Science and Technology EducationDocument Series No. 47

SIGNIFICANT INFLUENCESON CHILDREN'S LEARNING OF MATHEMATICS

by

Alan J. BishopKathleen Hart

Stephen LermanTerezinha Nunes

Education Sector

UNESCOParis, 1993ED/93WS-20

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Influences from society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Alan J.Bishop

The socio-cultural context of mathematical thinking: research findings and

educational implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Terezinha Nunes

The influences of teaching materials on the learning of mathematics . . . . . . . . . . . . . 43

Kathleen Hart

The role of the teacher in children's learning of mathematics . . . . . . . . . . . . . . . . . . . 61

Stephen Lerman

References in PME proceedings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Other references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Mathematics has become one of the mostimportant subjects in the school curriculumduring this century. As modern societies haveincreased in complexity and as that complexityhas accompanied rapid technological develop-ment, so the teaching of mathematics has comeunder increased scrutiny.

Research in mathematics education gene-rally has also become more and more importantduring that time, but has struggled to keep upwith the increasingly complex questions askedof it. There was a time when mathematics lear-ning could be conceptualised simply as theresult of teaching the subject called mathema-tics to a class of children in the privacy of theirclassroom. As long as the concerns were large-ly quantitative, such as how to provide moremathematics teachers, more mathematics class-rooms, and more mathematics teaching formore students, that picture appeared to be ade-quate. We now know that that simplistic pictu-re will not help in answering any of the signifi-cant questions of recent decades which predo-minantly concern the quality of the mathema-tics learning achieved.

Of course it is the case that in many, oreven most, countries in the world, there are stillquantitative issues surrounding mathematicslearning. Indeed, in the developing world thereis often little opportunity for the luxury ofengaging with subtle issues of quality in theface of horrendous quantitative problems ofeducational provision. Nevertheless, as theUNESCO publication Mathematics for allshowed (Damerow et al.,1984), one cannotdivorce the problems of mathematics learningfrom the wider problems of society, and thusthere is every reason for all mathematics edu-cators to become aware of the factors whichhave the potential to affect the quality ofmathematics learning taking place in schools.This book is intended to help bring to a wideraudience research which informs us about thesignificant influences on that quality.

In particular, the authors have identifiedfour groups of influences which

appear to be of crucial importance for learnersof mathematics. Firstly there are the demands,constraints and influences from the society inwhich the mathematics learning is taking place.These, in a sense, set the knowledge and emo-tional context within which the meaning andimportance of teaching and learning mathema-tics are established. Recent research hasdemonstrated that it no longer makes sense totry to consider mathematics learning as abstractand context-free, essentially because the lear -ner cannot be abstract or context-free. Theresearch problems centre on which of the manysocietal aspects are of particular significance,on how to study their influences, and on whatif anything to do about them.

The second set of influences concern theknowledge, skills and understanding which thelearners develop outside the school setting andwhich have significance for their learning insi-de the school. This topic for research has onlydeveloped relatively recently, but its findingshave surprised many mathematics teacherswith its implications for their work. One of thegreat educational challenges of the present timeconcerns how school mathematics teachingshould take learners' out-of-school knowledgeinto account.

The third set of influences on children'slearning of mathematics come from the tea-ching materials and aids to learning in theclassroom. These have become more subtle andvaried - from the textbook to the computer -and have increased in number and importanceconsiderably over the last decades. In the faceof such development, the need for continuingresearch and analysis concerning the signifi-cance of these influences has become evenmore important.

The fourth and final influence is not a newone at all - indeed it could be thought of as theoldest influence on mathematics learning - t h et e a c h e r. Every learner can quote the memory of aparticularly influential teacher, whether good orbad, and every teacher knows the feeling ofinfluence which the position gives them. T h e r ehas in the last decade been a growth of interest in

1

Introduction

research into the influence which teachers can,and do, have on the mathematics learners intheir charge, and it is important to bring theseideas to a wider audience.

The four authors have drawn on recentlypublished research which has been reported inpublications and at academic conferences, butin particular, as we are all long-standing mem-bers of the International Group for thePsychology of Mathematics Education(PME),we have made sustantial reference to ideaswhich have developed within that context. Themajor research strands of PME have been fullydocumented in the following book:

Mathematics and cognition, edited byPearla Nesher and Jeremy Kilpatrick,Cambridge University Press, Cambridge, 1990

under the headings of: epistemology, lear-ning early arithmetic, language, learningg e o m e t r y, learning algebra, and advancedmathematical thinking. In addition, the

references at the back of this book are in twosections - the first, to particular papers in cer-tain Proceedings of the annual conferences ofPME, which are referred to in the text as(xxxx,PME,1988) and the second, to otherpublications.

Interested colleagues can obtain copies ofthe Proceedings of previous PME conferencesby writing to:

Alan Bell,Shell Centre for Mathematics Education,University of Nottingham,Nottingham NG7 2RD,UK.

We would like to record our thanks toUNESCO for making this publication possible,and to our many colleagues around the worldwho share in the development and communica-tion of knowledge in this field.

Finally, we dedicate this book to learners ofmathematics everywhere.

2

Schools and individual learners exist withinsocieties and in our concern to ensure the maxi-mal effectiveness of school mathematics tea-ching, we often ignore the educational influenceof other aspects of living within a particulars o c i e t y. It is indeed tempting for m a t h e m a t i c seducators particularly to view the task of deve-loping mathematics teaching within their parti-cular society as being similar to that of col-leagues elsewhere, largely because of their sha-red beliefs about the nature of mathematicalideas. In reality such tasks cannot deal withmathematics teaching as if it is separable fromthe economic, cultural and political context ofthe society. Any analyses which are to have anychance of improving mathematics teaching mustdeal with the people - parents, teachers,employers, Government officials etc. - and musttake into account the prevailing attitudes,beliefs, and aspirations of the people in thats o c i e t y. The failure of the New Math revolutionin the 60's and the early 70's was a good exampleof this phenomenon (see Damerow andWe s t b u r y, 1984).

It is therefore my task in this first chapter toexplore those aspects of societies which mayexert particular influences on mathematics lear-ning. These may happen either intentionally oru n i n t e n t i o n a l l y. Societies establish educationalinstitutions for intentional reasons - and formalmathematics education is directly shaped andinfluenced by those institutions in different waysin different societies. A d d i t i o n a l l y, a society isalso composed of individuals, groups and insti-tutions, which do not have any formal or inten-tional responsibility for mathematics learning.They may nevertheless frame expectations andbeliefs, foster certain values and abilities, ando ffer opportunities and images, which willundoubtedly affect the ways mathematics is vie-wed, understood and ultimately learnt by indivi-dual learners.

Coombs (1985) gives us a useful frame-work here. In discussing various 'crises ineducation' he argues that education should beconsidered as a very broad phenomenon,

rather than being a narrow one, and that there ared i fferent kinds of education. In his work he sepa-rates Formal Education (FE) from Non-f o r m a lEducation (NFE), and Informal Education (IFE).

Formal Education, he says, "generallyinvolves full-time, sequential study extendingover a period of years, within the framework of arelatively fixed curriculum" and is "in principle,a coherent, integrated system, (which) lendsitself to centralized planning, management andfinancing" (p.24), and is essentially intended forall young people in society.

In contrast NFE covers "any organised, sys-tematic, educational activity, carried on outsidethe framework of the formal system, to provideselected types of learning to particular subgroupsin the population, adults as well as children"(p.23). In contrast to FE, NFE programs "tend tobe part-time and of shorter duration, to focus onmore limited, specific, practical types of know-ledge and skills of fairly immediate utility to par-ticular learners" (p.24).

Finally IFE refers to "The life-long processby which every person acquires and accumulatesknowledge, skills, attitudes and insight fromdaily experiences and exposure to the environ-ment.... Generally informal education is unorg a-nized, unsystematic and even unintentional attimes, yet it accounts for the great bulk of anyperson's total lifetime learning - including that ofeven a highly 'schooled' person" (p.24).

We shall organise this chapter's contributionaround these three different kinds of education,looking particularly at the influences from socie-ty on the three kinds of mathematical education.Finally there will be a discussion of some of thesignificant implications which result from thisa n a l y s i s .

Societal influences through formal mathe-matics education (FME)

It seems initially obvious that any societyinfluences mathematics learning through theformal and institutional structures which itintentionally establishes for this purpose.

3

Influences from Society

Alan J. Bishop

P a r a d o x i c a l l y, at another level of thinking, it willnot be clear to many people just what influenceany particular society could have on its m a t h e -matics learners, in terms of how this will diff e rfrom that which any other society might have.Indeed, mathematics and possibly science have, Isuspect, been the only school subjects assumedby many people to be relatively unaffected by thesociety in which the learning takes place.Whereas for the teaching of the language(s) ofthat society, or its history and geography, its artand crafts, its literature and music, its moral andsocial customs, which all would probably agreeshould be considered specific to that society,mathematics (and science) has been considereduniversal, generalised, and therefore in someway necessarily the same Tom society to society.So, is this a tenable view? What evidence ist h e r e ?

The formal influences on the mathematicallearners will come through four main agents -the intended mathematics curriculum, the exa-mination and assessment structures, the tea-chers and their teaching methods, and the lear-ning materials and resources available. The lasttwo aspects which are part of the implementedcurriculum (Travers and Westbury, 1989) willbe specifically considered in the final twochapters of this book and therefore I willconcentrate here on the first two, the intendedmathematics curriculum, and the examinations.

The intended mathematics curr i c u l u m

If we consider the mathematics curricula in dif-ferent countries, our first observation will be thatthey do appear to be remarkably similar acrossthe world. Howson and Wilson (1986) talk of the"canonical curriculum" (p.19) which appears toexist in many countries. They describe "the fami-liar school mathematics curriculum (which) wasdeveloped in a particular historical and culturalcontext, that of Western Europe in the aftermathof the Industrial Revolution". They point out that"In recent decades, what was once provided forthe few has now been made available to - i n d e e d ,forced upon - all. Furthermore, this same curri-culum has been exported, and to a large extentvoluntarily retained, by other countries across theworld. The result is an astonishing uniformity ofschool mathematics curricula world-wide" (p.8).

This fact has made it possible to conductlarge-scale multi-country surveys and compari-

sons of mathematical knowledge, skills andunderstanding such as those by theInternational Association for EducationAchievement (IEA) whereas such comparisonswould probably be unthinkable in a subject likehistory. Indeed, one of the main research issuesin the Second International MathematicsSurvey (SIMS) was whether the 'same' mathe-matics curricula were being compared (seeTravers and Westbury, 1989). Some idea of theextent of the agreements found can be gainedfrom Table 1 (below) which shows the relativeimportance accorded by different countries topotential topics and behavioural categories tobe included in the Population A (13 year olds)assessment. Travers and Westbury concludedfrom their further analysis of the data concer-ning Table 1 that "The only topic for whichthere appears to be a substantial problem ofmismatch is Geometry. Indeed a major findingof the Study proved to be the great diversity ofcurricula in geometry for Population A aroundthe world" (p.32).

It could be argued that such internationalsurveys, and the comparative achievement datathey generate, encourage the idea that themathematics curricula in different countriesshould be the same, particularly when, as inthis case, they are seeking the highest commonfactors of similarity. At the very least, thisapproach could well lead to mathematics edu-cators in many countries anxiously lookingover their shoulders at their colleagues in othercounties to see what their latest curriculartrends are. One wonders what would resultfrom a research study which sought to find thed i fferences between mathematics curriculaexisting in different countries. I suspect wemight well find a different picture from thatportrayed in Table 1. The recent book byHowson (1991) which documents the mathe-matics curricula in 14 countries, is therefore awelcome addition to our sources.

Indeed societies may well not only influencethe national intended curriculum in ways in whichSIMS would not reveal, but may also influencethe intended curriculum at more local levels.

4

5

Table 1

Moreover in the SIMS study there were stri-king differences between the intended and theimplemented curricula in various countries,and also striking variations in the latter withinnational systems. Some of that variation couldundoubtedly be due to local intended varia-tions: "With the significant exception of Japan,all systems have relatively high indices ofdiversity and it may be that such diversity is areflection of the responses that teachers maketo the overall variations of readiness forAlgebra that they find at this level" (Traversand Westbury, 1989, p.131).

It is well known that the intended mathematicscurricula certainly do vary between schools indifferentiated systems, i.e. where there is not acomprehensive (single) school structure. Insome highly differentiated systems inW.Europe for example the grammar school, thegymnasium and the lycée have very differentcurricula from the other schools in those sys-tems. The SIMS study showed (Travers andWe s t b u r y, 1989) the following data whichgives the 'Opportunity to Learn' figures forArithmetic and Algebra in the different sectorsof the differentiated systems in Finland,Netherlands and Sweden (Opportunity to Learnis a % measure of the availability of that topicin the curriculum).

As an example of how such curricular differen-tiation can influence performance, a Hungarianstudy (Klein and Habermann, PME, 1988) pro-duced the following data:

Type of school and curriculum Mean testscore

Grammar schoolsCurriculum Mathematics II"(special) 13.41

Grammar schoolsCurriculum Mathematics Is(special) 8.68

Specialised vocational secondaryschools (SVSS) Curriculum F(Direction Computing services) 6.03

Grammar schoolsBasic curriculum 5.41

SVSS Curriculum group A/B/E(Direction Industrial andagricultural professions) 4.87

SVSS Curriculum C (Direction Healthand medical professions) 3.39

SVSS Curriculum D (DirectionKindergarten nursing) 2.66

There is however little evidence that thedifferent curricula operating within nationalsystems are anything other than merely subsetsof the 'total' national curriculum. One readsphrases like "watered-down", "simpler ver-sion" etc. which suggest that differentiated sys-tems will tend towards what Keitel (1986) sayswas the situation "in England now at the secon-dary level: a sophisticated, highly demandingmathematics course for those who continuetheir studies, and next to nothing for the rest"(p.32). The differentiated mathematics curricu-lum thereby reflects the differentiation whichthe society appears to want, and which it seeksto perpetuate through its formal educationalstructures.

In fact there is no a priori reason at all whymathematics curricula should be the same in allcountries. Mathematics in schools can beconsidered in similar ways to art, history, andreligion, where there is no necessity for curri-cula to be the same in different societies. Wenow know from the ethnomathematics researchliterature which has been gathered in the lasttwenty years, about the enormous range ofmathematical techniques and ideas which havebeen devised in all parts of the world (seeAscher, 1991). We have evidence from allcontinents and all societies of symbolic sys-tems of arithmetical and geometrical naturesdevised to help humans extend their activitieswithin the physical and social environmentsthat have grown to be ever more complex.From the Incas with their quipus to help withtheir accounting (Ascher and Ascher, 1981), tothe Chinese with their detailed geomanticknowledge for designing cities (Ronan, 1981),or from the Igbo's sophisticated counting sys-tem (Zaslavsky, 1973) to the A b o r i g i n a lAustralian's supreme spatial and locationalsense (Lewis, 1976) the known world is full ofexamples of rich and varied indigenous mathe-matical knowledge systems.

6

FinlandLong courseShort course

NetherlandsVWO/HAVOAVOLTOLHNO

SwedenAdvanced General

Arithmeticmedian

7474

87837470

7061

Algebramedian

7367

83805347

5725

So the question needs to be asked: Wheredoes the expectation of a universal mathema-tics curriculum come from? Of course wewould expect there to be recognisable similari -ties, as Travers and Westbury show, just as werecognise similarities between the curricula ofother subjects across different societies. Butrecognising similarities is totally different fromexpecting universality. In part the expectationderives from the fact that countries and socie-ties are not isolated; there is much inter-socie-tal communication and there has been for manycenturies.

Societies influence the intended mathema-tics curriculum in most countries through cer-tain nationally or regionally structured organi-sations. In the case of strong centralizedgovernments the curriculum will be the respon-sibility of the education ministry, while in moredecentralised systems such as the USA,Australia and Canada the power for decidingon the intended curriculum rests with the stateor local government. For example, in theUnited Kingdom the present government hasrecently instituted a national curriculum, adevelopment in which the highly political natu-re of national curricular decision-making hasbeen rather obviously demonstrated. We havetherefore seen the typical political pressuregroups being very active - the 'back to basics'groups led by traditionalists amongst theemployers and the government, the teachersand educators concerned about the erosion oftheir influence by the central government's'interference', the more progressive industria-lists who want to ensure that school leavers cancompete with the 'best of the rest of Europe',allied with parents whose genuine concernsabout their children's futures are coloured bylurid media reporting about the poor compara-tive performance by UK pupils in various (andsometimes suspect) international surveys.

Governments, education civil servants,political academics and others with educationalpower are thus nowadays very aware of what ishappening in other countries because they feelthat they have to be. The competitive economicand political ethic demands it. Internationalconferences, 'expert' visits, exchanges, articles,books and reports, all enable ideas from onecountry to be available to others. But there ismore to it than mere communication however.

The expectation of a universal mathematicscurriculum has another basis. As well as com-munication, there has been gradual accultura-tion by dominant cultures, there has been theassimilation of new ideas believed to be moreimportant than traditional ones, and there hasalso been cultural imperialism, practised ofcourse on a large scale by colonial govern-ments (see Bishop, 1990 and Clements, 1989).There has in particular been the widespreaddevelopment of a belief in the desirability oftechnological and industrial growth.Underlying this technological revolution hasbeen the mathematics of decontextualised abs-traction, the mathematics of system and struc-ture, the mathematics - of logic, rationality andproof, and therefore the mathematics of univer-sal applicability, of prediction and control.

What we have witnessed during the lasttwo centuries is nothing less than the growth ofa new cultural form, which can be thought of asa Mathematico-Technological (MT) culture(see Bishop, 1988). The growth of the idea ofuniversally applicable mathematics has gonehand-in-hand with the growth of universallyapplicable technology. Developments withineach have fed the other, and with the inventionand increased sophistication of computers, thenexus is truly forged.

It is difficult sometimes to understand theextent of the influence of this MT culture, sofirmly embedded has it become in modernsocieties' activities, structures and thinking. Wenow are in danger of taking the ideas andvalues of universally applicable mathematicsso much for granted that we fail to notice them,or to question them, or to see the possibility ofdeveloping alternatives. The acceptance of uni-versally applicable mathematics has had a par-ticularly profound impact on mathematics cur-ricula in all countries and in all societies. Theimpetus to become ever more industrialisedand technologically developed in the so-called"under-developed" countries has been under-pinned by the belief in the importance of adop-ting the mathematics and science curricula ofthe more - industrialised societies. It is thebelief in the power of the ideas of universallyapplicable mathematics which has created theexpectation of the universal mathematics curri-culum.

7

So, all around the world, in societies withd i fferent economic structures (see Dawe,1989), in societies with different political struc-tures (see Swetz, 1978), and in societies withdifferent religious bases (see Jurdak, 1989),learners find themselves grappling, not alwayssuccessfully, with the intricacies of arithmeti-cal algorithms, algebraic symbolisations, geo-metrical theorems. Later on, if they pass therequired examinations, they meet infinitesimalcalculus, abstract algebraic structures, othergeometries and applicable mathematics.

Moreover, until perhaps ten years ago, thissituation was largely unquestioned. Now it isbeing seriously challenged from various quar-ters and we can see different groups withinsocieties trying to exert rather diff e r e n tinfluences on their school curricula. The majorchallenges which we can identify so farconcern the irrelevance of an industrialised andtechnological cultural basis for the curriculumin predominantly rural societies, and the rise ofcomputer education in industrialised societies.

The first of these challenges comes princi-pally from within societies having a predomi-nantly rural economy. The growth of the awa-reness of the economic and societal bases ofmuch of the universal mathematics curriculumhas not been as clear as the development of'intermediate technology' or 'appropriate tech-nology' in many of these countries. Howeverwe can now begin to recognise a desire for amore relevant school curriculum which is lea-ding mathematics educators in those countriesto search for a more 'appropriate mathematics'curriculum. Desmond Broomes, in Jamaica, isa mathematics educator responding to this per-ceived concern in the rural society of the WestIndies, and his ideas represent this view well.In Broomes (1981) he argues that:

"Developing countries and rural com-munities have, therefore, to reexaminethe curriculum (its objectives and itscontent). The major problem is to rura-lize the curriculum. This does not meanthe inclusion of agriculture as anothersubject on the programme of schools.Ruralizing the curriculum means incul-cating appropriate social attitudes forliving and working together in ruralcommunities. Ruralizing the curricu-lum must produce good farmers, but itmust also produce persons who wouldc o-operatively become economic

communities as well as social andeducational communities." (p.49)

In Broomes and Kuperus (1983) thisapproach means developing activities which:

"a) bring the curriculum of schools clo-ser to the activities of communitylife and to the needs and aspirationsof individuals;

b) integrate educational institutions,vertically and horizontally, into thecommunity so that outputs of suchinstitutions are better adapted to thelife and work in the community;

c) redistribute teaching in space, time,and form and so include in the edu-cation process certain living: expe-riences that are found in the com-munity and in the lives of persons;

d) broaden the curriculum of schoolsso that it includes, in a meaningfulway, socioeconomic, technical andpractical knowledge and skills, thatis, activities that allow persons tocombine mental and manual skillsto create and maintain and promotes e l f-s u fficiency as members oftheir community." (p.710)

In implementing this strategy, according toBroomes (1989) the teacher should pose pro-blems to the learner and these problems should:

- "be replete with the cultural expe-riences of the learner;

- be, for the most part but not solely,practical and utilitarian;

- Illustrate mathematics in use;

- be capable of being tackled profita-bly using mathematics;

- not make excessive mathematicaldemands (in terms of the learner'slevel of mathematical sophistica-tion);

- provide ample scope for cooperati-ve activity among persons at diffe-rent levels of mathematics compe-tence."(p.20)

We can find some of these same sentimentsshared by educators in parts of Africa, inSouthern Asia, in Papua New Guinea and in ruralparts of South America and Australia. In some ofthese countries there is a particularly strong desi-re to represent the growth of indigenous cultural

8

awareness through the mathematics curricu-lum. That is, what we are seeing is not just arejection of the MT culture which underlies the'universal' curriculum and underpins modernindustrialised societies, but a replacement ofthat cultural ideology with a rediscovered, indi-genous, cultural heritage. Gerdes (1985) inMozambique and Fasheh (1989), thePalestinian educator, are particularly eloquentin expressing this perspective on the mathema-tics curriculum, and Nebres (1988) is also ada-mant about its importance. The new mathema-tics textbooks in Mozambique, for example,perhaps demonstrate more than any other coun-try's do, just how one can reconstruct themathematical curriculum within a rural andnon-western society.

What is most significant from this kind ofcurricular activity is not, however, the particu-lar mathematics curriculum developed in anyone country, but rather the richer understandingwhich we can all now share that it is possible,and probably desirable in certain situations, toconstruct and implement alternatives to thecanonical universal curriculum. These deve-lopments encourage mathematics educatorseverywhere to explore ways in which the curri-culum can become a more responsive andappropriate agent in the mathematics educationof the society's learners. The time has passedwhen it was considered sensible to merelyimport another society's curriculum into one'sown society. It is however sensible to 'import'the idea of an alternative mathematics curricu-lum, as well as the kind of background resear-ch necessary to identify the appropriate bases,the strategies for development, and theapproaches to implementation. From a ruralsociety's perspective, of course, these ideas andapproaches would be most likely to be found inother rural societies.

The second major challenge to the myth ofthe universal mathematics curriculum is, para-doxically perhaps, coming from within indus-trialised societies themselves, as computer tech-nology grows in its range of influence. The jus-tification for much of the existence of particulartopics in the mathematics curriculum camefrom their assumed usefulness, both to the indi-viduals and to society as a whole. This wassimilar to the justification which lay the behindthe hack to basics' movement spearheaded byindustrialists and governments who were

reacting to the uselessness, as they and manyparents saw it, of the New Math of the previousgeneration. However the validity and relevanceof this 'utility' argument is now being seriouslydisputed within mathematics education as itbecomes clear that computers and calculators cantake over many of the previously needed skills.

Howson and Wilson (1986) describe someof the features of the new curricular situationbrought about by computers:

"a) algorithms

Algorithmic processes lie at theheart of mathematics and alwayshave done. Now, however, there isa new emphasis on algorithmicmethods and on comparing the effi-cacy of different algorithms for sol-ving the same problem e.g. sortingnames into alphabetical order orinverting a matrix.

b) Discrete mathematics

Computers are essentially discretemachines and the mathematics thatis needed to describe their func-tions and develop the software nee-ded to use them is also discrete. Asa result, interest in discrete mathe-matics Boolean algebra, differenceequations, graph theory, ... - hasincreased enormously in recentyears: so much so that the traditio-nal emphasis given to the calculus,both at school and university, hasbeen called into question.

c) Symbolic manipulation

The possibility of using the compu-ter to manipulate symbols ratherthan numbers was envisaged in theearly days of computing. Now,however, software is available formicros which will effectively carryout all the calculus techniquestaught at school - differentiation,integration by parts and by substi-tution, expansion in power series -and will deal with much of thepolynomial algebra taught therealso. Is it still necessary to teachstudents to do what can be done ona computer?" (p.69).

Fey (1988) concurs with these views andsummarises the situation thus:

“ the most prominent technolo-gy-motivated suggestions forchange in content/process goalsfocuses on decreasing attentionto those aspects of mathematicalwork that are readily done

9

by machines, and increasing emphasison the conceptual thinking and plan-ning required in any 'tool environment'.Another family of content recommen-dations focus on ways to enhance andextend the current curriculum to mathe-matical ideas and applications of grea-ter complexity than those accessible tomost students via traditional methods. Idistinguish:

1. Numerical computation, 2. Graphic computation,3. Symbolic computation,4 Multiple representations of infor-

mation,5. Programming and the connections

of Computer Science withMathematics curricula,

6. Artificial intelligences and machinetutors." (p.235)

It is not just the topics within the curricu-lum which are being questioned or proposed,the whole purpose and aims of mathematicseducation are now under scrutiny. Dorfler andMcLone (1986) present us with the followingargument:

"The dilemma for mathematical educa-tors in considering the place of mathe-matics in the school curriculum can bedescribed thus. On the one hand, theincreased technological demands ofsociety and the development of sciencerequire highly trained mathematicianswho can apply themselves to a varietyof problems; they also require profes-sional scientists, engineers and othersto have a greater acquaintance with andcompetence in mathematical technique.In addition, whilst the increased use ofcomputational aids is likely to lead toless demand for routine mathematicalskill from the general workforce, a lar-ger number of technical managers willbe needed who can interpret the use ofthese computational aids for generalapplication. On the other hand the basicmathematical requirement for employ-ment is unlikely to grow beyond gene-ral arithmetic skill (often with the aidof calculator) and the interpretation ofcharts, tables, graphs, etc.; indeed, notmuch beyond the needs of everydaylife mentioned earlier in this section.

A rguments for school mathematicsbased on its ability to develop powersof logical thinking do not provide suffi-cient justification as it cannot be clear-ly shown that such powers are unique-ly developed through the study ofmathematics" (p.57).

Keitel (1986) argues the same thing fromanother point of view. She is concerned in thatpaper with the 'social needs' argument formathematics teaching:

"By 'social needs' demands I unders-tand here the pressures urging schoolmathematics to comply with the needsfor certain skills and abilities requiredin social practice. Mathematics educa-tion should qualify the students inmathematical skills and abilities so thatthey can apply mathematics appropria-tely and correctly in the concrete pro-blem situations they may encounter intheir lives and work. Conversely, socialusefulness has been the strongest argu-ment in favour of mathematics as aschool discipline, and the prerequisiteto assigning mathematics a highlyselective function in the school sys-tem." (p.27)

She argues in her paper that computer edu-cation is now taking over this argument frommathematics education, and that this shouldrelease mathematics education from thatdemand. We need therefore to consider justwhat the intentions of future mathematics cur-ricula should be. Should mathematics notcontinue to be a core subject within the schoolcurriculum, but become optional? Should itbecome more of a critical and politically infor-med subject serving the needs of a concernedsociety faced with an environmental holocaust?Should it become more of a vehicle for develo-ping democratic values?

These kinds of curricular debates demons-trate that if we wish to take society's influenceson mathematical learning seriously then acade-mic considerations and criteria are not suffi-cient. In modern industrialised societies mathe-matics is so deeply embedded in the cultural,economic and knowledge fabric of that societythat the mathematics curriculum must alsoreflect the political, economic and socialconcerns of that society.

Sociologists concerned with educationhave long argued about the relationships bet-ween educational practices and societalstructures. Bowles and Gintis (1976) forexample argued that the evidence showedthat education, rather than eradicating socialinequality (one of the aspirations of educa-tion in a democracy), tends to reinforce it.Giroux (1983) however demonstrated thatalthough schools are tied to particular social

10

features of the society in which they exist, theycan also be places for "emancipatory teaching".Bourdieu (1973) distinguishes cultural repro-duction from social reproduction, and arguesthat the first is what is mainly transmitted bythe home and the school, with the curriculumand the examinations as the principle instru-ments for cultural reproduction.

There is however little dispute that theintended mathematics curriculum is an impor-tant vehicle for the induction and socialisationof young people into their culture and theirsociety, and as such it tends to reproduce andreinforce the complex of values which are ofsignificance in that society. It is only recentlyhowever that such social and political valueswithin the mathematics curriculum have beenrecognised and discussed (see, for example,Mellin-Olsen, 1987; Bishop 1988;Frankenstein, 1989). Hitherto, mathematics,due to the assumptions of universality, wasthought to be neutral - i.e.. culture-free, socie-tally-free and value-free. That stance is now nolonger valid and those who work in mathema-tics education everywhere need to recognisethe importance of revealing and debating thevalues aspects of the curriculum which havebeen largely implicit for many years. Someyears ago Swetz (1978) made us aware of somefeatures of the political embeddedness ofmathematics education in certain socialistcounties, but more recent developments byother researchers are provoking mathematicseducators in many other non-socialist countriesto consider just how their mathematics curricu-lum should respond to these different concernswithin their particular societies (see, forexample, the writings of several people inKeitel et al., 1989). Thus we can see from allthese diverse developments that the concept ofthe universal mathematics curriculum is losingits credibility both in theory and in practice.The intended mathematics curriculum has alsobeen shown to be a vehicle capable of respon-ding to the political and social needs of society.

It has therefore become an object of seriouspolitical concern itself - a state of affairs oftendeplored by many mathematicians. In a sensethough this was to be expected. Any subjectwhich becomes as important as mathematics hasbecome in society, cannot be immune to the dif-ferent pressure groups within that society. T h e

mathematics curriculum is clearly far tooimportant an instrument to be determined bymathematicians alone. Whether it can be as res-ponsive as it needs to be, to accommodate allthe influences from society, is however a mootpoint, and is one which will be taken up in thefinal section of this chapter.

The examinations

The second major influence from societyon the mathematics learners comes through theformal examinations, and the examination sys-tem. The examinations seem to the learners todefine operationally what is to be required ofthem as a result of their mathematical educa-tion - no matter what their teachers, parents,and other formal and informal educators maysay to the contrary. The old adage 'education iswhat remains with you after you've forgotteneverything you learnt at school' may have acertain amount of truth in it, but the fact is thatany learner on any formal educational coursewill be acutely aware of the essential demandsof their particular forthcoming examination, orassessment. In particular, as the examinationsoperationalise the significant components ofthe intended mathematics curriculum, so theytend to determine the implemented curriculum.

Just as was the case with the idea of theuniversal mathematics curriculum, so it is withmathematics examinations - they look verysimilar from country to country. As long asthere existed international consensus about thecontent of the intended curriculum, it also see-med sensible to have similar examinations.There are however variations between coun-tries, and although there has been no systema-tic research into the variety of examinationpractices around the world, it appears that thefollowing represent the major aspects of diffe-rence:

- proportion of short to long examinationquestions. Some education systems empha-sise short questions more, and may castthem in a multiple-choice format, whileothers prefer long questions to predomina-te. The issue seems to relate to how largethe content sample is for the examination -short questions allow a greater sampling,long questions less;

- proportion of 'content' to 'process'questions. This relates to the previousaspect, but refers essentially to whe-ther the examination is intended for t he

11

assessment of knowledge and skills whichhave been 'covered' in the teaching, or forthe assessment of processes which areassumed to have been developed;

- extent of teacher role in the examinationprocess. Some systems merely expect theteacher to be the administrator of the exa-minations produced by 'outsiders', whileothers take the view that the teacher'sassessment is a key part of the examinationprocess. In the latter case there will bemore of a moderating role played by 'outsi-ders';

- proportion of oral and practical work inclu-ded in the examination. In some systemsthe examinations are entirely written, whil-st in others practical materials will be avai-lable and in others pupils will give theiranswers mainly orally.

- amount of examined work completedduring the course of teaching, rather than ina final 'paper'. As more process aspects ofmathematics teaching are emphasised, sothe final paper has decreased in importance;

- extent of pupil choice. In some examina-tions all questions and papers are compul-sory, while in others there are choices to beexercised by the candidates. Such choicesallow pupils to emphasise their strengthsand minimise their weaknesses therebyallowing pupils to show what they knowrather than what they don't know.Individual attributes are valued by somesocieties more than by others;

- the extent to which the examination isn o r m-referenced or criterion-r e f e r e n c e d .While all examinations inevitably containboth aspects, the emphasis is a matter ofdecision, and relates particularly to theunderlying purpose of the assessment.

Thus even within the framework of the canoni-cal universalist mathematics curriculum therecan be marked variations in examinationcontent, procedure and emphasis, related to theparticular goals of the society and to the socie-tal demands being made of the assessment.

Moreover as the development of alter-native curricula increases so we can

expect to see yet more different problems andtasks presented, and different issues raised, andthese may well enter the formal examinationstructure and process. When this happens, it islikely to be a politically sensitive matter, as wasdemonstrated by an incident in a recent publicmathematics examination in the UK. In 1986,candidates in one examination for 16 year oldswere presented with some information on mili-tary spending in the world, which was compa-red with a statement from a journal (NewInternationalist) that "The money required toprovide Adequate food, water, education, heal-th and housing for everyone in the world hasbeen estimated at $17 billion a year". The can-didates were asked "How many weeks ofNATO and Warsaw Pact military spendingwould be enough to pay for this? (show allyour working)". The outcry from the nationalnewspapers and from society's establishmentwas remarkable, with one headline asking"What has arms spending to do with a mathsexam?"(Daily Mail, 14 June, 1986) and publiccorrespondence on this issue continued forsome time, illustrating well the societalcontrols felt to be important on the examina-tions and the examination system.

Here we are of course talking about sum-mative, rather than formative assessment.Society has little interest in the latter, seeing itas merely a part of the teaching process.Summative assessment, however, usually inthe form of an examination, is very much felt tobe society's concern and various tensions canbe seen to exist in all societies over the formalschool examinations. These tensions seem tobe particularly sharply defined in mathematicseducation perhaps because mathematics itselfcan appear to be a sharply definable subject,but also because of its role in selection for futu-re education or careers.

Amajor tension is between examinations asindicators of educational achievement, and asinstruments for academic control. In a schoolsubject like mathematics, in which there ismuch belief in the importance of the logicalsequences within mathematics itself, the prevai-ling image projected by the curriculum is of astrongly hierarchical subject - certain topicsmust necessarily precede others. With such animage, academic progression and control can befelt to be tightly determined, with, for example,performance at one level being seen to be achie-ved before the next topics in the logical

12

sequence can be attempted. This imagecontrols much of the current practice in mathe-matical assessment, with only those who havedemonstrated mastery of the required knowled-ge being allowed access to higher level lear-ning in the subject. It is an image which isfavoured by many academic mathematicians,and by those 'meritocratic' traditionalistsamongst employers, parents, and governmentofficials.

In contrast to this image is the belief thatthe examinations should as far as possibleenable the learners to demonstrate what theypersonally have learnt as a result of theirmathematical education. A consequence of thisbelief is that, given the variety of school lear-ning contexts for mathematics, the variety ofteachers and materials influencing learners,and the variety amongst the learners them-selves, the examinations should be as broadlypermissive as possible. As well as perhapsincluding what might be called standard mathe-matical questions, the learners should, therefo-re, be allowed to submit projects, investiga-tions, essays, computer programs, models andother materials, and to be assessed orally aswell as in written form. All of these types ofassessment do exist in different countries as Ihave suggested, but their acceptance in an exa-mination system by a society depends stronglyon beliefs concerning educational access, therecognition and celebration of individual diffe-rences, and the encouragement of individualdevelopment through a broad mathematicaleducation.

Under this view, there is no necessary rea-son why examinations in mathematics atschool should be the same from one society toanother. Each society makes its choice for itsown political, educational and social reasons.This view tends to have strongest supportamongst the more progressive educators,parents, and politicians of a more sociallydemocratic persuasion. It is a view which alsoseems to be gaining strength as 'personal tech-nology' - h a n d-held calculators of increasingsophistication, and personal computers -becomes broadly accepted within mathematicsteaching. Personal technology has negated manyof the traditional logical sequences thought to beso essential to mathematical development e.g.calculators are challenging the ideas of sequen-cing of arithmetic learning, and symbolic mani-pulation computer software is doing the samefor algebra and calculus as we have

seen. In addition, personal graphic calculatorscan now present geometrical displays whichare forcing mathematics educators to rethinkthe orders of geometrical and graphicalconstructions. Educationally, these personaltechnologies are putting so much potentialmathematical power in the hands of individuallearners that developing and examining indivi-dual, and particularly creative, talents becomesmuch more of a general possibility. As therange of possibilities of mathematical activitiesincreases so the necessary ordering of mathe-matical mastery becomes less clear and the for-mer view becomes less tenable.

In the debate between these two views, onesuspects that what is really at stake is the morefundamental issue of social control versus edu-cational opportunity, and the role which exami-nations and the examination systems play inthis political struggle. In all industrialised anddeveloping countries, access to the higher andfurther levels of education is not just academi-cally controlled but is socially controlled aswell. The established social order within anysociety has a vested interest in controllingmobility within that society and academic edu-cational control is an increasingly powerfulvehicle for achieving this (see Giroux, 1983 fora lucid discussion of writings on this issue).

Moreover academic control, through theuse of mathematical examinations is a particu-larly well-known phenomenon. It happensthroughout Europe, for examples and in manyother countries. Revuz (1978) described thephenomenon (mainly from the French perspec-tive) in these terms:

"Moreover, the quite proper demandfor the wide dissemination of a funda-mental mathematical culture has boo-meranged, and been transformed into amultiplicity of mathematical hurdles atthe point of entry into various profes-sions and training courses. A studentteacher is regarded as of more or lessvalue in accordance with his level inmathematics. Subject choices whichmake good sense in terms of their rele-vance to various jobs have been redu-ced to the level of being 'suitable onlyfor pupils who are not strong enough inmathematics'; instead of attractingpupils by their intrinsic interest, theycollect only those who could not fol-low, or who were not thought capableof following, a richer programme inmathematics. Success in mathematics

13

has become the quasi-unique criterionfor the career choices and the selectionof pupils. Apologists for the spread ofmathematical culture have no cause torejoice at this; this unhealthy prestigehas effects diametrically opposed tothose which they wish for.

The cause of this phenomenon isundoubtedly the need of our society, asat present constituted, to try to reprodu-ce a selective system which runs coun-ter to attempts at democratisationwhich are going on simultaneously.Latin having surrendered its role as asocial discriminator, mathematics wassummoned involuntarily to assume it. Itallowed objective assessment to bemade of pupils' ability, and it seemed,amidst the upheaval of secondary edu-cation in general, to be one of the sub-jects which knew how to reform itself,which stood firm and whose usefulness(ill understood, truth to tell) was wide-ly proclaimed." (p.177, translated byD.Quadling)

A similar perspective on the phenomenonis offered by Howson and Mellin-Olsen (1986).They say:

"Mathematics, as we have written ear-lier, enjoys a high status within societyat large. This may be due in part to its'difficulty' and its utility, but it has alsobeen argued by sociologists, such asWeber (1952) and Young (1971), thatits status is also due to the way in whichit can be 'objectively assessed'. Thepublic, rightly or wrongly, has faith inthe way in which mathematics is exa-mined and contrasts the apparent objec-tivity of mathematics examinationswith the 'softer', more subjective formsof assessment used for many other sub-jects, e.g. in the humanities. Youngadvanced the view that three criteriawhich helped determine a subject's sta-tus in the curriculum were:

1 the manner in which it was asses-sed the greater the formality thehigher the status;

2 whether or not it was taught to the'ablest' children;

3 whether or not it was taught inhomogeneous ability groups.

C e r t a i n l y, mathematics would fulfil allYoung's conditions. It is important, then,when considering possible changes inmethods of assessment to realise thattheir chances of acceptance will be grea-ter if they do not contradict society's

expectations of what a mathematics

examination should be." (p.22)

One example of this 'social filter' role ofexaminations is that in many European coun-tries it is extremely difficult for some immi-grant groups to gain access to higher education- not because of overt racist practices butbecause of academic control practices whichhave the effect of restricting access to certaingroups, through language and cultural aspectsfor example.

Where these practices have been recogni-sed by certain immigrant communities oneinteresting effect has been to concentrate theireducational and parental attention on themathematical performances demanded by the'social filter'. In the UK, for example, the AsianIndian students achieve significantly higher inmathematics examinations than many otherimmigrant groups, and they thereby progressmore successfully to the higher levels of edu-cation (see Verma and Pumfrey, 1988). Onesuspects that the large numbers of orientalAsian students succeeding in mathematics athigher levels in American universities is a simi-lar response to a perceived societal situation byan ethnic minority community (see, forexample, National Research Council, 1989,and Tsang, 1988).

H o w e v e r, generally the control throughmathematics examinations still prohibits accessto higher educational opportunities for manyother minority groups and indigenous peoplesin societies like Europe, Australia and USA.Furthermore where the Western Europeanmodel of examinations has either been adoptedby developing countries or in some caseswhere their students still take European exami-nations, the same problems occur. Forexample, Isaacs' (1985) description of some ofthe changes which the CaribbbeanExamination Council is making to its examina-tions, indicates the problems well:

“ The examinations will probablycontinue to undergo modifications toensure that they reflect more accuratelythe abilities, interests, and needs ofCaribbean students. It is hoped thatthey will help reduce the number ofstudents who see mathematics as a ter-rifying trial in their rite of passage toadulthood, and, at the same time,increase substantially the number whoperceive and use mathematics as a toolfor effective living. ” (p.234)

14

The other example of social control bymathematics and its examinations which isnow recognised concerns the barriers to thecontinuing educational opportunities for girls.There are strong views expressed to the effectthat girls and women are penalised in manycountries through the use of certain mathema-tics examinations (see Sells, 1980). Also, sincemathematics is not felt to be as popular a sub-ject as others, by girls, to the extent to whichmathematics is used for control, it restrictsgirls' possibilities in education more than boys'.

Educational opportunity could be expectedto be a strong goal in any democratic societybut, even in such societies, either the examina-tion structure, or the actual examinations, ensu-re that not all children have the same educatio-nal opportunities. Insofar as societies containinequalities, those will tend to be reflected inunequal educational opportunities, despitesteps being taken to remove the barriers andobstacles. The mathematical examination is awell-used obstacle, for example, but it can bealtered, its effects can be researched and publi-cised, and its influence therefore either be redu-ced or exploited. One suspects however that allthat will happen will be that unless fundamen-tal changes take place in the structuring ofsocieties, those societies will create newer obs-tacles - perhaps indeed the rise in importanceof computer education will increasingly cometo mean that computer competence will replacemathematical competence as the filter of socialand academic control.

Societal influences on children thro u g hinformal mathematics education (IFME)

It has been important to explore fairly tho-roughly the formal and intentional influenceswhich society exerts on mathematics learningthrough the curriculum and the examinations,because to a large extent it is the public know-ledge, or at least perception, of these whichdetermine to a large degree the characteristicsof the informal education which societalinfluences bring to bear on young people. Fewadults have access to within-school or within-classroom information except through occasio-nal visits either as a parent, or as an 'officialvisitor' e.g. as a school governor, or prospecti-ve employer. Also, Chevallard (1988) makes usaware t)f a contradiction within modern indus-trial societies that is:

"1 No modern society can live withoutmathematics .

2 In contradistinction to societies asorganised bodies, all but a few oftheir members can and do live agentle, contented life without anymathematics whatsoever." (p.49)

Mathematics is becoming, therefore, an 'invi-sible' form of knowledge to many people insociety.

On the other hand, most adults will havememories of their own experiences of schoollearning, including mathematics, which may ormay not be pleasant, or accurate, or even rele-vant to today's situation. Of course in mostsocieties it is still the case that not all adultshave been to school, that even those who havemay not have stayed very long, and even if theydid they may not have experienced muchmathematics beyond arithmetic. Neverthelessit would be rare at least to find a parent who didnot feel that they had some experiences andbeliefs about the school subject with which toinfluence the young learners. This is of courseparticularly the case for all adults' experiencesand understanding of various mathematicalideas gained through their work, their leisureand through their own informal education, asNunes will discuss in the next chapter. Thatcultural and societal heritage provides aninfluential source of knowledge, particularly insocieties where formal education facilities areunder pressure.

There are many individuals in a societywho could potentially influence a young per-son, but if we are considering young learners asa whole, the two principal groups of peoplewho seem to have the greatest potential forIFME are the adult members of the family andof the immediate social community, and peopleworking in what I shall term 'the media'. Fromthe young person's perspective, the only othermajor source of influence, apart from adultsexercising a formal educational role, is that oftheir peer group, and their role will be mentio-ned in a subsequent section.

Turning first to the immediately avai-lable adults for the young learner of mathe-matics, we can surmise at once that it istheir perception, memory and image of howmathematics in school was for them whichwill colour their influence. As well as there-fore being a traditional image, it alsoappears to be a mainly negative image. T h e

15

evidence, such as it is, documents the feelingsof inadequacy and inferiority felt by manyparents about their knowledge and image ofmathematics. In the UK, for example, theinfluential Cockcroft Committee (1982) com-missioned a study into the mathematical know-ledge of adults in UK society. The findingswere dramatic, as they revealed not just thewidespread inability to do what could be called'simple' mathematical tasks but also the frustra-ting, unpleasant and generally negative emo-tions felt by many people about their mathema-tical experiences. Here are some of the state-ments made by respondents:

"I get lost on long sums and neverknow what to do with the 'leftovers'."

"My mind boggles at the arithmetic inestimation."

"I'm hopeless at percentages really."

"I'm afraid I have to write it down. Mybrother can do it in his head."

"My husband says I'm stupid."

Even in a study which involved only UKadults who had already obtained first degrees atUniversity (excluding mathematics degrees),many of the same difficulties and negative atti-tudes emerged (Quilter and Harper, 1988). Inboth studies the reasons for these problemswere mixed but memories of poor teaching anduncaring teachers figured prominently, as didan image of mathematics as a 'rigid' subject,lacking relevance to their personal lives, andhaving correct procedures which needed to beperformed accurately. Strangely it also appearsthat for many adults, the negative experience isassumed to be so widespread that to claimmathematical ignorance and inadequacy issocially acceptable, however unpleasant it maybe personally.

From the perspective of the previous sec-tion also, we can well understand the guiltassociations which usually accompany suchfeelings. If, as was described there, mathema-tics has been used as a strong academic andsocial filter by society, many people will haveexperienced failure as a result. Even those whosurvived the filtering process can be expectedto have some negative feelings about mathe-matics - they know that they had to do well init to progress in their education or their work,and that that incentive was, in many cases, theironly motivation for continuing to study it.

If the collective parental and adult memoryof school mathematics is in fact a largely nega-tive one, this memory can so easily be trans-mitted as a negative image to the next genera-tion, thereby influencing the mathematicalexpectations of the children, their motivationsfor studying mathematics and their predisposi-tions for continuing, or not, to study the sub-ject. For example the evidence from the genderresearch demonstrates the strength of parentalinfluences (Fox, Brody and Tobin, 1980).While the patterns of influence may not be thesame across societies, the influences them-selves clearly exist.

However we should not assume that allparental influence is of a negative kind, nor thatthere is nothing to be done about it. The FAMI-LY MATH project (Thompson, 1989) is a pro-ject concerned to help parents to help their chil-dren with their school mathematics. A sThompson says: "The EQUALS educatorswork to encourage all students to continue withmath courses when they become optional inhigh school in the United States - particularlythose students from groups that areu n d e r-represented in math-based careers.These educators asked EQUALS to develop aprogram that would also involve parents inaddressing this issue" (p.62). The reports aboutFAMILY MATH indeed suggest that it is pos-sible to affect and mediate parental influenceson their children's attitudes and achievementsin mathematics.

Turning now to what I call 'the media', weneed first to clarify what this might mean indifferent societies. It refers to the people res-ponsible for disseminating information,images, beliefs, and ideas in general, producedby individuals and groups within society, butreceived by the young through different media- newspapers, circulars, books, radio, films,TV, advertisements etc. Occasionally, as withfor example an academic visitor to the school,a young learner will have a chance to engagedirectly with that individual, but such opportu-nities are rare, and in general their ideas aremediated by 'the media'.

One important difference between tradi-tional village societies and modern indus-trial societies is the relative proportion ofinfluence coming from immediate commu-nity adults compared with that from themedia. Whereas in traditional village

16

societies the local adults can exert maximalinfluence over the young learners (see forexample, Gay and Cole, 1967), in modernindustrial societies the amount of informationand imagery coming from the media is muchgreater and more complex, and has an all-per-vading influence both on the young and on theadults themselves. In these societies, the infor-mation and images available can include joband career information, the latest scientific andengineering ideas presented through 'popularscience' television programmes and in maga-zines, games and puzzles of a mathematicalnature, as well as facts, figures, charts, tablesand graphs on every conceivable aspect ofhuman activity. The learners will see mathema-tical ideas and activities appearing in a varietyof contexts; and they will see people engagingwith those ideas in various ways and withvarious feelings. However 'the media' do nottake it upon themselves to represent the wholespectrum of human engagement with mathe-matical ideas - their task is often one of satis-fying a customer or client need, as in selling amagazine or advertising a new computer sys-tem.

Generally therefore, and unlike the perso-nal knowledge one has from known adults, themedia will tend to project 'typical' images andthis typicality has become a source of greatconcern amongst those educational and socialactivists seeking to increase the educationaland employment possibilities of groups withintheir society who do not enjoy equal opportu-nities under the present systems. For example,the research on gender, on social class, and onethnic and political minorities in industrialisedsocieties has often pointed to biased stereoty-pical images portrayed by the media. In termsof influences on the learner of mathematics,these images often relate to the prestige andimportance industrialised societies attach tomathematical, technological, scientific andc o m p u t e r-related careers, to the need for highlevels of qualification and skill in these sub-jects, and to the predominance in these careersof mainly men, who tend to come from themore powerful ethnic or social groups in thatparticular society (see, for example, Holland,1991; Harris and Paechter, 1991). Thus themessages which are received by the youngpeople are that mathematics is a very impor-tant subject, that it is a difficult subject,

and that only certain people in society will beable to achieve well in it.

A recent movement to overcome this large-ly unhelpful influence is known as the'Popularization of Mathematics', a collection ofattempts to 'improve' the image of mathematicsas a subject, by using media of all kinds (TV,radio, newspapers, games, etc.) to reveal themore pleasurable faces of mathematics and togenerate interest, enthusiasm and positiveintentions, rather than failure, fear and avoi-dance. The conference of that title, organisedby the International Commission onMathematics Instruction, and its report(Howson and Kahane, 1990) involved educa-tors from many countries in the issues of whypopularize, how to do it successfully, and whoare the 'populace', amongst others. The ideasranged from Mathematics Trails (walks invol-ving mathematical problems) through citycentres, to mathematical game shows producedfor prime-time national television, from breath-takingly beautiful films and videos tomind-bending wooden puzzles, from 'popular'mathematical articles in national papers to spe-cial mathematical magazines for children.

The conference, and the report, left one inno doubt that the mathematics education com-munity believes that the influences coming atpresent from the media are not generally help-ful and need to be educated, and enriched.

Societal influences through non-f o r m a lmathematics education (NFME)

Adapting Coombs' (1985) definition, non-formal mathematics education would involveany organised, systematic, mathematics educa-tion activity, carried on outside the frameworkof the formal system, in order to provide selec-ted types of learning to particular subgroups inthe population, adults as well as children.

We can easily recognise NFME in provi-sion such as adult numeracy programmes,out-of-school 'gifted children' courses, televi-sed learning courses, correspondence learningactivities, and vocational training courses ofmany kinds. As at previous congresses, forexample, the Action Group 7 on "Adult, tech-nical and vocational education" held at theInternational Congress on MathematicsEducation (ICME6) in Hungary in 1988,

17

contained many examples of interesting deve-lopments grouped under the subheadings of' Vo c a t i o n a l / Technical Education', 'AdultEducation' and 'Distant Education', (Hirst andHirst, 1988).

At that congress there was also a SpecialDay devoted to the topic "Mathematics,Education, and Society" (Keitel et al., 1989) atwhich various ideas concerning NFME werepresented. For example, Smart and Isaacson(1989) described a course for nonscience qua-lified women who wished to change to a careerin technology and science, and which involvedmany mathematical 'conversions'. Sanchez(1989) talked about a radio programme seriesin Spain designed to popularise mathematics.Also Volmink (1989) described diff e r e n tapproaches to "Non-school alternatives inMathematics Education" which the black com-munities in South Africa have turned to becau-se of their frustration with the FME provisionat present: e.g. private commercial colleges,supplementary programs, and the Peoples'Education Movement which is articulatinggeneralised concepts which "aim to transformthe educational institution and will find finalembodiment only within a new political struc-ture" (p.60). Thus NFME not only off e r sopportunities for a different context for lear-ning mathematics, but in some situations it isclearly intended to challenge the precepts andpractice of FME. In the same vein, and in thesame country, South Africa, Adler (1988) des-cribes a project concerned with" n e w s p a p e r-based mathematics for adults"who had had little access to mathematics edu-cation because of the inadequate provisionswithin the apartheid system.

N o n-formal education has in some way beena marker of educational development in a socie-t y, and Freire (1972) showed the developingworld the way in which it could - play a signifi-cant role in a society's development. T h ePeoples' Education Movement in South Africa isa good recent example of this, and it is clearlyvery important that mathematics is not left out ofthe whole process. In industrialised societiesalso there are significant NFME developments,and a good example is given by Frankenstein's(1990) book which describes a 'radical' mathe-matics course intended mainly for adult studentswho have been made to feel a failure at maths. Itis a textbook which tries to overcome learning

obstacles by developing methods that helpempower students. More than most textbooks,it uses examples, illustrations, activities andtasks drawn from everyday experiences andcontexts not just to develop mathematicalknowledge, but also to develop a criticalunderstanding in her adult students.

Furthermore many of the participants at thepreviously mentioned ICMI conference on 'ThePopularization of Mathematics' (Howson andKahane, 1990) spoke of mathematics clubs,a f t e r-school activities, master classes, andexhibitions, all of which contribute importantinfluences of a nonformal mathematics educa-tion kind.

Volume 6 (1987) of the UNESCO series"Studies in Mathematics Education" is alsoentirely devoted to the theme of "Out-of-schoolmathematics education" and contains manyexamples including mathematical clubs,camps, contests, olympiads, and distance edu-cation courses. Losada and Marquez (1987)document in a detailed way the manyo u t-o f-school mathematics activities inColombia and demonstrate how significant thatform of education can be. As they say "whilethe teacher-student ratio in schools ranges from1 to 20 to 1 to 60, traditional classroom educa-tion cannot be expected to be particularly effec-tive... In such circumstances out-o f-s c h o o lmathematics programmes can provide moreopportunities for more young people" (p.117).

Clearly the borderline between IFME andNFME becomes blurred at times, particularlywhen there are considerable attempts to enrichboth at the same time by, for example, enlistingthe help and support of adults and other signi-ficant people in the overall mathematical edu-cation of the young people. That does not real-ly matter of course. What is important is thateducators recognise as educational, theinfluences which come either informally or in amore structured way, non-formally, from thevarious segments of society.

What then can be said about suchinfluences and what they affect? If we reflect alittle more on the informal and nonformalinfluences described so far, we can detecteffects at two levels, both of which have impor-tant consequences for mathematics learners;the level of mathematical ideas, and the level oflearning mathematics.

18

Influences on mathematical ideas and on thelearning of mathematics

At the first level, there are clearly influencesregarding mathematical concepts and skills.Before children come to school, their parentswill often have taught them to count, to beginto measure, to talk about shapes, time, direc-tions etc. Neither will this kind of parent's talkand activity cease when the children beginschool. Within the immediate adult communityparticular knowledge, often related to mathe-matics, will be shared, as Nunes documents inthe next chapter. The newspapers and othermedia reports involving money, charts, tables,percentages will all inform and educate thelearner, complementing and supplementing theinformation and ideas being learnt in school.Now, with sophisticated calculators and homecomputers becoming more widely available,the young mathematical learner may well beracing ahead of the formal mathematics curri-culum and will often outstrip the teacher'sknowledge in some specific domains.

Equally the sources for mathematical intui-tion are frequently images from society, ratherthan from within school. Popular images andbeliefs, for example, about statistical and pro-babilistic ideas seem not only to come fromsociety but also seem to be relatively imper-vious to formal educational influences.(Tversky and Kahnemann, 1982; Fischbein,1987).

Geometrical images will come from physi-cal aspects of the environment, although whatis important is how the individual interactswith that environment. Thus, once psycholo-gists thought that living in a 'carpentered envi-ronment' (with straight sided houses, rectangu-lar shapes, right angles etc) was very importantfor learning geometry and for developing spa-tial ability. Now we know that what is impor-tant is how individuals interact with their envi-ronment. So for example, to an urbanisedEuropean the desert of central Australia mayseem to be devoid of anything which could aidspatial development, but the spatial ability ofthe Aborigines who live and work there isknown to be exceptional because of what theyhave to do to survive there (Lewis, 1976). Thisis equally the case with Polynesian navigators(Lewis, 1972) and Kalahari nomads (Lea,FME, 1990). interactions with the physical

environment of a society undoubtedly give riseto many geometrical images and intuitions.

The other major source of societal influenceon the learner's knowledge of mathematicalideas is the language used. The relationshipsbetween language and mathematics are of cour-se extremely complex, and there is no space hereto cover all the ground which has in any casealready been analysed by others (Pimm, 1987;Zepp, 1989; Durkin and Shire, 1991).

From the perspective of this chapter thetwo most important aspects for mathematicseducators to be aware of seem to be:

- the fact that mathematics is notlanguage-free,

- not all languages are capable of expressingthe mathematical concepts of MT culture.

The first point may seem obvious, but it hasprofound implications. Mathematical knowled-ge, as it is developed in any society relates tothe language of communication in that society.As has already been shown, the mathematicscurriculum in many countries has been basedlargely on the Western-European model and ithas a certain cultural, and therefore, linguisticbasis. Though this basis is an amalgam of dif-ferent languages, the principal linguistic root isbelieved to be Indo-European (Leach, 1973).That particular 'shorthand' omits the importantGreek and Arabic connections in the develop-ment of universally applicable mathematics,and we should perhaps consider theIndian-Greek-Arabic-Latin chain as being itsoriginal language base. From this base, Italian,Spanish, French, German, and English develo-ped its language repertoire during the 17th,18th and 19th centuries, and it is probably thecase that nowadays English is the principalmedium for international mathematical resear-ch developments. This is an important problemfor any country, researcher, or student, forwhom English is not the first or preferred lan-guage.

In relation to the second point, in manycountries of the world there are several lan-guages used, but for national and political rea-sons, one (or some) are specifically chosen asthe national language(s). It is likely that not allthe languages being used in a society willnecessarily be capable of expressing the

19

concepts and structures of the MT culture - thiswill largely depend on the roots of the langua-ge. European-based languages, those withIndian roots, and the Arabic family of lan-guages appear to have the least structural diffe-rences, although there are always particularvocabulary gaps as international mathematicsdevelops.

Other languages, in rural A f r i c a ,Australasia, and those used amongst indige-nous American peoples, are being studied anddemonstrate their difficulties in expressingboth the structures and the vocabulary of theMT culture’s version of mathematics (see, forexample, Zepp, 1989 and Harris, 1980). This isnot of course to say that these languages areincapable of expressing any mathematicalideas - they will certainly be capable of expres-sing the mathematical ideas which their cul-tures have devised. This is the important lin-guistic relationship with ethnomathematics -and another reason for seeking to create moreindependent societally-based mathematics cur-ricula rather than relying on the model fromMT-based societies. Thus in this way the socie-tal language(s) can reinforce the societalmathematics which can offer the bases foralternative curricula.

But language issues are extremely com-plex, particularly from a societal perspective,and the political and social conflicts which dif-ferent language use can cause, can seem to beof a different order from those which shouldconcern mathematics educators. Neverthelessso much damage has been done to cultural andsocial structures in many countries by assu-ming the universal validity of MT-b a s e dmathematics that we cannot ignore the langua-ge aspects of this cultural imperialism (seeBishop, 1990). If countries, and societieswithin countries, are to engage in the process ofcultural reconstruction then the language ele-ment in relation to informal, non-formal, andformal mathematics education is critical.

A final point concerning the informal andn o n-formal influences on mathematical ideas isthat they have a cumulative effect. They build upinto an image of 'mathematics' as a subject itself.For example, we have already noted that it isprojected as being an important and prestigioussubject in both industrial and developing socie-ties and is thereby projected as being essentially a

benign subject. Little mention is publicly madeof its extensive association, through fundamen-tal research, with the armaments industry, withespionage and code breaking, and with econo-mic and industrial modelling of apolitically-partial nature. Little public debateoccurs about the questionable desirability offostering yet more mathematical research tomake our societies yet more dependent on evenmore complex mathematically-based technolo-gy (see however, Davis, 1989 and Hoyrup,1989). The reaction of 'the media' to the exami-nation question about costs of armamentsshows the extent of public ignorance of thesematters.

Equally mathematics in society is typified,and imagined by most people, as the most secu-re, factual and deterministic subject. There islittle public awareness of the disputes, thepower struggles, or the social arenas in whichmathematical ideas are debated and construc-ted. Descartes' dream still rules the generalsocietal image of mathematics. For example, inthe study by Bliss et al. (1989) concerning chil-dren's beliefs about "what is really true" inscience, religion, history and mathematics, themajority of children in England, Spain andGreece considered both mathematics andscience to be truer than history or religion. AsHowson and Wilson (1986) put it "Only inmathematics is there verifiable certainty. tell aprimary child that World War 2 lasted for tenyears, and he will believe it; tell him that twofours are ten, and there will be an argument"(p.12).

At a second level the informal and non-formal societal influences concern the l e a r n i n gof mathematics. We have already noted thebeliefs about its difficulty and its motivations,but there are also more fundamental and signi-ficant beliefs about how mathematics is learnt.Paralleling the popular image of mathematics assecure factual knowledge is the widespreadbelief that mathematical procedures need to bepractised assiduously and over-learnt so thatthey become routine, and that this should goh a n d-i n-hand with the memorising of thevarious conceptual ideas and their representa-tions. Another popular belief concerns 'understan-ding' as being an all-o r-nothing experience, ratherthan a gradual increasing of meaning and construc-ted connections. Overall the popular image isof a received, objective, form of knowledge,

20

rather than of a constructed subject, runningcontrary to what we know from recent cogniti-ve research. (Lave, 1988; Schoenfeld, 1987).

There is also, as has been mentioned, theimpression that because it is reputed to be a dif-ficult subject, only some learners will be ableto make progress with it. Thus, rather like artis-tic or musical ability, young people perceivethemselves as either having mathematical abi-lity or not. The research on mathematical gif-tedness does in fact demonstrate that it clearlyis a precocious talent, appearing early ratherlike musical giftedness, but one suspects thatthe image from society about mathematicalability is rather more broad in its reference thanjust to giftedness.

The concept of 'ability' does seem to be apervasive one in society outweighing 'environ-ment' as the cause of achievement, particularlyin the 'clear-cut' subject of mathematics. Theaccompanying belief is that one is fortunate tobe born with this ability since it is (clearly!)innate. The pride of a parent on discoveringthat their child is a gifted mathematician is pro-bably only clouded by the popular image fromsociety of mathematical geniuses being slight-ly odd characters living in a remote and esote-ric inner world and unable to socialise withother people - a theme to which several contri-butors to the ICMI conference on'Popularization' referred.

Equally parents, although not necessarilyrating their son's or daughter's abilities any dif-ferently, do apparently believe that mathema-tics is harder for girls and requires more effortto succeed in (Fox, Brody and Tobin, 1980).This belief undoubtedly helps to shape the fee-ling, prevalent in different societies, thatmathematics is not such an important subjectfor girls to study. Fortunately many womens'groups are now hard at work dispelling thisimage, and IOWME (the InternationalO rganization of Women and MathematicsEducation) has been particularly active. Recentresearch (Hanna, 1989) for example shows thatthe creation of a more positive and favourableimage for girls who are mathematically able isapparently having some beneficial effects.

These then are some of the most signifi-cant influences which society exerts on thelearners of mathematics. The 'messages' thelearners receive are many, and often

conflict. Some are intentionally influential,while others are merely accidental. But they allhelp to shape important images and intuitionsin the young learners' minds, which then act asthe personal cognitive and affective 'filter' forsubsequently experienced ideas. Let us thenmove on to consider how the learners makesense of, or cope with, these various societalinfluences which they experience. Are thereany research ideas which can help us to unders-tand and interpret the learners' situation?

The competing influences on the individuallearners

Considering first of all the area of motivation,this is perhaps the most significant aspect nee-ding to be richly understood by mathematicseducators, because it provides the essentialdynamic for the mathematics learning processwherever and whenever it takes place. A classicdichotomy is usually made between extrinsicand intrinsic motivation, referring to the loca-tion of the influence for motivation. In onesense we can think of societal influences asbeing essentially extrinsic whereas, forexample, the 'puzzling' nature of a mathemati-cal problem can seem to be located internallyto the learner. The interaction between intrinsicand extrinsic motivation though relates both tothe internalisation of the extrinsic, and also tothe perception of a familiar external target forthe intrinsic. It is clear that societal influencesinitially exist 'outside' the learner, but theextent of their influence will depend on howinternalised they become, and, therefore, thatinternalisation process is the key to understan-ding motivation for the learners.

We have claimed in this chapter thatsocieties formally influence mathematics lear-ning through their school curricula and theirexamination structures. From the learner'sperspective, therefore, those influences willmotivate to the extent that they are internali-sed and help to shape the learner's goal struc-tures. There appear to be two significant andinteracting aspects to attend to here - the 'mes-sages' being communicated to the learner andthe people conveying those messages. Interms of the mathematics curriculum, theprinciple message will concern the values -explicit or implicit (the 'hidden' curriculum) -which the curriculum embodies, mediated bythe teacher, and which may, or may not, be seen

21

as significant by the learner. The area of valuesin mathematics curricula has not been wellexplored in research, but the clues we havefrom research on attitudes and beliefs certainlysupports the above analysis (see for example,McLeod, PME, 1987).

However, the extent to which the mathema-tics curriculum embodies the values of MT cul-ture (for examples will only partly determinethe motivational power of those values. Thereal determining factor will be each indivi-dual's acceptance of those values as being wor-thwhile or not, which will depend on how theyrelate to the learner's goal structure, and howdifferent people figure in that structure. To talkof goal structures is not however to suggest thatthey are fixed and immutable. One recentlydeveloping research area - that of situatedcognition (Lave, 1988; Saxe, 1990) - revealsthe phenomenon of 'emergent' goals. If oneunderstands mathematics learning as some-thing which develops from a social mathemati-cal activity, then through that activity willemerge mathematical goals, to be achieved,which would not necessarily have been appa-rent before the start of that activity.

Thus, one corollary is that as one learnsabout mathematical ideas through doingmathematical activities, one also learns aboutgoals, and in a social context. A statement bythe teacher to the effect that an important goalof mathematics learning is to become systema-tic in one's thinking, for example, may well benegated by the learners' shared experiences oftrial-and-error approaches which neverthelesssucceed in finding appropriate solutions to pro-blems (see Booth, 1981). Moreover, it maywell be that the latter message, learnt in activi-ty and in the context of one's peers, or one'sadults, will be internalised more significantlythan the teacher's message.

This kind of gulf, between what learners aretold that they should do and what they actuallydo to be successful, may well be one source ofthe well-documented dislike of mathematics,and even of the phenomenon of mathophobia,the fear of mathematics. It is the kind of expe-rience which, if repeated often enough, willlead learners to believe that mathematics, evenif it is an important subject, may not be whatthey personally want to invest their energies in.More research on learners' emergent goals,

particularly in the school learning contextcould provide some very powerful ideas formathematics educators.

A more significant contributor to the lear-ner's mathematical motivation may well be themessages concerning the examinations, whichcome both formally and informally from socie-ty. The within-school 'messages' concerningfinal examinations, combined with the expe-riences of immediate adults will be highlysignificant in creating a schema of beliefs andexpectations with which the learner's ownexperiences of mathematical success or failurewill be interpreted. The internalisation ofbeliefs and expectations, as with any aspect ofmotivation, will be dependent on the signifi-cance to the learner of the people mediatingthose beliefs and expectations. Sullivan (1955)refers to 'Significant Others' i.e. people whocan exercise that kind of influence - they maybe role models, advisers, counsellors, objectsof worship and awe, or of scorn and dislike.Their significance is a personally attributablecharacteristic and will be likely to vary both asthe richness of the individual learners variesand as the variety of 'types' in the societyvaries. From this analysis we can see that thelearner's mathematics teachers, parents, mathe-matical peers and real or mediated 'mathemati-cians', are likely to be the major sources ofmathematical motivation influence.

The learner's interpretation of success andfailure does seem to be an important researchsite for understanding their motivation, particu-larly in a subject like mathematics, which isbelieved by many learners to be 'clear-cut',where one always knows (they think) whetherthe answer is right or wrong. Moreover this actof 'interpretation' can be seen as part of thewider cognitive task for the learner, of unders-tanding mathematical activity as a societallydefined phenomenon. Attributing success orfailure in the subject relates strongly for thelearner to personal questions and concerns like"How will I fit into society - what job will I do- how mathematically qualified should I beco-me?" and "How do I relate to other people insociety - am I more or less competent at mathe-matics than they are?"

Attribution theory can help us here.Wiener's (1986) work is being explored inrelation to mathematics learning and the fin-dings are interesting. For example Pedro et

22

al. (1981) reported that, in US society, malesare more likely to attribute their success to theirability than females, and that females are morelikely than males to attribute their successes totheir efforts. So far, however, we have noresearch which systematically explores mathe-matical attributions in relation to the influencesof significant others, although the role of theteacher does appear to be fundamental, asmight be expected (see Lerman, this book).

Attributional understandings and interpre-tations like these appear also to be stronglyembedded in the learner's metacognition aboutmathematics. Because mathematics at all levelsis learnt in socially defined activities, thepeople who share in influencing those mathe-matical activities will also play a role in sha-ping each learner's total perspective on mathe-matics. Gay and Cole (1967) in their study ofthe Kpelle, were clearly taken by surprise byone of the college students they were testing:

A Kpelle college student accepted allthe following statements: (1) the Bibleis literally true, thus all living thingswere created in the six days describedin Genesis; (2) the Bible is a book likeother books, written by relatively pri-mitive peoples over a long period oftime, and contains contradiction anderror; (3) all living things have gradual-ly evolved over millions of years fromprimitive matter; (4) a "spirit" tree in anearby village had been cut down, hadput itself back together, and had grownto full size again in one day. He hadlearned these statements from hisFundamentalist pastor, his collegeBible course, his college zoology cour-se, and the still-pervasive animist cultu-re. He accepted all, because all weresanctioned by authorities to which hefeels he must pay respect. (p35)

The researchers seemed to expect logicalconsistency to be an over-riding criterion forthat student, and they used that kind of eviden-ce to infer that mathematical reasoning (as theyknew it) had little place in Kpelle life. We nowknow of course that mathematics is not the uni-versal and independent form of knowledgewhich they assumed. Moreover, and perhaps forus now more importantly, they also assumedthat such socially-bound cognitive behaviourwas not typical in America (their home). Lave(1988) has certainly demonstrated that not to

be the case, and in general the growing aware-ness of the socially constructed nature of allhuman knowledge makes us realise that all lear-ners have the task of making sense of other peo-ple's messages in a variety of social situations.

Thus although perhaps mathematics educa-tors would like to think that in some way socie-tal influences from the immediate adults andfrom the media will enhance, support and gene-rally reinforce the messages which the childreceives from school, the reality is more likelyto be one of conflict. The variety of peopleinvolved makes it unlikely that there will be akind of benign consistency.

The learner is therefore rather like a bilin-gual, or multi-lingual, child who must learn touse the appropriate 'language' in the appropria-te social context. In fact it would be moreappropriate to describe the mathematics learneras bi-cultural, or multi-cultural, since so muchmore than language is involved in learningmathematics. We have already noted theinfluence of interactions with the physicalenvironment which create so much of animpact on geometrical and spatial intuitions.Also social customs and habits, which give riseto expectations, are not always expressedthrough language but are learnt through socialinteractions while engaged in particular activi-ties.

Halliday's (1974) work on 'linguistic distan-ce', for considering the cognitive task for bilin-gual learners, is an interesting construct here.Dawe's (1983) research used Halliday's idea toconjecture that bilingual learners of mathema-tics in English at school would have more diff i-culty the 'further away' the structure of theirhome language was from English. So, for hiss t u d y, the order: Italian, Punjabi, Mirpuri andJamaican Creole, was hypothesised as being thed i fficulty dimension, with Italian being the 'clo-sest' to English. For logical reasoning in Englishthis proved to be the case, but not for mathema-tical reasoning, where a gender effect interactedwith the language factor.

It is possible to broaden the idea of linguisticdistance to that of 'sociocultural distance' betweentwo different principal social situations experien-ced by the learner. Thus it seems reasonable toconjecture that mathematics learners whoseimmediate home cultures relate more closely tothe structure and character of the school culture,

23

will have less difficulty in reconciling the mes-sages coming from the two cultures, than willlearners whose home cultures are a long 'dis-tance' from their school culture.

Perhaps though we will make more pro-gress in educational terms if we attend less tothe "messages received" metaphor whichemphasises the role of the learners as predomi-nantly passive receivers, and more to the ideaof the learners as constructors of their own per-sonal and cultural knowledge. The essentialcognitive task for any learner is making mea-ning, by creating and constructing sensibleconnections between different phenomena, andbecause most mathematical activity takes placein a social context, this mathematical construc-tion will take place interactively. Each newgeneration of young mathematical learners isactively reconstructing the varied societalknowledge of mathematics with which theycome into contact, and in their turn they (asadults) will influence the social context withinwhich the next young generation will engage intheir own reconstruction.

Of particular relevance to this point isSaxe's recent research (Saxe, 1990) whichdemonstrates three aspects of importance to usas educators. Firstly his street candy-sellers inBrazil learned as much from older children asthey did from immediate adults in the commu-nities. So we should include 'older children' in'the categories of 'immediate adult' in our pre-vious discussions, as they are likely to play akey role in any socialisation process. Also thatknowledge was validated and developed withina strong peer-group structure, and it is likelythat peer-groups influence far more than hasbeen recognised so far. Finally although therewas some evidence of school mathematicalideas and techniques being used out of school,it was very clear that the street selling expe-riences had furnished a rich schema whichinformed the children's learning within theschool situation. This finding supports, interes-tingly, the cognitive instructional research fin-dings (see Silver 1987 for a summary) that themost significant contributor to the learning ofnew information is the extent of the previouslylearnt information. However it should alsoencourage educators to realise that the mostimportant prior knowledge may well have beenlearnt outside the school context, and will the-refore be embedded in a totally different socialstructure.

My own example of this, already extensi-vely quoted (see Bishop, 1979), concerns aninterview with a student in Papua New Guinea,a situation where the home/school socio-cultu-ral distance is large. I asked him "How do youfind the area of this (rectangular) piece ofpaper?" He replied "Multiply the length by thewidth". I continued "You have gardens in yourvillage. How do your people judge the area oftheir gardens?" "By adding the length andwidth". Taken rather aback I asked "Is that dif-ficult to understand?" "No, at home I add, atschool I multiply". Not understanding thesituation I pursued the point: "But they bothrefer to area". "Yes, but one is about the area ofa piece of paper and the other is about a gar-den". So I drew two (rectangular) gardens onthe paper, one bigger than the other, and thenasked "If these were two gardens which wouldyou rather have?" to which he quickly replied"It depends on many things, I cannot say. Thesoil, the shade ..." I was then about to ask thenext question "Yes, but it they had the samesoil, shade ..." when I realised how silly thatwould sound in that context!

Clearly his concern was with the two pro-blems: size of gardens, which was a problemembedded in one context rich in tradition,folk-lore and the skills of survival. The otherproblem, area of rectangular pieces of paperwas embedded in a totally different context.How crazy I must be to consider them as thesame problem!

So how did he reconcile the conflict whichI could see? I cannot answer that questionbecause I firmly believe that for him there wasno conflict. They were two different problemsset in two entirely different contexts, and it wasonly I who felt a conflict. As with Gay andCole above, my cultural background encoura-ged me to believe that logical consistencydemanded a resolution of the conflict whichwould arise if one were to attempt to generali-se the two procedures. If one were not interes-ted in doing that however, there is no problem.

Educational implications

What, then, are the educational implications thatcan be drawn from all of this evidence, analysisand conjecture? The first thing to ask is "Whatdo we mean by education?" since it is now clearthat the learner is subject to many influences ofa potentially educating kind. Also we can nowsee that if 'education' is only considered to be

24

what happens in school then any potentiallypowerful out-o f-school experiences will bedenied a legitimacy in that society. As was saidearlier, at present the school curriculum andexaminations contain what most adults thinkshould be learnt through mathematics teachingand any other mathematical knowledge theyhave learnt through practice is not “ real ”mathematics. In fact it would be easier to arguethat totally the opposite is true: school mathe-matics, because it is learnt in the necessarilyartificial world of school cannot be 'real'mathematics!

But the issue is not whether one context isreal and another not. Both clearly are real, butthey are also very different. Therefore theconceptual task is not to create an artificialdichotomy but to begin by looking for the simi-larities and Coombs (1985) has offered us animportant way forward. We can now considerformal mathematics education (FME), infor-mal mathematics education (IFME) andnon-formal mathematics education (NFME).

In certain cases the developments we cansee in NFME and IFME are motivated by thedesire to complement, supplement and general-ly extend the FME 'diet' offered by the formaleducational institutions, while in others there isan objective of challenging the assumptions,the structures or the practices of FME. Thesekinds of developments will surely haveinfluences on FME to the extent that they candemonstrate their successes to the educationalprofessionals and to the wider society.

From the formal educational viewpointthese developments in IFME and NFME couldbe considered as unwanted challenges to theeducational status quo, and which should there-fore be resisted by whatever political and socialmeans are possible. However more positivelyone can consider these developments as impor-tant educational experiments whose effective-ness, if proven, and whose validity, if accepted,compel them to be considered as viable possi-bilities within the formal educational provi-sion.

This is not to propose their demise asvalid educational agents in their own rightbut to recognise that formal mathematicseducation can learn from these 'experiments',

and can be modified. In some cases they aredeveloping new methods, new materials andnew practices, which could be used to extendthe range of ideas reaching teachers, and tea-ching material developers. In others, therecould be influences on the intended formal cur-riculum of the society and on the examinationprocedures.

The increases, both quantitative and quali-tative, in NFME and IFME provision in bothindustrial and developing societies, certainlydemonstrate that FME at present is not meetingthe demands which different societies aremaking. It should therefore be a continual sour-ce of challenge to those of us who are respon-sible for the character of FME in any society toexplore its potential for responding to the diffe-rent demands and influences coming froms o c i e t y. For example, television and otherpopular media can raise the awareness aboutcertain ideas, but they cannot develop theknowledge of those ideas systematically.Neither can they develop skills effectively. Onthe other hand skills, such as key-typing, canbe learnt effectively through non-formal situa-tions, which could have much pay-off for for-mal educational work with computers andword-processors. So how should FME be sha-ped, in the context of rapidly expanding andinfluential NFME and IFME developments?

Trying to answer that question means thatthose of us who work in mathematics educationmore generally need to be much more awarethan we have been of developments in NFMEand IFME, to accept them as valid educationalconcerns, to stimulate their active growth andto recognise their growing power and influencenot just on FME but on societies' developmentgenerally.

This chapter has demonstrated not only therange of influences which society brings tobear on mathematics learners, but also how thelearners try to deal with these and how the dif-ferent educational agents respond to societies'demands. It is a chapter predicated on the beliefthat a societal perspective on mathematics lear-ning is essential in framing the more narrowand specific concerns of research, develop-ment, and practice within formal mathematicseducation.

25

26

Alan Bishop is currently University Lecturer at the Department of Education in the Universityof Cambridge, England, where he is a specialist in mathematics education. His educationalbackground includes three years as a graduate student at Harvard University, USA, and he hasworked in several other countries as a visiting lecturer or professor since then. He has a stronginterest in social and cultural aspects of mathematics education, and is a co-convenor of theSocial Psychology of Mathematics Education working group. He was for eleven years theEditor of Educational Studies in Mathematics, and is still the Managing Editor of theMathematics Education Library, published by Kluwer of Dordrecht, The Netherlands.

R: 8 years old, is solving a problem aboutgiving change in a simulated shop situationwhich is part of an experiment. He writesdown 200 35 properly aligning units andtens and proceeds as follows:

R: Five to get to zero, nothing (writes downzero); three to get to zero, nothing (writesdown zero); two, take away nothing, two(writes down two).

E: Is it right?

R: No! So you buy something from me and itcosts 35. You pay with a 200 cruzeiros noteand I give it back to you?

After another unsuccessful attempt on the samecomputation, the experimenter asks:

E: Do you know how much it is?

R: If it were 30, then I'd give you 170.

E: But it is 35. Are you giving me a discount?

R: One hundred and sixty five.

From Carraher, Carraher &Schliemann, 1987, p.95

Much research has shown that mathemati-cal activity can look very different when it isembedded in different socio-cultural contexts.For example, we calculate the amount of chan-ge due in school word problems by writingdown the numbers and using the subtractionalgorithm. Outside school in a shop we mayfigure out change diff e r e n t l y, using oralmethods and decomposing numbers (as R. did,decomposing 35 into 30 and 5 and solving thetwo problems sequentially) or even by addingon from the value of the purchase to theamount given as the money is handed to thecustomer. It is my purpose in this paper to dis-cuss the fact that mathematical activities lookdifferent in different contexts and to explorethe implications of these differences for mathe-matics education. The review presented here isselective both due to space limitations andtomy choice of depth rather than breadth ofcoverage. Only two topics in mathematics

education will be reviewed: arithmetic andgeometry. For a broad coverage of topics, otherworks can be consulted (Keitel, et al., 1989;Educational Studies in Mathematics, 1988;Nunes, in press; Stigler and Baranes, 1988).

This paper is divided into four main sec-tions. The first section relates to arithmetic andanalyzes two types of difference between arith-metic in and out of school, differences in suc-cess rates and in the activity itself. The secondsection relates to space and geometry and looksat the interplay between logic and conventionin geometrical activities. In the third section,two views of how to bring out-o f-s c h o o lmathematics into the classroom are contrasted.Finally, the last section briefly summarizes themain issues in the study of the socio-culturalcontext of mathematical thinking and outlinesideas which can be explored and investigatedin the classroom.

Arithmetic in and out of school.

Human activity in everyday life is not randombut organized, or structured, to use a moremathematical term. Even the simplest interac-tion, like that between two friends meeting onthe street, can be shown not to be totally spon-taneous but rather structured and predictable. Iwill distinguish two types of organisation forthe purposes of this discussion, social and logi-c o-mathematical organization. Social org a n i z a-tion is prescriptive and often implicit; it has todo with what people should do in certain typesof situation without their being necessarily ableto know why they behave in those particularways. Logico-mathematical organisation isdeductive and can often be made explicit; itallows people to go beyond the informationgiven at the moment and to know why the dedu-ced information must be correct (and that holdseven in the cases when an error has been made).The logico-mathematical structure pertains tothe actions and situations as such while the

27

The Socio-cultural Context of Mathematical Thinking:

Research Findings and Educational Implications

Terezinha Nunes

social organization pertains to the actions asinterpersonal events, that is as interactions.

In any mathematical activity, both socialand logico-mathematical organization areinvolved, regardless of whether it takes place inthe classroom or outside (for a different view,which rejects the idea that mathematical skillsare based on logical structures, see Stigler &Baranes, 1988). However, it must be emphasi-zed that distinguishing between the social andthe logico-mathematical organization for ana-lytical purposes does not assume the indepen-dence of these two aspects of organisation inhuman life. In any instance of mathematicalactivity, be it in the classroom or outside, bothforms of organisation come into play.

Mathematical activities carried out in andout of school have different social organisa-tions but are based on the same logico-mathe-matical principles. The kernel of the socialorganisation differences between mathematicalactivity in and out of school appears to be thateveryday activities involve people in mathema-tizing situations while traditional mathematicsteaching focuses on the results of other people'smathematical activities (Bishop, 1983). Thus inschool, teachers expect that students will pro-duce a particular solution (related to the appli-cation of an algorithm, for example) right fromthe moment a problem is posed. In contrast, aneveryday problem may be correctly solvedthrough many different routes and no particularroute is prescribed from the outset.

To take a simple example, Scribner and hercollaborators (Scribner, 1984; Farhmeier,1984) have analyzed how inventory takers sol-ved the problem of finding out quickly andaccurately how many cartons of milk were inthe ice-box often from viewing points whichrequired them to fill in information about invi-sible cases in stacks. No particular route is"correct" or "expected" from the outset. In theclassroom, a similar problem would be phrasedas "how many cartons of milk are there if thereare 38 cases and each case holds 16 cartons".The school problem requires in principle coun-ting cases and multiplying number of cases bynumber of cartons in a case. Scribner and hercollaborators found that in real stock-taking,counting single cases to multiply by the num-ber of cartons was not the only strategy

available. Several other strategies were usedespecially because piles were not neat andcases might not be complete. Other methodsincluded the use of volume concepts (forexample, counting stacks of cases of knownheight by looking only at the top of the stacks),skip-counting cartons in incomplete cases (forexample, 8, 16, 24, 30 etc), compensatingacross cases and stacks (for example, countinga half case as full and then compensating forthis when coming across another incompletecase), keeping track of partial totals per area inthe ice-box etc. In short, actual inventorytaking does not have a preestablished path:inventory takers tend to draw freely on knowninformation combined in several ways. Incontrast, a school word problem of the samenature is most likely to be used in order to prac-tice the multiplication algorithm, a result ofother people's mathematical activity which is tobe learned in school.

Does this central difference in the socialstructure of mathematical activities in and outof school affect the way people represent andunderstand the logico-mathematical structures?This question will be looked at here from twoperspectives. The first relates to the rates ofsuccess in mathematical activities carried outin and outside school. The second relates to theway the activity is carried out, its characteris-tics and methods.

Differences in rates of success.

Two studies have documented systematicallyvery important differences in rates of successwhen the same people carry out basically thesame mathematical calculations in and out ofschool.

The first of these studies was by Carraher,Carraher, and Schliemann (1985), who inter-viewed five street vendors (9 to 15 years old) inRecife, Brazil. The youngsters sold small itemslike fruits, vegetables, or sweets in street cor-ners and markets. The study started with theinvestigators approaching the youngsters ascustomers and proposing different purchases tothe children, asking them about the total costsof purchase and the change that would be givenif different notes were used for payment. Thestudy was summarized by Carraher (PME,1988) as follows:

“ Below is a sequence taken from thisstudy which exemplifies the procedure:

28

Customer/experimenter: How much isone coconut?

Child/vendor: Thirty-five.

Customer/experimenter: I'd like three.How much is that?

Child/vendor: One hundred and five.

Customer/experimenter: I think I'd liketen. How much is that?

Child/vendor: (Pause) Three will beone hundred and five; with three more,that will be two hundred and ten.(Pause) I need four more. That is...(pause) three hundred and fifteen... Ithink it is three hundred fifty.

Customer/experimenter: I'm going togive you a five-hundred note. Howmuch do I get back?

Child/vendor: One hundred and fifty.

When engaged in this type of interac-tion, children were quite accurate intheir calculations: out of 63 problemspresented in the streets, 98% were cor-rectly solved. We then told the childrenwe worked with mathematics teachersand wanted to see how they solved pro-blems. Could we come back and askthem some questions? They agreedwithout hesitation. We saw the samechildren at most one week later andpresented them with problems usingthe same numbers and operations but ina school-like manner. Two types ofschool-like exercises were presented:word problems and computation exer-cises. Children were correct 73% of thetime in the word problems and 37% ofthe time in the computation exercises.The difference between everyday per-formance and performance on compu-tation exercises was significant. ”(Carraher, PME 1988, p.34).

Thus, young street vendors were more suc-cessful in their computations outside schoolthan in their efforts to solve school-like exer-cises presented to them by the same experi-menters.

Similar results were obtained by Lave(1988) in a study with 35 adults (21 to 80years) in California. All of the subjects in thisstudy had higher levels of instruction (range 6to 23 years) than the Brazilian youngsters.Lave and her colleagues observed adultsengaged in shopping in supermarkets andtrying to decide which was a better buy com-paring two quantities of a product with

two varying prices. An example of a problemof this type is transcribed below.

A shopper compared two boxes ofsugar, one priced at $2.16 for 5 pounds, theother $4.30 for 10 pounds. She explains,´The five pounds would be four dollars and32 cents. I guess I'm going to have to buythe 10-pound bag just to save a few pen-nies." (Murtaugh, 1985, p.35)

Problems of this nature involve compari-sons of ratio-- which one is larger, 2.16/5 or4.30/10? Lave and her associates observed98% correct responses in the ratio-comparisonscarried out in the supermarket by the adults; incontrast, only 57% of the responses given bythe same adults in a maths test of comparisonsof ratios were correct. Thus, the results obser-ved by Carraher, Carraher, and Schliemann(1985) for Brazilian street vendors are replica-ted among American adults: mathematics out-side school shows better results than inschool-like situations for the same subjectsworking on the same types of problems.

Differences in the social organization of streetand school mathematics and their similaritiesin logico-mathematical structuring.

Several authors have tried to analyze theactivities which are carried out when peopleare engaged in mathematical problems in andout of school. Nesher (PME, 1988) summari-zed some of the contrasts between mathematicsin and out of school. Citing Resnick (1987),Nesher pointed out the following differences:

"(1) schooling focuses on the indivi-dual's performance, whereas outsi-de school mental work is oftensocially shared;

(2) school aims to foster unaidedthought, whereas mental work out-side school usually involves cogni-tive tools;

(3) school cultivates symbolic thinking,whereas mental activity outsideschool engages directly withobjects and situations;

(4) schooling aims to teach general skillsand knowledge, whereas situa-t i o n-specific competence domi-nates outside." (p.56)

To these, Nesher added also that:

"Learning outside school is part of theimmediate social and economic system.The goal on the part of the trainer is to

29

put the trainee as soon as possible onthe production line" (p.56)

while

"This is not exactly the case at school.Schools aim to pass on knowledge tostudents to partly be employed atschool in further training, but mainly tobe employed elsewhere, after leavingschool (...) schools have to deal withquestions of motivation or with ques-tions of rewarding procedures" (p.57)

which may not come up outside school.

Nesher's contrast of learning in and out ofschool, however, does not cover what manyauthors seem to regard as perhaps the two mostimportant differences. These are: (1) that themathematics used outside school is a tool in theservice of some broader goal, and not an aim initself as it is in school (see Lave, 1988;Murtaugh, 1985); and (2) that the situation inwhich mathematics is used outside schoolgives it meaning, making mathematics outsideschool a process of modelling rather than amere process of manipulation of numbers.

Carraher (PME, 1988) expanded on thislast aspect of the difference by arguing that

"Mathematics outside school is condu-cive to the development of problemsolving strategies which reveal a repre-sentation of the problem situation. Thechoice of models used in problem sol-ving and the interval of responses areusually sensible [when mathematics iscarried out outside school] even thoughnot always correct. Students usingschool mathematics often do not seemto keep in mind the meaning of the pro-blem, displaying problem solving stra-tegies which have little connection withthe problem situation and coming upwith and accepting results which wouldbe rejected as absurd by anyoneconcentrating on meaning." (p.19)

Carraher supported this analysis by refer-ring to a study by Grando (1988), in whichfarmers' and students' responses to a series ofproblems were contrasted both in terms ofthe strategies used and in terms of how sen-sible the responses provided were in view ofthe problem. The farmers (n = 14) in thestudy had little or no school instruction andhad learned most of their mathematics outsi-de school; the students (20 in seventh and 40in fifth grade), even though belonging to thesame rural communities, had learned

most of their mathematics in school. They wereasked, for example, how many pieces of wirewith 1.5 meters of length could be obtained bydividing into pieces a roll of wire 7 m long.

"The farmer's responses, obtainedthrough oral calculation Typically anout-of-school method], fell between 4and 7 pieces with 93% giving the cor-rect answer. The students' responsesfell between .4 and 413 pieces. Theseextreme answers were given by stu-dents who carried out the algorithm fordivision (correctly or incorrectly) anddid not know where to place the deci-mal point" (p.12)

a difficulty not faced by farmers, who conti-nuously thought about the meaning of the ques-tion and would neither accept as an answer tothis problem a number which indicated less thanone piece of wire nor a number as high as 413.

The role of modelling in the farmers' pro-cedures was clearly identified through theirverbalizations, which indicated that they simul-taneously kept track of the number of piecesand the amount of wire used up. Their res-ponses could be obtained by a process like thefollowing: "one piece, one meter and a half;two pieces, three meters; four pieces, sixmeters--then, it's four pieces".

The argument being presented here is notthat farmers were more able than students inmathematics. It is possible that students couldhave solved the problem correctly if they hadused the same method as the farmers. All of thestudents attempted division in this problem, amethod which was adequate and perhaps trig-gered by the idea of dividing the roll of wireinto pieces. However, faced with the difficultyof operating with decimals, they tried no otherroute to solution. Farmers, in contrast, were notrestricted to division by any particular set.They were consequently able to avoid the deci-mal division by adding the pieces up to thedesired total length, thereby preserving theirability to monitor their response as solutionwas approached.

D i fferences in methods for solving pro-blems in and out of school are also veryclearly observed in problems involving morethan one variable, like proportions pro-blems. From the mathematical point of view,all proportions problems involve the samestructure. Thus it is possible to develop a

30

general algorithm which can be applied in anyproportions problem independently of the typeof variable involved, be it money, length,weight or whatever. This is in fact what the tea-ching of proportions in school aims at: thetransmission of a general algorithm, whichtakes the form a/x = b/c. However, outsideschool the nature of the variables and the rela-tionships which connect them are not set asidefor the sake of the development of a generalalgorithm. The routes to solution tend to reflectin some sense the relationship between thevariables, as we can see by contrasting the fin-dings reported by Carraher (PME, 1986) withforemen and by Schliemann and Nunes (1990)with fishermen solving proportions problemsin Brazil.

Carraher (PME, 1986) observed thatconstruction foremen solving proportions pro-blems about scale drawings preserved throu-ghout calculation the notion of scale--that is,the notion that a certain unit drawn on papercorresponds to a certain unit of "real life" wall.The 1 7 foremen interviewed in this study weresuccessively shown four blueprints and askedto figure out the real-life size of one or morewalls in each blueprint drawing. The data theyused to solve the problem were obtained fromthe blueprint: a pair of data on one wall (thereal-life size and the size on the blueprint) andthe size on the blueprint of the target wall. Allsuccessful solutions with unknown scales wereobtained by foremen who first simplified theinitial ratio (size on scale to real-life size) to aunit value and then used this ratio in solvingthe problem. When the original pair of data onone wall was, for example, 5 cm on the blue-print corresponding to 2 m in real-life, foremenfirst simplified this ratio in order to find thebasic correspondence--"2.5 cm on paper isworth 1 m of wall", as a foreman said--and thenused this simplified ratio to find the length ofthe target wall. This strategy of finding the sim-plified ratio allowed foremen to carry out thearithmetic in a way that reflected the relation-ship between the variables, value on paper andreal-life size.

Foremen's strategies build an interestingcontrast to those observed among fishermenby Schliemann and Nunes (1990) becausefishermen hardly ever used as a method theidentification of a simplified relationshipbetween the variables. Schliemann andNunes interviewed 16 fishermen who had to

deal with the price-weight relationship in theireveryday life in the context of selling the pro-ducts of their fishing. On the basis of their ana-lysis of the fishermen's everyday activities,Schliemann and Nunes expected that fishermenwould develop methods of solution of the typewhich has been termed "scalar solution" byNoelting (1980) and Vergnaud (1983). Scalarsolutions to proportions problems involve car-rying out parallel transformations on the twovariables (such as doubling, trebling or halvingthe values) without calculating across variables(and thus finding the functional factor).Schliemann and Nunes observed that scalarsolutions were preferred by fishermen evenwhen these- solutions became cumbersome dueto the fact that the target value was neither amultiple nor a divisor of the data given in theproblem (like finding the price of 10 kilosgiven the price of 3 kilos of shrimp). Thestrong preference observed among fishermenfor scalar solutions cannot be explained interms of their school instruction for two rea-sons: (1) scalar solutions are not taught inBrazilian schools, where the Rule of Three (a/x+ b/c) is traditionally taught; and (2) 14 of the16 fishermen had less than 6 years of schoolingand the proportions algorithm is taught in 6thor 7th grade in the area where the study wasconducted.

The preservation of meaning inout-of-school mathematics is clear not only inthe modelling strategies involved in problemsolving but also in the calculation proceduresused. Reed and Lave (1981), who first descri-bed the difference between oral and writtenarithmetic in greater detail, made this pointquite clearly by characterizing oral arithmetic,typical of unschooled Liberian taylors, as a"manipulation of quantities procedure" andwritten arithmetic as a "manipulation of sym-bols procedure". Carraher and Schliemann(1988) further expanded this analysis by poin-ting out both differences and similarities bet-ween oral and written arithmetic. Oral arithme-tic preserves the meaning of quantities in thesense that hundreds are treated as hundreds,tens as tens, and units as units, as can be seenin the two examples below, one involving sub-traction and one involving division.

In the first example, the child wassolving the problem 252 - 57 usingoral arithmetic in a simulatedshop. The child said: "Just

31

take the two hundred. Minus fifty, onehundred and fifty. Minus seven, onehundred and forty three. Plus the fiftyyou left aside, the fifty two, one hun-dred ninety three, one hundred ninetyfive" (from Carraher, PME,1988, p.10).

In the second example, the childwas solving a word problem whichasked about the division of 75 marbleamong five boys. He said: "If you giveten marbles to each one, that's fifty.There are twenty-five left over. To dis-tribute to five boys, twenty five, that'shard. (Experimenter: That's a hardone.) That's five more each. That's fif-teen" (from Carraher, PME,1988, p.7).

It can be easily recognized that the relativevalue of numbers is preserved throughout inoral computation--for example, the childrensaid "two hundred minus fifty" and "give tenmarbles to each boy". In contrast, in the schoolalgorithm for computation children would betaught to say "seven divided by five, one"(when dividing seventy by five and obtainingten) and "two minus one" (when subtractingone hundred from two hundred), ignoring therelative value of the digits.

Carraher and Schliemann (1988), however,pointed out not only differences but also simi-larities between oral and written arithmetic,arguing for the coordination of social and logi-cal structuring in any context in which mathe-matical activities are carried out. The similari-ties were related to the logico-mathematicalprinciples that constitute the basis for the arith-metic calculations. Detailed analysis of theprocesses of calculation showed that the pro-perties of the operations used in oral and writ-ten computations are the same: associativity foraddition and subtraction and distributivity formultiplication and division. Neither users oforal nor users of written arithmetic name theseproperties of the operations. However, whenthey understand the procedures they use, theyexplain the steps in calculation drawing on theproperties of the arithmetic operations.

To sum up the contrasts presentedhitherto between mathematics in schooland out of school, we saw that mathematicsoutside school is a tool to solve problemsand understand situations while schoolmathematics involves learning the resultsof other people's mathematics. As a conse-quence of this difference, mathematics

outside school tends to be more like modelling,in which both the logic of the situation and themathematics are considered simultaneously bythe problem solver. In contrast, school mathe-matics typically focuses on mathematics per se,resorting to applications not as a basis for thedevelopment of understanding but as occasionsfor practising specific procedures. Despite thedifferences, arithmetic in and out of schoolrelies on the same properties of operations. Thesocial organization of the mathematical activi-ties varies in and out of school while the logi-co-mathematical principles which come intoplay remain invariant.

Logic and convention in geometrical activi-ties.

Arithmetic is a type of mathematical activityrelated to the quantification of variables andoperations involving quantities. Geometry canbe seen as the mathematization of space--or the"grasping of space" through mathematics(Freudenthal, 1973). It involves a whole rangeof activities which can be carried out in and outof school like determining positions, identi-fying similarities (in shape, for example), ana-lyzing perspectives, producing displacements,quantifying space, and deducing new informa-tion through operations. Some of these activi-ties are involved in cultural practices outsideschool. These practices may have to do witheveryday needs of whole populations or theymay be specific of subgroups carrying out par-ticular activities. Determining position is asimple example of a geometry-type activity ineveryday life which may be shared by wholepopulations. In order to walk about in a villageor a city, everyone needs some ability to deter-mine one's position in relation to the place onewants to go to - especially if the place to befound was never visited before or if one isusing a map. Orientation in space appears to beaccomplished by a mixture of logical ands o c i o-culturally determined ways of repre-senting space.

The logic of determining positions involvesrelating objects to each other according to animaginary system of axes which can be appliedin any situation. Piaget and his collaborators(Piaget and Inhelder, 1956; Piaget, Inhelder &Szeminska, 1960) have made a significant

32

contribution to our understanding of this logicof space by tracing the development of theunderstanding of vertical and horizontal inSwiss children. They were able to show thatchildren's understanding of the vertical andhorizontal axes does not stem directly from theempirical experience of standing upright andlying horizontally. Piaget's work suggests thatthis under-standing is constructed as childrenrelate objects to one another in space.

Piaget and his colleagues did not concernthemselves with variations in the use of thesesystems of coordinates either within a cultureacross different situations or across cultures.More recently other authors have looked atthese variations and pointed out interesting cul-tural inputs into this logical system. Forexample, Harris (1989) notices that in mostWestern cultures we use the framework of"left" and "right", "in front of" and "behind","above" and "below" when describing posi-tions. "Left" and "right" are descriptions deri-ved from an imaginary axis related to the twosides of the body as "in front" and "behind" arerelated to the front and the back of the body;"above" and "below" are related to high andlow, as head and feet relate to each other. Whenthis organization is represented on paper,conventions are that the left-right axis on papercorresponds to a horizontal line while the per-pendicular axis from the nearest to the furthestpoint on the page corresponds to a vertical line.

H o w e v e r, as Harris further points out, othercultures may not readily use the body-r e l a t e dsystem of axes (like left-right) either for every-day practices or when space is represented onp a p e r. Several Aboriginal groups in A u s t r a l i aseem to prefer to use compass points even oververy small distances. "For example, when oneAboriginal woman was looking for a particularpicture on a wall covered with photos, her com-panion directed her to look to the west side of thewall" (Harris, 1989, p.12) - a situation in whichWesterners would have used "left/right" instead.S i m i l a r l y, when teachers were giving instruc-tions on how to do letters in a handwriting les-son, they would use the compass points; "to writethe letter 'd' the instructions (spoken in Wa l p i r i )might be: 'Start here on the north side, go acrosssouth like this and down across to the north andup, and then make a stick like this" (Harris,

1989, p.12). Bishop (1983) also reports the des-cription of a system of sloping axes by a stu-dent from Manu, an island in PapuaNew-Guinea, as "a line drawn horizontal in thenorthwest direction". The description of a slo-ping line as "horizontal" is incongruent withWestern conventions and could be interpretedas incorrect. However, if we consider it toge-ther with the description of the slope as "in thenorthwest direction", this student's descriptionof the diagram reveals the use of compasspoints in the representation of relationships in aplane.

Piaget and his collaborators (1960) stressedthe importance of this system of axes not onlyfor locating a point in space but also as part ofdevising a means for measuring angles anddefining figures in a plane. They describedchildren's success at copying angles as a func-tion of their understanding that the slope of aline could be described by two points simulta-neously related to a horizontal and a verticalaxis. Similarly, Carraher and Meira (1989)observed how youngsters learning to useLOGO employed a system of coordinates whentrying to determine the angle to be turned bythe turtle in order to copy a target figure. Theimposition of this imaginary set of axes on themanipulations in space with LOGO is remar-kable because no such a system had occurred tothe youngsters' instructors at the outset. TheLOGO learners seemed to be bringing to thisnew situation a way of organising space whichthey had developed independently of that parti-cular experience. The use of such static refe-rence systems is even more interesting whenone reflects upon the fact that this is not theonly way that the youngsters could haveapproached the metric of angles in a dynamicsituation. As Magina (1991) suggests, such ametric is available in our culture in the readingof traditional clocks but not a single subject inthe Carraher and Meira study used the clockanalogy when quantifying angles in LOGO.

A d i fferent type of geometric activitywhich was also analyzed by Piaget and hisc o-workers deals with the concept of thestraight line as "the line of sight", a conceptwhich is involved in projective geometry. A sFreudenthal (1973) points out, this is a com-plex definition of a straight line and may bepreceded by other ways of understanding thestraight line. It may also be developed moreclearly in connection with certain activitiesof specific groups of people who

33

have to take into account perspective in dra-wing or who have to carry out navigation.

While interviewing fishermen, I obtainedthe following explanation for how they foundtheir way in the narrow channel which theytook through the barrier of reefs off the coastwhere they fished:

"You just have to look at the church andthe tall coconut tree (a particularly talltree which stands out in the region).When the church runs in front of thetree, that's how you get into the chan-nel".

When I asked what he meant by "the chur-ch running in front of the tree", the fishermanexplained:

“ The things that are closer to the oceanmove faster as you move and look atthem. They don't move, you move, butwhat is behind them changes and makesit look like they move. That's what wemean when we say that the church runsin front of the coconut tree ” .

Despite the anecdotal nature of thisexample, it helps us understand why it is thatpeople concerned with navigation turn out todo so well in projective geometry tasks: theyrely on coordinations of lines of sight in orderto determine their course. In contrast, when weorient our routes on land we rely more on roadsand distances than on the line of sight.

Bishop's (1983) results comparing threegroups of students in Papua New-Guinea seemto support the idea that projective geometry isrelated to specialised activities. He showed thestudents a series of photographs of small abs-tract objects and asked them to identify theplace from which the camera took the photo-graph. This is reportedly a difficult task, whichwas performed much better by the students whocame from an island (who averaged 51 correctresponses) than by the students who came fromthe highland or the urban environments (whoaveraged 34 and 26 correct responses, respecti-vely). The specificity of the activity is evenmore clearly understood when one looks atthe results of some of the other tasks that werepresented to the students. These results do notreveal a general superiority of the islanders inall spatial tasks. For example, the islandersshowed a more restricted spatial vocabularythan the other two groups of students anddrew maps picturing the route from their ownroom to the university office which were less

accurate than the maps drawn by urban stu-dents.

In conclusion, similarly to what was obser-ved in the field of arithmetic, everyday prac-tices involving the mathematization of spacereflect both socio-cultural experiences andinvariant logical relations. Locating objects ona plane may vary both within a culture andacross cultures in the use of body-anchored orcompass-related systems of axes. However, itis possible to identify an invariant logic onwhich these variations are based, which isconstituted by the very notion that an imagina-ry system of axes can be imposed on theto-be-represented space. Everyday practices ofparticular groups can also give rise to specificways of understanding space which may not beas important to other people not involved insuch practices. The conception of a straight lineas "the line of sight" seems to emerge more rea-dily among people involved in navigation thanamong people who do not concern themselveswith navigation. This concept of the straightline does not appear to be a routine that peoplememorize but a real activity because the sameability can be used in rather difficult and diffe-rent tasks such as finding the position fromwhich a picture was taken.

Bringing street mathematics into the class-room.

In the previous sections we have discussedexamples of mathematics learned outsideschool without discussing how to build connec-tions between everyday mathematics and tea-ching. In this section we will explore how eve-ryday practices can be brought into the class-room and what effects this may have on schoollearning.

The idea of bringing out-of-school mathe-matics into the classroom is not new. The fol-lowing arguments, amazingly current both intheir nature and in the goals proposed for arith-metic teaching, are taken from a manual prepa-red for teachers by Brideoake and Groves asearly as 1939:

We felt that in the past, many childrenwho failed to achieve success in "sumlessons" showed considerable grasp ofthe subject when shopping in the HighStreet or taking the milk. They were notdevoid of number sense, but the schoolapproach seemed to be faulty (p.5).

34

In the earlier period it was "sums"which mattered. Class teaching was themethod, with its visualising and memo-rising, and the ideal was that the chil-dren should learn the four rules and beable to get the daily "six sums" right.Then a reaction in favour of individualwork set in. This was led by the greateducationist, Madame Montessori, andresulted in a flood of "number appara-tus", extensive exercises in counting,and pretentious written sums; but stillthere was, very often, little mentalalertness or real number power. Nowmany teachers are feeling their way to arealistic approach, with its underlyingidea that development of number senseis what matters, and that "sums" areonly the expression of this (Brideoakeand Groves,1939, pp.9-10).

To want to use a realistic approach or bringchildren's number sense into the classroom isone thing; to be successful at it is quite another.In this context, an informal observation by aneducator, Herndon (1971), is very informative.He describes how he met in a bowling alley astudent who had difficulty in school arithmeticbut who could keep track of eight bowlingscores at once in the alley. He then realized thatall of his students had some use of mathematicsoutside school in which they were successfuland tried to make them solve exercises inschool which were similar to what they coulddo outside school. This is how he describes hisattempt:

Back in eighth period I lectured him[the boy from the bowling alley] onhow smart he was to be a league scorerin bowling. I pried admissions from theother boys, about how they had paperroutes and made change. I made thegirls confess that when they went tobuy stuff they didn't have any difficultydeciding if those shoes cost $10.95 orwhether it meant $109.50 or whether itmeant $1.09 or how . much changethey'd get back from a twenty.Naturally I then handed outbowling-score problems, and naturallyeveryone could choose which ones theywanted to solve, and naturally the resultwas that all the dumb kids immediatelyrushed me yelling. "Is this right? I don'tknow how to do it! What's the answer?This ain't right, is it?" and "What's mygrade?". The girls who bought shoesfor £10.95 with a $20 bill came up with$400.15 for change and wanted toknow if that was right? (Herndon,pp.94-95).

Bringing out-of-school mathematics intothe classroom is not simply a matter of findingan everyday problem and presenting it as aword problem for the application of a formulaor an algorithm already taught. This approachdoes not change the main difference betweenmathematics in and out of school pointed outearlier because students would still be in thesame position of trying to learn the products ofother people's mathematical activities.Bringing out-of-school mathematics into theclassroom means giving students problemswhich they can mathematize in their own waysand, in so doing, come up with results(methods, generalizations, rules etc.) whichapproach those already discovered by others .

D i fferent PME authors have exploredroutes to mathematics teaching related to theidea that out-of-school mathematics can bebrought into the classroom with positiveresults. Some of these studies will be reviewedhere and can be used as starting points for fur-ther explorations. I would like to distinguishtwo types of teaching approaches in whicho u t-o f-school mathematics is brought intoschool.

The first approach starts from a particularaspect of mathematics which one wants thestudents to learn--notation systems, methods,theorems etc.--and then searches for everydayproblems which instantiate that aspect ofmathematics. In this case, the teacher will crea-te constraints in order to ensure that the speci-fic aspect of mathematics comes up in the ana-lysis of the everyday situation and will notconsider teaching successful unless the specificgoal is achieved .

The second approach involves bringingsensible problems from everyday life into theclassroom without a pre-established ideaabout which particular method of mathe-matizing the situation is to be the end-result ofthe lesson. Although the teacher chooses theproblem because there are interesting mathe-matical relations which can be explored in thesituation, there may be no ready-made solu-tion process which the students are expectedto learn by analyzing the problem situationand many ways of solving the problem will beconsidered legitimate. Teaching will be consi-dered "successful" if students analyze thesituations, use mathematical concepts in

35

their analyses, build relationships between dif-ferent concepts and improve on their ownmethods as they repeatedly employ them onsolving further related problems. Teachers mayintroduce mathematical notations as the pupilsdevise their solutions but the specific teachingof these notations is not the aim of the activitiesfrom the outset.

In short, the first approach is task oriented.It aims at teaching concepts, notations,methods etc. and can be evaluated on the basisof whether or not pupils accomplished the spe-cific target. The second approach is global andopen-ended. It aims at developing "mathemati-cal minds". Its evaluation must take intoaccount pupils' progress over time and determi-ne whether or not they become more sophisti-cated in their mathematical analyses of situa-tions as they progress in school.

This section is divided into three parts. Inthe first two, I will present work and ideas fromboth of these approaches. I leave to the readerthe task of evaluating them and deciding whe-ther they can be combined into a unified philo-sophy for the teaching of mathematics or not.In the third part, some of the teachers' reactionsto bringing out-of-school mathematics into theclassroom are presented.

Teaching specify aspects of mathematics withthe support of everyday life experiences.

We have seen that people learn much aboutmathematical concepts outside school.However, it does not follow that school mathe-matics can profit from children's knowledge ofout-of-school mathematics. It is quite possibleto recognize that mathematical ideas are oftenlearned outside school but still be scepticalabout the importance of bringing out-of-schoolmathematics into the classroom. The need forstudies which show whether bringing everydayconcepts into the classroom can actually contri-bute to the development of school mathematicswas clearly argued by Janvier (PME, 1985).His arguments are summarised in the followingpoints:

"(1) certain mathematical ideas havenatural phenomenological counter-part since they result by a processof abstraction from objects andobservable entities" (p.135);

(2) these ideas formed from everydayexperience represent models

which give meaning to the mathe-matical ideas;

(3) two positions can be formed aboutthe role that these models may havein mathematical education: (a)models are restricted in their gene-ralizability and should therefore beforgotten as soon as correct andefficient handling of abstract sym-bols is achieved; and (b) a model isirreversibly a model for life and canbe brought into play at any momentwhen there is a need for the mathe-matical relationships and rules tobe rediscovered.

The question posed by Janvier is thusessentially whether everyday mathematics is sobound up with the context in which it was lear-ned that it cannot be "pulled up" from thecontext and transformed into a more generalmodel. Models for understanding money, forexample, could remain defined as such and therelationships between numbers which werelearned in money contexts would not be reco-gnized in other contexts. A different but relatedconcern is expressed by Booth (PME, 1987),who raises the possibility that students may notlink up the specific representations used in tea-ching with the more general mathematicalones, and will thus not profit from the expe-riences with the particular situations. Below wewill review a sample of studies in which chil-dren learned mathematics in specific contextsand were tested for their learning in more gene-ral ways. They do not represent an exhaustivereview of the literature; they are described hereas samples of attempts to teach particularaspects of mathematics on the basis of situa-tions which pupils may already understandfrom their everyday experience.

Higino (1987) analyzed the use of chil-dren's knowledge of money as a support forlearning place value notation and the writtenalgorithms for addition and subtraction. Sheworked with a task previously used byCarraher (PME, 1985; Carraher andSchliemann, 1990) in which children play ashopping game and are asked to pay differentamounts of money for the items they purchase.The children are given nine tokens of two dif-ferent colours which represent coins of diffe-rent denominations, one and ten. They are thenasked to pay for items they purchase in the playshop using these tokens. The amounts ofmoney can be obtained (a) only with singles

36

(for example, six); (b) only with tens (forexample, 30); and (c) mixing singles and tens(for example, 17, a sum which can only be paidif the child uses one 10 and 7 singles). The 60children in Higino's study were 7 to 9 years ofage and were attending second grade in stateschools in Recife, Brazil. They were matchedat the outset of the study with respect to theirknowledge of place value notation and thenrandomly assigned to one of four groups.

(1) an unseen control group that received onlythe regular classroom instruction on placevalue notation and the addition/ subtractionalgorithms;

(2) a symbolic teaching group, that receivedexperimental instruction on place value andthe algorithms without recourse to anymaterials other than explanations about thedecades and the rules of the addition/ sub-traction algorithms;

(3) a sequential everyday-symbolic group, thatfirst practised counting money and workingout change in the pretend shop and thenreceived instruction on place value and onthe addition/subtraction algorithm;

(4) a parallel everyday-symbolic group, thatpractised counting money and giving chan-ge and received parallel instruction onplace value notation and the addition/ sub-traction algorithms.

Results showed that all four groups progressedwith instruction according to an immediatepost-test. However, in a delayed post-test givenapproximately one month later, children fromthe symbolic teaching group no longer differedsignificantly from the unseen control groupwhile the other two experimental groups stilldid significantly better. Moreover, childrenfrom the parallel everyday-symbolic groupshowed a small improvement in their meannumber of correct responses to theaddition/subtraction problems after one monthof instruction while the other groups showedslight decrements. Thus Higino was able toshow that connecting everyday experienceswith classroom learning of addition and sub-traction produces better results than teachingwithout regard for children's previous know-ledge.

Another study which attempted to analyzethe role of everyday situations in the deve-lopment of basic concepts of additio nand subtraction was carried out by De Corteand Verschaffel (PME, 1985). They argued that

most instructional approaches use word pro-blems as applications of addition and subtrac-tion but it is questionable whether word pro-blems representing real situations only havethis role in elementary arithmetic.

"Indeed, recent work on addition andsubtraction word problems has produ-ced strong evidence that young chil-dren who have not yet had instructionin formal arithmetic, can neverthelessalready solve those problems success-fully using a wide range of informalstrategies that model closely thesemantic structure and meaning of thedistinct problem types." (p.305)

Working from this assumption, De Corteand Verschaffel compared a control group, ins-tructed in the traditional way described above,to an experimental group, who learned additionand subtraction concepts mostly from solvingword problems and learning several ways ofrepresenting the semantic relations in the pro-blems. They observed that the experimentalgroup made twice as much progress onword-problem solving than the control groupafter one year of instruction, a difference whichwas statistically significant. De Corte andVerschaffel concluded that this line of investi-gation is worth pursuing further although theyrecognize that the problems used in the experi-mental program were relatively poor in contentand did not approach outside-school situationsas much as desired. Similar positive resultswere reported also by van den Brink (PME,1988) after a year-long experiment on chil-dren's learning of addition and subtractionconcepts either by modelling from everydayexperiences or from teaching with-in a traditio-nal approach.

Looking at somewhat more complex situa-tions, Janvier (PME, 1985) analyzed the role ofmodels in teaching negative numbers bycontrasting two groups of students, one who hadlearned negative numbers from a symbolicmodel and a second one which had learned fromwhat he calls a "mental image" model based onparticular experiences. The symbolic-m o d e lgroup learned negative numbers by having thesymbolism introduced only loosely related to aninitial situation and then concentrating on rulesfor manipulating numbers. Different colours

37

were used for negative and positive numbersand rules were of the type: "when you add twonumbers of the same colour you make an addi-tion and keep the same colour; when you addtwo numbers of different colours, you subtractthe small one from the big one and take thecolour of the big one" (Janvier, 1985, p.137).

The mental-image group attempted to solveproblems about a hot-air balloon to which bagsof sand or helium balloons could be attached,the first having the effect of lowering the bal-loon while the latter brought it higher. Balloonsand bags were initially of one unit but severalof either could be used at once so that symbo-lism could be developed for larger numbers.Adding meant tying an extra-element and sub-tracting was identified with removing an ele-ment from the dirigible. Janvier found that themental-image group did significantly better onaddition items than the symbolic-model group;no significant difference was found for subtrac-tion. Furthermore, when students were inter-viewed later, those from the mental-i m a g egroup could correct the mistakes they had ini-tially made by going back to the model; howe-ver, the symbolic-model students simply repea-ted within the model the same mistakes andwere unable to revise their wrong answers.

Other studies which must be mentioned andwhich deal with more advanced mathematicalconcepts are in the realm of functions and alge-bra. Janvier (1980) and Nonnon (PME, 1987)have investigated the use of situations represen-ted graphically in the teaching of functions.Both studies analyze students' difficulties withgraphic representation and improvement afterteaching. However, no systematic comparisonsbetween the experimental groups and otherstaught without the support of the everydaysituations represented graphically are availableand clear conclusions are thus precluded. Filloy(PME, 1985) studied the use of addition/sub-traction of packages of unknown weights ontot w o- plate balance scales as images for alge-braic manipulations with unknowns. A l t h o u g hhe reports some success, he seems to be cau-tious about the results of this experience in alater paper. Filloy and Rojano (PME, 1987)a rgued that subjects could achieve a good hand-ling of the concrete model but developed a ten-dency to stay and progress within this contexto n l y, showing difficulty in abstracting the

operations to a syntactic level and in breakingwith the semantics of the concrete model.

To summarize, it seems possible to use eve-ryday situations in the classroom as instancesof particular aspects of mathematics and obtainbetter learning of this mathematics in contextthan in a traditional teaching schema relyingprimarily on symbolic manipulations. Furtherresearch is undoubtedly still necessary sinceonly a very narrow range of mathematics hasbeen systematically investigated.

Real-life mathematics used in an open-endedfashion .

Streefland (PME, 1987) presented in very clearterms the instructional principles which guidethe open-ended type of realistic mathematicseducation which we are distinguishing from themodelling of specific mathematical aspects dis-cussed above. These instructional principlesare:

a ) context problems (or situations) occupy adominant role in mathematics education,serving both as a source and as a field ofapplication of mathematical concepts;

b) a great amount of attention must be paid tothe development of situation models, sche-mas and symbolising;

c) children can make significant contributionsto classroom work through their own pro-ductions and constructions in the process ofmoving from informal to formal methods;

d) this learning process is of an interactivenature;

e) different concepts become more clearlyinterrelated in this progressive mathemati-zation of situations.

Streefland (PME, 1987; PME, 1988) hasproduced several examples of how children canlearn mathematical concepts in the progressivemathematization of real situations. In one PMEresearch report, Streefland (PME, 1987) des-cribes the process of mathematization leadingfrom division in the sense of equal sharing tofractions. The children start from a situation inwhich they are asked to work out how to sharethree candy bars among 4 children. Examplesof how the distribution is performed and what

38

children learn from this distribution in this firstphase of teaching involve:

dividing each bar of chocolate into 4equal parts and taking one part for eachchild, a procedure which leads childrento describe each child's share as 3 x 1/4or 3/4;

realizing that each child could have ahalf chocolate, using up two wholechocolates, and the last chocolatewould then be cut into for equal parts,each child's share being described as1/2 + 1/4 (this description helps chil-dren understand that 1/2 and 2/4 are thesame amount).

In the second phase of teaching, "concretestories of fair sharing gradually melt into thebackground (...). The activities change intocomposition and decomposition of real frac-tions" (Streefland, PME, 1987, p.407) usingmethods of decomposing fractions into unitsand finding equivalences, but it is still best "notto wipe out all the traces of the real situationsreferred to even at this stage" (p.407). Forexample, 5/6 can be decomposed into 1/6 + 1/6+ 1/6 + 1/6 + 1/6 and then other equivalencescan be found like 3/6 + 2/6 and then 1/2 + 1/3and so on.

In the third level of teaching, "the rules forthe composition and decomposition of frac-tions will become the objects of mathematicalthought ( . . . ) the methods of production whichturn out to be the most efficient might becomestandard procedures or algorithms" (p.407)."For example, children work with questionslike: 1/2 + 1/3 are probably pseudonyms forother fractions with a common name; producethese other fractions".

Streefland reports that children workingwith this approach to the teaching of fractionsshow increasing skills in producing equivalentfractions, in operations with fractions with theexception of division, and a general dominanceof the elementary laws for operation asmethods of production.

Streefland (PME, 1988) presented furtherexamples of his instructional approach inclu-ding topics like subtraction, division and fur-ther examples about fractions. He also streng-thened his theoretical principles on teaching,by arguing that:

- when students work with their ownconstructions for mathematizing reality,they have the opportunity of stronglyinterconnecting several topics which are

related but taught separately in otherapproaches to mathematics educationbecause "genuine reality can be organizedmathematically in various ways"(Streefland, PME, 1988, p.89);

- in this learning process, children acquire avariety of aids and tools in the progressivemathematization of situations becomingcompetent in the use of different terminolo-gies, symbols, notations, schemas, andmodels;

- organizing and structuring the mathematicsshould be as much as possible the businessof the children themselves, a principlewhich stresses the importance of interac-tion and cooperation in this way of approa-ching mathematics education.

Other significant aspects of this realisticapproach to the teaching of mathematics areemphasized by De Lange (1987; in press), whodiscusses the importance of using real mathe-matics also in the assessment of children's suc-cess in learning mathematics, and by Boero(PME, 1989), who points out that there aremisconceptions of real situations which maycreep up in mathematics classes when externalcontexts are used for conceptualising.According to Boero, these misconceptions donot necessarily hinder the process of concep-tualizing mathematics; they are rather pro-blems to be faced by students "if one wants thepupils to gradually understand that there arelevels of 'intuitive evidence' and 'intuitive'ways of thinking which must be exceeded if arational working command of certain pheno-mena is to be reached, and that mathematicsmay have an important role in this passagefrom intuition to rationalization" (Boero, PME,1989, p.69).

The evidence produced by Streefland onpupils progress in the process of mathemati-zing situations is provocative. Mathematicsappears as a way of representing and mentallymanipulating situations, bridging the gap bet-ween a problem and a solution and allowingchildren to go beyond the information theystarted the task with. Streefland's descriptionsof how children progress from close model-ling the situations to developing abbreviatedand general procedures as they interact witheach other and compare the efficiency of thed i fferent procedures suggest the importanceof exploring this approach further with a

39

new range of topics. It is especially interestingto note how children seem to understand quiteclearly the mathematical equivalence of theird i fferent procedures on the basis of theirmodelling properties but still go on to evaluatetheir efficiency. They do not stop workingwhen they "find the answer".

To summarize, the open-ended realisticapproach to mathematics teaching involves thesearch for everyday situations which can betreated as sensible problems in mathematicsclasses. The choice of situations is not gearedto the teaching of particular mathematicstopics. Pupils success is not evaluated on thebasis of whether a specific bit of mathematicalknowledge was accomplished. Instead, pupilssuccess is evaluated on the basis of the pro-gressive sophistication of their methods andtheir building of relationships betweenconcepts, symbols, notations and terminolo-gies. This is essentially the basis of Streefland's(PME, 1988) contrast between the teachingapproach developed by "realistic instruction"to the "structuralist approach" proposed byDienes. While the realistic approach starts frompupils good sense in a problem situation andprogresses to more formal procedures, in thestructuralist approach "vertical mathematizingis overstressed and formal procedures areimposed" (p.89). In other words, concretematerials devised for instruction within thestructuralist approach do not aim at bringingo u t-o f-school reality into the classroom.Instead, mathematical objects are devised forthe purpose of concretely exemplifying mathe-matical procedures and representations alreadyformalized. The mathematical objects thusdevised may have no meaning outside theclassroom. This means that pupils may nothave models which they can readily call uponto understand their experiences in the class-room.

This open-ended realistic approach is notrestricted to arithmetic but has also been triedout with geometry. Goddijn and Kindt (PME,1985) strongly criticized traditional geometryteaching as stereotyped and restricted to "theflat world of textbook". They propose to givegeometry teaching a new approach by workingfrom the three-dimensional world. Examplesof questions in their programme include wor-king with proportion and scale comprehensionas students look at pictures and take into

account distances and viewing points. Studentsattempt to find the position from which a pic-ture was taken and are encouraged to useexplanatory drawings in this process. In sodoing, they "discover" the line of sight and thechanges in perspective and size of figures asthe "viewer" moves about. They look at sha-dows and reflect on related phenomena such asthe phases of the moon. They watch a studentenact the movements of the moon around theearth in front of a bright light and ask eachother how is moon seen, lit-up or in shadow,from the perspective of pupils is different partsof the room. They use the "line of sight" fromthe point where they stand, imagine it from thepoint where someone stands, and try to figureout where someone must stand in order to havea particular viewpoint.

Goddijn and Kindt describe how students(and teachers alike!) become surprised at howwell their methods using lines of sight (whichmay even be concretely represented by strings)work in drawing and solving these problems.But the results of this exploration of space arenot restricted to perspective drawing; studentsfound, for example, a totally new manner ofdoing the classic assignment of constructingthe line through a given point which intersectstwo skew lines. Thus, by teaching geometry as"grasping space through mathematics",Goddijn and Kindt were able to observe a pro-gressive sophistication of students' methods fordealing with spacial and geometrical problems.

Teachers' reactions to bringing out-of-schoolmathematics into the classroom.

Many mathematics teachers seem to resist atleast initially, the suggestion of bringing infor-mal knowledge of mathematics into the class-room. Several reasons are pointed out in theliterature. First, Schultz (1989) observes thatteachers seem to expect that they will carry outthe same pedagogy they were themselves therecipients of. They learn from modelling theirteachers as much as (if not more than) theylearn from theories in teacher education pro-grammes. Second, they are usually concer-ned with covering the standard curriculummaterial, which is determined in terms of tra-ditional mathematics topics, and not in termsof mathematization of situations. To thisconcern is added the resistance to the use ofintuitive methods in the classroom because

40

these methods conflict with teachers' views ofmathematics as a formal system of knowledge(see Ernest, 1989) and as "something superior"(Rosenberg, 1989).

Strategies which have been used to dealwith such resistance include both teachers'observations of children's spontaneous mathe-matical activity outside the classroom and tea-chers' experience of the same teaching approa-ch when they themselves mathematize expe-riences of a more complex nature. Rosenberg(1989) reports that "Aha:!-experiences happe-ned very often when they (the teachers) startedfrom informal problems and finally got tofamiliar formal structures" (p.83).

A last source of resistance comes from tea-chers' beliefs about what students will/will notdo when faced with a new educational expe-rience. In this context, it is interesting to reportbriefly on an informal observation with agroup of Brazilian teachers, who found it hardto believe that pupils would experiment withand improve upon procedures and mathemati-cal models once they had already found ananswer to the problem. In traditional mathe-matics teaching, finding an answer usuallydetermines the end of the activity. For this rea-son, teachers expected that students will stopworking as soon as an answer is found. T h e ywere also convinced that some answers arecorrect and others are wrong and that shiftingmathematical models within a problem simplymeans "finding the right answer versus findingthe wrong answer". A little later in the day Iasked the teachers to develop a simple way ofdetermining the rate of inflation, a problem ofeveryday significance in Brazil. They quicklycame up with the idea that inflation could beviewed as the average of all price increases.They drew up a list of items, wrote down theirestimations of increases in prices in percen-tages and calculated the average increase.They were not satisfied with their solution tothis problem and did not stop -working. T h e yrealized that housing and grocery shoppingmight affect people's everyday expenses moremarkedly than entertainment, for example.One teacher suggested that they could giveweights to the items which would be propor-tional to the impact the item had on a worker'smonthly spending. They worked out that thewage could be represented as 100% and theycould then estimate what percentage of it

was spent per month on each item. They calcu-lated a weighted average of the increases astheir new solution. At this point, the teachersseemed to have an "Aha! experience". Theyrealized that, in situations such as the one theyhad just worked on, students might well look atthe same problem in different ways by tryingout different mathematical models on the sameproblem.

Finally, it is interesting to add Schultz's(1989) comments on the impact that the use of.real-life mathematics in the classroom had onthe student-teachers she worked with.Although no systematic data are presented, sheobserved that they became more aware of whatthey had learned and the way they learned andseemed to develop new methods and ideas forteaching their own students. If modelling fromreal-life situations turns out to have a positiveimpact on future mathematics teachers, thismay be just as important as the effects it hasdirectly on pupils. However, no research dataseem yet available on this issue.

Final comments

This paper reviews research which shows thatthe social context in which mathematics is car-ried out influences the way people approachmathematical problems. Mathematics in manyof today's classrooms is taught as an abstractform, a set of symbols, procedures, and defini-tions to be learned for perhaps some later appli-cation. Mathematics used outside school is atype of modelling: it is a way of representingreality so that further knowledge about the rea-lity can be obtained from the manipulation ofthe representations without the need to checkthese new results against reality (D'Ambrosio,1986). These differences in the social situationsand their corresponding mathematics have animpact on rates of success and types of proce-dures used by the problem solvers. Researchhas shown that bringing out-of-school pro-blems into school as a way of teaching mathe-matics is not only possible but also a beginningof a more successful story about mathematicsteaching. Further research is still needed andmore clear theorising so that successful isola-ted experiences can be transformed into aneffective educational theory.

Bringing out-o f-school mathematics intothe classroom means facing questions whichmay not be addressed in the traditional

41

forms of teaching. Some of the questions whichemerge within this approach to mathematicsteaching are presented below.

We know that many concepts can be lear-ned outside school and that this knowledge canbe useful in school. But we cannot be casualabout which concepts and situations we chooseto bring into school. An implicit route in thischoice appears so far to be to look at presentmathematics curricula and choose problemsituations appropriate for present day curricula.However, is it possible to identify some "core"concepts which will guide this choice in a newway?

A second question has to do with the intro-duction of mathematical representations.Pupils solving real problems do not need toreinvent mathematical representation. W h e nand how is mathematical representation to beintroduced?

A third question which remains to be inves-tigated is which factors affect the passage frominformal to more formal mathematical repre-sentation in the process of mathematizing realproblem situations. Much of the success of therealistic approach depends upon interaction bet-ween pupils and we have achieved relativelylittle understanding of what in these interactionsbrings about children's progress. Is it conflict

between different perspectives or is it theconvergence of different ways of representingthe same problem? This is not a trivial questionbut one of great theoretical significance, asBryant (1982) has pointed out.

Finally, it would be desirable to accompli-sh better descriptions of the process of progres-sive mathematization which takes place whenstudents interact and solve real mathmaticssproblems together. Is there a single route in thedevelopment of each particular concept or arethere many ways to get to the same place?Does natural language, which is of course usedin the students' interactions, play a part in thisprocess or not? Is the progressive mathemati-zation of situations a process of reconstructionof solutions at ever- higher levels of abstractionor does it involve dropping old models whichare then replaced by new ones? Is there amoment when real problems are no longerimportant or do they retain their role of givinginitial meaning to formalizations even at thehighest levels of abstraction?

H o p e f u l l y, further research and fruitfulcooperation between researchers and teacherswill bring clearer answers to many of thesequestions in the future.

42

Terezinha Nunes was Associate Professor at the Universidade Federal de Pernambuco, Brazil, andhas been recently working as Research Fellow at The Open University, U.K. From October 1991she will be a lecturer at the Institute of Education, University of London. Her studies of mathema-tical knowledge developed outside school (published mostly under her previous surname, Carraher)analyze children's and adults' understanding of the numeration system, arithmetic, and simplemathematical models such as ratio and proportions. She has sustained much interest in the educa-tional implications of these findings and worked closely with the educational authorities in Recife,Brazil, in the development of teacher education programmes.

This is an excerpt from a conversation betweena teenage girl and a mathematics educator:(I: Interviewer; A: Alison)

I: You need four cubes for eight people so forfour people...?

A: Yes, but then if it was soup cubes, you'dwant the same amount of taste wouldn'tyou so you would have to put the sameamount in the soup to get the same taste toit, if you put less it wouldn't taste as nice.

I: Really, even if you were only making halfas much soup?

A: Yes, 'cos you need the same amount don'tyou, else it would taste horrible, it wouldn'ttaste as strong.

I: But let's say you make enough soup for 8people and you put 4 cubes in, now the nextday you are making soup for 4 people. Howmany cubes would you put in then?

A: I would put in two but it would not taste asnice would it?

I: Don't you think so?

A: No.

I: You don't think it would taste the same?

A: Because if you are going to make soup youare going to have to use the cubes anywayto make it taste aren't you?

I: Yes

A: So if you want it to taste just as good as itdid when you made it for 8 you would haveto put the same number of cubes in it.Otherwise it won't taste the same.

I: I think it would. I think if you are going tomake half as much soup, you want half asmany cubes.

A: I suppose you could, but I wouldn't if I wasmaking the soup.

Frank is only ten years old and he and his classhave been 'taught' about the volume of a cuboid.

(I: Interviewer; F: Frank) I x b x h = V

F: Well,a,b,c,d,e,f,g,h,i,j,k,1-1,2,3, 4, 5, 6, 7,8, 9,10,11,12. That'll be 12 times 2 and ...

I: Can you tell me how you know that?

F: Well, 1 is ... um... the tenth letter of thealphabet .

I: Yes.

F: So, 10.

I: Ah... who told you that?

F: . . . (long pause). . .

I: Did you just make it up? F: Think so, yes.

I: Okay, and 'b' will be? And 'h'?

F: H... um... eh4... a,b,c,d,e,f,g,h, ...1,2,3,4,5,6,7,8,9,10 ... two's is twenty... eight,twenty eights will be one hundred and ...one hundred and . .. eight ... oh, two hun-dred and ... oh, one hundred and forty.

I: Alright, and then it says equals V.

F: Equals V? 140.

I: How do you know that that's not right, sha-king your head?

F: V isn't the 120th letter of the alphabet(laughs)...

Frank's common sense came to the rescue!

These children have been in schoolmathematics classes for six to ten years butthere are obvious misconceptions held bythem. This chapter is about research concer-ning materials which are available to the tea-cher to assist in the process of teaching.Materials can include books, computer soft-ware or languages, concrete materials (mani-pulatives) or other aids. The emphasis is onresearch which led to the production of thematerial or which sought to produce evidenceof its effectiveness. The main source of theresearch references is the work of members

43

The Influences of Teaching Materials

on the Learning of Mathematics

Kathleen Hart

of the international group Psychology ofMathematics Education.

Most children learn their mathematics inschool although they use it and obtain their firstideas of the topic in the world outside the class-room. Within the classroom they may be grou-ped with others of the same age, other pupils ofthe same attainment level or simply all theother children who live in that village or street.The teacher who provides the experiences fromwhich they are supposed to learn has a difficulttask no matter which way the children whoshare the classroom are selected. T h eresearches of PME take as a basic premise thatthe teacher is striving for the children tounderstand what they are doing. PME reportswhich describe teaching experiments in whicha form of written material, a set of concretemanipulatives or a computer program havebeen tried so that learning is more effective arenot numerous. Many are based on the desire toprovide empirical evidence of the validity of alearning or psychological theory and in only afew cases are the results overwhelmingly sup-portive. However, books are written and mate-rials produced and the research results mustshed some light on how this can be betterachieved so that learning takes place. Nobodyhas yet provided a solution for the question'how can I best teach this child'. Throughout theworld, with very few exceptions the perfor-mance of children on mathematics tasks isconsidered unsatisfactory by educators in theircountry. Is this because educators set unreaso-nable tasks in the discipline of mathematics orbecause the expectations are entirely reaso-nable but those who teach the subject s i n g u l a r l yinefficient?

Learning theories and their influences

Most research is carried out within a theoreticalframework. The choice of framework influencesthe assumptions, the premises suggested and tes-ted and the outcomes. Sometimes materials areinvented to conform to the theory and the resear-ch is perhaps to test their l effectiveness orindeed provide some evidence of behaviourwhich supports the theory itself. Often existingbooks or teaching aids are criticised becausetheir development contradicts a learning theory.At the moment most mathematics books for

young children which are based on languageand words only would be condemned as "tooabstract", developmental psychology havingprovided theories which state that a youngchild is incapable of abstract reasoning.

Piaget's developmental theory outlinedstages of mental development loosely tied toage. The theory divided intellectual develop-ment into four major periods: sensorimotor(birth to 2 years); pre-operational (2 years to 7years); concrete operational (7 years to 11years); and formal operational (11 years andabove).

Piaget's interest was in the psychology ofthe child but many educators have extendedand interpreted his writings in order to applythe theory to teaching as is shown here byG i n s b u rg and Opper (1969) in their bookPiaget's Theory of Intellectual Development:

"While Piaget has not been mainlyconcerned with schools, one can derivefrom his theory a number of generalprinciples which may guide educatio-nal procedures. The first of these is thatthe child's language and thought aredifferent from the adults The teachermust be cognizant of this and must the-refore attempt to observe children veryclosely in an attempt to discover theirunique perspectives. Second, childrenneed to manipulate things in order tolearn. Formal verbal instruction isgenerally ineffective, especially foryoung children. The child must physi-cally act on his environment. Such acti-vity constitutes a major portion ofgenuine knowledge; the mere passivereception of facts or concepts is only aminor part of real understanding."

The theory of Piaget has for many yearsinfluenced those who teach mathematics,mainly through the advice given to teachers intraining. The definition of the concrete opera-tional level and the features which distinguishit from the formal operational level have ledoften to a belief that children below the age ofeleven years should be given manipulatives toassist them in learning mathematics or indeedin some cases to replace the abstraction ofmathematics with only those experienceswhich could be regarded as 'real'.

This has further become the suggestionthat the use of concrete materials is good in

44

itself irrespective of the topic being taught.Researchers have started to discuss the gulfbetween manipulatives and formal schoolmathematics and PME members have conside-red this with respect to the learning of Algebra.

Ausubel (1963) rather than having anage-dependent theory provides a theory of lear-ning which emphasises the dependence of newknowledge on old knowledge and is perhapsbetter Suited to the needs of mathematics tea-ching which traditionally builds on previousteaching. As Novak (1980) says in quotinghim:

"However, we must also consider thecognitive functioning and psychologi-cal set of the learner as new knowledgeis internalised. Here we must distingui-sh between rote learning wherein newknowledge is arbitrarily incorporatedinto cognitive structure in contrast tomeaningful learning wherein newknowledge is assimilated into specifi-cally relevant existing concepts or pro-positions in cognitive structure. Sincethe nature and degree of differentiationof relevant concepts and propositionsvaries greatly from learner to learner, itfollows that the extent of meaningfullearning also varies along a continuumfrom almost pure rote to highlymeaningful".

In Europe successful learners of mathema-tics in schools and universities have alwaysbeen few in number, successful linguists orengineers are more numerous. This state ofaffairs did not improve proportionally with theintroduction of universal secondary education.This has often meant that those who succeedhave had the reputation of possessing a super-ior (or certainly different) type of intelligence,their success being judged not attributable tothe skill of the teacher or the suitability ofmaterials but to the make-up of theirbrains/intelligence. Psychologists and educa-tionalists can only measure performance ontasks and therefore a theory which discussesthe potential for learning as does that of 'infor-mation processing' is of value.

Those who have espoused this theorytended to produce tasks which they saidwere useful for judging the amount of'space' available in one's cognitive make up.More recently the emphasis in mathematics

education research has moved to the construc-tivist point of view.

A large number of PME members espouseConstructivism, indeed the XIth Conference inMontreal had Constructivism as its centraltheme. Kilpatrick (PME, 1987) describes somefundamental points:

"The constructivist view involves twoprinciples:

1. Knowledge is actively constructedby the cognizing subject, not passi-vely received from the environ-m e n t .

2. Coming to know is an adaptiveprocess that organises one's expe-riential worlds; it does not discoveran independent preexisting worldoutside the mind of the knower. "

'The first principle is one to which mostcognitive scientists outside the beha-viourist tradition would readily giveassent, and almost no mathematics edu-cator alive and writing today claims tobelieve otherwise. The second principleis the stumbling block for manypeople."

"Radical constructivism adopts a nega-tive feedback, or blind, view toward the'real world'. We never come to know areality outside ourselves. Instead, allwe can learn about are the world'sconstraints on us, the things not allo-wed by what we have experienced asreality, what does not work. Out of therubble of our failed hypotheses, wecontinually erect ever more elaborateconceptual structures to organize theworld of our experience."

"Von Glasersfeld has identified fiveconsequences for educational practicethat follow from a radical constructivistposition:

(a) teaching (using procedures thataim at generating understanding)becomes sharply distinguishedfrom training (using proceduresthat aim at repetitive behaviour);

(b) processes inferred as inside thestudent's head become more inter-esting than overt behaviour;

(c) linguistic communication becomesa process for guiding a student'slearning, not a process for trans-ferring knowledge;

45

(d) students' deviations from the tea-cher's expectations become meansfor understanding their efforts tounderstand; and

e) teaching interviews becomeattempts not only to infer cogniti-ve structures but also to modifythem. All five consequences fitthe constructivist stance, but theyappear to fit other philosophicalpositions as well."

At all levels of the constructivist belief thechild is active in the learning process and not apassive receiver of knowledge. Radicalconstructivists would advise teachers to provi-de rich learning experiences for the child fromwhich he can construct his own mathematics.This is a viewpoint different from that whichsays the teacher armed with materials is thesource of the major part of mathematics infor-mation in the classroom.

Ample evidence has been provided, overmany years, that children change the informa-tion they are given by adults and remember itin a different form. Children also invent theirown methods for carrying out mathematicaltasks. Street children in Brazil for examplehave adequate methods of calculation of profitson goods they sell. Many English childrenignore the generalised, formal methods of solu-tion they are taught in school, in favour of morespecific, invented methods (Booth, 1981).Whether this is because the child's unsulliedintuition is at work or whether it is because ofdesperation arising from a lack of understan-ding of the teacher's presentation, is unknown.We know that many children start school kno-wing how to count, to sort and to put objects inorder. Children are able to construct mathema-tics but nobody has yet shown that they canconstruct all that is needed to be a mathemati-cally competent adult in the year 2000.Research which shows a teacher can be mosteffective with talk and a blackboard is not oftencarried out in mathematics education. Brophy(1986), writing in the Journal for Research in lMathematics Education, quoting from thegeneral research on teaching said:

"Mere engagement in activities will notfacilitate learning, of course, if thoseactivities are nut appropriate to the stu-dents' needs. Thus, the teacher's desi-re to maximize content coverage by

pacing students briskly through the cur-riculum must be tempered by the needto see that the students make conti-nuous progress along the way, movingthrough small steps with high, or atleast moderate, rates of success andminimal confusion or frustration.Process outcome research suggests thatteachers who elicit greater achievementgains from their students address thisdilemma effectively partly by

selecting activities that are ofappropriate difficulty levels for theirstudents in the first place and partly bypreparing those students thoroughly forthe activities so that they can handlethem without too much confusion orfrustration.

This research also indicates that theacademic learning time that is mostpowerfully associated with achieve-ment gains is not mere 'time on task', oreven 'time on appropriate tasks', buttime spent being actively taught or atleast supervised by the teacher. Greaterachievement gains are seen in classesthat include frequent lessons (wholeclass or small group, depending ongrade level and subject matter) inwhich the teacher presents informationand develops concepts through lectureand demonstration, elaborates thisinformation in the feedback given fol-lowing response to recitation or discus-sion questions, prepares the studentsfor follow-up assignments by givinginstructions and working through prac-tice examples, monitors progress onthose assignments after releasing thestudents to work on them independent-ly, and follows up with appropriatefeedback and re-teaching when neces-sary. The teachers in such classes carrythe content to their students personallyrather than leaving it to the curriculummaterials to do so, although they usual-ly convey information in brief presen-tations followed by opportunities forrecitation or application rather thanthrough extended lecturing."

Generally both theory and research seem topoint to the importance of the teacher and herinteraction with the child. Materials cannot ofthemselves provide enough for the child tolearn. Increasingly the suggestion is that tea-chers adopt the methods of research and learnfrom listening to and observing children.

46

Progression in mathematical ideas

Written materials, textbooks, worksheets, com-puter programs are produced with a particularprogression in mind. The writer presupposes alogic which dictates what should precede whatother piece of mathematics teaching. The struc-ture of mathematics often dictates what mustbe considered a pre-requisite. For many yearsBloom's taxonomy provided a theoreticalunderpinning for progression in what teachersshould present to children. More recently thesuggestions of van Hiele concerning the struc-ture of Geometric progression in the schoolcurriculum has become popular in researchprojects and has been extended to other parts ofmathematics. It is quite general and the increa-sing complexity of demand is obvious. Thesedefinitions are from Shaunessy (1986):

"Level O. (Visualisation) The studentreasons about basic geometricconcepts, such as simple shapes,primarily by means of visualconsiderations of the concept as awhole without explicit regard toproperties of its components.

Level 1. (Analysis) The student reasonsabout geometric concepts bymeans of an informal analysis ofcomponent parts and attributes.Necessary properties of theconcept are established.

Level 2. (Abstraction) The student logi-cally orders the properties ofconcepts, forms abstract defini-tions, and can distinguish betweennecessity and sufficiency of a setof properties.

Level 3. (Deduction) The student rea-sons formally within the contextof a mathematical system, com-plete with undefined

terms, axioms, an under-lying logical system, definedterms and theorems.

Level 4. (Rigor) The student can com-pare systems based on differentaxioms and study various geome-tries in the absence of concretemodels."

Biggs and Collis (1982) have formulated ataxonomy to provide tasks which become structu-rally more difficult and to assess the quality of chil-dren's answers. The advantage of this is that the

child is not labelled and conveniently assignedto a level of performance as is the danger withneo-Piagetian assessment but the quality of res-ponse can be seen to vary depending on thetask.

Gagne was one of the first theoreticians toanalyse topics for the pre-requisites at eachstage (especially in mathematics). Such an ana-lysis is vital for any production of materials.

Curriculum developers have ample adviceon progression but little empirical evidence ofa teaching sequence which is more effectivethan another because of the order of presenta-tion. The research scene in education is woe-fully lacking in longitudinal surveys where thesame children are investigated as they growolder. We have details of the performance ofcohorts of children of different ages but a state-ment about the performance of a particular agegroup cannot reflect the diversity within thatgroup. Age is not a very good predictor of per-formance both because of the diversity in oneage level but also because there is ample evi-dence of young children being capable of lear-ning quite complicated mathematics in a novelsetting. Here the questions that must be askedare:(1) Although young children can learn this

topic, why should they?

(2) They learn in a research setting, but arethey going to learn with the ordinary tea-cher in the ordinary classroom?

Hierarchies in mathematics attainment

In the absence of a detailed longitudinalstudy following the same children for manyyears, the empirical evidence of progressionwe have is based on the performance of chil-dren of different ages on various mathemati-cal topics. These data take on greater signifi-cance if the questions are written according tosome postulated hierarchical demand.National and indeed international surveysmay provide information on levels of diff i c u l-ty but they are usually carried out in order tomonitor whether a representative sample ofthe nation's children are performing any diff e-rently to the representative sample tested in ad i fferent year. The testing of children using thevan Hiele levels in Geometry has given some

47

evidence of levels of difficulty in Geometry(often within the U.S. setting).

A progression of easy to hard items givesinformation on some aspects of children'sunderstanding but is of more value if there isevidence that children who can successfullydeal with items at a stated level of difficultycan also demonstrate success on those that areregarded as being at a lower level. Such acheck can be carried out on the single occasionof testing by written questions or interviews.Statistical techniques to quantify scalability areavailable. In the case of methods of solutionused by children to solve problems one caninvestigate their use at different times in thechild's school career. Karplus and Karplus(1972) was able to show that strategies used bychildren to solve Ratio and Proportion pro-blems (or in most cases fail to solve them) for-med a progression. In his longitudinal study, asthe sample grew older (and the pupils learnedmore mathematics) they moved from one stra-tegy to another. Those methods of solution thatwere abandoned were considered to be at alower level being replaced by more sophistica-ted ones.

For example he showed that children whoreasoned that one amount was an enlargementof another simply because it was bigger were,two years later, reasoning in a different fashion.Some incorrect methods used, however, werevery persistent and showed no change or asmany children moved into using them asmoved out and on to other strategies.

Concepts in Secondary Mathematics andScience

The aim of the CSMS research project (Hart,1981) was to inform teachers about what secon-dary school children found difficult in theirmathematics and what they found easy, withsome indication of why. The sample was larg e(1000 children aged 12-16) and the intention wasto provide teachers with a view of the mathema-tical performance of the vast majority of childrenover four years of their secondary schooling,rather than at 16 years of age, when they com-plete a national examination. The methodologywas to collect data from interviews and alsofrom written tests. The interviews providedinformation about how children attempted

to solve problems and about what errors theymade. They were used to inform and enhancethe written test results. Interviewing childrenone-to-one has become an increasingly popularresearch method during the last ten years. Thebelief is that a child will answer more freelyand truthfully if allowed to describe orally andin its own words what is being done. TheCSMS project formulated a set of ten subjecthierarchies in which groups of items wereshown to be scalable, in that success at harderlevels presupposed success at easier levels. Achild was deemed to have succeeded at a parti-cular level if it had correctly answeredtwo-thirds or more of the items in that level andin all easier levels.

It was apparent from the results that therewas more variation within an age group thanbetween age groups. The 14-year-olds' perfor-mance on a given item was only about five toten per cent better than that of the 13-year-olds.Nor did there appear to be any large jump inunderstanding at fifteen, although more of theolder children solved the harder items success-fully. A longitudinal survey of 600 childrenproduced evidence that certain errors commit-ted by pupils at age 12 years were likely to per-sist, and be still apparent at age 15. Two hun-dred children were tested on questions inAlgebra, Ratio and Graphs at the end of theirsecond, third and fourth years in secondaryschool.

The research is dependent on the items andis likely to be influenced by whatever mathe-matics the pupils had previously studied. Alldata are influenced by many variables - pre-vious mathematical experiences, teachers,books, socio-economic class, the items usedand the evaluation of the answers. The socialfactors were spread in the CSMS data in thatwith such a large sample chosen on the criterionof approximation to the normal distribution ofn o n-verbal I.Q. scores, the influence of any oneteacher or textbook was blurred. However, sim-ply because children are more successful oncertain mathematical problems than on othersdoes not mean that this is necessarily the bestorder for the presentation of material. In theabsence of research showing that there is a'best order', a case can be made for the premi-se: if the pupils score badly on this topic whichthey have normally been taught, then it is

48

harder than this other on which they did well,therefore in the material being written it shouldcome later. The alternative is to rely on an ana-lysis of the mathematics involved or to base theorder of presentation on intuition in whichreflects teaching experience.

The CSMS research was carried out inEnglish schools but the items have been usedby mathematics education researchers frommany other countries and appear to producevaluable information for them. Lin (1988) hasreplicated all the CSMS test papers in Taiwanwith large samples. His results tend to supportthe existence of the levels and show no greatdiscrepancy although Taiwanese children tendto be very successful or fail badly on the items.Lin's interviews show that the Ta i w a n e s esample do not resort to child methods but areusually seeking in their memory for an algo-rithm they have been taught.

The performance pattern of a representati-ve sample of Taiwanese children was very dif-ferent however, perhaps reflecting the two cul-tures, as shown in Figure 1.

Fig.-1 Ratio Results UK/Taiwan

Progression in number complexity

Children's skills with number usually start withthe ability to enumerate the contents of a set,whether it is simply by repeating number wordswhilst climbing the stairs or giving a number tothe contents of a group of toys, cakes or people.It has been long recognised that fractions anddecimals are much more difficult for childrenthan the counting numbers. When there is anon-integer value in a question it is made verymuch harder, not slightly more diff i c u l t .

On the current mathematical diet provided inschools many British children reject all butwhole numbers and exist throughout theirsecondary school years trying to make sense oftheir mathematics whilst using only wholenumbers. The nature of the understanding ofFractions has occupied many mathematics edu-cation researchers over many years. The sym-bolic representation a/b hides at least seven dif-ferent interpretations, the following is providedby Kieran (1976).

His seven meanings are listed below, wehave illustrated each with an example:

1. Rational numbers are fractions which canbe compared, added, subtracted, etc.

This aspect concentrates on the meaning of afraction, usually the model employed for tea-ching is a region or piece of a rectangle orcircle. The operations on fractions emphasiserecognition, conventions and rules such as 'youcan only add two fractions if they have thesame denominators'.

2. Rational numbers are decimal fractionswhich form a natural extension (via ournumeration system) to the whole numbers.

With the advent of metrication in Britain theuse of decimal fractions has increased and frac-tions as parts of a whole (5/8, 3/14) which werecommon in imperial measures have ceased tobe important. The calculator of course employsdecimal fractions and the increased usage ofthis aid in the classroom means a greateremphasis on decimal notation.

3. Rational numbers are equivalence classesof fractions. Thus {1/2, 2/4, 3/6 ...........}and {2/3, 4/6, 6/9 .................} are rationalnumbers.

The common use of equivalence is in the com-parison of two fractions, addition and subtrac-tion, e.g. 1/2 = 2/4, if the second form is moreapposite we use that rather than the first.Children often think the fraction has changedin size and refer to 2/4 as 'bigger' because 2 and4 are bigger than 1 and 2, whereas of course thetwo ratios name exactly the same amount.

4. Rational numbers are numbers of the formp/q, where p, q are integers and q is not 0.In this form, rational numbers are 'ratio'numbers.

49

This is a large step from meaning (1) as theratio a/b is not used to label a part of a whole.The form a/b is often used in mathematics pro-blems, e.g. in enlargement where a scale factorof 2/3 cannot be seen as part of a pie/circle atall.

5. Rational numbers are multiplicative opera-tors.

Interpreted by some as a function machine,others as a sharing or partitioning e.g. 1/6 of 12.

6. Rational numbers are elements of an infini-te ordered quotient field. They are numbersof the form x = p/q where x satisfies theequation qx = p.

In school mathematics this can be viewed asusing fractions to answer questions which areimpossible within the whole number system,e.g. 3/5 or 3 = 5n.

7. Rational numbers are measures or pointson a number line.

The number line is a model often used for tea-ching mathematics, fractions can be placed onthe number line just as can whole numbers andnegative numbers. In particular, the measure-ment of length essentially uses a number line.

All these meanings are at some timeemployed in school mathematics, the emphasisgiven to each in teaching varies considerably.Achild is dependent on teaching for the realisa-tion of the use and conventions and cannot beexpected to 'discover' the meaning of what is anabstraction. The results of the CSMS research(Hart, 1981) shows that although many chil-dren can give the fraction name of a region andcan cope with the introductory ideas of thetopic, at least half the secondary school popu-lation refuses to work with fractions as num-bers, e.g. with fraction dimensions in an areaproblem. This of course restricts the mathema-tical attainment of these children since theycannot solve ratio problems, fail to see the setof values which can be taken by a letter and arevery restricted in the use of area or volume for-mulae.

Such a range of interpretations of a/b is notadequately catered for by the simple introductionthrough "a" regions of a pie which has been sub-divided into "b" pieces. The model is of littleuse to illustrate the multiplication or division of

fractions. You cannot multiply a piece of pie byanother piece and you certainly cannot enlargea photograph by multiplying by a region. Wetend to teach the region fraction names whenthe child is quite young, eight or nine years oldand the image is powerful, reinforced in thecase of 'one half' by common usage. The factthat the operation of division is complete whenone has a/b is not recognised by at least half the11-12 year olds tested by the CSMS team.

The CSMS Fractions results and subse-quent interviews pointed to the child's refusal

1) to see a/b as the result of division,

2) to accept that a/b was a number, and

3) to appreciate the equivalence of fractionswhich were often seen as equivalent andmultiples of each other at the same time.

For example in the CMF research (Hart et al.,1989) a group of 12-year-olds were taught'equivalent fractions' by their teacher.

All the six children (boys) present at thedelayed interviews, three months after the tea-ching, were asked whether 10/14 was double5/7 or equal to it. Four children declared thetwo fractions equal, appealing to the fact thattwo times five gave ten, etc. Mick thoughtsaying they were equal was a better answer butone could be double the other. Matt howeverthought that 10/14 was both equal to anddouble 5/7. His reasons were interesting in thatthey did not include multiplication:

(I: Interviewer; M: Matt)

I: So they are both true are they: 10/14ths isdouble 5/7ths and 10/14ths is equal to 5/7ths?

M: Yes.

I: What about this one? 10/14ths is more than5 / 7 t h s ?

M: That's false.

I: How do you know that?

M: Because that can't be the same as that. Ifthat is true then that would be false.

I: What about this one?

M: 10/14ths, you just add them: add that andyou get 10 and add that you get 14.

50

There is a large gap between being able to givea name to a region and being able to manipula-te fractions with operations. The fraction 'onehalf' is used by young children and seems to beaccessible much earlier than any other. The abi-lity to compute one half of a quantity or enlar-ge by a factor of two are not indicative of anunderstanding of other fractions and ratios.

Other difficult topics

Other topics which have been shown to bemore difficult than has traditionally been belie-ved are multiplication, proof, ratio and propor-tion. The CSMS results demonstrated thatmany British pupils did not use the operation ofmultiplication, replacing it with repeated addi-tion. Fischbein et al. (1985) considered 'repea-ted addition' intuitive. Children who interpretmultiplication in this way, need to see a unitrepeated, so "l/3 x 3/8" does not fit happilywithin this format. PME researchers haveshown that the nature of the multiplier is animportant factor affecting the difficulty of theproblem. Children perform better if the multi-plier is an integer, less well if it is a decimal lar-ger than 1, a multiplier which was a decimalless than 1 was even more difficult. The natureof the multiplicand has scarcely any effect butfurther recent research has investigated whe-ther the demand of 'symmetric' problems suchas "If length is x metres and the breadth ymetres, what is the area?" is greater than 'asym-metric' ones such as "one litre of milk costs xfrancs, how much for y litres?". The evidencewas clouded by pupils' use of the formula forarea and so the investigation continues.

There has, for many years, been a PMEworking group looking at the problems chil-dren have in dealing with Ratio and Proportion.Piaget stated that proportional reasoning wasappropriate to the formal operational level, it isa topic often called upon by science teachersand has been a rich source of research pro-blems. As we have seen Multiplication andFractions have both been shown to be difficultideas for children, computations associatedwith Ratio and Proportion require the use ofboth. Children as young as six years of agehave a qualitative understanding of proportionand can provide for themselves a comparisonmeasure as reported by Streefland when hedescribed a young boy stating that a cinema

poster had an inaccurate picture of a whalebecause he had seen a whale the previous yearand the whale was not that much bigger than aman.

The quantification of these ideas is muchmore difficult and the type of number involvedis important. Enlargement by an integer scalefactor is easier than by a non-integer. Ad hocmethods are used by both adults and children tosolve proportion problems. One very prevalentincorrect method known as "the incorrect addi-tion strategy" accounts for up to 40 per cent ofthe errors in some (usually geometric) pro-blems in the CSMS Ratio study. It has beenshown to occur even more often in the USA.

This strategy (referred to as "the incorrectaddition strategy") stemmed from the beliefthat enlargement could be produced by theaddition of an amount rather than by theemployment of a multiplicative method. Thechild, in this example, reasons that since thebase line had been increased by two units somust the upright be so increased.

Enlarge with a base of 5

The error was persistent over time and seemednot to be part of a continuum which eventuallybrings success. A subsequent research project"Strategies and Errors in Secondary mathema-tics (SESM) found that those in the Britishsample who consistently used this method ten-ded to replace multiplication by repeated addi-tion on items which they solved successfully.An SESM teaching experiment showed that theerror can be eradicated in enlargement pro-blems and greater success attained withshort-term intervention which addresses thebasic problems:

(i) the recognition by the child that the'method' he uses produces strange figures

(ii) the need for multiplication

(iii) facility with fraction multiplication

51

23

(iv) the possession of a technique for findingthe scale factor.

A teaching scheme which included the tea-ching of multiplication of fractions was triedwith groups of pupils known to use the "incor-rect addition strategy". There was initiallygreat success with this but on delayed post teststhe Fraction manipulation proved to be the bar-rier. A modified version of the teaching modu-le, in which the Fraction work was replaced bythe use of the calculator proved to be more suc-cessful (Hart, 1984). The intervention materialwas designed to meet the perceived problemslisted above, and when used by classroom tea-chers with intact classes resulted in the aban-donment of the incorrect addition strategy.Successful performance on Proportion pro-blems was demonstrated by about 60 per centof the total sample (n= 80) three months afterthe teaching. The classes had been chosen bytheir teachers, as likely to benefit from thematerial, a fact which was borne out by thepre-tests. Of the children classified as persis-tent users of the incorrect addition strategy, 22out of 23 had abandoned it by the time of thedelayed post test. The CSMS longitudinal dataon the other hand had shown that half of the'adders' at the age 13 years were still using thisstrategy two years later. It does seem worthw-hile to identify an error and design correctivematerial which tackles those aspects sympto-matic of the erroneous reasoning.

Other research on Ratio and Proportion hasinvestigated whether the child, when facedwith a problem such as that below, uses theratio (from Vergnaud, 1983): (xb or xa).

Richard buys 4 cakes, at 15 cents each.

How much does he have to pay?

a = 15, b = 4, M1 = number of cakes,

Fig. 2

Vergnaud refers to the ratio xb as 'Scalar'and xa as 'Function'. Other authors call them'between' and 'within'. The scalar ratio seems tobe that most favoured by pupils in differentcountries but whether this is by choice or theeffect of teaching is not known. It would seemsensible to encourage children to use both sincethis gives them greater flexibility in choosingthe computation they intend to do.

Through research we have found that inproblems involving Ratio and Proportion,

(i) the nature of the scale factor is important,often methods other than the algorithm canbe used,

(ii) some errors are very common but can becorrected by intervention and

(iii)'within' and 'between' ideas are not equallyaccessible.

All such considerations should be taken intoaccount in the writing of materials. Firstly theexamples given to the child for solution shouldbe such that the 'within' view sometimesappears to be the most natural and on otheroccasions the 'between' numbers being compa-red are the easiest. One should certainly avoidexamples which require only doubling.Multiplication is such a neglected operation itseems sensible to deal with it again at seconda-ry level regarding it as 'related' to division andnot addition and viewed as the total of an arrayrather than repeated collections. Generally aneffort must be made to convince children thatthe method for solution taught by a teacher issupposed to cover numerous cases and begeneralisable. The project 'Children'sMathematical Frameworks' (CMF) and subse-quent research reported at PME in 1989 havemonitored teachers in their own classrooms. Atno time did the teachers in the sample (n = 36)tell the children why a generalisable methodwas important. It seems sensible for a teacher'to sell' what is very much more powerful in thecontext of a problem which cannot be solvedby more naive or ad hoc strategies. It is oftenthe case that a very powerful tool is introducedto solve a problem for which the child can seean answer immediately and being sensiblerejects the teachers' method.

52

From concrete materials to formalisation

Young children, particularly those under aboutthe age of 11 years, very often experience inschool a more practical or applied type ofmathematics than their older brothers and sis-ters. Influenced by the theories of Piaget, tea-chers in many countries have been advised tobase mathematics for young children on the useof concrete materials (manipulatives). Suchmanipulatives can be put to many uses. Theycan be used to provide the base from whichconcepts are developed. For example, to knowwhat is meant by the word 'triangle' one needsto have seen, touched and explored a lot of tri-angles. Manipulatives can be the essence of aproblem, for example real or plastic moneybeing used for shopping. One use of manipula-tives which has been advocated over a numberof years is the provision of well-structuredexperiences which lead to the 'discovery'(albeit guided discovery) of a generalisation orrule. This can then become part of the repertoi-re of the child as he moves into the more abs-tract or formal type of mathematics commonlyassumed and built on in the secondary school.An example of this type of teaching is when thechild is working towards the acquisition of theformula for finding the area of a rectangle. Theclassroom experiences recommended by manytexts and teacher trainers include:

1 ) Covering space with various two dimensio-nal shapes in order to be able to quantifyhow much areas there is.

2) Covering rectangles with squares.

3) Moving squares around to form differentlyshaped rectangles.

4) Drawing rectangles on squared paper.

These activities are supposed to be enoughto prompt the recognition that the area of anyrectangle can be found by multiplying the num-ber of units in the length by the number in thewidth. It is this rule, often written A=l x b,which is carried into the secondary school andwhich is then used whenever the area of a rec-tangle is sought.

The assumption that all secondary-agedchildren have available a 'formal' methodof finding the area of a triangle can

be shown to be false by the following examplefrom C.S.M.S.

Find the area

12+ 13+ 14+a): 76 % 87 % 91 %b): 31 % 39 % 48 %

Fig. 3 Area examples CSMS

The pupils tested were 986 in number andcame from different British schools. These twoquestions were attempted by the same childrenon the same occasion. The formula for the areaof a rectangle would have been taught in theirprimary school mathematics and most of thechildren would have met the extension to thearea of a triangle in their lessons.

The theory is that the child has a firmerunderstanding of the rule because s/he has seenfrom where it came and because the discoverywas his/hers, the idea will be retained. Thisview fits well with the constructivist philoso-phy. Many children throughout the world havelearned (or failed to learn) in a different envi-ronment as Dorfler (PME, 1989) says

Mathematics for many students nevergets their own activity, it remainssomething which others have done(who really know how to do it) anddevised and which can only be imitated(for instance by the help of automizedalgorithmic routines at which the stu-dent works more like a machine thanlike a conscious human being). In otherwords, mathematics mostly is not partof the personal experience and the rea-son for this very likely is that themathematical knowledge of the stu-dents was not (or only in an insufficientway) the result of structuring and orga-nising their own experience.

'Guided discovery' seems an attractivealternative, however, there are drawbackswhen the ordinary teacher puts theory andinterpreted theory into practice in the

53

classroom. There is also a shortage of detailedand long term monitoring of most forms of tea-ching, to see whether those that are attractive inthe abstract are effective with teachers and chil-dren as they are and not as we might wishthem. Evidence of the lack of effective learninghas become available from the research project'Children's Mathematical Frameworks' (Hart etal., 1989). In this research, volunteer teacherswere asked to prepare a scheme of work for theteaching of a rule or formula, in which thepupils started with concrete material and theformulation was the synthesis of the practicalwork. Six children in each class were intervie-wed by the researchers

(i) before the teaching started,

(ii) after the concrete experiences and justbefore a lesson(s) in which the pupils wereexpected to 'discover' or be guided to 'dis-cover' the formula, 0

(iii)just after this lesson(s) and then threemonths later.

The interviews were to find the child's know-ledge of the basic concept, whether he wasabout to discover (or had just discovered) theformula and then whether he retained for futu-re use a method with concrete materials or aknown algorithm. The form of the child'sknowledge is important since if given taskswhich assume too high a level of sophisticationthen he is doomed to fail.

The topics investigated were: EquivalentFractions, The Subtraction A l g o r i t h m ,Formulae for the Area of a rectangle, theVolume of a cuboid and that for the circumfe-rence of a circle as well as the solution ofsimple algebraic equations, all with childrenaged 8-13 years. The data were collected in theform of audiotape recordings and the resultsshowed that the transition from concrete mate-rial to formal mathematics was by no meanseasy or straightforward. Very often the concre-te materials did not mirror the mathematics andimperfectly modelled the ideas involved.Sometimes they were sufficient for the child tocarry out a computation in a "concrete" mannerbut did not in any way lead to the rule or algo-rithm intended as shown here:

Fig. 4

The child Anne used 'Unifix' blocks tocarry out the subtraction 65-29. Anne said"Then I would have taken two tens and anotherten, but on the other ten, the third ten, I wouldhave taken one unit off, so I would have 29, oh!- - - oh yes, so then I would take the 29 awayand add up all the others and see how much isleft. And its 36."

This is an effective way of carrying out asubtraction using bricks but such a methoddoes not lead to the algorithm (which was theintention). This rule requires the child to startwith the 'ones' and to decompose 'a ten' into'ones' thus:

T O

6 5 9 from 5 we cannot 'do'.

- 2 9 Go to the tens, change one ofthese into ten onesand collect these with the 5

____ ones.

3 6 Take 9 from 15, leaving 6ones.

_____ Take 2 from 5 tens, leaving 3tens.

The primary school teacher who taught'equivalent fractions' used Cuisenaire rods, discs,a fraction board and the children drew and cutout circular discs. The secondary teachers, in thesame investigation, used diagrams of regions. Ineach case the 'family' of fractions was builtaround the factors of 12 and the number of partsin a region was dictated by the teacher. T h e r ewas no general method that a child could adoptin order to provide himself with equivalentregions. Indeed, it is difficult to see how anybo-dy can choose the appropriate number of discs orbricks with which to work unless one alreadyknows the equivalent fractions. Asked to

54

use regions or discs to find an equivalent for3/7, how is the child to decide on the number ofdiscs to take from the box unless the rule isalready known? Terence in this interchangetries to use diagrams as he has seen his teacherdo.

a)

b)

Fig. 5 The drawings of Terence

I: You've drawn 12 circles.

T: I marked in 6/12. Three-eighths ... 1, 2, 3,4, 5, 6, 7, 8. So the whole one would be inbetween. These two are not the same.

I: They're not the same, right. 3/8 and 6/12 ...we've knocked out because ... now what didyou do? I have to tell the machine what youdid. You put your hand over the last four ofthose circles.

T: Yes, because there's 12 here, but if I put my... hand over there to show that was 8 and3/8 would be there and it isn't the same asthat [pointing to 6 circles]. Try 6/11, is thesame (counts from left hand side of circlesand gets to same point).

I: As what ...

T: Er ... 6/12.

A common exhortation of teachers, seenduring the research, was to those pupils whofound it difficult to find and remember the ruleor algorithm. They were told to return to theconcrete materials. This however requires themto invent the modelling procedure unaided.Urged to take out the bricks 'to help', those chil-dren who did so simply used them as objectsfor counting.

A significant part of the belief in this type ofteaching is that the child's understanding is stron-ger because the discovery is his. Three monthsafter the teaching the children in the CMF study,

when asked how they had come to know a for-mula, said that their teacher had told them orthat a clever person had invented it. They didnot say 'we discovered it'. They seemed tomake no connection between the work withconcrete materials and the formula they used incalculations. If one considers the most impor-tant features of a rod (a commonly used mani-pulative), they are: the material of which it ismade (wood or plastic), its colour, its weight,length or volume. To the teacher, who is usingthis rod for a particular mathematical purpose,its outstanding quality may be that when putend to end with another the two are the samelength as a third. The mismatch between thetwo conceptions is obvious.

Further research built on the CMF results,has pursued the idea that since the materialsand mathematical formalisation were so diffe-rent in nature, a transitional phase (a bridge)was needed. This 'bridge' could be in the formof diagrams, tables, graphs or discussion andshould be distinct from both types of experien-ce but linking them. More forms of the 'bridge'are being tried but the first results were disap-pointing because often the teachers who wereasked to put 'bridge' activities into their tea-ching gave very little time to them (five or sixminutes) or in some cases put this activity afterthe formalisation.

Concrete materials are of course used byteachers for other purposes. Teachers of youn-ger children very often use manipulatives toconvey the basic principles of a topic. Researchresults from the early 70's for example tendedto show that the idea of a fraction being aregion could be effectively taught when chil-dren cut and folded rectangles of paper.

Teaching experiments

The philosophic and theoretic frameworksadopted by most researchers within PMEhave led to an interest in monitoring 'pro-cesses' and problem-solving skills rather thaninvestigating performance in computation.Usually many variables are at work in theclassroom and it is difficult to apportione ffectiveness to the efforts of the teacher, thenature of the material (text, concrete mate-rials or computer) or the fact that the contentsare process-orientated. Harrison et al. (PME,1980) reported on teaching experiments in

55

Calgary in which 'process' enriched material onFractions was taught to 12-13 year olds. Theteaching took about eleven and a half weeksand control groups were

taught from the current textbooks in the usualmanner. The results are shown in fig. 6 above.

The experimental groups perform slightlybetter than the control groups at the end of theteaching. The enthusiasm of the children wasgreater after working with the process-enrichedmaterial, however. Perhaps what is more signi-ficant is how little gain there is after the tea-ching. If the teacher's best efforts over 11 1/2weeks still leaves the children performing atonly 60 per cent success level, perhaps weshould reconsider what we expect of the pupils.Research teaching experiments are of necessityshort-term and usually for less time than theCalgary study and often the teaching matter isgiven to children without consideration of theirprevious knowledge of pre-requisite ideas.Before the topic is taught the pre-test is admi-nistered in order to find out how much of thespecific teaching points they already know.This is very seldom nil. The pupils usuallyhave quite a lot of knowledge to start with butseem to acquire less than expected during afocussed teaching sequence. A w o r t h-w h i l elong term study which could be carried out byteachers in their own classrooms is that of ana-lysing exactly which sections of their teachingare successful. Do they always get poor resultswhen they teach addition of fractions and

always good results when they teach congruen-ce? Is this pattern of success the same for otherteachers? Is it the teacher or the topic which isthe crucial variable?

Should we not be more concerned about mat-ching the material to the child's level of know-ledge?

The computer

The computer used as a teaching aid in theclassroom has gained considerable prominencesince 1980, the emphasis in research being onhow children's thinking skills or deeper unders-tanding have been improved. Tall and Thomas(PME, 1988) for example quoted a study inwhich children aged 12-13 years were given a"dynamic algebra module". This provided a"maths machine" which evaluated formulae fornumerical values of the variables. The teacher,however, was a vital component of the work.Results tended to show that the Control groupperformed better on skill based questions, forexample:

Simplify 3a + 4 + aExp. 38% Control 78%

On questions which required higher levels ofunderstanding the experimental group perfor-med better, for example:

The perimeter of a rectangle 5 by F.Exp. 50%. C. 29%.

Again we see that the teaching experiment hasproduced success but half of the children arestill not successful.

56

Fig. 6 Teaching Experiment (Harrison et al., PME, 1980)

The use of LOGO in classroom computerwork has been increasingly reported. T h emicroworlds are often used as a basis for dis-cussion of what children say and the interpreta-tion they place on situations. Comparison ofperformance between children in a LOGOenvironment and those in a control group sel-dom provides an overwhelming success on theside of the experimental group. Often closerscrutiny of individual marks shows a greaterdegree of progress for some of the LOGOgroup. Hoyles (PME, 1987) in a review, ofPME presented papers described three areas ofresearch within the context of 'Geometry andthe Computer Environment'.

I. Research which explores thedevelopment of childrenís unders-tandings of geometrical and spa-tial meanings and how progres-sion (for example, from globalityto increased differentiation) mightbe affected by computer "treat-ments".

II. Research which investigates the"training" influence of computerenvironments on different spatialabilities.

III: Research which takes as a startingpoint the design of geometriccomputer-based situations whichconfront the students with specific"obstacles" and seeks to identifystudent/ computer strategies, themeanings students construct andhow these meanings relate to therepresentations made available forthe computer "tools". Such resear-ch more or less explicitly uses thecomputer to create didactical toolsto facilitate the acquisition of spe-cific mathematical conceptions orunderstandings."

More recently Hoyles' group have beenreporting on particular topic work carried outwith small groups of children, for example theconcept of a parallelogram. It seems possiblethat children are assisted in learning by the useof the computer but it is not a panacea.

Illustrations and their use

Textbooks used in mathematics lessons arenow extensively illustrated with pictures,diagrams and graphs. It is worthwhileconsidering why we use these devices andwhat benefit they are to the children. Do

teachers buy colourful, heavily illustratedbooks because their pupils learn from themmore effectively or because they, the teachers,like the look of them?

By 'illustration' we mean non-word mate-rial. This includes various categories:

1) Line diagrams which are:

a ) the objects themselves - atriangle;

b) representing an object so a conventionis involved;

c) representing actions or steps, such as aflow diagram;

d) representing by a picture.

2) Pictures which show reality as seen by theartist.

3) Photographs.

4) Recording diagrams, tables, graphs,matrices, Venn diagrams.

(The children will be required to draw thesethemselves later.)

The illustrations can be used:

1) To convey information which is not bestconveyed in any other way.

2) To present a problem in non-verbal form.

3) To convey information when the recipientcannot have access to knowledge throughwords, for example those who cannot read.

4) To provide a focus for discussion so thatthe children can all give their views aboutthe same thing. (It can also be used toassess).

5) To present the exercise in context e.g. shop-ping.

6) To form a bridge between reality and abs-traction.

7. To record data in a more effective way.

8. To make the books attractive.

9. To (perhaps) motivate.

Children do not automatically realise how touse a diagram or what its intended message is. Itsspecial features need to be taught just as otheraspects of mathematics need to be taught. The dis-tinction between diagrams which show a relationship(such as a graph) and a picture which represents

57

reality, is lost on many children. For exampletime-distance graphs such as that shown infigure 7 are interpreted not as representative ofa journey but as showing hills and plateaus.This was so in about 14 per cent of cases in theCSMS study. (Sample n = 1396)

Fig. 7

Often teachers use a diagram to add weightto an argument. Thus when teaching aboutfractions they are likely to draw, freehand onthe blackboard, a circle divided into sections.The purpose of this is difficult to see, since ifthe pupils were not already knowledgeable offractions and their equivalents, the diagramwould confuse rather than help. For example,unless drawn accurately it cannot provide evi-dence that 3/8 = 6/16.

The CMF researchers found that childrenprovided the same inaccurate diagrams whentrying to convince the interviewer of equiva-lence, as seen in this effort of Joe.

a) Matt's diagram for 3/4 = 6/8

b) Joe's diagram for 3/5 + 7/10 = 10/15

Fig 8

Textbooks are nowadays often lavishlyillustrated, yet there is scarcely any research onthe effect these pictures have on the learning ofthe mathematics. Shuard and Rothery (1984)provided a review of the existing research onreading mathematics. In so doing they analysedpages from textbooks where information waspartly given in the picture, partly in the text andsometimes on lists which appeared beside thepicture. For example in shopping questionswhich are essentially testing addition, there arepictures of the articles which can be bought,perhaps labelled with prices, a shopping listand then a sentence or two in the text. The eyemovement needed to obtain the necessaryinformation is excessive.

There is some research in science educa-tion on the use of illustration in text books andother types of teaching material but very littlein mathematics. Much of Geometry is said tobe concerned with the improvement of chil-dren's spatial visualisation, yet little of it posi-tively addresses this issue. There has beensome recent research, resulting in books forchildren, which attempts to improve pupils'view of three dimensional objects and theirinterpretation in two dimensions. One studywas part of 'The Middle Grades MathematicsProject' from Michigan State University(Winter et al., 1986).

The Spatial visualization instructionalmaterial includes ten carefully sequencedactivities. The activities involve

58

representing three-dimensional objects (buil-dings made from cubes) in two-dimensionaldrawings and vice versa, constructingt h r e e-dimensional objects from theirtwo-dimensional representations. Two differentrepresentation schemes are used for the twodimensional drawings. First an "architectural"scheme involving three flat views of the buil-ding-base, front view and right view. After stu-dents are comfortable with this scheme, theyare introduced to isometric dot paper and henceto a representation consisting of a drawing ofwhat one sees looking at a building from a cor-n e r. Similarities, differences, strengths andinadequacies of the two schemes are explored.

It was concluded from the analysis of thepre-post data that after the instruction:

(1) sixth, seventh, and eighth grade boys andgirls performed significantly higher on thespatial visualisation test; however, no chan-ge in attitudes toward mathematics occur-red;

(2) boys and girls gained similarly from theinstruction, in spite of initial sex diffe-rences;

(3) seventh grade students, regardless of sex,gained more from the instruction than sixthand eighth graders.

In addition, the retention of the effects ofthe instruction persisted; after a four-week per-iod, boys and girls performed higher on thespatial visualisation test than on the post test.

Conclusion

We now know, through research in mathe-matics education, that certain topics are diffi-cult for children to learn because they containmany different aspects each of which brings itsown degree of complexity so that the total iscomposed of layers of difficulty. Proportion issuch a topic. It seems wise to write materialswhich build up the different understandingsrather than produce the topic all at once. It isalso wise to illustrate and give examples whichrequire the use of the idea or method and can-not be easily solved by lesser and more naivestrategies. There is then an incentive for thechild to take on the more powerful tool.

59

Kathleen Hart is visiting professor of Mathematics Education at King's College, London, England.She directs Nuffield Secondary Mathematics which is a curriculum development project producingmathematics materials for the age range 11-16 years. Her mathematics education career started as ateacher, then she was a teacher-trainer for ten years, a researcher for ten years, and then just prior totaking the post of director of the Nuffield project she was an inspector of schools. She has two doc-torates in mathematics education, one from Indiana, and the other from London. Her first degree isin mathematics, and she also has a London M.Phil obtained from research with seven year-olds whowere said to have a mental block against mathematics.

A founder member of PME, Dr. Hart is currently its President.

There is no doubt that the teacher plays a majorpart in creating the environment in which chil-dren can best learn mathematics. This is not toclaim too much for the role of the teacher.Stronger claims have been made, for examplethe linguistic philosophers of education wouldwant to say that unless learning is taking placeone cannot be said to be teaching. Certainlygovernments in many countries count teachersas solely responsible if there is evidence ofchildren not learning. Thus the statement thatbegins this chapter is relatively minimal anduncontroversial. Nevertheless, there are manyissues to be investigated, and the questionsbegin when one attempts to establish what partthe teacher plays, what is the best environmentand how can one create it, and what teachersunderstand about children's learning of mathe-matics. In this context, therefore, it is interes-ting that it is only in recent years that one findsa growing body of research that focuses on theteacher. For example, at the Tenth Meeting ofPME in 1986, there were no more than a smallhandful of research reports concerning tea-chers, but by the Fourteenth Meeting in 1990there were three Working Groups and aDiscussion Group, and at least 20 researchreports.

There are perhaps two major reasons forthe relatively late interest in the role of the tea-c h e r. The first is the dominance of the notionthat if we can specify how children learn, andproduce the most appropriate learning mate-rials informed by that knowledge the teacher'srole becomes largely that of an intermediary inthe process of transmission. If, further, we canspecify what the classroom should look like,how long children should study, and variousa ffective factors, we can tell the teacher whatto do to achieve the goal of children learningmathematics. Many would claim that a para-digm which incorporates these is no longertenable. The transmission metaphor is nolonger seen as adequate, and, indeed, the

nature of the learning process is seen as lessand less determinate. The affective and thecognitive may not be simply separable catego-ries as was previously claimed, and this has ledto research on the social context of the class-room which has considerably confused andenriched our observations and ideas. In all ofthese issues, the teacher is a central figure.

The second reason that research on tea-chers has only recently playing a significantpart on the agenda, is the nature of that resear-ch. This is an issue that is the subject ofon-going debate (Scott-Hodgetts, in press, b).Most research questions that are concernedwith teachers will not be analysable by signifi-cance tests. As Mason and Davis (PME, 1989)suggest, "As with our previous work, validityin our study lies, for us, in the extent to whichit resonates with experience, and to which itawakens awareness of issues which they mightotherwise have overlooked" (p.280-281).

There are some studies that draw on reaso-nably sized populations for quantitative analy-sis, such as those examining pre-service tea-chers' knowledge of aspects of mathematics(e.g. Vinner and Linchevski, PME, 1988). Inthe main, however, the research methods mostused are qualitative, based often on only asmall number of teachers (Jaworski, PME,1988, Lerman and Scott-Hodgetts, PME, 1991,in press), and the methodology and generali-sable outcomes of such research are not accep-ted by all in the mathematics education com-munity.

This is a theme which will arise frequently inthis chapter, namely the kinds of methods that aremost informative for research on the teacher'srole. Two aspects in particular will be emphasi-sed, firstly that a holistic perspective of interac-tions in the classroom is more fruitful than a frag-mented one, and secondly, that case studies, per-haps stimulated by findings from quantitative

61

The Role of the Teacher in Children's Learning of Mathematics

Stephen Lerman

studies, provide information that resonateswidely with teachers and other researchers.

Expectations of the fruitfulness of researchon teaching vary considerably. In a recent volu-me (Grouws et al., 1988) one finds both thepositivist expectations of Grouws: "No longercan we uncritically rely on the folklore of theclassroom; rather we must sort out fact frommyth and begin to build conceptual networksthat will help us to understand and improve thecomplex process of teaching" (Grouws, 1988p.1), as well as the sceptical thoughts ofBauersfeld: "... there seems no prospect of arri-ving at universal or overall theory of teaching-learning processes that we will all accept"(Bauersfeld, 1988 p.27).

However, as can be seen by the growinginterest in the role of the mathematics teacherin research reports at PME meetings, theresearch community is now engaging withthese issues, and the themes which will bereviewed in this chapter will indicate some ofthe important ideas emerging.

Following this introduction, the chapter isdivided into four sections. In the first sectionwe will examine some aspects of the teacher'swork, including notions of effective teaching,the teaching of problem-solving processes andthe role of meta-cognition, the power relation-ships in the classroom, some remarks on text-books, and on what constructivism, an alterna-tive theory of the way people learn, mightimply for teaching. As with the remainder ofthe chapter, the review will draw, in the main,on research reported at PME meetings as wellas on other widely available literature.

One of the major themes that emerg e sfrom the review is the significance of tea-chers' beliefs about mathematics, and aboutmathematics teaching, and the influences ofschool and society on teachers' actions in theclassroom. In some senses, this theme is fun-damental. Whether one is considering socie-ty's influences on the classroom, textbooks orother teaching materials, cultural influences,curriculum change, technology or whatever,they will all be mediated through the teacher,and specifically through the teacher's beliefsabout her role in her students' learning of

mathematics. Work in this area will be revie-wed in the second section.

At the same time, the teacher is an actor ina particular setting, within which the relation-ships and dynamics are constructed by theactors, but in turn the actors and their roles areconstructed by the classroom relationships anddynamics. Therefore, in the third section wewill look at the teaching and learning of mathe-matics in the context of the language and prac-tices of the classroom.

In the final section, we review research onteacher education, including action research,and teachers' knowledge of mathematics. Theconcluding remarks will return to the issue ofthe nature and role of research on teaching, andwill attempt to highlight major themes for futu-re research.

Aspects of the work of the mathematics tea-cher

Effective teaching

It may seem that the most appropriate place tobegin any survey of the actions of the teacher isto attempt to characterise what is effective tea-ching, as well as which aspects of teachingmake the teacher ineffective. If this is possible,one could design pre-service and in-servicecourses so as to bring about development andchange in those areas. Albert and Friedlander(PME, 1988) reported on an in-service coursefocused on counselling, and commenced theirpaper with an analysis of the flaws in unsuc-cessful teaching and a definition of effectiveteaching, the latter including "good class mana-gement, clear and correct mathematicalcontent, well-planned lessons resulting in afeeling of learning, good learning environment..." etc. (p.103). There is the danger that an ana-lysis of the actions that identify an effectiveteacher can become merely a list of desirableattributes, without any real evidence that suchattributes contribute significantly to school stu-dents' learning of mathematics. Also, phrasessuch as "good learning environment" are usedwhich themselves require elaboration.

Some researchers have thought it moreappropriate to make one's characterisationmore general, and to offer it in terms of broadgoals. Peterson (1988), in discussing teachingfor higher-order thinking, offers such a list:

62

"(a) a focus on meaning and unders-tanding mathematics and on thelearning task;

b) encouragement of student autono-my, independence, self-directionand persistence in learning, and

(c) teaching of higher-level cognitiveprocesses and strategies" (p.21)

and in the same volume Berliner et al. (1988)go even further in broadening the characterisa-tion of effective teaching, almost to the point ofstating the obvious:

"... there are two separate domains ofknowledge that require blending inorder for expertise in teaching to occur.These are (1) subject matter knowledgeand.(2) knowledge of classroom orga-nisation and management, which wecall pedagogic knowledge." (p.92)

It may be a more fruitful approach to focus onwhat happens in the classroom, in the contextof the goals set. Hanna (PME, 1987)approaches the problem by attempting to iden-tify those teaching strategies "that seemed tohave contributed most to greater achievementgains" (p.274) amongst eighth grade students,and concludes that these were:

"(a) an extremely organised approachto teaching, wherein material istaught until the teacher feels it ismastered, thus reducing the needfor frequent review, and

b) an approach in which every pre-sentation of material is followedby extensive practice in applyingthe new material to new situa-tions." (p.274)

Steinbring and Bromme (PME, 1988) havedeveloped a technique to monitor the processof knowledge development in the classroom,and present this pattern graphically, enablingcomparisons over time, student group, teachersetc. Kreith (1989) presents the idea of masterteachers acting as role models and workingwith intending teachers as interns, an issue towhich we will return when considering teachereducation.

There are dangers of hidden assumptionswhen one focuses on the effective teacher, or

the master teacher. Nickson (1988), forexample, suggests that "the notion of the'expert' is a contentious one and means diffe-rent things to different people." (p.247) Onecannot talk about 'effective' without determi-ning what it is that one values in the teaching ofmathematics. Classroom management, forexample, is one kind of notion when one is tea-ching a piece of content, in a transmissionmodel, and quite another when the focus is ongroup problem-posing work. Again it meansquite different things if one is teaching a classof 60, with little or no resources, and quite ano-ther with a group of 20 in a well-resourcedsituation.

We may also, in the end, have doubts aboutthe value of breaking down classroom activi-ties into teachers' actions, students' actions andaffective conditions, when what is taking placeis a complex process of interactions, intentions,relationships of power, expectations, and so on(Porter et al., 1988). Bishop and Goffree (1986)make a similar point when they argue for a shiftfrom what they call the 'lesson frame' to the'social construction frame'. The former is cha-racterised as follows:

"(the mathematics lesson) is construedas an 'event' with a definite beginning,an elaboration and a definite end. It hasa fixed tinge duration. Typically allchildren will be engaged in the sameactivities which are planned, initiatedand controlled by the teacher." (p.311)

They indicate that research, dissatisfactionwith outcomes, and alternative constructivisttheories of the way children learn have led to awidening of the frame:

"This orientation (social constructionframe) views mathematics classroomteaching as controlling the organisationand dynamics of the classroom for thepurposes of sharing and developingmathematical meaning." (p.314)

They include in their list of features of theframe notions such as "(2) it emphasises thedynamic and interactive nature of teaching;...(4) it recognises the 'shared' idea of knowingand knowledge, reflecting the importance ofboth content and context;" (p.314).

63

Studies based in the classroom provide per-haps the most fruitful approach for research inthe area of effective teaching, since an essentialcomponent is the recognition of thetheory-laden nature of describing an effectiveteacher, and thus of the need to build suitableresearch methods in the light of this.

For instance, the teacher might develop hernotion of student achievement, and criteria ofsuccessful teaching, relevant to the specificcontext. Through action research, or a collabo-rating observer in the classroom, teacher strate-gies that appear to help the students and theteacher to realise that achievement might thenbe identified. What is important here is that'desirable attributes' become something speci-fic to the particular teacher, in relation to herown values, and change is seen as an indivi-dual's progress in her own terms, rather thansomething that teacher-educators or resear-chers do to others.

An illustration of this approach can be seenin the following extract from the reflections ofa teacher, who had been critically examininghis teaching in the light of some reading he haddone, and some subsequent observations of hisstudents (Lerman and Scott-Hodgetts, PME,1991, in press):

"So my reflections led me to this point.I began to believe that if the didacticapproach was not good enough forClaire, it was not good enough for anypupil; that instruction and explanationby the teacher had its place in outliningthe problem, but not somehow givingmathematical ideas."

Another perspective on the issue of effecti-ve teaching, is given by Scott-Hodgetts (1984),(discussed also in Hoyles et al., 1985), in apaper that describes her research on childrens'ideas of what is a good teacher. In apaired-comparison test of characteristics thatgo to make a perfect teacher, the ranking emer-ged as follows:

"is able to explain the maths clearly andthoroughly;

has a patient and understanding attitudewhen people find the work difficult;

is prepared to spend a lot of time withpeople who need more help;

has a relaxed friendly relationship withthe pupils;

keeps up to date with each pupil's work,and lets them know how they are get-ting on;

makes the lesson interesting andenjoyable;

knows the subject really well;

has a good sense of humour."

Alibert (PME, 1988) found that first yearuniversity students valued aspects of the tea-chers' work that differ from those that the tea-chers might value. In her study, 52% of the stu-dents considered that a good mathematics tea-cher is first "interesting, convincing, clear,quiet" and in only 11% of answers were theresuch things as "helps students to reflect, to par-ticipate" (p.114). Two of the most interestingaspects of these studies are the following: first,students emphasise quite different aspects of'good' teaching than teachers themselves mightemphasise, and this information could be avaluable stimulus for reflection and research byteachers, and second, that the hidden messagesof one's beliefs are often conveyed to students,and reflected in what they value most in theirteacher.

M e t a-cognition and teaching mathematical'processes '

One of the major changes in the role of theteacher during the last decade or so, has beenbrought about by the shifting of the focus ofattention to the introduction of problem-sol-ving work into the classroom. It has raisedmany questions and engendered many debates.An early question was: what are problems, andhow do they differ from puzzles and investiga-tions? In fact we often use all three wordsduring any particular piece of work. We maystart by being interested in a problem, andbegin to investigate it. On the way we solveseveral puzzles, become stuck on many more,and generate new problems to investigate!D i fferent interpretations of the term 'pro-blem-solving' remain, however, as shown inthe study described below.

The issue of 'process' versus 'content' hasbeen discussed, in the literature and at confe-rences, and we have become aware that

64

we are at least partly discussing the philosophyof mathematics. To some extent, for the forma-lists, mathematics is its content and its structu-re, whereas for the quasi-empiricists( Tymoczko, 1985) mathematics is its pro-cesses. In recent years we have been develo-ping ways of assessing students' work on inves-tigations, in particular in Britain where schoolstudents have to submit a number of individualcourse works for the General Certificate ofSecondary Education (GCSE), a national exa-mination for 16 year olds. There is still, howe-ver, the issue of how one avoids assessment-ledinvestigation work in classrooms (Lerman,1989a), that is whether the potential for creati-vity and engagement disappear under the needto have problems whose answers can be 'mar-ked'.

Problem-solving has led on to problem-posing (Brown, 1984), and opened up the acti-vity to the wide and stimulating possibilities ofethnomathematics, empowerment and criticalmathematics (Frankenstein, 1990). But wheredoes it fit in to the school mathematics curricu-lum? Is it to be added on at times, or can oneteach the main part of the curriculum throughsuch work? The nature of mathematics is theessence of the problem here, and it findsexpression through the process/content debate.All these questions are the foci of currentresearch work in mathematics education, andwe will examine some of them here, as theyimpinge on the role of the teacher.

In a study of teachers' ideas of what is pro-blem-solving and how to teach it, Grouws et al.(PME, 1990) found that teachers were usingfour categories of meanings:

"(1) P r o b l e m-solving is word pro-blems;

(2) P r o b l e m-solving is finding thesolutions to problems;

(3) Problem-solving is solving practi-cal problems; and

(4) Problem-solving is solving thin-king problems." (p.136)

In their survey of 25 teachers, they foundthat 6 identified with the first definition, 10with the second, 3 with the third and 6 thefourth. However, their lesson goals and rela-ted instructional methods did not relate

to their definitions, and neither did the formatof their lessons. Teachers felt that there was notenough time for problem-solving instruction,standardised testing being a major interference.Students had low success and little self-confi-dence. Grouws et al. conclude:

"We now know that we must carefullydescribe what is meant when a teachergives critical importance toproblem-solving and its instruction inher classroom." (p.142)

To some extent, aspects of the study areprobably culture-specific since in many coun-tries problem-solving would not be consideredto mean word problems. Teachers may wellrespond by offering the other definitions,however, and many of the other findings maywell be supported widely. The final commentsof Grouws et al. are an important caution toresearchers and a reminder of the influence ofteachers' beliefs about mathematics and mathe-matics teaching. One must also take intoaccount the tendency to respond to certain jar-gon words, particularly 'problem-s o l v i n g ' ,which has been a major theme of the 1980's. Inthis context it is therefore particularly signifi-cant to find both that teachers have quite diffe-rent ideas of what it is, and that their teachingoften bears little relationship to those ideas.

Flener (PME, 1990) reported on a study ofthe consistency among teachers in evaluatingsolutions to mathematical problems, and oftheir reaction to solutions showing insight.Hypothetical students' solutions to mathemati-cal problems were sent to 2200 teachers, ofwhom 446 responded. Flener found a widerange of evaluations, that teachers creditedmethods that were taught in school and not pro-blem-solving ability at all, and in general tea-chers gave no credit for insight or creativesolutions. M o rgan (1991), presented experien-ced teachers with three hypothetical solutionsto a particular problem: the first, (labelled PV)used diagrams and full verbal description,using spoken rather than written language; thesecond (ND) used numbers and diagrams anda table, with text that presented the work inthe order in which it was done, and the third(BS) was concise and symbolic, with words

65

only as labels, without sentence structure. Allthe teachers placed ND the highest, and mostplaced PV second. This is in marked contrastwith the usual presentation of pure mathema-tics in academic journals, and suggests that tea-chers are developing particular and specificways of assessing school mathematics pro-blem-solving. She suggests that teachers aredeveloping criteria for what constitutes schoolmathematics, rather than mathematics in gene-ral, and that these criteria are not made expliciteither amongst teachers or to school students.

Again, we are led to consider what teachersunderstand by mathematics, in the context ofproblem-solving. This issue will be discussedfurther below, but whatever teachers andresearchers understand by problem-solving, amajor interest for mathematics educators is thequestion of whether problem-solving skills canbe taught. There are two inter-related and inter-acting levels to be considered, heuristicmethods and meta-cognition. Hirabayashi andShigematsu (PME, 1986, PME, 1987, PME,1988) have developed a powerful metaphor formeta-cognition: that of the Inner Teacher. Theysuggest that the teacher acts as the inner self ofthe students and thus the task of teachers is toencourage the growth of students' own innerteacher. Mason and Davis (PME, 1989) indica-te how useful they find this metaphor for thin-king about the shifts in attention required inproblem-solving and in mathematics educationin general. In discussing some theoreticalaspects of teaching problem-solving, Rogalskiand Robert (PME, 1988) suggest that it is pos-sible to "design methods related to a specificconceptual field", and they go on to outlinehow and when such instruction is appropriate:

"... such methods can be taught to stu-dents as soon as they have some avai-lable knowledge and the ability tomake explicit meta-cognitive activitiesin a precise way, and to take them asobject for thought, and (3) that studentsbenefit from such a teaching.Didactical situations which appear asgood 'candidates' for supporting such amethodological strategy involve: workin small groups, open and sufficientlycomplex problems and a didacticalenvironment giving a large place to

students' meta-cognitive activities suchas discussion about knowledge andheuristics, and elicitation ofm e t a-cognitive representations onmathematics, problem-solving, on lear-ning and teaching mathematics."(p.534)

The importance of making knowledge ofheuristics and self-reflection explicit to stu-dents is a theme developed by a number ofresearchers. In a study specifically on heuris-tics teaching, van Streun (PME, 1990)concludes that "Explicit attention for heuristicmethods and gradual and limited formulatingof mathematical concepts and techniques inmathematical education achieve a higher pro-blem-solving ability than implicit attention forheuristic methods and late and little formula-ting of mathematical concepts and techniques."(p.99) Mason and Davis (PME, 1987) argue forthe introduction of particular and specific voca-bulary, such as 'being stuck' and 'specialising',in helping the process of learning mathematics.They also claim a connection between "theeffectiveness of teachers' discussions of theirteaching" and "of students' discussions of theirlearning" (p.275). This notion of the analogy ofthe teacher's reflections and the student's willbe developed in the final section of this chap-ter.

Lester and Krol (PME, 1990) examine the"relative effectiveness of various teacher rolesin promoting meta-cognitive behaviour in stu-dents and the potential value of instructioninvolving a wide range of types ofp r o b l e m-solving activities" (p.151). T h e s eroles are: the teacher as external monitor, asfacilitator and as model. They observed the tea-ching and learning of students in seventh gradeclasses, and drew the following conclusions:

"Observation 1: Control processes andawareness of cognitive processesdevelop concurrently with thedevelopment of an understandingof mathematical concepts.

Observation 2: Problem-solving ins-truction, meta-cognitive instruc-tion in particular, is likely to bemost effective when it is providedin a systematically org a n i s e dmanner, on a regular basis, andover a prolonged period of time.

66

Observation 3: In order for students toview being reflective as impor-tant, it is necessary to use evalua-tion techniques that reward suchbehaviour.

Observation 4: The specific relation-ship between teacher roles andstudent growth as problem solversremains an open question.

Observation 5: Willingness to bereflective about one'sproblem-solving is closely linkedto one's attitude and beliefs."(p.156-157)

These studies highlight many importantissues, such as how to evaluate problem-sol-ving work and how to develop the inner tea-cher. They also leave open many questions,including what teachers understand by 'pro-b l e m-solving' and how their interpretationinfluences the way they teach, and how pro-blem-solving fits into the whole task of tea-ching mathematics. Answers to these questionsare often implicit unstated assumptions of tea-chers and researchers, but they are crucial fac-tors. For example, a study designed to examinestudents' feelings about problem-s o l v i n g ,where that activity is an occasional one in theclassroom and when regular textbook-drivenmathematical work is set aside, is likely to leadto one set of responses from students. Wherethat activity is the style of learning that isusually used by the students in their learning ofmathematics, one may well find quite differentresponses. Similarly, concerning teachers' atti-tudes to problem-solving, the 'hidden mes-sages' conveyed by the teacher of the impor-tance or significance of problem-solving arepicked up by students (Lerman, 1989a).

All the studies emphasise making heuris-tics and reflection explicit to students, whichcan be expressed in a rich way as the deve-lopment of the Inner Te a c h e r. It is interestingto reflect on the role of the teacher in this. Ithas been argued (Kilpatrick, PME, 1987) thatthe constructivist view of children's learningdoes not offer anything new in the sense ofresearch methods, or teaching methods. T h i smay be a difficult point to argue when onefocuses on childrens' learning of particularcontent of mathematics, [although both sidesare argued (e.g. Kilpatrick, PME, 1987; Steff e

and Killion, PME, 1986)] but an interpretationof each student's construction of her own InnerTeacher, in its fullest sense, may be seen asmore difficult to formulate outside of aconstructivist programme. For example, if oneis considering a child learning the procedure ofadding two fractions, it could be described asthe child constructing concepts and processesthat 'fit' the situation, i.e. give the expected ans-wers, make sense in relation to other concepts,can be coherently explained to others etc., or itcould be described as the child internalisingsomething taught by the teacher and therefore'matching' an established schema. Both inter-pretations could be descriptions of what takesplace, and the relative merits of the two des-criptions could be argued. From a constructi-vist view, the 'Inner Teacher' notion will by itsvery nature have a different meaning for eachindividual; it has no explicit identity that theteacher could teach, and it will be a meaningfulnotion to the extent to which it works for theindividual. It is a very 'fuzzy' notion, involvingspecifiable aspects, such as generalising, butalso unspecifiable aspects, such as recognisingthat one is tiring, or stuck, etc. The InnerTeacher notion can of course be specified in acognitive psychology perspective, and measu-red against developing schema, as a sort of'external' teacher replaying its voice in the indi-vidual's head. However, if a teacher is conside-ring how to develop the inner teacher in herstudents, seeing learning as a constructive pro-cess may well provide a depth of meaning thatother theories do not adequately offer.

In the research literature, one does comeacross doubts about whether problem-s o l-ving methods as such can be 'taught'. Suchdoubts are expressed often by researchersfrom outside of mathematics education wheninvestigating mathematical problem-solving.The recent debate in the Journal ForResearch in Mathematical Education is anillustration of those concerns (Owen andS w e l l e r, 1989, Lawson, 1990, Sweller,1990). Owen and Sweller (1989) make a casefor the lack of evidence of the eff e c t i v e n e s sof teaching heuristics. They suggest that thefailure of students to use newly learned prin-ciples intelligently to solve problems

67

may not be due to their lack of problem-solvingstrategies, but what they call "a lack of suitableschemas or rule automation" (p.326). In anyevent, they claim that there is no evidence thatlearning problem-solving strategies actuallyhelps in further problem-solving, i.e. are trans-ferable. Lawson (1990) argues to the contrary,that there are small but growing signs of evi-dence to support the "consideration of differenttypes of problem-solving strategies ... inmathematics classes" (p.409).

In his reply, Sweller (1990) reveals whatmay be the essence of the differences expressedin the debate, in reviewing the work of Charlesand Lester (l984). Sweller is looking for thetransfer of problem-solving strategies, andwhere researchers such as Charles and Lesterhave found enhanced problem-solving abilityafter instruction, but have not looked forSweller's interpretation of transfer, he consi-ders his case reinforced. The issue seems to bewhat is meant by domain-specific knowledgeand skills, and transferability. We would proba-bly argue that the strategies Charles and Lesterworked upon, "such as trying simple cases,creating a table, drawing diagrams, looking forpatterns, or developing general rules" (Lawson,1990, p.404) are broad enough to apply to thedomain of mathematical problem-solving andat the same time focused enough to be achie-vable. Since Sweller is dissatisfied with thetransferability of these skills, one can onlyassume that he expects transfer outside ofmathematics, to everyday life, or to otherschool subjects. However, mathematics is alanguage game, with its own meanings, styles,concepts and so on. One can perhaps articulatea notion of general problem-solving skills inthis wide interpretation, but only in the sense oftask orientation strategies, or executive strate-gies (Lawson, 1990, p.404). We would supportLawson in claiming that there is a growingbody of research suggesting that within mathe-matics, problem-solving strategies are a fruitfulfocus for work in the classroom, and have apositive effect at least on further problem-sol-ving work in mathematics.

In the problem-solving literature, there issome mention of the role of algorithmicmethods, that is to say precisely defined

procedures for solving problems. A typicalexample would be solving linear equations byappropriate steps such as 'cross-multiply' or'take to the other side and change the sign'. VanStreun (PME, 1990) suggests that "Being moresuccessful in problem solving is attended bymore frequently employing algorithmicmethods" (p.98). Sfard (PME, 1988) discussesthe distinction between operational and structu-ral methods:

"People who think structurally refer toa formally defined entity as if it were areal object, existing outside the humanmind. Those who conceive it operatio-nally, speak about a kind of processrather than a static construct." (p.560)

Sfard demonstrates that teaching conceptsthrough operational methods via algorithmscan have some success e.g. with the learning ofinduction. There are dangers in the reliance onalgorithmic teaching (Lerman, 1988, Kurth,PME, 1988, Steinbring, 1989) but its role as anheuristic is in need of further research.

Finally, there remains the question of howproblem solving should be integrated into thewhole mathematics curriculum. To some extentthis is a curriculum issue, which may be mani-fested in the particular textbook or schemeused in the school, and there are many coun-tries and schools where the individual teacherdoes not have a choice. In those countries andschools where perhaps the content is specifiedbut not the way it is taught, or where there isthe goal of a standard to be reached, but thestyle of work is dependent upon the teacher,how and when problem-solving appears is amatter of choice. As with so many questions inmathematics education this really depends onthe teacher's view of the nature of mathematics,and of the process of learning. If the teachersees mathematics as identified by ways ofthought, ways of looking at the world, as pro-cesses, then mathematics should be learnedthrough problem solving. If on the other hand,mathematics is thought of as a body of know-ledge, a specified amount of which studentsmust acquire, and then apply in problem sol-ving situations, then the teacher will have tofind a suitable time and place to teach problemsolving processes.

68

An important element in any discussionabout the role of problem-solving in learningmathematics, and one that will perhaps ensurethat the debate will remain, is the difficulty ofcomparison. It is fundamental that if oneintends to design some research to compareachievement in mathematics through the twodifferent approaches, one has to make the testsappropriate to the kind of mathematical work.Lester and Krol (PME, 1990) emphasise thisissue of the difficulty of comparing differentstyles of learning in their third observation,precisely because they require different stylesof assessment. Again, the most fruitful researchmay well come from studies by teachers, ofchildren engaged in problem solving work ofdifferent kinds, in the context of the learninggoals set by the teachers and the school.

The power relationships in the classroom

The classroom is characterised by powerstruggles and domination. This is not always inthe teacher's favour, as most starting teacherswill report. Indeed Walkerdine (1989) reportson a situation in which a woman teacher is sud-denly dominated by two little boys aged 4 or 5,using sexually abusive language:

"Annie takes a piece of Lego to add toa construction she is building. Terrytries to take it away from her to usehimself ... The teacher tells him to stopand Sean tries to mess up another chil-d's construction. The teacher tells himto stop. Then Sean says: 'Get out of it,Miss Baxter Baxter.'

TERRY: Get out of it, knickers MissBaxter.

SEAN: Get out of it, Miss Baxter pax-ter.

TERRY: Get out of it, Miss Baxter theknickers paxter knickers, bum.

SEAN: Knickers, shit, bum.

MISS BAXTER: Sean, that's enough.You're being silly.

SEAN: Miss Baxter, knickers, showyour knickers .

TERRY: Miss Baxter, show off yourbum (they giggle).

MISS BAXTER: I think you're beingvery silly.

TERRY: Shit, Miss Baxter, shit MissBaxter.

SEAN: Miss Baxter, show your knic-kers your bum off.

SEAN: Take all your clothes off, yourbra off. . ." (p.65 66)

Walkerdine adds: "People who have read thistranscript have been surprised and shocked tofind such young children making explicitsexual references and having so much powerover the teacher. What is this power and how isit produced?" (p.66)

In the main, however, the imbalance is inthe teacher's favour, and as Bishop (1988) com-ments:

"The first and most obvious principle isthat the teacher's power and influencemust be legitimately used - perhaps wecan say that it should be used and notabused" (p.130).

Hoyles (1982) for example gives instancesof both:

"I: What happened then?

P: Well the teacher was always pickingon me.

I: Picking on you?

P: Yes, and in one lesson she jumped onme; I wasn't doing anything but shesaid come to the board and do this sum- fractions it was. My mind went blank.Couldn't do nothing, couldn't evenbegin.

I: What did you feel then?

P: Awful, shown up. All my mates waslaughing at me and calling out. I wasstuck there. They thought it was greatfun. I felt so stupid I wanted the floor toopen up and swallow me. It was easyyou know. The teacher kept me thereand kept on asking me questions infront of the rest. I just got worse. I canremember sweating all over."

P: Yes, once, in the second year (and)we had this teacher, she was a reallygood teacher, maths it was, and I'venever been any good at maths. Shenever pushed you or nothing but let meget on with it at my own pace.

I: What do you mean exactly when yousay she never pushed you?

69

P. Well, she was nice. I had tried andshe realised it and didn't keep pic-king on me. I used to try reallyhard in her lessons and just get onwith it...

I: Can you tell me how you feltduring her lessons? What did youfeel inside?

P: Well really good, it was reallynice to be there." (p.353)

It is of course too simplistic to present theclassroom situation as one where the autono-mous teacher exercises control, and can choosethe power to be benevolent or despotic. Theextent to which the discursive practices of theclassroom construct the relationships of poweris discussed more fully in the section on thelanguage and practices of the classroom below.The relationship to mathematical knowledge isthe key factor that will be discussed here.

Mathematics has a special role in society. Itis seen as cultural capital for the individual, thatis, success in mathematics suggests financialsuccess in the future. It is also used by societyto legitimate policies and decisions. Whether itbe the billions of dollars of international debt ofone nation, or the high rate of inflation of ano-t h e r, the figures become indisputable and can beused to justify policies that make the lives of themajority miserable. This places teachers ofmathematics in a unique position, one that canresult in reinforcing the powerlessness of theindividual in modern society, or can enablepeople to question and challenge informationgiven The alternatives can be called 'empower-ment' and 'disempowerment'. Cooper (1989)claims that "There is evidence that teachers seemathematics as a crucial subject for reprodu-cing existing social values, and that they modi-fy mathematics curriculum material accordin-gly" (p.150). Cooper's point is that teachersaccept the status quo tacitly, and this results innegative power "where actions that are not inthe interest of those in charge are suppressed,thwarted, and prevented from being airedwithout the elite having to initiate or supportany actions or exercise any powers of veto"(p.153). Perhaps the most developed alterna-tive curriculum for mathematics, one thataims to empower students, is that byFrankenstein (1990). She draws on Freire's

distinction between a problem-posing curricu-lum and a 'banking' metaphor, where the indi-vidual is thought to store knowledge for retrie-val. She offers situations, such as statisticaldata on employment according to ethnic groupin the USA, and invites students to pose ques-tions to investigate mathematically.

Robinson (1989) takes the notion of empo-werment into the area of teacher education, andcompares two competing models for bringingabout teacher-change, the management para-digm and the empowerment paradigm:

"Rather than seeing change in schoolsas a finite process with externally spe-cified objectives, as the managementparadigm does, the empowerment para-digm sees change as an on-going acti-vity generated within the school by tea-chers, parents and students as part of anorganic process of professional rene-wal." (p.274)

Teachers are significant people in the livesof children, and the effects of teachers' expec-tations on individual children is an aspect ofthe teacher's power that is a focus for research,particularly in the field of gender, ethnicity orclass bias. In this context, an important aspectof the impact of teachers' expectations of chil-dren is the range of factors to which teachersattribute students' success or failure in mathe-matics. Fennema et al. (1990), for example,make some strong claims about such impact:

"(1) A teacher's causal attributions areimportant because perceptions ofwhy his/her students succeed orfail in achievement situations hasan impact on the teacher's expec-tancies for students' future achie-vement success.

(2) Teachers' attributions influencestudents' attributions through tea-cher behaviour". (p.57)

There has been one study (Kuyper and vander Werf, PME, 1990) reported at a meeting ofPME which 'releases' teachers from the respon-sibility for such influence. They claim:

" ...the differences in achievement, atti-tudes and participation cannot be attri-buted to (characteristics of) individual

70

math teachers. There is some eviden-ce,: however, that the gender of the tea-cher influences the perception by girlsand boys of their teacher's behavioursin a way that might be labelled 'ownsex favouritism'. The observed teacherbehaviours do not influence this per-ception." (p.150)

They end their study by suggesting "In ouropinion the results fit very nicely into a generalpattern, which can be verbalized as follows: thegender differences in math are not the teacher'sfault" (p.150). Their study involved more than5800 students and teachers, and clearly thelimitations on the size of papers for PME pro-ceedings did not permit a full description oftheir methodology and results. Their claim thatthe teacher behaviours do not influence stu-dents perceptions, however, is focused on theindividual. Some of their own evidence high-lights factors that are significant for teachers ingeneral. They found that teachers attribute tidi-ness more to girls, for example, and indus-triousness too, whereas 'disturbing order' ismore typical of boys in their view.Scott-Hodgetts (PME, 1987) points out thatthese teacher perceptions and the resulting tea-cher behaviours have significant cognitiveeffect, and are not merely affective observa-tions. They serve to reinforce serialist strate-gies, i.e. a-step-by-step approach, which moregirls than boys are predisposed to adopt. Theresult is that more boys tend to develop a broadrange of learning styles, whereas more girlsremain as serialists. There is substantial evi-dence in support of the Fennema et al. (1990)claims, as Hoyles et al. (1984) describe (p.26).

This brief discussion is an example of anissue to which we will return in the final sec-tion, namely to what extent does research reachteachers and offer them the possibility of chan-ging their work? Some teachers, having readliterature that suggests that teachers, irrespecti-ve of their gender, spend much more time ans-wering and dealing with the boys in their classthan the girls, have been stimulated to examinetheir own practice (Burton 1986). It should beof some concern if such research remains inlearned journals to which most teachers do nothave access (Lerman 1990a).

Textbooks

The school mathematics text book is a familiarelement in our work, as is well illustrated inHart's chapter in this volume. However, the tea-cher's use and dependence on the textbook is awell known phenomenon, but is under-resear-ched in the teaching of mathematics. Laborde(PME, 1987) has looked at students readingtexts, and Grouws et al. (1990) mention that thetextbook is the most important factor influen-cing students' attitudes. Van Dormolen (19B9)demonstrates how texts can play a role inreflecting real life situations in the classroom,and Frankenstein (1990) takes this further,focusing on the potentially emancipatory roleof the material offered in class. But the effect ofthe dominance of textbooks in mathematicsteaching has not been well investigated. Whenthe standard mathematics lesson begins withsome initial teacher exposition and is followedby the students working through an exercise intheir textbook, and homework is a further exer-cise from the book, the text, the textbook andthe textbook writers constitute an authority inthe classroom. Teachers often talk of "they areasking you to..." when attempting to clarify aquestion that appears in the text, and that stu-dents do not understand, and it is interesting toconsider what the relationship is between theteacher and the text that is conveyed by thisexpression.

Social messages that are hidden in texts areunquestioned by teachers and students, partlybecause the textbook is an illustration andmanifestation of the authority implicit in theclassroom. This is particularly the case inmathematics, perhaps because the sterile axio-matic form of the presentation of academicmathematics papers reinforces the authorityand status of the mathematical text. In thestudy of history, for example, students areencouraged to draw from a number of sources,highlight the ways in which those textbookwriters disagree with each other, and even toexplain the context in which different writershold competing views (this is not the case inall countries of course!). It is quite the opposi-te in the mathematics classroom. There arealso more overt reasons for textbooks beingunchallenged in the mathematics classroom,

71

as mathematics is seen by many teachers as"... a crucial subject for reproducing existingsocial values" (Cooper, 1989 p.150).

An illustration of the hidden messages ofschool mathematics textbooks, and a sugges-tion of ways that teachers could challengethose assumptions, was given by Lerman(1990b). There are different ability level textsin the most popular series in use in Britishschools, the School Mathematics Project,where the same topic, paying income tax, isdealt with by asking the top ability students tocalculate tax on an income of £50,000 whereasthe bottom ability text requires calculations foronly £9,000. The authors are assuming, andthus are conveying the message to the studentsthat low ability in mathematics, whatever thatmay mean, correlates with low intelligence,and/or poor career prospects etc. Lerman sug-gests that the dynamics of the classroom maywell change if students were encouraged to cri-tically examine their textbooks, and theexamples used, by comparing the pages fromthe two books. The author has offered such anotion to teachers on an in-service course, andone teacher commented that it would be toodangerous (Lerman, 1990b), perhaps because itthreatens the safety of the authority which thetextbook establishes in the classroom.

Is the relationship to 'authority' in mathe-matics any different if the teacher uses a work-card individualised scheme, or individualworksheets? What would be the implicationsand outcomes of developing materials as andwhen they are needed, perhaps by the studentsthemselves? Mellin-Olsen (1987) proposesclassroom work based on projects:

" A dormitory town outside Berg e n .High, grey concrete blocks of flats. Thearea was declared ready for occupationas soon as the garages had been built. Itwas not thought scandalous until later itwas discovered that they had forgottento provide the children with leisureareas.

The teachers of three classes (agegroup 10) prepared a project. What canwe do about the situation?" (p.218)

There are of course implications for theempowerment of students in an example like

this, especially when contrasted with a com-mon investigative task on a similar theme, tomake a model of a bedroom or classroom.Frankenstein gives many examples of projectsfor use in mathematics classrooms(Frankenstein 1990). Brown and Dowling(1989) propose what they call a research-basedapproach as an alternative to the textbook, forteachers to use in the classroom:

"Our method has been to propose aquestion - say "who does the best atschool?" - as the basis for a researchproject." (p.37)

These aspects of the use of texts and theirinfluence on students, and the dependence ofteachers on texts, is in need of considerableresearch and investigation. It will be interestingto follow reports of teachers' use of the innova-tive, perhaps revolutionary ideas ofM e l l i n-Olsen, Frankenstein, Brown andDowling and others, in changing the functionof the text.

Constructivism and the teacher

As a learning theory, constructivism is descri-bed in Hart's chapter in this volume, and thereare many important research programmes thatdraw on the paradigm of constructivism, thathave been reported in the literature. In this sec-tion we are concerned with what the theorymight suggest for the role of the teacher.

The Eleventh Annual Conference of PME,held in Montreal in 1987, has been the onlysuch annual meeting to be centred on a theme,that of constructivism. The term 'constructi-vism' had appeared in earlier meetings, and itwas presumably felt that the mathematics edu-cation research community in general wereunsure of the meaning and of the relevanceand/or implications for mathematics education.Consequently that theme was chosen for thePME meeting, to encourage debate on theissue. In his plenary presentation, Kilpatrick(PME, 1987) challenged the claim thatconstructivism is an alternative paradigm forlearning, and suggested that the putative out-comes of constructivist research and teachingare consistent with cognitive psychology. Sincethat meeting, many papers presented at PME

72

conferences have been couched in the languageof constructivism, and drawn on methodolo-gies of research and teaching, as well as theo-retical frameworks, that claim to have distinctinterpretations in the constructivist paradigm.The critique, however, has been only infre-quently continued (Goldin, PME, 1989), andthis is perhaps to the detriment of the debate,since we can only develop our understandingof the links between theory and practice, and inparticular alternative learning theories and theconsequences for teaching mathematics,through critical discussion.

In relation to the role of the teacher, therehave been a number of research reports concer-ned with the preparation of constructivist tea-chers (e.g. Simon, PME, 1988). However, theimplications of constructivist learning theoriesfor the role of the teacher, and for the kinds ofactivities that the teacher might initiate as aconsequence of a constructivist view of lear-ning, have still not been clearly elaborated, andcertainly not well tested, either through theore-tical critique or through classroom case studiesof teachers. This makes talk of 'constructivistteachers' less meaningful than might be thecase were there to be some elaboration of whatthat implies. There is also the question of thedistinction that has been made between 'weakconstructivists' and 'strong constructivists', theformer being those that subscribe only to thefirst hypothesis quoted by Kilpatrick (PME,1987), and the latter being those that subscribeto both:

"(1) Knowledge is actively constructedby the cognizing subject, not pas-sively received from the environ-ment.

(2) Coming to know is an adaptiveprocess that organizes one's expe-riential world; it does not discoveran independent, preexisting worldoutside the: mind of the knower."(p.3)

It was suggested at the PME meeting in1987 that everyone could subscribe to hypo-thesis (1). There have been some attempts todiscuss the implications of both positions.Lerman (1989b) and Scott-Hodgetts andLerman (PME, 1990) have suggested that

it is difficult to see how one can accept the firsthypothesis and not the second. They also main-tain that the second hypothesis, far from lea-ving one unable to say anything about any-thing, as Kilpatrick suggested in his presenta-tion (PME, 1987), is an empowering position,in the sense that ideas are continually open tonegotiation and development and we do notexpect to arrive at ultimate truths. However, itis clear that more debate and research are nee-ded.

Teachers' beliefs about mathematics

Ideas such as 'the teachers' values', 'what theteacher believes about problem solving' and soon, have occurred often in the review above.The issue of the influence of teachers' beliefson their actions has been and continues to be ofgreat interest and importance in mathematicseducation. Nesher (PME, 1988) discusses theeffects of teachers' actions and also reveals herown beliefs, in her plenary lecture at the twelf-th meeting of PME. In one part of her paper shefocuses on the ways that the behaviour of theteacher can have an effect on the child'sconception of the nature of mathematics, andshe warns of the danger of the teacher being theultimate judge of whether the child is correct ornot:

"This does not let the child constructfor himself the mathematical notionsand concepts. Nor does it enable him torealize that the truths of mathematicsare objective and necessary." (p.63)

Thom's often quoted claim that "all mathe-matical pedagogy, even if scarcely coherent,rests on a philosophy of mathematics" (1972,p.204) was one of the factors that stimulatedinterest amongst educators in epistemologies ofmathematics and their influence on teaching,and in the past ten years there have been a num-ber of studies in this area. There have been twodifferent directions of research: a 'top-downapproach', so-called because it starts from aconsideration of the current state in the philo-sophy of mathematics and the possible alterna-tive perspectives of the nature of mathematics(e.g. Lerman, 1983; Ernest, 1989), and a 'bot-tom-up' approach, which begins with teachers'views and behaviour, as in the work ofThompson (1984) for example.

73

The 'top-down' approach recognises therole of implicit theories, not only in teacherbehaviour, curriculum development etc., butalso in research perspectives, hypotheses andmethods. Thus it aims to reveal the implicitepistemological perspectives which underliecurriculum decisions (e.g. Nickson, 1981),influence the practice of teaching mathematics(e.g. Lerman, 1986), and also determine resear-ch directions. As Hersh (1979) wrote:

“ The issue then, is not, what is the bestway to teach, but what is mathematicsreally all about ... controversies abouthigh school teaching cannot be resol-ved without confronting problemsabout the nature of mathematics. ”(p.33)

Scott-Hodgetts (PME, 1987), in her discus-sion of holist and serialist learning strategies,demonstrates the significance of the teacher'sphilosophy of mathematics to the extension ofvarious learning strategies. In their study ofmathematicians and mathematics teachersviews, Scott-Hodgetts and Lerman (PME,1990) set their analysis of psychological/philo-sophical beliefs in a radical constructivist pers-pective, and propose that it provides a powerfulexplanatory potential:

"What A has been engaged in, withoutdoubt, is what is described as "an adap-tive process that organises one's expe-riential world". In doing this he hasbrought to bear different models of thenature of mathematics, picking andchoosing in order best to “ fit ” his par-ticular experiences at different times."(p.204)

Siemon (1989a) suggests that "what wedo ... is very much dependent on what weknow and believe about mathematics, aboutthe teaching and learning of mathematics, andabout the nature of our particular task asmathematics educators" (p.98). Dougherty(PME, 1990), in an on-going study of theinfluence of teachers' beliefs on problem-sol-ving instruction, characterises teachers' cogni-tive levels, a set of psychological attributes, ona concrete-abstract continuum, from A, themost rigid, concrete formalism, through B.social pluralism, and C, integrated pluralism,to D abstract constructivism. In interviews andobservations of 11 teachers, all on problem-

solving lessons so as to standardise the lessoncontent, Dougherty placed 8 in A and 1 in eachof the other categories, and she found strongconnections between these positions andconceptions of problem solving.

Methods of analysing and interpreting tea-chers' actions in the classroom, that engagewith the issue of teachers' beliefs about mathe-matics, have been reported in the literature.Jaworski (PME, 1989) discusses students deve-loping a 'principled' understanding of mathe-matical concepts, when comparing the inculca-tion of knowledge, as against its elicitation.She says:

"Successful teaching of mathematicsinvolves a teacher in intentionally andeffectively assisting pupils to construe,or make sense of mathematical topics."(p.147)

She demonstrates the dangers of 'ritualknowledge' in a wonderful example of the rea-soning of a child who regularly achieves goodmarks in mathematics.

"I know what to do by looking at theexamples. If there are two numbers Isubtract. If there are lots of numbers Iadd. If there are just two numbers andone is smaller than the other it is a hardproblem. I divide to see if it comes outeven and if it doesn't I multiply. "(p.148)

Ainley (PME, 1988) has looked at teachers'questioning styles, noting the power relation-ships, by analogy with the parent/child or drillsergeant/recruit. She suggests that assumingmathematics answers are right or wrong is adanger for the teacher in using questioning, andthat teachers assume that questioning is betterthan exposition. In her study, she looked at thedifferent perceptions of the children and theteachers, of the teachers' questions. She classi-fied questions into: pseudo-questions; genuinequestions; testing questions, and directingquestions, the latter sub-divided into structu-ring, opening-up and checking. Her focus onthe language of the classroom is further discus-sed in the following section.

Various studies show that there are a num-ber of variables that mediate, in an investiga-tion of the influences of teachers' beliefs.Thompson (1984) and Lerman (1986)

74

and other researchers report how the ethos ofthe school is a major element in the teacher'sapproach in the classroom. Siemon (1989b)illustrates this with comments from teacherswho, when asked "What are the main pressureson you as a mathematics teacher?" replied:

"School policy...you can't just be theone to break the system ... if you giveall your students an A you will be ques-tioned on it...you can't. You've got towork within the system as such.

The kids have their activity book. If theparents look at that at the end of theyear and they haven't done one activity... they're going to question it and thePrincipal's going to question whetheryou are following the program."(p.261)

Noss et al. (PME, 1990), in discussing astudy of teachers on an in-service course incomputer-based mathematics learning, com-ment that:

"... the extent to which a participantwas able to integrate the computer intohis or her mathematical pedagogy(theoretically and/or practically) appea-red more related to the direction inwhich a participant's thinking wasalready developing and with his or hercommitment to change, rather than thestyle of teaching approach, view ofmathematical activity, or rationale forattending the course." (p.180)

A global view of factors affecting the tea-ching of mathematics reveals clearly the majorinfluence of the conditions and resources in theschools in different countries. Nebres (1989)calls these the Macro problems, as against theMicro problems which are internal to mathe-matics education:

"... in many developing countries, theproblems that merit most thought andresearch are those due to pressuresfrom outside society. The purpose ofstudy regarding these pressures is toprovide some scope and freedom forthe educational system so that it canattend to the internal problems ofmathematics education." (p.12)

Nebres emphasises the importance of therole of cultural values, in understanding boththe Macro and Micro environments in mathe-matics education, and suggests that his aware-ness of that perspective is focused by

the "sharp contrast of traditions and cultures inEast Asia" (p.20). He writes:

"... we have realised that questions likethe status and salaries of teachers arenot simply a function of the economybut also of cultural values. Similarlythe rise of what we might call the inter-nal force of genius, wherein mathema-tics talent springs up even under diffi-cult economic circumstances, seems tobe fostered by the cultural environ-ment." (p.20)

Suffolk (1989) gives a detailed study of thesituation in Zambia, as seen through the role ofthe teacher. Most such studies, quite naturally,focus either on the children or on the situationof the schools, and more studies of the effect onthe teacher are needed. One approach, as aconsequence of a recognition of thesevariables, as reported by Nolder (PME, 1990),is to undertake a detailed analysis of the mathe-matics teachers in one particular school whichis undergoing some curriculum changes, in anattempt to reveal the nature of these influences.She found, for instance, that such changes ledto the teachers experiencing a good deal ofuncertainty, which affected their perceptions oftheir competence and confidence. Stephens etal. (1989) report on mathematics educationreform programmes, and state the followingprinciple for involving teachers:

"Teachers involved in change initia-tives need to know where they havecome from, where they are going, whythey are going there, and should haveaccess both to the best professionalthinking of their colleagues, and totheoretically sound, supportive envi-ronments." (p.245)

These studies indicate that 'change' isalways socially embedded, and cannot be isola-ted from the cultural context, from the politicalmilieu, from the social context of schools andcertainly not from the teachers in the class-room. Ignoring any of these elements mitigatesagainst any improvement in the teaching ofmathematics.

A stimulus in the development of a socialperspective of the nature of mathematics hascome from teachers working in situations ofdeprivation and/or oppression. If one tends tothink of mathematics as value free, and muchthe same all over the world, the work of some

75

of these researchers and teachers provide astrong case for rethinking. Fasheh (1989), indescribing mathematics education in schoolson the West Bank in Israel, argues that whatmatters is "... whether math is being learnedand taught within the perspective of perceivingeducation as praxis or within the perspective ofperceiving it as hegemony" (p.84). Gerdes(1985) demonstrates how mathematics educa-tion in Mozambique changed from pre-revolu-tionary colonial domination to post-indepen-dence construction of a socialist society:

"During the Portuguese domination,mathematics was taught, in the interestof colonial capitalism, only to a smallminority of African children ... to beable to calculate better the hut tax to bepaid and the compulsory quota of cot-ton every family had to produce ... to bemore lucrative "boss-boys" in SouthAfrican mines ... Post-independence ...Mathematics is taught to "serve theliberation and peaceful progress of thepeople". (p.15)

D'Ambrosio (1985), in developing the ideaof ethnomathematics, sets the growth of mathe-matics in a social context, and aligns the deve-lopment of modern science and, in particular,mathematics and technology with colonialism.There is a growing body of work in the area ofethnomathematics, at the macro level and at themicro level, with profound implications forteachers in mathematics classrooms, both inthe so-called developed world, and also theunderprivileged world, that engages with ideasof empowerment and liberation.

The language and the practices of the class-room

R e c e n t l y, there has been an interest in thediscursive practices of mathematics teaching(e.g. Keitel, et al., 1989; We i n b e rg and Gavalek,PME, 1987), influenced in the main by thework of Walkerdine (1988, 1989, PME, 1990).The focus of this critique is a shift away fromthe individual to the social, and is thus a critiqueof psychology, which can be characterised as anindividualistic way of interpreting people'sactions. Investigating the interactions in theclassroom (aspects such as discussion betweenstudents, teachers' questions, open-e n d e d

questions, the teacher's response to novel inter-ventions from students), leads one rapidly intoaspects of the power relationships of the class-room. Pimm (1984), for example, talks aboutthe use of the word 'we', and indicates just howmuch is actually being said and construed andstructured by the use of that small word:

"Given a class of pupils all doing some-thing incorrectly, a teacher can still say,'No, what we do is ...' Who is the com-munity to whom the teacher is appea-ling in order to provide the authority toimpose the practices which are about tobe exemplified?" (p.40)

Of course the dominant paradigm withinwhich research, testing, and theorising on themeaning of understanding takes place is a psy-chological one. Understanding is seen as someprocess that the individual undergoes, sometransition from not understanding a concept tounderstanding that concept. There are two ele-ments in this transformation from a pre-unders-tanding state to a state of understanding, theindividual and the concept. The latter is seen asan objective, external element which in theeducational environment is usually 'possessed'or held in some way, by the teacher. The tea-cher is seen to perform the intermediary role ofintroducing the concept, attempting to enableand facilitate its transmission from teacher tostudent, and measuring the outcome, - that iswhether the state of 'understanding' has beenachieved. The former element, the individual,is largely a closed unit to the teacher, makingthe business of determining whether the trans-formation has taken place or not, a very diffi-cult one. As Balacheff (1990) points out:

"It is not possible to make a directobservation of pupils' conceptions rela-ted to a given mathematical concept;one can only infer them from the obser-vation of pupils' behaviours in specifictasks, which is one of the more difficultmethodological problems we have toface." (p.262)

Research in this area in education thusfocuses either on interpretations of behaviourthat might constitute evidence of the processesof concept acquisition, or on the conditions thatwill bring about 'understanding' .

76

Mathematics is seen as hierarchical, andconflated with this is the notion that learningmathematics is hierarchical too, from the basicnotions of one-to-one correspondence, settingand combining sets, to more abstract levels of,for example, the calculus. It is seen by educa-tionalists as 'very difficult' (Hart, 1981). Otherresearch shows, however, that children's unres-tricted and undirected thinking does not followthe supposed hierarchy (e.g. Lave, 1988;Carraher, PME, 1987). It may just be the casethat our assumptions about the nature of mathe-matics and the nature of the learning processbecome self-fulfilling. We interpret, theorise,teach, test, assume, expect, measure and thusconfirm our initial expectations of children.

The psychological model of learning,which can be characterised as a process oftransformation from a pre-understanding stateto a state of understanding, determines the styleand intention of the various stages of the edu-cational process. The process begins with theteacher assessing the initial cognitive state ofthe individual. This is followed by the preparedlesson, whose content is determined by resear-ch into hierarchies of mathematical knowledgeand levels of difficulty of understanding, andwhose method is determined by large-scaletests using well-established quantitative proce-dures. Research on affective factors influencethe classroom setting, the appearance of thetext, etc. On-going assessment enables theidentification of misconceptions, misunders-tandings or partial understandings. These canbe corrected by suitable re-explanations, rein-forcement materials and so on. The final stageis the test that measures the successful acquisi-tion of the particular skill or concept, that isinterpreted as understanding, and again thattest will have been trialled over large groups ofstudents, in well-established ways.

Whilst the above is somewhat of a carica-ture, the intention is to highlight the manner inwhich the psychological paradigm dominatesand determines the whole structure of the edu-cational process. It is the essentially privatenature of the psychological interpretation ofunderstanding that pervades the educational

process from this perspective. Teachers are leftto try as best they can to recognise when thatprocess has occurred, and in the case of therecent development of a National Curriculumin Britain, with a long list of attainment targetsand levels to be assessed, tick the appropriatebox to register that achievement target for thestudent.

At the same time, there is an assumption,usually implicit, that we all arrive at the sameunderstanding of particular things, because ofthe absolute nature of those things, whetherphysical objects, such as tables, and unicorns,or mental objects such as circles, or 'three'. Acritical analysis, therefore, needs to work onboth aspects of the understanding process inthe psychological paradigm, namely the notionof the individual, and also the notion of know-ledge and its posited absolutist nature.

In the psychological paradigm, the majorfocus of attention is in how the individual, thesubject, functions in discourses. However, sub -ject is simultaneously the apparently autono-mous self-constituted individual who articu-lates a discourse and also the subject of thatdiscourse. That is to say, the individual isconstituted by a discourse. Thus, for example,in the classroom we are simultaneously the tea-cher, who initiates and perpetuates a discourse,and at the same time we are constituted by thatdiscourse, and by the relations of power thatpertain. To some extent this has been looked atabove, specifically related to the power of theteacher. It may appear, though, from that sec-tion, that teachers consciously choose positionsof power and can reject them. In analysing thediscourse of the classroom one can begin to seethe subtle historically-rooted relationships andmeanings carried by the interactions in theclassroom. Pimm's (1984) discussion of the useof 'we', and the position in which the acceptedand unchallenged role of the mathematics text-book places the teacher and students areexamples of the insights that can be gainedfrom an analysis of the discursive practices ofthe classroom.

An analysis of how languages operatewithin discursive practices indicates the way inwhich knowledge, understanding, meaning andourselves as subjects are constituted in and

77

bounded by our language, and that language isspecific to discursive practices. Meaning isabout use, no less but also no more. InWalkerdine's analysis of the home practices inwhich the notions of 'more' and 'less' occur( Walkerdine, 1988), she demonstrates thecontext in which those words have meaning forthose children, and the practices through whichthose meanings are constructed. The classroomteacher who is unaware of the meanings thechildren are bringing with them into the class-room may well make judgements about chil-dren's 'abilities' that are entirely inappropriate.

The alternative view of 'understanding' isone that sees the process as social, rather thanpsychological. That is to say, the individualdoes not go through some transformation fromare to post state of understanding in isolation,acquiring a given well-formed concept in somehidden way. It is necessarily an interaction ofthe child and his/her multitude of meanings,through language, with others and with expe-riences. That process of interaction may be lar-gely silent, a shift taking place without beingopenly articulated, but this does not mean thatit is in some way 'private'. Sometimes theunderstandings do not fit with those of others,including the teacher, and only talk, discussion,suggestions and conjectures and refutations,through cognitive conflict, or shifts of thoughtthrough resonance, enable further growth.

In the literature of mathematics education,one finds an emphasis on providing a contextfor mathematical concepts, and clearly this isessential if mathematics is to be about some-thing. There are two aspects to 'context', howe-ver. It is a matter, not only of attempting toembed mathematics in situations that may be'relevant' or 'meaningful' for students, but alsoan analysis of the discursive practices withinwhich mathematical terms and conceptsappear, both for students in their own expe-riences and for the mathematical community.

The language used in the mathematicsclassroom, particularly in relation to bilin-gualism, has been an important focus ofstudy for a number of years (Dawe, 1983;Zepp, 1989). Most such studies have been

concerned with children, and their learning.Once again, the role of the teacher, and hereparticularly the language of the teacher, hasonly recently become an issue, as researchersand teachers have come to realise the functionthat language serves in deep aspects of class-room interactions. Khisty et al. (PME, 1990),for example, reported on an exploratory studyof the language of the teacher, in a bilingualclassroom, to investigate the reasons for theunderachievement of Hispanic students in theUSA. They review the language factors thatmight play a part, emphasising an interactionistperspective, which attempts to bring togetherthe cognitive and social dimensions of lear-ning. They refer to how the use of languagestructures the classroom, the development of amathematics register, syntactic and semanticstructures in mathematics, and context. Theyconclude their introduction as follows:

"The importance of these points is thatlanguage issues in the teaching andlearning of mathematics may be morecrucial than previous research wouldsuggest." (p.107)

As reported above, Ainley (PME, 1988)has looked at the implications of teachers'questions, and revealed both a wide range ofpurposes, and also a discrepancy between chil-dren's perceptions of the teacher's question andthat of the teacher. Wood (PME, 1990) focusedon discussion in the classroom, and highlightsthe all too frequent consequences of what maybe called 'teacher-centred' learning:

"The patterns of interaction becomeroutinized in such a way that the stu-dents do not need to think about mathe-matical meaning, but instead focustheir attention on making sense of theteacher's directives." (p.148)

In a different direction of attention, Blissand Sakonidis (PME, 1988) looked at the tech-nical vocabulary of algebra, and the everydaylanguage that is sometimes substituted forthese terms, in an attempt to discover whetherit helps or hinders communication of algebraicideas.

Analyses of the language of the classroomalso make connections between the concepts

78

within the 'language game' of mathematics andin everyday use. Pimm (PME, 1990) discussesmetonymic and metaphoric connections, and indoing so highlights a central aspect of the waysmathematical language and thought work. Forexample, metaphoric links are those whichwould highlight the meaning of 50% in diffe-rent contexts, whereas metonymic links wouldbe those between 50% and 0.5 and 1/2 and 4/8etc.

"Being aware of structure is one part ofbeing a mathematician. A l g e b r a i cmanipulation can allow some new pro-perty to be apprehended that was notvisible before - the transformation wasnot made on the meaning, but only onthe symbols and that can be verypowerful. Where are we to look formeaning? Mathematics is at least asmuch in the relationships as in theobjects, but we tend to see (and lookfor) the objects ...

Part of what I am arguing for is a farbroader concept of mathematical mea-ning, one that embraces both of theseaspects (the metaphoric and the meto-nymic foci for mathematical activity)and the relative independence of thesetwo aspects of mathematical meaningwith respect to acquisition."(p.134-135)

One can expect significant further researchin the deconstruction and reconstruction of thelanguage and practices of the mathematicsclassroom.

Teacher education

Many educational researchers in mostcountries of the world are involved in teachereducation, either pre-service or in-service orboth, and many, if not all of the issues concer-ning teaching apply equally to teacher educa-tors. Students on a SummerMath programmewere reported by Schifter (PME, 1990) asdemanding: "We need a math course taught theway you're teaching us to teach." (p.198) Jaji etal. (1985), in an article whose main focus is onproviding pre-service teachers in Zimbabwewith a repertoire of roles from which to choo-se, rather than replicate the ways in which theywere taught, offer this transcript of a teachereducator in action:

"Now what I am about to say is veryimportant. It will almost certainly comeup in the Examinations, so I suggestthat you write it down.

(The group took out their pens ...).

In the new, modern approach to tea-ching ...

(They wrote it down as she spoke ...).

In the new, modern approach to tea-ching, we, as teachers, no longer dicta-te notes to children. Instead we arrangeresources in such a way as to enablechildren to discover things for them-selves." (p.153)

If we are to suggest that teachers shouldbecome involved with research on their practi-ce, as will be discussed below, teacher educa-tors are teachers too and one would expect thatteacher educators would be involved in resear-ch on their own practice. One can in fact seeevidence of a small but growing interest inresearch on teacher education in recent mee-tings of PME.

A further consequence of the analogy bet-ween students in schools learning mathematicsand research on that, and student teachers lear-ning to teach mathematics and research in thatarea, is that just as we cannot talk of makingstudents learn mathematics, so we cannot talkof making teachers change their teaching. Onedoes however come across such discourse inthe literature, in particular in relation to in-service courses. As teacher educators we canoffer ideas and possibilities, with activities thatencourage reflection and awareness and perso-nal growth (Jaworski and Gates, PME, 1987;Lerman and Scott-Hodgetts, PME, 1991, inpress), but to try to impose teacher change isboth unrealistic and perhaps arrogant.

There are other reasons for reflecting on thepractice of teacher education. In Britain, andprobably many other parts of the world too, tea-cher education programmes are at a turningpoint, under the influence of political pressures.There is a strongly held view that teachers canbest learn "on the job" rather than in college,with the metaphor of apprenticeship in trainingheld to be more appropriate than the combina-tion of theory and practice that might be impliedby the metaphor of a profession. According tothis view, at secondary age in Britain, (students

79

aged 11 to 18) anyone can teach, provided onehas a degree in a suitable subject, given anappropriate system of apprenticeship. T h a tsuch a system is vastly less expensive thanspending between one and four years in a col-lege is almost certainly the main but unstatedrationale. Whatever the motives, teachers andteacher-educators must be clear about the natu-re of the process of teaching, and therefore ofteacher education. There is no doubt that onecan learn a great deal by watching other tea-chers, and not necessarily 'master-teachers',which is a designation used in some countries.Given a theoretical framework in which to cri-tically evaluate and assimilate one's observa-tions, observing others can be an essential andvaluable aspect of learning to become a tea-cher. But learning to become a successful tea-cher is not a matter of copying someone whohas been designated an expert. It is a matter offinding one's own way of teaching, developingways in which one can identify what is happe-ning in the classroom, and of drawing on expe-rience and theory to make decisions on action,followed by reflective evaluation of thoseactions. There are frequently conflictingdemands on the teacher (Underhill, PME,1990), and research on factors that help pros-pective teachers to develop their own unders-tanding of teaching (e.g. Brown, PME, 1986;Dionne, PME,1987) are needed.

In what follows we will look first at themathematical content of teacher educationcourses, and the content/process debate as itmanifests itself at this level. A brief review ofsome of the main issues that arise in the deve-lopment of in-service courses appropriate tothe needs of teachers will then be followed bya development of the notions of the teacher asreflective practitioner and as researcher.

There have been a number of studies thathave focused on prospective teachers' orpractising teachers' misconceptions of topicsin mathematics. Concerning teachers of ele-mentary age children, Linchevski and Vi n n e r(PME, 1988) looked at naive concepts ofsets, Vinner and Linchevski (PME, 1988) onthe understanding of the relationship bet-ween division and multiplication, Tierney etal. (PME, 1990) on area, Simon (PME, 1990)on division and Harel and Martin (PME,

1986) on the concept of proof held by elemen-tary teachers. In the high school age range,Even (PME, 1990) has examined mis-conceptions in understanding of functions.

What are we to make of these findings, andwhat might be the possibilities for improvingthe situation? It is probably the case that mostpeople have negative experiences of learningmathematics at school, and as in the main,prospective elementary teachers are not mathe-matics specialists, they too are generally atleast unconfident about mathematics. If pre-service courses teach mathematics in the sameway that those people learned mathematics atschools, patterns of lack of confidence,under-achievement and feelings of failure arelikely to remain into their teaching career(Blundell et al., 1989). Since children's earlyexperiences of mathematics occur in interac-tion with elementary teachers, this is a majorproblem for mathematics education. Ways ofbreaking this cycle must clearly be developed,and one significant element, as mentionedabove, is for students to be taught the way thatwe expect them to teach. Schifter (PME, 1990),who came across that demand from her stu-dents, reports on a programme of in-serviceeducation which attempts to effect such a trans-formation:

“ ... the notion of "mathematicscontent" as the familiar sequence ofcurricular topics is reconceived as"mathematics process": at once theactive construction of some mathemati-cal concepts - e.g. fractions, exponents- and reflection on both cognitive andaffective aspects of that activity. Thework of the course is organised aroundexperiences of mathematical explora-tion, selected readings, and, perhapsmost importantly, journal keeping. ”(p.191)

The journal writing is a most effective wayof demonstrating the changes that the teachersfelt themselves to have gone through, duringthe course. The main intention of the journals,however, is to encourage and enable the reflec-tive process that is an essential component ofbreaking out of the repetitive cycle of lack ofsuccess. It is not that such reflection will gua-rantee that no misconceptions in mathematicalunderstanding will occur, but that where

80

cognitive conflicts arise in teachers' experienceof mathematical activities and concepts, or arecognition of aspects of mathematics that theydo not understand, they are more likely to beable to confront them, rather than remain withtheir misconceptions.

Blundell et al. (1989) also draw on stu-dents' reflections as expressed in journals, intheir evaluation of the nature of their students'experiences and learning, through the course.In this study the course was a preservice onefor prospective primary (elementary) schoolteachers, and as in the Schifter (PME, 1990)study, the focus was on mathematical processesand learning processes. They describe the aimsas follows:

"The fundamental belief which under-lies this course structure is that a reco-gnition and mastery of these generalconcepts and strategies will enable stu-dents to approach any new content areawith confidence and competence, andthat the efficient acquisition of mea-ningful mathematical knowledge isthus facilitated. Furthermore, thisacquisition is not dependent upon theskills of the "teacher", but rather uponthose of the "learner" - leading to muchgreater independence and self suffi-ciency." (p.27)

Thus the other aspect of 'being taught theway we want them to teach' is that when stu-dent teachers focus on their own learningreflectively, during the course, they becomeaware of what their school students are goingthrough in their learning too (Waxman &Zelman, PME, 1987). As the student teachersgain confidence, independence and self-suffi-ciency, they can recognise the potential fortheir own school students to do the same. Thestudents commented on their early learningexperiences, and then on their recent growth asmathematics learners:

"The only maths teacher I remember,although only vaguely, is the one whokindly allowed me to spend most of mytime outside the classroom door ..."

"September 1965.1 hate maths ... Irecall heading a page of my maths book"funny sums . For this I was sent tostand outside the class, told I was a"bloody imbecile" and that I was theworst student he had ever had the mis-fortune to teach ..." (p.27)

"We're never afraid of anything noware we? Whatever you give us we'lljump in and have a go."

"I really love doing maths. I really lookforward to Tuesdays and Fridays. Allthis time I've been going on thinkingI'm no good at maths - and now I'mdoing it." (p.29)

The authors reflect on their own learning asteacher educators, from this analysis:

"Clearly the students have gained inconfidence by this stage of the course,but it is not yet an unequivocal confi-dence. It seems as though they are stillambivalent, feeling confident whenthey focus on their process abilities, butdoubting when concentrating oncontent in mathematics ... However,they do not overtly acknowledge andvalue the acquisition of process skillsto the extent to which we might havehoped." (p.30)

Journal writing as an empowering activityin reflection, whether it be on one's mathemati-cal work or on one's learning during a course isitself the subject of study (e.g. Brandau, PME,1988; Hoffman and Powell, 1989; Frankensteinand Powell, 1989).

In-service courses provide the opportunityfor teachers to meet with others, reflect on theirexperiences, re-interpret their work in the lightof theories which they may meet during thecourse and develop areas of their expertise.Fahmy and Fayez (1985) attempt to outlineguidelines for in-service education based ontheir work in Egypt. They emphasise the pro-fessionalisation of teacher education, orienta-tion towards self-education and research activi-ties and careful consideration of the mathema-tical needs of the teachers. There are situationsin which the in-service programme has to begeared towards major curriculum develop-ments, and thus are firmly focused on theseneeds. Xiangming et al. (1985) describe a net-work of in-service courses in institutions throu-ghout China to implement a 12 year program-me from a 10-year one in schools. S i m i l a r l y,Roberts (1984) describes a programme ofs c h o o l-based teacher education in Swaziland,balanced with in-college sessions, and Vila andLima (1984) a distance-learning programme inBrazil. Eshun et al. (1984) pick up on the problems

81

that these latter experience, and emphasise theconstraints for in-service courses for primaryteachers that result from the large numbersinvolved, and the few higher education institu-tions that can provide that support.

An increasingly recurring theme in theseand other literature concerning teacher educa-tion is the notion of teachers as 'reflective prac-titioners'. Lerman and ScottHodgetts (PME,1991, in press) discuss Schön's characterisationof the nature of teaching and develop his inter-pretation:

"Schön ... maintains that teaching isreflective practice, in his discussion ofthe unsatisfactory distinction drawnbetween theoreticians and practitio-ners.

In describing his notion ofr e f l e c t i o n-inaction, Schön ... startsfrom what he calls the practitioner'sknowledge-in-action:

"It can be seen as consisting of strate-gies of action, understanding of pheno-mena, ways of framing the problematicsituations encountered in day-t o-d a yexperience."

When surprises occur, leading to" u n c e r t a i n t y, uniqueness, value-conflict", the practitioner calls on whatSchön terms reflection-i n-action, aquestioning and criticising function,leading to on the spot decision-making,which is "at least in some degreeconscious".

In our situation, we would suggest thatwe are extending the idea of the reflec-tive practitioner. We would agree withSchön that recognising that a teacher ismuch more than a practitioner is essen-tial, both for teachers themselves (andourselves), and for teacher educators,administrators and others. However,when we use the term reflective practi-t i o n e r, we are also describingm e t a-cognitive processes of, for ins-tance, recording those special incidentsfor later evaluation and self-c r i t i c i s m ,leading to action research; consciouslysharpening one's attention in order tonotice more incidents; finding one'sexperiences resonating with others,and/or the literature, and so on. We areconcerned with the transition from thereflective practitioner in Schön's sense,to the researcher (Scott-H o d g e t t s1990), who has a developed criticalattention, noticing interesting andsignificant incidents, and turningthese into research questions. By

analogy with some recent views of thenature of mathematics (e.g. Lerman1986; Scott-Hodgetts PME, 1987),'Mathematics Education' is most use-fully seen, not as a body of externalknowledge, recorded in articles, papersand books, that one reads and uses, butas an accumulation of work uponwhich a teacher can critically draw, toengage with those questions thatconcern and interest her/him(Scott-Hodgetts, in press, a)."

The notion of teacher as researcher,although around the education world for someyears, has only recently appeared in the litera-ture of mathematics education (e.g. ATM 1987;Scott-Hodgetts 1988; Fraser et al., 1989). Anearly statement of the potential of bringingtogether research and teaching was given bythe Theme Group on the Professional Life ofTeachers at the Fifth International Congress onMathematical Education (Cooney et al., 1986):

"It was emphasised that the desirabilityof having teachers participate in resear-ch activities may extend beyond thequestion of what constitutes researchand relate to the professional develop-ment of the teacher by virtue of enga-ging in such activity. That is, not onlycan the teacher contribute to the crea-tion of grounded theory but the teachercan mature and elevate his/her ownexpectations and professional aspira-tions by participating in a reflectiveresearch process." (p.149)

Research is often seen as the prerogative offull-time people based in institutions of highereducation. This is to misunderstand the natureof teaching and the educational experience.There is no doubt that full-time researchershave had the opportunity to develop skills andperspectives in research that class teachershave not had, but the separation of researchfrom practice is one of the major factors thatleads to suspicion of researchers by teachers,and the well-known phenomenon mentionedearly in this chapter, that the findings of resear-ch do not often reach teachers (Scott-Hodgetts,1988; Lerman, 1990a). In a comparison studyof teachers' perceptions of their classrooms,and pupil perceptions of the same classrooms,Carmeli et al. (PME, 1989) found that therewere statistically significant dif f e r e n c e s ,with the teachers being far more positive

82

that their pupils. There was no mention in thereport of whether the teachers were shown theoutcome of the study, and what effect that mayhave had on their own practice, although thatmay of course have taken place later. Researchon teachers, carried out by outsiders, and thenreported in the academic literature, is of muchless value than research in which the teachersare involved, and which they can use in theirown growth as teachers.

Conclusion

There are of course many questions aboutthe influence of the role of the teacher in chil-drens' learning of mathematics, and in thischapter we have described attempts to engagewith some of those questions, and answerthem. Two fundamental problems remain,which were referred to in the introduction tothis chapter: what do 'answers' look like, in thisdomain, and what kinds of research methodo-logies are most suitable to investigate thosequestions?

As with so many issues in mathematicseducation, the place to commence this discus-sion is with attitudes and beliefs. For example,one of the most systematic and developedmethodologies for examining the practices ofthe teacher, in order to achieve children'sunderstanding of mathematics is that develo-ped by the group engaged in recherches endidactigue des mathematiques in France. It isinteresting to examine the assumptions aboutthe nature of classroom interactions that under-pin their system, and to see how the alternativeview described in this chapter offers anothermethodological approach. Brousseau,Chevallard, Balacheff and others have develo-ped notions of the didactical process, theory ofdidactical situations, didactical transpositionetc. Balacheff (1990) describes his basicassumptions and the fundamental problem, ashe conceptualises it:

"So if pupils' conceptions have all theproperties of an item of knowledge,we have to recognise that it might bebecause they have a domain of validi-t y. These conceptions have not beentaught as such, but it appears thatwhat has been taught opens the possi-bility for their existence. Thus, thequestion is to know whether it is

possible to avoid a priori any possibili-ty of pupils constructing unintendedconceptions...

I have suggested that pupils' uninten-ded conceptions can be understood asproperties of the content to be taught orthe way that it is taught." (p.262-263)

This approach to research on teaching is apositivist one, which anticipates the possibilityof solving these problems. It is one whichfocuses on the individual's construction, in thelight of the content and what the teacher does.At the same time, it appears that the construc-tion is to be of an object that exists in some apriori sense. The methodology that Balacheffdescribes assumes that it is possible to specifyall the conditions of the classroom, the mathe-matical content, and so on. Brousseau (1984)recognises the enormity of the task and des-pairs of the possibility of success:

"Research into didactics is itself doo-med to naiveté and/or failure, since it isbecoming more and more difficult, asthe body of knowledge concerned withdidactics grows steadily more compli-cated and technical, for a good studentof mathematics to devote enough timeto absorbing it before he sets out tobreak fresh ground in this field."(p.250)

A methodology of research on teachingwould look different from a perspective whichfocuses on the construction of knowledgewithin an analysis of discursive practices,knowledge as signs that are slippery, not fixed,that is signs that appear clear and certain, butshift when analysed and then shift again. If wecannot specify all aspects of the teaching situa-tion in advance, it is essentially because theactors in that setting bring with them their cul-tural experiences, in the widest sense. Theseexperiences are not 'private', just not complete-ly specifiable. And of course the classroomitself is a whole new social situation with all itsown practices. As a consequence, the teachingprocess is best seen as one which creates situa-tions in which students construct their ownknowledge, and where students' constructionsare valued, even if they are unintended. There areconceptions and misconceptions, but there arealso partial-conceptions and other-c o n c e p t i o n s

83

too. Valuing these contributions recognises thatlearning is what an individual does for herself,not something the teacher does to the student,and also places responsibility on the student forher own learning. It also avoids the studentdependence on the teacher as judge, andenables the development of self-critical skillsin students. All conceptions offered can bebrought into the 'objective' arena of the class-room for examination, to search for 'proofs andrefutations'.

In the context of research methods, onelearns something important about mathematicsteaching when one discovers that in a survey of32 teachers and 1338 pupils the teachers had asignificantly more positive view of a number ofelements of the classroom environment(Carmeli et al., PME, 1989). One learns muchmore, perhaps, in a case study of a teacher whotakes that information and attempts to modifyand develop the relationships, communication,discussion and other interactions in her class-room, and develop ways of evaluating thechanges and the consequences for herself andher students, through action research. Thus,qualitative research is perhaps most usefulwhen it provides insights that stimulate resear-ch by teachers themselves, and where the gene-ralisability is found through other teachers andresearchers recognising, and reflecting on, theimplications for their own work. That is to say,'proof by resonance, where an idea or an analy-sis of a classroom experience or event 'reso-nates' with the experience of the reader or liste-ner, may be more fruitful and more widelyapplicable than 'proof' by significance testing,at least in relation to teaching.

As mentioned above, there are many ques-tions for research and investigation about tea-ching that have been identified in this chapterthat can generate valuable insights into the tea-ching of mathematics. Some recommendationsof important areas for research and investiga-tion follow, and pursuing the theme of the finalsection, there is an overlap of the concerns ofthe teacher, the researcher and the teacher edu-cator. These concerns will be emphasised bypeople according to their needs and interests aswell as circumstances, rather than their particular

task in mathematics education. As they havebeen elaborated in the body of the chapter, theywill be merely listed here.:

(1) the influence of the language and the prac-tices of the mathematics classroom on lear-ning;

(2) ways of rooting the mathematical activitiesof the classroom in the experiences of thechildren, whilst at the same time findingways of drawing them into mathematicaldiscourse;

(3) identifying teachers' beliefs and theinfluence of those beliefs on practice.There are methodological problems, suchas the impact of society's demands, the par-ticular school ethos, mathematics educa-tion jargon, and there are theoretical frame-works being developed in which to set suchstudies;

(4) developing the Inner Teacher, the meta-cognitive functioning that is an essentialpart of learning and anathematising;

(5) the nature of problem-solving and pro-blem-posing, the influence of teachers' per-ceptions of problem-solving, the teachingof problem-solving processes and the rela-tionship between the teacher's role and stu-dent growth as problem-solvers;

(6) classroom-based studies focusing on howteachers define and attempt to realize thegoals that they set, through action research;

(7) investigations on the implications of alter-native learning theories for teaching andteacher education;

(8) the role of the textbook in the social rela-tions of the classroom, and the relation-ships to mathematical knowledge, andalternative forms of text;

(9) the study of the practice of teacher educa-tion in an analogous way to that of the tea-cher in the classroom.

The mathematics classroom is a confu-sing but rich environment. Students bringtheir worlds into it, and express those worldsthrough language. Our role as teachers is toenable them to interact with and integratethe language game of mathematics, in a

84

manner that empowers rather than disempowers,and that enables them to continue to keep mathe-matics as their own, not to see it as belonging tous, the teachers and the authority. It can be main-tained that not being able to arrive at absoluteand certain answers to how that can be done,

makes the science of mathematics teaching justlike all sciences, an open-ended and excitingprocess of experimentation and theory-buil-ding.

85

Stephen Lerman taught mathematics in secondary schools in Britain for a number of years and fora short time in a kibbutz school in Israel. He was Head of Mathematics in a London comprehensi-ve school for 5 years after which he began research in mathematical education whilst 'role-swap-ping' with his wife and looking after two young daughters. His research was on teachers' attitudesto the nature of mathematics, and how that might influence their teaching, and he completed his doc-torate in 1986. Since then he has pursued his interest in theoretical frameworks of mathematical edu-cation/ working in the areas of philosophy of mathematics, equal opportunities, constructivism, andaction research. He is Senior Lecturer in Mathematical Education at South Bank Polytechnic,London, where he teaches on initial and in-service teacher education courses and undergraduatemathematics courses and supervises research students. He is also director of a Curriculum Studiesin Mathematics project. He is co-convenor of a British research group on Research into SocialPerspectives of Mathematics Education which has been meeting since 1986 and co-convenor of aPME Working Group "Teachers as Researchers in Mathematics Education".

1980 R.Karplus (Ed.) Proceedings of the Fourth International Conference for the Psychology ofMathematics Education. Berkeley, California: Lawrence Hall of Science, University ofCalifornia.

1985 L.Streefland (Ed.) Proceedings of the Ninth International Conference for the Psychology ofMathematics Education. Utrecht, The Netherlands: State University of Utrecht, Subfacultyof Mathematics, OW & OC.

1986 Proceedings of the Tenth International Conference for the Psychology of MathematicsEducation. London: University of London, institute of Education.

1987 J.C.Bergeron, N.Herscovics & C.Kieran (Eds) Proceedings of the Eleventh InternationalConference for the Psychology of Mathematics Education. Montréal, Canada: Universitéde Montréal.

87

References in PME Proceedings

Harrison, B., Brindley, S. & Bye, M.P. Effectsof a process-enriched approach to frac-tions (pp.75-81).

C a r r a h e r, T. N. The decimal system.Understanding and notation, vol. 1,(pp.288-303).

De Corte, E. & Verschaffel, L. Working withsimple word problems in early mathemati-cal instruction, vol. 1, (pp.304-309).

Filloy, E. & Rojano, T. Obstructions to theacquisition of elemental algebraic conceptsand teaching strategies, vol. 1,(pp.154-158).

Goddijn, A. & Kindt, M. Space geometrydoesn't fit in the book, vol. 1,(pp.171-182).

Janvier, C. Comparison of models aimed at tea-ching signed integers, vol. 1,(pp.135-140).

A c i o l y, N. & Schliemann, A. D. Intuitivemathematics and schooling in under-standing a lottery game (pp.223-228).

Brown, C.A. A study of the socialization toteaching of a beginning secondary mathe-matics teacher (pp.336-341).

C a r r a h e r. T. N. Rated addition. A c o r r e c tadditive solution to proportions problems(pp.247-252).

Harel, G. & Martin, G. The concept of proofheld by pre-service elementary teachers:Aspects of induction and deduction(pp.386-391).

Hirabayashi, I. & Shigematsu, K.Metacognition: The role of the "innerteacher" (pp.165-170).

S t e ffe, L.P. & Killion, K. Mathematicsteaching: A specification in a constructivist

Booth, L. R. Equations revisited, vol. 1(pp.282-288).

C a r r a h e r, T. N. & Schliemann, A. D.Manipulating equivalences in the marketand in maths, vol. 1 (pp.289-294.)

Dionne, J.J. (1987) Elementary schoolteachers' perceptions of mathematics andmathematics teaching and learning: Twelvecase studies, vol. 2(pp.84-92).

Filloy, E. Modelling and the teaching ofalgebra, vol. 1, (pp.285-300).

Hanna, G. Instructional strategies andachievement in grade 8,vol. 3 (pp.269-274).

Hirabayashi, I. & Shigematsu, K.Metacognition: The role of the "innerteacher" (2),vol. 2(pp.243-249).

Hoyles, C. Geometry and the computerenvironment, vol. 2 (pp.60-66).

Jaworski, B. & Gates, P. Use of Classroomvideo for teacher inservice education,vol. 2(pp.93-99).

1988 A.Borbas (Ed.) Proceedings of the Twelfth International Conference for the Psychology ofMathematics Education. Veszprem, Hungary: Ferenc Genzwein, O.O.K.

88

Kilpatrick, J. What constructivism might be inmathematics education, vol. 1 (pp.3-27).

Laborde, C. Pupils reading mathematics: Anexperimentation, vol. 3(pp.194-200).

Mason, J. & Davis, J. The use of explicitlyintroduced vocabulary in helping studentsto learn, and teachers to teach in mathema-tics, vol. 3(pp.275-281).

McLeod, D.B. Beliefs, attitudes and emotions:affective factors in mathematics learning,vol 1 (pp.170-180).

Nonnon, P. Acquisition d'un langage graphiquede codage par la modélisation en temps réeldes données d'expériences,vol.l,(pp.228-234).

Scott-Hodgetts, R. Preferred learning strategiesand educational/mathematical philoso-phies: An holistic study, vol.3(pp.339-345).

Streefland, L. Free production of fractionmonographs, vol. 1, (pp.405-410).

Waxman, B. & Zelman, S. Children's and tea-chers' mathematical thinking: Helpingmake the connections, vol. 2 (pp.142-148).

Weinberg, D. & Gavalek, J. A social construc-tivist theory of instruction and the develop-ment of mathematical cognition, vol.3(pp.346-352).

A i n l e y, J. Perceptions of teachers' questioningstyles, vol 1 (pp.92-9 9 ) .

Albert, J. & Friedlander, A. Teacher change as aresult of counselling, vol. 1 (pp.101-108).

Alibert, D. Codidactic system in the course ofmathematics: How to introduce it? vol 1( p p . 1 0 9-11 6 ) .

Bliss, J. & Sakonidis, H. Teachers' written expla-nations to pupils about algebra, voll ( p p . 1 3 9-1 4 6 ) .

Brandau, L. Mathematical vulnerability, voll ( p p . 1 9 3-2 0 0 ) .

C a r r a h e r, T. N. Street mathematics and schoolmathematics, vol. 1 (pp.1-2 3 ) .

Hirabayashi, I. & Shigematsu, K. Metacognition- The role of the inner teacher,v o l 2 ( p p . 4 1 0-4 1 6 ) .

Jaworski, B. One mathematics teacher, vol 2( p p . 4 2 5-4 3 2 ) .

Klein, S. and Habermann, G.M. A look at thea ffective side of mathematics learning inHungarian secondary schools (pp.24-5 3 ) .

Kurth, W. The influence of teaching on childrens'strategies for solving proportional and inver-sely proportional word problems, vol2 ( p p . 4 4 1-4 4 8 ) .

Linchevski, L. & Vi n n e r, S. The naive concepts ofsets in elementary teachers,v o l 2 ( p p . 4 7 1-4 7 8 ) .

Mason, J. & Davis, P.J. Cognitive and metaco-gnitive shifts, vol 2(pp.487-494). Ve s z p r e m ,H u n g a r y.

Nesher, P. Beyond constructivism, vol 1( p p . 5 4-7 4 ) .

N e s h e r, P. Beyond constructivism. Learningmathematics at school, vol. 1 (pp.54-7 4 ) .

Rogalski, J. & Robert, A. Teaching and learningmethods for problem-solving: Some theoreti-cal issues and psychological hypotheses, vol2 ( p p . 5 2 8-5 3 5 ) .

Sfard, A. Operational vs structural method of tea-ching mathematics - A case study, vol2 ( p p . 5 6 0-5 6 7 ) .

Simon, M.A. Formative evaluation of a construc-tivist mathematics teacher inservice program,vol 2(pp.576-5 8 3 ) .

Steinbring, H. & Bromme, R. Graphical lessonpatterns and the process of knowledge deve-lopment in the mathematics classroom, vol.2 ( p p . 5 9 3 6 0 0 ) .

Streefland, L. Reconstructive learning, vol. 1( p p . 7 5-9 1 ) .

Tall, D. & Thomas, M. Longer-term conceptualbenefits from using a computer in algebra tea-ching, vol. 2 (pp.601-6 0 8 ) .

van den Brink, J. Cognitive psychology andmechanistic versus realistic arithmetic, vol1 ( p p . 2 0 1-2 0 6 ) .

Vinner, S. & Linchevski, L. Is there any rela-tion between division and multiplication?Elementary teachers' ideas about division,vol 2(pp.625-632).

1989 G.Vergnaud, J.Rogalski & M.Artigne (Eds) Proceedings of the Thirteenth InternationalConference for the Psychology of Mathematics Education. Paris, France: Laboratoire dePsychologie du Développement et de l'Education de l'Enfant.

1990 G.Booker, P.Cobb & T.N.de Mendicuti (Eds) Proceedings of the Fourteenth InternationalConference for the Psychology of Mathematics Education. Oaxtapec, Mexico: ProgramCommittee.

89

Boero, P. Mathematical literacy for all expe-riences and problems, vol. 1 (pp.62-76).

Carmeli, M., Benchaim, D. & Fresko, B.Comparison of teacher and pupil percep-tions of the learning environment in mathe-matics classes, vol. 1 (pp.157-163).

Dorfler, W. Protocols of actions as a cognitivetool for knowledge construction, vol. I(pp.212-219).

Goldin, G.A. Constructivist epistemology anddiscovery learning in mathematics, vol. 2(pp.15-22).

Jaworski, B. To inculcate versus to elicit know-ledge, vol. 2 (pp.147-154).

Mason, J.H. & Davis, P.J. The inner teacher,The didactic tension and shifts of attention,vol. 2 (pp.274-281).

Dougherty, B.J. Influences of teacher cogniti-ve/conceptual levels on problem-solvinginstruction, vol. 1 (pp.119-126).

Even, R. The two faces of the inverse function:Prospective teachers' use of "undoing", vol.1 (pp.37-44).

Flener, F.O. Can teachers evaluate problem sol-ving ability? vol. 1 (pp.127-134).

Grouws, D.A., Good, T.A., & Dougherty, B.J.Teacher conceptions about problem solvingand problem solving instruction, vol. 1,135-142.

Khisty, L.L., McLeod, D.B. & Bertilson, K.Speaking mathematically in bilingualclassrooms: An exploratory study of tea-cher discourse, vol.3 (pp.93-99).

Kuyper, H. & Werf, M.P.C. van der, Math tea-chers and gender differences in- m a t hachievement, math participation and atti-tudes towards math, vol. 1 (pp.143150).

Lea, H. Spatial concepts in the Kalahari(pp.259-266).

Lester, F.K. & Krol, D.L. Teaching students tobe reflective: A study of two grade sevenclasses, vol.1 (pp.151-158). Oaxtepec,Mexico.

Nolder, R. Accommodating curriculum changein mathematics: Teachers' dilemmas, vol. 1(pp.167-174).

Noss, R., Hoyles, C. & Sutherland, R.Teachers' characteristics and attitudes asmediating variables in computer-b a s e dmathematics learning, vol. 1 (pp.175182).

Pimm, D.J. Certain metonymic aspects ofmathematical discourse, vol. 3 (pp.129-136).

Schifter, D. Mathematics process as mathe-matics content: A course for teachers, vol. 1(pp.191-198).

Scott-Hodgetts, R. & Lerman, S. Psychological/ philosophical aspects mathematical acti-vity: Does theory influence practice? vol. 1(pp.199-206).

Simon, M.A. Prospective elementary teachers'knowledge of division, vol. 3 (pp.313-320).

Streun, A. van, The teaching experiment 'heu-ristic mathematics education', vol. 1(pp.93-99).

Tierney, C., Boyd, C. & Davis, G. Prospectiveprimary teachers' conceptions of area, vol.2 (pp.307-315). Oaxtepec, Mexico.

Underhill, R. A. Web of beliefs: Learning toteach in an environment with conflictingmessages, vol.1 (pp.207-213).

Walkerdine, V. Difference, cognition andmathematics education, plenary paper.Published in For the Learning ofMathematics,10(3), (pp.51-56).

Wood, T. The development of mathematicaldiscussion, vol.3 (pp.147-156).

Lerman, S. & Scott-Hodgetts, R. 'Critical inci-dents' in classroom learning - their role indeveloping classroom practice. A s s i s s i ,Italy (in press).

90

1991 Proceedings of the Fifteenth International Conference for the Psychology of MathematicsEducation. Genova, Italy: University of Genova.

Adler, J. (1988). Newspaper-based mathema-tics for adults in South Africa. EducationalStudies in Mathematics, 19, 59-78.

Ascher, M. (1991). Ethnomathematics: a mul -ticultural view of mathematical ideas.California: Brooks Cole.

Ascher, M. & Ascher R. (1981). Code of quipu.Ann Arbor: University of Michigan Press.

ATM (1987). Teacher is (as) researcher. Derby,UK: Association of Teachers ofMathematics.

Ausubel, D.P. (1963). The psychology of mea -ningful verbal learning. New York: Grune& Stratton.

Balacheff, N. (1990). Towards a problématiquefor research on mathematics teaching.Journal for Research in MathematicsEducation, 21 (4), 258-272.

Bauersfeld, H. (1988). Interaction, constructionand knowledge: Alternative perspectivesfor mathematics education. In D.Grouws,T. Cooney, and D.Jones,(Eds.) Perspectiveson research on effective mathematics tea -ching vol. 1, (27-46). Lawrence ErlbaumAssociates for National Council ofTeachers of Mathematics, Reston, VA.

Berliner, D.C., Stein, P., Sabers, D., BrownClaridge, P., Cushing, K. & Pinnegar,S.(1988). Implications of research on peda-gogical expertise and experience formathematics teaching. In D.Grouws,T.Cooney and D.Jones, (Eds.) Perspectiveson research on effective mathematics tea -ching vol. 1 (67-95). Reston, VA: LawrenceErlbaum Associates for National Councilof Teachers of Mathematics.

Biggs, J.B. & Collis, K.F. (1982). Evaluatingthe quality of learning: The SOLO taxono -my. New York: Academic Press.

Bishop, A.J. (1979). Visualising and mathema-tics in a pre-technological culture.Educational Studies in Mathematics, 10,135-146.

Bishop, A. J. (1983). Space and geometry. In R.Lesh & M. Landau (Eds.) Acquisition ofmathematics concepts and pro c e s s e s(pp.175-203). New York: Academic Press, .

Bishop, A.J. (1988). Mathematical encultura -tion: A cultural perspective on theMathematics Education. Dordrecht,Holland: Kluwer.

Bishop, A.J. (1990). Western mathematics: thesecret weapon of cultural imperialism.Race and Class, 32(2), 5165.

Bishop, A.J. & Goffree, F. (1986). Classroomo rganisation and dynamics. InB.Christiansen, A.G.Howson, & M.Otte(Eds.) Perspectives on mathematics educa -tion (pp.309-365). Dordrecht, Holland: D.Reidel (Kluwer).

Bliss, J., Guttierez, R., Koulaidis, V., Ognoru,J. & Sakonidis, H.N. (1989). A cross-cultu-ral study of children's ideas about 'What isreally true' in four curricula subjects:Science, Religion, History andMathematics. In C.Keitel (Chief Ed.)Mathematics, education, and society,( p p . 1 4 1-142). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Blundell, D., Scott-Hodgetts, R. & Lerman, S.(1989). Mathematising the hard stuff. Forthe Learning of Mathematics, 9 (2), 27-31.

Booth, L.R. (1981). Child-methods in seconda-ry mathematics. Educational Studies inMathematics, 12, 29-41.

Bourdieu, P. (1977). Outline of a theory ofp r a c t i c e. Cambridge, UK.: CambridgeUniversity Press.

Bowles, S. & Gintis, H. (1976). Schooling incapitalist America. New York: Routledgeand Kegan Paul.

Brideoake, E. & Groves, I. D. (1939).Arithmetic in action. A practical approachto infant number work. London: Universityof London.

Broomes, D. (1981). Goals of mathematics forrural development. In R.Morris (Ed.)Studies in mathematics education, vol.2,41-59. Paris: UNESCO.

91

Other References

Broomes, D. (1989). The mathematicaldemands of a rural economy. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, ( p p . 1 9-21). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Broomes, D. & Kuperus, P.K. (1983).Problems of defining the mathematicscurriculum in rural communities. InM.Zweng, T.Green, J.Kilpatrick,H.Pollak & M.Suydam (Eds)P roceedings of Fourth Internationalc o n g ress on Mathematical Education,(pp.708-711). Boston: Birkhauser.

B r o p h y, J. (1986). Teaching and learningmathematics: where research should begoing. Journal for Research inMathematics Education, 17(5), 323-346.

Brousseau, G. (1984). The IREMs' Role inHelping Elementary School Teachers. InR.Morris, (Ed.).Studies in mathematicseducation, Volume 3: The MathematicalEducation of Primary-School Te a c h e r s(235-751) Paris: UNESCO.

Brown, A. & Dowling, P. (1989). Towards acritical alternative to inter-nationalism andmonoculturalism in mathematics educa-tion. Centre for Multicultural EducationWorking Paper No. 10. London: Institute ofEducation University of London.

Brown, S. (1984). The logic of problem gene-ration: from morality and solving to posingand rebellion. For the Learning ofMathematics, 4 (1), 9-20

Bryant, P.E. (1982). The role of conflict andagreement between intellectual strategiesin children's ideas about measurement.British journal of DevelopmentalPsychology, 73, 242-251.

Burton, L. (Ed.) (1986). Girls into maths cango. London: Holt, Rinehart & Winston.

C a r r a h e r, T. N. & Meira, L. L. (1989). Learning computer languages andconcepts. The Quarterly Newsletter of theL a b o r a t o ry of Comparative HumanCognition, 11, 1-7.

Carraher, T. N. & Schliemann, A. D. (1988).Culture, arithmetic, and mathematicalmodels. Cultural Dynamics, 1, 180-194.

Carraher, T. N. & Schliemann, A. D. (1990).Knowledge of the numeration systemamong pre-schoolers. In L. Steffe & T.Wood (Eds.)Transforming early childhoodeducation: International perspectives( p p . 1 3 5- 141). Hillsdale, NJ: LawrenceErlbaum.

Carraher, T. N., Carraher, D. W. & Schliemann,A. D. (1985). Mathematics in the streetsand in schools. British Journal ofDevelopmental Psychology, 3, 21-29.

Carraher, T. N., Carraher, D. W. & Schliemann,A. D. (1987). Written and oral mathema-tics. Journal for Research in MathematicsEducation, 18, 83-97.

Charles, R., & Lester F. (1984). An evaluationof a process-oriented instructional programin mathematical problem solving in grades5 and 7. Journal for Research inMathematics Education,15 (1), 15-34.

Chevallard, Y. (1988). Implicit mathematics:their impact on societal needs anddemands. In J.Malone, H.Burkhardt,C.Keitel (Eds) The mathematics curricu -lum: Towards the year 2000, (pp.49-57).Western Australia: Curtin University ofTechnology.

Clements, M.A. (1989). Mathematics for them i n o r i t y. Geelong, Australia: DeakinUniversity Press.

Cockcroft Committee (1982). M a t h e m a t i c scounts. London: Her Majesty's StationeryOffice.

Coombs, P.H. (1985). The world crisis in edu -cation: the . view from the eighties. NewYork: Oxford University Press.

Cooney, T., Goffree, F. & Southwell, B. (1986).Report on Theme Group 2: T h eProfessional Life of Teachers. In M.Carss(Ed.) Proceedings of the Fifth InternationalC o n g ress on Mathematical Education,(pp.146-158). Boston: Birkhauser.

Cooper, T. (1989). Negative power, hegemony,and the primary mathematics classroom: As u m m a r y. In C.Keitel (Chief Ed.),Mathematics, education, and society,( p p . 1 5 0-154). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

92

D'Ambrosio, U. (1985) Ethno-mathematicsand its place in the history andpedagogy ofmathematics. For the Learning ofMathematics, 5 (1), 44-48.

D'Ambrosio, U. (1986). Da realidade aacao.Reflexoes sobre educacao e matema -tica. Campinas (Brazil): Summus Editorial.

D a m e r o w, P. & We s t b u r y, I.- ( 1 9 8 4 ) .Conclusions drawn from the experiences ofthe New Mathematics movement. InP.Damerow, M.E.Dunkley, B.F.Nebres &B . Werry (Eds) Mathematics for all,(pp.22-25). Paris: UNESCO.

D a m e r o w, P., Dunkley, M.E., Nebres, B.F.&Werry, B. (Eds) (1984). Mathematics forall. Paris: UNESCO.

Davis, P.J. (1989). Applied mathematics associal contract. In C.Keitel (Chief-E d . )Mathematics, education, and society,( p p . 2 4-27). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Dawe, L. (1983). Bilingualism and mathemati-cal reasoning in English as a second lan-guage. Educational Studies inMathematics, 14, 325-353.

Dawe, L. (1989). Mathematics teaching and . .learning in village schools of the SouthPacific. In C.Keitel (Chief Ed.),Mathematics, education, and society,( p p . 1 9-21). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

De Lange, J. (1987). Mathematics, insight, andmeaning. Utrecht, Holland: OW & OC.

De Lange, J. (in press). Real tasks and real . .assessment. In R. Davis & C. Maher (Eds.)Relating schools to reality in mathematicslearning.

D o r f l e r, W. and McLone, R.R. (1986).Mathematics as a school subject. InB.Christiansen, A.G.Howson & M.Otte(Eds) Perspectives on mathematics educa -t i o n, (pp.49-97). Dordrecht, Holland:Reidel (Kluwer).

Dormolen, J. van (1989). Values of texts forlearning mathematics for real life. InC.Keitel (Chief Ed.), Mathematics, educa -tion, and society, (pp.94-97). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Durkin, K. and Shire, B. (1991). Language inmathematical education. Milton Keynes:Open University Press.

Educational Studies in Mathematics, (1988).Vol.19, No.2 (the whole issue). (Reprintedas A.J.Bishop (Ed.) (1988). Mathematics,education and culture. Dordrecht, Holland:Kluwer.

Ernest, P. (1989). The impact of beliefs on theteaching of mathematics. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, (pp.99-101). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Eshun, B., Galda, K. & Saunders P. (1984).Support for mathematics teachers: Teacherassociations and radio. In R.Morris (Ed.),Studies in Mathematics Education Volume3: The mathematical education of prima -ry-school teachers, (pp.219-228). Paris:UNESCO.

Fahmi, M.I. & Fayez, M.M. (1985). The in-service education of secondary schoolmathematics teachers. In R.Morris (Ed.),Studies in Mathematics Education Volume4: The education of secondary school tea -chers of mathematics, (pp.141-152). Paris:UNESCO.

F a h r m e i e r, E. (1984). Taking inventory:Counting as problem solving. T h eQuarterly Newsletter of the Laboratory ofComparative Human Cognition, 6, 6-10.

Fasheh, M. (1989). Mathematics in a socialcontext: Math within education as praxisversus within education as hegemony. InC.Keitel (Chief Ed.), Mathematics, educa -tion, and society (pp.84-86) Science andTechnology Education, Document Seriesno. 35. Paris: UNESCO.

Fennema, E., Peterson, P.L., Carpenter, T.P. &Lubinski, C.A. (1990). Teachers' attribu-tions and beliefs about girls, boys andmathematics. Educational Studies inMathematics, 21, 55-69.

Fey, J. (1988). The computer and the teachingof mathematics. In A.Hirst & K.Hirst (Eds)P roceedings of the Sixth InternationalC o n g ress on Mathematical Education,( p p . 2 3 4-236). Budapest: Janos BolyaiMathematical Society.

Fischbein, E. (1987). Intuition in science andmathematics. Dordrecht, Holland: Reidel(Kluwer).

93

Fischbein, E., Deri, M., Nello, M. & Marino,M. (1985). The role of implicit models insolving verbal problems in multiplicationand division. Journal for Research inMathematics Education, 16, 3-17.

Fox, L.H., Brody, L. & Tobin, D. (1980).Women and the mathematical mystique.Baltimore: Johns Hopkins UniversityPress.

Frankenstein, M. (1990). Relearning maths: Adifferent third R-radical maths. London:Free Association Books.

Frankenstein, M. & Powell, A.B. (1989).Mathematics, education and society: empo-wering non-traditional students. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, (pp.157-159). Science andTechnology Education, Document Seriesno.35, Paris: UNESCO.

F r a s e r, B.J., Malone, J.A. & Neale, J.M.(1989). Assessing and improving the psy-chological environment of mathematicsclassrooms. Journal for Research inMathematics Education, 20 (2), 191-201.

Freire, P. (1972). Pedagogy of the oppressed.Harmondsworth: Penguin.

Freudenthal, H. (1973). Mathematics as aneducational task. Dordrecht, Holland:Reidel (Kluwer).

Gay, J. & Cole, M. (1967). The new mathema -tics in an old culture. New York: Holt,Rinehart and Winston.

Gerdes, P. (1985). Conditions and strategies foremancipatory mathematics education inunderdeveloped countries. For theLearning of Mathematics, 5 (1), 15-20.

Ginsburg, H. & Opper, S. (1969). Piaget'stheory of intellectual development. NewJersey: Prentice-Hall Inc.

Giroux, H. (1983). Theory and resistance ineducation. London: Heinemann.

Grando, N. 1. (1988). A matematica no meiorural: pequenos agricultores, estudantes eprofessores. Master's Thesis, Mestrado emPsicologia, Universidade Federal dePernambuco, Recife, Brazil.

Grouws, D. (1988). Introduction. In D.Grouws,T.Cooney, & D.Jones, (Eds.) Perspectiveson research on effective mathematics tea -ching vol. 1. Reston, VA: LawrenceErlbaum Associates for National Councilof Teachers of Mathematics.

Grouws, D., Cooney, T. & Jones, D. (Eds.)(1988). Perspectives on research on effecti -ve mathematics teaching vol. 1. Reston,VA: Lawrence Erlbaum Associates forNational Council of Teachers ofMathematics.

Halliday, M.A.K. (1974). Aspects of sociolin-guistic research. In Interactions betweenlinguistics and mathematics education.Nairobi, Kenya: UNESCO symposium.

Hanna, G. (1989). Girls and boys about equalin mathematics achievement in eighthgrade: results from twenty countries. InC.Keitel (Chief Ed.), Mathematics, educa -tion, and society, (pp.134-137). Scienceand Technology Education, DocumentSeries no.35. Paris: UNESCO.

Harris, M. & Paechter, C. (1991). Work reclai-med: status mathematics in non-elitistcontext. In M.Harris, (Ed.) Schools, mathe -matics and work, (pp.277-283). London:Falmer Press.

Harris, P. (1980). Measurement in tribal abori -ginal communities. D a r w i n, Australia:Northern Territory Department ofEducation.

Harris, P. (1989). Teaching the space strand inaboriginal schools. Darwin, A u s t r a l i a :Northern Territory Department ofEducation.

Hart, K. (Ed.) (1981). C h i l d ren's under -standing of mathematics: Z1-16. London:John Murray.

Hart, K. (1984). Ratio: Children's strategiesand errors. Windsor: NFER-Nelson.

Hart, K., Johnson, D.C., Brown, M, Dickson,L., Clarkson, R. (1989). Children 's mathe -matical frameworks 8-13: A study of class -room teaching. Windsor: NFER-Nelson.

Herndon, J. (1971). How to survive in yournative land . New York: Simon andSchuster.

Hersh, R. (1979). Some proposals for revivingthe philosophy of mathematics. Advancesin Mathematics, 31(1), 31-50.

Higino, Z. M. M. (1987). Por que e dificil paraa crianca aprender a fazer continhas nop a p e l ? Master's Thesis, Mestrado emPsicologia, Universidade Federal dePernambuco, Recife, Brazil.

94

Hirst, A. & Hirst, K. (Eds) (1988). Proceedingsof the Sixth International Congress onMathematical Education. Budapest: JanosBolyai Mathematical Society.

H o ffman, M.R. & Powell, A.B. (1989).Mathematical and commentary writing:Vehicles for student reflection and empo-werment. In C.Keitel (Chief Ed.),Mathematics, education, and society ,( p p . 1 3 1-133). Science and Te c h n o l o g yEducation, Document Series no. 35. Paris:UNESCO.

Holland, J. (1991). The gendering of work. InM.Harris (Ed.) Schools, mathematics andwork, (pp.230-252). London, Falmer Press.

Howson, A.G. (1991). National curricula inm a t h e m a t i c s. Leicester, U.K.: T h eMathematical Association.

Howson, A.G. & Kahane J.P. (Eds) (1990). Thepopularization of mathematics. Cambridge,U.K.: Cambridge University Press.

Howson, A.G. & Mellin-Olsen, S. (1986).Social norms and external evaluation. InB.Christiansen, A.G. Howson and M.Otte(Eds) Perspectives on mathematics educa -tion, (pp.1-48). Dordrecht, Holland: Reidel(Kluwer).

Howson, A.G. & Wilson, B. (1986). Schoolmathematics in the 1990s. Cambridge,U.K.: Cambridge University Press.

Hoyles, C. (1982). The pupil's view of mathe-matics learning. Educational Studies inMathematics,13, 340-372.

Hoyles, C., Armstrong, J., Scott-Hodgetts, R.& Taylor, L. (1984). Towards an understan-ding of mathematics teachers and the waythey teach. For the Learning ofMathematics, 4 (2), 25-32.

Hoyles, C., Armstrong, J., Scott-Hodgetts, R.& Taylor, L. (1985). Snapshots of a mathe-matics Teacher: Some preliminary datafrom the mathematics teacher project. Forthe Learning of Mathematics, 5(2), 46-52.

Hoyrup, J. (1989). On mathematics and war. InC.Keitel (Chief Ed.), Mathematics, educa -tion, and society, (pp.30-32). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Isaacs, I. (1986) National assessment projects:the Caribbean. In M.Carss (Ed.)P roceedings of the Fifth InternationalC o n g ress on Mathematical Education,(p.234). Boston: Birkhauser.

Jaji, G., Nyagura, L. & Robson, M. (1985).Zimbabwe: Support for tutors utilizingvideo and printed modules. In R.Morris,(Ed.) Studies in Mathematics EducationVolume 4: The Education of SecondarySchool Teachers of Mathematics,(pp.153-158). Paris: UNESCO.

Janvier, C. (1981). Use of situations in mathe-matics education. Educational Studies inMathematics, 12, 113-122.

Jurdak, M. (1989). Religion and languages ascultural carriers and barriers in mathema-tics education. In C.Keitel (Chief Ed.),Mathematics, education, and society,( p p . 1 2-14). Science and Te c h n o l o g yEducation, Document Series no. 35. Paris:UNESCO.

Karplus, R. & Karplus, E. (1972). Intellectualdevelopment beyond elementary school 3.Ratio, a longitudinal survey. B e r k e l e y,California: Lawrence Hall of Science,University of California.

Keitel, C. (1986). Social needs in secondarymathematics education. For the Learningof Mathematics, 6 (3), 27-33.

Keitel, C., Damerow, P., Bishop, A. & Gerdes,P. (Eds) (1989). Mathematics, education,and society. Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Kieran, T.E. (1976). Research on rational num -ber learning. Athens, Georgia: Universityof Georgia.

Kreith, K. (1989). The recruitment and trainingof master teachers. In C.Keitel (Chief Ed.),Mathematics, education, and society,( p p . 1 0 1-103). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Lave, J. (1988). Cognition in practice: Mind,mathematics and culture in everyday life.New York: Cambridge University Press.

95

Lawson, M.J. (1990). The case for instructionin the use of general problem-solving stra-tegies in mathematics teaching: A commenton Owen and Sweller. Journal forResearch in Mathematics Education, 21(4),403-410.

Leach, E. (1973). Some anthropological obser-vations on number, time and commonsen-se. In A.G.Howson (Ed.) Developments inmathematics education, (pp.136-1 5 3 ) .Cambridge, U.K.: Cambridge UniversityPress.

Lerman, S. (1983). Problem-solving or know-ledge-centred: the influence of philosophyon mathematics teaching. I n t e r n a t i o n a ljournal of Mathematical Education inScience and Technology,14(1), 59-66.

Lerman, S. (1986). Alternative views of thenature of mathematics and their possibleinfluence on the teaching of mathematics.Unpublished PhD dissertation. London:King's College.

Lerman, S. (1988). The dangers of drill.Mathematics Teacher, 81(6), 412-413.

Lerman, S. (1989a). Investigations, where tonow? In P.Ernest, (Ed.) Mathematics tea -ching: The state of the art, (pp.73-80).London:FalmerPress.

Lerman, S. (1989b). Constructivism, mathema-tics and mathematics education.Educational Studies in Mathematics, 20,211 -223.

Lerman, S. (1990a). The role of research in thepractice of mathematics education. For theLearning Of Mathematics, 10(2),25-28.

Lerman, S. (1990b). Changing focus in themathematics classroom. Paper presented atthe 8th meeting of the group for Researchinto Social Perspectives of MathematicsEducation, South Bank Polytechnic,London.

Lewis, D. (1972). We the navigators. Honolulu,University of Hawaii Press.

Lewis, D. (1976). Observations on route-findingand spatial orientation among the A b o r i g i n a lpeoples of the Western desert region of cen-tral Australia. Oceania, XLV I, 249-2 8 2 .

Lin, F.L. (1988). Societal differences and theirinfluences on children's mathematicalunderstanding. Educational Studies inMathematics, 19, 471-497.

Losada, M.F. de, 8r Marquez, R.L. (1987).Out-of-school mathematics in Colombia.In R.Morris (Ed.) Studies in MathematicsEducation, Vo l . 6, (pp.11 7-132). Paris:UNESCO.

Magina, S. (1991). Understanding angles. Talkpresented at the Institute of Education,London.

Mellin-Olsen, S. (1987). The politics of mathe -matics education. Dordrecht, Holland:Reidel (Kluwer).

Morgan, C. (1991). The form of pupils' writtenlanguage and teachers' judgements ofmathematical work. Unpublished paper.London, UK: South Bank Polytechnic.

Murtaugh, M. (1985). The practice of arithme-tic by American grocery shoppers.A n t h ropology and Education Quart e r l y,16(3), 186-192.

National Research Council (1989). Everybodycounts: A report to the nation on the futureof mathematics education . Wa s h i n g t o n :National Academy Press.

Nebres, 8.F. (1988). School mathematics in the1990's: recent trends and the challenge todeveloping countries. In A.Hirst andK.Hirst (Eds), Proceedings of the SixthInternational Congress on MathematicalEducation (pp.1 1-28). Budapest: JanosBolyai Mathematical Society.

Nickson, M. (1981). Social foundations of themathematics curriculum. Unpublished PhDdissertation. University of London Instituteof Education.

Nickson, M. (1988). Pervasive themes andsome departure points for research intoe ffective mathematics teaching. InD.Grouws, T.Cooney & D.Jones, (Eds.)Perspectives on re s e a rch on effectivemathematics teaching Vol. 1, (pp.245-252).Reston, VA: Lawrence Erlbaum Associatesfor National Council of Teachers ofMathematics.

Noelting, G. (1980). The development of pro-portional reasoning and the ratio concept:Problem structure at successive stages: pro-blem-solving strategies and the mechanismof adaptive restructuring (Part 2).Educational Studies in Mathematics, 11,331-363.

96

Novak, J.D. (1980). Methodological issues ininvestigating meaningful learning. In W . F.Archenhold, R. H. Driver, A . Orton & S.Wood-Robinson (Eds.), Cognitive develop -ment research in science and mathematics,(129-148). Proceedings of an InternationalSeminar. Leeds, U.K.: University of Leeds.

Nunes, T. (in press). Ethnomathematics andeveryday cognition . In D. Grouws (Ed.)Handbook of research in mathematics edu -cation.

Owen, E. & Sweller, J. (1989). Problem sol-ving as a learning device? Journal forResearch in Mathematics Education, 20(3),322-328.

Pedro, J.D., Wolleat, P., Fennema, E.- &B e c k e r, A.D. (1981). Election of highschool mathematics by females and males:attributions and attitudes. A m e r i c a nEducational Research Journal, 18,207-218.

Peterson, P.L. (1988). Teaching for higher-order thinking in mathematics: The chal-lenge for the next decade. In D.Grouws,T.Cooney, & D.Jones, (Eds.) Perspectiveson research on effective mathematics tea -ching Vol. 1, (pp.2-26). Reston, VA :Lawrence Erlbaum Associates for NationalCouncil of Teachers of Mathematics.

Piaget, J. & Inhelder, B. (1956). The child'sconception of space. London: Routledgeand Kegan Paul.

Piaget, J., Inhelder, B. & Szeminska, A. (1960).The child's conception of geometry.London: Routledge and Kegan Paul.

Pimm, D. (1987). Speaking mathematically.London: Routledge and Kegan Paul.

Pimm, D. J. (1984). Who is We? MathematicsTeaching, No. 107, 3942.

Porter, A., Floden, R., Freeman, D., Schmidt,W. & Schwille, J. (1988). Content determi-nants in elementary school mathematics. InD.Grouws, T.Cooney, & D.Jones, D. (Eds.)Perspectives on re s e a rch on effectivemathematics teaching Vol. 1, (pp.96-113).Reston, VA: Lawrence Erlbaum Associatesfor National Council of Teachers ofMathematics.

Quilter, D. and Harper, E. (1988). Why wedidn't like mathematics and why we can'tdo it. Educational Research, 30( 2 )121-129.

Reed, H. & Lave, J. (1981). Arithmetic as a toolfor investigating relationships between cul-ture and cognition. In R. W. Casson (Ed.)Language, culture, and cognition:A n t h ropological perspectives ( p p . 4 3 7-4 5 5 ) .New York: McMillan.

Resnick, L. (1987). Learning in school and out.Educational Researcher, 16, 13-21.

Revuz, A. (1978). Change in mathematics edu-cation since the late 1950s - ideas and rea-lisations, France. Educational Studies inMathematics, 9, 171-181.

Roberts, B. (1984) The in-service educationproject in Swaziland (1973-77). InR.Morris, (Ed.) Studies in mathematicseducation vol. 3: The mathematical educa -tion of primary-school teachers,(pp.205-212). Paris: UNESCO.

Robinson, I. (1989). The empowerment para-digm for the professional development ofteachers of mathematics. In N. F.Ellerton &M . A .Clements (Ed s . ) School mathema -tics: The challenge to change ,(pp.269-283). Geelong, Australia: DeakinUniversity.

Robitaille, D.F. & Garden, R.A. (Eds) (1989).The IEA study of mathematics II: Contextsand outcomes of school mathematics.Oxford: Pergamon.

Ronan, C.A. (1981). The shorter science andcivilization in China: vol. 2. Cambridge,U.K.: Cambridge University Press.

Rosenberg, D. C. (1989). Knowledge transferfrom one culture to another. HEWET fromthe Netherlands to Argentina. In C. Keitel(Chief Ed.), Mathematics, education, ands o c i e t y ( p p . 8 2-84). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Sanchez, S.R. (1989). The necessity of popula-rizing mathematics via radio programmes.In C.Keitel (Chief Ed.), Mathematics, edu -cation, and society, (pp.61-62). Scienceand Technology Education, DocumentSeries no. 35. Paris: UNESCO.

Saxe, G.B. (1990). Culture and cognitive deve -lopment: Studies in mathematical unders -tanding. Hillsdale, N.J.: Erlbaum.

97

Schliemann, A. D. & Nunes, T. N. (1990). Asituated schema of proportionality. BritishJournal of Developmental Psychology, 8,259-268.

Schoenfeld, A.H. (1987). Cognitive scienceand mathematics education. Hillsdale,N.J.:Erlbaum.

Schultz, D. 1989. PRIMITE-projects inreal-life integrated mathematics in teachereducation. In C. Keitel (ChiefEd.)Mathematics, education, and society( p p . 9 1-94). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Scott-Hodgetts, R. (1984). Mathematics tea-chers: Pupils' views and their attributionalinterpretations of classroom episodes. InProceedings of the British Society for thePsychology of Learning Mathematics,Newsletter No. 13, 9-11.

Scott-Hodgetts, R. (1988). Why should tea-chers be interested in research in mathema-tics education? In D. Pimm (Ed.)Mathematics, teachers and childre n ,( p p . 2 6 3-272). London: Hodder andStoughton.

Scott-Hodgetts, R. (in press, a). Mathematicslearners and learning . Milton Keynes:Open University Press.

Scott-Hodgetts, R. (in press, b). Qualified, qua-litative research: The best hope in our questfor quantifiable improvements. Paper pre-sented at the F o u rth Conference onSystematic Co-operation Between Theoryand Practice in Mathematics Education.Brakel, Germany.

S c r i b n e r, S. (1984). Product assembly:Optimizing strategies and their acquisition.The Quarterly Newsletter of theL a b o r a t o ry of Comparative HumanCognition, 6 (1-2), 11- 19.

Sells, L.W. (1980). The mathematics filter andthe education of women and minorities. InL.H.Fox, L.Brody & D.Tobin (Eds),Women and the mathematical mystique( p p . 6 6-76). Baltimore: Johns HopkinsUniversity Press.

S h a u g h n e s s y, J.M. & Burg e r, W. F. (1986).Assessing children's intellectual growth ingeometry. Final report. Oregon: OregonState University.

Shuard, H. & Rothery, A. (1984) Children rea -ding mathematics. London: John Murray.

Siemon, D. (1989a). How Autonomous is theTeacher of Mathematics? In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, ( p p . 9 7-99). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Siemon, D. (1989b). Knowing and believing isseeing: A constructivist's perspective ofchange. In N.F. Ellerton & M.A. Clements(Eds.) School mathematics: The challengeto change , (pp.250-268). Geelong,Australia: Deakin University.

Silver, E.A. (1987). Foundations of cognitivetheory and research for mathematics pro-b l e m-solving. In A.H.Schoenfeld (Ed.),Cognitive science and mathematics educa -tion. Hillsdale, N.J.: Erlbaum.

Smart, T. & Isaacson, Z. (1989) "It was nicebeing able to share ideas": women learningmathematics. In C.Keitel (Chief Ed.),Mathematics, education, and society,( p p . 11 6-118). Science and Te c h n o l o g yEducation, Document Series no.35. Paris:UNESCO.

Steinbring, H. (1989). Routine and meaning inthe mathematics classroom. For theLearning of Mathematics, 9(1), 24-33.

Stephens, W.M., Lovitt, C., Clarke, D. &Romberg, T.A. (1989). Principles for theprofessional development of teachers ofmathematics. In N.F. Ellerton &M.A.Clements (Eds.) School mathematics:The challenge to change, (pp.220-2 4 9 ) .Geelong, Australia: Deakin University

Stigler, J. W. & Baranes, R. (1988). Culture andmathematics learning. In E. Z. Rothkopf(Ed.)Review of research in education (vol.1 5 , p p . 2 5 3-306). Washington, DC:American Educational ResearchAssociation.

Suffolk, J. (1989). The role of the mathematicsteacher in developing countries. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, (pp.103-105). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Sullivan, H.S. (1955). Conceptions of modernpsychiatry. London: Tavistock.

98

Sweller, J. (1990). On the limited evidence forthe effectiveness of teaching general pro-b l e m-solving strategies. Journal ForResearch in Mathematics Education, 21(4),411-415.

Swetz, F.J. (Ed.) (1978). Socialist mathematicse d u c a t i o n. Southampton, Pa: Burg u n d yPress.

Thom, R. (1972). Modern mathematics: Doesit exist? In A.G.Howson, (Ed.) D e v e l -opments in mathematics education.Cambridge, UK: Cambridge UniversityPress.

Thompson, A.G. (1984). The relationship ofteachers' conceptions of mathematics andmathematics teaching to instructional prac-tice. Educational Studies in Mathematics,15, 105-127.

Thompson, V. (1989). FAMILY MATH: lin-king home and mathematics. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, ( p p . 6 2-65). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Travers, K.J. & Westbury, I. (Eds) (1989). TheIEA study of mathematics I: Analysis ofmathematics curr i c u l a. Oxford. U.K.:Pergamon.

Tsang, S.-L. (1988). The mathematics achieve-ment characteristics of A s i a n - A m e r i c a nstudents. In R.R. Cocking & J.P.Mestre(Eds), Linguistic and cultural influences onlearning mathematics ( p p . 1 2 3-1 3 6 ) .Hillsdale, N.J.: Erlbaum.

Tversky, A. & Kahneman (1982). Judgementunder uncertainty: heuristics and biases. InD.Kahneman, P.Slovic & A.Tversky (Eds),Judgement under uncertainty. Cambridge,U.K.: Cambridge University Press.

Tymoczko, T. (Ed.) (1985). New directions inthe philosophy of mathematics. Boston:Birkhauser.

Vergnaud, G. (1983). Multiplicative structures.In R. Lesh & M. Landau (Eds.)Acquisitionof mathematics concepts and pro c e s s e s(pp.125-174). New York: Academic Press.

Verma, G. & Pumfrey, P. (1988). Educationalattainments - Issues and outcomes in multi -cultural education. London: Falmer.

Vila, M. do Carmo & Lima, R. M. de Souza(1984). A case-study of an in-service trai-ning programme for practising teacherswithout degrees. In R.Morris, (Ed.) Studiesin mathematics education volume 3: Themathematical education of primary-schoolteachers, (pp.213-218). Paris: UNESCO.

Volmink, J.D. (1989). Non-school alternativesin mathematics education. In C.Keitel(Chief Ed.), Mathematics, education, ands o c i e t y, (pp.59-61). Science andTechnology Education, Document Seriesno.35. Paris: UNESCO.

Walkerdine, V. (1988). The mastery of reason.London: Routledge.

Walkerdine, V. (1989). Counting girls out.London: Virago.

Weber, M. (1952). Essays in sociology (trans).Cambridge: MIT Press.

Wiener, B. (1986). An attributional theory ofmotivation and emotion. New Yo r k :Springer.

Wi n t e r, M.J., Lappan, G., Phillips, E. &Fitzgerald, W. (1986). Middle gradesmathematics project: spatial visualisation.New York: Addison Wesley.

Xiangming, M., Fangjuan, T., Daosheng, M. &Ghangfeng, J. (1985). A survey of seconda-ry school mathematics teaching in China.In R.Morris, (Ed.) Studies in mathematicseducation volume 4: The education ofsecondary school teachers of mathematics,(pp.159-169). Paris: UNESCO.

Young, M.F.D. (Ed.) (1971). Knowledge andcontrol. London: Collier-Macmillan.

Zaslavsky, C. (1973). Africa counts. Boston:PWS Publishers.

Zepp, R. (1989). Language and mathematicseducation. Hong Kong: API Press.

Zepp, R. (1989). Language and mathematicseducation. Hong Kong: API Press.

Zweng, M., Green, T., Kilpatrick, J., Pollak, H.& Suydam, M. (1983). Proceedings of theF o u rth International Congress onMathematical Education. Boston:Birkhauser.

99