implications of research for mathematics teacher education

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  • 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

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    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 As a social and historical document it is magnificent, but as an aid toteacher education it leaves much to be desired, not only in terms of what itmakes available but how it makes it available.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

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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

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