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Becoming a Teacher of MathematicsLeone Burton aa Professor of Education (Mathematics and Science) ,University of BirminghamPublished online: 02 Aug 2006.

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Cambridge Journal of Education, Vol. 22, No. 3, 1992 377

Becoming a Teacher of MathematicsLEONE BURTONProfessor of Education (Mathematics and Science), University of Birmingham

INTRODUCTION

This paper draws attention to inconsistencies in educational purposes which liebehind much decision-making in education generally and teacher education inparticular. It highlights an objective view of knowledge which emphasises fragmen-tation and creates experts and novices. This is contrasted with a constructivist viewof knowledge, termed that of the 'polymath', which is more holistic and respectsindividual differences. These two approaches, expert and polymath, are discerniblein the published documentation of the English/Welsh National Curriculum and,depending on how they are viewed, pose either opportunities or constraints onteachers and teacher educators. The central argument is that such confusion makesit possible to identify the gap which exists between the two different knowledgeparadigms, allowing pedagogic measures to be taken to address it in classrooms.However, current initiatives with respect to initial teacher education might becounter-productive and, far from grasping these necessary opportunities teachers,including teacher educators, could be left feeling highly constrained and conse-quently devalued.

The model of medical training is being offered to education as one worthy ofemulation. Most doctors are trained 'in' a number of discrete, subject disciplinessuch as anatomy and physiology and a quick extrapolation might lead some toconclude that this subject-based approach is efficacious. On what is such a conclu-sion based? I believe it springs from a philosophy of knowledge paradigm which haspervaded formal education since the explosion of disciplines led to a competitivedemarcation which replaced the seventeenth century notion of a polymath, a highlyrespected, broadly educated person of varied learning, with the idea of an expert in aclearly defined area.

The expert model relies on a conception of discrete knowledge together with itsown identifiable patterns of knowledge investigation and collection—the methodo-logy of doing science, say, which has, until recently, been reified and unquestionedas the subject has established its credentials. (For criticisms see, for example,Harding & Hintikka, 1983, and Keller, 1985.) Attempts to suggest changes in eithercontent or style meet criticisms rooted in the history of experience of that particulardiscipline. But, as John Elliott (1991) has pointed out, the expert model also invokesdependency by setting up rules for uni-directional communication from expert toclient. The nature and style of communication are embedded within the rules for the

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knowledge specialism and do not permit the inclusion of categories deemed to lieoutside the area nor of perspectives lying outside the 'case lore'. Hence, to return tomedicine, expertness resides in the personage of the consultant, whose patronageand patronisation are demonstrated in assumptions of patient ignorance. Challengeoften evokes attempts at mystification by, for example, using complex terminologymeant to expose ignorance whereas all it exposes is the vulnerability of the 'experts',reliant as they often are on a narrow and constrained knowledge base. This is, in myview, a reflection of the misuse of male power within the expert model with theintention of both intimidating the recipient and protecting himself, a phenomenonnot unknown in mathematics classrooms and to which reference will be made againbelow.

Criticisms of this model in medical education have, however, come from twodifferent perspectives. First, there is the criticism that a specific discipline training,especially one which leads towards a speciality approach to a medical condition, canimpose such blinkers on the practitioners that they are prohibited from making acorrect diagnosis because of their intent in finding reasons for the appeal to their"expertness" (see Abercrombie, 1960). A test-dependent diagnostic methodologycan likewise lead to a reliance on the 'reliability' of tests even when they contradictobservable phenomena or common sense.

A second criticism is that although disease might be discipline specific, thehuman sufferer is not and the doctor ignores the complex interaction of the physical,psychological, social and spiritual in the treatment to the jeopardy of the patient,and of the treatment of the condition (see, for example, Pelletier, 1978). Interest in'the whole person' has led, for example, to research on the impact of stress onphysical functoning which has exposed linkages between physiology and feelings,and consequent disease. Acknowledgement of feelings in the aetiology of disease,and its treatment, has led to medical schools including in their training programmesan exposure of medical students to simulated situations which are personallystressful, such as sharing with a patient a diagnosis of a life-threatening illness,sharing with a member of a family news of a death, and so on.

These criticisms are, it seems to me, embedded in a different paradigm ofknowledge and one which is consistent with the notion of breadth and interconnec-tedness of learning, rather than fragmentation, that is with the polymath, rather thanthe expert. Within this view, the power of knowledge lies in its interrelatedness, itsholistic nature. The polymath is respected precisely because s/he has a broad baseof learning and an ability to link different disciplinary perspectives in order to arriveat a new synthesis. Just as the 'expert' model brings with it methodologies andexpectations of practice which are consistent with its images, so the 'polymath'model embodies methodologies which acknowledge ownership and respect (seeHerbart, 1989) and practices which are built around reflection, interdependentcommunication (see, for example, Edwards & Brunton, 1992) and a challenge tostereotypes (Elliott, 1991, p. 311).

How does this analysis apply to the education of a teacher, and a teacher ofmathematics in particular? The conception of the English/Welsh National Curricu-lum with which teachers, parents and children are presented is fragmented. Each

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Becoming a Teacher of Mathematics 379

subject has its own document, its own set of Attainment Targets, and its own Non-statutory Guidance to its implementation. The syllabus content is the responsibilityof the National Curriculum Council, which, however, has no jurisdiction over theassessment. This is controlled by the Schools Examination and Assessment Council.If the teacher has any autonomy at all in this situation, theoretically it rests in thepedagogical stance which s/he adopts within the classroom. However, the frequentchanges in demands on teachers have induced justifiable fears. Their resultantvulnerability, especially as they are being inducted into a new appraisal systemconcurrently with the introduction of the National Curriculum, leads many to feelmore secure in adopting a textbook, allowing its authors to dictate syllabus andpedagogy, than in taking on the responsibility themselves.

In the particular case of the application of the expert model to subject areaswhich have a gender bias towards the male, the power of the discipline in atechnological society, together with the coercive power of the male expert, iscombined in some classrooms of mathematics, science and computer science, andsome technology classrooms, to convince female pupils of the inadvisability, orunacceptability of their success, or of their inability to succeed. On the other hand,the movement in schools to the incorporation of course work and the relocation ofcontrol in pupils' rather than teachers' hands, that is to working in a polymathcontext, has resulted in considerable changes in attitude and performance by girlstaking GCSE mathematics (see Stobart et ah, 1992). Further evidence of theteaching/learning gap provided by the different paradigmatic conceptions is offeredin the next section.

THE TEACHING/LEARNING GAP

Teachers have received two messages. The curriculum (or rather the syllabus)consists of knowledge and skills defined by 'experts'. You, the teacher, are notexpert enough to be trusted with more than the 'delivery' of the package. The viewof knowledge as being 'deliverable' is, I believe, a commodity view—the knowledgeis manufactured elsewhere, acquisition is tested by another statutory agency, and therole of the teacher is seen as simply to act as the mediator. However, there is ampleevidence from Her Majesty's Inspectorate (HMI), Government reports and com-missions, and teacher educators' and teachers' research and experience which addsup to dissatisfaction with this model as one that provokes, never mind ensures,learning. And, as Ball & Bowe have pointed out, teachers rarely "simply" act asmediators but their "priorities, experience and professional expertise" are "set overand against the structure, content and progression of subject knowledge presented inthe National Curriculum documents" (1992, p. 105). Furthermore, the NationalCurriculum itself presents a conflict with the alternative polymath model, theprocess expectations of which have been written in to expected curricular practice.For example, contrast the mathematics Statements of Attainment Target 2, number,which include: "understand the relationship between place values in whole num-bers"; "use fractions, decimals or percentages as appropriate to describe situations";with the following statements in the Non-statutory Guidance to the Mathematics

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National Curriculum: "each person's 'map' of the network and of the pathwaysconnecting different mathematical ideas is different, thus people understand mathe-matics in different ways" (para 2.1, p. Cl); "the acquisition of knowledge and skillsin mathematics should not, however, be seen as an end in itself" (para 1.2, p. Dl) ;"In life, experiences do not come in separate packages with subject labels. As weexplore the world around us and live our day-to-day lives, mathematical experiencespresent themselves alongside others" (para 1.1, p. F l ) .

Becoming a teacher calls out very different expectations, needs, and perspectivesdepending upon which paradigm is governing the prospective teacher's learning,teaching and experiences in classrooms. From the above it can be seen that thecurricular messages offered to teachers, whether experienced or prospective, areconfused. The power carried by the statutory instrument cannot be ignored byteachers. At the same time, many schools have been experimenting with differentstyles of engaging with the GCSE course work requirements, themselves set into a'polymath', rather than an 'expert' expectation. Schools have, for example, beenexploiting the possibility of a student submitting one piece of course work for twosubject areas, for example mathematics and geography, or mathematics and art. Thepotentiality for all pupils to acquire a sense of ownership of their material and of thisaffecting their motivation towards learning the subjects has not been lost on teachers.

An additional hazard for prospective teachers is the experience which they havehad in school, which is likely to exert considerable influence on their expectations oftheir role in the classroom. If they come from successful experience of didacticlearning, very likely for example in the case of the Postgraduate Certificate ofEducation (PGCE) student who has already gained a degree in mathematics, andtheir teaching practices are within the same mode, any questioning of its efficacy bythe teacher education institution is more likely to fall on deaf ears. Again, it is notthe syllabus which is likely to challenge expectations as the National Curriculumsyllabus is as familiar as the textbook learning which preceded it. The challenge ismore likely to arise from questions about effective learning, that is pedagogicquestions, and from issues of evaluation and reflection which have largely beenunencountered in mathematics classrooms until recently. If the 'expert' paradigm ofknowledge is acceptable, such questions remain outside the realms of considerationof teachers and students since the syllabus, and the assessments, are dictated. Ifteachers and/or students are beginning to move into a 'polymath' stance, verydifficult and challenging issues are encountered almost immediately. These include:

—resources for learning;—style of classroom organisation;—role of the teacher;—role of the pupils; and—the nature of evaluation.

ADDRESSING THE GAP

It has been my argument so far that the United Kingdom National Curriculum is

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Becoming a Teacher of Mathematics 381

experienced by teachers in a confusing way because different messages are carriedby different parts of the documentation and by ministerial statements. Because ofthe confusion, these messages can only be partially consistent with the approach tolearning found in the school, or in the classroom of particular teachers. The kind ofteacher education which emerges from an 'expert' view of knowledge is alsofragmented and based on locating expertise. Currently, the United KingdomGovernment is locating the expertise of teacher education within schools while,simultaneously and again confusedly, proclaiming that schools have failed genera-tions of pupils and 'standards must rise'. This overly simplistic view fails to accountfor the complexities of organising and managing a classroom for learning except byutilising a mode of teaching which is didactically-based, i.e. within the expertparadigm and which has been shown to be dysfunctional for many pupils.

In presenting a different paradigm for understanding learning, including thelearning about teaching by prospective teachers, I have identified five areas, each inits own way difficult and challenging, with which teachers must engage. I amsuggesting that the gap can be addressed but it requires a different view of learningand consequently of teacher education from the one that is currently fashionablewith politicians.

Resources for Learning

In order to move from a textbook and teacher dependent classroom which iscomparatively inactive towards learning which is predicated on pupil enquiry, themajor resource base shifts. Pupils become active learners and, consequently, teach-ers become resource providers and resources themselves. The classroom needs to beas rich a learning environment as can be created and pupils should develop theexpectation that they can represent the mathematical situation into which they areenquiring by using concrete materials, pictures, computers, calculators, symbols, orwhatever is the appropriate medium.

Making this shift is dependent upon a good choice of activity and the provisionof appropriate resources. In this respect, the teacher education institution is central asit is likely to be offering students resource-based activities in the context of adiscussion about their rationale. Carefully choosing these activities so that theyaddress the needs of the pupils in a way that the students encounter aspects oflearning which surprise them is an important part of the construction of such teachereducation courses. One of the most challenging aspects of such courses is to close thegap between the experiences of the students in the institution and the situations theyconfront in the schools. Active learning imposes upon the prospective teacher therequirement to consider and re-consider the nature of their oral interactions withpupils. Do they only 'question'? Are their questions 'closed'? Do they provide answerswhere these can shift the pupils' thinking? Do they accept the responsibility forlearning or return it to the pupils? Do they experiment with different styles ofinteracting to see different outcomes and consider their effects? These, and manyother questions, can be posed to prospective teachers to engage them with thecomplexity and difficulty of the profession that they are proposing to enter.

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Style of Classroom Organisation

The pervasive style of classroom organisation for the teaching of mathematics inprimary and secondary schools has been individualised for a considerable amount oftime. Most schools have chosen to work with individualised booklets which permitdifferentiation of learning and are a considerable aid to discipline and control. Suchbooklets are also quite motivating to pupils, not because of their content so much asbecause of what they permit in terms of engagement. If in a class, pupils areworking at their own booklets, as long as they are apparently working, the pace ofwork, and the learning, are outside the immediate control of the teacher. Thissituation has become an acceptable one for all concerned and forms what ourcolleagues in France refer to as the 'didactic contract'. To interfere with this is torenegotiate this contract and many students attempt to do so without taking thepupils into their confidence. Such efforts almost invariably end up with difficultieswhich are then blamed on the style of organisation rather than on the failure tonegotiate.

It has become increasingly apparent in recent years that this individualisedstyle of learning is not functional. It induces learner dependency on the materialwhich is still presented in the conventional mode of such textbooks. First the pupilsare informed, then a practice example is offered, and then the pupils are told topractice themselves. This fails to allow time to make the question the pupils' own;to set up a representation; to ask questions and to hear the questions of others; andto fit all these into a meaningful structure. The alternative is to allow a chosenactivity to offer the opportunity for learning to a group of pupils who are requiredto define their questions and to construct a way, or ways, of answering them and,finally, to assess what they have found out in the process. Such group-basedactivities are fairly rare in mathematics classrooms, although some materials (see,for example, Kerslake et ah, 1990) do now exist for the primary school and somesecondary mathematics schemes are beginning to engage with the need to providealternative modes of learning. Using materials of this kind does impose upon theteacher/student that they consider the classroom implications of deciding thatmathematics should be learnt in a context, that mathematics can contribute to linksacross the curriculum, that activity-based learning imposes particular learningrequirements, and especially, the effects of asking pupils to work together. Thisapproach, which I have called the approach of the 'polymath' not only challenges thecontent but also the pedagogical style of the curriculum.

The Role of the Teacher

Shifting away from policing the classroom is very acceptable to prospective teacherssince their major fear is the loss of discipline and control. However, such a shift isdependent upon the pupils accepting the new learning style and acknowledging thatan intensive 10 minute discussion with the teacher is of greater value to them thanthe fragments of brief exchanges which researchers report to be the substance ofmost classroom interactions. The teacher, also, has to feel confident that, in

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Becoming a Teacher of Mathematics 383

releasing the control of the learning to the pupils, s/he is not abandoning her/hisprofessional role. The easiest way I have found to convince teachers of this is toraise with them the kinds of information which they observe they can obtain oncethey are focusing on learning, rather than on disciplining. In particular, they notetheir surprise at what the pupils do not know, that they expect them to know, andwhat they do know which they have not been 'taught'. Once this first surprise isencountered, the teacher can shift into an information-gathering stance in whichs/he attempts to discover something about the learning of each pupil in the class. Ofcourse this is only possible over an extended period, say a week, rather than a daywhere the impossibility of obtaining anything other than trivial information, forexample from a test, is recognised. This then has knock-on implications for record-keeping and reporting as the teacher is obtaining far more detailed and incisivelearning information than can be gained in other, more conventional, ways. Teachershave always, of course, engaged in on-the-spot information gathering about pupils.The difference here is that such information is being obtained systematically andrecorded. I suggest to prospective teachers that they identify, say, five pupils astargets for information gathering in one lesson and that they carry a clip-board withthem for this purpose. They report back on the wealth and unexpectedness of whatthey find.

Why is this a 'polymath' rather than an 'expert' approach to teaching? Becausethere is no expectation that the information which is gathered by the teacher is set inconcrete. On the next occasion when s/he observes a particular pupil, differences inactivity, or experiences, or feelings, might throw up very different behaviour. Thisshifts the classroom from being a place where information is exchanged, to a placewhich is more experimental and where certain variables might provoke differentoutcomes. The teacher needs to be aware of these differences, and to what they arelinked, so that s/he can experiment differently next time. Hence the differencebetween student teachers' lesson evaluations recording that 'the lesson went well'and evaluations that attempt to match lesson objectives with outcomes in order tomake judgements about future lessons.

The Role of the Pupils

I am accustomed to asking pupils who have raised their hands with a query to tellme what their question is. This frequently obtains the response: 'Question? Wearen't allowed to have questions. We are only allowed to find answers'. I believe thatwithout personal questions, there is a very low likelihood of learning taking place.However, expectations of finding answers in the absence of personal questions has,for a long time, been integral to many mathematics classrooms operating on a modelof 'delivery'. In my experience, it is comparatively easy to get pupils thinking interms of questions. What is more difficult is to ensure that they accept theirresponsibility to pursue those questions and that learning is dependent upon thatprocess. Such an outcome is a function of the locus of control in the classroom—does it radiate from the teacher, or from the pupil? What do the pupils understandas their classroom responsibilities and those of their teacher and how flexible are

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they? Once again, this is a matter of negotiation between individuals, a negotiationwhich is often avoided by prospective teachers because of the perceived challenge itoffers to discipline. Without such overtly acknowledged negotiation, I believe thatthe implicit lack of respect for the learner takes over and the student can be facedwith potential for classroom confrontation.

The Nature of Evaluation

I have already indicated that learning without reflection is, in my view, inadequate.Mathematics classrooms are particularly prone to rushing on through classroomcontent without allowing any time to stop and consider such questions as:

What do I know, now, that I did not know before?What strategies were effective in my learning that?Are there some things that I still only partially understand, or don'tunderstand, that I want to mark for future learning?What surprises did I have?What disappointments did I have?Have I learnt 'what' or 'how to'?Could I ask another pupil a question which would challenge them in thesame direction as I have been working?

Many different reflective styles are being used in mathematics classrooms aroundthe world to encourage pupils not only to learn mathematics, but also to learn abouttheir learning of mathematics. I have encouraged primary teachers to ask theirpupils, in the last 10 minutes of the day, to close their eyes and think of one thingthat they have learnt that day and to tell that one thing to a partner and listen totheirs. Groups of children who have been engaged on an extended piece of work canbe asked to list their outstanding questions, as well as to identify the mathematicsthat came out of their project. One use of the National Curriculum is to identifyStatements of Attainment that are the focus of a sequence of activities, with therequest to pupils that they have a discussion with the teacher when they think thatthey have achieved one of the statements. This imposes a further discipline on theteacher—to identify the mathematical boundaries around the Statement of Attain-ment in order to be able to challenge the pupil not only with validating, but alsofalsifying, questions. Some teachers are experimenting with mathematical diaries inwhich pupils record their responses to the learning upon which they have engagedand which from time to time they share with their teachers. On a teacher educationcourse, I asked students to keep a diary over a year and, at the end of the year, touse a highlighter pen to chart their personal learning over the course. All of thesetechniques, and many more, are representative of a view of learning which ispersonal, as well as social, and in the control of the learner rather than the teacher.

However, it should have become clear that the shift which is being outlined isnot easy. Nor can it be made in an unsupportive context and many schools arecurrently unsupportive of their own staff, never mind their student teachers. "In allthe case study schools the implementation of the National Curriculum took its toll

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Becoming a Teacher of Mathematics 385

in terms of morale, commitment and energy" (Ball & Bowe, 1992, p. 106). Many ofour student teachers are reporting the surprise of the teachers in their practiceschools that anyone should want to enter the teaching profession at the current time.Many colleagues in teacher education are recognising that the motivation for doingso is currently more to do with threatened graduate unemployment than a highcommitment to the learning of young people. Within the past few days a pupil inschool has told me that offering pupils mathematical "challenges" (the pupil'sword) might go some way towards helping them to "put up with the other stuffand, in another school, a student teacher told me how much "easier" it is to teachmathematics didactically without worrying about the impact of the lesson on thelearning of the pupils. This is the dilemma which I have been addressing. The pupilsrespond positively to a 'polymath' approach, one which challenges them to locatetheir learning, make links with what they know and expand that base, and whichtrusts them to be in control of that learning. The teachers, and student teachers, aremore likely to use a teaching approach which reduces the stress on them, emanatesfrom an 'expert' view, and is more to do with discipline and control than it is withlearning. The one contradicts the other and possible room for negotiation towardspartial accommodation is reduced by the pressures and stresses currently permeatingthe education system.

Effective teacher education has to be more concerned with ways of addressingthe gap, than with the socialising of new teachers into current practices. Currentinitiatives in teacher education which require a PGCE student to spend 24 weeksout of the 36 week course in school and which are likely to be extended to include asimilar requirement of other teacher education courses, do not take into account themany pressures under which schools are currently operating, the area of profession-alism which teachers claim, and reject (for example, most teachers with whom Ihave spoken claim no knowledge of teacher education), and the particularities whichare a natural part of a teacher educator's professional life, such as research,curriculum development, and familiarity with state-of-the-art literature.

CONCLUSION

All educational systems claim to be in existence in order to ensure growth anddevelopment. However, it is important to remember that we have little evidence oftheir success in so doing. One explanation for this that I have offered is that there isconfusion within the system between two conflicting views on learning, one which Ihave labelled the 'expert' view, and the other which I have called the 'polymath'.Although I have painted these as two extremes, in most classrooms the conflictbetween them becomes apparent fairly quickly as teachers and pupils struggle toidentify the purposes, relationships, links which do, or do not exist, as the contextfor learning. In a resource-restricted era, using the 'expert' model appears to be anefficient and cheaper way of coping with the complexity. However, I believe thatalthough it might appear to cope, the results will not be more effective learning.

To offer learning which empowers individuals in their lives, giving them alanguage with which to think about their experiences, knowledge and skills to apply

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to their life-style needs, and the qualifications to continue to the next stage ofeducation is, in my view, the only reason for compulsory schooling. This requirescoming to terms with the current situation, as well as having objectives, and ideals,for the future. It is unreasonable to expect anyone to have thought through all ofthese aspects prior to commencing on a teacher education course, or even once in apost in school. Learning is about living, and not about completing one particularcourse.

In these circumstances, it seems to me that becoming a teacher is aboutengaging with the complexity of learning; not with a view to working withinsimplistic, or even simple, solutions but with a recognition of just how little isknown about how we learn, and how experimental is each learning situation. Inthese circumstances, the best that a teacher can do is to offer pupils the opportunityto embark upon their own learning in such a way that they feel more competent andmore confident at the end of each experience. Setting up the conditions in teachereducation courses to allow for small experiences within which this happens so thatthe prospective teacher begins to feel, and respect, the power of being a teacher isthe first step along the way. Such small experiments have always been part ofeffective institution-based teacher education courses. They are one way of linkingthe research and curriculum development interest of staff with their teachingresponsibilities on teacher education courses. This is not a function of resources. Inmy view, it is a function of the philosophy within which teaching and learning isconceptualised.

Correspondence: Leone Burton, School of Education, University of Birmingham,PO Box 363, Birmingham B15 2TT, United Kingdom.

REFERENCES

ABERCROMBIE, M.J.L. (1960) The Anatomy of Judgment (London, Hutchinson).BALL, S.J. & BOWE, R. (1992) Subject departments and the 'implementation' of National Curriculum

policy: an overview of the issues, Journal of Curriculum Studies, 24(2), pp. 97-115.EDWARDS, A. & BRUNTON, D. (1992) Supporting reflection in teachers' learning, in: J. CALDERHEAD

(Ed.) (1992) Conceptualising Reflection in Teacher Development (London, Falmer Press).ELLIOTT, J. (1991) A model of professionalism and its implications for teacher education, British

Educational Research Journal, 17(4), pp. 309-318.HARDING, S. & HINTIKKA, M. (1983) Discovering Reality (London, Reidel).HERBART, C. (1989) Talking of Silence (Brighton, Falmer Press).KELLER, E.F. (1985) Reflections on Gender and Science (New Haven, CT, Yale University Press).KERSLAKE, D. et al. (1990) HBJ Mathematics (London, Harcourt Brace Jovanovich).PELLETIER, K.R. (1978) Mind as Healer; Mind as Slayer (London, Alan & Unwin).STOBART, G., ELWOOD, J. & QUINLAN, M. (1992) Gender bias in examinations: how equal are the

opportunities? British Educational Research Journal, 18(3), pp. 261-276.

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