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Implications of research formathematics teacher educationA. J. Bishop aa Department of Education , University of Cambridge , UKPublished online: 07 Jul 2006.

To cite this article: A. J. Bishop (1982) Implications of research for mathematics teachereducation, Journal of Education for Teaching: International research and pedagogy, 8:2,118-135, DOI: 10.1080/0260747820080202

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Implications of research formathematics teacher education

A. J. BISHOPDepartment of Education, University of Cambridge, UK

This paper reviews the research areas considered to be significant for mathematicsteacher education. The research is reviewed in several sections: construingchildren's thinking, affective ideas, teaching methods, the mathematics class-room, and curriculum analyses. In the final section ways of using ideas from thisresearch are discussed. The review seeks to reflect recent research developmentsand to indicate promising avenues for future study.

INTRODUCTION

It was not so long ago that the only source of ideas for teacher education was'advice' from experienced teachers. An excellent example of this type of sourceis a book called Problems in Classroom Method by Waples (1927), which containsa collection of over 140 incidents-plus-advice. Problems like 'How to handle thepupil who parodies the teacher's questions' are each followed by a paragraphor two of condensed wisdom from experienced teachers, supervisors and'experts'. As a social and historical document it is magnificent, but as an aid toteacher education it leaves much to be desired, not only in terms of what itmakes available but how it makes it available. How any aspiring, or practising,teacher was supposed to digest such advice is anyone's guess - or perhaps itwas a teacher's reference book to be kept on the classroom shelf and consultedin moments of crisis!

In our present era we are in a much more fortunate position. Not only hasresearch made great advances in both focus and methodology, but also weknow much more about the process of teacher education. The following reviewattempts to delineate those areas of research that, in my view, have significantimplications for mathematics teacher education.

The review will not be exhaustive nor can it be objective. As a teachereducator concerned primarily with initial teacher training, my own bias istowards the teacher as a learner, learning from the children, from classrooms,from materials, from us, about the teacher's professional task and how to carryit out. My personal orientation is towards constructive alternativism (Kelly,1955) and I therefore prefer to view the contributions of educational researchnot as 'results', which should be 'applied'. I look for helpful constructs thathave been generated by researchers, for a sensitization towards problems and

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May 1982 Research and mathematics teacher educat on 119

for ways of generating experiences that will help the student teache .viden hispossibilities for action.

Research is not the only source for ideas for teacher education but it israpidly becoming a highly significant source. In particular there has been agrowth in research that focuses on mathematics education (see JET 8:1), and itseemed appropriate for this paper to bring to a wider audience ideas fromrecent research in our field. If, therefore, there appears to be little reference tomore familiar educational research work this is due not necessarily to ignor-ance but to choice. Methods work in teacher-training courses has traditionallybeen construed, particularly by those outside it, as the application (or transfer)of ideas from general educational research and theory to the problems ofmathematics teaching. Indeed there are those who feel that methods work isnot a profitable area for research. If this paper can help to dispel that myth thenthe author will be partially satisfied, but the real point is that in order to preparebetter mathematics teachers it is necessary that research into mathematicseducation be encouraged and stimulated. Promising avenues will therefore beindicated where appropriate.

CONSTRUING CHILDREN'S THINKING

Traditionally this has been the most commonly researched area in mathematicseducation - the learner coming to grips with new mathematical ideas and withusing his ideas in the solution of problems - and for this review, the researchwill be considered in two parts: work that considers children's meanings andunderstandings, and work on processes and abilities.

Ausubel (1968) made the very helpful distinctions between 'rote' and'meaningful' learning, and 'discovery' and 'reception' learning, and Shulman(1970) first brought this analysis to the mathematics-education community'sattention. The idea of meaningful learning is a significant and sensitive one formathematics education because of the fact that the abstractness of mathe-matical ideas can make them potentially meaningless. The archetypal 'bad'mathematics lesson consists of the children moving around meaninglesssymbols in a ritualistic performance of an apparently arbitrary rule laid downby the teacher.

The 'meaningful' construct is not new however, and has been of interestsince the early research of Piaget (at least) and the general picture of thelearning child trying to make sense of his world is a fairly well-accepted view.Many other researchers also use children's errors as their data base but a veryimportant change of perspective for student teachers occurs when these errorsare not viewed as 'wrong' ideas, which must be 'corrected', but as sources ofinformation concerning the child's interpretation and understanding of themathematical ideas. For example, Kent (1979) analyses meanings that underpinsome children's errors and clearly shows the need to try and uncover the child's

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120 Journal of Education for Teaching Vol.8 No.2

view. Again this is a particularly important point in mathematics where somuch emphasis has traditionally been placed on right answers and correctmethods.

Different kinds of understandings and meanings have come to light fromthe developmental work of Piaget and Inhelder (1969) and from Bruner (1966),and also from more recent work by Mellin-Olsen (1976) and Skemp (1976) whodistinguish relational understanding from instrumental understanding. Otherwork by Bell (1976) uses children's explanations of mathematical ideas to revealtheir meanings.

The discovery-reception dimension concerns the role the learner plays ingaining new information. The principal power of discovery, it is argued, is thatthe learner is using his own understanding to guide his search therebyensuring that the discovery will result in meaningful learning. However, aconcern to save time and to avoid the pursuit of 'irrelevant' discoveries oftenresults in extremely guided discovery experiences being offered by teachers. Itmight be more profitable to consider this as reception learning where the onusis firmly on the teacher to offer the handholds of meaning that will enable thelearner to grasp the new ideas. Of significance once again, then, is the teacher'sknowledge of the learner's understanding in order to ensure that the newmaterial can be successfully assimilated or accommodated. Recent work onintuitions by Fischbein, Tirosh and Hess (1979) and Tall and Vinner (1981)support the idea of the controlling effects of the child's previous learning. Asanother example the Concepts in Secondary Mathematics and Science (CSMS)team (1981) found how reluctant children are to accept multiplication byfractions when they have developed a strong intuition, or understanding, that'multiplication makes things bigger'.

What of the child's use of mathematical ideas? Traditionally factor-analyticresearch has furnished us with our ideas concerning mathematical ability andresearchers such as Wrigley (1958) and Hamza (1952) have contributed inter-pretations. Recently, however, the work of Krutetskii (1976), translated fromRussian, has given a tremendous boost to the study of mathematical abilitiesand clinical methodology. The shift in construct from 'ability' to 'abilities' issignificant and seems to convey a change from something that differentiatesbetween pupils (high and low ability) to something that they all share in someform, from something which may be inherent, to something which is poten-tially developable in all pupils. Some of Krutetskii's 'abilities' are a striving forclarity, simplicity, economy and rationality of solutions, the ability to curtailthe process of mathematical reasoning, and the ability to switch from a direct toa reverse train of thought.

Another contribution to our construction of children's mathematicalactivity comes from those who study 'problem solving'. Much of the work isderived from Polya's (1945) seminal writing, and appeals to those who feel thatat the heart of mathematical activity is the ability to solve problems, usually

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May 1982 Research and mathematics teacher education 121

defined as a task for which the learner can see no immediately obvious orstandard algorithmic solution. Burton's (1980) work continues the Polyatradition and focuses on heuristics and their uses, while in Australia Newman(1977) has stimulated much development in the analysis of the errors childrenmake while trying to solve problems, revealing once again the child's inter-pretation of the situation.

This type of research is not just of interest in the problem-solving orstrategy context but is also revealing concerning children's performance ofwhat are thought of as routine skills or algorithms. For example, in the recentCSMS research (1981), which used both group and individual testing, muchwas revealed about what Booth (1981) calls 'child-methods' - those techniqueschildren use in what are supposedly routine tasks like subtraction. The tasksmay be routine but the individual processes used may differ markedly, asothers like Plunkett (1979) and Mclntosh (1978) have also found.

This also relates to work on individual differences. Hadamard's (1945)classic work created much interest in the relative uses of visual imagery byproblem solvers, an idea extended by Krutetskii who distinguished 'analytic'and 'geometric' types among his subjects. More recent analysis by Bishop(1981) and influences from sex-difference research (Fennema and Sherman,1977) seems likely to continue to focus research on the processes children use inoperating mathematically.

The mathematics-education community is being forced to consider itsown criteria for judgement as a result of this type of research. Traditional viewsof right answers and correct methods, which have fed student teachers' per-ceptions of children's learning, are being replaced by children's meanings,understandings, interpretations, processes and 'child-methods', which canoffer much more for the interpretation of classroom behaviour.

AFFECTIVE IDEAS

Traditional attitude research told us very little about children's feelingsregarding mathematics, apart from supporting the general 'folk-lore' that it's ahard subject and children on the whole don't like it. Paralleling the develop-ments outlined in the previous section, there has recently been an increasinginterest in understanding more sympathetically the child's feelings.

Duckworth and Entwistle (1974) using a Kelly-type repertory grid tech-nique, modified for a group approach, informed us a little more aboutchildren's perceptions of mathematics as a school subject, showing thatchildren perceived it not only as a difficult subject but also as an importantsubject to study. The data from various large-scale surveys (for exampleAssessment of Performance Unit (APU), 1980) confirm the children's feelingsabout the difficulty of the subject. There are two corollaries to this point.Firstly, when dealing with the less-able children (who, perhaps, are better

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considered as slower learners given the often limited range of teachingmethods used in most schools), it is crucial not to make the subject appear tooeasy, otherwise it will be rejected as not being 'proper' mathematics.Mathematics does apparently need to be seen as difficult for its own credibility.

The other corollary concerns the anxiety surrounding the problem oflearning a subject perceived simultaneously as difficult yet important. In theUSA it is called 'mathophobia' and there are many attempts to analyse it andcounteract it (see, for example, Resek and Rupley, 1980). In the UK the work ofHoyles (1975) illustrates the problems faced by children coming to terms withtheir perceived inability, and shows furthermore the crucial role of the teacherin the classroom. As will be seen in a later section, the 'atmosphere' of themathematics classroom created by the teacher can have a significant effect onthe pupils, and in the context of this section the analogy of the dentist's waiting-room springs to mind! Ray's (1975) study of the attitudes of student teachers totheir own school experience shows clearly that they brought favourableattitudes towards mathematics as a school subject, but not towards theteaching of it.

To counterbalance this slightly pessimistic perception of mathematicsteaching, reference should also be made to attempts to understand more whatdoes interest pupils in the subject (see, for example, Backhouse, 1980 andPreston, 1973) and to emphasize the increasing importance of studying thechild's understanding of the mathematical situation, where distinctionsbetween cognitive and affective aspects often seem extremely artificial.'Meaning' and 'understanding' are increasingly being perceived as socialphenomena, a construction that in part shifts the attention from the learner tothe classroom, the teacher and the teaching.

TEACHING METHODS

The research which focuses more on aspects of teaching is best considered intwo sections. In this first section the more traditional type of research onteaching methods will be discussed while the next section will review morerecent research developments, which take the classroom as the 'research site'.

Traditional teaching-method research in fact locates itself within a'development-and-research' sequence, in contrast to the often erroneouslydescribed Research and Development picture of traditional educationalresearch. It is usually stimulated by the creation of new methods and the desireto evaluate their effectiveness. The two most significant developments inmathematics teaching methods have concerned the mode-of-presentationaspect and discovery sequences, and these have therefore provided the foci formost teaching-method research.

The mathematics teacher today has a wide range of modes available withwhich to engage his pupils with mathematical ideas, from structural

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apparatus and films, to games and workcards. Research has studied manyaspects and there has been some concern over the quality and value of theseexperiments. Their value for the student teacher, however, lies not in anypotential answer to the question 'What is the best way to teach?' but rather inassisting in the elaboration and analysis of the various components of teachingmethods. Biggs (1967) and Williams (1971) for example have clarified manyaspects of the use of structural apparatus, and the work of Land and Bishop(1966-9) contributed significantly to an understanding of the roles of diagramsin mathematics teaching. Dienes (1963) for many years has explored thepotential of games, with their strategic and 'local' logical rules, while Janvier(1981) analyses the use of real-life situations as motivating contexts for mathe-matics learning. The role of language is of course critical and Austin andHowson's (1979) review of that research indicates the variety of aspects thatneed to be considered.

However there does seem to be more research interest in methods thatstimulate the visual imagery of the learner. Bishop (1973) showed how structuralapparatus can affect spatial ability, while Kent and Hedger (1980) demon-strated the power of visual imagery and how this can be exploited in themathematics classroom. Marriott's research method (1978) modernized thetraditional teaching-method research paradigm and looked at the types oferrors rather than global scores occurring after some experimental teaching. Forexample, after using a 'visual' fraction kit he found more pupils making errorsindicating that they were visualizing rather than computing, for example 1/4 +1/8 = 1/3.

The development of teaching methods that facilitate pupils' discovery hasled to a great deal of research into the whole aspect of sequence in mathematicsteaching. Many conflicting findings emerged from the early literature butAnthony's (1973) analysis seems most helpful in reconciling the findings -showing that the power of the discovery sequence is only realized when thelearner actually achieves the discovery. This is difficult to arrange in large-group teaching but is relatively easily built into individualized teachingschemes using workcards or programmed material.

Ausubel's (1968) interest in reception learning caused him to considermethods of achieving meaningful learning that were not based on the conceptof discovery. One result has been the idea of advance organizers - 'introductorymaterial at a higher level of abstraction, generality, and inclusiveness than thelearning passage itself (Ausubel, 1978; p. 252). The necessary corollary forachieving meaningful learning is ensuring that the advance organizer is itselfrelatable to ideas that the learners already possess. Lesh (1976) explored thevalue of organizers in the context of mathematics teaching and indeed showedthat an appropriate organizer presented in advance of the material did havesignificance over one presented after the material.

However, as every teacher knows, and as student teachers soon discover,

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children are individuals, and a method which works for some won't neces-sarily work for others. Hence there has been a growth in research that looks atindividualized teaching and Miller's (1976) review of much of this researchshowed the variety of concerns explored. However, consideration of researchthat studies individualized teaching is not the same as considering researchthat contributes implications for individualizing teaching, and this reviewerfinds more of interest in the latter category. One approach, called Aptitude-Treatment-Interaction (ATI) research, studies the effects methods have onparticular learner characteristics. Whilst interactions have proved difficult tolocate when aptitudes have been considered (see, for example, Eastman andCarry, 1975), there seems to be more promise with learning styles and learnerpersonality (Radatz, 1979). Trown and Leith (1975) have also located inter-actions between discovery and expository sequences, and pupil anxiety.

Another approach to the individualizing of teaching comes from thediagnostic and remediation school, which tends to pursue a pseudo-medicaland clinical paradigm. Engelhardt (1976) has produced an extensive review ofthis literature, which is growing rapidly. The essential difference betweenthese two approaches is that whereas ATI considers pupil characteristics thatare taken as 'given', the remediation model is all about change and 'correction'.Much error research feeds this school and one can see the importance of thepoint made earlier that errors offer the teacher insights into the pupil'sunderstanding and meaning. It may indeed be useful to consider how tocorrect that error but the error may only be an overt symptom of a deeper seatedmisunderstanding, which may prove much more difficult to modify.

The danger of the medical analogy is that it encourages the teacher toconsider the learner as a passive recipient of treatment and remediation. Case(1975) argued convincingly for a method that creates 'cognitive conflict' in thelearner thereby creating favourable conditions for the establishing of moreappropriate meanings. Additionally, a pupil's error or weakness may in fact bea symptom of a very limited teaching methodology. For instance, learners whoare subjected to an unremitting diet of orally presented mathematical rules thatthey must routinely follow could well be found to be lacking in their ability touse visual imagery in their solution of mathematical problems. A change inmethods could possibly reveal strengths which had hitherto lain dormant.

Also, teaching that focuses solely on abstract symbol manipulation, at theexpense of understanding and meaning, may well be responsible for variousresearch findings which show weakness in solving non-routine tasks andproblems which demand the application of taught material (APU, 1980; Rees,1973; CSMS, 1981).

THE MATHEMATICS CLASSROOM

The value of the research in the previous section lies with its ideas concerningthe engagement of the child with mathematical ideas. The fact that experi-

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merits are often done with groups of pupils doesn't mean that notice is taken ofthe interactions within that grouping. Indeed, in some strange way theclassroom of the teaching-method experiment is an extension of a learninglaboratory. It is no accident that much of the research is conceptualized withinindividualized learning settings and various instructional technology 'solu-tions', where the teacher is considered as just another 'instructional deliverysystem' (Merrill and Wood, 1974; p. 2).

However a real classroom is a social arena more than anything else and anyteacher knows that he ignores that fact at his peril. Fortunately more re-searchers are choosing the classroom as their research 'site' and ways ofconceptualizing classroom phenomena are increasing. Language has tradi-tionally been of interest and researchers have studied aspects from the level ofquestions being used (Friedman, 1976) to the whole logic of classroom dis-course (Smith and Meux, 1970), an area which could have tremendous value formathematics education but which has not yet been exploited systematically.Good and Grouws (1979), also in the USA, used process-product techniques tointerpret the 'significant' behaviours of 'successful' teachers. They alsoachieved success in training these behaviours in their student teachers.

The mathematics classroom can be dominated by the power of the teacher,perhaps because of the abstract and possibly 'meaningless' nature of thesubject, and this point emerges from the work of Yates (1978) who studied fourclassrooms in depth. The advent of workcards and individualized schemes has,however, changed dramatically the teacher's role in the classroom and to a largeextent has undermined the teacher's power and authority. Morgan (1977)sensitively documented the effect one scheme had on the teachers and thepupils.

Being in an arena, the classroom's participants exist in an atmosphere ofmutual evaluation, with the criteria for judgement of behaviour being derivednot only from the perceived roles but also from the shared 'community ofknowledge' as Nash (1973) put it. The mathematics classroom is a potentiallyhostile environment because of the relatively easy evaluation by the teacherand the other pupils of any pupil's contribution or performance. This fact couldlie behind many pupils' feelings of anxiety about mathematics and adds yetanother dimension to the way a teacher interprets, and then deals with, apupil's error. Relationships, as in any classroom, are crucial and MacPherson's(1973) account of his deliberate attempts to improve his relationships with hispupils can be of great interest to student teachers.

Other classroom research indicates that an 'official' mathematics curri-culum doesn't actually exist - what one has is the teacher's mediated version ofit, the pupils' perception of it and a negotiated reality. Yates (1978), Bauersfeld(1980) and Kemme (1981) all offer us excellent examples from the classroom ofsuch curricular negotiations. As Bauersfeld said 'The student's reconstructionof mathematical meaning is a construction via social negotiation about what is

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meant and about which performance of meaning gets the teacher's (or thepeers') sanction' (p. 35).

This research development is potentially the most promising from thepoint of view of teacher education. The more a student teacher understands ofthe dynamics of the mathematics classroom the better able he is to survive, andto learn how to affect those dynamics.

CURRICULUM ANALYSES

The curriculum has been analysed at various levels, all of which can contributeto the student teacher's understanding. At one level the APU (1980) distin-guishes concepts, skills, applications and problem-solving, largely from theviewpoint of testing, but with an implication that there are likely to bedifferences in teaching methods relating to these different components of thecurriculum. At another level Gagne's (1968) work on task analysis can suggestsome worthwhile exercises for student teachers. More recently the CSMS team(1981) has explored the stage notion in secondary-school mathematics bystudying the difficulty levels of tasks within specified curriculum areas.

However, another approach to the curriculum was suggested towards theend of the previous section.

Curriculum in the classroom is indeed a negotiated reality and studentteachers need to clarify their own position in order to engage in that nego-tiation. Mathematics is not a subject 'out there' to be merely transmitted. It is ahuman enterprise shaped by cultural, historical and political forces. Curri-culum development in mathematics has taken many forms (see Howson, 1979,for an excellent analysis) and the student teacher needs to recognize the'political' nature of that exercise. Examinations, textbooks, syllabuses, work-card schemes are the overt products of a complex process, as are nationalassessment programmes such as that run by the APU (1980). In a decentralizedsystem like ours an understanding of the teacher's role in the process is critical.

Part of the growth of that understanding can come from analyses ofsystems different from ours in some way, and it is pleasing to note thathistorical analyses of mathematics education are increasing (see, for example,Howson, 1973 and Brock and Price, 1980). So too are cultural analyses. Gay andCole (1967) studied the problem of an African culture confronted with modernmathematics and Lancy (1978) updated this issue by analysing the work of theIndigenous Mathematics Project in Papua New Guinea. Harris (1980) reportedon work with Aboriginal tribal communities while Swetz (1978) considered thespecial features that mathematics education appears to have in socialistcountries, thereby focusing clearly on the political context. Research such asthis can only have a beneficial effect on teacher education, enabling us torecognize more clearly the nature of our own system and curriculum.

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USING IDEAS FROM RESEARCH

As a concluding section to this review it is worth considering how the ideasthat have emerged from the research can be used in mathematics teachereducation. A number of different aspects can be identified.

Supplying constructs

The first use is to help student teachers understand, and construe meaning in,the various phenomena they must deal with: the classroom, the children, thesubject, the textbooks, tests and examinations, concrete materials. Thephenomena can be introduced to the students 'in the flesh' so to speak, or viasound tapes, videotapes, transcripts of lessons, and actual materials. Thematerials of the Mathematics Teacher Education Project (MTEP) (Wain andWoodrow, 1980) include many suitable examples. Discussion, interpretationand analysis are the appropriate modes of teaching within which the tutor canoffer alternative constructs from research to widen perspectives and to contrastvalue positions.

Perhaps the most obvious use of this technique is with data aboutchildren's mathematical thinking and abilities. Tapes of discussions bychildren solving problems in groups, completed homeworks, answers to tests,and essays about mathematics, are all excellent materials with which to beginclarifying what mathematical activity is, what sorts of meanings children giveto mathematical ideas, and how they think about the different procedures androutines. Into such 'clarification' can be injected constructs from the psycho-logical and other research literatures.

Constructs are provoked by the need to describe contrasts, so as much aspossible should be done to present the student teacher, and indeed theexperienced teacher, with contrasts. For example, one very valuable exercise isto arrange the students in groups of five with a mathematical topic chosen foreach group. Each member of the group then has a school textbook that dealswith that topic. The students must compare books and describe the differencesbetween them in terms of how they treat the topic and then report theirfindings to the other groups. Many ideas, and constructs, can emerge fromsuch an exercise: for example, language levels, use of examples, illustrations,sequences of ideas, activities generated, logic of explanations, and so on. Thestudents can also begin to clarify their relationship with the textbook, or moregenerally with textual material.

Other examples of stimulating 'contrast' material are photographs ofdifferent classroom layouts, the students' own subtraction and long-divisionalgorithms, different types of games, CSE (Certificate of Secondary Education),'O'-level and 'A'-level papers (different grades of the UK General Certificate of

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Education), videotapes of different types of lessons, and school mathematicssyllabuses.

Less obvious than these examples, perhaps, are those of mathematicalsystems other than our own, as reported in Gay and Cole (1967), Lancy (1978)and Harris (1980). All of these studies have data on the language/mathematicsinterface, bringing a perspective on the subject which is often very new.Lancy's project, furthermore, collected much material on different classifi-cation and counting systems, and it is extremely stimulating for studentteachers to have to accommodate their schema to the idea of the existence ofover 200 different counting systems (and still increasing!). On the teachingside, contrasting examples from Swetz (1978) on Socialist MathematicsEducation can be found that point up many features of our own mathematics-teaching system, such as departmental procedures for grouping children, thedevelopment of values of discovery and independent thought, and the decen-tralization of decision making. A secondary benefit lies in showing that con-structs need to be analysed, if they are to be understood - for example,Krutetskii (1976) demonstrated the socialists' approach to the whole area of'abilities' and an examination of his analyses can be extremely beneficial, forinstance, in the general setting of a discussion about mixed-ability teaching inmathematics.

Sensitization towards problem areas

Another use of ideas from research emphasizes the more personal, active, andprocess aspects of knowledge gaining, rather than the more impersonal,passive, and 'learning other people's constructs' flavour of the previoussection. Here, two types of teaching approaches can be used with eitherstudent teachers or teachers on in-service courses.

First, the student can become, for a time, a researcher and can undertake alimited piece of empirical enquiry. This enquiry can be entirely created,structured and performed by the student alone, or can be guided in many waysby the tutor. The research literature abounds with problems that are appro-priate for empirical enquiry, ranging from children's learning difficulties totheir problem-solving strategies, or from comparisons of teaching approachesto analyses of verbal interactions in the classroom. Many research instrumentscan be exploited, like written tests, clinical interview protocols, observationschedules, and attitude questionnaires. Different researcher 'stances' can beadopted, such as experimental teacher, interviewer, observer, recorder.

Perhaps the important points to focus on, however, are the choice of asignificant problem area for the student and the choice of a research method-ology and stance that does not 'distance' the student too far from the problem -for example, a clinical interview might be preferable to a sophisticated psycho-metric technique such as factor analysis. The aim is not to further the general

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body of knowledge of the field but to engage the student in the active enquiryof a significant educational problem in mathematics.

Such an enquiry can form an assessable project in the course, if desired, inwhich case the write-up and analysis would be emphasized as well, or it couldjust be an 'experience' for each student, perhaps focusing on a commonproblem in a different (research) way, to be discussed and 'debriefed' in agroup, oral, mode. For example, if a topic like 'sex-role stereotyping inmathematics teaching' was chosen for a student group, then individualstudents could test children on a written-attitude instrument, interviewindividual children using photographs as stimuli, interview teachers aboutdifferences between the boys and girls they teach, observe and record class-room interactions between pupils or between pupils and the teacher, observenon-verbal behavioural patterns in class, and experiment with the use ofreverse sex-role mathematics problems. After a time for reflection and analysisthe group discussion of different findings can be a most stimulating experiencefor all concerned (including of course the tutor).

The value of individual research activity has been appreciated for manyyears and by many people, and it is clear that both initial and in-service teachereducation can benefit from an injection of individual research on the part of thestudents. A good example of this value for an experienced teacher is shown byMacPherson (1973) in a revealing investigation of his own effect on relation-ships in his classes. The impact of such personally gained knowledge cannot beoverestimated, and goes beyond the bounds of that immediate situation. Theteacher-researcher is sensitized to a range of 'peripheral' problems and to thepersonal-value dimension of other people's constructs and theories.

A second way of sensitizing people to problems is to engage them insimulations or role-play activities. Teaching is a social activity and isinfluenced by social and interpersonal forces. It is as much to do with feelings,perceptions and emotions as it is with understandings, constructs and theory,and research is making us increasingly aware of these aspects. Simulations canease the student teacher into these pressures and role-play can allow for theexperimentation with different roles which 'reality' does not always permit.

Two examples from the author's own work will illustrate the potential ofthis type of activity. In the first, a group of thirty (plus) students are involved ina simulated Parent-Teacher Association (PTA) meeting. The tutor plays thepresiding headmaster, there 'merely' to chair the meeting, and the students aredivided into groups -maths teachers, science and other 'user' subject teachers,parents, employers and technical college and polytechnic lecturers. The subjectfor the meeting is the school's 11-16 mathematics curriculum, and possiblechanges to it. The students are given time to discuss and clarify their positions,in groups; the PTA meeting lasts for about forty-five minutes, and then thewhole exercise is debriefed.

On one occasion this simulation has been carried out using not only

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students but also their own teaching-practice supervising teachers. Theincrease in the 'political' character of the meeting that this addition broughtforth had dramatic effects on some of the students, who undoubtedly had theirperceptions of the teacher's role altered by the experience!

A different type of role play focuses more on children in classrooms. Forthis eight students are chosen to be a 'class' and are each given a 'thumb-nail'sketch of a child who they are asked to play. The other students observeparticular 'children' and after the 'lesson' (taught by the tutor) these observers'perceptions are compared with the thumb-nail sketches. Also the 'children' areencouraged to describe their experiences in class, where they would like to sit,who to sit next to, who they would not like to sit next to, what sort of teachingand teacher they prefer, and generally to describe the feelings created by thepresence of the other 'children' and the 'teacher'. Ideally all the students shouldexperience this role play, though time is always a problem, because playing therole of someone else, in this case a child, has a heightening and sensitizingeffect on the student which no amount of discussion and reading can reallyemulate.

Increasing the student's range of action

Many students initially appear to believe that the teacher's job is essentially toexplain the subject to others who don't yet understand it. An importantcomponent of their education is therefore to widen their range of action frommerely 'explaining', and research like that by Good and Grouws (1979) canpoint to many other teacher behaviours that need to be developed. Forexample, the posing of suitable questions is one very necessary skill not easilylearnt, possibly because the many roles and functions of teacher's questionsare not well understood by the students. The complementary skill is theteacher's answering ability, revealed when dealing with pupils' questions.'What would you do if I said I didn't know?' is an example of an answeringtechnique that can often widen discussion of this aspect. Research can alsooffer much in the general area of diagnosis by offering examples of diagnosticinterviews for analysis and by guiding the students to particular kinds ofproblems.

Whilst those skills can be practised with individuals, some require a groupcontext. Leading and controlling discussions is a much needed skill, as ispractising the teacher's role in negotiating mathematical meaning withlearners. The teacher as a model may well play a crucial part in successfulnegotiation and students should learn to verbalize and generally to externalizetheir thinking - particularly the more usually 'hidden' aspects like guessing,estimating, hunch-playing, strategy-planning, monitoring, judging andevaluating. As much use as possible should therefore be made of group workwith the students and, again, the MTEP source-book (Wain and Woodrow,

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1980) contains many examples of suitable group activities.Another approach to helping the students broaden their range of actions is

to focus on the pre-lesson and within-lesson decisions that the teacher mustmake. The research literature abounds with ideas for mathematics-teachingprocedures, particularly in the mode-of-presentation and sequence areas, anddecisions regarding approaches are critical, as Clark and Yinger (1979) showedfrom their studies of teacher planning. Students clearly need much practice atexploring the range of possible approaches. This aspect of teacher educationhas a rather creative flavour about it with activities like 'List the mathematicalideas which you could explore with a pack of cards', 'Suggest ten differenttypes of material with which you could introduce "area"', and 'Describe tenexamples, from the child's world, of operations which do have an inverse'.

The period of teaching practice is, however, the time when the quality ofthe pre-lesson decision making can really be judged, and the criteria forchoosing become more apparent. Another aspect of reality impinges thenwhich is that of prior departmental and school decisions about the curriculumand approaches. The model of the autonomous teacher is one that does nothold up in reality, and the student teacher, given the books he must use, thesyllabus he must follow and the classes he must teach (often in a carefullycontrolled way), may feel he is in no position to make any pre-lessondecisions. Even experienced teachers exist within departments and withinschools, and must learn to make their decisions within that social context,and Arfwedson (1976) pointed clearly to the mis-match of 'goals' and 'rules' inschools, which can cause much conflict.

Nevertheless, even if the student teacher feels constrained with hisplanning activities, he will quickly realise the manifold choices existing withinthe lessons. His planning can help with some of the decisions, but no pre-lesson planning can predict and circumscribe every incident that may occur(see, for example, Bishop, 1976). This was the area that the Waples (1927) book(referred to in the Introduction) was presumably aimed at.

The range of within-lesson decisions is vast, from micro-curricula adapt-ations and method variation for particular individuals, through relationship-developing, to discipline and control aspects. Simulations through work withstop-action videotapes, and written incidents, as in Bishop and Whitfield(1972), can help to prepare the student for the classroom, but once again thedynamic nature of this type of decision making will only really be sharpenedby the reality of actual teaching.

CONCLUSION

It is therefore clear that with the increase in research into mathematicseducation has come a comparable increase in the type, and the content, of ideasfor teacher training. It is also clear that much of the work in teacher training that

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was undertaken in the general educational theory context can now beapproached much more meaningfully through the mathematics-educationcontext. One would therefore expect, and hope, to see the future mathematicsteacher spending more training time in specifically mathematics-educationactivities and less in the areas of the individual educational disciplines.

Furthermore, at a time of much public pressure on mathematics teachingand much discussion of curricula and methods, it is important to recognize thecontributions of the many serious studies of problems and issues in the field.The 'growth' areas appear to be: understanding the child's version of reality,studies of classroom phenomena and analyses of the social context of thecurriculum. All of these developments are to be welcomed as they can only helpthe continual improvement of mathematics teacher education.

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