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AUTHOR Bishop, Alan J.; And OthersTITLE Significant Influences on Children's Learning of
Mathematics. Science and Technology EducationDocument Series, No. 47.
INSTITUTION United Nations Educational, Scientific, and CulturalOrganization, Paris (France).
REPORT NO ED-93/WS-20PUB DATE 93NOTE 102p.PUB TYPE Viewpoints (Opinion/Position Papers, Essays, etc.)
(120) -- Collected Works Serials (022)
EARS PRICE MF01/PC05 Plus Postage.DESCRIPTORS Community Influence; Cultural Context; Educational
Quality; Elementary Secondary Education;*Instructional Materials; *Mathematics Education;*Prior Learning; *Social Influences; StudentExperience; *Teacher Influence; Teacher Role
IDENTIFIERS Mathematics Education Research; *Psychology ofMathematics Education
ABSTRACTThis book is intended to bring to a wider audience
research concerned with the significant influences on the quality ofmathematics learning taking place in schools. The four essays are:(1) "Influences From Society" (Alan Bishop); (2) "The Socio-culturalContext of Mathematical Thinking: Research Findings and EducationalImplications" (Terezinha Nunes); (3) "The Influences of TeachingMaterials on the Learning of Mathematics" (Kathleen Hart); and (4)"The Role of the Teacher in Children's Learning of Mathematics"(Stephen Lerman). A list of references from the International Groupfor the Psychology of Mathematics Education conference proceedingscontains 77 citations and a list of other references contains 194citations. (MKR)
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Cover photos
1. Photo UNESCO/Paul Almasy
2. Photo UNATIONS
3. Photo UNESCO/D. Bahrman
4. Photo rights reserved
Paris, 1993
ED-93PNS-20
Science and Technology EducationDocument Series No. 47
SIGNIFICANT INFLUENCESON CHILDREN'S LEARNING OF MATHEMATICS
by
Alan J. BishopKathleen Hart
Stephen LermanTerezinha Nunes
Education Sector
UNESCO
4
Table of Contents
Introduction 1
Influences from society
Alan J.Bishop
3
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
5
Introduction
Mathematics has become one of the mostimportant subjects in the school curriculumduring this century. As modern soceties haveincreased in complexity and as thatcomplexity has accompanied rapidtechnological development, so the teachingof mathematics has come under increasedscrutiny.
Research in mathematics educationgenerally has also become more and moreimportant during that time, but has struggledto keep up with the increasingly complexquestions asked of it. There was a time whenmathematics learning could be conceptualisedsimply as the result of teaching the subjectcalled mathematics to a class of children inthe privacy of their classroom. As long as theconcerns were largely quantitative, such ashow to provide more mathematics teachers,more mathematics classrooms, and moremathematics teaching for more students, thatpicture appeared to be adequate. We nowknow that that simplistic picture will nothelp in answering any of the significantquestions of recent decades whichpredominantly concern the quality of themathematics learning achieved.
Of course it is the case that in many, oreven most, countries in the world, there arestill quantitative issues surroundingmathematics learning. Indeed, in thedeveloping world there is often littleopportunity for the luxury of engaging withsubtle issLes of quality in the face ofhorrendous quantitative problems ofeducational provision. Nevertheless, as theUNESCO publication Mathematics for allshowed (Damerow et al.,1984), one cannotdivorce the problems of mathematicslearning from the wider problems of society,and thus there is every reason for all mathe-matics educators to become aware of thefactors which have the potential to affectthe quality of mathematics learning takingplace in schools. This book is intended to helpbring to a wider audience research whichinforms us about the significant influences onthat quality.
In particular, the authors haveidentified four groups of influences which
6
appear to be of crucial importance for learnersof mathematics. Firstly there are thedemands, constraints and influences from thesociety in which the mathematics learning istaking place. These, in a sense, set theknowledge and emotional context withinwhich the meaning and importance ofteaching and learning mathematics areestablished. Recent research hasdemonstrated that it no longer makes sense totry to consider mathematics learning asabstract and context-free, essentially becausethe learner cannot be abstract or context-free.The research problems centre on which of themany societal aspects are of particularsignificance, on how to study their influences,and on what if anything to do about them.
The second set of influences concern theknowledge, skills and understanding whichthe learners develop outside the schoolsetting and which have significance for theirlearning inside the school. This topic forresearch has only developed relativelyrecently, but its findings have surprised manymathematics teachers with its implicationsfor their work. One of the great educationalchallenges of the present time concerns howschool mathematics teaching should takelearners out-of-school knowledge intoaccount.
The third set of influences on children'slearning of mathematics come from theteaching materials and aids to learning inthe classroom. These have become more subtleand varied - from the textbook to thecomputer - and have increased in number andimportance considerably over the lastdecades. In the face of such development, theneed for continuing research and analysisconcerning the significance of these influenceshas become even more important.
The fourth and final influence is not anew one at all - indeed it could be thought ofas the oldest influence on mathematicslearning - the teacher. Every learner canquote the memory of a particularlyinfluential teacher, whether good or bad, andevery teacher knows the feeling of influencewhich the position gives them. There has inthe last decade been a growth of interest in
1
research into the influence which teacherscan, and do, have on the mathematicslearners in their charge, and it is importantto bring these ideas to a wider audience.
The four authors have drawn on recentlypublished research which has been reportedin publications and at academic conferences,but in particular, as we are all long-standingmembers of the International Group for thePsychology of Mathematics Education(PME),we have made sustantial reference to ideaswhich have developed within that context.The major research strands of PME have beenfully documented in the following book:
Mathematics and cognition, edited byPear la Nesher and Jeremy Kilpatrick,Cambridge University Press, Cambridge, 1990
under the headings of: epistemology, learningearly arithmetic, language, learninggeometry, learning algebra, and advancedmathematical thinking. In addition, the
2
references at the back of this book are in twosections - the first, to particular papers incertain Proceedings of the annual conferencesof PME, 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 publicationpossible, and to our many colleagues aroundthe world who share in the development andcommunication of knowledge in this field.
Finally, we dedicate this book to learnersof mathematics everywhere.
Influences from Society
Alan J. Bishop
Schools and individual learners exist withinsocieties and in our concern to ensure themaximal effectiveness of school mathematicsteaching, we often ignore the educationalinfluence of other aspects of living within aparticular society. It is indeed tempting formathematics educators particularly to viewthe task of developing mathematics teachingwithin their particular society as beingsimilar to that of colleagues elsewhere,largely because of their shared beliefs aboutthe nature of mathematical ideas. In realitysuch tasks cannot deal with mathematicsteaching as if it is separable from theeconomic, cultural and political context of thesociety. Any analyses which are to have anychance of improving mathematics teachingmust deal with the people - parents,teachers, employers, Government officialsetc. and must take into account theprevailing attitudes, beliefs, and aspirationsof the people in that society. The failure ofthe New Math revolution in the 60's and theearly 70's was a good example of thisphenomenon (see Damerow and Westbury,1984).
It is therefore my task in this firstchapter to explore those aspects of societieswhich may exert particular influences onmathematics learning. These may happeneither intentionally or unintentionally.Societies establish educational institutionsfor intentional reasons - and formalmathematics education is directly shapedand influenced by those institutions indifferent ways in different societies.Additionally, a society is also composed ofindividuals, groups and institutions, whichdo not have any formal or intentionalresponsibility for mathematics learning.They may nevertheless frame expectationsand beliefs, foster certain values andabilities, and offer opportunities and images,which will undoubtedly affect the waysmathematics is viewed, ui.derstood andultimately learnt by individual learners.
Coombs (1985) gives us a usefulframework here. In discussing various 'crisesin education he argues that education shouldbe considered as a very broad phenomenon,
rather than being a narrow one, and thatthere are different kinds of education. In hiswork he separates Formal Education (FE)from Non-formal Education (NFE), andInformal Education (IFE).
Formal Education, he says, "generallyinvolves full-time, sequential studyextending over a period of years, within theframework of a relatfvely fixed curriculum"and is "in principle, a coherent, integratedsystem, (which) lends itself to centralizedplanning, management and financing" (p.24),and is essentially intended for all youngpeople in society.
In contrast NFE covers "any organised,systematic, educational activity, carried onoutside the framework of the formal system,to provide selected types of learning toparticular subgroups in the population, adultsas well as children" (p.23). In contrast to FE,NFE programs "tend to be part-time and ofshorter duration, to focus on more limited,specific, practical types of knowledge andskills of fairly immediate utility toparticular learners" (p.24).
Finally IFE refers to "The life-longprocess by which every person acquires andaccumulates knowledge, skills, attitudes andinsight from daily experiences and exposureto the environment.... Generally informaleducation is unovganized, unsystematic andeven unintentional at times, yet it accounts forthe great bulk of any person's total lifetimelearning including that of even a highly'schooled' person" (p.24).
We shall organise this chapter'scontribution around these three differentkinds of education, looking particularly atthe influences from society on the three kindsof mathematical education. Finally therewill be a discussion of some of the significantimplications which result from this analysis.
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
Paradoxically, at another level of thinkingit will not be clear to many people just whatinfluence any particular society could haveon its mathematics learners, in terms of howthis will differ from that which any othersociety might have. Indeed, mathematicsand possibly science have, I suspect, been theonly school subjects assumed by many peopleto be relatively unaffected by the society inwhich the learning takes place. Whereas forthe teaching of the language(s) of thatsociety, or its history and geography, its artand crafts, its literature and music, its moraland social customs, which all would probablyagree should be considered specific to thatsociety, mathematics (and science) has beenconsidered universal, generalised, andtherefore in some way necessarily the samefrom society to society. So, is this a tenableview? What evidence is there?
The formal influences on themathematical learners will come throughfour main agents - the intended mathematicscurriculum, the examination and assessmentstructures, the teachers and their teachingmethods, and the learning materials andresources available. The last two aspectswhich 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, theintended mathematics curriculum, and theexaminations.
The intended mathematics curriculum
If we consider the mathematics curricula indifferent countries, our first observation willbe that they do appear to be remarkablysimilar across the world. Howson and Wilson(1986) talk of the "canonical curriculum"(p.19) which appears to exist in manycountries. They describe "the familiar schoolmathematics curriculum (which) wasdeveloped in a particular historical andcultural context, that of Western Europe inthe aftermath of the Industrial Revolution".They point out that "In recent decades, whatwas once provided for the few has now beenmade available to - indeed, forced upon - all.Furthermore, this same curriculum has ',eenexported, and to a large extent voluntarilyretained, by other countries across the world.The result is an astonishing uniformity ofschool mathematics curricula world-wide"(p.8).
4
This fact has made it possible to conductlarge-scale multi-country surveys andcomparisons of mathematical knowledge,skills and understanding such as those by theInternational Association for EducationAchievement (LEA) whereas such com-parisons would probably be unthinkable in asubject like history. Indeed, one of the mainresearch issues in the Sec.,nd InternationalMathematics Survey (SIMS) was whetherthe 'same' mathematics curricula were beingcompared (see Travers and Westbury, 1989).Some idea of the extent of the agreementsfound can be gained from Table 1 (below)which shows the relative importanceaccorded by different countries to potentialtopics and behavioural categories to beincluded in the Population A (13 year olds)assessment. Travers and Westbury concludedfrom their further analysis of the dataconcerning Table 1 that 'The only topic forwhich there appears to be a substantialproblem of mismatch is Geometry. Indeed amajor finding of the Study proved to be thegreat diversity of curricula in geometry forPopulation A around the world" (p.32).
It could be argued that such internationalsurveys, and the comparative achievementdata they generate, encourage the idea thatthe mathemadcs curricula in differentcountries should be the same, particularlywhen, as in this case, they are seeking thehighest common factors of similarity. At thevery least, this approach could well lead tomathematics educators in many countriesanxiously looking over their shoulders attheir colleagues in other counties to see whattheir latest curricular trends are. Onewonders what would result from a researchstudy which sought to find the differencesbetween mathematics curricula existing indifferent countries. I suspect we might wellfind a different picture from that portrayedin Table 1. The recent book by Howson (1991)which documents the mathematics curriculain 14 countries, is therefore a welcomeaddition to our sources.
Indeed societies may well not onlyinfluence the national intended curriculum inways in which SIMS would not reveal, butmay also influence the intended curriculum atmore local levels.
Table 1
Content topics
Behavioral Categories'Computation
ComprehensionApplication
Analysis
OCO Arithmetic001 Natural numbers and whole numbers V V V I
002 Common fractions V VI I
003 Decimal fractions V V VI004 Ratio, proportion, percentage V VI I
005 Number theory I I
006 Powers and exponents I I
007 Other numeration systems008 Square roots I I
009 Dimensional analysis I I
too Algebra101 Integers V VI I
102 Rationals103 Integer exponents Is
1C4 Formulas and algebraic expressions105 Polynomials and rational expressions I Is106 Equations and inequations (linear only) V I I Is107 Relations and functions I I I108 Systems of linear equations109 Finite systems110 Finite sets I I I111 Flowcharts and programming112 Real numbers
200 Geometry201 Classification of plane figures I V I Ls
202 Properties of plane figures I V I I
203 Congruence of plane figures I I I Is204 Similarity of plane figures I I I Is205 Geometric constructions Is Is Is206 Pythagorean triangles Is Is Is
207 Coordinates I 1 I Is208 Simple deductions Is I I I
209 Informal transformations in geometry I I I
210 Relationships between lines andplanes in space
211 Solids (symmetry properties) Ls Is Is -212 Spatial visualization and
representation Ls Is -213 Orientation (spatial) Is214 Decomposition of figures215 Transformational geometry Is Is Is
3oo Statistics301 Data collection J5 I I
302 Organization of data I I I Is303 Representation of data I I I Is304 Interpretation of data (mean,
median, mode) I I I
305 L:ombinatoric306 Outcomes, sample spaces and events Is
307 Counting of sets, P(AB), P(AB),independent events
308 Mutually exclusive events309 Complementary events
400 Measurement401 Standard units of measure V V V -402 Fstirna lion I I I
403 Approximation I I I
404 Determination of measures:areas, volumzs, etc. V VI I
'Rating scale: V = very important; I - important; Is - important for some systems.A dash (-) = not Important.
ST WA
Moreover in the SIMS study there werestriking differences between the intended andthe implemented curricula in variouscountries, and also striking variations in thelatter within national systems. Some of thatvariation could undoubtedly be due to localintended variations: "With the significantexception of Japan, all systems haverelatively high indices of diversity and itmay be that such diversity is a reflection ofthe responses that teachers make to theoverall variations of readiness for Algebrathat they find at this level" (Travers andWestbury, 1989, p.131).
It is well known that the intendedmathematics curricula certainly do varybetween schools in differentiated systems,i.e. where there is not a comprehensive(single) school structure. In some highlydifferentiated systems in W.Europe forexample the grammar school, the gymnasiumand 'he lycée have very different curriculafrom the other schools in those systems. TheSIMS study showed (Travers and Westbury,1989) the following data which gives the'Opportunity to Learn' figures for Arithmeticand Algebra in the different sectors of thedifferentiated systems in Finland,Netherlands and Sweden (Opportunity toLearn is a % measure of the availability ofthat topic in the curriculum).
Finland
Arithmeticmedian
Algebramedian
Long course 74 73Short course 74 67
NetherlandsVWO/HAVO 87 83MAVO 83 80LTO 74 53LHNO 70 47
SwedenAdvanced 70 57General 61 25
As an example of how such curriculardifferentiation can influence performance, aHungarian study (Klein and Habermann,PME, 1988) produced the following data:
Type of school and curriculum Mean test score
Grammar schoolsCurriculum "Mathematics 11"(special)
Grammar schoolsCurriculum "Mathematics I'(special)
6
13.41
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 sub-sets of the 'total' national curriculum. Onereads phrases like "watered-down", "simplerversion" etc. which suggest that different-iated systems will tend towards what Keitel(1986) says was the situation "in England nowat the secondary level: a sophisticated,highly demanding mathematics course forthose who continue their studies, and next tonothing for the rest" (p.32). The different-iated mathematics curriculum therebyreflects the differentiation which the societyappears to want, and which it seeks toperpetuate through its formal educa:ionalstructures.
In fact there is no a priori reason at allwhy mathematics curricula should be thesame in all countries. Mathematics in schoolscan be considered in similar ways to art,history, and religion, where there is nonecessity for curricula to be the same indifferent societies. We now know, from theethnomathematics research literature whichhas been gathered in the last twenty n!ars,about the enormous range of mathematicaltechniques and ideas which have beendevised in all parts of the world (see Ascher,1991). We have evidence from all continentsand all societies of symbolic systems ofarithmetical and geometrical naturesdevised to help humans extend theiractivities within the physical and socialenvironments that have grown to be ever morecomplex. From the Incas with their quipus tohelp with their accounting (Ascher andAscher, 1981), to the Chinese with theirdetailed geomantic knowledge for designingcities (Ronan, 1981), or from the Igbo'ssophisticated counting system (Zaslavsky,1973) to the Aboriginal Australian's supremespatial and locational sense (Lewis, 1976) theknown world is full of examples of rich andvaried indigenous rnathe-matical knowledgesystems.
11
So the question needs to be asked: Wheredoes the expectation of a universalmathematics curriculum come from? Of coursewe would expect there to be recognisablesimilarities, as Travers and Westbury show,just as we recognise similarities between thecurricula of other subjects across differentsocieties. But recognising similarities istotally different from expecting universality.In part the expectation derives from the factthat countries and societies are not isolated;there is much inter-societal communicationand there has been for many centuries.
Societies influence the intendedmathematics curriculum in most countriesthrough certain nationally or regionallystructured organisations. In the case of strongcentralized governments the curriculum willbe the responsibility of the educationministry, while in more decentralizedsystems such as the USA, Australia andCanada the power for deciding on theintended curriculum rests with the state orlocal government. For example, in the UnitedKingdom the present government has recentlyinstituted a national curriculum, adevelopment in which the highly politicalnature of national curricular decision-makinghas been rather obviously demonstrated. Wehave therefore seen the typical politicalpressure groups being very active - the 'backto basics groups led by traditionalistsamongst the employers and the government,the teachers and educators concerned aboutthe erosion of their influence by the centralgovernment's 'interference', the moreprogressive industrialists who want to ensurethat school leavers can compete with the'best of the rest of Europe', allied withparents whose genuine concerns about theirchildren's futures are coloured by lurid mediareporting about the poor comparativeperformance by UK pupils in various (andsometimes suspect) international surveys.
Governments, education civil servants,political academics and others witheducational power are thus nowadays veryaware of what is happening in othercountries because they feel that they have tobe. The competitive economic and politicalethic demands it. International conferences,'expert' visits, exchanges, articles, books andreports, all enable ideas from one country tobe available to others. But there is more to itthan mere communication however.
The expectation of a universalmathematics curriculum has another basis.As well as communication, there has beengradual acculturation by dominant cultures,there has been the assimilation of new ideasbelieved to be more important thantraditional ones, and there has also beencultural imperialism, practised of course on alarge scale by colonial gow:rnments (seeBishop, 1990 and Clements, 1989). There hasin particular been the widespreaddevelopment of a belief in the desirability oftechnological and industrial growth.Underlying this technological revolution hasbeen the mathematics of decontextualisedabstraction, the mathematics of system andstructure, the mathematics of logic,rationality and proof, and therefore themathematics of universal applicability, ofprediction and control.
What we have witnessed during the lasttwo centuries is nothing less than the growthof a new cultural form, which can be thoughtof as a Mathematico-Technological (MT)culture (see Bishop, 1988). The growth of theidea of universally applicable mathematicshas gone hand-in-hand with the growth ofuniversally applicable technology.Developments within each have fed theother, and with the invention and increasedsophistication of computers, the nexus is trulyforged.
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.We now are in danger of taking the ideas andvalues of universally applicable mathe-matics so much for granted that we fail tonotic- them, or to question them, or to see thepossibility of developing alternatives. Theacceptance of universally applicablemathematics has had a particularly pro-found impact on mathematics curricula in allcountries and in all societies. The impetus tobecome ever more industrialised andtechnologically developed in the so-called"under-developed" countries has been under-pinned by the belief in the importance ofadopting the mithematics and sciencecurricula of the more - industrialisedsocieties. It is the belief in the power of theideas of universally applicable mathematicswhich has created the expectation of theuniversal mathematics curriculum.
12 7
So, all around the world, in societieswith different economic structures (see Dawe,1989), in societies with different politicalstructures (see Swetz, 1978), and in societieswith different religious bases (see Jurdak,1989), learners find themselves grappling, notalways successfully, with the intricacies ofarithmetical algorithms, algebraic symbol-isations, geometrical theorems. Later on, ifthey pass the required examinations, theymeet infinitesimal calculus, abstract alge-braic structures, other geometries and appli-cable mathematics.
Moreover, until perhaps ten years ago,this situation was largely unquestioned. Nowit is being seriously challenged from variousquarters and we can see different groupswithin societies trying to exert ratherdifferent influences on their school curricula.The major challenges which we can identifyso :ar concern the irrelevance of an indust-rialised and technological cultural basis forthe curriculum in predominantly ruralsocieties, and the rise of computer educationin industrialised societies.
The first of these challenges comesprincipally from within societies having apredominantly rural economy. The growth ofthe awareness of the economic and societalbases of much of the universal mathematicscurriculum has not been as clear as thedevelopment of 'intermediate technology' or'appropriate technology' in many of thesecountries. However we can now begin torecognise a desire for a more relevant schoolcurriculum which is leading mathematics
educators in those countries to search for amore 'appropriate mathematics' curriculum.Desmond Broomes, in Jamaica, is a mathe-matics educator responding to this perceivedconcern in the rural society of the WestIndies, and his ideas represent this viewwell. In Broomes (1981) he argues that:
"Developing countries and ruralcommunities have, therefore, to re-examine the curriculum (its objectivesand its content). The major problem is toruralize the curriculum. This does notmean the inclusion of agriculture asanother subject on the programme ofschools. Ruralizing the curriculummeans inculcating appropriate socialattitudes for living and working togetherin rural communities. Ruralizing thecurriculum must produce good farmers,but it must also produce persons whowould co-operatively become economic
8
com-munities as well as social and educ-ational communities." (p.49)
In Broomes and Kuperus (1983) thisapproach means developing activitieswhich:
"a) bring the curriculum of schoolscloser to the activities of com-munitylife 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 com-munity;
c) redistribute teaching in space, time,and form and so include in theeducation process certain livingexperiences that are found in thecommunity and in the lives ofpersons;
d) broaden the curriculum of schools sothat it includes, in a meaningful way,socioeconomic, technical andpractical knowledge and skills, thatis, activities that allow persons tocombine mental and manual skills tocreate and maintain and promoteself-sufficiency as members of theircommunity." (p.7l0)
In implementing this strategy, accordingto Broomes (1989) the teacher should poseproblems to the learner and these problemsshould:
"be replete with the culturalexperiences of the learner;
be, for the most part but not solely,practical and utilitarian;
Illustrate mathematics in use;
be capable of being tackledprofitably using mathematics;
not make excessive mathematicaldemands (in terms of the learner's
mathematicallevel o fsophistication);
provide ample scope for cooperativeactivity among persons at differentlevels o f mathematicscompetence."(p.20)
We can find some of these samesentiments shared by educators in parts ofAfrica, in Southern Asia, in Papua NewGuinea and in rural parts of South Americaand Australia. In some of these countriesthere is a particularly strong desire torepresent the growth of indigenous cultural
13
awareness through the mathematicscurriculum. That is, what we are seeing is notjust a rejection of the MT culture whichunderlies the 'universal' curriculum andunderpins modern industrialised societies, buta replacement of that cultural ideology witha rediscovered, indigenous, cultural heritage.Gerdes (1985) in Mozambique and Fasheh(1989), the Palestinian educator, areparticularly eloquent in expressing thisperspective on the mathematics curriculum,and Nebres (1988) is also adamant about itsimportance. The new mathematics textbooksin Mozambique, for example, perhapsdemonstrate more than any other country'sdo, just how one can reconstruct themathematical curriculum within a rural andnon-western society.
What is most significant from this kindof curricular activity is not, however, theparticular mathematics curriculumdeveloped in any one country, but rather thericher understanding which we can all nowshare that it is possible, and probablydesirable in certain situations, to constructand implement alternatives to the canonicaluniversal curriculum. These developmentsencourage mathematics educatorseverywhere to explore ways in which thecurriculum can become a more responsive andappropriate agent in the mathematicseducation of the society's learners. The timehas passed when it was considered sensible tomerely import another society's curriculuminto one's own society. It is however sensibleto 'import' the idea of an alternativemathematics curriculum, as well as the kindof background research necessary to identifythe appropriate bases, the strategies fordevelopment, and the approaches toimplementation. From a rural society'sperspective, of course, these ideas andapproaches would be most likely to be foundin other rural societies.
The second major challenge to the mythof the universal mathematics curriculum is,paradoxically perhaps, coming from withinindustrialised societies themselves, ascomputer technology grows in its range ofinfluence. The justification for much of theexistence of particular topics in themathematics curriculum came from theirassumed usefulness, both to the individualsand to society as a whole. This was similar tothe justification which lay the behind the'back to basics' movement spearheaded byindustrialists and governments who were
reacting to the uselessness, as they and manyparents saw it, of the New Math of theprevious generation. However the validityand relevance of this 'utility' argument isnow being seriously disputed withinmathematics education as it becomes clearthat computers and calculators can take overmany 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 the heartof mathematics and always havedone. Now, however, there is a newemphasis on algorithmic methodsand on comparing the efficacy ofdifferent algorithms for solving thesame problem e.g. sorting namesinto alphabetical order or inverting amatrix.
b) Discrete mathematics
Computers are essentially discretemachines and the mathematics thatis needed to describe their functionsand develop the software needed touse them is also discrete. As a result,interest in discrete mathematics -Boolean algebra, differenceequations, graph theory, ... - hasincreased enormously in recentyears: so much so that the traditionalemphasis given to the calculus, bothat school and university, has beencalled into question.
c) Symbolic manipulation
The possibility of using the computerto manipulate symbols rather thannumbers was envisaged in the earlydays of computing. Now, however,software is available for micros whichwill effectively carry out all thecalculus techniques taught at school- differentiation, integration by partsand by substitution, expansion inpower series - and will deal withmuch of the polynomial algebrataught there also. Is it still necessaryto teach students to do what can bedone on a computer?" (p.69).
Fey (1988) concurs with these views andsummarises the situation thus:
14
"The most prominent technology-motivated suggestions for change incontent/process goals focuses ondecreasing attention to those aspects ofmathematical work that are readily done
9
by machines, and increasing emphasison the conceptual thinking and planningrequired in any 'tool environment'.Another family of contentrecommendations focus on ways toenhance and extend the currentcurriculum to mathematical ideas andapplications of greater complexity thanthose accessible to most students viatraditional methods. I distinguish:
1. Numerical comput-ation,2. Graphic computation,3. Symbolic computation,4. Multiple representations of
information,5. Programming and the connections of
Computer Science with Mathematicscurricula,
6. Artificial intelligences and machinetutors." (p.235)
It is not just the topics within thecurriculum which are being questioned orproposed, the whole purpose and aims ofmathematics education are now underscrutiny. Dorf ler and Mc Lone (1986) presentus with the following argument:
"The dilemma for mathematicaleducators in considering the place ofmathematics in the school curriculumcan be described thus. On the one hand,the increased technological demands ofsociety and the development of sciencerequire highly trained mathematicianswho can apply themselves to a variety ofproblems; they also require professionalscientists, engineers and others to have agreater acquaintance with andcompetence in mathematical technique.In addition, whilst the increased use ofcomputational aids is likely to lead to lessdemand for routine mathematical skillfrom the general workforce, a largernumber of technical managers will beneeded who can interpret the use ofthese computa-tional aids for generalapplication. On the other hand the basicmathematical requirement foremployment is unlikely to grow beyondgeneral arithmetic skill (often with the aidof calculator) and the interpretation ofcharts, tables, graphs, etc.; indeed, notmuch beyond the need5 of everyday lifementioned earlier in this section.
Arguments for school mathematicsbased on its ability to develop powers oflogical thinking do not provide sufficientjustification as it cannot be clearly shownthat such powers are uniquely developedthrough the study of mathematics" (p.57).
10
Keitel (1986) argues the same thing fromanother point of view. She is concerned inthat paper with the 'social needs' argumentfor mathematics teaching:
"By 'social needs' demands I understandhere the pressures urging schoolmathematics to comply with the needsfor certain skills and abilities required insocial practice. Mathematics educationshould qualify the students inmathematical skills and abilities so thatthey can apply mathematicsappropriately and correctly in theconcrete problem situations they mayencounter in their lives and work.Conversely, social usefulness has beenthe strongest argument in favour ofmathematics as a school discipline, andthe prerequisite to assigningmathematics a highly selective functionin the school system." (p.27)
She argues in her paper that computereducation is now taking over this argumentfrom mathematics education, and that thisshould release mathematics education fromthat demand. We need therefore to considerjust what the intentions of futuremathematics curricula should be. Shouldmathematics not continue to be a core subjectwithin the school curriculum, but becomeoptional? Should it become more of a criticaland politically informed subject serving theneeds of a concerned society faced with anenvironmental holocaust? Should it becomemore of a vehicle for developing democraticvalues?
These kinds of curricular debatesdemonstrate that if we wish to take society'sinfluences on mathematical learningseriously then academic considerations andcriteria are not sufficient. In modernindustrialised societies mathematics is sodeeply embedded in the cultural, economicand knowledge fabric of that society that themathematics curriculum must also reflect thepolitical, economic and social concerns of thatsociety.
Sociologists concerned with educationhave long argued about the relationshipsbetween educational practices and societalstructures. Bowles and Gintis (1976) forexample argued that the evidence showedthat education, rather than eradicatingsocial inequality (one of the aspirations ofeducation in a democracy), tends to reinforceit. Giroux (1983) however demonstrated thatalthough schools are tied to particular social
15
features of the society in which they exist,they can also be places for "emancipatoryteaching". Bourdieu (1973) distinguishescultural reproduction from socialreproduction, and argues that the first iswhat is mainly transmitted by the home andthe school, with the curriculum and theexaminations as the principle instruments forcultural reproduction.
There is however little dispute that theintended mathematics curriculum is animportant vehicle for the induction andsocialisation of young people into theirculture and their society, and as such it tendsto reproduce and reinforce the complex ofvalues which are of significance in thatsociety. It is only recently however that suchsocial and political values within themathematics curriculum have been recognisedand discussed (see, for example, Me Ilin-Olsen, 1987; Bishop1988; Frankenstein, 1989).Hitherto, mathematics, due to theassumptions of universality, was thought tobe neutral i.e.. culture-free, societally-freeand value-free. That stance is now no longervalid and those who work in mathematicseducation everywhere need to recognise theimportance 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-socialistcountries to consider just how theirmathematics curriculum should respond tothese different concerns within theirparticular societies (see, for example, thewritings of several people in Keitel et al.,1989). Thus we can see from all these diversedevelopments that the concept of theuniversal mathematics curriculum is losing itscredibility both in theory and in practice.The intended mathematics curriculum hasalso been shown to be a vehicle capable ofresponding to the political and social needs ofsociety.
It has therefore become an object ofserious political concern itself - a state ofaffairs often deplored by manymathematicians. In a sense though this wasto be expected. Any subject which becomes asimportant as mathematics has become insociety, cannot be immune to the differentpressure groups within that society. The
16
mathematics curriculum is clearly far tooimportant an instrument to be determined bymathematicians alone. Whether it can be asresponsive as it needs to be, to accommodateall the influences from society, is however amoot point, and is one which will be taken upin the final section of this chapter.
The examinations
The second major influence from society on themathematics learners comes through theformal examinations, and the examinationsystem. The examinations seem to thelearners to define operationally what is to berequired of them as a result of theirmathematical education - no matter whattheir teachers, parents, and other formal andinfornil educators may say to the contrary.The old adage 'education is what remainswith you after you've forgotten everythingyou learnt at school' may have a certainamount of truth in it, but the fact is that anylearner on any formal educational course willbe acutely aware of the essential demands oftheir particular forthcoming examination, orassessment. In particular, as theexaminations operationalise the significantcomponents of the intended mathematicscurriculum, so they tend to determine theimplemented curriculum.
Just as was the case with the idea of theuniversal mathematics curriculum, so it iswith mathematics examinations - they lookvery similar from country to country. As longas there existed international consensus aboutthe content of the intended curriculum, it alsoseemed sensible to have similarexaminations. There are however variationsbetween countries, and although there hasbeen no systematic research into the varietyof examination practices around the world, itappears that the following represent themajor aspects of difference:
proportion of short to long examinationquestions. Some education systemsemphasise short questions more, and maycast them in a multiple-choice format,while others prefer long questions topredominate. The issue seems to relate tohow large the content sample is for theexamination - short questions allow agreater sampling, long questions less;
proportion of 'content' to 'process'questions. This relates to the previousaspect, but refers essentially to whetherthe examination is intended for the
11
assessment of knowledge and skillswhich have been 'covered' in theteaching, or for the assessment ofprocesses which are assumed to have beendeveloped;
extent of teacher role in the examinationprocess. Some systems merely expect theteacher to be the administrator of theexaminations produced by 'outsiders',while others take the view that theteacher's assessment is a key part of theexamination process. In the latter casethere will be more of a moderating roleplayed by 'outsiders';
proportion of oral and practical workincluded in the examination. In somesystems the examinations are entirelywritten, whilst in others practicalmaterials will be available and in otherspupils will give their answers mainlyorally.
amount of examined work completedduring the course of teaching, rather thanin a final 'paper'. As more process aspectsof mathematics teaching areemphasised, so the final paper hasdecreased in impartance;
extent of pupil choice. In someexaminations all questions and papers arecompulsory, while in others there arechoices to be exercised by the candidates.Such choices allow pupils to emphasisetheir strengths and minimise theirweaknesses thereby allowing pupils toshow what they know rather than whatthey don't know. Individual attributesare valued by some societies more than byothers;
the extent to which the examination isnorm-referenced or criterion-referenced.While all examinations inevitablycontain both aspects, the emphasis is amatter of decision, and relatesparticularly to the underlying purpose ofthe assessment.
Thus even within the framework of thecanonical universalist mathematicscurriculum there can be marked variations inexamination content, procedure andemphasis, related to the particular goals ofthe society and to the societal demands beingmade of the assessment.
Moreover as the development ofalternative curricula increases so we can
12
expect to see yet more different problems andtasks presented, and different issues raised,and these may well enter the formalexamination structure and process. When thishappens, it is likely to be a politicallysensitive matter, as was demonstrated by anincident in a recent public mathematicsexamination in the UK. In 1986, candidates inone examination for 16 year olds werepresented with some information on militaryspending in the world, which was comparedwith a statement from a journal (NewInternationalist) that "The money required toprovide adequate food, water, education,health and housing for everyone in the worldhas been estimated at $17 billion a year".The candidates were asked "How manyweeks of NATO and Warsaw Pact militaryspending would be enough to pay for this?(show all your working)". The outcry fromthe national newspapers and from society'sestablishment was remarkable, with oneheadline asking "What has arms spending todo with a maths examr(Daily Mail, 14 June,1986) and public correspondence on this issuecontinued for some time, illustrating well thesocietal controls felt to be important on theexaminations and the examination system.
Here we are of course talking aboutsummative, rather than formativeassessment. Society has little interest in thelatter, seeing it as merely a part of theteaching process. Summative assessment,however, usually in the form of anexamination, is very much felt to be society'sconcern and various tensions can be seen toexist in all societies over the formal schoolexaminations. These tensions seem to beparticularly sharply defined in mathematicseducation perhaps because mathematicsitself can appear to be a sharply definablesubject, but also because of its role in selectionfor future education or careers.
A major tension is between examinationsas indicators of educational achievement,and as instruments for academic control. In aschool subject like mathematics, in whichthere is much belief in the importance of thelogical sequences within mathematics itself,the prevailing image projected by thecurriculum is of a strongly hierarchical subject- certain topics must necessarily precedeothers. With such an image, academicprogression and control can be felt to betightly determined, with, for example,performance at one level being seen to beachieved before the next topics in the logical
17
sequence can be attempted. This imagecontrols much of the current practice inmathematical assessment, with only thosewho have demonstrated mastery of therequired knowledge being allowed access tohigher level learning in the subject. It is animage which is favoured by many academicmathematicians, and by those 'meritocratic'traditionalists amongst employers, parents,and government officials.
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 ofthis belief is that, given the variety ofschool learning contexts for mathematics, thevariety of teachers and materials influencinglearners, and the variety amongst thelearners themselves, the examinationsshould be as broadly permissive as possible.As well as perhaps including what might becalled standarl mathematical questions, thelearners should, therefore, be allowed tosubmit projects, investigations, essays,computer programs, models and othermaterials, and to be assessed orally as wellas in written form. All of these types ofassessment do exist in different countries as Ihave suggested, but their acceptance in anexamination system by a society dependsstrongly on beliefs concerning educationalaccess, the recognition and celebration ofindividual differences, and theencouragement of individual developmentthrough a broad mathematical education.
Under this view, there is no necessaryreason 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 whichalso seems to be gaining strength as 'personaltechnology' - hand-held calculators ofincreasing sophistication, and personalcomputers - becomes broadly accepted withinmathematics teaching. Personal technologyhas negated many of the traditional logicalsequences thought to be so essential tomathematical development e.g. calculatorsare challenging the ideas of sequencing ofarithmetic learning, and symbolicmanipulation computer software is doing thesame for algebra and calculus as we have
seen. In addition, personal graphiccalculators can now present geometricaldisplays which are forcing mathematicseducators to rethink the orders of geometricaland graphical constructions. Educationally,these personal technologies are putting somuch potential mathematical power in thehands of individual learners that developingand examining individual, and particularlycreative, talents becomes much more of ageneral possibility. As the range ofpossibilities of mathematical activitiesincreases so the necessary ordering ofmathematical mastery becomes less clear andthe former view becomes less tenable.
In the debate between these two views,one suspects that what is really at stake isthe more fundamental issue of social controlversus educational opportunity, and the rolewhich examinations and the examinationsystems play in this political struggle. In allindustrialised and developing countries,access to the higher and further levels ofeducation is not just academically controlledbut is socially controlled as well. Theestablished social order within any societyhas a vested interest in controlling mobilitywithin that society and academiceducational control is an increasinglypowerful vehicle for achieving this (seeGiroux, 1983 for a lucid discussion of writingson this issue).
Moreover academic control, through theuse of mathematical examinations is aparticularly well-known phenomenon. Ithappens throughout Europe, for example, andin many other countries. Revuz (1978)described the phenomenon (mainly from theFrench perspective) in these terms:
"Moreover, the quite proper demand forthe wide dissemination of a fundamentalmathematical culture has boomeranged,and been transformed into a multiplicityof mathematical hurdler at the point ofentry into various professions andtraining courses. A student teacher isregarded as of more or less value inaccordance with his level in mathematics.Subject choices which make good sensein terms of their relevance to various jobshave been reduced to the level of being'suitable only for pupils who are notstrong enough in mathematics'; insteadof attracting pupils by their intrinsicinterest, they collect only those who couldnot follow, or who were not thoughtcapable of following, a richer programmein mathematics. Success in mathematics
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has become the quasi-unique criterionfor the career choices and the selection ofpupils. Apologists for the spread ofmathematical culture have no cause torejoice at this; this unhealthy prestige haseffects diametrically opposed to thosewhich they wish for.
The cause of this phenomenon isundoubtedly the need of our society, as atpresent constituted, to try to reproduce aselective system which runs counter toattempts at democratisation which aregoing on simultaneously. Latin havingsurrendered its role as a socialdiscriminator, mathematics wassummoned involuntarily to assume it. Itallowed objective assessment to be madeof pupils' ability, and it seemed, amidstthe upheaval of secondary education ingeneral, to be one of the subjects whichknew how to reform itself, which stoodfirm and whose usefulness (illunderstood, truth to tell) was widelyproclaimed." (p.177, translated byD.Quadling)
A similar perspective on the phenomenonis offered by Howson and Mel lin-Olsen(1986). They say:
"Mathematics, as we have written earlier,enjoys a high status within society atlarge. 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), that itsstatus is also due to the way in which itcan be 'objectively assessed'. The public,rightly or wrongly, has faith in the way inwhich mathematics is examined andcontrasts the apparent objectivity ofmathematics examinations with the'softer', more subjective forms ofassessment used for many other subjects,e.g. in the humanities. Young advancedthe view that three criteria which helpeddetermine a subject's status in thecurriculum were:
1 the manner in which it was assessed -the greater the formality the higherthe status;
2 whether or not it was taught to the'ablest' children;
3 whether or not it was taught inhomogeneous ability groups.
Certainly, mathematics would fulfil allYoung's conditions. It is important, then,when considering possible changes inmethods of assessment to realise thattheir chances of acceptance will begreater if they do not contradict society's
14
expectations of what a mathematicsexamination should be." (p.22)
One example of this 'social filter' role ofexaminations is that in many Europeancountries it is extremely difficult for someimmigrant groups to gain access to highereducation - not because of overt racistpractices but because of academic controlpractices which have the effect of restrictingaccess to certain groups, through language andcultural aspects for example.
Where these practices have beenrecognised by certain immigrant communitiesone interesting effect has been to concentratetheir educational and parental attention onthe mathematical performances demandedby the 'social filter'. In the UK, for example,the Asian Indian students achievesignificantly higher in mathematicsexaminations than many other immigrantgroups, and they thereby progress moresuccessfully to the higher levels of education(see Verma and Pumfrey, 1988). One suspectsthat the large numbers of oriental Asianstudents succeeding in mathematics at higherlevels in American universities is a similarresponse to a perceived societal situation byan ethnic minority community (see, forexample, National Research Council, 1989,and Tsang, 1988).
However, generally the control throughmathematics examinations still prohibitsaccess to higher educational opportunities formany other minority groups and indigenouspeoples in societies like Europe, Australiaand USA. Furthermore where the WesternEuropean model of examinations has eitherbeen adopted by developing countries or insome cases where their studerts still takeEuropean examinations, the same problemsoccur. For example, Isaacs' (1985) descriptionof some of the changes which the CaribbbeanExamination Council is making to itsexaminations, indicates the problems well:
"The examinations will probably continueto undergo modifications to ensure thatthey reflect more accurately the abilities,interests, and needs of Caribbeanstudents. It is hoped that they will helpreduce the number of students who seemathematics as a terrifying trial in theirrite of passage to adulthood, and, at thesame time, increase substantially thenumber who perceive and usemathematics as a tool for effectiveliving." (p.234)
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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 effeTtthat girls and women are penalised in manycountries through the use of certainmathematics examinations (see Sells, 1980).Also, since mathematics is not felt to be aspopular a subject as others, by girls, to theextent to which mathematics is used forcontrol, it restricts girls' possibilities ineducation more than boys'.
Educational opportunity could beexpected to be a strong goal in any democraticsociety but, even in such societies, either theexamination structure, or the actualexaminations, ensure that not all childrenhave the same educational opportunities.Insofar as societies contain inequalities, thosewill tend to be reflected in unequaleducational opportunities, despite steps beingtaken to remove the barriers and obstacles.The mathematical examination is a well-used obstacle, for example, but it can bealtered, its effects can be researched andpublicised, and its influence therefore eitherbe reduced or exploited. One suspectshowever that all that will happen will bethat unless fundamental changes take placein the structuring of societies, those societieswill create newer obstacles - perhaps indeedthe rise in importance of computer educationwill increasingly come to mean that computercompetence will replace mathematicalcompetence as the filter of social andacademic control.
Societal influences on children throughinformal mathematics education (IFME)
It has been important to explore fairlythoroughly the formal and intentionalinfluences which society exerts onmathematics learning through the curriculumand the examinations, because to a largeextent it is the public knowledge, or at leastperception, of these which determine to alarge degree the characteristics of theinformal education which societal influencesbring to bear on young people. Few adultshave access to within-school or within-classroom information except throughoccasional visits either as a parent, or as an'official visitor' e.g. as a school governor, orprospective employer. Also, Chevallard(1988) makes us aware of a contradictionwithin modern industrial 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'invisible' form of knowledge to many peoplein society.
On the other hand, most adults will havememories of their own experiences of schoollearning, including mathematics, which mayor may not be pleasant, or accurate, or evenrelevant to today's situation. Of course inmost societies it is still the case that not alladults have been to school, that even thosewho have may not have stayed very long,and even if they did they may not haveexperienced much mathematics beyondarithmetic. Nevertheless it would be rare atleast to find a parent who did not feel thatthey had some experiences and beliefs aboutthe school subject with which to influence theyoung learners. This is of course particularlythe case for all adults' experiences andunderstanding of various mathematical ideasgained through their work, their leisure andthrough their own informal education, asNunes will discuss in the next chapter. Thatcultural and societal heritage provides aninfluential source of knowledge, particularlyin societies where formal education facilitiesare under pressure.
There are many individuals in a societywho could potentially influence a youngperson, but if we are considering younglearners as a whole, the two principal groupsof people who seem to have the greatestpotential for IFME are the adult members ofthe family and of the immediate socialcommunity, and people working in what Ishall term 'the media'. From the youngperson's perspective, the only other majorsource of influence, apart from adultsexercising a formal educational role, is thatof their peer group, and their role will bementioned in a subsequent section.
Turning first to the immediatelyavailable adults for the young learner ofinathematics, 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 astherefore being a traditional image, it alsoappears to be a mainly negative image. The
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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 Comrnittee (1932)commissioned a study into the mathematicalknowledge of adults in UK society. Thefindings were dramatic, as they revealed notjust the widespread inability to do whatcould be called 'simple' mathematical tasksbut also the frustrating, unpleasant andgenerally negative emotions felt by manypeople about their mathematicalexperiences. Here are some of the statementsmade by respondents:
"I get lost on long sums and never knowwhat to do with the left-overs'."
"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 firstdegrees at University (excludingmathematics degrees), many of the samedifficulties and negative attitudes emerged(Quitter and Harper, 1988). In both studiesthe reasons for these problems were mixed,but memories of poor teaching and uncaringteachers figured prominently, as did animage of mathematics as a 'rigid' subject,lacking relevance to their personal lives; andhaving correct procedures which needed to beperformed accurately. Strangely it alsoappears that for many adults, the negativeexperience is assumed to be so widespreadthat to claim mathematical ignorance andinadequacy is socially acceptable, howeverunpleasant it may be personally.
From the perspective of the previoussection also, we can well understand the guiltassociations which usually accompany suchfeelings. If, as was described there,mathematics has been used as a strongacademic and social filter by society, manypeople will have experienced failure as aresult. Even those who survived the filteringprocess can be expected to have some negativefeelings about mathematics - they know thatthey had to do well in it to progress in theireducation or their work, and that thatincentive was,in many cases, their onlymotivation for continuing to study it.
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If the collective parental and adultmemory of school mathematics is in fact alargely negative one, this memory can soeasily be transmitted as a negative image tothe next generation, thereby influencing themathematical expectations of the children,their motivations for studying mathematicsand their predispositions for continuing, ornot, to study the subject. For example theevidence from the gender researchdemonstrates the strength of parentalinfluences (Fox, Brody and Tobin, 1980).While the patterns of influence may not bethe same across societies, the influencesthemselves clearly exist.
However we should not assume that allparental influence is of a negative kind, northat there is nothing to be done about it. TheFAMILY MATH project (Thompson, 1989) is aproject concerned to help parents to helptheir children with their schoolmathematics. As Thompson says: "TheEQUALS educators work to encourage allstudents to continue with math courses whenthey become optional in high school in theUnited States - particularly those studentsfrom groups that are under-represented inmath-based careers. These educators askedEQUALS to develop a program that wouldalso involve parents in addressing this issue"(p.62). The reports about FAMILY MATHindeed suggest that it is possible to affect andmediate parental influences on theirchildren's attitudes and achievements inmathematics.
Turning now to what I call 'the media',we need first to clarify what this mightmean in different societies. It refers to thepeople responsible for disseminatinginformation, images, beliefs, and ideas ingeneral, produced by individuals and groupswithin society, but received by the youngthrough different media - newspapers,circulars, books, radio, films, TV,advertisements etc. Occasionally, as with forexample an academic visitor to the school, ayoung learner will have a chance to engagedirectly with that individual, but suchopportunities are rare, and in general theirideas are mediated by 'the media'.
One important difference betweentraditional village societies and modernindustrial societies is the relative proportionof influence coming from immediatecommunity adults compared with that fromthe media. Whereas in traditional village
21
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-pervading influence both on the young and onthe adults themselves. In these societies, theinformation and images available can includejob and career information, the latestscientific and engineering ideas presentedthrough 'popular science' televisionprogrammes and in magazines, games andpuzzles of a mathematical nature, as well asfacts, figures, charts, tables and graphs onevery conceivable aspect of human activity.The learners will see mathematical ideasand activities appearing in a variety ofcontexts; and they will see people engagingwith those ideas in various ways and withvarious feelings. However 'the media' do nottake it upon themselves to represent thewhole spectrum of human engagement withmathematical ideas their task is often oneof satisfying a customer or client need, as inselling a magazine or advertising a newcomputer system.
Generally therefore, and unlike thepersonal knowledge one has from knownadults, the media will tend to project'typical' images and this typicality hasbecome a source of great concern amongst thoseeducational and social activists seeking toincrease the educational and employmentpossibilities of groups within their societywho do not enjoy equal opportunities underthe present systems. For example, theresearch on gender, on social class, and onethnic and political minorities inindustrialised societies has often pointed tobiased stereotypical images portrayed by themedia. In terms of influences on the learner ofmathematics, these images often relate tothe prestige and importance industrialisedsocieties attach to mathematical,technological, scientific and computer-related careers, to the need for high levels ofqualification and skill in these subjects, andto the predominance in these careers ofmainly men, who tend to come from the morepowerful 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 veryimportant subject, that it is a difficult subject,
and that only certain people in society willbe able to achieve well in it.
A recent movement to overcome thislargely unhelpful influence is known as thePopularization of Mathematics', a collectionof attempts to 'improve' the image ofmathematics as a subject, by using media ofall kinds (TV, radio, newspapers, games, etc.)to reveal the more pleasurable faces ofmathematics and to generate interest,enthusiasm and positive intentions, ratherthan failure, fear and avoidance. The con-ference of that title, organised by theInternational Commission on MathematicsInstruction, and its report (Howson andKahane, 1990) involved educators from manycountries in the issues of why popularize,how to do it successfully, and who are the'populace', amongst others. The ideas rangedfrom Mathematics Trails (walks involvingmathematical problems) through citycentres, to mathematical game showsproduced for prime-time national television,from breathtakingly beautiful films andvideos to mind-bending wooden puzzles, from'popular' mathematical articles in nationalpapers to special mathematical magazinesfor children.
The conference, and the report, left one inno doubt that the mathematics educationcommunity believes tha t the influencescoming at present from the media are notgenerally helpful and need to be educated,and enriched.
Societal influences through non-formalmathematics education (NFME)
Adapting Coombs' (1985) definition, non-formal mathematics education would involveany organised, systematic, mathematicseducation activity, carried on outside theframework of the formal system, in order toprovide selected types of learning toparticular subgroups in the population, adultsas well as children.
We can easily recognise NFME inprovision such as adult numeracyprogrammes, out-of-school 'gifted children'courses, televised learning courses,correspondence learning activities, andvocational training courses of many kinds. Asat previous congresses, for example, theAction Group 7 on "Adult, technical andvocational education" held at theInternational Congress on MathematicsEducation (ICME6) in Hungary in 1988,
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contained many examples of interestingdevelopments grouped under the sub-headings of 'Vocational/TechnicalEducation', 'Adult Education' and 'DistantEducation', (Hirst and Hirst, 1988).
At that congress there was also a SpecialDay devoted to the topic "Mathematics,Education, and Society" (Keitel et al., 1989)at which various ideas concernrng NFMEwere presented. For example, Smart andIsaacson (1989) described a course for non-science qualified women who wished tochange to a career in technology and science,and which involved many mathematical'conversions'. Sanchez (1989) talked about aradio programme series in Spain designed topopularise mathematics. Also Volmink(1989) described different approaches to"Non-school alternatives in MathematicsEducation" which the black communities inSouth Africa have turned to because of theirfrustration with the FME provision atpresent: e.g. private commercial colleges,supplementary programs, and the Peoples'Education Movement which is articulatinggeneralised concepts which "aim to transformthe educational institution and will findfinal embodiment only within a new politicalstructure" (p.60). Thus NFME not only offersopportunities for a different context forlearning mathematics, but in some situationsit is clearly intended to challenge theprecepts and practice of FME. In the samevein, and in the same country, South Africa,Adler (1988) describes 4 project concernedwith "newspaper-based mathematics foradults" who had had little access tomathematics education because of theinadequate provisions within the apartheidsystem.
Non-formal education has in some waybeen a marker of educational development ina society, and Freire (1972) showed thedeveloping world the way in which it couldplay a ::ignificant role in a society'sdevelopment. The Peoples' EducationMovement in South Africa is a good recentexample of this, and it is clearly veryimportant that mathematics is not left out ofthe whole process. In industrialised societiesalso there are significant NFMEdevelopments, and a good example is given byFrankenstein's (1990) book which describes a'radical' mathematics course intendedmainly for adult students who have beenmade to feel a failure at maths. It is atextbook which tries to overcome learning
18
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 atthe previously mentioned ICMI conference on'The Popularization of Mathematics'(Howson and Kahane, 1990) spoke ofmathematics clubs, after-school activities,master classes, and exhibitions, all of whichcontribute important influences of a non-formal mathematics education kind.
Volume 6 (1987) of the UNESCO series"Studies in Mathematics Education" is alsoentirely devoted to the theme of "Out-of-school mathematics education" and containsmany examples including mathematicalclubs, camps, contests, olympiads, anddistance education courses. Losada andMarquez (1987) document in a detailed waythe many out-of-school mathematicsactivities in Colombia and demonstrate howsignificant that form of education can be. Asthey say "while the teacher-student ratio inschools ranges from 1 to 20 to 1 to 60,traditional classroom education cannot beexpected to be particularly effective... Insuch circumstances out-of-school mathematicsprogrammes can provide more opportunitiesfor more young people" (p.117).
Clearly the borderline between IFME andNFME becomes blurred at times, particularlywhen there are considerable attempts toenrich both at the same time by, for example,enlisting the help and support of adults andother significant people in the overallmathematical education of the young people.That does not really matter of course. Whatis important is that educators recognise aseducational, the influences which comeeither informally or in a more structured way,non-formally, from the various segments ofsociety.
What then can be said about suchinfluences and what they affect? If wereflect a little more on the informal and non-formal influences described so far, we candetect effects at two levels, both of whichhave important consequences for mathematicslearners; the level of mathematical ideas,and the level of learning mathematics.
23
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,directions etc. Neither will this kind ofparent's talk and activity cease when thechildren begin school. Within theimmediate adult community particularknowledge, often related to mathematics,will be shared, as Nunes documents in thenext chapter. The newspz pers and othermedia reports involving money, charts,tables, percentages will all inform andeducate the learner, complementing andsupplementing the information and ideasbeing learnt in school. Now, withsophisticated calculators and homecomputers becoming more widely available,the young mathematical learner may well beracing ahead of the formal mathematicscurriculum and will often outstrip theteacher's knowledge in some specificdomains.
Equally the sources for mathematicalintuition are frequently images from society,rather than from within school. Popularimages and beliefs, for example, aboutstatistical and probabilistic ideas seem notonly to come from society but also seem to berelatively impervious to formal educationalinfluences. (Tversky and Kahnemann, 1982;Fischbein, 1987).
Geometrical images will come fromphysical aspects of the environment,although what is important is how theindividual interacts with that environment.Thus, once psychologists thought that livingin a 'carpentered environment' (with straightsided houses, rectangular shapes, rightangles etc) was very important for learninggeometry and for developing spatial ability.Now we know that what is important is howindividuals interact with their environment.So for example, to an urbanised European thedesert of central Australia may seem to bedevoid of anything which could aid spatialdevelopment, but the spatial ability of theAborigines who live and work there is knownto be exceptional because of what they haveto do to survive there (Lewis, 1976). This isequally the case with Polynesian navigators(Lewis, 1972) and Kalahari nomads (Lea,PME, 1990). Interactions with the physical
environment of a society undoubtedly giverise to many geometrical images andintuitions.
The other major source of societalinfluence on the learner's knowledge ofmathematical ideas is the language used.The relationships between language andmathematics are of course extremely complex,and there is no space here to cover all theground which has in any case already beenanalysed 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 ofexpressing the mathematical concepts ofMT culture.
The first point may seem obvious, but ithas profound implications. Mathematicalknowledge, as it is developed in any societyrelates to the language of communication inthat society. As has already been shown, themathematics curriculum in many countrieshas been based largely on the Western-European model and it has a certain cultural,and therefore, linguistic basis. Though thisbasis is an amalgam of different languages,the principal linguistic root is believed to beIndo-European (Leach, 1973). Thatparticular 'shorthand' omits the importantGreek and Arabic connections in thedevelopment of universally applicablemathematics, and we should perhapsconsider the Indian-Greek-Arabic-Latinchain as being its original language base.From this base, Italian, Spanish, French,German, and English developed its languagerepertoire during the 17th, 18th and 19thcenturies, and it is probably the case thatnowadays English is the principal mediumfor international mathematical researchdevelopments. This is an important problemfor any country, researcher, or student, forwhom English is not the first or preferredlanguage.
In relation to the second point, in manycountries of the world there are severallanguages used, but for national and politicalreasons, one (or some) are specifically chosenas the national language(s). It is likely thatnot all the languages being used in a societywill necessarily be capable of expressing the
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concepts and structures of the MT culture - thiswill largely depend on the roots of thelanguage. European-based languages, thosewith Indian roots, and the Arabic family oflanguages appear to have the least structuraldifferences, although there are alwaysparticular vocabulary gaps as internationalmathematics develops.
Other languages, in rural Africa,Australasia, and those used amongstindigenous American peoples, are beingstudied and demonstrate their difficulties inexpressing both the structures and thevocabulary of the MT culture's version ofmathematics (see, for example, Zepp, 1989and Harris, 1980). This is not of course to saythat these languages are incapable ofexpressing any mathematical ideas - theywill certainly be capable of expressing themathematical ideas which their cultureshave devised. This is the importantlinguistic relationship withethnomathematics - and another reason forseeking to create more independentsocietally-based mathematics curricularather than relying on the model from MT-based societies. Thus in this way the societallanguage(s) can reinforce the societalmathematics which can offer the bases foralternative curricula.
But language issues are extremelycomplex, particularly from a societalperspective, and the political and socialconflicts which different language use cancause, can seem to be of a different order fromthose which should concern mathematicseducators. Nevetheless so much damage hasbeen done to cultural and social structures inmany countries by assuming the universalvalidity of MT-based mathematics that wecannot ignore the language aspects of thiscultural imperialism (see Bishop, 1990). Ifcountries, and societies within countries, areto engage in the process of culturalreconstruction then the language element inrelation to informal, non-formal, and formalmathematics education is critical.
A final point concerning the informal andnon-formal influences on mathematical ideasis that they have a cumulative effect. Theybuild up into an image of 'mathematics' as asubject itself. For example, we have alreadynoted that it is projected as being animportant and prestigious subject in bothindustrial and deveioping societies and isthereby projected as being essentially a
20
benign subject. Little mention is publiclymade of its extensive association, throughfundamental research, with the armamentsindustry, with espionage and code breaking,and with economic and industrial modellingof a politically-partial nature. Little publicdebate occurs about the questionabledesirability of fostering yet moremathematical research to make our societiesyet more dependent on even more complexmathematically-based technology (seehowever, Davis, 1989 and Hoyrup, 1989).The reaction of 'the media' to theexamination question about costs ofarmaments shows the extent of publicignorance of these matters.
Equally mathematics in society istypified, and imagined by most people, asthe most secure, factual and deterministicsubject. There is little public awareness ofthe disputes, the power struggles, or thesocial arenas in which mathematical ideasare debated and constructed. Descartes'dream still rules the general societal imageof mathematics. For example, in the study byBliss et al. (1989) concerning children'sbeliefs about "what is really true" in science,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 a primary child that World War 2lasted for ten years, ad he will believe it;tell him that two fours are ten, and therewill be an argument" (p.12).
At a second level the informal and non-formal societal influences concern thelearning of mathematics. We have alreadynoted the beliefs about its difficulty and itsmotivations, but there are also morefundamental and significant beliefs abouthow mathematics is learnt. Paralleling thepopular image of mathematics as securefactual knowledge is the widespread beliefthat mathematical procedures need to bepractised assiduously and over-learnt so thatthey become routine, and that this should gohand-in-hand with the memorising of thevarious conceptual ideas and theirrepresentations. Another popular beliefconcerns 'understanding' as being an all-or-nothing experience, rather than a gradualincreasing of meaning and constructedconnections. Overall the popular image is ofa received, objective, form of knowledge,
0 5
rather than of a constructed subject, runningcontrary to what we know from recentcognitive research. (Lave, 1988; Schoenfeld,1987).
There is also, as has been mentioned, theimpression that because it is reputed to be adifficult subject, only some learners will beable to make progress with it. Thus, ratherlike artistic or musical ability, young peopleperceive themselves as either havingmathematical ability or not. The research onmathematical giftedness does in factdemonstrate that it clearly is a precocioustalent, appearing early rather like musicalgiftedness, but one suspects that the imagefrom society about mathematical ability israther more broad in its reference than just togiftedness.
The concept of 'ability does seem to be apervasive one in society outweighing'environment' as the cause of achievement,particularly in the 'clear-cut' subject ofmathematics. The accompanying belief isthat one is fortunate to be born with thisabiiity since ii is (clearly!) innate. The prideof a parent on discovering that their child isa gifted mathematician is probably onlyclouded by the popular image from society ofmathematical geniuses being slightly oddcharacters living in a remote and esotericinner world and unable to socialise withother people - a theme to which severalcontributors to the ICMI conference onPopularization' referred.
Equally parents, although not necessarilyrating their son's or daughter's abilities anydifferently, do apparently believe thatmathematics is harder for girls and requiresmore effort to succeed in (Fox, Brody andTobin, 1980). This belief undoubtedly helpsto shape the feeling, prevalent in differentsocieties, that mathematics is not such animportant subject for girls to study.Fortunately many wornens' groups are nowhard at work dispelling this image, andIOWME (the International Organization ofWomen and Mathematics Education) hasbeen particularly active. Recent research(Hanna, 1989) for example shows that thecreation of a more positive and favourableimage for girls who are mathematically ableis apparently having some beneficial effects.
These then are some of the mostsignificant influences which society exerts onthe learners of mathematics. The 'messages'the learners receive are many, and often
conflict. Some are intentionally influential,while others are merely accidental. But theyall help to shape important images andintuitions in the young learners' minds, whichthen act as the personal cognitive andaffective 'filter' for subsequently experiencedideas. Let us then move on to consider howihe learners make sense of, or cope with,these various societal influences which theyexperience. Are there any research ideaswhich can help us to understand and interpretthe learners' situation?
The competing influences on the individuallearners
Considering first of all the area ofmotivation, this is perhaps the mostsignificant aspect needing to be richlyunderstood by mathematics educators,because it provides the essential dynamic forthe mathematics learning process whereverand whenever it takes place. A classicdichotomy is usually made between extrinsicand intrinsic motivation, referring to thelocation of the influence for motivation. Inone sense we can think of societal influencesas being essentially extrinsic whereas, forexample, the 'puzzling' nature of amathematical problein can seem to be locatedinternally to the learner. The interactionbetween intrinsic and extrinsic motivationthough relates both to the internalisation ofthe extrinsic, and also to the perception of afamiliar external target for the intrinsic. Itis clear that societal influences initiallyexist 'outside' the learner, but the extent oftheir influence will depend on howinternalised they become, and, therefore,that internalisation process is the key tounderstanding motivation for the learners.
We have claimed in this chapter thatsocieties formally influence mathematicslearning through their school curricula andtheir examination structures. From thelearner's perspective, therefore, thoseinfluences will motivate to the extent thatthey are internalised and help to shape thelearner's goal structures. There appear to betwo significant and interacting aspects toattend to here - the 'messages' beingcommunicated to the learner and the peopleconveying those messages. In terms of themathematics curriculum, the principlemessage will concern the values - explicit orimplicit (the 'hidden' curriculum) - whichthe curriculum embodies, mediated by theteacher, and which may, or may not, be seen
21
as significant by the learner. The area ofvalues in mathematics curricula has not beenwell explored in research, but the clues wehave from research on attitudes and beliefscertainly supports the above analysis (see forexample, McLeod, PME, 1987).
However, the extent to which themathematics curriculum embodies the valuesof MT culture (for example) will only partlydetermine the motivational power of thosevalues. The real determining factor will beeach individual's acceptance of those valuesas being worthwhile or not, which willdepend on how they relate to the learner'sgoal structure, and how di.".erent peoplefigure in that structure. To talk of goalstructures is not however to suggest that theyare fixed and immutable. One recentlydeveloping research area that of situatedcognition (Lave, 1988; Saxe, 1990) - revealsthe phenomenon of 'emergent' goals. If oneunderstands mathematics learning assomething which develops from a socialmathematical activity, then through thatactivity will emerge mathematical goals, tobe achieved, which would not necessarilyhave been apparent before the start of thatactivity.
Thus, one corollary is that as one learnsabout mathematical ideas through doingmathematical activities, one also learnsabout goals, and in a social context. Astatement by the teacher to the effect that animportant goal of mathematics learning is tobecome systematic in one's thinking, forexample, may well be negated by thelearners' shared experiences of trial-and-error approaches which nevertheless succeedin finding appropriate solutions to problems(see Booth, 1981). Moreover, it may well bethat the latter message, learnt in activityand in the context of one's peers, or one'sadults, will be internalised moresignificantly than the teacher's message.
This kind of gulf, between what learnersare told that they should do and what theyactually do to be successful, may well be onesource of the well-documented dislike ofmathematics, and even of the phenomenon ofmathophobia, the fear of mathematics. It isthe kind of experience which, if repeatedoften enough, will lead learners to believethat mathematics, even if it is an importantsubject, may not be what they personallywant to invest their energies in. Moreresearch on learners' emergent goals,
22
particularly in the school learning contextcould provide some very powerful ideas formathematics educators.
A more significant contributor to thelearner's mathematical motivation may wellbe the messages concerning the examinations,which come both formally and informallyfrom society. The within-school 'messages'concerning final examinations, combined withthe experiences of immediate adults will behighly significant in creating a schema ofbeliefs and expectations with which thelearner's own experiences of mathematicalsuccess or failure will be interpreted. Theinternalisation of beliefs and expectations, aswith any aspect of motivation, will bedependent on the significance to the learnerof the people mediating those beliefs andexpectations. Sullivan (1955) refers to'Significant Others' i.e. people who canexercise that kind of influence - they may berole models, advisers, counsellors, objects ofworship and awe, or of scorn and dislike.Their significance is a personallyattributable characteristic and will be likelyto vary both as the richness of the individuallearners varies and as the variety of 'types'in the society varies. From this analysis wecan see that the learner's mathematicsteachers, parents, mathematical peers andreal or mediated 'mathematicians', arelikely 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,particularly in a subject like mathematics,which is believed by many learners to be'clear-cut', where one always knows (theythink) whether the answer is right or wrong.Moreover this act of 'interpretation' can beseen as part of the wider cognitive task forthe learner, of understanding mathematicalactivity as a societally defined phenomenon.Attributing success or failure in the subjectrelates strongly for the learner to personalquestions and concerns like "How will I fitinto society - what job will I do - howmathematically qualified should I become?"and "How do I relate to other people insociety - am I more or less competent atmathematics than they are?"
Attribution theory can help us here.Wiener's (1986) work is being explored inrelation to mathematics learning and thefindings are interesting. For example Pedro et
al. (1981) reported that, in US society, malesare more likely to attribute their success totheir ability than females, and that femalesare more likely than males to attribute theirsuccesses to their efforts. So far, however,we have no research which systematicallyexplores mathematical attributions inrelation to the influences of significantothers, although the role of the teacher doesappear to be fundamental, as might beexpected (see Lerman, this book).
Attributional understandings andinterpretations like these appear also to bestrongly embedded in the learner's meta-cognition about mathematics. Becausemathematics at all levels is learnt insocially defined activities, the people whoshare in influencing those mathematicalactivities will also play a role in shapingeach learner's total perspective onmathematics. Gay and Cole (1967) in theirstudy of the Kpelle, were clearly taken bysurprise by one of the college students theywere testing:
A Kpelle college student accepted all thefollowing statements: (1) the Bible isliterally true, thus all living things werecreated in the six days described inGenesis; (2) the Bible is a book like otherbooks, written by relatively primitivepeoples over a long period of time, andcontains contradiction and error; (3) allliving things have gradually evolved overmillions of years from primitive matter;(4) a "spirit" tree in a nearby village hadbeen cut down, had put itself backtogether, and had grown to full size againin one day. He had learned thesestatements from his Fundamentalistpastor, his college Bible course, hiscollege zoology course, and the still-pervasive animist culture. He acceptedall, because all were sanctioned byauthorities to. which he feels he must payrespect. (p35)
The researchers seemed to expect logicalconsistency to be an over-riding criterion forthat student, and they used that kind ofevidence to infer that mathematicalreasoning (as they knew it) had little placein Kpelle life. We now know of course thatmathematics is not the universal andindependent form of knowledge which theyassumed. Moreover, and perhaps for us nowmore importantly, they also assumed thatsuch socially-bound cognitive behaviour wasnot typical in America (their home). Lave(1988) has certainly demonstrated that not to
be the case, and in general the growingawareness of the socially constructed natureof all human knowledge makes us realisethat all learners have the task of makingsense of other people's messages in a varietyof social situations.
Thus although perhaps mathematicseducators would like to think that in someway societal influences from the immediateadults and from the media will enhance,support and generally reinforce the messageswhich the child receives from school, thereality is more likely to be one of conflict.The variety of people involved makes itunlikely that there will be a kind of benignconsistency.
The learner is therefore rather like abilingual, or multi-lingual, child who mustlearn to use the appropriate 'language' in theappropriate social context. In fact it would bemore appropriate to describe themathematics learner as bi-cultural, or multi-cultural, since so much more than language isinvolved in learning mathematics. We havealready noted the influence of interactionswith the physical environment which createso much of an impact on geometrical andspatial intuitions. Also social customs andhabits, which give rise to expectations, arenot always expressed through language butare learnt through social interactions whileengaged in particular activities.
Halliday's (1974) work on 'linguisticdistance', for considering the cognitive taskfor bilingual learners, is an interestingconstruct here. Dawe's (1983) research usedHalliday's idea to conjecture that bilinguallearners of mathematics in English at schoolwould have more difficulty the 'furtheraway' the structure of their home languagewas from English. So, for his study, theorder: Italian, Punjabi, Mirpuri and JamaicanCreole, was hypothesised as being thedifficulty dimension, with Italian being the'closest' to English. For logical reasoning inEnglish this proved to be the case, but not formathematical reasoning, where a gendereffect interacted with the language factor.
It is possible to broaden the idea oflinguistic distance to that of 'socioculturaldistance' between two different principalsocial situations experienced by the learner.Thus it seems reasonable to conjecture thatmathematics learners whose immediatehome cultures relate more closely to thestructure and character of the school culture,
2F) 23
will have less difficulty in reconciling themessages coming from the two cultures, thanwill learners whose home cultures are a long'distance from their school culture.
Perhaps though we will make moreprogress in educational terms if we attend lessto the "messages received" metaphor whichemphasises the role of the learners aspredominantly passive receivers, and more tothe idea of the learners as constructors oftheir own personal and cultural knowledge.The essential cognitive task for any learner ismaking meaning, by creating and constructingsensible connections between differentphenomena, and because most mathematicalactivity takes place in a social context, thismathematical construction will take placeinteractively. Each new generation of youngmathematical learners is activelyreconstructing the varied societal knowledgeof mathematics with which they come intocontact, and in their turn they (as adults)will influence the social context withinwhich the next young generation will engagein their own reconstruction.
Of particular relevance to this point isSaxe's recent research (Saxe, 1990) whichdemonstrates three aspects of importance tous as educators. Firstly his street candy-sellers in Brazil learned as much from olderchildren as they did from immediate adultsin the communities. So we should include'older children' in the categories of'immediate adult in our previous discussions,as they are likely to play a key role in anysocialisation process. Also that knowledgewas validated and developed within a strongpeer-group structure, and it is likely thatpeer-groups influence far more than has benrecognised so far. Finally although therewas some evidence of school mathematicalideas and techniques being used out of school,it was very clear that the street sellingexperiences had furnished a rich schemawhich informed the children's learningwithin the school situation. This findingsupports, interestingly, the cognitiveinstructional research findings (see Silver1987 for a summary) that the most significantcontributor to the learning of new informationis the extent of the previously learntinformation. However it should alsoencourage educators to realise that the mostimportant prior knowledge may well havebeen learnt outside the schooi context, andwill therefore be embedded in a totallydifferent social structure.
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My own example of this, alreadyextensively quoted (see Bishop, 1979),concerns an interview with a student in PapuaNew Guinea, a situation where thehome/school socio-cultural distance is large.I asked him "How do you find the area ofthis (rectangular) piece of paper?" Hereplied "Multiply the length by the width".I continued "You have gardens in yourvillage. How do your people judge the areaof their gardens?" "By adding the length andwidth". Taken rather aback I asked "Is thatdifficult to understand?" "No, at home I add,at school I multiply". Not understanding thesituation I pursued the point: "But they bothrefer to area". "Yes, but one is about the areaof a piece of paper and the other is about agarden". So I drew two (rectangular) gardenson the paper, one bigger than the other, andthen asked "If these were two gardens whichwould you rather have?" to which he quicklyreplied "It depends on many things, I cannotsay. The soil, the shade ..." I was then aboutto ask the next question "Yes, but it they hadthe same soil, shade ..." when I realised howsilly that would sound in that context!
Clearly his concern was with the twoproblems: size of gardens, which was aproblem embedded in one context rich intradition, folk-lore and the skills ofsurvival. The other problem, area ofrectangular pieces of paper was embedded ina totally different context. How crazy I mustbe to consider them as the same problem!
So how did he reconcile the conflictwhich I could see? I cannot answer thatquestion because I firmly believe that for himthere was no conflict. They were twodifferent problems set in two entirelydifferent contexts, and it was only I who felta conflict. As with Gay and Cole above, mycultural background encouraged me to believethat logical consistency demanded aresolution of the conflict which would arise ifone were to attempt to generalise the twoprocedures. If one were not interested in doingthat however, there is no problem.
Educational implications
What, then, are the educationalimplications that can be drawn from all ofthis evidence, analysis and conjecture? Thefirst thing to ask is "What do we mean byeducation?" since it is now clear that thelearner is subject to many influences of apotentially educating kind. Also we can nowsee that if 'education' is only considered to be
29
what happens in school then any potentiallypowerful out-of-school experiences will bedenied a legitimacy in that society. As wassaid earlier, at present the school curriculumand examinations contain what most adultsthink should be learnt through mathematicsteaching and any other mathematicalknowledge they have learnt through practiceis not "real" mathematics. In fact it would beeasier to argue that totally the opposite istrue: school mathematics, because it is learntin the necessarily artificial world of schoolcannot be 'real mathematics!
But the issue is not whether one context isreal and another not. Both clearly are real,but they are also very different. Thereforethe conceptual task is not to create anartificial dichotomy but to begin by lookingfor the similarities and Coombs (1985) hasoffered us an important way forward. We cannow consider formal mathematics education(FME), informal mathematics education(IFME) and non-formal mathematicseducation (NFME).
In certain cases the developments we cansee in NFME and IFME are motivated by thedesire to complement, supplement andgenerally extend the FME 'diet' offered bythe formal educational institutions, while inothers there is an objective of challenging theassumptions, the structures or the practices ofFME. These kinds of developments willsurely have influences on FME to the extentthat they can demonstrate their successes tothe educational professionals and to thewider society.
From the formal educational viewpointthese developments in IFME and NFME couldbe considered as unwanted challenges to theeducational status quo, and which shouldtherefore be resisted by whatever politicaland social means are possible. However morepositively one can consider thesedevelopments as important educationalexperiments whose effectiveness, if proven,and whose validity, if accepted, compelthem to be considered as viable possibilitieswithin the formal educational provision.
This is not to propose their demise asvalid educational agents in their own rightbut to recognise that formal mathematicseducation cm 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, andteaching material developers. In others,there could be influences on the intendedformal curriculum of the society and on theexamination procedures.
The increases, both quantitative andqualitative, in NFME and IFME provision inboth industrial and developing societies,certainly demonstrate that FME at present isnot meeting the demands which differentsocieties are making. It should therefore be acontinual source of challenge to those of uswho are responsible for the character of FMEin any society to explore its potential forresponding to the different demands andinfluences coming from society. For example,television and other popular media can raisethe awareness about certain ideas, but theycannot develop the knowledge of those ideassystematically. Neither can they developskills effectively. On the other hand skills,such as key-typing, can be learnt effectivelythrough non-formal situations, which couldhave much pay-off for formal educationalwork with computers and word-processors. Sohow should FME be shaped, in the context ofrapidly expanding and influential NFME and1FME developments?
Trying to answer that question meansthat those of us who work in mathematicseducation more generally need to be muchmore aware than we have been ofdevelopments in NFME and IFME, to acceptthem as valid educational concerns, tostimulate their active growth and torecognise their growing power and influencenot just on FME but on societies' developmentgenerally.
This chapter has demonstrated not onlythe range of influences which society brings tobear on mathematics learners, but also howthe learners try to deal with these and howthe different educational agents respond tosocieties' demands. It is a chapter predicatedon the belief that a societal perspective onmathematics learning is essential in framingthe more narrow and specific concerns ofresearch, development, and practice withinformal mathematics education.
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Alan Bishop is currently University Lecturer at the Department of Education in the University ofCambridge, 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 the SocialPsychology of Mathematics Education working group. He was for eleven years the Editor ofEducational Studies in Mathematics, and is still the Managing Editor of the MathematicsEducation Library, published by Kluwer of Dordrecht, The Netherlands.
26 31
The Socio-cultural Context of Mathematical Thinking:
Research Findings and Educational Implications
Terezinha Nunes
R: 8 years old, is solving a problem about givingchange in a simulated shop situation which ispart of an experiment. He writes down 200 -35 properly aligning units and tens andproceeds as follows:
R: Five to get to zero, nothing (writes down zero);three to get to zero, nothing (writes downzero); two, take away nothing, two (writesdown 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, CarraherSchliemann, 1987, p.95
Much research has shown thatmathematical activity can look verydifferent when it is embedded in differentsocio-cultural contexts. For example, wecalculate the amount of change due in schoolword problems by writing down the numbersand using the subtraction algorithm. Outsideschool in a shop we may figure out changedifferently, using oral methods anddecomposing numbers (as R. did, decomposing35 into 30 and 5 and solving the two problemssequentially) or even by adding on from thevalue of the purchase to the amount given asthe money is handed to the customer. It is mypurpose in this paper to discuss the fact thatmathematical activities look different indifferent contexts and to explore the implic-ations of these differences for mathematicseducation. The review presented here isselective both due to space limitations and tomy choice of depth rather than breadth ofcoverage. Only two topics in mathematic:
education will be reviewed: arithmetic andgeometry. For a broad coverage of topics,other works 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 mainsections. The first section relates to arith-metic and analyzes two types of differencebetween arithmetic in and out of school,differences in success rates and in theactivity itself. The second section relates tospace and geometry and looks at theinterplay between logic and convention ingeometrical activities. In the third section,two views of how to bring out-of-schoolmathematics into the classroom arecontrasted. Finally, the last section brieflysummarizes the main issues in the study ofthe socio-cultural context of mathematicalthinking and outlines ideas which can beexplored and investigated in the classroom.
Arithmetic in and out of school.
Hum.an activity in everyday life is notrandom but organized, or structured, to use amore mathematical term. Even the simplestinteraction, like that between two friendsmeeting on the street, can be shown not to betotally spontaneous but rather structured andpredictable. I will distinguish two types oforganization for the purposes of thisdiscussion, social and logico-mathematicalorganization. Social organization is pres-criptive and often implicit; it has to do withwhat people should do in certain types ofsituation without their being necessarilyable to know why they behave in thoseparticular ways. Logico-mathematicalorganization is deductive and can often bemade explicit; it allows people to go beyondthe information given at the moment and toknow why the deduced information must becorrect (and that holds even in the caseswhen an error has been made). The logico-mathematical structure pertains to theactions and situations as such while the
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social organization pertains to the actions asinterpersonal events, that is as interactions.
In any mathematical activity, bothsocial and logico-mathematical organizationare involved, regardless of whether it takesplace in the classroom or outside (for adifferent view, which rejects the idea thatmathematical skills are based on logicalstructures, see Stigler & Baranes, 1988).However, it must be emphasized thatdistinguishing between the social and thelogico-mathematical organization foranalytical purposes does not assume theindependence of these two aspects oforganization in human life. In any instance ofmathematical activity, be it in the classroomor outside, both forms of organization comeinto play.
Mathematical activities carried out inand out of school have different socialorganizations but are based on the samelogico-mathematical principles. The kernelof the social organization differencesbetween mathematical activity in and out ofschool appears to be that everydayactivities involve people in mathematizingsituations while traditional mathematicsteaching focuses on the results of otherpeople's mathematical activities (Bishop,1983) Thus in school, teachers expect thatstudents will produce a particular solution(related to the application of an algorithm,for example) right from the moment aproblem is posed. In contrast, an everydayproblem may be correctly solved throughmany different routes and no particular routeis prescribed from the outset.
To take a simple example, Scribner andher collaborators (Scribner, 1984; Farhmeier,1984) have analyzed how inventory takerssolved 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 aboutinvisible cases in stacks. No particular routeis "correct" or "expected" from the outset. Inthe classroom, a similar problem would bephrased as "how many cartons of milk arethere if there are 38 cases and each caseholds 16 cartons". The school problemrequires in principle counting cases andmultiplying number of cases by number ofcartons in a case. Scribner and hercollaborators found that in real stock-taking,counting single cases to multiply by thenumber of cartons was not the only strategy
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available. Several other strategies wereused especially because piles were not neatand cases might not be complete. Othermethods included the use of volume concepts(for example, counting stacks of cases ofknown height by looking only at the top ofthe stacks), skip-counting cartons inincomplete cases (for example, 8, 16, 24, 30etc), compensating across cases and stacks (forexample, counting a half case as full andthen compensating for this when comingacross another incomplete case), keepingtrack of partial totals per area in the ice-boxetc. In short, actual inventory taking does nothave a pre-established path: inventorytakers 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 topractice the multiplication algorithm, aresult of other people's mathematicalactivity which is to be learned in school.
Does this central difference in the socialstructure of mathematical activities in andout of school affect the way people representand understand the logico-mathematicalstructures? This question will be looked athere from two perspectives. The first relatesto the rates of success in mathematicalactivities carried out in and outside school.The second relates to the way the activity iscarried out, its characteristics and methods.
Differences in rates of success.
Two studies have documented systematicallyvery important differences in rates of successwhen the same people carry out basicallythe same mathematical calculations in andout of school.
The first of these studies was byCarraher, Carraher, and Schliemann (1985),who interviewed five street vendors (9 to 15years old) in Recife, Brazil. The youngsterssold small items like fruits, vegetables, orsweets in street corners and markets. Thestudy started with the investigatorsapproaching the youngsters as customers andproposing different purchases to thechildren, asking them about the total costs ofpurchase 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:
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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 be onehundred and five; with three more, thatwill be two hundred and ten. (Pause) I
need four more. That is... (pause) threehundred and fifteen... I think it is threehundred fifty.
Customer/experimenter: I'm going togive you a five-hundred note. How muchdo I get back?
Child/vendor: One hundred and fifty.
When engaged in this type ofinteraction, children were quite accuratein their calculations: out of 63 problemspresented in the streets, 98% werecorrectly solved. We then told thechildren we worked with mathematicsteachers and wanted to see how theysolved problems. Could we come backand ask them some questions? Theyagreed without hesitation. We saw thesame children at most one week laterand presented them with problems usingthe same numbers and operations but ina school-like manner. Two types ofschool-like exercises were presented:word problems and computationexercises. Children were correct 73% ofthe time in the word problems and 37% ofthe time in the computation exercises.The difference between everydayperformance and performance oncomputation exercises was significant."(Carraher, PME 1988, p.3-4).
Thus, young street vendors were moresuccessful in their computations outsideschool than in their efforts to solveschool-like exercises presented to them bythe same experimenters.
Similar results were obtained by Lave(1988) in a study with 35 adults (21 to 80years) in California. All of the subjects inthis study had higher levels of instruction(range 6 to 23 years) than the Brazilianyoungsters. Lave and her colleagues observedadults engaged in shopping in supermarketsand trying to decide which was a better buycomparing two quantities of a product with
two varying prices. An example of a problemof this type is transcribed below.
A shopper compared two boxes of sugar, onepriced at $2.16 for 5 pounds, the other $4.30for 10 pounds. She explains7The five poundswould be four dollars and 32 cents. I guessI'm going to have to buy the 10-pound bagjust to save a few pennies." (Murtaugh, 1985,P.35)
Problems of this nature involvecomparisons of ratio-- which one is larger,2.16/5 or 4.30/10? Lave and her associatesobserved 98% correct responses in the ratio-comparisons carried out in the supermarketby the adults; in contrast, only 57% of theresponses given by the same adults in amaths test of comparisons of ratios werecorrect. Thus, the results observed byCarraher, Carraher, and Schliemann (1985)for Brazilian street vendors are replicatedamong American adults: mathematicsoutside school shows better results than inschool-like situations for the same subjectsworking on the same types of problems.
Differences in the social organization ofstreet and school mathematics and theirsimilarities in logico-mathematical struc-turing.
Several authors have tried to analyze theactivities which are carried out when peopleare engaged in mathematical problems inand out of school. Nesher (PME, 1988)summarized some of the contrasts betweenmathematics in and out of school. CitingResnick (1987), Nesher pointed out thefollowing differences:
"(1) schooling focuses on theindividual's performance, whereasoutside school mental work isoften socially shared;
(2) school aims to foster unaidedthought, whereas mental workoutside school usually involvescognitive tools;
(3) school cultivates symbolicthinking, whereas mental activityoutside school engages directlywith objects and situations;
(4) schooling aims to teach generalskills and knowledge, whereassituation-specific competencedominates 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
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put the trainee as soon as possible on theproduction line" (p.56)
while'This is not exactly the case at school.Schools aim to pass on knowledge tostudents to partly be employed at schoolin further training, but mainly to beemployed elsewhere, after leaving school(...) schools have to deal with questions ofmotivation or with questions of rewardingprocedures' (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 twomost important differences. These are: (1)that the mathematics used outside school isa tool in the service of some broader goal, andnot an aim in itself as it is in school (seeLave, 1988; Murtaugh, 1985); and (2) that thesituation in which mathematics is usedoutside school gives it meaning, makingmathematics outside school a process ofmodelling rather than a mere process ofmanipulation of numbers.
Carraher (PME, 1988) expanded on thislast aspect of the difference by arguing that
"Mathematics outside school isconducive to the development ofproblem solving strategies which reveal arepresentation of the problem situation.The choice of models used in problemsolving and the interval of responses areusually sensible [when mathematics iscarried out outside school] even thoughnot always correct. Students using schoolmathematics often do not seem to keepin mind the meaning of the problem,displaying problem solving strategieswhich have little connection with theproblem situation and coming up withand accepting results which would berejected as absurd by anyoneconcentrating on meaning." (p.19)
Carraher supported this analysis byreferring to a study by Grando (1988), inwhich farmers and students' responses to aseries of problems were contrasted both interms of the strategies used and in terms ofhow sensible the responses provided were inview of the problem. The farmers (n = 14) inthe study had little or no school instructionand had learned most of their mathematicsoutside school; the students (20 in seventhand 40 in fifth grade), even though belongingto the same rural communities, had learned
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most of their mathematics in school. Theywere asked, for example, how many pieces ofwire with 1.5 meters of length could beobtained by dividing into pieces a roll ofwire 7 m long.
"The farmer's responses, obtainedthrough oral calculation [typically anout-of-school method], fell between 4 and7 pieces with 93% giving the correctanswer. The students' responses fellbetween .4 and 413 pieces. Theseextreme answers were given by studentswho carried out the algorithm for division(correctly or incorrectly) and did notknow where to place the decimal point"(p.12)
a difficulty not faced by farmers. whocontinuously thought about the meaning ofthe question and would neither accept as ananswer to this problem a number whichindicated less than one piece of wire nor anumber as high as 413.
The role of modelling in the farmers'procedures was clearly identified throughtheir verbalizations, which indicated thatthey simultaneously kept track of the numberof pieces and the amount of wire used up.Their responses could be obtained by a processlike the following: "one piece, one meter anda half; two pieces, three meters; four pieces,six meters--then, it's four pieces".
The argument being presented here is notthat farmers were more able than students inmathematics. It is possible that studentscould have solved the problem correctly ifthey had used the same method as thefarmers. All of the students attempteddivision in this problem, a method whichwas adequate and perhaps triggered by theidea of dividing the roll of wire into pieces.However, faced with the difficulty ofoperating with decimals, they tried no otherroute to solution. Farmers, in contrast, werenot restricted to division by any particularset. They were consequently able to avoid thedecimal division by adding the pieces up tothe desired total length, thereby preservingtheir ability to monitor their response assolution was approached.
Differences in methods for solvingproblems in and out of school are also veryclearly observed in problems involving morethan one variable, like proportions problems.From the mathematical point of view, allproportions problems involve the samestructure. Thus it is possible to develop a
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general algorithm which can be applied inany proportions problem independently ofthe type of variable involved, be it money,length, weight or whatever. This is in factwhat the teaching of proportions in schoolaims at: the transmission of a generalalgorithm, which takes the form a/x = b/c.However, outside school the nature of thevariables and the relationships whichconnect them are not set aside for the sake ofthe development of a general algorithm.The routes to solution tend to reflect in somesense the relationship between thevariables, as we can see by contrasting thefindings reported by Carraher (PME, 1986)with foremen and by Schliemann and Nunes(1990) with fishermen solving proportionsproblems in Brazil.
Carraher (PME, 1986) observed thatconstruction foremen solving proportionsproblems about scale drawings preservedthroughout calculation the notion ofsealc--that is, the notion that a certain unitdrawn on paper corresponds to a certain unitof "real life" wall. The 17 foremeninterviewed in this study were successivelyshown four blueprints and asked to figure outthe real-life size of one or more walls in eachblueprint drawing. The data they used tosolve the problem were obtained from theblueprint: a pair of data on one wall (thereal-life size and the size on the blueprint)and the size on the blueprint of the targetwall. All successful solutions with unknownscales were obtained by foremen who firstsimplified the initial ratio (size on scale toreal-life size) to a unit value and then usedthis ratio in solving the problem. When theoriginal pair of data on one wall was, forexample, 5 cm on the blueprint correspondingto 2 m in real-life, foremen first simplifiedthis ratio in order to find the basiccorrespondence--"2.5 cm on paper is worth 1 mof wall", as a foreman said--and then usedthis simplified ratio to find the length ofthe target wall. This strategy of finding thesimplified ratio allowed foremen to carry outthe arithmetic in a way that reflected therelationship between the variables, value onpaper and real-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. Sehliemann andNunes interviewed 16 fishermen who had to
deal with the price-weight relationship intheir everyday life in the context of sellingthe products of their fishing. On the basis oftheir analysis of the fishermen's everydayactivities, Sehliemann and Nunes expectedthat fishermen would develop methods ofsolution of the type which has been termed"scalar solution" by Noelting (1980) andVergnaud (1983). Scalar solutions toproportions problems involve carrying outparallel transformations on the twovariables (such as doubling, trebling orhalving the values) without calculatingacross variables (and thus finding thefunctional factor). Schliemann and Nunesobserved that scalar solutions were preferredby fishermen even when these solutionsbecame cumbersome due to the fact that thetarget value was neither a multiple nor adivisor of the data given in the problem(like finding the price of 10 kilos given theprice of 3 kilos of shrimp). The strongpreference observed among fishermen forscalar solutions cannot be explained in termsof their school instruction for two reasons: (1)scalar solutions are not taught in Brazilianschools, where the Rule of Three (a/x + b/e)is traditionally taught; and (2) 14 of the 16fishermen had less than 6 years of schoolingand the proportions algorithm is taught in6th or 7th grade in the area where the studywas conducted.
The preservation of meaning inout-of-school mathematics is clear not onlyin the modelling strategies involved inproblem solving but also in the calculationprocedures used. Reed and Lave (1981), whofirst described the difference between oraland written arithmetic in greater detail,made this point quite clearly bycharacterizing oral arithmetic, typical ofunschooled Liberian taylors, as a"manipulation of quantities procedure" andwritten arithmetic as a "manipulation ofsymbols procedure". Carraher andSchliemann (1988) further expanded thisanalysis by pointing out both differences andsimilarities between oral and writtenarithmetic. Oral arithmetic preserves themeaning of quantities in the sense thathundreds are treated as hundreds, tens astens, and units as units, as can be seen in thetwo examples below, one involvingsubtraction and one involving division.
In the first example, the child was solvingthe problem 252 - 57 using oral arithmeticin a simulated shop. The child said: "Just
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take the two hundred. Minus fifty, onehundred and fifty. Minus seven, onehundred and forty three. Plus tha fiftyyou left aside, the fifty two, one llundredninety three, one hundred ninety five"(from Carraher, PME, 1988, p.10).
In the second example, the child wassolving a word problem which askedabout the division of /5 marble amongfive boys. He said: "If you give tenmarbles to each one, that's fifty. Thereare twenty-five left over. To distribute tofive boys, twenty five, that's hard.(Experimenter: That's a hard one.) That'sfive more each. That's fifteen" (fromCarraher, PME, 1988, p.7).
It can be easily recognized that therelative value of numbers is preservedthroughout in oral computation--forexample, the children said "two hundredminus fifty" and "give ten marbles to eachboy". In contrast, in the school algorithm forcomputation children would be taught to say"seven divided by five, one" (when dividingseventy by five and obtaining ten) and "twominus one" (when subtracting one hundredfrom two hundred), ignoring the relativevalue of the digits.
Carraher and Schliemann (1988), how-ever, pointed out not only differences but alsosimilarities between oral and written arith-metic, arguing for the coordination of socialand logical structuring in any context inwhich mathematical activities are carriedout. The similarities were related to thelogico-mathematical principles that con-stitute the basis for the arithmeticcalculations. Detailed analysis of theprocesses of calculation showed that theproperties of the operations used in oral andwritten computations are the same:associativity for addition and subtractionand distributivity for multiplication anddivision. Neither users of oral nor users ofwritten arithmetic name these properties ofthe operations. However, when they under-stand the procedures they use, they explainthe steps in calculation drawing on the pro-perties of the arithmetic operations.
To sum up the contrasts presentedhitherto between mathematics in school andout of school, we saw that mathematicsoutside school is a tool to solve problems andunderstand situations while schoolmathematics involves learning the results ofother people's mathematics. As aconsequence of this difference, mathematics
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outside school tends to be more likemodelling, in which both the logic of thesituation and the mathematics areconsidered simultaneously by the problemsolver. In contrast, school mathematicstypically focuses on mathematics per se,resorting to applications not as a basis for thedevelopment of understanding but asoccasions for practising specific procedures.Despite the differences, arithmetic in andout of school relies on the same properties ofoperations. The social organization of themathematical activities varies in and out ofschool while the logico-mathematicalprinciples which come into play remaininvariant.
Logic and convention in geometricalactivities.
Arithmetic is a type of mathematicalactivity related to the quantification ofvariables and operations involvingquantities. Geometry can be seen as themathematization of space--or the "graspingof space" through mathematics(Freudenthal, 1973). It involves a wholerange of activities which can be carried outin and out of school like determiningpositions, identifying similarities (in shape,for example), analyzing perspectives,producing displacements, quantifying space,and deducing new information throughoperations. Some of these activities areinvolved in cultural practices outside school.These practices may have to do witheveryday needs of whole populations or theymay be specific of subgroups carrying outparticular activities. Determining positionis a simple example of a geometry-typeactivity in everyday life which may beshared by whole populations. In order towalk about in a village or a city, everyoneneeds some ability to determine one'sposition in relation to the place one wants togo to - especially if the place to be found wasnever visited before or if one is using a map.Orientation in space appears to beaccomplished by a mixture of logical andsocio-culturally determined ways of repre-senting space.
The logic of determining positionsinvolves relating objects to each otheraccording to an imaginary system of axeswhich can be applied in any situation.Piaget and his collaborators (Piaget andInhelder, 1956; Piaget, lnhelder &Szeminska, 1960) have made a significant
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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 fromthe empirical experience of standing uprightand lying horizontally. Piaget's worksuggests that this under-standing isconstructed as children relate objects to oneanother in space.
Piaget and his colleagues did not concernthemselves with variations in the use ofthese systems of coordinates either within aculture across different situations or acrosscultures. More recently other authors havelooked at these variations and pointed outinteresting cultural inputs into this logicalsystem. For example, Harris (1989) noticesthat in most Western cultures we use theframework of "left" and "right", "in front of"and "behind", "above" and "below" whendescribing positions. "Left" and "right" aredescriptions derived from an imaginary axisrelated to the two sides of the body as "infront" and "behind" are related to the frontand the back of the body; "above" and"below" are related to high and low, as headand feet relate to each other. When thisorganization is represented on paper,conventions are that ihe left-right axis onpaper ccrresponds to a horizontal line whilethe pupendicular axis from the nearest tothe furthest point on the page corresponds toa vertical line.
However, as Harris further points out,other cultures may not readily use thebody-related system of axes (like left-right)...ither for everyday practices or when spaceis represented on paper. Several Aboriginalgroups in Australia seem to prefer to usecompass points even over very smalldistances. "For example, when oneAboriginal woman was looking for aparticular picture on a wall covered withphotos, her companion directed her to look tothe west side of the wall" (Harris, 1989,p.12) - a situation in which Westernerswould have used "left/right" instead.Similarly, when teachers were givinginstructions on how to do letters in ahandwriting lesson, they would use thecompass pints; "to write the letter 'd' theinstructionf (spoken in Walpiri) might be:'Start here on the north side, go across southlike this and down across to the north and up,and then make a stick like this" (Harris,
1989, p.12). Bishop (1983) also reports thedescription of a system of sloping axes by astudent from Manu, an island in PapuaNew-Guinea, as "a line drawn horizontal inthe northwest direction". The description ofa sloping line as "horizontal" is incongruentwith Western conventions and could beinterpreted as incorrect. However, if weconsider it together with the description ofthe slope as "in the northwest direction",this student's description of the diagramreveals the use of compass points in therepresentation of relationships in a plane.
Piaget and his collaborators (1960)stressed the importance of this system of axesnot only for locating a point in space but alsoas part of devising a means for measuringangles and defining figures in a plane. Theydescribed children's success at copying anglesas a function of their understanding that theslope of a line could be described by twopoints simultaneously related to a horizontaland a vertical axis. Similarly, Carraher andMeira (1989) observed how youngsterslearning to use LOGO employed a system ofcoordinates when trying to determine theangle to be turned by the turtle in order tocopy a target figure. The imposition of thisimaginary set of axes on the manipulations inspace with LOGO is remarkable because nosuch a system had occurred to the youngsters'instructors at the outset. The LOGO learnersseemed to be bringing to this new situation away of organizing space which they haddeveloped independently of that particularexperience. The use of such static referencesystems is even more interesting when onereflects upon the fact that this is not the onlyway that the youngsters could haveapproached the metric of angles in adynamic situation. As Magina (1991)suggests, such a metric is available in ourculture in the reading of traditional clocksbut not a single subject in the Carraher andMeira study used the clock analogy whenquantifying angles in LOGO.
A different type of geometric activitywhich was also analyzed by Piaget and hisco-workers deals with the concept of thestraight line as "the line of sight", a conceptwhich is involved in projective geometry. AsFreudenthal (1973) points out, this is acomplex definition of a straight line and maybe preceded by other ways of understandingthe straight line. It may also be developedmore clearly in connection with certainactivities of specific groups of people who
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have to take into account perspective indrawing or who have to carry out navigation.
While interviewing fishermen, I
obtained the following explanation for howthey found their way in the narrow channelwhich they took through the barrier of reefsoff the coast where 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 channel".
When I asked what he meant by "thechurch running in front of the tree", thefisherman explained:
"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 isthat people concerned with navigation turnout to do so well in projective geometry tasks:they rely on coordinations of lines of sight inorder to determine their course. In contrast,when we orient our routes on land we relymore on roads and distances than on the lineof sight.
Bishop's (1983) results comparing threegroups of students in Papua New-Guinea seemto support the idea that projective geometryis related to specialized activities. Heshowed the students a series of photographsof small abstract objects and asked them toidentify the place from which the cameratook the photograph. This is reportedly adifficult task, which was performed muchbetter by the students who came from anisland (who averaged 51 correct responses)than by the students who came from thehighland or the urban environments (whoaveraged 34 and 26 correct responses,respectively). The specificity of theactivity is even more clearly understoodwhen one iooks at the results of some of theother tasks that were presented to thestudents. These results do not reveal ageneral superiority of the islanders in allspatial 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
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accurate than the maps drawn by urbanstudents.
In conclusion, similarly to what wasobserved in the field of arithmetic,everyday practices involving themathematization of space reflect bothsocio-cultural experiences and invariantlogical relations. Locating objects on a planemay vary both within a culture and acrosscultures in the use of body-anchored orcompass-related systems of axes. However,it is possible to identify an invariant logic onwhich these variations are based, which isconstituted by the very notion that animaginary system of axes can be imposed onthe to-be-represented space. Everydaypractices of particular groups can also giverise to specific ways of understanding spacewhich may not be as important to otherpeople not involved in such practices. Theconception of a straight line as "the line ofsight" seems to emerge more readily amongpeople involved in navigation than amongpeople who do not concern themselves withnavigation. This concept of the straight linedoes not appear to be a routine that peoplememorize but a real activity because thesame ability can be used in rather difficultand different tasks such as finding theposition from which 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 buildconnections between everyday mathematicsand teaching. In this section we will explorehow everyday practices can be brought intothe classroom and what effects this mayhave on school learning.
The idea of bringing out-of-schoolmathematics into the classroom is not new.The following arguments, amazingly currentboth in their nature and in the goalsproposed for arithmetic teaching, are takenfrom a manual prepared for teachers byBrideoake and Groves as early as 1939:
We felt that in the past, many children%siho 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).
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In the earlier period it was "sums" whichmattered. Class teaching was themethod , with its visualising andmemorising, and the ideal was that thechildren should learn the four rules andbe able 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 "numberapparatus", extensive exercises incounting, and pretentious written sums;but still there was, very often, little mentalalertness or real number power. Nowmany teachers are feeling their way to ar:alistic approach, with its underlyingidc that development of number senseis woat matters, and that "sums" are onlythe expression of this (Bridcoake andGroves, 1939, pp.9-10).
To want to use a realistic approach orbring children's number sense into theclassroom is one thing; to be successful at it isquite another. In this context, an informalobservation by an educator, Herndon (1971),is very informative. He describes how hemet in a bowling alley a student who haddifficulty in school arithmetic but who couldkeep track of eight bowling scores at once inthe alley. He then realized that all of hisstudents had some use of mathematicsoutside school in which they were successfuland tried to make them solve exercises inschool which were similar to what theycould do outside school. This is how hedescribes his attempt:
Back in eighth period I lectured him [theboy from the bowling alley) on how smarthe was to be a league scorer in bowling. I
pried admissions from the other boys,about how they had paper routes andmade change. I made the girls confessthat when they went to buy stuff theydidn't have any difficulty deciding ifthose shoes cost $10.95 or whether itmeant $109.50 or whether it meant $1.09or how much change they'd get backfrom a twenty. Naturally I then handedout bowling-score problems, andnaturally everyone could choose whichones they wanted to solve, and naturallythe result was that all the dumb kidsimmediately rushed me yelling. "Is thisright? I don't know how to do it! What'sthe answer? This ain't right, is it?" and"What's my grade?". The girls whobought saaoes for £10.95 with a $20 billcame up with $400.15 for change andwanted to know if that was right?(Herndon, pp.94-95).
Bringing out-of-school mathematics intothe classroom is not simply a matter offinding an everyday problem and presentingit as a word problem for the application of aformula or an algorithm already taught.This approach does not change the maindifference between mathematics in and out ofschool pointed out earlier because studentswould still be in the same position of tryingto learn the products of other people'smathematical activities. Bringingout-of-school mathematics into theclassroom means giving students problemswhich they can mathematize in their ownways and, in so doing, come up with results(methods, generalizations, rules etc.) whichapproach those already discovered byothers.
Different PME authors have exploredroutes to mathematics teaching related tothe idea that out-of-school mathematics canbe brought into the classroom with positiveresults. Some of these studies will bereviewed here and can be used as startingpoints for further explorations. I would liketo distinguish two types of teachingapproaches in which out-of-schoolmathematics is brought into school.
The first approach starts from aparticular aspect of mathematics which onewants the students to learn--notationsystems, methods, theorems etc.--and thensearches for everyday problems whichinstantiate that aspect of mathematics. Inthis case, the teacher will create constraintsin order to ensure that the specific aspect ofmathematics comes up in the analysis of theeveryday situation and will not considerteaching successful unless the specific goal isachieved.
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-resultof the lesson. Although the teacher choosesthe problem because there are interestingmathematical relations which can beexplored in the situation, there may be noready-made solution process which thestudents are expected to learn by analyzingthe problem situation and many ways ofsolving the problem will be consideredlegitimate. Teaching will be considered"successful" if students analyze thesituations, use mathematical concepts in
4 fr)35
their analyses, build relationships betweendifferent concepts and improve on their ownmethods as they repeatedly employ them onsolving further related problems. Teachersmay introduce mathematical notations asthe pupils devise their solutions but thespecific teaching of these notations is not theaim of the activities from the outset.
In short, the first approach is taskoriented. It aims at teaching concepts,notations, methods etc. and can be evaluatedon the basis of whether or not pupilsaccomplished the specific target. The secondapproach is global and open-ended. It aimsat developing "mathematical minds". Itsevaluation must take into account pupils'progress over time and determine whether ornot they become more sophisticated in theirmathematical analyses of situations as theyprogress in school.
This section is divided into three parts.In the first two, I will present work and ideasfrom both of these approaches. I leave to thereader the task of evaluating them anddeciding whether they can be combined intoa unified philosophy for the teaching ofmathematics or not. In the third part, someof the teachers' reactions to bringing out-of-school mathematics into the classroom arepresented.
Teaching specific aspects of mathematicswith the support of everyday lifeexperiences.
We have seen that people learn much aboutmathematical concepts outside school.However, it does not follow that schoolmathematics can profit from children'sknowledge of out-of-school mathematics. Itis quite possible to recognize thatmathematical ideas are often learnedoutside school but still be sceptical about theimportance of bringing out-of-school mathe-matics into the classroom. The need forstudies which show whether bringing every-day concepts into the classroom can actuallycontribute to the development of schoolmathematics was clearly argued by Janvier(PME, 1985). His arguments are summarizedin the following points:
"(1) certain mathematical ideas havenatural phenomenologicalcounterpart since they result by aprocess of abstraction from objectsand observable entities" (p.135);
(2) these ideas formed from everydayexperience represent models
36
which give meaning to themathematical ideas;
(3) two positions can be formed aboutthe role that these models mayhave in mathematical education:(a) models are restricted in theirgeneralizability and should there-fore be forgotten as soon as correctand efficient handling of abstractsymbols is achieved; and (b) amodel is irreversibly a model forlife and can be brought into play atany moment when there is a needfor the mathematical relationshipsand rules to be rediscovered.
The question posed by Janvier is thusessentially whether everyUay mathematicsis so bound up with the context in which itwas learned that it cannot be "pulled up"from the context and transformed into a moregeneral model. Models for understandingmoney, for example, could remain defined assuch and the relationships between numberswhich were learned in money contexts wouldnot be recognized in other contexts. Adifferent but related concern is expressed byBooth (PME, 1987), who raises thepossibility that students may not link up thespecific representations used in teachingwith the more general mathematical ones,and will thus not profit from the experienceswith the particular situations. Below wewill review a sample of studies in whichchildren learned mathematics in specificcontexts and were tested for their learning inmore general ways. They do not represent anexhaustive review of the literature; they aredescribed here as samples of attempts toteach particular aspects of mathematics onthe basis of situations which pupils mayalready understand from their everydayexperience.
Higino (1987) analyzed the use ofchildren's knowledge of money as a supportfor learning place value notation and thewritten algorithms for addition andsubtraction. She worked with a taskpreviously used by Carraher (PME, 1985;Carraher and Schliemann, 1990) in whichchildren play a shopping game and areasked to pay different amounts of money forthe items they purchase. The children aregiven nine tokens of two different colourswhich represent coins of differentdenominations, one and ten. They are thenasked to pay for items they purchase in theplay shop using these tokens. The amounts ofmoney can be obtained (a) only with singles
(for example, six); (b) only with tens (forexample, 30); and (c) mixing singles and tens(for example, 17, a sum which can only bepaid if the child uses one 10 and 7 singles).The 60 children in Higino's study were 7 to 9years of age and were attending second gradein state schools in Recife, Brazil. They werematched at the outset of the study withrespect to their knowledge of place valuenotation and then randomly assigned to oneof four groups:
11) an unseen control group that received onlythe regular classroom instruction on placevalue notation a n d t h eaddition/subtraction algorithms;
(2) a symbolic teaching group, that receivedexperimental instruction on place valy..and the algorithms without recourse toany materials other than explanationsabout the decades and the rules of theaddition/subtraction algorithms;
(3) a sequential everyday-symbolic group,that first practised counting money andworking out change in the pretend shopand then received instruction on placevalue and on the addition/subtractionalgori thm;
(4) a parallel everyday-symbolic group, thatpractised counting money and givingchange and received parallel instructionon place value notation and theaddition/subtraction algorithms.
Results showed that all four groupsprogressed with instruction according to animmediate post-test. However, in a delayedpost-test given approximately one monthlater, children from the symbolic teachinggroup no longer differed significantly fromthe unseen control group while the other twoexperimental groups still did significantlybetter. Moreover, children from the paralleleveryday-symbolic group showed a smallimprovement in their mean number of correctresponses to the addition/subtractionproblems after one month of instruction whilethe other groups showed slight decrements.Thus Higino was able to show thatconnecting everyday experiences withclassroom learning of addition andsubtraction produces better results thanteaching without regard for children'sprevious knowledge.
Another study which attempted toanalyze the role of everyday situations inthe development of basic concepts of addition
and subtraction was carried out by De Corteand Verschaffel (PME, 1985). They arguedthat most instructional approaches use wordproblems as applications of addition andsubtraction but it is questionable whetherword problems representing real situationsonly have this role in elementaryarithmetic.
"Indeed, recent work on addition andsubtraction word problems has producedstrong evidence that young children whohave not yet had instruction in formalarithmetic, can nevertheless alreadysolve those problems successfully using awide range of informal strategies thatmodel closely the semantic structure andmeaning of the distinct problem types."(p.305)
Working from this assumption, De Corteand Verschaffel compared a control group,instructed in the traditional way describedabove, to an experimental group, wholearned addition and subtraction conceptsmostly from solving word problems andlearning several ways of representing thesemantic relations in the problems. Theyobserved that the experimental group madetwice as much progress on word-problemsolving than the control group after one yearof instruction, a difference which wasstatistically sig-nificant. De Corte andVerschaffel concluded that this line ofinvestigation is worth pursuing furtheralthough they recognize that the problemsused in the experimental program wererelatively poor in content and did notapproach outside-school situations as muchas desired. Similar positive results werereported also by van den Brink (PME, 1988)after a year-long experiment on children'slearning of addition and subtrac-tion conceptseither by modelling from everydayexperiences or from teaching with-in atraditional approach.
Looking at somewhat more complexsituations, Janvier (PME, 1985) analyzed therole of models in teaching negative numbersby contrasting two groups of students, one whohad learned negative numbers from asymbolic model and a second one which hadlearned from what he calls a "mental image"model based on particular experiences. Thesymbolic-model group learned negativenumbers by having the symbolism introducedonly loosely related to an initial situationand then concentrating on rules formanipulating numbers. Different colours
37
were used for negative and positive numbersand rules were of the type: "when you addtwo numbers of the same colour you make anaddition and keep the same colour; when youadd two numbers of different colours, yousubtract the small one from the big one andtake the colour of the big one" (Janvier, 1985,p.137).
The mental-image group attempted tosolve problems about a hot-air balloon towhich bags of sand or helium balloons couldbe attached, the first having the effect oflowering the balloon while the latterbrought it higher. Balloons and bags wereinitially of one unit but several of eithercould be used at once so that symbolism couldbe developed for larger numbers. Addingmeant tying an extra-element and subtractingwas identified with removing an elementfrom the dirigible. Janvier found that themental-image group did significantly betteron addition items than the symbolic-modelgroup; no significant difference was found forsubtraction. Furthermore, when studentswere interviewed later, those from themental-image group could correct themistakes they had initially made by goingback to the model; however, thesymbolic-model students simply repeatedwithin the model the same mistakes andwere unable to revise their wrong answers.
Other studies which must be mentionedand which deal with more advancedmathematical concepts are in the realm offunctions and algebra. Janvier (1980) andNIonnon (PME, 1987) have investigated theuse of situations represented graphically inthe teaching of functions. Both studiesanalyze students difficulties with graphicrepresentation and improvement afterteaching. However, no systematic com-parisons between the experimental groupsand others taught without the support of theeveryday situations represented graphicallyare available and clear conclusions are thusprecluded. Filloy (PME, 1985) studied theuse of addition/subtraction of packages ofunknown weights onto two- plate balancescales as images for algebraic manipulationswith unknowns. Although he reports somesuccess, he seems to be cautious about theresults of this experience in a later paper.Filloy and Rojano (PME, 1987) argued thatsubjects could achieve a good handling of theconcrete model but developed a tendency tostay and progress within this context only,showing difficulty in abstracting the
38
operations to a syntactic level and inbreaking with the semantics of the concretemodel.
To summarize, it seems possible to useeveryday situations in the classroom asinstances of particular aspects ofmathematics and obtain better learning ofthis mathematics in context than in atraditional teaching schema relyingprimarily on symbolic manipulations.Further research is undoubtedly stillnecessary since only a very narrow range ofma thema tics ha s been systema ticallyinvestigated.
Real-life mathematics used in an open-endedfashion.
Streefland (PME, 1987) presented in veryclear terms the instructional principleswhich guide the open-ended type of realisticmathematics education which we aredistinguishing from the modelling of specificmathematical aspects discussed above.These instructional principles are:
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 paidto the development of situation models,schemas and symbolising;
c) children can make significantcontributions to classroom work throughtheir own productions and constructions inthe process of moving from informal toformal methods;
d) this learning process is of an interactivenature;
e) different concepts become more clearlyinterrelated in this progressivemathematization of situations.
Streefland (PME, 1987; PME, 1988) hasproduced several examples of how childrencan learn mathematical concepts in theprogressive mathematization of realsituations. In one PME research report,Streefland (PME, 1987) describes the processof mathematization leading from division inthe sense of equal sharing to fractions. Thechildren start from a situation in which theyare asked to work out how to share threecandy bars among 4 children. Examples ofhow the distribution is performed and what
43
children learn from this distribution in thisfirst phase 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 a halfchocolate, using up two whole chocolates,and the last chocolate would then be cutinto for equal parts, each child's sharebeing described as 1 /2 + 1/4 (thisdescription helps children understandthat 1/2 and 2/4 are the same amount).
In the second phase of teaching, "concretestories of fair sharing gradually melt intothe background (...). The activities changeinto composition and decomposition of realfractions" (Streefland, PME, 1987, p.407)using methods of decomposing fractions intounits and finding equivalences, but it is stillbest "not to wipe out all the traces of the realsituations referred to even at this stage"(p.407). For example, 5/6 can be decomposedinto 1/6 + 1/6 + 1/6 + 1/6 + 1/6 and then otherequivalences can be found like 3/6 + 2/6 andthen 1/2 + 1/3 and so on.
In the third level of teaching, "the rulesfor the composition and decomposition offractions will become the objects of mathe-matical thought (...) the methods ofproduction which turn out to be the mostefficient might become standard proceduresor algorithms" (p.407). "For example,children work with questions like: 1/2 + 1/3are probably pseudonyms for other fractionswith a common name; produce these otherfractions".
Streefland reports that children workingwith this approach to the teaching offractions show increasing skills in producingequivalent fractions, in operations withfractions with the exception of division, anda general dominance of the elementary lawsfor operation as methods of production.
Streefland (PME, 1988) presented furtherexamples of his instructional approachincluding topics like subtraction, divisionand further examples about fractions. Healso strengthened his theoretical principleson 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 acquirea variety of aids and tools in theprogressive mathematization ofsituations becoming competent in the useof different terminologies, symbols,notations, schemas, and models;
organizing and structuring themathematics should be as much aspossible the business of the childrenthemselves, a principle which stressesthe importance of interaction andcooperation in this way of approachingmathematics education.
Other significant aspects of this realisticapproach to the teaching of mathematics areemphasized by De Lange (1987; in press),who discusses the importance of using realmathematics also in the assessment ofchildren's success in learning mathematics,and by Boero (PME, 1989). who points outthat there are misconceptions of realsituations which may creep up inmathematics classes when external contextsare used for conceptualizing. According toBoero, these misconceptions do notnecessarily hinder the process ofconceptualizing mathematics; they arerather problems to be faced by students "ifone wants the pupils to gradually understandthat there are levels of 'intuitive evidence'and 'intuitive' ways of thinking which mustbe exceeded if a rational working command ofcertain phenomena is to be reached, and thatmathematics may have an important role inthis passage from intuition to ration-alization" (Boero, PME, 1989, p.69).
The evidence produced by Streefland onpupils' progress in the process ofma thematizing situations is provocative.Mathematics appears as a way ofrepresenting and mentally manipulatingsituations, bridging the gap between aproblem and a solution and allowing childrento go beyond the information they startedthe task with. Streefland's descriptions ofhow children progress from close modellingthe situations to developing abbreviated andgeneral procedures as they interact witheach other and compare the efficiency of thedifferent procedures suggest the importanceof exploring this approach further with a
39
new range of topics. It is especiallyinteresting to note how children seem tounderstand quite clearly the mathematicalequivalence of their different procedures onthe basis of their modeling properties butstill go on to evaluate their efficiency. Theydo not stop working when they "find theanswer".
To summarize, the open-ended realisticapproach to mathematics teaching involvesthe search for everyday situations which canbe treated as sensible problems in mathe-matics classes. The choice of situations is notgeared to the teaching of particularmathematics topics. Pupils success is notevaluated on the basis of whether a specificbit of mathematical knowledge wasaccomplished. Instead, pupils success isevaluated on the basis of the progressivesophistication of their methods and theirbuilding of relationships between concepts,symbols, notations and terminologies. This isessentially the basis of Streefland's (PME,1988) contrast between the teachingapproach developed by "realisticinstruction" to the "structuralist approach"proposed by Dienes. While the realisticapproach starts from pupils good sense in aproblem situation and progresses to moreformal procedures, in the structuralistapproach "vertical mathematizing isoverstressed and formal procedures areimposed" (p.89). In other words, concretematerials devised for instruction within thestructuralist approach do not aim at bringingout-of-school reality into the classroom.Instead, mathematical objects are devisedfor the purpose of concretely exemplifyingmathematical procedures and represent-ations already formalized. The mathe-matical objects thus devised may have nomeaning outside the classroom. This meansthat pupils may not have models whichthey can readily call upon to understandtheir experiences in the classroom.
This open-ended realistic approach isnot restricted to arithmetic but has also beentried out with geometry. Goddijn and Kindt(PME, 1985) strongly criticized traditionalgeometry teaching as stereotyped andrestricted to "the flat world of textbook".They propose to give geometry teaching anew approach by working from thethree-dimensional world. Examples ofquestions in their programme include workingwith proportion and scale comprehension asstudents look at pictures and take into
40
account distances and viewing points.Students attempt to find the position fromwhich a picture was taken and areencouraged to use explanatory drawings inthis process. In so doing, they "discover" theline of sight and the changes in perspectiveand size of figures as the "viewer" movesabout. They look at shadows and reflect onrelated phenomena such as the phases of themoon. They watch a student enact themovements of the moon around the earth infront of a bright light and ask each otherhow is moon seen, lit-up or in shadow, fromthe perspective of pupils is different parts ofthe room. They use the "line of sight" fromthe point where they stand, imagine it fromthe point where someone stands, and try tofigure out where someone must stand in orderto have a particular viewpoint.
Goddijn and Kindt describe how students(and teachers alike!) become surprised athow well their methods using lines of sight(which may even be concretely representedby strings) work in drawing and solving theseproblems. But the results of this explorationof space are not restricted to perspectivedrawing; students found, for example, atotally new manner of doing the classicassignment of constructing the line through agiven point which intersects two skew lines.Thus, by teaching geometry as "graspingspace through mathematics", Goddijn andKindt were able to observe a progressivesophistication of students methods fordealing with spacial and geometricalproblems.
Teachers' reactions to bringing out-of-schoolmathematics into the classroom.
Many mathematics teachers seem to resist atleast initially, the suggestion of bringinginformal knowledge of mathematics into theclassroom. Several reasons are pointed out inthe literature. First, Schultz (1989) observesthat teachers seem to expect that they willcarry out the same pedagogy they werethemselves the recipients of. They learnfrom modelling their teachers as much as (ifnot more than) they learn from theories inteacher education programmes. Second, theyare usually concerned with covering thestandard curriculum material, which isdetermined in terms of traditionalmathematics topics, and not in terms ofmathematization of situations. To thisconcern is added the resistance to the use ofintuitive methods in the classroom because
45
these methods conflict with teachers' viewsof mathematics as a formal system ofknowledge (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 spontaneousmathematical activity outside the classroomand teachers' experience of the sameteaching approach when they themselvesmathematize experiences of a more complexnature. Rosenberg (1989) reports that"Ah&-experiences happened very oftenwhen they (the teachers) started frominformal problems and finally got tofamiliar formal structures" (p.83).
A last source of resistance comes fromteachers' beliefs about what studentswill/will not do when faced with a neweducational experience. In this context, it isinteresting to report briefly on an informalobservation with a group of Brazilianteachers, who found it hard to believe thatpupils would experiment with and improveupon procedures and mathematical modelsonce they had already found an answer tothe problem. In traditional mathematicsteaching, finding an answer usuallydetermines the end of the activity. For thisreason, teachers expected that students willstop working as soon as an answer is found.They were also convinced that some answersare correct and others are wrong and thatshifting mathematical models within aproblem simply means "finding the rightanswer versus finding the wrong answer". Alittle later in the day I asked the teachers todevelop a simple way of determining therate of inflation, a problem of everydaysignificance in Brazil. They quickly came upwith the idea that inflation could be viewedas the average of all price increases. Theydrew up a list of items, wrote down theirestimations of increases in prices inpercentages and calculated the averageincrease. They were not satisfied with theirsolution to this problem and did not stopworking. They realized that housing andgrocery shopping might affect people'severyday expenses more markedly thanentertainment, for example. One teachersuggested that they could give weights tothe items which would be proportional tothe 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. Theycalculated a weighted average of theincreases as their new solution. At thispoint, the teachers seemed to have an "Aha!experience". They realized that, insituations such as the one they had justworked on, students might well look at thesame problem in different ways by trying outdifferent mathematical models on the sameproblem.
Finally, it is interesting to add Schultz's(1989) comments on the impact that the use ofreal-life mathematics in the classroom hadon the student-teachers she worked with.Although no systematic data are presented,she observed that they became more awareof what they had learned and the way theylearned and seemed to develop new methodsand ideas for teaching their own students. Ifmodelling from real-life situations turns outto have a positive impact on futuremathematics teachers, this may be just asimportant as the effects it has directly onpupils. However, no research data seem yetavailable on this issue.
Final comments
This paper reviews research which showsthat the social context in whichmathematics is carried out influences theway people approach mathematicalproblems. Mathematics in many of today'sclassrooms is taught as an abstract form, a setof symbols, procedures, and definitions to belearned for perhaps some later application.Mathematics used outside school is a type ofmodelling: it is a way of representing realityso that further knowledge about the realitycan be obtained from the manipulation of therepresentations without the need to checkthese new results against reality(D'Ambrosio, 1986). These differences in thesocial situations and their correspondingmathematics have an impact on rates ofsuccess and types of procedures used by theproblem solvers. Research has shown thatbringing out-of-school problems into school asa way of teaching mathematics is not onlypossible but also a beginning of a moresuccessful story about mathematics teaching.Further research is still needed and moreclear theorizing so that successful isolatedexperiences can be transformed into aneffective educational theory.
Bringing out-of-school mathematics intothe classroom means facing questions whichmay not be addressed in the traditional
46 41
forms of teaching. Some of the questionswhich emerge within this approach tomathematics teaching are presented below.
We know that many concepts can belearned outside school and that thisknowledge can be useful in school. But wecannot be casual about which concepts andsituations we choose to bring into school. Animplicit route in this choice appears so far tobe to look at present mathematics curriculaand choose problem situations appropriatefor present day curricula. However, is itpossible to identify some "core" conceptswhich will guide this choice in a new way?
A second question has to do with theintroduction of mathematical represent-ations. Pupils solving real problems do notneed to reinvent mathematical represent-ation. When and how is mathematicalrepresentation to be introduced?
A third question which remains to beinvestigated is which factors affect thepassage from informal to more formalmathematical representation in the processof mathematizing real problem situations.Much of the success of the realistic approachdepends upon interaction between pupils andwe have achieved relatively littleunderstanding 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 trivialquestion but one of great theoreticalsignificance, as Bryant (1982) has pointedout.
Finally, it would be desirable toaccomplish better descriptions of the processof progressive mathematization which takesplace when students interact and solve realmathmaticss problems together. Is there asingle route in the development of eachparticular concept or are there many ways toget to the same place? Does naturallanguage, which is of course used in thestudents interactions, play a part in thisprocess or not? Is the progressivemathematization of situations a process ofreconstruction of solutions at ever- higherlevels of abstraction or does it involvedropping old models which are thenreplaced by new ones? Is there a momentwhen real problems are no longer importantor do they retain their role of giving initialmeaning to formalizations even at thehighest levels of abstraction?
Hopefully, further research and fruitfulcooperation between researchers andteachers will bring clearer answers to manyof these questions in the future.
Terezinha Nunes was Associate Professor at the Universidade Federal de Pernambuco, Brazil,and has been recently working as Research Fellow at The Open University, U.K. From October1991 she will be a lecturer at the Institute of Education, University of London. Her studies ofmathematical knowledge developed outside school (published mostly under her previoussurname, Carraher) analyze children's and adults' understanding of the numeration system,arithmetic, and simple mathematical models such as ratio and proportions. She has sustainedmuch interest in the educational implications of these findings and worked closely with theeducational authorities in Recife, Brazil, in the development of teacher education programmes.
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The Influences of Teaching Materials
on the Learning of Mathematics
Kathleen Hart
This is an excerpt from a conversationbetween a teenage girl and a mathematicseducator: (I: Interviewer; A: Alison)
You need four cubes for eight people sofor four 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 tasteto it, if you put less it wouldn't taste asnice.
Really, even if you were only makinghalf as much soup?
A: Yes, 'cos you need the same amount don'tyou, else it would taste horrible, itwouldn't taste as strong.
But let's say you make enough soup for 8people and you put 4 cubes in, now thenext day you are making soup for 4people. How many cubes would you putin then?
A: I would put in two but it would not tasteas nice would it?
I : Don't you think so?
A: No.
I: You don't think it would taste thesame?
A: Because if you are going to make soupyou are going to have to use the cubesanyway to make it taste aren't you?
I: Yes
A: So if you want it to taste just as good asit did when you made it for 8 you wouldhave to put the same number of cubes init. Otherwise it won't taste the same.
I think it would. I think if you are goingto make half as much soup, you wanthalf as many cubes.
A: I suppose you could, but I wouldn't if Iwas making the soup.
Frank is only ten years old and he andhis class have been 'taught' about thevolume of a cuboid.
(I: Interviewer; F: Frank) 1 xbxh= V
F:
F:
F:
F:
F:
F:
Well, a, h, c, d, e, f, g, h, i, j, k, I - 1, 2, 3,4, 5, 6, 7, 8, 9, 10, 11, 12. That'll be 12times 2 and ...
Can you tell me how you know that?
Well, 1 is ... urn... the tenth letter of thealphabet.
Yes.
So, 10.
Ah... who told you that?
... (long pause)...
Did you just make it up?
Think so, yes.
Okay, and 'b' will be? And 'h'?
H... urn... eh 4... 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 onehundred and ... one hundred and ... eight... oh, two hundred and ... oh, onehundred and forty.
I: Alright, and then it says equals V.
F: Equals V? 140.
I: How do you know that that's not right,shaking 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 researchconcerning materials which are availableto the teacher to assist in the process ofteaching. Materials can include books,computer software or languages, concretematerials (manipulatives) or other aids.The emphasis is on research which led tothe production of the material or whichsought to produce evidence of itseffectiveness. The main source of theresearch references is the work of members
43 43
of the international group Psychology ofMathematics Education.
Most children learn their mathematicsin school although they use it and obtaintheir first ideas of the topic in the worldoutside the classroom. Within theclassroom they may be grouped with othersof the same age, other pupils of the sameattainment level or simply all the otherchildren who live in that village or street.The teacher who provides the experiencesfrom which they are supposed to learn hasa difficult task no matter which way thechildren who share the classroom areselected. The researches of PME take as abasic premise that the teacher is strivingfor the children to understand what theyare doing. PME reports which describeteaching experiments in which a form ofwritten material, a set of concretemanipulatives or a computer program havebeen tried so that learning is more effectiveare not numerous. Many are based on thedesire to provide empirical evidence of thevalidity of a learning or psychologicaltheory and in only a few cases are theresults overwhelmingly supportive.However, books are written and materialsproduced and the research results must shedsome light on how this can be betterachieved so that learning takes place.Nobody has yet provided a solution for thequestion 'how can I best teach this child'.Throughout the world, with very fewexceptions the performance of children onmathematics tasks is consideredunsatisfactory by educators in their country.Is this because educators set unreasonabletasks in the discipline of mathematics orbecause the expectations are entirelyreasonable but those who teach the subjectsingularly inefficient?
Learning theories and their influences
Most research is carried out within atheoretical framework. The choice offramework influences the assumptions, thepremises suggested and tested and theoutcomes. Sometimes materials areinvented to conform to the theory and theresearch is perhaps to test theireffectiveness or indeed provide someevidence of behaviour which supports thetheory itself. Often existing books orteaching aids are criticised because theirdevelopment contradicts a learning theory.At the moment most mathe-matics books for
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young children which are based on languageand words only would be condemned as "tooabstract", developmental psychologyhaving pro-vided theories which statethat a young child is incapable of abstractreasoning.
Piaget's developmental theory outlinedstages of mental development loosely tiedto age. The theory divided intellectualdevelopment into four major periods:sensorimotor (birth to 2 years); pre-operational (2 years to 7 years); concreteoperational (7 years to 11 years); andformal operational (11 years and above).
Piaget's interest was in the psychologyof the child but many educators haveextended and interpreted his writings inorder to apply the theory to teaching as isshown here by Ginsburg and Opper (1969) intheir book Piaget's Theory of IntellectualDevelopment:
"While Piaget has not been mainlyconcerned with schools, one can derivefrom his theory a number of generalprinciples which may guide educationalprocedures. The first of these is thatthe child's language and thought aredifferent from the adult's. The teachermust be cognizant of this and musttherefore attempt to observe childrenvery closely in an attempt to discovertheir unique perspectives. Second,children need to manipulate things inorder to learn. Formal verbalinstruction is generally ineffective,especially for young children. The childmust physically act on his environment.Such activity con-stitutes a majorportion of genuine knowledge; themere passive recep-tion of facts orconcepts is only a minor part of realunderstand ing."
The theory of Piaget has for many yearsinfluenced those who teach mathematics,mainly through the advice given toteachers in training. The definition of theconcrete operational level and the featureswhich distinguish it from the formaloperational level have led often to a beliefthat children below the age of eleven yearsshould be given manipulatives to assistthem in learning mathematics or indeed insome 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
4.9
itself irrespective of the topic being taught.Researchers have started to discuss the gulfbetween manipulatives and formal schoolmathematics and PME members haveconsidered this with respect to the learningof Algebra.
Ausubel (1963) rather than having anage-dependent theory provides a theory oflearning which emphasises the depen-dence of new knowledge on old knowledgeand is perhaps better suited to the needs ofmathematics teaching which traditionallybuilds on previous teaching. As Novak(1980) says in quoting him:
"However, we must also consider thecognitive functioning and psychologicalset of the learner as new knowledge isinternalised. Here we must distinguishbetween rote learning wherein newknowledge is arbitrarily incorporatedinto cog-nitive structure in contrast tomeaningful learning wherein newknowledge is assimilated intospecifically relevant existing conceptsor propositions in cognitive structure.Since the nature and degree ofdifferentiation of relevant concepts andpropositions varies greatly from learnerto learner, it follows that the extent ofmeaningful learning also varies along acontinuum from almost pure rote tohighly meaningful".
In Europe successful learners ofmathematics in schools and universitieshave always been few in number, successfullinguists or engineers are more numerous.This state of affairs did not improveproportionally with the introduction ofuniversal secondary education. This hasoften meant that those who succeed havehad the reputation of possessing a superior(or certainly different) type of intelligence,their success being Ndged not attributable tothe skill of the teacher or the suitability ofmaterials but to the make-up of theirbrains/intelligence. Psychologists andeducationalists can only measureperformance on tasks and therefore a theorywhich discusses the potential for learningas does that of 'information processing' is ofvalue.
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. Morerecently the emphasis in mathematics
education research has moved to theconstructivist point of view.
A large number of PME members espouseConstructivism, indeed the XIth Conferencein Montreal had Constructivism as itscentral theme. Kilpatrick (PME, 1987)describes some fundamental points:
"The constructivist view involves twoprinciples:
1. Knowledge is actively constructedby the cognizing subject, notpassively received from theenvironment.
2. Coming to know is an adaptiveprocess that organizes one'sexperiential worlds; it does notdiscover an independent pre-existing world outside the mind ofthe knower."
'The first principle is one to which mostcognitive scientists outside thebehaviourist tradition would readilygive assent, and almost no mathe-matics educator alive and writing todayclaims to believe otherwise. The secondprinciple is the stumbling block formany people."
'Radical constru,tivism adopts anegative feedback, or blind, view towardthe 'real world'. We never come toknow a reality outside ourselves.Instead, all we can learn about are theworld's constraints on us, the things notallowed 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 constructivist
position:
(a) teaching (using procedures that aimat generating understanding)becomes sharply distinguishedfrom training (using proceduresthat aim at repetitive behaviour);
(b) processes inferred as inside thestudent's head become moreinteresting 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 theteacher's expectations becomemeans for understanding theirefforts to understand; and
e) teaching interviews become attemptsnot only to infer cognitivestructures but also to modify
them. All fiveconsequences fit the constructiviststance, but they appear to fit otherphilosophical positions as well."
At all levels of the constructivist beliefthe child is active in the learning processand not a passive receiver of knowledge.Radical constructivists would adviseteachers to provide rich learningexperiences for the child from which he canconstruct his own mathematics. This is aviewpoint different from that which saysthe teacher armed with materials is thesource of the major part of mathematicsinformation in the classroom.
Ample evidence has been provided,over many years, that children change theinformation they are given by adults andremember it in a different form. Childrenalso invent their own methods for carryingout mathematical tasks. Street children inBrazil for example have adequate methodsof calculation of profits on goods they sell.Many English children ignore thegeneralised, formal methods of solutionthey are taught in school, in favour of morespecific, invenled methods (Booth, 1981).Whether this is because the child'sunsullied intuition is at work or whether itis because of desperation arising from a lackof understanding of the teacher'spresentation, is unknown. We know thatmany children start school knowing how tocount, to sort and to put objects in order.Children are able to construct mathematicsbut nobody has yet shown that they canconstruct all that is needed to be amathematically competent adult in theyear 2000. Research which shows a teachercan be most effective with talk and ablackboard is not often carried out inmathematics education. Brophy (1986),writing in the Journal for Research inMathematics Education, quoting from thegeneral research on teaching said:
"Mere engagement in activities will notfacilitate learning, of course, if thoseactivities are not appropriate to thestudents' needs. Thus, the teacher'sdesire to maximize content coverage by
46
pacing students briskly through thecurriculum must be tempered by theneed to see that the students makecontinuous progress along the way,moving through small steps with high,or at least moderate, rates of successand minimal 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 achievementgains is not mere 'time on task', or even'time on appropriate tasks', but timespent being actively taught or at leastsupervised by the teacher. Greaterachievement gains are seen in classesthat include frequent lessons (wholeclass or small group, depending ongrade level and subject matter) in whichthe teacher presents information anddevelops concepts through lecture anddemonstration, elaborates this infor-mation in the feedback given followingresponse to recitation or discussionquestions, prepares the students forfollow-up assignments by givinginstructions and working throughpractice examples, monitors progresson those assignments after releasingthe students to work on themindependently, and follows up withappropriate feedback and re-teachingwhen necessary. The teachers in suchclasses carry the content to theirstudents personally rather than leavingit to the curriculum materials to do so,although they usually conveyinformation in brief presentationsfollowed by opportunities for recitationor application rather than throughextended lecturing."
Generally both theory and researchseem to point to the importance of theteacher and her interaction with the child.Materials cannot of themselves provideenough for the child to learn. Increasinglythe suggestion is that teachers adopt themethods of research and learn fromlistening to and observing children.
5 "1
Progression in mathematical ideas
Written materials, textbooks, worksheets,computer programs are produced with aparticular progression in mind. The writerpresupposes a logic which dictates whatshould precede what other piece ofmathematics teaching. The structure ofmathematics often dictates what must beconsidered a pre-requisite. For many yearsBloom's taxonomy provided a theoreticalunderpinning for progression in whatteachers should present to children. Morerecently the suggestions of van Hie leconcerning the structure of Geometricprogression in the school curriculum hasbecome popular in research projects and hasbeen extended to other parts ofmathematics. It is quite general and theincreasing complexity of demand is obvious.These definitions are from Shaunessy(1986):
"Level 0. (Visualization) 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 I. (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 studentlogically orders the properties ofconcepts, forms abstractdefinitions, and can distinguishbetween necessity and sufficiencyof a set of properties.
Level 3. (Deduction) The studentre2sons formally within the contextof a mathematical system,complete with undefined
terms, axioms, anunderlying logical system, definedterms and theorems.
Level 4. (Rigor) The student cancompare systems based ondifferent axioms and study variousgeometries in the absence ofconcrete models."
Biggs and Collis (1982) haveformulated a taxonomy to provide taskswhich become structurally more difficultand to assess the quality of children'sanswers. The advantage of this is that the
child is not labelled and convenientlyassigned to a level of performance as is thedanger with neo-Piagetian assessment butthe quality of response can be seen to varydepending on the task.
Gagne was one of the first theoreticiansto analyse topics for the pre-requisites ateach stage (especially in mathematics).Such an analysis is vital for any productionof materials.
Curriculum developers have ampleadvice on progression but little empiricalevidence of a teaching sequence which ismore effective than another because of theorder of presentation. The research scene ineducation is woefully lacking inlongitudinal surveys where the samechildren are investigated as they growolder. We have details of the per-formanceof cohorts of children of different ages but astatement about the performance of aparticular age group cannot reflect thediversity within that group. Age is not avery good predictor of performance bothbecause of the diversity in one age level butalso because there is ample evidence ofyoung children being capable of learningquite complicated mathematics in a novelsetting. Here the questions that must beasked are:(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 ordinaryteacher in the ordinary classroom?
Hierarchies in mathematics attainment
In the absence of a detailedlongitudinal study following the samechildren for many years, the empiricalevidence of progression we have is based onthe performance of children of differentages on various mathematical topics. Thesedata take on greater significance if thequestions are written according to somepostulated hier-archical demand.National and indeed international surveysmay provide information on levels ofdifficulty but they are usually carried out inorder to monitor whether a representativesample of the nation's children areperforming any differently to therepresentative sample tested in a differentyear. The testing of children using the vanHide levels in Geometry has given some
52 47
evidence of levels of difficulty in Geometry(often within the U.S. setting).
A progression of easy to hard itemsgives information on some aspects ofchildren's understanding but is of morevalue if there is evidence that childrenwho can successfully deal with items at astated level of difficulty can alsodemonstrate success on those that areregarded as being at a lower level. Such acheck can be carried out on the singleoccasion of testing by written questions orinterviews. Statistical techniques toquantify scalability are available. In thecase of methods of solution used by childrento solve problems one can investigate theiruse at different times in the child's schoolcareer. Karplus and Karplus (1972) wasable to show that strategies used bychildren to solve Ratio and Proportionproblems (or in most cases fail to solvethem) formed a progression. In hislongitudinal study, as the sample grewolder (and the pupils learned moremathematics) they moved from one strategyto another. Those methods of solution that
'were abandoned were considered to be at alower level being replaced by moresophisticated ones.
For example he showed that childrenwho reasoned that one amount was anenlargement of another simply because itwas bigger were, two years later, reasoningin a different fashion. Some incorrectmethods used, however, were verypersistent and showed no change or as manychildren moved into using them as movedout and on to other strategies.
Concepts in Secondary Mathematics andScieace
The aim of the CSMS research project(Hart, 1981) was to inform teachers aboutwhat secondary school children founddifficult in their mathematics and whatthey found easy, with some indication ofwhy. The sample was large (1000 childrenaged 12-16) and the intention was toprovide teachers with a view of themathematical performance of the vastmajority of children over four years of theirsecondary schooling, rather than at 16 yearsof age, when they complete a nationalexamination. The methodology was tocollect data from interviews and also fromwritten tests. The interviews providedinfomiation about how children attempted
48
to solve problems and about what errorsthey made. They were used to inform andenhance the written test results.Interviewing children one-to-one hasbecome an increasingly popular researchmethod during the last ten years. Thebelief is that a child will answer morefreely and truthfully if allowed to describeorally and in its own words what is beingdone. The CSMS project formulated a set often subject hierarchies in which groups ofitems were shown to be scalable, in thatsuccess at harder levels presupposed successat easier levels. A child was deemed tohave succeeded at a particular level if ithad correctly answered two-thirds or moreof the items in that level and in all easierlevels.
It was apparent from the results thatthere was more variation within an agegroup than between age groups. The 14-year-olds performance on a given item wasonly about five to ten per cent better thanthat of the 13-year-olds. Nor did thereappear to be any large jump i t
understanding at fifteen, although more ofthe older children solved the harder itemssuccessfully. A longitudinal survey of 600children produced evidence that certainerrors committed by pupils at age 12 yearswere likely to persist, and be still apparentat age 15. Two hundred children weretested on questions in Algebra, Ratio andGraphs at the end of their second, third andfourth years in secondary school.
The research is dependent on the itemsand is likely to be influenced by whatevermathematics the pupils had previouslystudied. All data are influenced by manyvariables - previous mathematical exper-iences, teachers, books, socio-economic class,the items used and the evaluation of theanswers. The social factors were spread inthe CSMS data in that with such a largesample chosen on the criterion ofapproximation tc the normal distribution ofnon-verbal I.Q. scores, the influence of anyone teacher or textbook was blurred.However, simply because children are moresuccessful on certain mathematical problemsthan on others does not mean that this isnecessarily the best order for thepresentation of material. In the absence ofresearch showing that there is a 'bestorder', a case can be made for the premise:if the pupils score badly on this topic whichthey have normally been taught, then it is
53
harder than this other on which they didwell, therefore in the material beingwritten it should come later. Thealternative is to rely on an analysis of themathematics involved or to base the orderof presentation on intuition in whichreflects teaching experience.
The CSMS research was carried out inEnglish schools but the items have beenused by mathematics education researchersfrom many other countries and appear toproduce valuable information for them. Lin(1988) has replicated all the CSMS testpapers in Taiwan with large samples. Hisresults tend to support the existence of thelevels and show no great discrepancyalthough Taiwanese children tend to bevery successful or fail badly on the items.Lin's interviews show that the Taiwanesesample do not resort to child methods butare usually seeking in their memory for analgorithm they have been taught.
The performance pattern of arepresentative sample of Taiwanesechildren was very different however,perhaps reflecting the two cultures, asshown in Figure 1.
501-
40 r
30
20r--
101
00
'.>
El
TAIWAN
U K
Fig. 1 Ratio Results UK/Taiwan
Progression in number complexity
Children's skills with number usually startwith the ability to enumerate the contentsof a set, whether it is simply by repeatingnumber words whilst climbing the stairs orgiving a number to the contents of a group oftoys, cakes or people. It has been longrecognised that fractions and decimals aremuch more difficult for children than thecounting numbers. When there is a non-integer value in a question it is made verymuch harder, not slightly more difficult.
On the current mathematical diet providedin schools many British children reject allbut whole numbers and exist throughouttheir secondary school years trying to makesense of their mathematics whilst usingonly whole numbers. The nature of theunderstanding of Fractions has occupiedmany mathematics education researchersover many years. The symbolicrepresentation a /b hides at least sevendifferent interpretations, the following isprovided by Kieran (1976).
His seven meanings are listed below, wehave illustrated each with an example:
1. Rational numbers are fractions whichcan be compared, added, subtracted, etc.
This aspect concentrates on the meaning of afraction, usually the model employed forteaching is a region or piece of a rectangle orcircle. The operations on fractionsemphasise recognition, conventions andrules such as 'you can only add two fractionsif they have the same denominators'.
2. Rational numbers are decimal fractionswhich form a natural extension (via ournumeration system) to the wholenumbers.
With the advent of metrication in Britainthe use of decimal fractions has increasedand fractions as parts of a whole (5/8, 3/14)which were common in imperial measureshave ceased to be important. The calculatorof course employs decimal fractions and theincreased usage of this aid in the classroommeans a greater emphasis on decimalnota tion.
3. Rational numbers are equivalenceclasses of fractions. Thus (1/2, 2/4, 3/6 .
) and (2/3, 4/6, 6/9are rational numbers.
The common use of equivalence is in thecomparison of two fractions, addition andsubtraction, e.g. 1/2 = 2/4, if the second formis more apposite we use that rather thanthe first. Children often think the fractionhas changed in size and refer to 2/4 as'bigger' because 2 and 4 are bigger than 1 and2, whereas of course the two ratios nameexactly the same amount.
4. Rational numbers are numbers of theform p/q, where p, q are integers and q isnot 0. In this form, rational numbers are'ratio' numbers.
5ek49
This is a large step from meaning (1) as theratio a/b is not used to label a part of awhole. The form a/b is often used inmathematics problems, e.g. in enlargementwhere a scale factor of 2/3 cannot be seen aspart of a pie/circle at all.
5. Rational numbers are multiplicativeoperators.
Interpreted by some as a function machine,others as a sharing or partitioning e.g. 1/6of 12.
6. Rational numbers are elements of aninfinite ordered quotient field. Theyare numbers of the form x = p/q where xsatisfies the equation qx = p.
In school mathematics this can be viewed asusing fractions to answer questions whichare impossible within the whole numbersystem, 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 forteaching mathematics, fractions can beplaced on the number line just as can wholenumbers and negative numbers. Inparticular, the measurement of lengthessentially uses a number line.
All these meanings are at some timeemployed in school mathematics, theemphasis given to each in teaching variesconsiderably. A child is dependent onteaching for the realisation of the use andconventions and cannot be expected to'discover' the meaning of what is anabstraction. The results of the CSMSresearch (Hart, 1981) shows that althoughmany children can give the fraction name ofa region and can cope with the introductoryideas of the topic, at least half thesecondary school population refuses to workwith fractions as numbers, e.g. withfraction dimensions in an area problem.This of course restricts the mathematicalattainment of these children since theycannot solve ratio problems, fail to see theset of values which can be taken by a letterand are very restricted in the use of area orvolume formulae.
Such a range of interpretations ofa /b is not adequately catered for by thesimple introduction through "a" regions of apie which has been subdivided into "b"pieces. The model is of little use toillustrate the multiplication or division of
50
fractions. You cannot multiply a piece of pieby another piece and you certainly cannotenlarge a photograph 'by multiplying by aregion. We tend to teach the region fractionnames when the child is quite young, eightor nine years old and the image is powerful,reinforced in the case of 'one half' bycommon usage. The fact that the operationof division is complete when one has a/b isnot recognised by at least half the 11-12year olds tested by the CSMS team.
The CSMS Fractions results andsubsequent interviews pointed to the child'srefusal
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 sametime.
For example in the CMF research (Hart etal., 1989) a group of 12-year-olds weretaught 'equivalent fractions' by theirteacher.
All the six children (boys) present atthe delayed interviews, three months afterthe teaching, were asked whether 10/14was double 5/7 or equal to it. Four childrendeclared the two fractions equal, appealingto the fact that two times five gave ten, etc.Mick thought saying they were equal was abetter answer but one could be double theother. Matt however thought that 10/14was both equal to and double 5/7. Hisreasons were interesting in that they did notinclude multiplication:
(I: Interviewer; M: Matt)
I: So they are both true are they: 10/14thsis double 5/7ths and 10/14ths is equal to5 /7ths?
M: Yes.
1: What about this one? 10/14ths is morethan 5/7ths?
M: That's false.
I: How do you know that?
M: Because that can't be the same as that.If that is true then that would be false.
I: What about this one?
M: 10/14ths, you just add them: add thatand you get 10 and add that you get 14.
55
There is a large gap between being able togive a name to a region and being able tomanipulate fractions with operations. Thefraction 'one half' is used by young childrenand seems to be accessible much earlier thanany other. The ability to compu e. one halfof a quantity or enlarge by a factor of twoare not indicative of an understanding ofother fractions and ratios.
Other difficult topics
Other topics which have been shown to bemore difficult than has traditionally beenbelieved are multiplication, proof, ratioand proportion. The CSMS resultsdemonstrated that many British pupils didnot use the operation of multiplication,replacing it with repeated addition.Fischbein et al. (1985) considered 'repeatedaddition' intuitive. Children who interpretmultiplication in this way, need to see aunit repeated, so "1/3 x 3/8" does not fithappily within this format. PMEresearchers have shown that the nature ofthe multiplier is an important factoraffecting the difficulty of the problem.Children perform better if the multiplier isan integer, less well if it is a decimal largerthan 1, a multiplier which was a decimalless than 1 was even more difficult. Thenature of the multiplicand has scarcely anyeffect but further recent research hasinvestigated whether the demand of'symmetric' problems such as "If length is xmetres and the breadth y metres, what isthe area?" is greater than 'asymmetric' onessuch as "one litre of milk costs x francs, howmuch for y litres?". The evidence wasclouded by pupils' use of the formula forarea and so the investigation continues.
There has, for many years, been a PMEworking group looking at the problemschildren have in dealing with Ratio andProportion. Piaget stated that proportionalreasoning was appropriate to the formaloperational level, it is a topic often calledupon by science teachers and has been a richsource of research problems. As we haveseen Multiplication and Fractions haveboth been shown to be difficult ideas forchildren, computations associated withRatio and Proportion require the use of both.Children as young as six years of age have aqualitative understanding of proportion andcan 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 previousyear and the whale was not that muchbigger than a man.
The quantification of these ideas ismuch more difficult and the type of numberinvolved is important. Enlargement by aninteger scale factor is easier than by a non-integer. Ad hoc methods are used by bothadults and children to solve proportionproblems. One very prevalent incorrectmethod known as "the incorrect additionstrategy" accounts for up to 40 per cent of theerrors in some (usually geometric) problemsin the CSMS Ratio study. It has been shownto occur even more often in the USA.
This strategy (referred to as "theincorrect addition strategy") stemmed fromthe belief that enlargement could beproduced by the addition of an amountrather than by the employment of amultiplicative method. The child, in thisexample, reasons that since the base linehad been increased by two units so must theupright be so increased.
Enlarge with a base of 5
The error was persistent over time andseemed not to be part of a continuum whicheventually brings success. A subsequentresearch project "Strategies and Errors inSecondary mathematics (SESM) found thatthose in the British sample whoconsistently used this method tended toreplace multiplication by repeatedaddition on items which they solvedsuccessfully. An SESM teaching experimentshowed that the error can be eradicated inenlargement problems and greater successattained with short-term interventionwhich addresses the basic problems:
(i) the recognition by the child that the'method' he uses produces strangefigures
(ii) the need for multiplication
(i ) facility with fraction multiplication
5' 51
(iv) the possession of a technique forfinding the scale factor.
A teaching scheme which included theteaching of multiplication of fractions wastried with groups of pupils known to use the"incorrect addition strategy". There wasinitially great success with this but ondelayed post tests the Fractionmanipulation proved to be the barrier. Amodified version of the teaching module, inwhich the Fraction work was replaced bythe use of the calculator proved to be moresuccessful (Hart, 1984). The interventionmaterial was designed to meet theperceived problems listed above, and whenused by classroom teachers with intactclasses resulted in the abandonment of theincorrect addition strategy. Successfulperformance on Proportion problems wasdemonstrated by about 60 per cent of thetotal sample (n= 80 ) three months after theteaching. 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 aspersistent users of the incorrect additionstrategy, 22 out of 23 had abandoned it bythe time of the delayed post test. TheCSMS longitudinal data on the other handhad shown that half of the 'adders' at theage 13 years were still using this strategytwo years later. It does seem worthwhile toidentify an error and design correctivematerial which tackles those aspectssymptomatic of the erroneous reasoning.
Other research on Ratio and Proportionhas investigated whether the child, whenfaced with a problem such as that below,uses the ratio (from Vergnaud, 1983): (xb orx a ) .
Richard buys 4 cakes, at 15 cents each.
How much does he have to pay?
a = 15, b = 4, M1 = number of cakes,M2 = cost.
M1
1
x b
Fig. 2
52
M2 M1 M2
1 a
x b x a
a
x
x a
Vergnaud refers to the ratio xb as'Scalar' and xa as 'Function'. Other authorscall them 'between' and 'within'. Thescalar ratio seems to be that most favouredby pupils in different countries but whetherthis is by choice or the effect of teaching isnot known. It would seem sensible toencourage children to use both since thisgives 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 isimportant, often methods other thanthe algorithm can be used,
( i i ) some errors are very common but can becorrected by intervention and
(iii) 'within' and 'between' ideas are notequally accessible.
All such considerations should be taken intoaccount in the writing of materials. Firstlythe examples given to the child for solutionshould be such that the 'within' viewsometimes appears to be the most naturaland on other occasions the between numbersbeing compared are the easiest. One shouldcertainly avoid examples which requireonly doubling. Multiplication is such aneglected operation it seems sensible to dealwith it again at secondary level regardingit as 'related' to division and not additionand viewed as the total of an array ratherthan repeated collections. Generally aneffort must be made to convince childrenthat the method for solution taught by ateacher is supposed to cover numerous casesand be generalisable. The project'Children's Mathematical Frameworks'(CMF) and subsequent research reported atPME in 1989 have monitored teachers intheir own classrooms. At no time did theteachers in the sample (n = 36) tell thechildren why a generalisable method wasimportant. It seems sensible for a teacher'to sell' what is very much more powerful inthe context of a problem which cannot besolved by more naive or ad hoc strategies. Itis often the case that a very powerful tool isintroduced to solve a problem for which thechild can see an answer immediately andbeing sensible rejects the teachers' method.
5 7
From concrete materials to formalisation
Young children, particularly those underabout the ar of 11 years, very oftenexperience in school a more practical orapplied type of mathematics than theirolder brothers and sisters. Influenced by thetheories of Piaget, teachers in manycountries have been advised to basemathematics for young children on the useof concrete materials (manipulatives). Suchmani-pulatives can be put to many uses.They can be used to provide the base fromwhich concepts are developed. For example,to know what is meant by the word'triangle' one needs to have seen, touchedand explored a lot of triangles.Manipulatives can be the essence of aproblem, for example real or plastic moneybeing used for shopping. One use ofmanipulatives which has been advocatedover a number of years is the provision ofwell-structured experiences which lead tothe 'discovery' (albeit guided discovery) ofa generalisation or rule. This can thenbecome part of the repertoire of the child ashe moves into the more abstract or formaltype of mathematics commonly assumed andbuilt on in the secondary school. Anexample of this type of teaching is whenthe child is working towards theacquisition of the formula for finding thearea of a rectangle. The classroomexperiences recommended by many texts andteacher trainers include:
1) Covering space with various twodimensional shapes in order to be ableto quantify how much areas there is.
2) Covering rectangles with squares.
3) Moving squares around to formdifferently shaped rectangles.
4) Drawing rectangles on squared paper.
These activities are supposed to beenough to prompt the recognition that thearea of any rectangle can be found bymultiplying the number of units in thelength by the number in the width. It isthis rule, often written A=1 x b, which iscarried into the secondary school and whichis then used whenever the area of arectangle is sought.
The assumption that all secondary-aged children have available a 'formal'method of finding the area of a triangle can
be shown to be false by the followingexample from C.S.M.S.
Find the area
a):b):
b)
4cm
3cm\
12+ 13+ 14+76 % 87 % 91 %31 % 39 % 48 %
Fig. 3 Area examples CSMS
The pupils tested were 986 in numberand came from different British schools.These two questions were attempted by thesame children on the same occasion. Theformula for the area of a rectangle wouldhave been taught in their primary schoolmathematics and most of the childrenwould have met the extension to the area ofa triangle in their lessons.
The theory is that the child has afirmer understanding of the rule becauses/he has seen from where it came andbecause the discovery was his/hers, theidea will be retained. This view fits wellwith the constructivist philosophy. Manychildren throughout the world havelearned (or failed to learn) in a differentenvironment as Dorfler (PME, 1989) says
Mathematics for many students nevergets their own activity, it remainssomething which others have done (whoreally know how to do it) and devisedand which can only be imitated (forinstance by the help of automizedalgorithmic routines at which thestudent works more like a machine thanlike a conscious human being). In otherwords, mathematics mostly is not partof the personal experience and thereason for this very likely is that themathematical knowledge of thestudents was not (or only in aninsufficient way) the result ofstructuring and organizing their ownexperience.
'Guided discovery' seems an attractivealternative, however, there are drawbackswhen the ordinary teacher puts theory andinterpreted theory into practice in the
5853
classroom. There is also a shortage ofdetailed and long term monitoring of mostforms of teaching, to see whether those thatare attractive in the abstract are effectivewith teachers and children as they are andnot as we might wish them. Evidence of thelack of effective learning has becomeavailable from the research project'Children's Mathematical Frameworks'(Hart et al., 1989). In this research,volunteer teachers were asked to prepare ascheme of work for the teaching of a rule orformula, in which the pupils started withconcrete material and the formulation wasthe synthesis of the practical work. Sixchildren in each class were interviewed bythe researchers
(i) before the teaching started,
(ii) after the concrete experiences and justbefore a lesson(s) in which the pupilswere expected to 'discover' or beguided to 'discover' the formula,
(iii ) just after this lesson(s) and thenthree months later.
The interviews were to find the child'sknowledge of the basic concept, whether hewas about to discover (or had justdiscovered) the formula and then whetherhe retained for future use a method withconcrete matcrials or a known algorithm.The form of the child's knowledge isimportant since if given tasks which assumetoo high a level of sophistication then he isdoomed to fail.
The topics investigated were:Equivalent Fractions, The SubtractionAlgorithm, Formulae for the Area of arectangle, the Volume of a cuboid and thatfor the circumference of a circle as well asthe solution of simple algebraic equations,all with children aged 8-13 years. Thedata were collected in the form of audio-tape recordings and the results showed thatthe transition from concrete material toformal mathematics was by no means easyor straightforward. Very often the concretematerials did not mirror the mathematicsand imperfectly modelled the ideasinvolved. Sometimes they were sufficientfor the child to carry out a computation in a"concrete" manner but did not in any waylead to the rule or algorithm intended asshown here:
54
Fig. 4
The child Anne used 'Unifix' blocks tocarry out the subtraction 65-29. Anne said"Then I would have taken two tens andanother ten, but on the other ten, the thirdten, I would have taken one unit off, so Iwould have 29, oh! - - oh yes, so then Iwould take the 29 away and add up all theothers and see how much is left. And its36."
This is an effective way of carrying outa subtraction using bricks but such a methoddoes not lead to the algorithm (which wasthe intention). This rule requires the childto start with the 'ones' and to decompose 'aten' into 'ones' thus:
T 06 5 9 from 5 we cannot 'do'.
- 2 9 Go to the tens, change one ofthese into ten ones andcollect these with the 5ones.
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 childrendrew and cut out circular discs. Thesecondary teachers, in the sameinvestigation, used diagrams of regions. Ineach case the 'family' of fractions was builtaround the factors of 12 and the number ofparts in a region was dictated by theteacher. There was no general method thata child could adopt in order to providehimself with equivalent regions. Indeed, itis difficult to see how anybody can choosethe appropriate number of discs or brickswith which to work unless one alreadyknows the equivalent fractions. Asked to
59
use regions or discs to find an equivalent for3/7, how is the child to decide on thenumber of discs to take from the box unlessthe rule is already known? Terence in thisinterchange tries to use diagrams as he hasseen his teacher do.
a )
h)
C=3 0 0 CDOoo on oo 000
0/o o 000 oo °
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 onewould be in between. These two are notthe same.
I: They're not the same, right. 3/8 and6/12 ... we've knocked out because ...now what did you do? I have to tell themachine what you did. You put yourhand over the last four of those circles.
T: Yes, because there's 12 here, but if I putmy ... hand over there to show that was8 and 3/8 would be there and it isn't thesame as that [pointing to 6 circles]. Try6/11, is the same (counts from left handside of circles and gets to same point).
I: As what ...
T: Er ... 6/12.
A common exhortation of teachers, seenduring the research, was to those pupilswho found it difficult to find and rememberthe rule or algorithm. They were told toreturn to the concrete materials. Thishowever requires them to invent themodelling procedure unaided. Urged to takeout the bricks 'to help', those children whodid so simply used them as objects forcounting.
A significant part of the belief in thistype of teaching is that the child'sunderstanding is stronger because thediscovery is his. Three months after theteaching the children in the CMF study,
when asked how they had come to know aformula, said that their teacher had toldthem or that a clever person had inventedit. They did not say 'we discovered it'.They seemed to make no connection betweenthe work with concrete materials and theformula they used in calculations. If oneconsiders the most important features of arod (a commonly used manipulative), theyare: the material of which it is made(wood or plastic), its colour, it; weight,length or volume. To the teacher, who isusing this rod for a particularmathematical purpose, its outstandingquality may be that when put end to endwith another the two are the same lengthas a third. The mismatch between the twoconceptions is obvious.
Further research built on the CMFresults, has pursued the idea that since thematerials and mathematical formalisationwere so different in nature, a transitionalphase (a bridge) was needed. This 'bridge'could be in the form of diagrams, tables,graphs or discussion and should be distinctfrom both types of experience but linkingthem. More forms of the 'bridge' are beingtried but the first results weredisappointing because often the teacherswho were asked to put 'bridge' activitiesinto their teaching gave very little time tothem (five or six minutes) or in some casesput this activity after the formalisation.
Concrete materials are of course used byteachers for other purposes. Teachers ofyounger children very often usemanipulatives to convey the basicprinciples of a topic. Research results fromthe early 70's for example tended to showthat the idea of a fraction being a regioncould be effectively taught when childrencut and folded rectangles of paper.
Teaching experiments
The philosophic and theoretic frameworksadopted by most researchers within PMEhave led to an interest in monitoring'processes' and problem-solving skillsrather than investigating performance incomputation. Usually many Variables areat work in the classroom and it is difficultto apportion effectiveness to the efforts ofthe teacher, the nature of the material(text, concrete materials or computer) or thefact that the contents are process-orienta ted. Harrison et al. (PME, 1980)reported on teaching experiments in
G,)55
Calgary in which 'process' enrichedmaterial on Fractions was taught to 12-13year olds. The teaching took about elevenand a half weeks and control groups were
always good results when they teachcongruence? Is this pattern of success thesame for other teachers? Is it the teacher orthe topic which is the crucial variable?
Test or Subtest Max.Score
ExperimentalMean Score
RegularMean Score
Probability(Evs R)
Number of Students N=207 N=179
CSMS Fractions Pre 84 35.85 (43%) 34.75 (41%)
Post 84 51.57 (61%) 47.76 (57%) <0.01
Problems Pre 46 9.95 (21%) 9.35 (20%)
Post 46 16.40 (36%) 14.39 (31%) 0.01
Computation Pre 12 3.15 (26%) 2.43 (20%)
Post 12 6.68 (56%) 5.68 (47%) 034
Fig. 6 Teaching Experiment (Harrison et al., PME, 1980)
taught from the current textbooks in theusual manner. The results are shown in fig. 6above.
The experimental groups performslightly better than the control groups atthe end of the teaching. The enthusiasm ofthe children was greater after workingwith the process-enriched material,however. Perhaps what is more significantis how little gain there is after theteaching. If the teacher's best efforts over11weeks still leaves the childrenperforming at only 60 per cent success level,perhaps we should reconsider what weexpect of the pupils. Research teachingexperiments are of necessity short-term andusually for less time than the Calgary studyand often the teaching matter is given tochildren without consideration of theirprevious knowledge of pre-requisite ideas.Before the topic is taught the pre-test isadministered in order to find out how muchof the specific teaching points they alreadyknow. This is very seldom nil. The pupilsusually have quite a lot of knowledge tostart with but seem to acquire less thanexpected during a focussed teachingsequence. A worth-while long term studywhich could be carried out by teachers intheir own classrooms is that of analysingexactly which sections of their teaching aresuccessful. Do they always get poor resultswhen they teach addition of fractions and
56
Should we not be more concerned aboutmatching the material to the child's levelof knowledge?
The computer
The computer used as a teaching aid in theclassroom has gained considerableprominence since 1980, the emphasis inresearch being on how children's thinkingskills or deeper understanding have beenimproved. Tall and Thomas (PME, 1988) forexample quoted a study in which childrenaged 12-13 years were given a "dynamicalgebra module". This provided a "mathsmachine" which evaluated formulae fornumerical values of the variables. Theteacher, however, was a vital component ofthe work. Results tended to show that theControl group performed better on skillbased questions, for example:
Simplify 3a + 4 + aExp. 38% Control 78%
On questions which required higher levelsof understanding the experimental groupperformed better, for example:
The perimeter of a rectangle 5 by F.Exp. 50%. C. 29%.
Again we see that the teaching experimenthas produced success but half of thechildren are still not successful.
o6 4-
The use of LOGO in classroom computerwork has been increasingly reported. Themicroworlds are often used as a basis fordiscussion of what children say and theinterpretation they place on situations.Comparison of performance betweenchildren in a LOGO environment and thosein a control group seldom provides anoverwhelming success on the side of theexperimental group. Often closer scrutiny ofindividual marks shows a greater degree ofprogress for some of the LOGO group.Hoy les (PME, 1987) in a review, of PMEpresented papers described three areas ofresearch within the context of 'Geometryand the Computer Environment'.
I. Research which explores thedevelopment of children'sunderstandings of geometrical andspatial meanings and howprogression (for example, fromglobality to increased differ-entiation) might be affected bycomputer "treatments".
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". Suchresearch more or less explicitlyuses the computer to createdidactical tools to facilitate theacquisition of specificmathematical conceptions orunderstandings."
More recently Hoy les' group have beenreporting on particular topic work carriedout with small groups of children, forexample the concept of a parallelogram. Itseems possible that children are assisted inlearning by the use of the computer but it isnot 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, theteachers, like the look of them?
By 'illustration' we mean non-wordmaterial. This includes various categories:
1) Line diagrams which are:
a ) the objects themselves - atriangle;
b) representing an object so a conven-tion is involved;
c) representing actions or steps, suchas a flow diagram;
d) representing by a picture.
2) Pictures which show reality as seen bythe artist.
3) Photographs.
4) Recording diagrams, tables, graphs,matrices, Venn diagrams.
(The children will be required to drawthese themselves 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 therecipient cannot have access toknowledge through words, for examplethose who cannot read.
4) To provide a focus for discussion so thatthe children can all give their viewsabout the same thing. (It can also beused to assess).
5) To present the exercise in context e.g.shopping.
6) To form a bridge between reality andabstraction.
7. To record data in a more effective way.
8. To make the books attractive.
9. To (perhaps) motivate.
Children do not automatically realisehow to use a diagram or what its intendedmessage is. Its special features need to betaught just as other aspects of mathematicsneed to be taught. The distinction betweendiagrams which show a relationship (suchas a graph) and a picture which represents
57
3
reality, is lost on many children. Forexample time-distance graphs such as thatshown in figure 7 are interpreted not asrepresentative of a journey but as showinghills and plateaus. This was so in about 14per cent of cases in the CSMS study.(Sample n = 1396)
Distancefrom homein miles
6.30 7.00 7 30 8. )0 S.3(J 9.00
Time
Fig. 7
Often teachers use a diagram to addweight to an argument. Thus when teachingabout fractions they are likely to draw,freehand on the blackboard, a circledivided into sections. The purpose of this isdifficult to see, since i; the pupils were notalready knowledgeable of fractions andtheir equivalents, the diagram wouldconfuse rather than help. For example,unless drawn accurately it cannot provideevidence that 3/8 = 6/16.
The CMF researchers found thatchildren provided the same inaccuratediagrams when trying to convince theinterviewer of equivalence, as seen in thiseffort of Joe.
a) Matt's diagram for 3/4 = 6/8
58
b) Joe's diagram for 3/5 + 7/10 = 10/15
Fig. 8
Textbooks are nowadays often lavishlyillustrated, yet there is scarcely anyresearch on the effect these pictures have onthe learning of the mathematics. Shuardand Rothery (1984) provided a review ofthe existing research on readingmathematics. In so doing they analysedpages from textbooks where informationwas partly given in the picture, partly inthe text and sometimes on lists whichappeared beside the picture. For examplein shopping questions which are essentiallytesting addition, there are pictures of thearticles which can be bought, perhapslabelled with prices, a shopping list andthen a sentence or two in the text. The eyemovement needed to obtain the necessaryinformation is excessive.
There is some research in scienceeducation on.the use of illustration in textbooks and other types of teaching materialbut very little in mathematics.Much ofGeometry is said to be concerned with theimprovement of children's spatialvisualisation, yet little of it positivelyaddresses this issue. There has been somerecent research, resulting in books forchildren, which attempts to improvepupils' view of three dimensional objectsand their interpretation in two dimensions.One study was part of 'The Middle GradesMathematics Project' from Michigan StateUniversity (Winter et al., 1986).
The Spatial visualization instructionalmaterial includes ten carefully sequencedactivities. The activities involve
63
representing three-dimensional objects(buildings made from cubes) in two-dimensional drawings and vice versa,constructing three-dimensional objects fromtheir two-dimensional representations.Two different representation schemes areused for the two dimensional drawings.First an "architectural" scheme involvingthree flat views of the building-base, frontview and right view. After students arecomfortable with this scheme, they areintroduced to isometric dot paper and henceto a representation consisting of a drawingof what one sees looking at a building from acorner. Similarities, differences, strengthsand inadequacies of the two schemes areexplored.
It was concluded from the analysis ofthe pre-post data that after the instruction:
(1) sixth, seventh, and eighth grade boysand girls performed significantlyhigher on the spatial visualizationtest; however, no change in attitudestoward mathematics occurred;
(2) boys and girls gained similarly from theinstruction, in spite of initial sexdifferences;
(3) seventh grade students, regardless ofsex, gained more from the instructionthan sixth and eighth graders.
In addition, the retention of the effectsof the instruction persisted; after a four-week period, boys and girls performedhigher on the spatial visualization testthan on the post test.
Conclusion
We now know, through research inmathematics education, that certain topicsare difficult for children to learn becausethey contain many different aspects each ofwhich brings its own degree of complexity sothat the total is composed of layers ofdifficulty. Proportion is such a topic. Itseems wise to write materials which buildup the different understandings rather thanproduce the topic all at once. It is also wiseto illustrate and give examples whichrequire the use of the idea or method andcannot be easily solved by lesser and morenaive strategies. There is then an incentivefor the child to take on the more powerfultool.
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 startedas a teacher, then she was a teacher-trainer for ten years, a researcher for ten years, and then justprior to taking the post of director of the Nuffield project she was an inspector of schools. She hastwo doctorates in mathematics education, one from Indiana, and the other from London. Her firstdegree is in mathematics, and she also has a London M.Phil obtained from research with sevenyear-olds who were said to have a mental block against mathematics.
A founder member of PME, Dr. Hart is currently its President.
gel59
The Role of the Teacher in Children's Learning of Mathematics
Stephen Lerman
There is no doubt that the teacher plays amajor part in creating the environment inwhich children can best learn mathematics.This is not to claim too much for the role ofthe teacher. Stronger claims have been made,for example the linguistic philosophers ofeducation would want to say that unlesslearning is taking place one cannot be said tobe teaching. Certainly governments in manycountries count teachers as solely responsibleif there is evidence of children not learning.Thus the statement that begins this chapteris relatively minimal and uncontroversial.Nevertheless, there are many issues to beinvestigated, and the questions begin whenone attempts to establish what part theteacher plays, what is the best environmentand how can one create it, and what teachersunderstand about children's learning ofmathematics. In this context, therefore, it isinteresting that it is only in recent years thatone finds a growing body of research thatfocuses on the teacher. For example, at theTenth Meeting of PME in 1986, there were nomore than a small handful of research reportsconcerning teachers, but by the FourteenthMeeting in 1990 there were three WorkingGroups and a Discussion Group, and at least 20research reports.
There are perhaps two major reasons forthe relatively late interest in the role of theteacher. The first is the dominance of thenotion that if we can specify how childrenlearn, and produce the most appropriatelearning materials informed by thatknowledge, the teacher's role becomeslargely that of an intermediary in the processof transmission. lf, further, we can specifywhat the classroom should look like, howlong children should study, and variousaffective factors, we can tell the teacherwhat to do to achieve the goal of childrenlearning mathematics. Many would claimthat a paradigm which incorporates these isno longer tenable. The transmission metaphoris no longer 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 separablecategories as was previously claimed, andthis has led to research on the social contextof the classroom which has considerablyconfused and enriched our observations andideas. In all of these issues, the teacher is acentral figure.
The second reason that research onteachers has only recently playing asignificant part on the agenda, is the natureof that research. This is an issue that is thesubject of on-gobg debate (Scott-Hodgetts, inpress, b). Most research questions that areconcerned with teachers will not beanalysable by significance tests. As Masonand Davis (PME, 1989) suggest, "As with ourprevious work, validity in our study lies, forus, in the extent to which it resonates withexperience, and to which it awakensawareness of issues which they mightotherwise have overlooked" (p.280-281).
There are some studies that draw onreasonably sized populations for quantitativeanalysis, such as those examining pre-serviceteachers' knowledge of aspects ofmathematics (e.g. Vinner and Linchevski,PME, 1988). In the main, however, theresearch method-, most used are qualitative,based often on only a small number of teachers(Jaworski, PME, 1988, Lerman and Scott-Hodgetts, PME, 1991, in press), and themethodology and generalisable outcomes ofsuch research are not accepted by all in themathematics education community.
This is a theme which will arisefrequently in this chapter, namely the kindsof methods that are most informative forresearch on the teacher's role. Two aspects inparticular will be emphasised, firstly that aholistic perspective of interactions in theclassroom is more fruitful than a fragmentedone, and secondly, that case studies, perhapsstimulated by findings from quantitative
6561
studies, provide information that resonateswidely with teachers and other researchers.
Expectations of the fruitfulness ofresearch on teaching vary considerably. In arecent volume (Grouws et al., 1988) one findsboth the positivist expectations of Grouws:"No longer can we uncritically rely on thefolklore of the classroom; rather we must sortout fact from myth and begin to buildconceptual networks that will help us tounderstand and improve the complex processof teaching" (Grouws, 1988 p.1), as well asthe sceptical thoughts of Bauersfeld: "...there seems no prospect of arriving atuniversal 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 mathematicsteacher in research reports at PME meetings,the research community is now engaging withthese issues, and the themes which will bereviewed in this chapter will indicate someof the important ideas emerging.
Following this introduction, the chapteris divided into four sections. In the firstsection we will examine some aspects of theteacher's work, including notions of effectiveteaching, the teaching of problem-solvingprocesses and the role of meta-cognition, thepower relationships in the classroom, someremarks on textbooks, and on whatconstructivism, an alternative theory of theway people learn, might imply for teaching.As with the remainder of the chapter, thereview will draw, in the main, on researchreported at PME meetings as well as on otherwidely available literature.
One of the major themes that emergesfrom the review is the significance ofteachers' beliefs about mathematics, andabout mathematics teaching, and theinfluences of school and society on teachers'actions in the classroom. In some senses, thistheme is fundamental. Whether one isconsidering society's influences on theclassroom, textbooks or other teachingmaterials, cultural influences, curriculumchange, technology or whatever, they willall be mediated through the teacher, andspecifically through the teacher's beliefsabout her role in her students' learning of
62
mathematics. Work in this area will bereviewed in the second section.
At the same time, the teacher is an actorin a particular setting, within which therelationships and dynamics are constructedby the actors, but in turn the actors and theirroles are constructed by the classroomrelationships and dynamics. Therefore, inthe third section we will look at the teachingand learning of mathematics in the context ofthe language and practices of the classroom.
In the final section, we review researchon teacher education, including actionresearch, and teachers' knowledge ofmathematics. The concluding remarks willreturn to the issue of the nature and role ofresearch on teaching, and will attempt tohighlight major themes for future research.
Aspects of the work of the mathematicsteacher
Effective teaching
It may seem that the most appropriate placeto begin any survey of the actions of theteacher is to attempt to characterise what iseffective teaching, as well as which aspectsof teaching make the teacher ineffective. Ifthis is possible, one could design pre-serviceand in-service courses so as to bring aboutdevelopment and change in those areas.Albert and Friedlander (PME, 1988) reportedon an in-service course focused on counselling,and commenced their paper with an analysisof the flaws in unsuccessful teaching and adefinition of effective teaching, the latterincluding "good class management, clear andcorrect mathematical content, well-plannedlessons resulting in a feeling of learning, good.earning environment ..." etc. (p.103). Thereis the danger that an analysis of the actionsthat identify an effective teacher can becomemerely a list of 4.1.ssirable attributes, withoutany real evidence that such attributescontribute significantly to school students'learning of mathematics. Also, phrases suchas "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:
6i"
"(a) a focus on meaning andunderstanding mathematics and onthe learning task;
(b) encouragement of studentautonomy, independence, sel f-direction and 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 thecharacterisation of effective teaching,almost to the point of stating the obvious:
"... there are two separate domains ofknowledge that require blending in orderfor expertise in teaching to occur. Theseare (1) subject matter knowledge and (2)knowledge of classroom organization andmanagement, which we call pedagogicknowledge." (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 toidentify those teaching strategies "thatseemed to have contributed most to greaterachievement gains" (p.274) amongst eighthgrade students, and concludes that thesewere:
"(a) an extremely organized approach toteaching, wherein material is taughtuntil the teacher feels it is mastered,thus reducing the need for frequentreview, and
(b) an approach in which everypresentation of material is followedby extensive practice in applying thenew material to new situations."(p.274)
Steinbring and Bromme (PME, 1988) havedeveloped a technique to monitor the processof knowledge development in the classroom,and present this pattern graphically,enabling comparisons over time, studentgroup, teachers etc. Kreith (1989) presentsthe idea of master teachers acting as rolemodels and working with intending teachersas interns, an issue to which we will returnwhen considering teacher education.
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 meansdifferent things to different people." (p.247)One cannot talk about 'effective' withoutdetermining what it is that one values in theteaching of mathematics. Classroommanagement, for example, is one kind ofnotion when one is teaching a piece of content,in a transmission model, and quite anotherwhen the focus is on group problem-posingwork. Again it means quite different things ifone is teaching a class of 60, with little or noresources, and quite another with a group of20 in a well-resourced situation.
We may also, in the end, have doubtsabout the value of breaking down classroomactivities into teachers' actions, students'actions and affective conditions, when whatis taking place is a complex process ofinteractions, intentions, relationships ofpower, expectations, and so on (Porter et al.,1988). Bishop and Goffree (1986) make asimilar point when they argue for a shiftfrom what they call the 'lesson frame' to the'social construction frame'. The former ischaracterised as follows:
'(the mathematics lesson) is construed asan 'event' with a definite beginning, anelaboration and a definite end. It has afixed time duration. Typically all childrenwill be engaged in the same activities whichare planned, initiated and controlled by theteacher." (p.311)
They indicate that research, dissatisfactionwith outcomes, and alternative constructivisttheories of the way children learn have ledto a widening of the frame:
'This orientation (social construction frame)views mathematics classroom teaching ascontrolling the organisation and dynamicsof the classroom for the purposes of sharingand developing mathematical 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 ofknowing and knowledge, reflecting theimportance of both content and context;"(p.314).
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Studies based in the classroom provideperhaps the most fruitful approach forresearch in the area of effective teaching,since an essential component is the recognitionof the theory-laden nature of describing aneffective teacher, and thus of the need tobuild suitable research methods in the lightof this.
For instance, the teacher might developher notion of student achievement, andcriteria of successful teaching, relevant to thespecific context. Through action research, or acollaborating observer in the classroom,teacher strategies that appear to help thestudents and the teacher to realise thatachievement might then be identified.What is important here is that 'desirableattributes' become something specific to theparticular teacher, in relation to her ownvalues, and change is seen as an individual'sprogress in her own terms, rather thansomething that teacher-educators orresearchers do to others.
An illustration of this approach can beseen in the following extract from thereflections of a teacher, who had beencritically examining his teaching in the lightof some reading he had done, and somesubsequent observations of his students(Lerman and Scott-Hodgetts, PME, 1991, inpress):
"So my reflections led me to this point. I
began to believe that if the didacticapproach was not good enough for Claire, itwas not good enough for any pupil; thatinstruction and explanation by the teacherhad its place in outlining the problem, butnot somehow giving mathematical ideas."
Another perspective on the issue ofeffective teaching, is given by Scott-Hodgetts(1984), (discussed also in Hoy les et al., 1985),in a paper that describes her research onchildrens' ideas of what is a good teacher. Ina paired-comparison test of characteristicsthat go to make a perfect teacher, theranking emerged as follows:
"is able to explain the maths clearly andthoroughly;
has a patient and understanding attitudewhen people find the work difficult;
64
is prep,,red to spend a lot of time withpeople who need more help;
has a relaxed friendly relationship with thepupils;
keeps up to date with each pupil's work, andlets them know how they are getting on;
makes the lesson interesting and enjoyable;
knows the subject really well;
has a good sense of humour."
Alibert (PME, 1988) found that first yearuniversity students valued aspects of theteachers work that differ from those thatthe teachers might value. In her study, 52%of the students considered that a goodmathematics teacher is first "interesting,convincing, clear, quiet" and in only 11% ofanswers were there such things as "helpsstudents to reflect, to participate" (p.114).Two of the most interesting aspects of thesestudies are the following: first, studentsemphasise quite different aspects of 'good'teaching than teachers themselves mightemphasise, and this information could be avaluable stimulus for reflection and researchby teachers, and second, that the hiddenmessages of one's beliefs are often conveyed tostudents, and reflected in what they valuemost in their teacher.
Meta-cognition and teaching mathematical'processes'
One of the major changes in the role ofthe teacher during the last decade or so, hasbeen brought about by the shifting of the focusof attention to the introduction of problem-solving work into the classroom. It has raisedmany questions and engendered many debates.An early question was: what are problems,and how do they differ from puzzles andinvestigations? In fact we often use all threewords during any particular piece of work.We may start by being interested in aproblem, and begin to investigate it. On theway we solve several puzzles, become stuck onmany more, and generate new problems toinvestigate! Different interpretations of theterm 'problem-solving' remain, however, asshown in the study described below.
The issue of 'process' versus 'content' hasbeen discussed, in the literature and atconfewhces, and we have become aware that
we are at least partly discussing thephilosophy of mathematics. To some extent,for the formalists, mathematics is its contentand its structure, whereas for the quasi-empiricists (Tymoczko, 1985) mathematics isits processes. In recent years we have beendeveloping ways of assessing students' workon investigations, in particular in Britainwhere school students have to submit anumber of individual course works for theGeneral Certificate of Secondary Education(GCSE), a national examination for 16 yearolds. There is still, however, the issue ofhow one avoids assessment-led investigationwork in classrooms (Lerman, 1989a), that iswhether the potential for creativity andengagement disappear under the need to haveproblems whose answers can be 'marked'.
Problem-solving has led on to problem-posing (Brown, 1984), and opened up theactivity to the wide and stimulatingpossibilities of ethnornathematics,empowerment and critical mathematics(Frankenstein, 1990). But where does it fit into the school mathematics curriculum? Is it tobe added on at times, or can one teach themain part of the curriculum through sochwork? The nature of mathematics is theessence of the problem here, and it findsexpression through the process/contentdebate. All these questions are the foci ofcurrent research work in mathematicseducation, and we will examine some of themhere, as they impinge on the role of theteacher.
In a study of teachers' ideas of what isproblem-solving and how to teach it, Grouwset al. (PME, 1990) found that teachers wereusing four categories of meanings:
"(1) Problem-solving is word problems;
(2) Problem-solving is finding thesolutions to problems;
(3) Problem-solving is solving practicalproblems; and
(4) Problem-solving is solving thinkingproblems." (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 andrelated instructional methods did not relate
to their definitions, and neither did theformat of their lessons. Teachers felt thatthere was not enough time for problem-solving instruction, standardized testingbeing a major interference. Students had lowsuccess and little self-confidence. Grouws etal. conclude:
"We now know that we must carefullydescribe what is meant when a teachergives critical importance to problem-solvingand its instruction in her classroom." (p.142)
To some extent, aspects of the study areprobably culture-specific since in manycountries problem-solving would not beconsidered to mean word problems. Teachersmay well respond by offering the otherdefinitions, however, and many of the otherfindings may well be supported widely. Thefinal comments of Grouws et al. are animportant caution to researchers and areminder of the influence of teachers' beliefsabout mathematics and mathematicsteaching. One must also take into account thetendency to respond to certain jargon words,particularly 'problem-solving', which hasbeen a major theme of the 1980's. In thiscontext it is therefore particularly significantto find both that teachers have quitedifferent ideas of what it is, and that theirteaching often bears little relationship tothose 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 tomathematical problems were sent to 2200teachers, of whom 446 responded. Flenerfound a wide range of evaluations, thatteachers credited methods that were taughtin school and not problem-solving ability atall, and in general teachers gave no credit forinsight or creative solutions. Morgan (1991),presented experienced teachers with threehypothetical solutions to a particularproblem: the first, (labelled PV) useddiagrams and full verbal description, usingspoken rather than written language; thesecond (ND) used numbers and diagrams and atable, with text that presented the work inthe order in which it was done, and the third(BS) was concise and symbolic, with words
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only as labels, without sentence structure. Allthe teachers placed ND the highest, andmost placed PV second. This is in markedcontrast with the usual presentation of puremathematics in academic journals, andsuggests that teachers are developingparticular and specific ways of assessingschool mathematics problem-solving. Shesuggests that teachers are developing criteriafor what constitutes school mathematics,rather than mathematics in general, andthat these criteria are not made expliciteither amongst teachers or to school students.
Again, we are led to consider whatteachers understand by mathematics, in thecontext of problem-solving. This issue will bediscussed further below, but whateverteachers and researchers understand byproblem-solving, a major interest formathematics educators is the question ofwhether problem-solving skills can betaught. There are two inter-related andinteracting levels to be considered, heuristicmethods and meta-cognition. Hirabayashiand Shigematsu (PME, 1986, PME, 1987, PME,1988) have developed a powerful metaphorfor meta-cognition: that of the Inner Teacher.They suggest that the teacher acts as theinner self of the students and thus the task ofteachers is to encourage the growth ofstudents own inner teacher. Mason andDavis (PME, 1989) indicate how useful theyfind this metaphor for thinking about theshifts in attention required in problem-solving and in mathematics education ingeneral. In discussing some theoreticalaspects of teaching problem-solving,Rogalski and Robert (PME, 1988) suggest thatit is possible to "design methods related to aspecific conceptual field", and they go on tooutline how and when such instruction isappropria te:
"... such methods can be taught to studentsas soon as they have some availableknowledge and the ability to make explicitmeta-cognitive activities in a precise way,and to take them as object for thought, and(3) that students benefit from such ateaching. Didactical situations whichappear as good 'candidates' for supportingsuch a methodological strategy involve:work in small groups, open and sufficientlycomplex problems and a didacticalenvironment giving a large place to
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students' meta-cognitive activities such asdiscussion about knowledge and heuristics,and elicitation of meta-cognitiverepresentations on mathematics, problem-solving, on learning and teachingmathematics." (p.534)
The importance of making knowledge ofheuristics and self-reflection explicit tostudents is a theme developed by a number ofresearchers. In a study specifically onheuristics teaching, van Streun (PME, 1990)concludes that "Explicit attention forheuristic methods and gradual and limitedformulating of mathematical concepts andtechniques in mathematical educationachieve a higher problem-solving abilitythan implicit attention for heuristic methodsand late and little formulating ofmathematical concepts and techniques."(p.99) Mason and Davis (PME, 1987) argue forthe introduction of particular and specificvocabulary, such as 'being stuck' and'specialising', in helping the process oflearning mathematics. They also claim aconnection between "the effectiveness ofteachers' discussions of their teaching" and"of students' discussions of their learning"(p.275). This notion of the analogy of theteacher's reflections and the student's will bedeveloped in the final section of thischapter.
Lester and Krol (PME, 1990) examine the"relative effectiveness of various teacherroles in promoting meta-cognitive behaviourin students and the potential value ofinstruction involving a wide range of types ofproblem-solving activities" (p.151). Theseroles are: the teacher as external monitor, asfacilitator and as model. They observed theteaching and learning of students in seventhgrade classes, and drew the followingconclusions:
"Observation 1: Control processes andawareness of cognitive processesdevelop concurrently with thedevelopment of an understanding ofmathematical concepts.
Observation 2: Problem-solving instruction,meta-cognitive instruction in partic-ular, is likely to be most effective whenit is provided in a systematicallyorganized manner, on a regular basis,and over a prolonged period of time.
Observation 3: In order for students to viewbeing reflective as important, it isnecessary to use evaluation techniquesthat reward such behaviour.
Observation 4: The specific relationshipbetween teacher roles and studentgrowth as problem solvers remains anopen question.
Observation 5: Willingness to be reflectiveabout one's problem-solving is closelylinked to one's attitude and beliefs."(p.156-157)
These studies highlight many importantissues, such as how to evaluate problem-solving work and how to develop the innerteacher. They also leave open manyquestions, including what teachersunderstand by 'problem-solving' and howtheir interpretation influences the way theyteach, and how problem-solving fits into thethe whole task of teaching mathematics.Answers to these questions are often implicitunstated assumptions of teachers andresearchers, but they are crucial factors. Forexample, a study designed to examinestudents' feelings about problem-solving,where that activity is an occasional one inthe classroom and when regular textbook-driven mathematical work is set aside, islikely to lead to one set of responses fromstudents. Where that activity is the style oflearning that is usually used by the studentsin their learning of mathematics, one maywell find quite different responses.Similarly, concerning teachers' attitudes toproblem-solving, the 'hidden messages'conveyed by the teacher of the importance orsignificance of problem-solving are picked upby students (Lerman, 1989a).
All the studies emphasise makingheuristics and reflection explicit to students,which can be expressed in a rich way as thedevelopment of the Inner Teacher. It isinteresting to reflect on the role of theteacher in this. It has been argued(Kilpatrick, PME, 1987) that theconstructivist view of children's learning doesnot offer anything new in the sense ofresearch methods, or teaching methods. Thismay be a difficult point to argue when onefocuses on childrens' learning of particularcontent I. mathematics, [although both sidesare argued (e.g. Kilpatrick, PME, 1987; Steffe
and Killion, PME, 1986)] but an interpretationof each student's construction of her own InnerTeacher, in its fullest sense, may be seen asmore ciifficult to formulate outside of aconstructivist programme. For example, if oneis considering a child learning the procedureof adding two fractions, it could be describedas the child constructing concepts andprocesses that 'fit' the situation, i.e. give theexpected answers, make sense in relation toother concepts, can be coherently explained toothers etc., or it could be described as thechild internalising something taught by theteacher and therefore 'matching' anestablished schema. Both interpretationscould be descriptions of what takes place, andthe relative merits of the two descriptionscould be argued. From a constructivist view,the 'Inner Teacher' notion will by its verynature have a different meaning for eachindividual; it has no explicit identity thatthe teacher could teach, and it will be ameaningful notion to the extent to which itworks for the individual. It is a very 'fuzzy'notion, involving specifiable aspects, such asgeneralising, but also unspecifiable aspects,such as recognising that one is firing, or stuck,etc. The Inner Teacher notion can of course bespecified in a cognitive psychologyperspective, and measured againstdeveloping schema, as a sort of 'external'teacher replaying its voice in theindividual's head. However, if a teacher isconsidering how to develop the inner teacherin her students, seeing learning as aconstructive process may well provide adepth of meaning that other theories do notadequately offer.
In the research literature, one does comeacross doubts about whether problem-solvingmethods as such can be 'taught'. Such doubtsare expressed often by researchers fromoutside of mathematics education wheninvestigating mathematical problem-solving. The recent debate in the Journal ForResearch in Mathematical Education is anillustration of those concerns (Owen andSweller, 1989, Lawson, 1990, Sweller, 1990).Owen and Sweller (1989) make a case for thelack of evidence of the effectiveness ofteaching heuristics. They suggest that thefailure of students to use newly learnedprinciples intelligently to solve problems
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may not be due to their lack of problem-solving strategies, but what they call "a lackof suitable schemas or rule automation"(p.326). In any event, they claim that thereis no evidence that learning problem-solvingstrategies actually helps in further problem-solving, i.e. are transferable. Lawson (1990)argues to the contrary, that there are smallbut growing signs of evidence to support the"consideration of different types of problem-solving strategies ... in mathematics classes"(p.409).
In his reply, Sweller (1990) reveals whatmay be the essence of the differencesexpressed in the debate, in reviewing thework of Charles and Lester (1984). Sweller islooking for the transfer of problem-solvingstrategies, and where researchers such asCharles and Lester have found enhancedproblem-solving ability after instruction, buthave not looked for Sweller's interpretationof transfer, he considers his case reinforced.The issue seems to be what is meant bydomain-specific knowledge and skills, andtransferability. We would probably arguethat the strategies Charles and Lesterworked upon, "such as trying simple cases,creating a table, drawing diagrams, lookingfor patterns, or developing general rules"(Lawson, 1990, p.404) are broad enough toapply to the domain of mathematicalproblem-solving and at the same time focusedenough to be achievable. Since Sweller isdissatisfied with the transferability of theseskills, one can only assume that he expectstransfer outside of mathematics, to everydaylife, or to other school subjects. However,mathematics is a language game, with itsown meanings, styles, concepts and so on. Onecan perhaps articulate a notion of generalproblem-solving skills in this wideinterpretation, but only in the sense of taskorientation strategies, or executive strategies(Lawson, 1990, p.404). We would supportLawson in claiming that there is a growingbody of research suggesting that withinmathematics, problem-solving strategies area fruitful focus for work in the classroom, andhave a positive effect at least on furtherproblem-solving work in mathematics.
In the problem-solving literature, thereis some mention of the role of algorithmicmethods, that is to say precisely defined
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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'.Van Streun (PME, 1990) suggests that "Beingmore successful in problem solving is attendedby more frequently employing algorithmicmethods" (p.98). Sfard (PME, 1988) discussesthe distinction between operational andstructural methods:
"People who think structurally refer to aformally defined entity as if it were a realobject, existing outside the human mind.Those who conceive it operationally, speakabout a kind of process rather than a staticconstruct." (p.540)
Sfard demonstrates that teac},ing conceptsthrough operational methods via algorithmscan have some success e.g. with the learningof induction. There are dangers in thereliance on algorithmic teaching (Lerman,1988, Kurth, PME, 1988, Steinbring, 1989) butits role as an heuristic is in need of furtherresearch.
Finally, there remains the question ofhow problem solving should be integratedinto the whole mathematics curriculum. Tosome extent this is a curriculum issue, whichmay be manifested in the particular textbookor scheme used in the school, and there arerr, 1, countries and schools where theircidual teacher does not have a choice. Inthose countries and schools where perhapsthe content is specified but not the way it istaught, or where there is the goal of astandard to be reached, but the style of workis dependent upon the teacher, how and whenproblem-solving appears is a matter ofchoice. As with so many questions inmathematics education this really dependson the teacher's view of the nature ofmathematics, and of the process of learning.If the teacher sees mathematics as identifiedby ways of thought, ways of looking at theworld, as prccesses, then mathematics shouldbe learned through problem solving. If on theother hand, mathematics is thought of as abody of knowledge, a specified amount ofwhich students must acquire, and then applyin problem solving situations, then theteacher will have to find a suitable time andplace to teach problem solving processes.
An important element in any discussionabout the role of problem-solving in learningmathematics, and one that will perhapsensure that the debate will remain, is thedifficulty of comparison. It is fundamentalthat if one intends to design some research tocompare achievement in mathematicsthrough the two different approaches, onehas to make the tests appropriate to the kindof mathematical work. Lester and Krol(PME, 1990) emphasise this issue of thedifficulty of comparing different styles oflearning in their third observation, preciselybecause they require different styles ofassessment. 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 alwaysin the teacher's favour, as most startingteachers will report. Indeed Walkerdine(1989) reports on a situation in which awoman teacher is suddenly dominated by twolittle boys aged 4 or 5, using sexually abusivelanguage:
"Annie takes a piece of Lego to add to aconstruction she is building. Terry tries totake it away from her to use himself ... Theteacher tells him to stop and Sean tries tomess up another child's construction. Theteacher tells him to stop. Then Scan says:'Get out of it, Miss Baxter paxter.'
TERRY: Get out of it, knickers Miss Baxter.
SEAN: Get out of it, Miss Baxter paxter.
TERRY: Get out of it, Miss Baxter theknickers paxter knickers, bum.
SEAN: Knickers, shit, bum.
MISS BAXTER: Sean, that's enough. You'rebeing silly.
SEAN: Miss Baxter, knickers, show yourknickers.
TERRY: Miss Baxter, show off your bum(they giggle).
MISS BAXTER: I think you're being verysilly.
TERRY: Shit, Miss Baxter, shit MissBaxter.
SEAN: Miss Baxter, show your knickersyour bum off.
SEAN: Take all your clothes off, yourbra off..." (p.65-66)
Walkerdine adds: "People who have readthis transcript have been surprised andshocked to find such young children makingexplicit sexual references and having so muchpower over the teacher. What is this powerand how is it produced?" (p.66)
In the main, however, the imbalance is inthe teacher's favour, and as Bishop (1988)comments:
"The first and most obvious principle is thatthe teacher's power and influence must belegitimately used - perhaps we can say thatit should be used and not abused" (p.130).
Hoy les (1982) for example gives instancesof both:
"I: What happened then?
P: Well the teacher was always picking onme.
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 sumfractions 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) wehad this teacher, she was a really goodteacher, maths it was, and I've neverbeen any good at maths. She neverpushed you or nothing but let me geton with it at my own pace.
I: What do you mean exactly when yousay shc never pushed you?
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P. Well, she was nice. I had tried and sherealised it and didn't keep picking onme. I used to try really hard in herlessons and just get on with it...
I: Can you tell me how you felt during herlessons? What did vou feel inside?
P: Well really good, it was really nice to bethere." (p.353)
It is of course too simplistic to present theclassroom situation as one where theautonomous teacher exercises control, and canchoose the power to be benevolent or despotic.The extent to which the discursive practicesof the classroom construct the relationships ofpower is discussed more fully in the section onthe language and practices of the classroombelow. The relationship to mathematicalknowledge is the key factor that will bediscussed here.
Mathematics has a special role insociety. It is seen as cultural capital for theindividual, that is, success in mathematicssuggests financial success in the future. It isalso used by society to legitimate policies anddecisions. Whether it be the billions ofdollars of international debt of one nation, orthe high rate of inflation of another, thefigures become indisputable and can be used tojustify policies that make the lives of themajority miserable. This places teachers ofmathematics in a unique position, one thatcan result in reinforcing the powerlessness ofthe individual in modern society, or canenable people to question and challengeinformation given. The alternatives can becalled 'empowerment' and 'disempowerment'.Cooper (1989) claims that "There is evidencethat teachers see mathematics as a crucialsubject for reproducing existing social values,and that they modify mathematicscurriculum material accordingly" (p.150).Cooper's point is that teachers accept thestatus quo tacitly, and this results in negativepower "where actions that are not in theinterest of those in charge are suppressed,thwarted, and prevented from being airedwithout the elite having to initiate orsupport any actions or exercise any powers ofveto" (p.153). Perhaps the most developedalternative curriculum for mathematics, onethat aims to empower students, is that byFrankenstein (1990). She draws on Freire's
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distinction between a problem-posingcurriculum and a 'banking' metaphor, wherethe individual is thought to store knowledgefor retrieval. She offers situations, such asstatistical data on employment according toethnic group in the USA, and invites studentsto pose questions to investigate mathe-matically.
Robinson (1989) takes the notion ofempowerment into the area of teachereducation, and compares two competingmodels for bringing about teacher-change, themanagement paradigm and the empowermentparadigm:
"Rather than seeing change in schools as afinite process with externally specifiedobjectives, as the management paradigmdoes, the empowerment paradigm seeschange as an on-going activity generatedwithin the school by teachers, parents andstudents as part of an organic process ofprofessional renewal." (p.274)
Teachers are significant people in thelives of children, and the effects of teachers'expectations on individual children is anaspect of the teacher's power that is a focusfor research, particularly in the field ofgender, ethnicity or class bias. In thiscontext, an important aspect of the impact ofteachers' expectations of children is therange of factors to which teachers attributestudents success or failure in mathematics.Fennema et al. (1990), for example, makesome strong claims about such impact:
"(1) A teacher's causal attributions areimportant because perceptions ofwhy his/her students succeed or failin achievement situations has animpact on the teacher's expectanciesfor students' future achievementsuccess.
(2) Tcachers' attributions influencestudents' attributions throughteacher behaviour". (p.57)
There has been one study (Kuyper andvan der Werf, PME, 1990) reported at ameeting of PME which 'releases' teachersfrom the responsibility for such influence.They claim:
...the differences in achievement,attitudes and participation cannot beattributed to (characteristics of) individual
math teachers. There is some evidence,however, that the gender of the teacherinfluences the perception by girls and boysof their teacher's behaviours in a way thatmight be labelled 'own sex favouritism'.The observed teacher behaviours do notinfluence this perception." (p.150)
They end their study by suggesting "In ouropinion the results fit very nicely into ageneral pattern, which can be verbalized asfollows: the gender differences in math arenot the teacher's fault" (p.150). Their studyinvolved more than 5800 students andteachers, and clearly the limitations on thesize of papers for PME proceedings did notpermit a full description of theirmethodology and results. Their claim thatthe teacher behaviours do not influencestudents perceptions, however, is focused onthe individual. Some of their own evidencehighlights factors that are significant forteachers in general. They found thatteachers attribute tidiness more to girls, forexample, and industriousness too, whereas'disturbing order' is more typical of boys intheir view. Scott-Hodgetts (PME, 1987)points out that these teacher perceptions andthe resulting teacher behaviours havesignificant cognitive effect, and are notmerely affective observations. They serve toreinforce serialist strategies, i.e. a step-by-step approach, which more girls than boysare predisposed to adopt. The result is thatmore boys tend to develop a broad range oflearning styles, whereas more girls remain asserialists. There is substantial evidence insupport of the Fennema et al. (1990) claims,as Hoy les et al. (1984) describe (p.26).
This brief discussion is an example of anissue to which we will return in the finalsection, namely to what extent does researchreach teachers and offer them the possibilityof changing their work? Some teachers,having read literature that suggests thatteachers, irrespective of their gender, spendmuch more time answering and dealing withthe boys in their class than the girls, havebeen stimulated to examine their ownpractice (Burton 1986). It should be of someconcern if such research remains in learnedjournals to which most teachers do not haveaccess (Lerman 1990a).
Textbooks
The school mathematics text book is a
familiar element in our work, as is wellillustrated in Hart's chapter in this volume.However, the teacher's use and dependenceon the textbook is a well known phenomenon,but is under-researched in the teaching ofmathematics. Laborde (PME, 1987) haslooked at students reading texts, and Grouwset al. (1990) mention that the textbook is themost important factor influencing students'attitudes. Van Dormolen (1989) demonstrateshow texts can play a role in reflecting reallife situations in the classroom, andFrankenstein (1990) takes this further,focusing on the potentially emancipatory roleof the material offered in class. But the effectof the dominance of textbooks in mathematicsteaching has not been well investigated.When the standard mathematics lessonbegins with some initial teacher expositionand is followed by the students workingthrough an exercise in their textbook, andhomework is a further exercise from the book,the text, the textbook and the textbookwriters constitute an authority in theclassroom. Teachers often talk of "they areasking you to..." when attempting to clarify aquestion that appears in the text, and thatstudents do not understand, and it isinteresting to consider what the relationshipis between the teacher and the text that isconveyed by this expression.
Social messages that are hidden in textsare unquestioned by teachers and students,partly because the textbook is an illustrationand manifestation of the authority implicitin the classroom. This is particularly thecase in mathematics, perhaps because thesterile axiomatic form of the presentation ofacademic mathematics papers reinforces theauthority and status of the mathematicaltext. In the study of history, for example,students are encouraged to draw from anumber of sources, highlight the ways inwhich those textbook writers disagree witheach other, and even to explain the context inwhich different writers hold competingviews (this is not the case in all countries ofcourse!). It is quite the opposite in themathematics classroom. There are also moreovert reasons for textbooks beingunchallenged in the mathematics classroom,
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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 asuggestion of ways that teachers couldchallenge those assumptions, was given byLerman (1990b). There are different abilitylevel texts in the most popular series in use inBritish schools, the School MathematicsProject, where the same topic, paying incometax, is dealt with by asking the top abilitystudents to calculate tax on an income of£50,000 whereas the bottom ability textrequires calculations for only £9,000. Theauthors are assuming, and thus are conveyingthe message to the students that low abilityin mathematics, whatever that may mean,correlates with low intelligence, and/or poorcareer prospects etc. Lerman suggests that thedynamics of the classroom may well change ifstudents were encouraged to criticallyexamine their textbooks, and the examplesused, by comparing the pages from the twobooks. The author has offered such a notionto teachers on an in-service course, and oneteacher commented that it would be toodangerous (Lerman, 1990b), perhaps becauseit threatens the safety of the authoritywhich the textbook establishes in theclassroom.
Is the relationship to 'authority' inmathematics any different if the teacher usesa workcard individualised scheme, orindividual worksheets? What would be theimplications and outcomes of developingmaterials as and when they are needed,perhaps by the students themselves? Me Ilin-Olsen (1987) proposes classroom work basedon projects:
"A dormitory town outside Bergen. High,grey concrete blocks of flats. The area wasdeclared ready for occupation as soon asthe garages had been built. It was notthought scandalous until later it wasdiscovered that they had forgotten toprovide the children with leisure areas.
The teachers of three classes (age group 10)prepartd a project. What can we do aboutthe situation?" (p.218)
There are of course implications for theempowerment of students in an example like
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this, especially when contrasted with acommon investigative task on a similartheme, to make a model of a bedroom orclassroom. Frankenstein gives manyexamples of projects for use in mathematicsclassrooms (Frankenstein 1990). Brown andDowling (1989) propose what they call aresearch-based approach as an alternative tothe textbook, for teachers to use in theclassroom:
"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 beinteresting to follow reports of teachers' useof the innovative, perhaps revolutionaryideas of Mel lin-Olsen, Frankenstein, Brownand Dowling and others, in changing thefunction of the text.
Constructivism and the teacher
As a learning theory, constructivism isdescribed in Hart's chapter in this volume,and there are many important researchprogrammes that draw on the paradigm ofconstructivism, that have been reported inthe literature. In this section we areconcerned with what the theory mightsuggest for the role of the teacher.
The Eleventh Annual Conference of PNIE,held in Montreal in 1987, has been the onlysuch annual meeting to be centred on a theme,that of constructivism. The term'constructivism' had appeared in earliermeetings, and it was presumably felt that themathematics education research communityin general were unsure of the meaning and ofthe relevance and/or implications formathematics education. Consequently thattheme was chosen for the PME meeting, toencourage debate on the issue. In his plenarypresentation, Kilpatrick (PME, 1987)challenged the claim that constructivism isan alternative paradigm for learning, andsuggested that the putative outcomes ofconstructivist research and teaching areconsistent with cognitive psychology. Sincethat meeting, many papers presented at PME
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conferences have been couched in thelanguage of constructivism, and drawn onmethodologies of research and teaching, aswell as theoretical frameworks, that claimto have distinct interpretations in theconstructivist paradigm. The critique,however, has been only infrequentlycontinued (Goldin, PME, 1989), and this isperhaps to the detriment of the debate, sincewe can only develop our understanding of thelinks between theory and practice, and inparticular alternative learning theories andthe consequences for teaching mathematics,through critical discussion.
In relation to the role of the teacher,there have been a number of research reportsconcerned with the preparation ofconstructivist teachers (e.g. Simon, PME,1988). However, the implications ofconstructivist learning theories for the role ofthe teacher, and for the kinds of activitiesthat the teacher might initiate as aconsequence of a constructivist view oflearning, have still not been clearlyelaborated, and certainly not well tested,either through theoretical critique orthrough classroom case studies of teachers.This makes talk of 'constructivist teachers'less meaningful than might be the case werethere to be some elaboration of what thatimplies. There is also the question of thedistinction that has been made between'weak constructivists' and 'strongconstructivists', the former being those thatsubscribe only to the first hypothesis quotedby Kilpatrick (PME, 1987), and the latterbeing those that subscribe to both:
"(1) Knowledge is actively constructed bythe cognizing subject, not passivelyreceived from the environment.
(2) Coming to know is an adaptiveprocess that organizes one'sexperiential world; it does notdiscover an independent, pre-existing world outside the mind ofthe knower." (p.3)
It was suggested at the PME meeting in1987 that everyone could subscribe tohypothesis (1). There have been someattempts to discuss the implications of bothpositions. Lerman (1989b) and Scott-Hodgettsand Lerman (PME, 1990) have suggested that
it is difficult to see how one can accept thefirst hypothesis and not the second. Theyalso maintain that the second hypothesis,far from leaving one unable to say anythingabout anything, as Kilpatrick suggested inhis presentation (PME, 1987), is anempowering position, in the sense that ideasare continually open to negotiation anddevelopment and we do not expect to arrive atultimate truths. However, it is clear thatmore debate and research are needed.
Teachers' beliefs about mathematics
Ideas such as 'the teachers' values', 'whatthe teacher believes about problem solving'and so on, have occurred often in the reviewabove. The issue of the influence of teachers'beliefs on their actions has been and continuesto be of great interest and importance inmathematics education. Nesher (PME, 1988)discusses the effects of teachers' actions andalso reveals her own beliefs, in her plenarylecture at the twelfth meeting of PME. In onepart of her paper she focuses on the waysthat the behaviour of the teacher can havean effect on the child's conception of thenature of mathematics, and she warns of thedanger of the teacher being the ultimatejudge of whether the child is correct or not:
"This does not let the child construct forhimself the mathematical notions andconcepts. Nor does it enable him to realizethat the truths of mathematics are objectiveand necessary." (p.63)
Thom's often quoted claim that "allmathematical pedagogy, even if scarcelycoherent, rests on a philosophy ofmathematics" (1972, p.204) was one of thefactors that stimulated interest amongsteducators in epistemologies of mathematicsand their influence on teaching, and in thepast ten years there have been a number ofstudies 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 thephilosophy of mathematics and the possiblealternative perspectives of the nature ofmathematics (e.g. Lerman, 1983; Ernest,1989), and a 'bottom-up' approach, whichbegins with teachers' views and behaviour,as in the work of Thompson (1984) forexample.
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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 teachingmathematics (e.g. Lerman, 1986), and alsodetermine research directions. As Hersh(1979) wrote:
'The issue then, is not, what is the best wayto teach, but what is mathematics really allabout ... controversies about high schoolteaching cannot be resolved withoutconfronting problems about the nature ofmathematics." (p.33)
Scott-Hodgetts (PME, 1987), in herdiscussion of holist and serialist learningstrategies, demonstrates the significance ofthe teacher's philosophy of mathematics tothe extension of various learning strategies.In their study of mathematicians andmathematics teachers views, Scott-Hodgettsand Lerman (PME, 1990) set their analysis ofpsychological/philosophical beliefs in aradical constructivist perspective, andpropose that it provides a powerfulexplanatory potential:
"What A has been engaged in, withoutdoubt, is what is described as "an adaptiveprocess that organises one's experientialworld". In doing this he has brought to beardifferent models of the nature ofmathematics, picking and choosing in orderbest to 'fit' his particular experiences atdifferent 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,and about 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-solving instruction, characterises teachers'cognitive levels, a set of psychologicalattributes, on a concrete-abstract continuum,from A, the most rigid, concrete formalism,through B, social pluralism, and C,integrated pluralism, to D abstractconstructivism. In interviews andobservations of 11 teachers, all on problem-
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solving lessons so as to standardise the lessoncontent, Dougherty placed 8 in A and 1 ineach of the other categories, and she foundstrong connections between these positions andconceptions of problem solving.
Methods of analysing and interpretingteachers' actions in the classroom, thatengage with the issue of teachers' beliefsabout mathematics, have been reported inthe literature. Jaworski (PME, 1989)discusses students developing a 'principled'understanding of mathematical concepts,when comparing the inculcation ofknowledge, as against its elicitation. Shesays:
"Successful teaching of mathematicsinvolves a teacher in intentionally andeffectively assisting pupils to construe, ormake sense of mathematical topics."(p.147)
She demonstrates the dangers of 'ritualknowledge' in a wonderful example of thereasoning of a child who regularly achievesgood marks in mathematics.
"I know what to do by looking at theexamples. If there are two numbers Isubtract. If there are lots of numbers I add.If there are just two numbers and one issmaller than the other it is a hard problem.I divide to see if it comes out even and if itdoesn't I multiply." (p.148)
Ain ley (PME, 1988) has looked atteachers' questioning styles, noting the powerrelationships, by analogy with theparent/child or drill sergeant/recruit. Shesuggests that assuming mathematics answersare right or wrong is a danger for the teacherin using questioning, and that teachers assumethat questioning is better than exposition. Inher study, she looked at the differentperceptions of the children and the teachers,of the teachers' questions. She classifiedquestions into: pseudo-questions; ,genuinequestions; testing questions, and directingquestions, the latter sub-divided intostructuring, opening-up and checking. Herfocus on the language of the classroom isfurther discussed in the following section.
Various studies show that there are anumber of variables that mediate, in aninvestigation of the influences of teachers'beliefs. Thompson (1984) and Lerman (1986)
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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 mainpressures on you as a mathematics teacher?"replied:
"School policy...you can't just be the one tobreak the system ... if you give all yourstudents an A you will be questioned onit...you can't. You've got to work within thesystem as such.
The kids have their activity book. If theparents look at that at the end of the yearand they haven't done one activity ... they'regoing to question it and the Principal'sgoing to question whether you are followingthe program." (p.261)
Noss et al. (PME, 1990), in discussing astudy of teachers on an in-service course incomputer-based mathematics learning,comment that:
"... the extent to which a participant wasable to integrate the computer into his orher mathematical pedagogy (theoreticallyand/or practically) appeared more relatedto the direction in which a participant'sthinking was already developing and withhis or her commitment to change, ratherthan the style of teaching approach, view ofmathematical activity, or rationale forattending the course." (p.180)
A global view of factors affecting theteaching of mathematics reveals clearly themajor influence of the conditions and resourcesin the schools in different countries. Nebres(1989) calls these the Macro problems, asagainst the Micro problems which areinternal to mathematics education:
"... in many developing countries, theproblems that merit most thought andresearch are those due to pressures fromoutside society. The purpose of studyregarding these pressures is to providesome scope and freedom for theeducational system so that it can attend tothe internal problems of mathematicseducation." (p.12)
Nebres emphasises the importance of therole of cultural values, in understanding boththe Macro and Micro environments inmathematics education, and suggests that hisawareness of that perspective is focused by
the "sharp contrast of traditions and culturesin East Asia" (p.20). He writes:
"... we have realized that questions like thestatus and salaries of teachers are notsimply a function of the economy but alsoof cultural values. Similarly the rise of whatwe might call the internal force of genius,wherein mathematics talent springs upeven under difficult economiccircumstances, seems to be fostered by thecultural environment." (p.20)
Suffolk (1989) gives a detailed study ofthe situation in Zambia, as seen through therole of the teacher. Most such studies, quitenaturally, focus either on the children or onthe situation of the schools, and more studiesof the effect on the teacher are needed. Oneapproach, as a consequence of a recognition ofthese variables, as reported by Nolder (PME,1990), is to undertake a detailed analysis ofthe mathematics teachers in one particularschool which is undergoing some curriculumchanges, in an attempt to reveal the nature ofthese influences. She found, for instance, thatsuch changes led to the teachers experiencinga good deal of uncertainty, which affectedtheir perceptions of their competence andconfidence. Stephens et al. (1989) report onmathematics education reform programmes,and state the following principle forinvolving teachers:
"Teachers involved in change initiativesneed to know where they have come from,where they are going, why they are goingthere, and should have access both to thebest professional thinking of theircolleagues, and to theoretically sound,supportive environments." (p.245)
These studies indicate that 'change' isalways socially embedded, and cannot beisolated from the cultural context, from thepolitical milieu, from the social context ofschools and certainly not from the teachers inthe classroom. Ignoring any of these elementsmitigates against any improvement in theteaching of mathematics.
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
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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 ofperceiving -cation as praxis or within theperspective of perceiving it as hegemony"(p.84). Gerdes (1985) demonstrates howmathematics education in Mozambiquechanged from pre-revolutionary colonialdomination to post-independence constructionof a socialist society:
"During the Portuguese domination,mathematics was taught, in the interest ofcolonial capitalism, only to a small minorityof African children ... to be able to calculatebetter the hut tax to be paid and thecompulsory quota of cotton every familyhad to produce ... to be more lucrative"boss-boys" in South African mines ... Post-independence ... Mathematics is taught to"serve the liberation and peaceful progressof the people". (p.15)
D'Ambrosio (1985), in developing theidea of ethnomathematics, sets the growth ofmathematics in a social context, and alignsthe development of modern science and, inparticular, mathematics and technologywith colonialism. There is a growing body ofwork in the area of ethnomathematics, at themacro level and at the micro level, withprofound implications for teachers inmathematics classrooms, both in the so-called developed world, and also the under-privileged world, that engages with ideas ofempowerment and 1 ibera ti on.
The language and the practices of theclassroom
Recently, there has been an interest inthe discursive practices of mathematicsteaching (e.g. Keitel, et al., 1989; Weinbergand Gavalek, PME, 1987), influenced in themain by the work of Walkerdine (1988, 1989,PME, 1990). The focus of this critique is ashift away from the individual to the social,and is thus a critique of psychology, whichcan be characterised as an individualisticway of interpreting people's actions.Investigating the interactions in theclassroom (aspects such as discussion betweenstudents, teachers' questions, open-ended
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questions, the teacher's response to novelinterventions from students), leads onerapidly into aspects of the powerrelationships of the classroom. Pimm (1984),for example, talks about the use of the word'we', and indicates just how much is actuallybeing said and construed and structured by theuse of that small word:
"Given a class of pupils all doing somethingincorrectly, a teacher can still say, 'No, whatwe do is ...' Who is the community to whomthe teacher is appealing in order to providethe authority to impose the practices whichare about to be exemplified?" (p.40)
Of course the dominant paradigm withinwhich research, testing, and theorising onthe meaning of understanding takes place is apsychological one. Understanding is seen assome process that the individual undergoes,some transition from not understanding aconcept to understanding that concept. Thereare two elements in this transformation froma pre-understanding state to a state ofunderstanding, the individual and theconcept. The latter is seen as an objective,external element which in the educationalenvironment is usually 'possessed' or held insome way, by the teacher. The teacher isseen to perform the intermediary role ofintroducing the concept, attempting to enableand facilitate its transmission from teacherto student, and measuring the outcome, thatis whether the state of 'understanding' hasbeen achieved. The former element, theindividual, is largely a closed unit to theteacher, making the business of determiningwhether the transformation has taken placeor not, a very difficult one. As Balacheff(1990) points out:
"It is not possible to make a directobservation of pupils' conceptions relatedto a given mathematical concept; one canonly infer them from the observation ofpupils' behaviours in specific tasks, which isone of the more difficult methodologicalproblems we have to face." (p.262)
Research in this area in education thusfocuses either on interpretations of behaviourthat might constitute evidence of theprocesses of concept acquisition, or on theconditions that will bring about'understanding'.
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Mathematics is seen as hierarchical, andconflated with this is the notion thatlearning mathematics is hierarchical too,from the basic notions of one-to-onecorrespondence, setting and combining sets, tomore abstract levels of, for example, thecalculus. It is seen by educationalists as 'verydifficult' (Hart, 1981). Other researchshows, however, that children's unrestrictedand undirected thinking does not follow thesupposed hierarchy (e.g. Lave, 1988;Carraher, PME, 1987). It may just be the casethat our assumptions about the nature ofmathematics and the nature of the learningprocess become self-fulfilling. We interpret,theorise, teach, test, assume, expect, measureand thus confirm our initial expectations ofchildren.
The psychological model of learning,which can be characterised as a process oftransformation from a pre-understandingstate to a state of understanding, determinesthe style and intention of the various stagesof the educational process. The process beginswith the teacher assessing the initialcognitive state of the individual. This isfollowed by the prepared lesson, whosecontent is determined by research intohierarchies of mathematical knowledge andlevels of difficulty of understanding, andwhose method is determined by large-scaletests using well-established quantitativeprocedures. Research on affective factorsinfluence the classroom setting, theappearance of the text, etc. On-goingassessment enables the identification ofmisconceptions, misunderstandings or partialunderstandings. These can be corrected bysuitable re-explanations, reinforcementmaterials and so on. The final stage is thetest that measures the successful acquisitionof the particular skill or concept, that isinterpreted as understanding, and again thattest will have been trialled over large groupsof students, in well-established ways.
Whilst the above is somewhat of acaricature, the intention is to highlight themanner in which the psychological paradigmdominates and determines the wholestructure of the educational process. It is theessentially private nature of thepsychological interpretation ofunderstanding that pervades the educational
process from this perspective. Teachers areleft to try as best they can to recognise whenthat process has occurred, and in the case ofthe recent development of a NationalCurriculum in Britain, with a long list ofattainment targets and levels to be assessed,tick the appropriate box to register thatachievement target for the student.
At the same time, there is an assumption,usually implicit, that we all arrive at thesame understanding of particular things,because of the absolute nature of those things,whether physical objects, such as tables, andunicorns, or mental objects such as circles, or'three'. A critical analysis, therefore, needsto work on both aspects of the understandingprocess in the psychological paradigm,namely the notion of the individual, and alsothe notion of knowledge and its positedabsolutist nature.
In the psychological paradigm, the majorfocus of attention is in how the individual,the subject, functions in discourses. However,subject is simultaneously the apparentlyautonomous self-constituted individual whoarticulates a discourse and also the subject ofthat discourse. That is to say, the individualis constituted by a discourse. Thus, forexample, in the classroom we aresimultaneously the teacher, who initiatesand perpetuates a discourse, and at the sametime we are constituted by that discourse, andby the relations of power that pertain. Tosome extent this has been looked at above,specifically related to the power of theteacher. It may appear, though, from thatsection, that teachers consciously choosepositions of power and can reject them. Inanalysing the discourse of the classroom onecan begin to see the subtle historically-rootedrelationships and meanings carried by theinteractions in the classroom. Pimm's (1984)discussion of the use of 'we', and the positionin which the accepted and unchallenged roleof the mathematics textbook places theteacher and students are examples of theinsights that can be gained from an analysisof the discursive practices of the classroom.
An analysis of how languages operatewithin discursive practices indicates the wayin which knowledge, understanding, meaningand ourselves as subjects are constituted in and
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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 practicesin which the notions of 'more' and 'less' occur(Walkerdine, 1988), she demonstrates thecontext in which those words have meaningfor those children, and the practices throughwhich those meanings are constructed. Theclassroom teacher who is unaware of themeanings the children are bringing withthem into the classroom may well makejudgernents about children's 'abilities' thatare entirely inappropriate.
The alternative view of 'understanding'is one that sees the process as social, ratherthan psychological. That is to say, theindividual does not go through sometransformation from pre to post state ofunderstanding in isolation, acquiring a givenwell-formed concept in some hidden way. Itis necessarily an interaction of the child andhis/her multitude of meanings, throughlanguage, with others and with experiences.That process of interaction may be largelysilent, a shift taking place without beingopenly articulated, but this does not meanthat it is in some way 'private'. Sometimesthe understandings do not fit with those ofothers, including the teacher, and only talk,discussion, suggestions and conjectures andrefutations, through cognitive conflict, orshifts of thought through resonance, enablefurther growth.
In the literature of mathematicseducation, one finds an emphasis on providinga context for mathematical concepts, andclearly this is essential if mathematics is tobe about something. There are two aspects to'context', however. It is a matter, not only ofattempting to embed mathematics insituations that may be 'relevant' or'meaningful' for students, but also an analysisof the discursive practices within whichmathematical terms and concepts appear,both for students in their own experiences andfor the mathematical community.
The language used in the mathematicsclassroom, particularly in relation tobilingualism, has been an important focus ofstudy for a number of years (Dawe, 1983;Zepp, 1989). Most such studies have been
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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 thefunction that language serves in deep aspectsof classroom interactions. Khisty et al. (PME,1990), for example, reported on anexploratory study of the language of theteacher, in a bilingual classroom, toinvestigate the reasons for theunderachievement of Hispanic students in theUSA. They review the language factors thatmight play a part, emphasising aninteractionist perspective, which attempts tobring together the cognitive and socialdimensions of learning. They refer to how theuse of language structures the classroom, thedevelopment of a mathematics register,syntactic and semantic structures inmathematics, and context. They concludetheir 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, Ain ley (PME, 1988)has looked at the implications of teachers'questions, and revealed both a wide range ofpurposes, and also a discrepancy betweenchildren's perceptions of the teacher'squestion and that of the teacher. Wood(PME, 1990) focused on discussion in theclassroom, and highlights the all toofrequent consequences of what may be called'teacher-centred' learning:
"The patterns of interaction becomeroutinized in such a way that the studentsdo not need to think about mathematicalmeaning, but instead focus their attentionon making sense of the teacher'sdirectives." (p.148)
In a different direction of attention, Blissand Sakonidis (PME, 1988) looked at thetechnical vocabulary of algebra, and theeveryday language that is sometimessubstituted for these terms, in an attempt todiscover whether it helps or hinderscommunication of algebraic ideas.
Analyses of the language of the classroomalso make connections between the concepts
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within the 'language game' of mathematicsand in everyday use. Pimm (PME, 1990)discusses metonymic and metaphoricconnections, and in doing so highlights acentral aspect of the ways mathematicallanguage and thought work. For example,metaphoric links are those which wouldhighlight the meaning of 50% in differentcontexts, whereas metonymic links would bethose between 50% and 0.5 and 1/2 and 4/8etc.
"Being aware of structure is one part ofbeing a mathematician. Algebraicmanipulation can allow some new propertyto be apprehended that was not visiblebefore the transformation was not madeon the meaning, but only on the symbolsand that can be very powerful. Where arewe to look for meaning? Mathematics is atleast as much in the relationships as in theobjects, but we tend to see (and look for) theobjects ...
Part of what I am arguing for is a farbroader concept of mathematical meaning,one that embraces both of these aspects(the metaphoric and the metonymic foci formathematical activity) and the relativeindependence of these two aspects ofmathematical meaning with respect toacquisition." (p.134-135)
One can expect significant furtherresearch in the deconstri.ction andreconstruction of the langeage and practices ofthe mathematics classroom.
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 issuesconcerning teaching apply equally to teachereducators. Students on a Summer Mathprogramme were reported by Schifter (PME,1990) as demanding: "We need a math coursetaught the way you're teaching us to teach."(p.198) Jaji et al. (1985), in an article whosemain focus is on providing pre-serviceteachers in Zimbabwe with a repertoire ofroles from which to choose, rather thanreplicate the ways in which they weretaught, offer this transcript of a teachereducator in action:
"Now what I am about to say is veryimportant. It will almost certainly come upin the Examinations, so I suggest that youwrite it down.
(The group took out their pens ...).
In the new, modern approach to teaching ...
(They wrote it down as she spoke ...).
In the new, modern approach to teaching,we, as teachers, no longer dictate notes tochildren. Instead we arrange resources insuch a way as to enable children to discoverthings for themselves." (p.153)
If we are to suggest that teachers shouldbecome involved with research on theirpractice, as will be discussed below, teachereducators are teachers too and one wouldexpect that teacher educators would beinvolved in research on their own practice.One can in fact see evidence of a small butgrowing interest in research on teachereducation in recent meetings of PME.
A further consequence of the analogybetween students in schools learningmathematics and research on that, andstudent teachers learning to teachmathematics and research in that area, isthat just as we cannot talk of making studentslearn mathematics, 60 we cannot talk ofmaking 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 activitiesthat encourage reflection and awareness andpersonal growth (Jaworski and Gates, PME,1987; Lerman and Scott-Hodgetts, PME, 1991,in press), but to try to impose teacher changeis both unrealistic and perhaps arrogant.
There are other reasons for reflecting onthe practice of teacher education. In Britain,and probably many other parts of the worldtoo, teacher education programmes are at aturning point, under the influence of politicalpressures. There is a strongly held view thatteachers can best learn "on the job" ratherthan in college, with the metaphor ofapprenticeship in training held to be moreappropriate than the combination of theoryand practice that might be implied by themetaphor of a profession. According to thisview, at secondary age in Britain, (students
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aged 11 to 18) anyone can teach, provided onehas a degree in a suitable subject, given anappropriate system of apprenticeship. Thatsuch a system is vastly less expensive thanspending between one and four years in acollege is almost certainly the main butunstated rationale. Whatever the motives,teachers and teacher-educators must be clearabout the nature of the process of teaching,and therefore of teacher education. There isno doubt that one can learn a great deal bywatching other teachers, and not necessarily'master-teachers', which is a designationused in some countries. Given a theoreticalframework in which to critically evaluateand assimilate one's observations, observingothers can be an essential and valuable aspectof learning to become a teacher. But learningto become a successful teacher is not a matterof copying someone who has been designatedan expert. It is a matter of finding one's ownway of teaching, developing ways in whichone can identify what is happening in theclassroom, and of drawing on experience andtheory to make decisions on action, followedby reflective evaluation of those actions.There are frequently conflicting demands onthe teacher (Underhill, PME, 1990), andresearch on factoi that help prospectiveteachers to develop their own understandingof 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 reviewof some of the main issues that arise in thedevelopment of in-service coursesappropriate to the needs of teachers willthen be followed by a development of thenotions of the teacher as reflectivepractitioner and as researcher.
There have been a number of studies thathave focused on prospective teachers' orpractising teachers' misconceptions of topicsin mathematics. Concerning teachers ofelementary age children, Linchevski andVinner (PME, 1988) looked at naive conceptsof sets, Vinner and Linchevski (PME, 1988) onthe understanding of the relationshipbetween division and multiplication, Tierneyet al. (PME, 1990) on area, Simon (PME, 1990)on division and Harel and Martin (PME,
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1986) on the concept of proof held byelementary teachers. In the high school agerange, Even (PME, 1990) has examined mis-conceptions in understanding of functions.
What are we to make of these findings,and what might be the possibilities forimproving the situation? It is probably thecase that most people have negativeexperiences of learning mathematics atschool, and as in the main, prospectiveelementary teachers are not mathematicsspecialists, they too are generally at leastunconfident about mathematics. If pre-service courses teach mathematics in thesame way that those people learnedmathematics at schools, patterns of lack ofconfidence, under-achievement and feelingsof failure are likely to remain into theirteaching career (Blundell et al., 1989). Sincechildren's early experiences of mathematicsoccur in interaction with elementaryteachers, this is a major problem formathematics education. Ways of breakingthis cycle must clearly be developed, and onesignificant element, as mentioned above, isfor students to be taught the way that weexpect them to teach. Schifter (PME, 1990),who came across that demand from herstudents, reports on a programme of in-serviceeducation which attempts to effect such atransformation:
"... the notion of "mathematics content" asthe familiar sequence of curricular topics isreconceived as "mathematics process": atonce the active construction of somemathematical concepts - e.g. fractions,exponents - and reflection on both cognitiveand affective aspects of that activity. Thework of the course is organized aroundexperiences of mathematical exploration,selected readings, and, perhaps mostimportantly, journal keeping." (p.191)
The journal writing is a most effectiveway of demonstrating the changes that theteachers felt themselves to have gonethrough, during the course. The mainintention of the journals, however, is toencourage and enable the reflective processthat is an essential component of breaking outof the repetitive cycle of lack of success. It isnot that such reflection will guarantee thatno misconceptions in mathematicalunderstanding will occur, but that where
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cognitive conflicts arise in teachers'experience of mathematical activities andconcepts, or a recognition of aspects ofmathematics that they do not understand,they are more likely to be able to confrontthem, rather than remain with theirmisconceptions.
Blundell et al. (1989) also draw onstudents' reflections as expressed in journals,in their evaluation of the nature of theirstudents' experiences and learning, throughthe course. In this study the course was a pre-service one for prospective primary(elementary) school teachers, and as in theSchifter (PME, 1990) study, the focus was onmathematical processes and learningprocesses. They describe the aims as follows:
"The fundamental belief which underliesthis course structure is that a recognitionand mastery of these general concepts andstrategies will enable students to approachany new content area with confidence andcompetence, and that the efficientacquisition of meaningful mathematicalknowledge is thus facilitated. Furthermore,this acquisition is not dependent upon theskills of the "teacher", but rather upon thoseof the "learner" - leading to much greaterindependence and self sufficiency." (p.27)
Thus the other aspect of 'being taught theway we want them to teach' is that whenstuden. 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 selfsufficiency, they can recognise the potentialfor their own school students to do the same.The students commented on their earlylearning experiences, and then on their recentgrowth as mathematics learners:
"The only maths teacher I remember,although only vaguely, is the one who kindlyallowed me to spend most of my timeoutside the classroom door ..."
"September 1965. I hate maths ... I recallheading a page of my maths book "funnysums". For this I was sent to stand outsidethe class, told I was a "bloody imbecile" andthat I was the worst student he had ever hadthe misfortune to teach ..." (p.27)
"We're never afraid of anything now arewe? Whatever you give us we'll jump inand have a go."
"I really love doing maths. I really lookforward to Tuesdays and Fridays. All thistime I've been going on thinking I'm nogood at maths - and now I'm doing it."(p.N)
The authors reflect on their own learningas teacher educators, from this analysis:
"Clearly the students have gained inconfidence by this stage of the course, but itis not yet an unequivocal confidence. Itseem,., as though they are still ambivalent,feeling confident when they focus on theirprocess abilities, but doubting whenconcentrating on content in mathe-matics
However, they do not overtlyacknowledge and value the acquisition ofprocess skills to the extent to which wemight have hoped." (p.30)
Journal writing as an empoweringactivity in reflection, whether it be on one'smathematical work or on one's learningduring a course is itself the subject of study(e.g. Brandau, PME, 1988; Hoffman andPowell, 1989; Frankenstein and Powell, 1989).
1n-service courses provide theopportunity for teachers to meet with others,reflect on their experiences, re-interpret theirwork in the light of theories which they maymeet during the course and develop areas oftheir expertise. Fahmy and Fayez (1985)attempt to outlihe guidelines for in-serviceeducation based on their work in Egypt. Theyemphasise the prof,!ssionalization of teachereducation, orientation toww:ds self-educationand recearch activities and carefulconsideration of the mathematical needs ofthe teachers. There are situations in whichthe in-service programme has to be gearedtowards major curriculum developments, andthus are firmly focused on these needs.Xiangming et al. (1985) describe a network ofin-service courses in institutions throughoutChina to implement a 12 year programmefrom a 10-year one in schools. Similarly,Roberts (1984) describes a programme ofschool-based teacher education inSwaziland, balanced with in-collegesessions, and Vila and Lima (1989) adistance-learning programme in Brazil.Eshun et al. (1984) pick up on the problems
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that these latter experience, and emphasisethe constraints for in-service courses forprimary teachers that result from the largenumbers involved, and the few highereducation institutions that can provide thatsupport.
An increasingly recurring theme in theseand other literature concerning teachereducation is the notion of teachers as'reflective practitioners'. Lerman and Scott-Hodgetts (PME, 1991, in press) discuss Schön'scharacterisation of the nature of teachingand develop his interpretation:
"Schein ... maintains that teaching isreflective practice, in his discussion of theunsatisfactory distinction drawn betweentheoreticians and practitioners.
In describing his notion of reflection-in-action, Schön ... starts from what he callsthe practitioner's knowledge-in-action:
"It can be seen as consisting of strategies ofaction, understanding of phenomena, waysof framing the problematic situationsencountered in day-to-day experience."
When surprises occur, leading to"un......tainty, uniqueness, value-conflict",the practitioner calls on what Schön termsreflection-in-act-on, a questioning andcriticising function, leading to on the spotdecision-making, which is "at least in somedegree conscious".
In our situation, we would suggest that weare extending the idea of the reflectivepractitioner. We would agree with Schönthat recognising that a teacher is muchmore than a practitioner is essential, bothfor teachers themselves (and ourselves),and for teacher educators, administratorsand others. However, when we use the termreflective practitioner, we are alsodescribing meta-cognitive processes of, forinstance, recording those special incidentsfor later evaluation and self-criticism,leading to action research; consciouslysharpening one's attention in order tonotice more incidents; finding one'sexperiences resonating with others, and/orthe literature, and so on. We arecor cerned with the transition from thereflective practitioner in Schön's sense, tothe researcher (Scott-Hndgetts 1990), whohas a developed critical attention, noticinginteresting and significant incidents, andturning these into research questions. By
82
analogy with some recent views of thenature of mathematics (e.g. Lerman 1986;Scott-Hodgetts PME, 1987), 'MathematicsEducation' is most usefully seen, not as abody of external knowledge, recorded inarticles, papers and books, that one readsand uses, but as an accumulation of workupon which a teacher can critically draw, toengage with those questions that concernand interest her/him (Scott-Hodgetts, inpress, a)."
The notion of teacher as researcher,although around the education world forsome years, has only recently appeared in theliterature of mathematics education (e.g.ATM 1987; Scott-Hodgetts 1988; Fraser et al.,1989). An early statement of the potential ofbringing together research and teaching wasgiven by the Theme Group on the ProfessionalLife of Teachers at the Fifth InternationalCongress on Mathematical Education (Cooneyet al., 1986):
"It was emphasised that the desirability ofhaving teachers participate in researchactivities may extend beyond the questionof what constitutes research and relate tothe professional development of theteacher by virtue of engaging in suchactivity. That is, not only can the teachercontribute to the creation of groundedtheory but the teacher can mature andelevate his/her own expectations andprofessional aspirations by participating ina reflective research process." (p.149)
Research is often seen as the prerogativeof full-time people based in institutions ofhigher education. This is to misunderstandthe nature of teaching and the educationalexperience. There is no doubt that full-timeresearchers have had the opportunity todevelop skills and perspectives in researchthat class teachers have not had, but theseparation of research from practice is one ofthe major factors that leads to suspicion ofresearchers by teachers, and the well-knownphenomenon mentioned early in this chapter,that the findings of research do not oftenreach teachers (Scott-Hodgetts, 1988;Lerman, 1990a). In a comparison study ofteachers' perceptions of their classrooms, andpupil perceptions of the same classrooms,Carmeli et al. (PME, 1989) found that therewere statiFtically significant differences,with the teachers being far more positive
8 6
that their pupils. There was no mention inthe report of whether the teachers wereshown the outcome of the study, and whateffect that may have had on their ownpractice, although that may of course havetaken place later. Research on teachers,carried out by outsiders, and then reported inthe academic literature, is of much less valuethan research in which the teachers areinvolved, and which they can use in theirown growth as teachers.
Condusion
There are of course many questions aboutthe influence of the role of the teacher inchildrens' learning of mathematics, and inthis chapter we have described attempts toengage with some of those questions, andanswer them. Two fundamental problemsremain, which were referred to in theintroduction to this chapter: what do'answers' look like, in this domain, and whatkinds of research methodologies are mostsuitable to investigate those questions?
As with so many issues in mathematicseducation, the place to commence thisdiscussion is with attitudes and beliefs. Forexample, one of the most systematic anddeveloped methodologies for examining thepractices of the teacher, in order to achievechildren's understanding of mathematics isthat developed by the group engaged inrecherches en didactique des mathématiquesin France. It is interesting to examine theassumptions about the nature of classroominteractions that underpin their system, andto see how the alternative view described inthis chapter offers another methodologicalapproach. Brousseau, Chevallard, Balacheffand others have developed notions of thedidactical process, theory of didacticalsituations, didactical transposition etc.Balacheff (1990) describes his basicassumptions and the fundamental problem, ashe conceptualizes it:
"So if pupils' conceptions have all theproperties of an item of knowledge, we haveto recognize that it might be because theyhave a domain of validity. Theseconceptions have not been taught as such,but it appears that what has been taughtopens the possibility for their existence.Thus, the question is to know whether it is
8
possible to avoid a priori any possibility ofpupils constructing unintendedconceptions...
I have suggested that pupils' unintendedconceptions can be understood asproperties of the content to be taught or theway that it is taught." (p.262-263)
This approach to research on teaching isa positivist one, which anticipates thepossibility of solving these problems. It isone which focuses on the individual'sconstruction, in the light of the content andwhat the teacher does. At the same time, itappears that the construction is to be of anobject that exists in some a priori sense. Themethodology that Balacheff describesassumes that it is possible to specify all theconditions of the classroom, the mathe-matical content, and so on. Brousseau (1984)recognises the enormity of the task anddespairs of the possibility of success:
"Research into didactics is itself doomed tonaiveté and/or failure, since it is becomingmore and more difficult, as the body ofknowledge concerned with didactics growssteadily more complicated and technical,for a good student of mathematics todevote enough time to absorbing it beforehe sets out to break fresh ground in thisfield." (p.250)
A methodology of research on teachingwould look different from a perspectivewhich focuses on the construction ofknowledge within an analysis of discursivepractices, knowledge as signs that areslippery, not fixed, that is signs that appearclear and certain, but shift when analysedand then shift again. If we cannot specify allaspects of the teaching situation in advance,it is essentially because the actors in thatsetting bring with them their culturalexperiences, in the widest sense. Theseexperiences are not 'private', just notcompletely specifiable. And of course theclassroom itself is a whole new socialsiruation with all its own practices. As aconsequence, the teaching process is best seenas one which creates situations in whichstudents construct their own knowledge, andwhere students' constructions are valued, evenif they are unintended. There are conceptionsand misconceptions, but there are alsopartial-conceptions and other-conceptions
83
too. Valuing these contributions recognisesthat learning is what an individual does forherself, not something the teacher does to thestudent, and also places responsibility on thestudent for her own learning. It also avoidsthe student dependence on the teacher asjudge, and enables the development of self-critical skills in students. All conceptionsoffered can be brought into the 'objective'arena of the classroom for examination, tosearch for 'proofs and refutations'.
In the context of research methods, onelearns something important aboutmathematics teaching when one discoversthat in a survey of 32 teachers and 1338pupils the teachers had a significantly morepositive view of a number of elements of theclassroom environment (Carmeli et al., PME,1989). One learns much more, perhaps, in acase study of a teacher who takes thatinformation and attempts to modify anddevelop the relationships, communication,discussion and other interactions in herclassroom, and develop ways of evaluatingthe changes and the consequences for herselfand her students, through action research.Thus, qualitative research is perhaps mostuseful when it provides insights thatstimulate research by teachers themselves,and where the generalisability is foundthrough other teachers and researchersrecognising, and reflecting on, theimplications for their own work. That is tosay, 'proof' by resonance, where an idea or ananalysis of a classroom experience or event'resonates with the experience of the readeror listener, may be more fruitful and morewidely applicable than 'proof' bysignificance testing, at least in relation toteaching.
As mentioned above, there are manyquestions for research and investigation aboutteaching that have been identified in thischapter that can generate valuable insightsinto the teaching of mathematics. Somerecommendations of important areas forresearch and investigation follow, andpursuing the theme of the final section, thereis an overlap of the concerns of the teacher,the researcher and the teacher educator.Th;!se concerns will be emphasised by peopleaccording to their needs and interests as wellas circumstances, rather than their particular
84
task in mathematics education. As theyhave been elaborated in the body of thechapter, they will be merely listed here.:
(1) the influence of the language and thepractices of the mathematics classroomon learning;
(2) ways of rooting the mathematicalactivities of the classroom in theexperiences of the children, whilst at thesame time finding ways of drawing theminto mathematical discourse;
(3) identifying teachers' beliefs and theinfluence of those beliefs on practice.There are methodological problems, suchas the impact of society's demands, theparticular school ethos, mathematicseducation jargon, and there aretheoretical frameworks being developedin which to set such studies;
(4) developing the Inner Teacher, the meta-cognitive functioning that is an essentialpart of learning and mathematising;
(5) the nature of problem-solving andproblem-posing, the influence of teachers'perceptions of problem-solving, theteaching of problem-solving processes andthe relationship between the teacher'srole and student growth as problem-solvers;
(6) classroom-based studies focusing on howteachers define and attempt to realizethe goals that they set, through actionresearch;
(7) investigations on the implications ofalternative learning theories forteaching and teacher education;
(8) the role of the textbook in the socialrelations of the classroom, and therelationships to mathematicalknowledge, and alternative forms of text;
(9) the study of the practice of teachereducation in an analogous way to that ofthe teacher in the classroom.
The mathematics classroom is a confusingbut rich environment. Students bring theirworlds 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
83
manner that empowers rather thandisempowers, and that enables them tocontinue to keep mathematics as their own,not to see it as belonging to us, the teachersand the authority. It can be maintained thatnot being able to arrive at absolute andcertain answers to how that can be done,
makes the science of mathematics teachingjust like all sciences, an open-ended andexciting process of experimentation andtheory-building.
Stephen Lerman taught mathematics in secondary schools in Britain for a number of yearsand for a short time in a kibbutz school in Israel. He was Head of Mathematics in a Londoncomprehensive school for 5 years after which he began research in mathematical educationwhilst 'role-swapping' with his wife and looking after two young daughters. His researchwas on teachers' attitudes to the nature of mathematics, and how that might influencetheir teaching, and he completed his doctorate in 1986. Since then he has pursued hisinterest in theoretical frameworks of mathematical education, working in the areas ofphilosophy of mathematics, equal opportunities, constructivism, and action research. He isSenior Lecturer in Mathematical Education at South Bank Polytechnic, London, where heteaches on initial and in-service teacher education courses and undergraduate mathematicscourses and supervises research students. He is also director of a Curriculum Studies inMathematics 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-convenorof a PME Working Group "Teachers as Researchers in Mathematics Education".
83 REST fiVMABli85
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