Implications of research for mathematics teacher education

Download Implications of research for mathematics teacher education

Post on 17-Mar-2017

220 views

Category:

Documents

7 download

TRANSCRIPT

  • This article was downloaded by: [University of Chicago Library]On: 14 November 2014, At: 03:38Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

    Journal of Education for Teaching:International research andpedagogyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/cjet20

    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

    To link to this article: http://dx.doi.org/10.1080/0260747820080202

    PLEASE SCROLL DOWN FOR ARTICLE

    Taylor & Francis makes every effort to ensure the accuracy of all the information(the Content) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed by Taylor& Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francisshall not be liable for any losses, actions, claims, proceedings, demands, costs,expenses, damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

    This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly

    http://www.tandfonline.com/loi/cjet20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/0260747820080202http://dx.doi.org/10.1080/0260747820080202

  • forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

    Dow

    nloa

    ded

    by [

    Uni

    vers

    ity o

    f C

    hica

    go L

    ibra

    ry]

    at 0

    3:38

    14

    Nov

    embe

    r 20

    14

    http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions

  • 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

    Dow

    nloa

    ded

    by [

    Uni

    vers

    ity o

    f C

    hica

    go L

    ibra

    ry]

    at 0

    3:38

    14

    Nov

    embe

    r 20

    14

  • 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

    Dow

    nloa

    ded

    by [

    Uni

    vers

    ity o

    f C

    hica

    go L

    ibra

    ry]

    at 0

    3:38

    14

    Nov

    embe

    r 20

    14

  • 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

    Dow

    nloa

    ded

    by [

    Uni

    vers

    ity o

    f C

    hica

    go L

    ibra

    ry]

    at 0

    3:38

    14

    Nov

    embe

    r 20

    14

  • 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

    Dow

    nloa

    ded

    by [

    Uni

    vers

    ity o

    f C

    hica

    go L

    ibra

    ry]

    at 0

    3:38

    14

    Nov

    embe

    r 20

    14

  • 122 Journal of Education for Teaching Vol.8 No.2

    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...

Recommended

View more >