538-4544. Brimer, M. A. and P. E. Griffin: 1985, Mathematics Achievement in Hong Kong Secondary

Schools,Centreof Asian Studies,Universityof Hong Kong. Chesler, M. A. and W. M. Caves: 1981, A Sociology of Education - Access to Power and

Privilege,MacmillanPublishingCompany.

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Lau, M. Y.: 1986, The Relationship of Parental Attitudes and of the Socio-Economic Status of Students to Students' Attitudes towards Mathematics, Unpublished M.Ed. Dissertation,

Universityof Hong Kong. Mayeske,G. W.: 1970,'Teacherattributesandschoolachievement',in officeof Education(ed.), Do Teachers Make a Difference, Government Printing Office, Washington, D.C.

Noonan, R. and H. Wold: 1983,'Evaluatingschoolsystemsusingpartialleastsquares',Evaluation in Education: An International Review Series, Vol. 7, No. 3.

Department of Education, University of Hong Kong, Hong Kong.

THOMAS S. POPKEWITZ

INSTITUTIONAL

ISSUES IN THE STUDY OF SCHOOL

MATHEMATICS:

CURRICULUM

RESEARCH'

ABSTRACT.Mathematicscannot be treatedsolely as a logicalconstructionor a matterof Whatis definedas schoolmathematicsis shapedand fashionedby psychologicalinterpretation. social and historicalconditionsthat have little to do with the meaningof mathematicsas a disciplineof knowledge.To understandschoolconditions,theessayconsiders(1) the socialand culturalissuesthat underliethe patternsof schooling;(2) the assumptionsand implicationsof curriculumlanguagesfor teachingmathematics,and (3) the contradictorymeaningof change and reformthat underliecurrenteffortsto improveinstruction.

This essay was originallywritten for the National Science Foundationto addressthe issue of assessingthe impactof the currentreformprograms.It respondsto an increasedconcernwithinthe United Statesabout the quality of mathematicseducationand the meansto improveinstruction.The intellectual concern of the essay is: How might the monitoringof educational reformsin mathematicsbe approached?What features of schooling and teachingshould be given priorityin the discussionof curriculumpractices? The approachis to considerthese questionsby viewing schoolingas an institutionalarrangement.The discussionassumesschoolingas an arbitary creation of society in which certain forms of knowledgeare vested with privilegewhile other knowledgeis omitted. That vestureof school knowledge has involved intense struggle,strainsand contradictionsas different socialinterestssoughtto bestowpreferenceto theirvisons of society.2From this perspective,the teaching of mathematicsis not a "natural"or "inevitable"to human progressor enlightenment,but a socially constructed enterprisein which its status and selection is derivedfrom the particular functionsof schoolingas an institutionof upbringingand labor selection. What becomesthe learningof mathematicalknowledgein schoolingmay have little to do with the formal logic elements of the discipline.School mathematicsinvolvesnot only acquiringcontent;it involvesparticipatingin a social world that containsstandardsof reason,rulesof practiceand conceptionsof knowledge.The social patternsof schoolconductarenot neutral but relatedto the largersocial and culturaldifferentiationthat exist in our societies. The problemof inquiryis to understandhow the teachingof mathematics is realizedwithina sociallyorganizedand constructedworldof schooling.It Educational Studies in Mathematics 19 (1988) 221-249. ? 1988 by Kluwer Academic Publishers.

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is to understand how our pedagogical signs, gestures and routines set conditions by which individuals are to be located in their social situation and events are to be interpreted. To understand the complex dynamics of pedagogical actions, three questions are given attention. They are: (1) What social and cultural issues underlie the institutional patterns of schooling? (2) What are the assumptions and implications of curriculum languages for teaching mathematics? (3) What do we mean by change and reform?How do educational change and reform practices illuminate or obscure the social conditions in which school knowledge is produced? The questions are posed in this manner in order to redirect attention to how we think about achievement, success and failure in mathematics education. This essay draws upon a "critical sociology." The scholarship seeks to develop a method of inquiry that involves an interplay of sociology, social philosophy and history.3 The concern is with knowledge as patterns of language, value and practice that help to shape our consciousness of the social world. The argument follows Durkheim's observation at the turn of the century that he knew of no instance in which theories of change have gone into practice without great modification, and unintended and unwilled consequences. 1. WHAT SOCIAL AND CULTURAL ISSUES UNDERLIE THE INSTITUTIONAL PATTERNS OF SCHOOLING?4

How can we think about the institutional rules that underline mathematics teaching? The concept of institution gives attention to the patterns of social conduct and value that give direction to school practices. Schools function according to rules and procedures that provide coherence and meaning to everyday activities and interaction. Such rules and procedures are embodied in the regularized patterns of behavior, specific vocabularies and particular roles that we associate with schooling. Teachers' social view of the children who come to school, their visons of child development, the collegial relations in the school as well as the norms of evaluation and school organization interrelate to define the patterns of teaching. Even language and terminology control the possible ways of viewing and interpreting what occurs around us. The language and other patterning mechanisms in school control even our preceptions of the issues. The significance of institutional settings is that they are so potent that the social structuring experienced in schools channels thought and actions of

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participants, giving definition and meaning to both school reform and pedagogical practices and knowledge. Institutional patterns evoke various symbol systems theories, folk knowledge, myth, and common sense ideas. These symbol systems make the ongoing activities, roles, and relationships appear as normal and reasonable ways to acting within the setting. The discourse about children's learning, the writing of classroom lessons and organizational procedures do not exist independently of a complex and ongoing social world of expectations, demands, attitudes, and emotions. The folk knowledge given to new teachers about the "practicality" of work with children establishes guidelines as to what is permissible and desirable in the role of teacher. The theories of learning are thought of as sensible statements because of the background organization, language and history about schools and classrooms.5 The activities help to establish criteria of relevance and value to the ongoing activities of schooling. Just as we have learned that altering the content of lessons does not necessarily alter what or how students learn, so must we pay attention to the interrelationship between content and the form of the institutions in which mathematics occur. School conduct is constrained not just by traditions in the mathematics field but also by the quality of other traditions that are given organization through the classroom. Changing school mathematics, from this perspective, involves more than merely incorporating some new practice and organizational relationship into an existing framework of mathematics education. Surface and UnderlyingMeanings Distinguishing between the surface and underlying meaning of schooling can clarify the importance of institutional patterns. The surface layer consists of the publicity accepted criteria or standards by which people judge success or failure. Writing a lesson objective, doing microteaching, or working in a team-teaching situation might provide such a public criterion. Simple seeming acts of classroom management and planning, however, are not isolated events. Presuppositions and "rules of the game" that form underlying layers of meaning give plausibility and legitimacy to the publically accepted criteria, existing prior to and defining the parameters of any specific new activity, such as an innovative program. Change requires an understanding of how the introduction of new practices interrelates with the existing structures of rules to challenge, modify or legitimate those arrangements. A single mathematics lesson observed in an U.S. inner-city elementary school can illustrate the relation of surface and underlying rules of

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institutionallife (Popkewitzet al., 1982). The students were black, from familiesof the industrialpoor and unemployed.6The stated and ostensible purpose of the lesson was to help students learn subtraction;the teacher wrote a lesson plan, constructedmaterials,and evaluatedaccordingto the previouslystated objectives.The lesson was justifiedfor differentreasons: subtractionis an importantelementof a mathematicscurriculum,and future lessons dependupon acquiringthe presentedknowledge.Duringthe lesson, the teacher explainedelements of subtraction,and studentsworked with textbooks and work sheets. Subjectmatter,however,was only one part of lesson content;the lesson carriedsocial messagesthat wereas importantas any overtinformation.The introductionto the lesson involved a discussionthat focused on the children's academicfailures.The discoursereflectedthe teacher'sfeeling that becauseof the studentswelfarestatus,theylikelypossessedundesirabletraits that neededto be overcomebeforeany achievementcould be obtained.The teacherstalked about the lack of learningas relatedto the welfarepsychology of the children'sfamilieswhereteachersperceivedno disciplineof hard work or value for school learning.Much of the classroominteractionwas relatedto the teacher'sbelief in the culturaland personalpathologyof the childrenratherthan to any textbooknotion of "learning." In this arithmeticlesson,we findnot only referencesto socialcircumstance but also a processof selectionof one of the possibledefinitionsof knowledge and childhood availablein schooling.The lesson on arithmeticwas based upon a "deficit"model of learning.Mathematicswas viewed as having a fixed and unyieldingdefinition.Teacherswere to fill the minds of the students, reflectingwhat Paulo Freire(1970) referredto as the "bankingconcept of education."Otherdefinitionsof knowledgeare possiblebut werenot available in the classroom discourse.For instance, one might identify a constructiveview, emphasizingknowledgeas emergingout of participation in communityand the activesymbolicmediationby individualsin the construction of knowledge. A view different from both constructivistand "deficit"modelsis a socialpsychologymodel,directionattentionto a dialectial relationamong culture,social settingand the developmentof the mind (Vygotsky, 1978). The discourseof this mathematicslesson can help us to unravelmore profound and complex relations that underlie the knowledge and social organizationof classroomlife. From the differentviewsof cognitionand the natureof knowledgeemergecontrastingviews of societyand polity (Popkewitz, 1983c).To treata childas a deficitis to defineknowledgeatomistically; the individualis an essentiallyreceptive,reflectiveorganismwhose qualities

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are shaped by the environmentover time. The epistemologyalso entails a political theory:the individualis denied the role of actor in the creation of history and culture. Social life is defined as fixed and unyielding to intervention. In contrast,a focus on negotiationin learninggives value to community and self as integrallyrelatedand mutuallyreinforcing.In mathematicseducation, for example, negotiation is seen as contributing to motivation (Hatano and Inagaki, 1988).The idea of negotiationis not neutralbut has particularsocial and culturalimplications.It emphasizes,in the U.S., an early twentiethcenturyliberalview which focuses upon the functionaladjustmentof individualsas they workedcollectivelyto improvetheir world. This notion of individualitywas an element of a political ideology that legitimatednew economicpatternsof professionalorganizationsand corporate industrialization(see Lukes, 1973). The psychologicalview of liberationthroughcommunityalso has its own limitations.The concernwith the presentobscuresthe place of history in fashioningconsciousnessitself.Ignoredarethe waysin whichour categories such as "learning,""individualization" and "community"are sociallyconstructedas responsesto changingpowerrelationsin societyand the articulation of new social agendas(see Braudel,1980;Napoli, 1981). Differingviews of cognition, society, and childhood allow us to understand differentlayers of the relationshipsamong mathematicspedagogy, psychologyand politicaltheoryin the discourseand practicesof classrooms. The examplesillustratethat whatseemsas simpleacts of classroomplanning or managementmay, in fact, contain profoundand complex principlesof authority,legitimacyand powerrelations.As our pedagogiesand psychologies about mathematicalreasoningand learningareincorporatedinto lenses of curriculum,they are not neutral:posited are relationsof individuality, knowledgeand society that reflectlargerconsiderationsof economics,culture and politics. The social patternsand discoursepracticesof schoolinghave important implicationsfor the conduct of mathematicseducation.The mathematics curriculumis not only aboutthe abstractionsof numbersor logicalstructure. Mathematicsis used as a part of a discourseof schoolingin whichthereare specificpatternsand practices.The importanceof the routinesand conversationshas little to do with mathematicseducation.The discursivepractices of the pedagogygive structureto whatstudentsareto learnand, at the same moment,givesorganizationto the mannerin whichteachersare to produce that learning.The topic, the organizationandthe social messagesall reflect assumptionsabout the natureof knowledgeas definedwithinthe confinesof

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schoolingwhicharenot necessarilythose of a mathematicsdiscipline(Donovan, 1983; Stephens, 1982). The sequencesgiven to lessons, the examples used to explaina concept,and the social/psychologicaltheoriesof children's growth embody epistemologicaland politicaltheoriesabout our world. An achievementof schooling is not necessarilythe formal mastery of content:it is givingdirectionto social thoughtand the formationof intelligence both for those who succeedand for those who fail. Whilecurriculum theorytends to view mathematicsas a universallanguagein whichthe logic of relationsto an answerbecomesparamount,such a conceptionobscures that manner in which the content is brought into a context of longterm patternsof institutionalizationand culturalhabits in schooling,translating and transformingthe content of mathematics.We can begin to understand from the urban elementaryschool mathematicslesson that the form and content of schooling are interrelated;they not only channel thought and action, but posit social values about authorityand control. Institutional Differentiation in Schooling

The importanceof socialprocessesin definingthe meaningand implications of school mathematicscan be consideredfurtherby focusingupon the rituals of homogeneityand differentiationin schooling.In most countries,pupils are taught mathematicsin ways that suggesta homogeneityof practiceand consensus of purpose. Theories and organizationof school mathematics implythat thereis a unified,universalpatternof behaviorand meaningthat underliesexperience.Everyoneis expected to go to school to be treated equallyand objectivelyin learningschool subjectmatter,and, if differentiation occurs, it is expected to be the result of merit rather than ascribed characteristicsof individuals.The problemof curriculumis what to select as content for all; in instruction,how to most effectivelyorganizethat content, or what technologiesshouldeveryonehave, such as the use of calculators or computers,"to insurethat everystudentbecomesfamiliarwith these importantprocesses"(The ConferenceBoardof the MathematicalSciences, 1982). While the ritualsand ceremoniesof schoolingcreatean illusionof homogeneity,the actual social transactionsin schools representdifferentiationin what is taught and learned(McLaren, 1986;DeLone, 1979). Rather than one commontype of school, therearedifferentformsof schoolingfor different people. These differentforms of schoolingemphasizedifferentways of consideringideas, contain differentsocial values, and maintain different principlesof legitimacyand forms of social control. Let me provide two

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examples, one historical, one contemporary. Each enables us to explore the issues of knowledge and social differentiation in mathematics education. Mathematics in the formation of the U.S. school. The public rhetoric about mathematics education suggests that the organization of knowledge gives emphasis to universal values of learning. Yet when considered historically, the actual construction of curriculum has reference to different social values. Stanic's (1987) discussion of the emergence of mathematics education in U.S. schools at the turn of the century places two types of instruction as central to the debate about curriculum purpose and organization. One concerned teaching children how to think and reason properly. This focus assumed that public school mathematics would provide the mental discipline and character appropriate for eventual leadership in social and economic institutions. The conception of mathematics education was elite and related to those who would go to college. A second curriculum orientation focused upon functional requirements for those who would never go to college. Mathematics education was to provide practice for managing everyday life, such as using arithmetic for household budgets. Each type of instruction involved research programs that justified and organized teaching, and in the process, issues of social differentiation were obscured as the problems of instruction were made to seem as scientific questions of individual development and learning. In each instance, the pedagogical arguments and science were related to issues of social transformation and institutional development. The period between 1880 and 1920 was a time when mass secondary schooling was becoming a reality in the United States (Popkewitz, 1987). The two strands of mathematics education contained different views of the probable destination of the child, the type of harmonious society that the school was to produce, and a conception of labor socialization. The two views of mathematics pedagogy represented competing views within Protestant middle classes and elites about the function of the schools. The new schools were incorporating immigrants from Southern and Eastern Europe who were viewed with suspicion because they did not speak English or practice the appropriate religion. Professional educators also were trying to attract the middle classes into the public schools (Peterson, 1985). Among segments of business groups, in the social gospel movement in Eastern U.S. Protestant churches and among certain progressive educators, the elite knowledge was to be taught to all to promote social mobility and upbringing that reinforced the views of the polity; for other elements of these groups, education was to promote social utility by helping people learn their place in life and work.

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The importance of the different curriculum approaches has at least two dimensions for consideration in our contemporary pedagogy and statements about curricular standards in mathematics education. First, the differentiation in the work and knowledge of schooling represents different sensibilities and awarenesses necessary for access to positions of privilege and status in society. The social organization of schooling transmits the cultural and social awareness appropriate for a society which has different roles, status positions and occupational tasks. Blue-collar workers, shopkeepers and scientists do not need, on the surface, the same knowledge or sensitivities to perform successfully in society. These divisions and distinctions are still with us in contemporary curriculum construction and statements of standards. Proposed indicators of change in the U.S. school mathematics, and guidelines for curriculum and evaluation contain distinctions between what is available for all students and what is available for college intending students (Murnane and Raizen, 1988). While the language of the late 19th century has been modified in current documents, social categories are carried into pedagogy as there are structural divisions concerning what knowledge is desirable for whom. The social values are made invisible by a language that makes curriculum choices seem universal. The application of standards and indicators, however, involve historical elements of discrimination that exist in school: Everyone knows who the "all" are and who are the college intending. Second, the differentiation in schools represent larger strains and struggles which not only reproduce culture but are dynamic elements within social structure. While knowledge differentiation may seem functional in society, the actual organization of school work may involve interests and dispositions in conflict with the functional requisites of the larger system. The U.S. civil rights movement of the 1960s which sought to eliminate racial discrimination and the feminist movement of the past decade impose pressures not only on who is taught mathematics (Fox, 1977), but also challenges some of the "rational" assumptions which underlie the selection of mathematics in school as valued knowledge within a hierarchy of human understanding (Beechey and Donald, 1985; Beck, 1979; Gilligan, 1982). The historical origins of what is selected as school knowledge was not administrative, technical or behavioral, but driven as subject matter related to social and political interest in schooling. Mathematics teaching as the social organization of knowledge. Differentiation in contemporary schooling provides a second example of how curriculum is rooted in social and cultural issues that are not readily

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apparent. The issue of differentiation is illustrated in a study of a reform program in elementary schools (Popkewitz et al., 1982). The program was to introduce a management system of curriculum objectives and assessment procedures that would individualize instruction for U.S. elementary schools. Six schools around the country were studied in depth to understand how the technologies of the program were realized in an institutional context. While one might expect variations of use within the general pattern given by the reform program, three different cultural patterns emerged for realizing the reform technologies; each contained different assumptions and implications for the work and knowledge of schooling. If we focus on the mathematics and science taught as one aspect of life in school, we can illuminate institutional differences in the meaning of practice. Teachers in three of the six schools taught mathematics as a maze of facts, and science as a body of predefined tasks and facts. The instructional focus was with measurable skills that could be placed in systems of hierarchially ordered learning objectives. Since only a limited range of skills in mathematics could be made measurable, most instruction gave attention to these skills. The work of teachers and students emphasized procedures for pacing students through the objectives and recording their achievement. These conditions came to be called Technicalschools as we considered the assumptions and implications of the ongoing patterns. The concern with problems of management of instruction assumed priority over the mathematical content in defining school work. The work of teachers was to find the most efficient ways to process children through the record-keeping procedures for children's math levels; the work of children was generally related to children's ability to look industrious. Achievement was often judged on the basis of hard and continuous work rather than on the quality of the results. The values that underlay teaching of mathematics were consistent with those of other subjects as institutional priorities formed a coherent pattern. The social processes tended to posit a view of knowledge as reified;fixed, unyielding and immutable to human thought and criticism. In another school using the same reform program, mathematics instruction emphasized children's playfulness with numbers, and science involved a tentativeness and skepticism towards the phenomena of the world. This school we called Constructive. While there was an objective based curriculum, teachers believed that the objectives set for each grade were easily learned and that there were more important purposes to instruction. Instruction encouraged children's facilities with language and responsibilities to the subtleties of interpersonal situations. School work included creative dramatics, music, art, as well as group activities in social studies, reading, science

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and mathematics.The knowledgeof curriculumwas treatedas permeable and provisional.Student'swork involvedfindinganswersto problemsthat werecreatedby theirinterestsas well as problemsthat emergeddirectlyfrom the curriculum.An overpopulationof chinchillain one classroomresultedin researchon the animals'environmentalhabits and incorporatinga metric unit into the constructionof a new cage. In two schools, called Illusory,therewere regularperiodsof instruction, textbooks and activitiesto symbolizecontent instruction.Examinationof the social patternsof conduct,however,revealedan emphasison the rituals of teachingand learningwithoutmuchfollow through.Whilechildrensat at their desks, they were taught little or no mathematics.The chalkboards would have long lists of activitiesbut a quick glance aroundthe room saw that both teachersand studentsweredoing other things. The teaching of mathematicsexisted in relation to a larger pattern of cultural meaning and values. The illusory quality was also a part of art, music and other divisions of time in school. When an instructional televisionprogramchangedto one that focusedupon the cultureof poverty, the teachersaid, "Put your head down and let you mind rest."At another part of the school was a special class for remediationwhich included a variety of machinesto help childrenlearn to read. When childrenentered the room, there was little instructionas the time was used as a form of recess.

The illusoryqualityof the school curriculumwas dramatizedby a professional language that had little referenceto practice but which helped to createimagesof a rational,technicaland efficientorganization.A principal referredto instructionalobjectivesas the "keyfactorin instruction;"but the languageof the school-wideobjectiveswas pecularlyopaque- "eachlearner will be able to state a school-wide objective;""there should be positive communicationand involvement."The languageof objectives,instructional programming,units, flexibility,and self pacing was a ceremoniallanguage that createdan image of schoolingand a vision of efficiencyand competent professionals,but the meaningswere ambiguous.In the context of these schools, the languagecould be interpretedas circularsemantic ballets in which languagebecameits own referent. While the aim of the reform was to create a single organizationand curricularform that was effectivein any community,the reform neither created a common condition of schooling nor freed schooling from the constraintsof differentsocial conditions.Technical,constructiveand illusory schools contained distinct assumptionsabout teaching,learningand schooling. Each institutionalconfigurationmaintaineddifferentbehaviors

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and competencies;containedparticularsystemsof meaning,patternsof social relationsand authorityin whichthe contentof mathematicsdeveloped. The Social Predicament of Schooling

While I have initiallyposed the problemof mathematicsteachingby focusing upon the internalorganizationof schooling,we also must considerthe relationof our institutionarrangementsto the social predicamentin which schools are located. U.S. educatorsare asked to respond to a variety of social,culturaland economicissues,rangingfrompoliticalsocializationand health education to the preventionof drug use and teenage pregnancy. School curriculaalso are to solve materialproblemsby giving priorityto certaineconomic/culturalforms, such as teachingscientificand technological knowledgeand rationalthought that, in the currentU.S. climate,is to give the United Statesa priviledgedpositionin the currentglobal industrial realignments(see e.g. NationalScienceBoard,InterimReport, 1983). The social predicament,however,is not only these externalpressuresto includecertainknowledge.School subjectsas scienceand mathematicsare culturalas well as factualactivities,involvingthe sensitivitiesand awareness that are not readilyaccessibleto all students.The "new"scienceand mathematicsof the U.S. curriculumreformsof the 1960sand the computer"literacy" movementof the 1980sinvolve styles of communicationfound in the professionalstrataof society,wherework dependsupon the abilityto play with words and communication(Bernstein,1977). The preferredform of mathematicseducationof the currentreformscontinuesto be those notions drawnfrom elite strata of society whichemphasizea tentativenesstoward ideas and the developmentof interpersonalskills (Popkewitzet al., 1986).7 How schoolingcan introduceequalityof conditionsin a world of unequal conditionsposes a continualtensionin the selection,organizationand evaluation of school knowledge. Responses to the social predicament. Elements of the social predicament do

not emergeas carefullyarticulatedargumentsor forcefullydocumentedconcerns. Culturaland professionalexpectationsprovidebackgroundassumptions for schoolingand are absorbedinto the discourseand practicesof the school in a varietyof ways. Classroompracticereflectsboth school traditions and socioculturalvalues,althoughtheydo not alwayscoincideand can involveconflict.Programsare interpreted,modifiedand used in relationto professionalideologies that make the ongoing patternsseem naturaland benign.Thesesocial tensionsin schoolcurriculumcan be givenattentionby

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furtheranalyzingthe differentinstitutionalpatternsof technical,constructive and illusoryschooling. The socializationpatterns in the constructiveschool can be viewed in relation to the professionalbackgroundof the childrenwho come to the school. The intellectualand socialpointsof view of professionaloccupations involve a complex division of labor relatedto achievedstatus ratherthan ascribedstatus(Bernstein,1977;Couldner,1979).The achievedstatus,however, is not derivedfrom "craft"skills or necessarilyownershipof property or capital;but from servicesthat involvethe creationand controlof systems of communication.Wordsare the currencyof exchange.Mathematicsin this context was both a skill and a way to reasonabout the world;readingwas not only to learn skills but to engagein interpretationand appreciationof literature. Illusoryschoolingwas a response,in part,to teachers'perceptionsthat the requisitedispositionsfor schoolingwere lackingin the childrenwho came from the poor communities.The logic of schoolingwas that the children come form broken homes, do not have adequatedisciplineor the correct attitudesfor schoolwork,and havefew or no educationalmaterialsavailable to themin the home;theseconditionsmakeit difficultor impossibleto learn properly.To teachmathematicshad less to do withlearningthe contentthan using school subjectsas a vehicle to establishan orderly,busy place where childrenare safe, and wherethey can learnthe "right"attitudesand behaviors that will help them when they get older. The illusoryqualityof these schools had a dual quality.It was a response to a seigementalityin whichmuchprofessionalenergywent into creatingan imageof schoolingthat will satisfycriticsof the U.S. urbanschoolsor defect adversecriticismof poor achievementin academicareas. The symbols of productivityand practicesproducedan illusion that pre-emptedthe attention of outsiders. The particularuse of ritual, ceremonyand languagewas also a form of control and authority.The formalroutinesof definingtime as periodsand organizedaroundsubjectsmaintainsa myth of homogeneity,formalequity and opportunitywithin the school setting. The celebrationsof the illusory schools are the sacredcharacterof certainculturallyagreedupon categories of knowledgeand the legitimacyof expertswho generatesuch knowledge. All childrenin the illusoryschoolswere taughtthat thereare importantand unquestionedcategoriesof knowledgeand skills. The differentiationritualof the school proclaimsthat most childrenwill not succeed in masteringthe sacred knowledge.While mathematicswas definedas importantto successin the world,the childrenin theseschoolson

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the whole, wouldnot succeedin its mastery.The ideologyof pathologymade that failurepersonaland communal,for the institutionalimage was a meritocraticcompetitionin which a few deserveto win and the losersdeserveto lose. The technicalconceptionof pedagogywhich broke learninginto discrete parts had differentsocial assumptionsand implications.By definingeach elementof mathematics,scienceand readinginto isolated, and fragmented elementsin an hierarchicalsystem,the curriculumcoincideswith a particular structureof consciousnessand workfoundin industrialsocieties(See Berger et al., 1973;Braverman,1974;Noble, 1977).It involvesthe social invention of bureaucracyas a cognitivestructureas well as a way of organizingpeople. Workis subdividedso that an individualno longerhas a conceptionof how the separateelementsof the work process relates to the total product of labor. Emphasison learninghow to decode a word may also mean that a personnever learnsabout literacy;or learningdiscreteskills of additionas particularbehaviorelementsmay desensitizeindividualsto how these skills are relatedto largersystemsof thoughtand practice. The relationof technicalschoolingto their social context illustratesthe complexarrangementsthat underliethe discourseof mathematicseducation as a social construction.Teachers'and administrators'perceptionsin two schools defined the school mandate as involving teaching the functional skills necessaryfor the anticipatedblue-collaror serviceoccupationsof the pupils.A thirdtechnicalschool was locatedin an affluentbusinesscommunity. The hierarchicalstyle of work and principlesof authorityseemedto expresslargersocial beliefs that tied the particularcommunityto a single religiousgroup that dominatedthe infastructureof the community. Other dynamicsalso intervenedin how curriculumwas interpreted.The constructive school involved conflict between the teachers' perceived mandateof its professionalobligationsand the districtadministrativeideologieswhichsoughtconsistencyand standardizationin schoolprogramsby incorporatingbusinessapproachesof accountabilityin education.A technical school was locatedin an U.S. southernruralcommunitywhose population (white and black) maintained a sense of continuity, community obligation and responsibility,and a familiaritybetweenthe people in the community and the school staff. The illusory schools, in contrast, were located in urban neighborhoodswhere there was little sense of history or community. The social predicamentof the school is often obscuredby traditionsthat give symboliccoherenceand reasonablenessto school practices.An U.S. belief is that social institutionsshouldbe rationallyorganized,for example,

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has led to the development of administrative theories about school organization, curriculum and evaluation. These administrative theories of behavioral objectives, criterion-referenced measures and competency testing, however, give little reference to the institutional rules of schooling. As a ritual of schooling, the theories of organization and learning help to project the image of a moder institution that is efficient and rational. The language directs attention to the surface qualities of our social patterns, leaving unscrutinized the rules of social relations and ways in which schooling articulates patterns of control and power.8 Mathematics as a category of schooling. Within this context of social predicament, we can return to the question of the social assumptions that underlie the teaching of mathematics.9 At least three dynamics of the insititutional pattern of school that have little to do with conventional definitions of learning give focus to the subject matter. First, mathematics instruction gives symbolic reference to the scientific and technological base of society. Mathematics can be viewed as representing the hope and challenge of an industrial and communication-based/ communication-controlled society. Its cognitive character signifies enlightenment, a path by which a rational, scientific and pragmatically organized society will bring progress to its material and social world. The symbolic function of mathematics education becomes more significant when we recognize that U.S. public school instruction rarely, if ever, goes beyond 19th century mathematics. The enlightenment belief introduces a second and related dynamic to mathematics' status. As a preferredcategory of understanding, mathematics is to be recognized as of value even for those who cannot master its codes. The curriculum category of mathematics carries the status differentials, social divisions and hierarchies found in the work in society. The curriculum establishes legitimacy for those experts who have acquired the knowledge, modes of interpretation and occupations in which mathematical knowledge is made a part of a professional mandate. A third social implication is the dual quality of mathematics in the construction of reality. Mathematics can enable us to understand relationships and guide interpretations in ways not available in other discourses. In this sense, it provides a form of knowledge that transcends our immediate situation and experiences. But the language of mathematics can also obscure and mystify our social conditions. It can refocus attention on our world in a way that deflects attention from how social patterns are humanly constructed. As mathematics is used to explain political elections, profit and debit, budgets,

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demography, etc., the numbers become part of a socially constructed reality. The historical manner in which people create institutions is buried in a presentation of knowledge that seems to express only the relationships between numbers. Social practices are made to seem beyond human intervention and individual agency. As a result, to draw upon Foucault's (1973) analysis of discourse, subjects become objects, power is hidden in the maze of numbers, and purpose is made to seem irrelevant to the constructions of social life. The possibly contradictory meaning of mathematics has little to do with the internal logic of the discipline and more with the social uses of knowledge in a complex and differentiated society. The social function of mathematics is a general issue of science and the secularization of our world; the function of mathematics in creating anonymous and abstract relations is part of a modern social consciousness that has been called, "the homeless mind," (see e.g. Berger et al., 1973). 2. WHAT ARE THE ASSUMPTIONS AND IMPLICATIONS OF CURRICULUM LANGUAGES FOR THE WORK AND KNOWLEDGE OF DISCIPLINED THOUGHT?

This issue of the language of mathematics as a constructed quality of social reality is a central issue in the construction of school curriculum. Mathematics has a dynamic and communal quality activity out of which mathematical knowledge emerges (Davis and Hersh, 1981; Ulam, 1976). Theories of pedagogy, in contrast, crystalize mathematics through linguistic inventions that make that knowledge seem objective and natural. Knowledge in schooling is conceptualized as specific qualities of learning, steps or stages of problem solving or formal mathematical equations or concepts. The focus on logical or psychological qualities obscures the interplay of the communal/craft quality of mathematics. Interwoven with personal skills and individual creativity are community patterns and norms that provide direction and self-correcting mechanisms for the generation of mathematical knowledge. The historical, social and personal dimensions are lost in curriculum design, (Popkewitz, 1977a, 1983b) as knowledge is defined as external to human involvement. To consider how mathematical knowledge is expressed in pedagogical theory and practice, explorations in the sociology and history can be helpful. Five observations about the problem of reason, thought and practice in mathematics are relevant. First, there is not a single method of inquiry, but many methods that scientists create as they confront the problems of their disciplines. Methods

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are influenced by communal standards, craft skills and imagination. Sociologists and historians have continually focused upon science and mathematics as methods of inquiry which produce knowledge (see Kuhn, 1970). The methods of inquiry are not carefully laid out prescriptions for action, but an interplay of orientations, dispositions and conceptual lenses which combine to give direction to knowledge production. Some sociologists have argued that scientific and mathematical creativity and imagination are best understood in relation to communal standards of recognition which bestow objective validity upon the particular results of research. While the material and work requirements of science and mathematics are different (Hagstrom, 1965), the role of others who have shared thought and outlook is also important (Davis and Hersh, 1981). From initial training, mathematicians learn what are the basic unanswered questions, the criteria of evaluation, and the particular thinking relevant to particular specialities. There is also a sense of competition, appropriate attitudes and dispositions that are learned with one's encounters with others (Ulam, 1976). The communal expectations, demands, attitudes and consistent attitudes and emotions become a cognitive structure by which individuals approach the tasks of problem-solving and reasoning. It is in the social structures of experiences that a psychology of mathematical thought and reason must be constructed rather than as cognitive structures abstracted and separated from the complex social, political and intellectual conditions in which knowledge is produced. Second, the concepts of science and mathematics are both answers and questions. Concepts are answers in that thc categories create boundaries by which scientists are to think about phenomena. But they are also questionprovoking words, suggesting that there are unknowns, mysteries and ambiguities in the world that need exploration. While we tend not to consider mathematics in this manner, mathematicians have thought of their work as imagining new possibilities rather than merely following specific lines of reasoning or making concrete calculations (Ulam, 1976). Third, many concepts are the subject of constant debate and exploration. The relations of order and randomness is an element of current debate in mathematics and science, having implications for the nature of geometric form and causality in science (Crutchfeld et al., 1986). At the cutting edge of science and mathematics is a conceptual playfulness and a competition among colleagues to generate knowledge. Such playfulness, skepticism and competition are central dynamics of inquiry. Fourth, concepts are at once affective and cognitive expressions about the world. Concepts contain root assumptions, such as that our social affairs

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work as a machine, an organism or dialectically. These values are reflected in the words of biology, sociology and educational psychology. Social science concepts, in particular, develop as responses to issues of social transformations and form part of a political agenda to respond to change (Popkewitz, 1984). Mathematic concepts may not have the same metaphoric quality as those of science and social science. But concepts in mathematics do presuppose relations and causal networks; mathematical models tend to emphasize linear rather than dialectical relations. Further, the concepts and relations expressed in mathematics are often used to explore human problems, thus providing an horizon by which possibilities are to be framed. The form of mathematics adopted can direct us to think about the world as harmonious and stable, or in flux. Understanding mathematics requires not only learning its logic but an understanding of the individual and collective consciousness by which the logic was realized (Davis & Hersh, 1981). For Archimedes, the sum of the angles of a triangle was not only 180?, it was a phenomenon of nature; to Newton, it was of deduction and application as well as bound to the universe that God set aside. Fifth, science and mathematics have both internal and external influences on knowledge growth. While school textbooks focus upon the accumulation of "facts" and technological development as a reason for the importance of contemporary science, external factors such as industrial growth and the formation of the modern state have produced "epistemic drift" in the organization of mathematicians and their knowledge (Elzinga, 1985). The current industrial and military uses of computers and the industrial funding of genetic engineering have influenced the problems and theory development in mathematics.?0 The current "growth" fields in science, technology and mathematics contain practices that give emphasis to utilitarian concepts of the disciplines (Ralston, 1986; Dickson, 1984; Popkewitz and Pitman, 1986). While an examination of mathematics as well as epistemological field directs attention to its dynamic quality, the language of curriculum transforms its practices into a crystallized form. The concepts of schooling are treated as objects with fixed parameters for children to internalize. Often, concepts are "proofs" to be mastered. The state of flux surrounding concepts and the debate among competing paradigms of inquiry are ignored." In many ways, the current U.S. interest in constructivist psychology maintains an uneasy alliance with a past borrowed from behavorist notions of knowledge: the content of learning is fixed, the processes of discovery defined by constructivist notions become "the intervening variables" to mastery.'2

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Yet, concepts are neither fixed nor neutral to our affairs. Mathematics is, at once, a "talent of drawing pictures of juggling symbols," with others, "picking the flow in an argument." It is "a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks" (Gian-Carlos Rotce in Davis and Hersh, 1981, p. xviii). The manner in which disciplined knowledge is brought into school, however, gives emphasis to consensus and stability. A result is a static conception of the knowledge, methods and values of science (Popkewitz, 1977b, 1983b). The decontextualization of interest can be found at a variety of layers of educational discourse. The structure-of-knowledge argument in curriculum, for example, defines concepts and generalizations as "things" to learn. Concepts are treated as objects whose definitions children are to internalize. The resulting curriculum emphasizes moral and political values under the guise of teaching technically neutral science, social sciences and mathematics (Popkewitz, 1983b). If one examines an U.S. mathematics education text as a cultural artifact, the myth structures of the polity and economy are asserted in the construction of problem-solving, including ideologies of individualism and economy. A decontextualized discourse is apparent in the general discussion of reform. United States school reforms focus upon the individual's ability to lear, instructional changes that increase learning effectiveness, and standards for evaluation that maintain the assumptions of possessive individualism. Children are to "learn" more or better, teachers are to be held "accountable," and so on. The problem becomes identifying common errors or misconceptions and identifying strategies to overcome the deficit (see Bell, 1986). Competency in science or mathematics is discussed as a narrow and restricted range of experience tied to some form of testing. Notions of problem-solving tend to be strategies to devise or obtain the answer. It is within this problem of decontextualization of knowledge that we can explore a social function of psychology in pedagogical inquiry. The use of psychology in education suggests teaching is objective and technical; evaluation is based upon efficiency. Much of the literature in mathematics education draws its base from the psychological literature, assuming that the issues of knowledge resides in the logic of presentation and the internal states or qualities of individuals. While U.S. have assigned priority to psychology as a discipline of curriculum and evaluation, the practices of inquiry are borrowed from a discipline created for purposes other than the understanding and transmission of mathematics and science. The history of educational psychology has little to

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do with understanding the relationship of the structure of mathematics as a field of knowledge and issues of pedagogy. Educational psychology involved the development of an academic discipline concerned with the successful adjustment of the individual to the environment (O'Donnell, 1985; Napoli 1981). U.S. psychology had twin tasks: it was to help mitigate the crisis of religion as late 19th century theology confronted evolutionary theory. The development of psychology was also designed to disseminate and advance a practical knowledge in an emerging industrial nation. The utilitarian focus had little to do with science itself, leading an historian of psychology to conclude that by the early 20th century "psychology in general would flourish neither as a mental discipline nor as a research science but as the intellectual underpinning and scientific legitimator of utilitarian pursuits, especially in the field of education" (O'Donnell, 1985, p. 37). The practical concerns of psychology gave focus to a discourse about schooling which was functional in nature, and objective in method, and which transformed moral, ethical and cultural issues into problems of individual differences. This argument is not about psychology as a disciplined form of inquiry, but its limitations for the field of educational inquiry in which it dominates. Psychology can tell us little about what is to be selected from mathematics, nor can it provide insight into the dynamics of knowledge production of that discipline. These tasks are of philosphy, sociology, political economy and history. A curriculum methodology and monitoring approach about mathematics needs a broad intellectual focus to create methods that consider the interplay communal/craft qualities of disciplinary knowledge and the institutional processes of schooling. 3. WHAT DO WE MEAN BY CHANGE EDUCATIONAL

CHANGE

AND REFORM? HOW DO

AND REFORM PRACTICES

OR OBSCURE THE SOCIAL CONDITIONS PRODUCE

ILLUMINATE

IN WHICH SCHOOLS

KNOWLEDGE?

In the previous discussion, I focused on certain dimensions of institutional life. Let me now proceed to the last question to be addressed. It can be rephrased in light of the previous discussion: To consider curriculum change, how might the institutional dynamics of schooling be considered? What notions of change illuminate the social complexities that inform the teaching of mathematics? To focus upon the issue of change or reform, we must return to two assumptions which frame the discussion: First, that which is defined as the curriculum of mathematics has more to do with the social history and

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imperatives of schooling than with the patterns of work, dispositions and knowledge found in the scholary community of mathematics. Second, the conduct of educational research is itself embedded in social contexts and contains values, providing not only descriptions of current status, but positing conceptions of progress. The prescriptive/descriptive quality of science is one we rarely consider, but it is an irony that linguistic and semiotic scholars have continually brought to our attention. Science is an abstraction of reality through the use of language; the languages of science enable us to categorize and classify events in ways that involve predispositions toward those solutions seen as appropriate. Strategies for collecting data about children's or teachers' performance, for example, creates boundaries about what is important and how it should be considered. The power of science to understand, and the limitations of the boundaries created for considering human possibilities, are always with us. Let me provide an example. Often we collect information about student achievement or the "effects" of teacher inservice programs. The acts of data creation/retrieval assume the likelihood of at least two related outcomes: First, variations will lead to conclusions about what should be modified. Existing research about the relation of teacher praise and student achievement, for example, leads to recommendations that teachers build more positive reinforcement into their lessons. Positive relations between school leadership and achievement, or "expert" teachers, produce similar results as educators consider how to create effective schools (Berliner, 1986). A second outcome of research, and to my mind more important for policy questions, is that our research models adopts prior assumptions about how we should think about the organization of ongoing relations in the world.'3 The praise/achievement example presupposes a positivistic notion of the world, defining the world as a system of discrete and separate things. Change, from this perspective, is additive. The increased quantity of one variable is supposed to influence directly the outcome of the other. To borrow from phenomonology, the only cause in the research is the "because of"; that is, prior behavior induces some change in current situations to bring about a more desirable outcome. To pose the question of monitoring, evaluating or studying mathematics education, then, is to choose, at some level, values and assumptions about change. These choices have tended to emerge from three sets of metaassumptions that underlie our research traditions. These are: (1) a purposeful-rational view in which research/change processes are designed to move the world closer to a prior schema or model; (2) an evolutionary model of

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change in which there is a slow and steady movement to change elements of a social system; (3) a dialectical view of social systems. In the latter instance, the problem is not one of progress, as we traditionally consider that notion, but of illuminating those elements which hinder or limit human possibilities. Any movement or change involves contradictions in which new barriers are created and unanticipated consequences occur. These three views of change have little to do with the specific techniques of inquiry that are used, be they survey, tests or "qualitative" approaches. Actual data collection with each variety of technique could be done within any paradigmatic pattern. The three possibilities or "models" of change also do not seek to exhaust the possible ideal types, but are to suggest that strategies of monitoring involve prior questions about the nature of the social world which act upon and have implications for understanding the problem of mathematics education itself. PURPOSEFUL-RATIONAL MODEL

One view of change is a rational model in which there is thought to be an isomorophic relationship between the model and the world. This is evident where people believe flow charts of change or the stage models of reform coincide with the dynamics of our ongoing real world. The problem of change focuses upon following a rational, orderly sequence to implement some identified goal that is presupposed by the system. A purposful-rational order is assumed. There is a definition of the world as logically ordered and rationally controllable through administrativechanges in the organization of daily life. Seeking to find the most efficient ordering of content or instruction in arithmetic lessons assumes that mathematics monitoring, for example, involves identifying inputs, leading to processes, then to outputs, such as achievement, participation or aspirations (Shavelson et al., 1987). Change involves delineating each step in a logical and orderly sequence. The order of the model of change is believed to be universal to all situations, institutions and organizational purposes. The model assumes that if people follow the correct stages or steps and are careful not to fall into the pitfalls of organizational resistance, the outcome will be the success of the proposed reform. Institutionalization, from this perspective, is the use of the prescriptions or strategies after the initial stages of dissemination and implementations are concluded. The strategies of change become, more often than not, questions of quantity: Do teachers ask more high order questions? Are there more computer programming courses than before? Is student achievement higher in mathematics than in previous years? Do teachers and administrators feel

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more satisfied and more professional? The isomorphic quality of the model is contained in the assumption that the world around us can fit into the stages or sequences identified in the model. In fundamental ways, the isomorphic model contains a one-dimensional conception of social existence. The "noise" of cultural and social interactions, the complexities of causation that involve nonrandom practice and relational dimensions as part of the social order, and the role of human purpose are lost. The problem seems rational; to improve mathematics education, the U.S. National Science Foundation is advised to develop an indicator system of reform, collect more data and procedures for analyzing and reporting which will lead to information to guide policy makers, (Shavelson et al., 1987). Practices of reform are made independent of the nonrational elements of politics, the ambiguities of social affairs, and structural conditions which provide a background by which choices are seen as relevant and reasonable. The view of the world is a reification of human existence itself. What is essentially a language of metaphor to enable us to suppose that things are "like this" or "like that" becomes what is and should be. The isomorphic model works against change as it focuses upon the facades of social life and crystallizes the status quo. The iconic visions are made literal and empirical attributes of reality. EVOLUTIONARY MODEL

Related to the isomorphic model is a second view of change that ascribes an evolutionary quality to social organizations. In this "process" model, the problem is to guide the evolution of the system, be it the mind of a child or a school organization. Strategies of evolution may involve the invention of a new element in staff organization, such as special career incentives to ensure greater professional esprit de corps. The task of change is to devise a way of helping teachers evolve working relations that incorporate the new into the old patterns. Sometimes this is labeled a problem-solving approach or a "seduction model" of change in mathematics. It is to get teachers to accept the need for changes in behavior and then for an expert to guide them through a gradual evolutionary approach to the desired mathematical teaching (Burkhardt et al., 1986). A local staff considers issues and problems, devises strategies of change through inservice programs, and implements the change strategies to bring forth the solutions. For children to learn, there is work related to stages of growth that lead to some perferred outcome. An assumption is that the incorporation of the new program will regulate the system, making it healthier and more progressive.

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The evolutionary view of change involves certain assumptions that need clarification. In rejecting the mechanical view of the isomorphic model, an assumption of organism is accepted (Nisbet, 1976). The analogy to an organism involves certain assumptions: (1) change has directionality, that is, there is a trend or longitudinal shape to movement; (2) growth of an organism is cumulative, i.e., what may be seen at any given moment is the cumulative result of what has gone before it in its life; (3) developmental change is irreversible:change has stages and these have genetic as well as sequential relations to one another; and (4) there is purpose to growth. St. Augustine saw purpose in the human drama that was transhistorical and spiritual. Purpose, for Marx, was entailed in the struggle towards a classless and just society. In its modern form, progress implies a belief in rational understanding, a possibility of deducing generalizations which remain valid for some time and, to a degree, a determinism in our social conditions. The ability to impose an ever-increasing control over both the natural and social environment is made central to social and moral life. The difficulty of an evolutionary view of change is its concern with harmony, consensus and stability; that is, change is explained through focusing upon the functional interdependence of the system. The relation to existing structures is stressed. Further, time is identified with social change. One presupposes that one can take a snapshot of a social system over time to reveal structure as one does of the architecture of a building. This assumption is misleading. Social systems have patterns of social relations that are inseparable from their continual reproduction over time. It is like redesigning a floor lay-out without having focused upon a building's structural arrangements; one needs first to posit a theory of structure to distinguish activity that contributes to stability from those practices which are in flux. The problem of adaptation related to evolution and function also is filled with ambiguity. Not all adaptations are functional or related to structure. Recent archaeological and anthropological evidence raises serious questions about adaptation as a way of explaining differences or progress. Darwin's view of the effect of variational evolution upon group change raises questions about adaptation as a metaphor for social theory. When adaptation is observed in a species, it can be explained by the differential survival and reproduction of variant types being guided and biased by their differential efficiency or resistance to environmental stress and dangers. But any use of differential survival and reproduction, even when it has nothing to do with the struggle for existence, will result in some evolution, not just adaptive evolution. The evolutionary model in social theory becomes a form of Panglossian biology, confusing the ideas of Darwin that all adaptation is a

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consequenceof variationalevolutionwith an assumptionthat all variational evolutionleads to adaptation. DIALECTICS AND CHANGE

The limitations of evolutionarysocial theory direct attention to a third notion of change,a concernwith dialectics.Here,thereis an assumptionthat all social processesinvolvean interactionbetweenthat whichseemsto be in opposition.Stabilityis alwaysjuxtaposedto changeand social transformation, traditionwith dynamics.Further,the interactionof traditionand transformations in social conditions produces changes in quality as well as quantity.More computersinvolveconsideringnot just more use of technology, but also the social relationsproducedas materialconditionsof schooling are altered. The dynamicsof social life are not orderlyand linear;interventiondoes not ensure progress. The industrialrevolution produced more material goods and more workercontrol over leisure time, yet the peasant of the MiddleAges had more"holidays"andleisuretime.The U.S. CivilWarfreed slaves in the South, but new forms of discriminationand racial bias were createdby the turn of the centurywith consequencesas seriousas those of slavery.The developmentof mass educationprovidedgreaterattentionto individualmerit and access to materialsuccess,while also providingmore effectivemeans for social reproductionand control in times of social and culturalstress. Theseexamplesillustratethe complexitiesand unforeseenconsequencesof social actionwhichmustbe attendedto whenconsideringissuesof monitoring. To change a mathematicscurriculum,the activity must be viewed as within a social system, requiringattention to the interactionof different "contexts"in schooling. One notion of context is the particulartime and space in which social action occurs. We assess teachers'attitudes about mathematicsteachingor observeparticularpracticesin schools. But in our desireto take into account the "environment,"we often ignore a broader notion of context.The most trivialexchangeof wordsin a classroomimplicates the speakerin a long-termhistoryof the languagein which the words are formed and, at the same time, in the continuingreproductionof that language.When teacherstalk about childrenas learners,mathematicsas a subjectmatterof school or teachingas a specificseriesof pedagogicalacts, these words contain assumptionsabout structure,function, agency and knowledge that have developed in the past and have become a part of common sense language(Stanic, 1987;O'Donnell, 1985).

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The present contains structures of social patterns, such as in U.S. schools, ideologies of individualism, notions of science as progress, beliefs about mathematics as producing social and material advancement. The difficulty of identifying change in school mathematics lies in considering what elements of our discourse are being produced by current interactions and what elements are derived from contexts that existed outside of schooling and prior to our participation in these social affairs. It is the interactions of "context" that produce a twin motif of change and stability for our endeavors to alter school practices. To introduce new mathematics courses involves realizing the program in an historical system that includes teacher behavioral patterns, cultural norms of the classroom and school organization, and the social conditions outside the school that interrelate to produce knowledge. CONCLUSIONS

The essay has focused upon three questions about mathematics education in the social context of schooling. In each question, analysis considered the complexities of curriculum in an ongoing, cultural world in which there are unequal social relations and different interests. What is transmitted as mathematical knowledge, it was argued, may have little to do with the disciplinary standards, expectations and understandings associated with the field of mathematics; rather the subject matter is shaped and fashioned by institutional imperatives and values that underlie schooling. In U.S. educational research, there has been an over-reliance on functional and positivistic models. It has been assumed that the issues of teaching are those of increased efficiency of learning or organization. This orientation may overly effect a mathematics education community because of a content that seems, on the surface, purely logical, stable and unambiguous in form. It may also be a legacy of U.S. social sciences in which the methodological assumptions of behavioralism and positivism dominate, even in those attempts to restructure a cognitive science. This orientation can be contrasted to literature about conduct of schooling, science and mathematical communities which indicates that the received rationality is not rational or reasonable for conveying the complexity of our social institutions. To study, evaluate or monitor mathematics education requires theoretical and methodological approaches that can more adequately express the complexities of schooling as a social institution. Movement and change are not linear, sequential or directional processes. Our social conditions contain a host of elements that interact in ways that are never fully specified,

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predetermined, anticipated or willed. Further, to add "new" elements to our situations is to add to the dynamics of those situations in ways that qualitatively and quantitatively change them. To describe these relationships in a more conventional language, all elements of a situation are, at the same time, independent and dependent variables. Each element is modified as it enters into a social situation in ways that alter not only its relationship with other elements, but also its own internal relations. NOTES I wish to thank Thomas Rombergand Anne Zarinniafor their helpful commentswhile thinkingabout the problemof this paper.The paperwas preparedfor the National Science Foundation, T. Romberg and D. Steward (eds.) The Monitoring of School Mathematics: Back-

groundPapers(Vol 3). Reportpreparedfor the NationalScienceFoundationfor theestablishment of a school mathematicsmonitoringcenter,WisconsinCenterfor EducationResearch, 1987,Madison,WI. chapter24. An earlierdraftof thispaperappearedin the SpanishMinistry of EducationJournal'La producci6ndel conocimientoy los lenguajescurriculares.Cuestiones institucioualesen el seguimientode las matematicasescolore',Revistade Education,282(EneroAbril), 1987,pigs 61-87. 2 See Durkheim's(1930/1977)discussionof the evolutionof schoolcurriculumand the debates that occurredwith the introductionof mathematicsas a school subject. 3 See, e.g. Bourdieu,1985;Foucault, 1973, 1979;Lundgren,1977, 1983;Whitty, 1985;and Habermas,1971, 1973. 4 For a more generaldiscussionof the conceptof institutionand the problemof reform,see Popkewitz,1979, 1983a. 5See O'Donnell,1985;Popkewitz,1984. 6 The discussionis basedon a compositeof a varietyof lessonsthat occurredover time. 7 An interestingdiscussionof the social conditions of being a mathematician,see Ulam (1976). 8 The use of administrative theoriesare not "only"rituals,they contain practicesthat have implicationsfor work of teachersand children.See Popkewitz(1987a). 9 I must again emphasizethat my focus is not upon the internalstructureof mathematical knowledge.Rather,the concernis upon the mannerin whichmathematicsbecomespart of a publicdiscourseabout knowledge.The latterfocus gives attentionto how disciplinaryknowledge enters into public institutionsin a mannerthat has differentimplicationsfrom those intendedin the formaldiscussionsof educationalpurposeand goals. 10 Internaldisciplinaryvaluesalso influencecareersin science.Thesemay referto the valueof "basic"vs. "applied"work. Diciplinaryvaluesmay be in conflictwith the externalpressures that move researchin certaindirections.Whilemathematicianstend to claimthat theirdiscipline is unrelatedto practicalproblems,the currentconcernfor "discrete"mathematicsillustrateselementsof the fieldthat expressinterestin utility,havinginfluenceupon the standards and careerdevelopments(Ralston, 1986). " See Popkewitz(1977a);Sloan(1983)discusseshow technicalnotionsof sciencebecomepart of our politicaland educationaldiscourse.Curriculummakes argumentsof armamentseem "merely"those of efficiencyand effectivenessratherthan of ethicaland moralissues. 12 Cobb (1986)discussesthe epistologicaldifficultiesof constructivisttraditionsin mathematics education. 13 For discussionof the assumptionsand implicationsof models for change,see Popkewitz, 1984,Ch. 6.

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REFERENCES Beechey, V. and J. Donald: 1985, Subjectivity and Social Relations, Opa University Press, Milton Keynes, Philadelphia. Bell, A.: 1986, 'Diagnostic teaching: 2. Developing conflict-discussion lessons', Mathematics Teaching 116, 26-29. Berger, P., B. Berger, and H. Kellner: 1973, The Homeless Mind: Modernization and Consciousness, Vintage Books, New York. Berliner, D.: 1986, 'In pursuit of the expert pedagogue', Educational Researcher 15(7), 5-13. Bernstein, B.: 1977, Class, Codes and Control: Towardsa Theory of Educational Transmissions, (2nd ed.), (Vol. 3). Routledge and Kegan Paul, London. Bourdieu, P.: 1985, Distinction: A Social Critiqueof the Judgment of Taste, Harvard University Press, Cambridge. Braudel, F.: 1980, On History, S. Matthews (trans.), The University of Chicago Press, Chicago. Braverman, H.: 1974, Labor and Monopoly Capital: The Degradation of Work in the Twentieth Century, Monthly Review Press, New York. Burkhardt, H., R. Fraser, and J. Ridgway: 1986, 'The dynamics of curriculum change', A Position Paper for the Mathematical Sciences Education Board Curriculum Frameworks Committee, Shell Centre for Mathematical Education, University of Nottingham. Durkheim, E.: 1977, The Evolution of Educational Thought: Lectures on the Formation and Development of Secondary Education in France, P. Collins (trans.), Routledge and Kegan Paul, London. Cobb, P.: 1986, 'Making mathematics: Children's learning and the constructivist tradition', Harvard Educational Review 56(3), 301-306. The Conference Board of the Mathematical Sciences: 1982, "The mathematical sciences curriculum K-12: What is still fundamental and what is not report to the NSB commission on precollege education in mathematics, science and technology", National Science Foundation, Washington, D.C. Crutchfeld, J., J. Farmer, N. Packard, and R. Shaw: 'Chaos', Scientific American 225(6), 46-57. Davis, P. and R. Hersh: 1981, The Mathematical Experience, Introduction by Gian-Carlos Rota, Houghton Mifflin Co., Boston. Dickson, D.: 1984, The New Politics of Science, Pantheon Books, New York. DeLone, R.: 1979, Small Futures: ChildrenInequityand the Limits of Liberal Reform, Harcourt, Brace and Jonovich, New York. Donovan, B.: 1983, Power and Curriculum Implementation:A Case Study of an Innovatory Mathematics Program, Unpublished Ph.D. Thesis, University of Wisconsin-Madison, Madison, Wisconsin. Elzinga, A.: 1985, 'Research, bureaucracy and the drift of epistemic criteria', in The University System: The Public Policies of the Home of Scientists, B. Wittrock and A. Elzinga (eds.), Almquist & Wiksell International, Stockholm, pp. 191-220. Freire, P.: 1970, Pedagogy of the Oppressed, M. Bergman Ramos (trans.), Herder and Herder, New York. Foucault, M.: 1973, The Order of Things: An Archaeology of the Human Sciences, Vintage Books, New York. Foucault, M.: 1979, Discipline and Punish: The Birth of the Prison, A. Sheridan (trans.), Vintage Books, New York. Fox, L. H., E. Fennema, and J. Sherman (eds.): 1977, 'Women and mathematics: Research perspectives for change', U.S. Department of Helath, Education and Welfare, National Institute of Education, Education and Work Group, Washington. Gilligan, C.: 1982, In a Different Voice: Psychological Theory and Women's Development, Harvard University Press, Cambridge.

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Gouldner, A.: 1979, The Future of the Intellectual and the Rise of the New Class, Seabury Press, New York. Habermas, J.: 1971, Knowledgeand Human Interest, J. Shapiro (trans.), Beacon Press, Boston. Habermass, J.: 1973, Legitimation Crisis, T. McCarthy (trans.), Beacon Press, Boston. Hagstrom, W.: 1965, The Scientific Community, Basic Books, New York. Hatano, G. and K. Inagaki: 1988,'A theory of motivation for comprehention and its application to mathematics instruction', in T. Romberg and D. Steward (eds.), The Monitoring of School Mathematics: Background Papers Vol. 2: Implicationsfrom Psychology: Outcomes of Instruction. pp. 27-46, Wisconsin Center for Educational Research, Madison. Kuhn, T.: 1970, The Structure of Scientific Revolutions, (2nd ed.), University of Chicago Press, Chicago. Lukes, S.: 1973, Individualism,Basil Blackwell, Oxford. Lundgren, U.: 1977, Model Analysis of Pedagogical Process, CWK/Gleerup, Lund. Lundgren, U.: 1983, Between Hope and Happening: Text and Context in Curriculum,Deakin University Press, Geelong, Australia. McLaren, P.: 1986, Schooling as a Ritual Performance, Routledge and Kegan Paul, Boston. Murnane, R. J. and S. S. Raizen (eds.): 1988, ImprovedIndicators of Science and Mathematics Education in Grades 1-12, National Research Council. Napoli, D.: 1981, Architects of Adjustment: The History of the Psychological Profession in the United States, Kennikat Press, Port Washington, N.Y. National Science Board Commission on Precollegiate Education in Mathematics, Science and Technology: 1983, 'Interim report to the National Science Board', National Science Board, National Science Foundation, Washington, D.C. Nisbet, R.: 1976, History and Social Change, Oxford University Press, New York. Noble, D.: 1977, American by Design: Science Technologyand the Rise of Corporate Capitalism, Knopf, New York. O'Donnell, J.: 1985, The Origins of Behaviorism:American Psychology, 1876-1920, N.Y. University, New York. Peterson, P.: 1985, The Politics of School Reform, 1870-1940, University of Chicago Press, Chicago. Popkewitz, T.: 1977a, 'Community and craft as metaphor of social inquiry curriculum', Educational Theory 5(1), 41-60. Popkewitz, T.: 1977b, 'The latent values of the discipline-centered curriculum', Theory and Research in Social Education 5, 41-60. Popkewitz, T.: 1979, 'Educational reform and the problem of institutional life', Educational Researcher 8(3), 3-8. Popkewitz, T. (eds): 1983a, Change and Stability in Schooling. The Dual Quality of Educational Reform, Deakin University, Geelong, Australia. Popkewitz, T.: 1983b, 'Methods of teacher education and culture codes', in P. Tamir (ed.), Preservice and Inservice Education of Science Teachers, Ballaban Press, Rehovot, Israel. Popkewitz, T.: 1983c, 'The sociological bases for individual differences, the relation of the solitude to the crowd', in G. Fenstermacher and J. Goodlad (eds.), Individualdifferencesand the common curriculum,National Society for the Study of Education, Chicago. Popkewitz, T.: 1984, Paradigm and Ideology in EducationalResearch: The Social Functions of the Intellectual, Falmer Press, London and New York. Popkewitz, T. and A. Pitman: 1986, 'The idea of progress and the legitimation of state agendas: American proposals for school reform, Curriculumand Teaching 1(1-2), 11-24. Popkewitz, T. (ed.): 1987a, Critical Studies in Teacher Education: Its Folklore, Theory and Practice, Falmer Press, New York and London. Popkewitz, T. (ed.): 1987b, The Formation of School Subjects: The Struggle to Form An American Institution, Falmer Press, New York and London.

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Popkewitz, T., A. Pitman, and A. Barry: 1986, 'Educational reform and its millennial quality: The 1980's', Journal of curriculumStudies 18(3) 267-284. Popkewitz, T., B. Tabachnick, and G. Wehlage: 1982, The Myth of Educational Reform: A Study of School Responses to a Program of Change, University of Wisconsin Press, Madison. Ralston, A.: 1986, 'Discrete mathematics: The new mathematics of science', American Scientist 74(6), 611-618. Shavelson, R., L. McDonnell, J. Oakes, N. Carey, and L. Picus: 1987, Indicator Systems for Monitoring Mathematics and Science Education, Rand Corporation, Santa Monica, CA. Sherman, J. A. and E. T. Beck, (eds.): 1979, The Prism of Sex: Essays in the Sociology of Knowledge,Proceedings of a symposium sponsored by WRI of Wisconsin, Inc., University of Wisconsin Press, Madison. Sloan, D. (ed.): 1983, Education of Peace and Disarmament: Towarda Living World, Teachers College Press, New York. Stanic, G.: 1987, 'Mathematics education in the United States at the beginning of the 20th Century', in T. Popkewitz, (ed.), The Formation of School Subjects: The Strugglefor Creating an American Institution, Falmer Press, London. Stephens, W. M.: 1982, Mathematical Knowledgeand School Work:A Case Study of the Teaching of Developing Mathematical Processes, Wisconsin Center for Educational Research, Madison. Ulam, S.: 1976, Adventures of a Mathematician, Charles Scribner's Sons, New York. Whitty, G.: 1985, Sociology and School Knowledge:CurriculumTheory, Research and Politics, Methuen, London. Vygotsky, L.: 1978, Mind in Society, the Developmentof Higher Psychological Process, Harvard University Press, Cambridge, Mass.

University of Wisconsin-Madison, 225 North Mills Street, Madison, Wisconsin 53706, U.S.A.

RICHARD

THE COMPUTER

NOSS

AS A CULTURAL

MATHEMATICAL

INFLUENCE

IN

LEARNING'

ABSTRACT. My starting point in this paper is that there is a cultural gap between the mathematics that children do as part of their everyday experience and the mathematics that they learn at school; my thesis is that the computer has (perhaps uniquely) the potential to bridge this divide. The paper will examine the cultural impact - both actual and potential - of the computer on children's mathematical education; at the ways in which the introduction of the computer does and will change the ambient space in which children learn mathematics. I begin with a brief discussion of the cultural context of mathematics learning and the relationship between informal, everyday mathematical activity, and formal, school mathematics. This perspective leads to a closer examination of what it means to do mathematics, and on the relationship of a technology to the mathematics embedded within a given culture. I discuss the issue of injecting meaning into mathematical activity, and then examine some ways in which the computer might offer a solution to this central problem. Next, I give some examples of the influence of the computer on the culture of the mathematics classroom. Finally, I suggest some of the outstanding issues of research and curriculum development which remain.

WHAT IS THE CULTURAL

CONTEXT OF MATHEMATICS?

Making sense of the advent of the computer into the mathematics classroom entails a cultural perspective, not least because of the ways in which children are developing the computer culture by appropriating the technology for their own ends. But what of the culture into which the computer is being introduced? Much illuminating comment in this area has come from researchers whose focus has been ostensibly with non-western cultures. For example, Gay and Cole's (1976) study of the culture and mathematics of the Kpelle of Liberia centres around the task of constructing bridges between the Kpelle's 'indigenous' mathematics and the 'new' mathematics of the school curriculum. It is the contention of this paper that the task of bridge-building is central, not just for introducing 'new mathematics into an old culture' but equally for introducing formal mathematics into technologically developed cultures such as our own. As Mellin-Olsen convincingly shows (Mellin-Olsen, 1987), many if not most of the children in Western classrooms, are confronted with the mathematics of a subculture of which they are not - and perhaps have no wish to be - members; where there is, in Papert's phrase, no 'cultural resonance' (Papert, 1980) between their own economic and social activities, and the activities in which they are invited to participate at school. While technological development may on one level appear to obscure the relationship Educational Studies in Mathematics 19 (1988) 251-268. ? 1988 by Kluwer Academic Publishers.

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between formal and informal mathematics (or in D'Ambrosio's (1985) terms - ethnomathematics), I propose that the technology itself-specifically the computer - can be the instrument for bridging the gap between the two. WHAT DOES IT MEAN TO DO MATHEMATICS?

It is clear that most children and adults are not aware that they are engaging in mathematical activity even when they are involved in quite complex numerical or geometric activities (Wolfe, 1984) and that for many, the very suggestion that they participate in mathematical activity is sufficient to induce panic (Buxton, 1981). Yet mathematical ideas and mathematical ways of thinking provide powerful means of making sense of our social, economic, and cultural environment. As Lancy puts it "Grouping, categorizing, generalizing etc. represent a fundamental human need every bit as basic as the need to eat, to drink, or to socialize" (Lancy, 1983, p. 64). In the sense that human beings are by definition, creatures who seek to explain and control their environment, everybody is a mathematician. Much the same could be said of any systematised way of thinking about the world, for example, in relation to philosophy: "It is essential to destroy the widespread prejudice that philosophy is a strange and difficult thing just because it is the specific intellectual activity of a particular category of specialists or of professional and systematic philosphers" (Gramsci, 1971, p. 323). If we consider substituting 'mathematics' for 'philosophy' in Gramsci's claim, we are forced to take seriously the problem of defining that which is special to mathematical as opposed to, say, philosophical activity. I suggest that it is useful to conceive of a difference - on the one hand to avoid the danger of subsuming under the title of mathematics any activity which involves abstraction and generalisation; on the other hand of defining mathematics out of existence as say, simply a way of thinking about relationships and structure. I think the difference lies in the formalism inherent in specifically mathematical activity (I do not want this to be confused with formal in the sense of a formal system - there is more than enough room in my meaning for the intuition and playing around that characterises the process of mathematical activity). And it is precisely this issue of formalism which I will argue, suggests a constructive role for the computer. The question of what it means to do mathematics is central to understanding the possible roles that the computer might play, particularly in contexts other than its direct employment as a tool for solving a predetermined problem. This issue is the subject of some debate among those who have

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considered the importance of culturally embedded mathematical activity. Gerdes (1986) for example, uses the example of the regular hexagonally patterned baskets used by Mozambiquan fishermen, to argue that there exists 'frozen' mathematics within a culture, which can serve as a starting point for mathematical activity within classrooms of that culture. He points out that the artisan who merely imitates the technique is not doing mathematics, in contrast to those who discovered the technique. Gerdes suggests that understanding the processes underlying the techniques is the crucial step in participating in mathematical activity. In doing so, he stops just short of defining such activity as mathematical in itself. In discussing Gerdes' description of the mathematics embedded within the basket-weaving Keitel (1986) comments: "It is not the point here whether or not this is mathematics - in my view it is very much what Freudenthal calls pre-mathematics (with repsect to cognitive levels of children in Western culture. And it is evident that such examples are excellent starting points for discovery learning, or - and that is Gerdes' concern, for embedding mathematics education in a peculiar culture environment.) - The point here is that human work structures reality according to regularities which potentially are accessible to mathematical analysis" (Keitel, 1986, p. 44). Keitel's observation states rather succinctly a role that 'spontaneous' mathematics is seen to play in the learning of mathematics - namely as a starting point for more formal learning. Keitel's point is that basket-weaving itself is not mathematics; it is what she calls the 'fore-stage' of mathematics, "a field of problems in social reality which immanently - at least partly - are organised in some correspondence with mathematical structures, and hence may better, or even exclusively, be solved by the employment of mathematical devices" (Keitel, 1986, p. 45). As Hoyles (1986) points out, participation in the activity itself is not mathematics unless the didactical context is such as to provoke reflection on and synthesis of the mathematical relationships embedded within the activity. Can we consider this kind of 'intellectual material' (Mellin-Olsen, 1986) culturally-embedded mathematical activity - as more than simply starting points for mathematical learning? In the first place, there is the question of formalisation. While it is perfectly possible to envisage all kinds of formalisations which can be generated by the basket-making activity, it is clear that the process of basket-making itself does not require any formalisation or abstraction. And yet it is that kind of abstraction (which is essentially algebraic in character) which lies at the core of official mathematics. I will use the term mathematisation to cover all the processes of mathematical formalisation and abstraction, including those which are preformal in the sense of

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being non-written, intuitive, and fragmented. Traditional attempts to use situations as the basis for subsequent mathematisation have been based on attempting to develop links between the activity and the mathematical abstraction. What we have not had at our disposal was the means for learners to engage in culturally embedded activities while simultaneously mathematising their activities. This option has been closed precisely because the technology at our disposal (books, pens, paper, etc.) has been inappropriate to construct mathematical environments in which mathematisation can naturally occur. The second problem is that which Gerdes refers to when he suggests that the person who discovers the technique is the one who is doing mathematics. Discovering is at best a haphazard affair and can hardly be relied on as a methodology for learning all of mathematics. Equally crucial (and related to discovery) is the learner's reflection on her own activities - again not unproblematic in conventional learning environments. Here again, it is possible that the available technology may be an important element. For the computer does contain the potential for focusing the learner's attention on selected ideas and concepts by providing feedback in an interactive way which is not available with other technology (but not, as I shall argue below, without careful intervention by teachers). THE RELATIONSHIP OF TECHNOLOGY TO MATHEMATICS EDUCATION

At this point I want to look more closely at technology and its role in mathematics education, on the understanding that technological development is merely one form - the material form - of a more general notion of culture that consists of the sum total of deposits in the consciousness of humans. For the technology which is at the disposal of a given culture ( and which, dialectically, has grown out of it) directly influences the kinds of mathematics which are indigenous, spontaneous or frozen into that culture. Bishop (1979) gives many examples of the ways in which technology and culture are related to mathematical development. He cites the case of two university students in Papua New Guinea who drew a map of the campus which contained no roads; they were born in the island region where roads did not exist. Lancy in his study of Papua New Guinea (Lancy, 1983) shows that culture and schooling do have an effect on cognitive development, and that despite differences in technology (between different Papuan cultures) which are reflected in the structure of language, "if you select the right domain and direct your questions to the right levels in the hierarchy, taxonomic behaviour will emerge, and at a very early age" (Lancy, 1983, p. 159).

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There is ample evidence - at least from third world countries - of the distance between everyday language and activities and those of school (see for example, Mitchelmore, 1983; Berry, 1985). For the Kpelle child: "the world remains a mystery to be accepted on authority, not a complex pattern of comprehensible regularities". (Gay and Cole, 1967, p. 94). Could not much the same be said for many of the children in our schools in the West? And what difference does technology make? The culture of a non-technological society contains a variety of contexts for the generation of mathematical abstraction, but it cannot be equally rich since it is the material culture which determines the complex organisation of society and the ideological and intellectual forms which accompany it. In technological cultures, practical activities have become increasingly complex and the sciences have become deeply interwoven with everyday life, and paradoxically, increasingly invisible. Let us consider the culture of Western mathematics, a culture in which children's schooling beyond a certain age (say 12 or 13) is based on the symbolic abstraction of algebra. To be sure, our culture, like all others, contains within it everyday situations in which algebra exists (albeit in a 'frozen' state) and many attempts have been made to draw links between children's experience and their mathematics. This store of situations is, compared to a non-technological society, quite rich. Could we say that algebra forms part of the indigenous mathematics of our culture? Hardly. In fact, quite the reverse. Precisely because of the ways in which the hitherto existing technology has proved a problematic vehicle for the introduction of algebraic abstraction, there are severe cultural obstacles to developing learning environments in which children can actively engage in formalisation (as opposed to those which may serve 'merely' as a foundation for formalisation). MEANING AND FUNCTIONALITY

I recently gave a lecture in which I presented a class of 40 pre-service students (not mathematics majors) with a number of mathematics examination questions which contained rather clear political, moral or other value judgements. The students agreed that one of these questions was both violent and racist (it contained references to 'Christians' throwing 'Turks' overboardsee Maxwell, 1985), and I suggested to them that they might like to recast the question in 'value-free' terms, to see if such an exercise was feasible or desirable. The results overwhelmingly conformed to 'text-book' mathematics problems, with reference to human beings expunged altogether and

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meaningless references to beads and other artefacts of the mathematics classroom replacing human beings. Just over half the replies simply replaced 'Christians' and 'Turks' with various kinds of confectionery (sweets and toffees predominating) and fruit (mostly apples and oranges). It seems that the price that is paid for removing value judgements from mathematical problems is to literally de-humanise them. In comparison, a recent examination question in the UK which invited examinees to compare the total amount spent on armaments (by the US and the USSR together) with the amount needed to feed the world's population, received a prolonged and unanimous attack in the national press. Small wonder then that mathematics lessons are so often 'not about anything' (D.E.S., 1982). Considerable insight into the question of the dehumanisation of the subject has recently been provided by considering differences in cognitive styles between men and women (Gilligan, 1982) and commented on from a mathematical perspective by Brown (1984) and more recently by Papert (1986). Mathematics - at least the mathematics of the school classroom - is typically seen as hard-edged; as a subject in which meaningless problems are posed (by others) about - at best - real but material objects (and often about unreal and meaningless objects).2 We need to be wary of accepting (worst of all by default), that mathematics needs to be hard-edged and dehumanised (there is an interesting debate to be had on whether abstraction and dehumanisation are necessarily linked; see Davis and Hersch (1986) for one viewpoint). I have argued that the central problem is thus to inject meaning - and in particular, personal meaning - into school mathematics (see Hoyles, 1985a). It is inconceivable that children will, in general, be able to utilise mathematical tools and concepts unless they feel personally involved in their use; unless, that is, mathematical concepts become functional tools embedded within children's own cultural environment. At the very least, we need to find objects other than sweets, toffees and fruit, with which it makes sense to undertake symbolic manipulation. Of the pedagogical and technological innovations available, it is far from obvious that the computer has a role to play. Indeed the cultural view of computers is precisely that they are deterministic, dehumanising and cold. The implications of this cultural perspective for education, and for mathematics education in particular have been far reaching, and the attack on computers as a culturally destructive medium has not been muted. As Papert (1985) points out, by focusing only on the machines rather than how they fit into the ambient culture, proponents of this view are no less guilty of a 'technocentric' perspective than the hackers they attack. The

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criticism of computers (sic) is exemplified by Davy (1985) who asks: "What kind of a culture are we developing if people have to meet its most powerful ideas through machines rather than through people?" (p. 554). This takes us to the central issue. For the thrust of this paper is that the powerful ideas of a culture are always mediated through the technology which is available to that culture. Is it conceivable that our culture's knowledge and understandings could be transmitted to new generations without the employment of pens, paper, publishing technology, and typewriters? How much of the science and history of human culture is frozen into the production of a single piece of paper? The question is what facets of the computational environment it is possible to exploit in order to realise a vision of the computer as a means of enriching the indigenous culture of children, rather than Davy's dehumanised vision. COMPUTER BASED LEARNING ENVIRONMENTS

The key property of the computer which I want to examine here is its ability to allow its user to explore, investigate and pose problems, and to offer flexible representations of situations, of which at least one is on the symbolic, formal level. It is interesting that the major area in which the computer has permeated children's culture - that of computer 'games' - is one area where, on the face of it, the player is denied access to this kind of power. But this is only half the picture. As Turkle (1985) points out in her insightful study of the computer culture, such games do offer the player a high degree of control within a limited domain. This ability of the computer (or rather sensitively written software) to allow users to interact in a personally powerful way is the common thread that runs through the various cultural manifestations of the computer in society. On the other hand, precisely because such games do not in general allow natural access to the symbolic, formal representation, they have little or no role in mathematics learning. I do not want to trivialise the potential for cultural resonance which this perspective indicates. Of course there is a resonance which comes about from the fact that computers are everywhere, that they are (sometimes) fun, that they are glamorous and so on. The key point is that children see computer screens as 'theirs', as a part of a predominantly adult culture which they can appropriate and use for their own ends. It may be that shooting aliens is less of an intellectually stimulating experience than writing a program, but it belongs to the child - it is something which they can take from- perhaps -ven use to subvert - adult culture.3

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Most of the computer culture is owned (in every sense) by official society; but just enough is appropriable by young people -just enough that is for young people to feel that they can control it. One of the best examples we have of this kind of appropriation (and certainly the best researched) is that of children learning to program in Logo. Logo is a computer programming language designed for learning - there is an intentional pun here. It allows the learner extremely straightforward access both to creating interesting screen effects and to the computational/mathematical ideas which underlie them. There have been a number of longitudinal studies which have sought to analyse the power of this environment from a mathematical perspective, and which have illustrated that children are able to explore and use a variety of mathematical ideas in a wide range of programming contexts (Papert et al., 1979; Hoyles et al., 1985; Noss, 1985). These studies have confirmed Papert's claim that by learning Logo, the child is behaving as a mathematician - is essentially doing mathematics. But the question remains as to what kind of mathematics? To what extent does the mathematics of the computer culture intersect with the broader mathematical culture? In outlining an answer to these questions, I want first to clarify what Logo is standing for in this discussion. It is intended as a placeholder for a certain kind of interaction with the computer; an interaction which allows for particular kinds of mathematical activity to take place but it is unlikely to provide such an environment uniquely. New computer-based environments are currently being designed and more established ones are being applied to create similar kinds of learning contexts. Logo however, does provide the most extensively researched example of this kind of work and it will form the basis for what follows. It is helpful to distinguish three ways in which the culture of children's mathematical learning may be influenced by interacting with the computer via Logo (or - and this is the last time I shall make this qualification - with any similarly powerful computer-based tool). The first is to examine how the mathematics that children can do is influenced; the second is to ask what implications this may hold for what children learn; and finally, to examine what may follow for what children may be taught. WHAT MATHEMATICS MAY CHILDREN DO?

I want to illustrate the ways in which the computer can influence children's mathematical activity by an example. The example concerns a group of seven 13-year old Logo-experienced children working on a structured and progressive set of tasks - both on and off-computer-which were designed to place

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them in a situation designed to allow them to 'bump up against' the ideas of ratio and proportion. The task involved designing a program for a generalised N-shaped figure, a problem whose complete 'solution' was certainly beyond their mathematical experience (as it would have involved the invention of trigonometry!). For details of the task and the findings, see Hoyles and Noss (in press). The topics of ratio and proportion have been well researched, in particular from a Piagetian perspective that understanding of proportionality becomes evident only in the formal stage; later work has seemingly confirmed that children of early secondary school age (13-15 years) find extreme difficulty in thinking of a relationship between two quantities as requiring anything other than an additive operation (Hart, 1980). Analysis of the strategies employed by the children engaged in this Logo task revealed a number of interesting characteristics. In the first place, none of the children adopted the 'additive' strategy on the computer which could have been predicted from existing research findings. Secondly, six of the seven children eventually proposed solutions which could be classified as involving a proportional strategy. Thirdly, when the same children were given a pencil-and-paper ratio test (Hart, 1980), their performance reflected the findings of Hart's study, with none of the children producing the correct answer to an item of roughly the same depth and content as the N-task which they had for the most part successfully tackled in the Logo context. The conclusion from this exploratory study is that the computer provided the support by which children could explore and develop relationships that were just beyond their grasp with traditional (i.e. pencil-and-paper) technology. For example Paul recognised quite early on in the task the need to find some kind of relationship between two crucial lengths, but was completely 'blocked' as to what to do about it when working on paper. Moving to the computer appeared to set him free to explore the range of possibilities, an opportunity offered - at least in part - by the interactive nature of the environment. I propose, however, that the feedback provided by the computer offers only a surface explanation of the results. The difficulty which children experience with multiplicative relationships is essentially a cultural one. Children's experience is largely multiplication-impoverished (witness the difficulty we have in thinking about images of multiplication for the classroom compared with addition), and contexts requiring non-trivial multiplication are not an everyday part of most children's experience. What does the computer bring to this situation? It would be incautious to propose far-reaching conclusions from such a small-scale study. Nevertheless, findings such as these are beginning to broadly converge (see, for

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example, Hoyles, 1985b; Noss, 1986a). What evidence we have seems to indicate that it is the need for formalisation, rather than merely the feedback involved, that is seminal in influencing learners' conceptions. In the example above, the computer is not teaching the child about ratio, it is enlarging the culture within which the child operates. The essence of the computer-child interaction is built on the synthesis between the child's need to formalise the relationship algebraically (i.e. to construct a program), and to receive confirmation (or otherwise) of her intuitions by perceiving the effect on the screen. In proposing this explanation, I am emphasising the opportunity afforded by the Logo environment to use symbols in a meaningful context - to pose and solve problems with symbols rather than to play with 'concrete' situations which subsequently (and often artificially) require symbolisation (see Pimm, 1986 for an enlightening discussion of this point in a general context). It is in this sense - that of offering a context in which mathematical formalisation is a necessary part of a system to be explored - that the culture of mathematical learning may be enlarged by the computer. WHAT MATHEMATICS MAY CHILDREN LEARN?

Being involved in an activity is not a sufficient condition for learning to take place. Or rather, to pose a slightly weaker version of the same statement, it does not guarantee a match between what the learner learns and what the teacher thinks she is learning. For the latter, reflection is required, as well as a conscious effort to draw the learner's attention to the 'important' relationships involved. However, there is a significant range of learning which is - in Papert's sense - Piagetian; learning which takes place through immersion in a culture (a culture which may contain important vehicles for 'teaching'), the most obvious being that of the acquisition of natural language. The attempt to locate learning of this kind which is generated by computer interaction has only recently begun (see for example, Lawler, 1985), and has been beset by studies which have employed poor research designs or untenable research hypotheses. As far as mathematical learning is concerned, some illumination of this problem can be given by starting from the observation that many children appear to harbour fundamental misconceptions about elementary geometrical concepts until quite an advanced age. For example, Hart's study (Hart, 1980) indicates that many children fail to appreciate that the length of an object is not changed by displacement; only 42% of first-year secondary

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students judged that two lines (one oblique and one horizontal with their end points aligned) were such that the oblique line was longer than the horizontal one. This kind of finding is remarkable in that it implies that there is often little shared meaning between teacher and learner of common terms such as 'length'. One explanation of this robust finding may be that the term 'length' in common usage has a much less precise meaning than that which is necessary to answer the above question correctly. The mathematical meaning of the word is usually not taught; it is somehow picked up (or not) from diverse settings within everyday activity. Unfortunately the two sets of meanings often do not correspond. In an exploratory study, Noss (1987) compared the responses of 84 children who had studied Logo for one year and 92 who had no computing experience, on a set of geometrical paper-and-pencil items designed to probe children's conceptions of length and angle. The children were aged between 8 and 11 years. The aim was to gain information about the kinds of primitive components of geometrical knowledge - such as length conservation - which might be mentally constructed during Logo work. Two further examples were the ability to distinguish the invariance of an angle under rotation and variation in the length of the rays which define it, and the recognition that the comparison of two lengths depends on the units of measurement employed. In all there were three categories of items for the concept of length, and three for angle. The question was whether, by enriching the learning culture in which the children were involved, they would more easily be able to match their different fragmented conceptions of the ideas of length and angle into a more conventional mathematical form. The findings of the study can be summarised as follows: (i) For all three angle categories there was a trend (significant in two categories) in favour of the Logo groups. (ii) In two out of the three length categories in which comparison with Hart's study was possible, the Logo group's performance was almost up to the level of those in Hart's study (despite a 1-3 year age difference), in contrast to the comparison group which was somewhat lower. (iii) In five out of six of the categories, the Logo girls were differentially successful in relation to the boys; in the non-Logo groups, this situation was reversed. What kind of explanation can be suggested for these findings in terms of the cultural perspective outlined above? That the effect on the pupils' conception of angle was more marked than for the concept of length is consistent with the findings of Papert et al. (1979), who suggest that new knowledge acquired in the Logo environment has to 'compete' with existing knowledge, and that the amount of time required to displace it would depend

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on how firmly rooted it was. Papert conjecturesthat knowledge about angles would be more easy to displacethan knowledgeabout length. This conjecturegoes some way toward explainingthe differentialeffect in favour of the girls. There is evidence of a deficit of spatial abilities among some girls, most probably accounted for (though not necessarily entirely)by a socio-culturalbias against spatial experienceswhich tend to be encouragedamongboys (Badger, 1981),but whose effecton mathematical attainment are remediablethrough appropriateactivities (Sherman, 1980). If this is the case, then it would follow that Logo experiencesare more likely to be helpful for girls to constructgeometricalconcepts than boys. The appropriatenessof the environmentto girlsis, of course,a key element in this chain of argument.On this point, Turkle(1984) has suggestedthat Logo offers a programmingenvironmentwhich is more appropriateto girls'cognitivestyles than other forms of computing. To summarise,there is evidence that some spontaneouslearning may take place in suitably designedcomputer-basedenvironments,and that a possiblemechanismfor understandingthis processis to focus on the ways in which the computer influencesthe cultural reservoirof mathematical ideas availablefor childrento draw on. WHAT MATHEMATICS MAY BE TAUGHT?

The introductionof the computer into the learningenvironmententails more than simplya technicalcomponent(hardware+ software).In the first instance, there is a pedagogicalcomponent which consists typically of a teacher,a curriculum,writtenmaterials,etc. - meansby which the interaction with the computer-basedmaterialcan be structuredand childrenencouragedto reflecton their activities(see Hoyles and Noss, 1987a, for an elaborationof these ideas). While theremay be - as suggestedin the previous section- a categoryof mathematicalknowledgewhich can be acquired spontaneously,it is equally the case that learningmathematicsrequiresa conscious awarenessof mathematicalstructuresby studentsand thus conscious interventionby teachers.A numberof researchershave pointed out that it is perfectlypossibleto remainunawareof the essentialmathematical ideas behind Logo programming(see for example, Hillel, 1984; Leron, 1985). In addition, we need to take account of the contextualcomponent- the settingin which the problemsor more generally,the learningsituationsare framed.It is clear that the distinctionbetweena problemand a situationis

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intimatelybound up with the context. For example,Lave et al. (1985) have shown how for people engagedin real-lifeproblem-solvingsituations,there exists a process of gap-closingbetweenthe expectedshape of the solution and the information and calculation devices available; that for real-life problem-solving,a situationbecomesa problemin the course of activityin a particularsetting;peopleand settingssimultaneouslycreateproblemsand solution shapes.4 A numberof researchershave illustratedhow the computercan be used to createan environmentwherethis kind of dialecticalrelationshipbetween problem generationand solution can take place in a reasonablynatural way. For example, Noss (1986b) has shown how children engaged in a Logo environmentswitchbetweenexploratoryand problem-solvingactivities and Hoyles and Sutherland(1986) have illustratedhow Logo provides both a rich environmentfor pupil-posedproblemsas well as a wide range of contextsfor spontaneousexperimentalactivityand collaboration. It should be clear that the computeris being cast in a rather special relationshipto the learningprocess,not simplyas anotherconcreteembodiment of an abstractmathematicalconcept. As Dorfler (1986) argues, the distinctionbetween concreteand abstractis artificialin any case, since it presupposesthat there exists an a priori union of actions and operations whichis fracturedin the courseof learning.The key idea is that of focusing attentionon the importantrelationshipsinvolved,a role in which- as Weir (1987) points out- the computeris ratherwell cast; but not without the conscious interventionof educators,and the careful developmentof an ambientlearningculture. This particularfacility of the computerto focus the learner'sattention and simultaneouslyto provide feedback seems to provide a promising frameworkfor thinkingabout teachingmathematicalideas in a computerbased context. It is helpful to considerthe strandof researchinitiatedby Vygotsky,who has emphasisedthat the key to collaborationis that it provides supportfor entryinto the cognitiveareain whichthe child would not be capable of solving problemsunaided- the 'zone of proximaldevelopment' (Vygotsky, 1978). In a study of pupils who have been allowed to explore and use mathematicalconcepts encapsulatedwithin Logo programs,Hoyles and Noss (1987b) have shown that the computeris capable of performinga similarscaffoldingrole5for developingunderstandingof those concepts.At the same time, we are beginningto understandthe role that the teachercan play directlyin the process- a role which, in a computer-basedenvironmentcertainlybecomesno less criticalbut considerably more subtle(see Noss and Hoyles, 1987).

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ISSUES FOR RESEARCH AND CURRICULUM DEVELOPMENT

I have arguedthat the computer- used in suitableways- can expandwhat it is possible to do, learn and teach in the mathematicsclassroom. This processis verymuchin its infancy,not least becauseof the limitationsof the hardwarewhich are only now graduallybeing overcome. One area which looks promisingfor the future is in the design of new computer-basedenvironmentswhichofferthe opportunityto intergratedisparateactivities,and thus for mathematisationto take place in a varietyof contexts.For example,'Logowriter'allows for an integrationof word-processing and programming,so that for example,childrenwill need to write Logo programmesto illustratetheirstories,and to tailortheirword-processor to theirown requirements.[Thiskind of integrationmay be particularly importantfor girls. For example,Taylor(1986) suggeststhat incorporation of constructionactivity into a story is a helpful way for girls to become engagedwith Logo activities.] Computer-basedenvironmentssuch as these are continuinga tradition withincomputersciencewhichbeganwith the artificialintelligencecommunity in the nineteen-sixties,who weremore concernedwith providingthemselveswith intellectualtools to help themdefineand exploresituations,than with rigorous algorithmdesign for the solution of well-definedproblems. The parallelwithmathematicseducationis reasonablyclear;it is no accident that the formercommunitywas the catalyst which providedLISP and its derivativeLogo, while the latterhas been responsiblefor languagessuch as Pascal (and of course BASIC). One of the processeswhich seems as if it might at last be happening,is that the increasingpowerof the hardwareand softwareis allowingthe explorersto displacethe educationalhegemonyof the problemsolversand algorithmdesigners(see Noss, 1986cfor a discussion of a culturalside-effectof this hegemony).This processis one in which, I suggest,mathematicseducatorsneed to play an active role. At the same time, therehas been a growthof interestin researchingthe ways in which existing computer-basedtools can be employed to create microworldsfor learningabout reasonablywell-definedsubsetsof the mathematicscurriculumin areassuch as probability,functionsand variables,the concept of limit, and 3-dimensionalgeometry (for a recent collection of paperson this researcheffort,see Hoyles, Noss and Sutherland,1986). Alongsidethesedevelopments,therehas emergeda widespreadconsensus on the importanceof the teacherin the learningprocess;a recognitionthat the computermay have an importantrole to play in influencingteacher attitudes towards mathematicsand its teaching, and to begin to opera-

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tionalise Papert's notion of a 'computer culture' within the mathematics classroom. Research is in progress at the Institute of Education in London6 to explore and evaluate the use of the computer in this way, and to work in conjunction with teachers on the creation of mathematical microworlds for use in the classroom. Almost all of the research and curriculum development currently taking place is necessarily locked into the framework of the existing mathematics curriculum. To be sure, those innovations which are successful will invariably have some success in eroding the traditional content of the syllabus. Nevertheless, I want to conclude with an eye on the future, by suggesting that a necessary task for mathematics education is to begin to turn towards examining how the emerging computer culture will materially alter the content of the mathematics we teach and the kind of mathematics children will be able to do and learn. To do this will require considerable human resources and a high density of machines; work which has begun in a Boston elementary school7 (not specifically focusing on mathematics), is indicating that the social, affective and cultural issues which follow are at least as interesting as the cognitive. It is only by researching these issues - in situ - that we may begin to genuinely glimpse the potential for cultural change that the computer might bring to the learning of mathematics. NOTES This paper is based on substantially the same data as is discussed in an article in Cultural Dynamics. 2 A recent example of the extent to which this myth is culturally accepted involved a TV interview with a sociology professor who had given birth to the notion of the QALY-a Quality Adjusted Life Year - which allowed him to judge the relative 'values' of two human lives so that scarce funds could be 'scientifically' allocated to the most deserving cases. The interviewer, seeking perhaps some rationale for the idea, asked the professor whether this was a helpful way to think about the value of human life, or perhaps 'only a mathematical formula'. 3 I see a parallel here with the development of the 'youth culture' of the nineteen-sixties and its influence on musical taste. Think back to the way in which the musical idols of the time were denigrated by adult society for their trivialisation of the official music culture, and the way in which many of those same idols are now seen as the representatives of the musical establishment. To be sure Elvis was not Bach - in an important sense the former did indeed trivialise our conception of music (no more so, of course, than existing popular music of the time); what happened was that young people claimed a part of that culture for themselves - and created it themselves. In doing so, they created new kinds of instruments and new musical technologies (some of which fed back to official music), and they profoundly changed and enlarged the conception of musical content itself. 4 In the light of Gilligan's findings, it is worth noting that Lave's study was undertaken exclusively with women. The possibility that her findings reflected- to some extent at least the preferences of female cognitive style cannot be ruled out.

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5 The notion of scaffoldingwas coined by Brunerand his colleaguesto describethe way in whichcollaborationcouldofferthe learnerjust enoughsupportto do thingswhichshecould not do independently. 6 With Celia Hoyles and RosamundSutherlandand Funded by the Economicand Social ResearchCouncil. 7 The 'HeadlightProject'whichbeganin 1985,is directedby SeymourPapertandhis colleagues at the MassachusettsInstituteof Technology. REFERENCES Research Badger,M.: 1981,'Whyaren'tgirlsbetterat maths?A reviewof research',Educational 24, 11-23. issues',Forthe Berry,J.: 1985,'Learningmathematicsin a secondlanguage:Somecross-cultural Learning of Mathematics 5(2), 18-23.

Bishop, A.: 1979, 'Visualisingand mathematicsin a pre-technologicalculture',Educational Studies in Mathematics 10, 135-146.

Brown,S.: 1984,'Thelogicof problemgeneration:frommoralityandsolvingto de-posingand rebellion', For the Learning of Mathematics 4(1), 9-20. Buxton, L.: 1981, Do You Panic about Maths?, Heinemann, London.

and its placein the historyand pedagogyof mathD'Ambrosio,U.: 1985,'Ethnomathematics ematics', For the Learning of Mathematics 5(1), 45-48. Davis, P. and R. Hersch: 1987, Descarte's Dream: The WorldAccording to Mathematics, Har-

vester,Brighton,UK. Davy, J.: 1985,'Mindstormsin the lamplight',in Sloane,D. (ed.), TheComputerin Education: A CriticalPerspective,pp. 11-20, TeachersCollegePress,Columbia. D.E.S. (Dept. of Educationand Science):1982,Mathematics from 5 to 16, HMSO, London. Dorfler,W.: 1986,'Thecognitivedistancebetweenmaterialactionsand mathematicaloperations', in Proceedings of the Tenth International Conferencefor the Psychology of Mathematics

Education,pp. 141-146,London. Gay, J. and M. Cole: 1967, The New Mathematics and an Old culture:A Study among the Kpelle

of Liberia,Holt, Rhinehartand Winston,New York. Gerdes,P.: 1986,'On culture.Mathematicsand curriculumdevelopmentin Mozambique',in Hoines, M. J. and Mellin-Olsen,S. (eds.), Mathematicsand Culture,pp. 15-41, Caspar Forlag,Radal,Norway. Gilligan,C.: 1982,In a DifferentVoice,HarvardUniversityPress,Cambridge,Mass. Gramsci,A.: 1971,PrisonNotebooks,Lawrenceand Wishart,London. Hart, K.: 1980, Secondary School Children's Understandingof Mathematics. A Report of the Mathematics Componentof the CSMS Programme, Chelsea College, University of London.

Hillel,J.: 1984'Mathematicaland programmingconceptsacquiredby childrenaged 8-9 in a

restricted Logo environment', Proceedings of the 9th InternationalConferencefor the Psychology of Learning Mathematics, Holland. Hoyles, C.: 1985a, Culture and Computers in the Mathematics Classroom, Inaugural Lecture,

Universityof LondonInstituteof Education. Hoyles C.: 1985b,'Developinga contextfor Logo in school mathematics',Journalof Mathematical Behaviour 4(3), 237-256.

and generalisationof Hoyles,C.: 1986,'Scalinga mountain- a studyof the use,discrimination some mathematicalconceptsin a Logo environment',EuropeanJournalof Psychologyof Education 1(2), 111-126.

Hoyles,C. and R. Noss: 1987a,'Synthesisingmathematicalconceptionsandtheirformalisation throughthe constructionof a Logo-basedschool mathematicscurriculum',International Journal of Mathematics Education in Science and Technology 18, 581-595.

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Hoyles, C. and R. Noss: 1987b, 'Children working in a structured Logo environment: From doing to understanding', Recherches en Didactique des Mathematiques 7. Hoyles, C. and R. Noss: (in press), 'The computer as a catalyst in children's proportion strategies', Journal of Mathematical Behavior 5. Hoyles, C., R. Noss, and R. Sutherland (eds.): 1986, Proceedings of the Second International Conferencefor Logo and Mathematics Education, University of London Institute of Education, London. Hoyles, C. and R. Sutherland: 1986, 'Peer interaction in a programming environment', in Proceedings of the Tenth International Conferencefor the Psychology of Learning Mathematics, pp. 354-359, London. Hoyles, C., R. Sutherland and J. Evans: 1985, 'The Logo maths project: A preliminary investigation of the pupil-centred approach to the learning of Logo in the secondary mathematics classroom, 1983-4', University of London Institute of Education. Keitel, C.: 1986, 'Cultural premises and presuppositions in psychology of mathematics education', Plenary Lectures, Proceedings of the Tenth InternationalConferencefor the Psychology of Mathematics Education, London. Lancy, D.: 1983, Cross Cultural Studies in Cognition and Mathematics, Academic Press, New York. Lave, J., M. Murtaugh and 0. de la Rocha: 1985, 'The dialectic of arithmetic and grocery shopping', in Rogoff, B. and Lave, J. (eds.), Everyday Cognition: Its Development in Social Context, pp. 67-94, Harvard University Press, Cambridge, Mass. Lawler, R.: 1985, Computer Experience and Cognitive Development, Ellis Horwood, Chichester, UK. Leron, U.: 1985, 'Logo today: Vision and reality', The Computing Teacher 12, 26-32. Maxwell, J.: 1985, 'Hidden messages', Mathematics Teaching, 111, 18-20. Mellin-Olsen, S.: 1986, 'Culture as a key theme for mathematics education', Hoines, M. U. and Mellin-Olsen, S. (eds.), Mathematics and Culture, pp. 99-121, Caspar Forlag, Radal, Norway. Mellin-Olsen, S.: 1987, The Politics of Mathematics Education, Reidel, Dordrecht, Holland. Mitchelmore, M.: 1983, 'Geometry and spatial learning: Some lessons from a Jamaican experience', For the Learning of Mathematics 3(3), 2-7. Noss, R.: 1985, Creating a MathematicalEnvironmentthroughProgramming:A Study of Young Children Learning Logo (Doctoral dissertation, Chelsea College, University of London), University of London Institute of Education. Noss, R.: 1986a, 'Constructing a conceptual framework for elementary algebra through Logo programming', Educational Studies in Mathematics 17(4), 335-357. Noss, R.: 1986b, 'What mathematics do children do with Logo?', Journal of ComputerAssisted Learning 3, 2-12. Noss, R.: 1986c, 'Is small really beautiful?', Micromath 2(1), 26-29. Noss, R.: 1987, 'Children's learning of geometrical concepts through Logo', Journal for Research in Mathematics Education 18, 343-362. Noss, R. and C. Hoyles: 1987, 'Structuring the mathematical environment: The dialectic of process and content', Proceedings of the Third InternationalConference of Logo and Mathematics Education, Plenary Lecture, Montreal. Papert, S.: 1980, Mindstorms: Children, Computers and Powerful Ideas, Harvester Press, Brighton, UK. Papert, S.: 1985, 'Computer criticism vs. technocentric thinking', Proceedings of Logo 85, Plenary Lectures, pp. 53-67, Cambridge, Mass. Papert, S.: 1986, 'Beyond the cognitive: the other face of mathematics', Plenary Lectures, Tenth International Conferencefor the Psychology of Mathematics Education, London. Papert, S., D. Watt, A. DiSessa and S. Weir: 1979, Final Report of the Brookline Logo Project, Part 2. Al Memo No. 545, Massachusetts Institute of Technology, Cambridge, Mass.

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Pimm,D.: 1986,'Beyondreference',MathematicsTeaching116, 48-51. Sherman,J.: 1980,'Mathematics,spatialvisualisationand relatedfactors:Changesin girlsand boys, grades 8-11', Journal of Educational Psychology 72, 476-482.

Taylor, H.: 1986, 'Experiencewith a primaryschool implementingan equal opportunityenquiry', in Burton L. (ed.), Girlsinto Maths Can Go, pp. 156-162, Holt, Rhinehartand Winston,London. Turkle, S.: 1985, The Second Self: Computersand the Human Spirit, Simon and Schuster, New

York. Vygotsky,L.: 1978,Mindin Society,HarvardUniversityPress,Harvard. Weir, S.: 1987, Cultivating Minds: A Logo Casebook, Harper and Row, New York.

Wolf,A.: 1984,'Practicalmathematicsat work:LearningthroughYTS', Research& Development ReportNo. 21, ManpowerServicesCommission,Sheffield,UK.

Institute of Education, University of London, 20 Bedford Way, London, WCIH OAL, U.K.

BOOK REVIEW

Erich Ch. Wittmann, Elementargeometrie und Wirklichkeit, Vieweg, Braunschweig/Wiesbaden, 1987. xiv + 467 pp. This book has the subtitle - an introduction to geometrical thinking. It is written for mathematics teachers and for teachers in training, and its aim is to counteract the current estrangement between mathematics and everyday life. The arguments are 'inhaltlich-anschaulich' rather than axiomatic, and although I can think of no English phrase which quite conveys the meaning of the German, English speaking readers will find the approach very familiar. To speak of intuitiveforms of argument is not enough, because this does not incorporate the additional shade of meaning of inhaltlich. Inhaltlich methods of demonstration are methods which call upon the meaning of the terms employed, as distinct from abstract methods, which escape from the interpretation of the terms and employ only the abstract relations between them. In the foreword the author argues for this approach in a way which suggests that it requires a more vigorous defence in his native country than is the case elsewhere. In the United Kingdom the Mathematical Association was founded to promote such an approach to the initial stages of geometry, and this became the established orthodoxy. We may wonder, in passing, whether the complementary aim of developing a pedagogically satisfying, precise logical formulation for later stages of school instruction has ever been successfully achieved. In the 1960s there was a strong movement to enhance the logical rigour of school mathematics, and some may feel that geometry should be presented in a rigorously deductive framework, making the best use of recent knowledge. The author does not hold this position, and he systematically avoids an axiomatic approach. He refers much to the history of the ideas and their applications, and his methods of demonstration are very close to the methods employed by the Greeks. Euclid's presentation of geometry is now known to be much less pure than tradition supposed, and his methods are known to involve an amalgam of logic and appeals to spatial intuition. In this book the intention is to develop an understanding of the relations of geometry with the material world by analysing sequences of problems. Educational Studies in Mathematics 19 (1988) 269-272.

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First there is a chapter presenting various problem situations which indicate the general flavour of what is to follow. There is then a chapter on the intuitive foundations of geometry, which identifies the objects and the operations which are the basis of the study. These are the familiar configurations of Euclidean geometry, and the fundamental mappings of the plane onto itself- reflection, translation and rotation, and later magnification and contraction. This provides a rich geometry, with powerful and exciting methods of proof. The approach is one in which the student is encouraged to harness a range of intuitions which are untapped by methods of proof which are restricted to arguments by congruence, and in which the structures of geometry have a more obvious relationship with those to be found in other parts of mathematics. But there are some attendant difficulties to which I will return later. In chapter three this knowledge of figures and their transformations is applied to a range of problems in the geometry of triangles and circles. The problems include reflection in curved mirrors, special points of the triangle, the theorem of Pythagoras, the golden ratio properties of the pentagon and the irrationality of phi, inversion and the Peaucellier linkage. These problems are for the most part familiar, but they are often insufficiently featured in contemporary textbooks. Chapter four gives the geometry of the heavenly bodies with much practical detail; and it makes a good case for including this material in school courses. Chapters follow on the symmetry of plane figures, ellipse constructions and the Platonic solids. Then there is a detailed discussion of length, area and volume: these are apparently simple matters, but many teacher training courses could give a more thoughtful treatment of these fundamental notions than they do. The chapter on plane trigonometry introduces the angular functions on the intuitively acceptable foundation of wrapping a line round the unit circle, and the essential formulae are related to the congruence of triangles in a very informative way. The final chapter on coordinate geometry covers a lot of ground. Vectors are introduced quite early on, and a careful distinction is made between position vectors and translations. Linear geometry follows, including Ceva's theorem, with applications to statics. Then come the conic sections, the ellipse and hyperbola being defined by the properties of their focal distances. As the ideas develop the relation of the scalar product to angular measure plays a key role. Thus the development and organisation of the material is in sympathy with some of the recent proposals for the reform of geometry teaching, but the linear algebra is kept firmly in its place. It gives support, without being in the centre of the stage. The book ends with some of the

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properties of the Platonic solids - and this reminder of Euclid's Elements cannot be coincidental. Teachers will find here much eminently useful lesson material. There are pertinent illustrations and practical applications which would enrich any course. I found the presentation of the theory of strip ornaments very attractive, and I liked the tap on the hydrant with a shaft in the form of a Reuleaux triangle. This is a defense against improper use, because it can only be turned by a spanner with a hole of the same shape. Other readers will certainly find favourites of their own. But persuasive as the individual chapters are, what of the overall shape of the book? When there is no global axiomatic system with its own natural lines of development to provide an overall structure, there are inevitably problems of selecting material. Teachers who wish to fit the ideas of this book into their courses will certainly find problems of selection and overall structure, and similar problems arise when we consider local questions of detail. The methods of demonstration employed in this book are methods with which most of us grew up, and we came to accept the style without looking deeply at its limitations. But when we return to geometry with our standards of logic sharpened by current presentations of arithemetic, algebra and logic we find it hard to know exactly what proves what. The inadequacies of the time-honoured forms of argument have been demonstrated. For example, they are known to embody unacknowledged appeals to the diagram rather than to logic. Attempts to devise school courses with rigorous approach in the style of Hilbert's axiomatics which are accessible to pupils with modest experience have been unsuccessful. But when we seek to develop geometrical thinking we have to remember the logical inadequacies of some of the arguments which satisfy the intuition, and we have to consider whether something might be done to reduce the gap between intuition and logic as now understood. The problem is illustrated in chapter two, which contains an informative discussion of the relations between geometrical configurations and mappings of the plane. It contains much worthwhile classroom material, but what is the chicken and what is the egg? People's intuitive perceptions vary, so what is clear to some is not clear to others. It seems inescapable, with this approach, that in some proofs the conclusions are as intuitive as the premises; so in these cases the student may wonder exactly what the function of proof is. Is it not possible in a course of this kind to include some items which approach those areas of geometry which are the key to later axiomatisation?

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One might (for example) consider one of the central issues - the development of coordinate systems from a foundation of the incidence axioms. To do this it is necessary to study the particular role of the theorems of Menelaus and Pappus, and the discussion can be enlivened (in the spirit of this book) by practical applications of incidence constructions in perspective and in nomography. Projective geometry is easier to put on a satisfactory foundation than Euclidean geometry. Should this influence the design of a course? How much projective geometry do we need to experience in order to appreciate what is involved? Emma Castelnuovo has shown in this journal that many valuable teaching points can be made with very little theory. When making these points I am no doubt expressing views with which the author has sympathy. He says in the second chapter that it was the original intention to have a second part of the book which would contain proof analyses, by means of which some of the inhaltlich-anschaulichproofs would be embedded in local axiomatic frameworks. We must hope that this plan will be carried out in some future publication. My second major point is a more personal one. Can computer graphics help with the teaching of geometry? There are references to computers in this book, but they are brief and in a low key. But today popular computer magazines accept matrix algebra as a matter of course, and in one evening's television entertainment there is often enough animated graphics to provide material for an entire course in transformation geometry. I have found computer graphics a splendid motivation for pupils, and without doubt it is one of the main areas of application of coordinate geometry at the present time. Nowadays the course which many students follow as they move through school and University does not provide a very substantial grounding in geometry, and teacher training courses which seek to resist this trend find problems. By no means the least of these is the problem of deciding what to put in courses when time may be very limited. This book is a welcome, thoughtful and interestingly individual contribution to the debate. It contains a great deal that I would wish to teach to the target audience - but many questions concerning the overall organisation of geometry and its place in courses for teachers of mathematics remain. 44 Cleveland Avenue, Darlington DL3 7HG, Great Britain.

T. J. FLETCHER

BOOK REVIEW

C. C. McKnight,F. J. Crosswhite,J. A. Dossey, E. Kifer,J. O. Swafford,K. J. Travers and T. J. Cooney, The UnderachievingCurriculum- Assessing US School Mathematicsfrom an International Perspective, Stipes Publish-

ing Company,Champaign,Illinois, January1987. 127 pp. $8.00. This book providesinformationon the United States performancein the Second InternationalMathematicsStudy (SIMS). The text is made up of small pieces of informationtaken from the survey data and illustrating performance.Interspersedthroughoutthe informativesections are statements of opinion by 'pundits'. The final section attemptsto apportionthe blame for the comparatively poor performanceof Americanstudents.The writersare obviouslydeeply worried by what they see as failure and seek to find clues which might accountfor it, particularlythroughclose inspectionof the responsesof the Japanese.SIMSis supposedlyto give informationon how differentcountries implementtheirmathematicscurriculumand how their pupils performon jointly agreedquestions.Thereis howevera strongelementof competition and the fact that Japan has 'won' in almost every section leads to much discussionon how we can emulatetheirpractices.This idea of 'winningand losing'is a boon to politiciansand the media.Wordslike 'fallingstandards' are bandiedaroundand teachersare yet again castigated.In Britain,propelledby the bandwagonof 'standardsmustbe improved'we are now faced with the introductionof a national curriculumaccompaniedby national testingthroughoutthe child'sschool career. Points of particularinterestand those whichmight be fruitfuldiscussion points with studentsand practisingteachersincludethe following: (1) A country'sview of post 16 yearseducation,revealedin the percentage of the populationstill at school at age 17 (92% in Japan,82% in USA and 17% in England and Wales). The perceivedimportanceof learning mathematicsmight be interpretedas the percentageof this populationstill studyingthe subject(100%in Hungary,15%in USA, 35%in Englandand Wales). (2) The perceivedrole of the teacherin differentcountriesand how it mighteffectthe way childrenare taughtis partlyillustratedby responsesto Educational Studies in Mathematics 19 (1988) 273-275.

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the question "is it easy to teach mathematics?" The Japanese teachers saw the teaching of mathematics as a difficult enterprise and lack of student achievement was blamed on the professional limitations of the teachers (themselves). In contrast the American teachers' responses showed that they thought teaching mathematics was relatively easy and student behaviour was the cause of lack of desired progress. The teaching load (at 8th grade) of the American teacher is heavy in comparison with most other countries, at the 12th grade Scotland's teachers had the highest teaching load. (3) The number of courses in mathematics and in mathematics pedagogy taken by mathematics teachers beyond high school varied from country to country. American teachers tended to take many more mathematics courses than they took courses in pedagogy, the Japanese concentrated more on methodology courses and the Swedes, surprisingly, seemed to take not many courses of either sort. (4) The nature of school Geometry appears to be a matter of dispute between countries. Of the internationally agreed Geometry items tested at 8th grade on average only half of them were taught in the participating countries. (5) Sometimes the report gives specific items, the percentage of classes who had been taught the mathematics tested by this item and the percentage passing it. There is often a large mismatch, resulting in many children failing what they have supposedly been taught. Do teachers of mathematics set problems which they know will be failed? On the other hand we might learn something about the transfer of knowledge, from those countries which seemingly did not teach the topic to classes which then proceeded to score highly on it (e.g. Belgium). The authors finally attack some of the 'sacred' beliefs of American teachers, e.g., size of class or/and universality of educational opportunities are causes for low performance, by quoting countries where the pupils perform well and are taught in large classes, etc. They attack the idea of the spiral curriculum for possibly causing fragmentation of mathematics and consequent lack of understanding. Lacking throughout, however, is a call for more research to provide data before we embark on the implementation of further theories of mathematics teaching without evidence of their effectiveness. We might take to heart the words of Brophy (1986) when he discussed in JRME the research on effective teaching (thus leavingthe educationof mathematicsteachersto be informedmostly by idiosyncratic personalexperiencesand untestedtheoreticalcommitmentsratherthan a relevantscientific knowledgebase).

What benefits have the mathematics education community world-wide gained from SIMS? Could the money spent on it, have been better used?

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Why are we preoccupied with testing to obtain scores which we can compare with scores in other countries, other years, other age groups? This movement is not only on the international front but also within countries (APU and NAEP for example). It is only after we have done this type of testing that we think of the questions we should have asked, such as, how and why. They are harder to answer but they will not go away simply because we ignore them or as is currently the trend, pretend they can be answered from data which was collected for an entirely different purpose. Director, Nuffield Secondary Mathematics, King's College, 552, King's Road, London SW 10 OUA, Great Britain.

K. M. HART

BOOK REVIEW

Louise Lafortune (ed.), Women and Mathematics, Editions Remue-M6nage, Montr6al,Qu6bec, 1986. ISBN 2-89091-0652. 260 pp., BF 590. This book is the result of the colloquium "Womenand Mathematics"which was held in Montreal in June 1986 at the initiative of MOIFEM (International Movement for Women and MathematicsTeaching).This movement was created as a result of the 1976 ICME Congress. An introductionby Louise Lafortuneis followed by six contributionsto the colloquium and the reports of two workshops: "A feminist view of mathematics" and "Means of action". It concludes with a catalogue of resources giving a detailed bibliographyof the subject as well as a list of associations in Canadaand the USA which are concernedwith women and mathematics. The first article, by Leone Burton, is entitled "Women and mathematics:is there an intersection?"This article gives her general thoughtson the relationship between women and mathematics.In it the authorsets two types of mathematics in juxtaposition: "public" mathematics, which is always presented in a stereotyped way as a rigorous, logical, abstract and objective science, and "private" mathematics, the mathematics of the mathematicianat work, nonobjective and subject to creativity and intuition as much as to strict rules. According to this article, the reconciliation between women and mathematics appearsprincipally to be a problem of the methodof teaching: not to restrictthe teaching of mathematicsto its "public world", with its masculine connotations, but to open up the teaching of the "private"world of mathematics.Whetheror not one favours improving the relationshipsof women with mathematics,one can only approve of this goal; since the public world of mathematics,in spite of its overtly masculine values, will be revealedas much to men as to women when it is not so obscure. In the article which follows, entitled"Mary,Sofia, Emmy, mathematiciansin history", Louise Lafortune introduces us to the life stories of three famous women mathematicians: Mary Fairfax Somerville, Sofia Vassilievna KrukovskayaKovalevskaya and Emmy Noether, and shows their difficulties in getting themselves recognised as mathematicians.Have these obstacles been overcome nowadays? On readingLouise Lafortune'sarticle it is to be regretted that they have not, when comparingthese three lives with those of present-day female mathematicians. The following three papers present research which is being carried out in Quebec: - 1. a study of the factors involved in the choice of subjects of pupils in three groups in the 5th year of secondaryschool (RobertaMura). 19 (1988)277-280. EducationalStudiesin Mathematics

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- 2. researchinto the psychologicalinfluenceson the success in mathematicsamonggirlsin the sixthyearof primaryschool(LiseLegault). - 3. the influenceof the family in the choice of a careerwhich is not feminineone (C61ineGuilbert). considereda traditional 1. Althoughgirls do as well as boys up to the end of their secondary schooling,thereis a lowerproportionof girls studyingmathematicsat college level (in Quebec).RobertaMura'sstudytriesto explainthisby investigatingthe variableswhichareconnectedbothwiththesex of the studentandherchoiceof subjects.Unfortunately, amongthe manyvariableslisted,only one seemsto fit the category,andthennot veryconvincingly;thatis, the girl'sconfidencein her mathematical for ability.In particular, girlsandboysofferdifferentexplanations theirsuccessor failurein maths- boys succeedbecausetheyaregiftedandfail becausetheydo notworkhardenough,whilein thecase of thegirlstheorderof explanationswould be the opposite.Accordingto the author,"If thereis the same attitudetowardsthe othersciences,one can assumethatgirls (morethan boys) see theirchoiceof scientificsubjectsas consistingof risksof failure(since they believe they are less capable)or even moreas a commitmenton theirpart to putin concentrated andunflaggingefforts." 2. In the nextarticle,Lise Legaultreportsresearchconductedwith20 girlsin the sixth yearof primaryschool,contrastingtheirsuccessin mathematics. She has triedto definethecognitiveandaffectiveinfluenceson successandfailurethe cognitivedimensionis measuredby Piaget tests administeredto the class andby theirclass ranking;the affectiveby projectivetests.The studyis short and lacks some preciseness,but the authormaintainsthat there would be a correlationbetweensuccessandthelevel of intelligence,while the influenceof theaffectivefactorwouldappearmorehazy. 3. C61ineGuilbert'sarticlewidensthe field by questioningthe influenceof the family on the choice of a careerwhichis not traditionally feminine.After reviewingresearchin this field, the authorgives theresultsof herown research conductedwith400 femalestudentsin differentfacultiesusinga questionnaire. These results confirm that certainfactors in the family circle do have an influence: parents' educationallevel; father's profession;mother's attitude towardswork;wheterthedominatinginfluenceis maleor female. Finally Lesley Lee's contributionreportson three experimentsaimed at explaining mathematicsto adult females. These were conductedin three differentframeworks: - three-hourworkshopsaimedat counsellorsand careersadvisers,both male andfemale,andat the participants in variousprogrammes of reintegration of womenstartingworkagain. - directteachingwhile the womenwereenrolledin a programme of reintegrationintothe worldof work. - a mathematics courseas a partof a programme of adulteducation. This attemptto clarify mathematicswas conductedtakinginto consideration

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both the cognitive and affective dimensions.On the affective level it "is primarilyconcernedwith destroyingthe myththatmathsis a neutral,objective, cold activitywithoutemotionality.The participants' emotions,negativeas well as positive, should play a part",thus alludingto the terminologyof Leone On the cognitivelevel, "it Burton,destroyingthe mythof publicmathematics. shouldshow thateach word,each symbol,everymathematical developmentis human a at a certain time. This producedby activity activity is constantly and often arises as a result of some kind of need which has been evolving from and also a balance of And experienced power". quite rightly, in our the author stresses both role the an opinion, playedby approachcombining historicalwithscientificelementsandalso thenecessityof workingto the level of the beliefs concernedwith thatmathematical activity- "beliefsystems"as demonstrated A. Schoenfeld. by In conclusion,this book in the Frenchlanguageis very readable(although readersmightbe put off by the authors'systematicfeminisationof words.)In particular,it gives a very good idea of the researchin this field which is currentlybeing done in Quebec. Overallthe studiesaim to understandthe of psychological,cognitiveor socialreasonsleadingto the under-representation womenin mathematicsand in the sciencesgenerally.They show clearlythe varietyof differentinfluencesand the complexinteractionof these factors. Althoughsome of the studiesare still in the exploratory stages,the analysisof theresultshasa genuinevalue. However,it would be an exaggerationto say thatthis collectionof studies presenta significantadvancein the field. Its interestlies in the clearpresentation, the large bibliography,and the present-daytheories,not to mentionthe official recognitionfor these ideas. The outcomeof this recognitionis the supportby the authoritiesfor the Colloquiumand for the publicationof its proceedings. In Europethere is the same problem- that of guidinggirls throughthe educationsystem- whichis beginningto receivea favourablereactionfromthe authorities.A declarationon the equalityof the educationof girlsandboys was adoptedby theEECin 1985. In France,just as elsewhere,womenare becomingbetterqualified- more than50%of high-schoolgraduatesandof collegestudentsarewomen- butvery few treada scientificpathway.Since 1985,regionalactionhas been takenon this,andin 1987 a nationalassociation,called"WomenandMathematics", was foundedwiththefollowingaims: - to encouragewomento enterthefieldof science,particularly mathematics. - to fosterthe interestof girls and womenin mathematics, sciencegenerally, andin technology. - to ensurecontactwith associationswith similaraims, especiallyin other Europeancountries. One of its firstbattleswill be to combatthe disastrouseffects of the recent

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establishment of a mixed entry examination for the Ecoles Normales Sup6rieures:which resultedin a huge fall in the numberof girls entering to read mathematicsat this level. On the other hand, specific research in this field is practically non-existent. However, we can mention here the work of C. Laville, Some Female Fantasies about Mathematics, published in the Cahiers of the University of Paris X; the current thesis by Mme P6gliasco, at the University of Paris 7, on differences between boys and girls of 11 to 15 years in regard to mathematicsat secondary school level; and the research/action group "Girls and Scientific Subjects", which was organised by IREM at Rennes with the participationof teachers of many subjects togetherwith careersadvisers. IREM, UniversitdParis 7, 2 Place Jussieu, 75005 Paris, France Universit6 de Rennes 1, Association "Femmeset Mathematiques"

M. ARTIGUE

M. F. COSTEROY

BOOK REVIEW

J. Dhombres, A. Dahan-Dalmedico, R. Bkouche, C. Houzel et M. Guillemot, Mathematiques aufil des ages, I.R.E.M. Groupe Epistemologie et Histoire, Gauthier-Villars, ? BORDAS, Paris, 1987, ISBN 2-04-016448-0. xiii + 327 pp. 150FF. The mathematicians worship the creative power of mind beyond measure and at the same time they disregard the creative power of mind beyond measure. Their adoration aims at the individual man, and their neglect falls upon the produced objects. Everybody agrees a new construction of say a proof to be a highly creative act; but who is willing to grant the rendering of this construction to require some sort of replication of this creativity? Two guilds offer themselves for this neglected task, teachers and historians. So first of all what we have to expect from teaching the history of mathematics is the exposition of this creative process of mind. The objects of this process, the objects of mathematics, are of course products of thinking. So the history of mathematics deals with products of thinking, more precisely with the changing of these products. That is why historical thinking about mathematics requires a meta-level of thought which allows the mathematical concepts to alter themselves. To say it precisely, the A Priori of history of mathematics is Old Nick of current mathematics: that there is never an absolute sense of a mathematical concept. (Obviously this is the huge stumbling-block which divides actual and historical minded mathematicians.) Now it is an astonishing fact that this objective dilemma is hardly realized, to say nothing of a running methodological debate. Historiography of mathematics today suffers from an intensive vacuum of methodological reflection. (By the way: This vacuum did not exist during the first decades of this century!) The consequences of this lack of methodological reflection are easily to be foreseen. Where there is no sensitivity for (the implications of) the variation of meaning, there the concepts are presented as fixed. Where there is no consciousness about the reciprocation between the local and the global (say the single concept and the embracing theory), there the mathematical entities are handled as eternal objects which only bear the property to be caught Educational Studies in Mathematics 19 (1988) 281-286.

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correctly or falsely. Where there is no readiness (or ability) to appreciate a mathematical reasoning for its own sake there always will arise an artificial judgement from hindside. Mathematical concepts as absolute, mathematical entities as eternal and independent from the changing of knowledge, mathematical facts as indubitably true or false according to their qualification in today's mathematics these are the omnipresent features of a historiography of mathematics which does not even realize to say nothing of reflecting its methodology. I like to call this kind of historiography resultism as it projects the whole past onto the present and so takes the stage of present day mathematics as definitive. Resultism declares all events of the past as mere precursors of today's facts, which means that those events of the past which are not reflected in some actual honoured result are plainly ignored. Resultism nevertheless has some tempting seductions. First of all resultism enables the mathematician to do the historical job in just the same way as his actual job. (The most brilliant pronounciations of resultism are the historical notes by Bourbaki.) Secondly resultism removes the complications which arise from the variability of its objects under inquiry and so trivializes the subject of history. In consequence, thirdly, resultism facilitates dramatically the learning of history because it enables the teacher to present any tiny piece of original mathematical reasoning expecting the scholar to understand it just correctly without knowing anything about the very historical situation of the Zeitgeist of that time. And last but not least resultism in sum contributes to the most popular illusion of a kingdom of mathematics, "where mathematical theories dwell like the seraphim, purged of all the impurities of earthly uncertainty" (Lakatos). Now the distinguished I.R.E.M. Groupe Epistemologie et Histoire presents a new book: Mathematiques aufil des ages. Mathematical text-extracts from antiquity till the turning of the last century (Peano, Borel, Einstein, and even Bourbaki 1948) are collected. The authors characterize their aim as "the wish to restitute this cultural dimension too often neglected in its scholastic presentation. We equally want to favour a better formation of the spirit of this science and to make the reader participate in the pleasure and the intellectual stimulation provoked by the texts" (p. XIII). The cultural dimension of mathematics is seen in its being undissolvably interwoven with the whole body of science in our societies by transmissions and diffusions. Really an ambitious scope! The self-imposed limitations follow from the intention to create small unities which permit a paper or a course of lessons at the lycees, especially in collaboration with the teachers from physics, history, or philosophy.

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The six chapters are chosen as mathematical subjects: Object and Utility of Mathematics - Arithmetic and Theory of Numbers - Algebra - Analysis - Calculus of Probability - Geometry. It is left to the reader to reconstitute the stage of mathematical knowledge at a certain epoch, helped by chronological advice. Each chapter presents in roughly but not always upheld chronological order citations of the most prominent geniuses. The text-fragments vary in length from one sentence to more than two pages. They are translated into French where necessary, and they are introduced, commented, or linked by often short commentaries. Sometimes parts of the sources are rendered in modern terms and symbols. The biographies at the end of the book, which dedicate two to ten (Diophant!) lines to each mathematician - nearly 150 entries on 16 pages and not yet complete - deserve no notice. Of course it is not possible to give here a detailed account of every atom of the book. So I will try two different glances, a global one and a local one, and concentrate on the most important aspects. At first the global one. Already an overlook shows that each chapter presents at least one citation from Euclid. (To honour to the truth, except the chapter on probability, but this might be a slight negligence by chance?) And also the other way round, besides the Babylonian Tablet 13 901 and the Egyptian Rhind papyrus (and Plato and Aristotle) there is no earlier illuminati than Euclid mentioned. It seems not to be boundless zeal to point out the inevitable consequence of this art of lecturing the coffee-grounds: The reader is secretly taught that the whole body of mathematics was founded by (or already known to) Euclid: Arithmetic, Algebra, Analysis (sic.), and - this the reader might yet have expected - Geometry!? So what is left from the full-toned announcement of the Avant-propos, mathematics "has a history as long-lasting as the history of mankind, being inscribed in our civilization and our cultures" (p. XIII)? Were our civilizations and our cultures only able to vary the appearances of mathematics, not to create its essence? Is mathematics never created (besides by Euclid) but only altered? Here the authors miss the most essential point in the history of a creative process: to work out the production of the new and to characterize the significance of these innovations. Without understanding these transformations the whole story is nothing more than a dull joining of items, exempted from any "intellectual stimulation" whatsoever. But this might be a too condescending judgement from a global glance, so let me look for the details somewhere. I open the chapter on Analysis (responsible are Dahan and Dhombres, the last one being the prime editor of

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the whole book), but I feel sure that there are no structural diversities to the other chapters. I leave out the remote antiquity (Rhind papyrus, Euclid, Aristotle), the beginners (Stevin, Descartes, Galilei, Cavalieri, Torricelli, Fermat, Newton, Leibniz, l'Hospital - no Bernoulli!-) and concentrate on the first great change of paradigm which took place between Euler and Cauchy. Euler's Introductio in analysin infinitorum(1748) appears under the heading "The reversion of the base of analysis", while Cauchy's Analyse algebrique (1821) is presented as "A new rigour in analysis". Euler is said to have made "the concept of function as the base of his exposition" (p. 193, my pronounciation). This however is far away from being true. Euler 1748 indeed describes, characterizes, and transforms the several kinds of functions - but clearly he is not able to work with his concept of function ("an analytic expression") as a tool, for it is lacking the required precision (see e.g. Euler's dealing with the intermediate value theorem in ?33 - not presented in the book under review). This precision however is found in Cauchy 1821 who explicitly demands a function to be one-valued, and that is why he now is capable of really deducing some properties from this new concept. But what do Dahan and Dhombres claim? They declare Cauchy to have "by an effort of rigour" elucidated "the concepts of the specific base of analysis: infinitely small, limit, continuity, convergence, etc." (p. 198). Unfortunately they fail to explain how the concept of infinitely small (which counts for the basic concepts of Calculus since Leibniz' invention - see even p. 173 of the reviewed book - but which nowadays seems very suspect) was able to serve as a new 'rigorous' specific basic concept. Even worse five lines downwards, Dahan and Dhombres state that Cauchy in defining continuity "however confounds two notions, continuity and uniform continuity"; the statement is repeated on p. 201. What a mess! Really an intellectual stimulation the reader is left alone with! If this statement were true, the authors would have to explain how Cauchy could be rigorous on the basis of one vague and one clearly confused notion. Or, as it has been discussed for at least one decade (but the authors consequently ignore the existing controversies among historians) Cauchy 1821 did not work with muddy notions but within a universe of discourse which cannot plainly be projected in our paradigm of real numbers which was established in 1872 and casted in axioms since 1899! Here it is clearly seen how the above deplored vacuum of methodological reflection necessarily leads the authors to helpless confusion. To be sure this confusion is not at all anew, but now the time should really have come to clear it up.

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The authors face an objective dilemma: First Cauchy was a great mathematician who installed a new fruitful paradigm in analysis; second he worked with the wrong concepts (and stated some "wrong theorems" - the reviewed book tacitly passes over this delicate fact). But Dahan and Dhombres decide to cheat the reader out of factual clarification and so they fail to teach him an important lecture on the history of mathematics. If they would have taken their introductory proclamation seriously, if they were really presenting an "intellectual stimulation" here would be a suitable place to stop the everlasting repetition of historical cock-and-bull stories and to supply intrinsic insights. These insights would consist (i) of a meaningful characterization of the changing of mathematics from Euler to Cauchy, (ii) of the perception that there indeed exist different but closely related universes of discourse in mathematics which cannot mechanically be translated into each other's language. Now (ii) clearly is not consistent with resultism as characterized above. So we reach the conclusion that it is resultism which prevents the authors to get deeper insights in the development of mathematics. Most unfortunately this basic feature shows up throughout the whole book. But resultism tends to trivialize past mathematics, and it breaks completely down if it is meant to capture some mathematical theory which cannot be projected in today's accepted mathematics without severe factual damage. It is a great pity that an illustrious circle of historians and epistemologists (ten authors are explicitely named as responsible for the six chapters, and the list of collaborators adds up to thirty-five) is not able to realize those shortcomings. After all it is now the time to overcome this historiography of mathematics which is chained by a complete lack of methodological reflection. It is time to start weaving a filigree to grasp the real historic process instead of mechanically repeating mathematics along a non-existing rosary of ages. Postscript. Some readers of this review might ask, if that's not too strong; if it is not overshooting the mark to criticise a book written for lycees by the highest methodological standards of historical research?My answer is a clear No! First of all we should be willing to teach only the best of our knowledge, especially to the growing generation, instead of putting them off with second hand insights. And secondly I really doubt if history of mathematics is a legitimate and meaningful topic for the curricula of our schools. One doubt results from the disastrous state of affairs in the History of Mathematics, but there is a second doubt. You cannot teach differential equations before or without a considerable amount of calculus - and analogously you cannot

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teach history of mathematics before or without a considerable amount of mathematics. As I explained in the beginning of the review, history of mathematics is a meta-thinking whose objects are the changing features of mathematical ideas. I know for myself that I have not been able to grasp some essential point of the history of mathematics at the time of my school days. Of course this might have been an idiosyncracy of my own, but this is not yet proved. So I stay with my conviction that the book under review is indeed not written for lycees in a realistic meaning of this phrase. And that is why I do not regard my criticisms as exaggerated. Of course someone might draw the conclusion from the above, that real history of mathematics indeed is no meaningful topic for teaching the lyceens, but therefore it should be presented only as some sort of fairy-tale. But then, I am sure, we would at once agree that the fragmentary texts of this book are missing dimensions of tension, excitement, and colours to bear even a tiny bit of attention by the pretended audience. AG Fachdidaktik, FB Mathematik, Technical University, Schlofigartenstr. 7 D-6100 Darmstadt, West-Germany.

DETLEF

D. SPALT