The effects of an overt constructivist approach to learning mathematics and its subsequent effects on classroom teaching

  • Published on
    24-Feb-2017

  • View
    212

  • Download
    0

Transcript

  • This article was downloaded by: [UTSA Libraries]On: 05 October 2014, At: 09:22Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

    Teacher Development: An international journal ofteachers' professional developmentPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/rtde20

    The effects of an overt constructivist approach tolearning mathematics and its subsequent effects onclassroom teachingM. Simmons aa University of Stirling , United KingdomPublished online: 19 Dec 2006.

    To cite this article: M. Simmons (1999) The effects of an overt constructivist approach to learning mathematics andits subsequent effects on classroom teaching, Teacher Development: An international journal of teachers' professionaldevelopment, 3:2, 173-196, DOI: 10.1080/13664539900200082

    To link to this article: http://dx.doi.org/10.1080/13664539900200082

    PLEASE SCROLL DOWN FOR ARTICLE

    Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

    This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

    http://www.tandfonline.com/loi/rtde20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/13664539900200082http://dx.doi.org/10.1080/13664539900200082http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions

  • Teacher Development, Volume 3, Number 2, 1999

    173

    The Effects of an Overt Constructivist Approach to Learning Mathematics and its Subsequent Effects on Classroom Teaching

    M. SIMMONS University of Stirling, United Kingdom

    ABSTRACT A Masters unit in mathematics education was given to three practising primary teachers. Constructivism as a theory on which to develop a flexible pedagogy was a central theme and the style of presentation of workshops was guided by constructivist principles. Contact time was 20 hours spread over 10 weeks. The course contained a number of different aspects relating to mathematics teaching and learning some of which provided research data: (1) discussion about the nature of constructivism in mathematics learning, and its implications for teaching; this centred around seminar papers on aspects of constructivism; (2) a workshop each week that focused on a rich mathematical topic designed to provoke mathematical activity and interaction and to encourage reflection on that activity; (3) an extended investigation of a mathematical situation; and (4) classroom implementation of some aspect of (3) together with a report of its implementation and an evaluation of childrens understanding. A combined individual and social constructivist approach to learning mathematics by primary teachers is described. some evidence is presented of changes in belief held by teachers about the nature of mathematics and the subsequent changes in classroom practice and expectations of childrens learning.

    Constructivist principles, whether they derive from a radical or social perspective, appear to have captured the imagination of educational researchers by providing a forum through which teaching and learning can be enhanced. Much philosophical debate has centred around the various forms of constructivism and their implications for educational practice. Radical constructivism, as a theory of knowing and in particular as this relates to mathematics education, has been thrown into the spotlight by von Glasersfeld (1991, 1995). Research on how mathematics pedagogy may by reconstructed

    Dow

    nloa

    ded

    by [

    UT

    SA L

    ibra

    ries

    ] at

    09:

    22 0

    5 O

    ctob

    er 2

    014

  • M. Simmons

    174

    from a constructivist perspective is also well documented (Steffe, 1983, 1991, 1995; Steffe & Wiegel, 1992; Steffe & Kieren, 1994; Simon, 1995).

    Constructivism works from the premise that constructing meaning and understanding are two sides of the same coin; that in making our own personal constructs and reflecting upon them, we build understanding. The theoretical debate has in part focused on how and under what conditions these personal constructs take place. From this debate (see, for example, Steffe & Gale, 1995) some common ground can be found between the radical constructivists and the social constructionists, which can guide good practice. Both, for example, recognise the difficulties in trying to tie the notion of understanding to some absolute mathematical concept. Both recognise the importance of accepting that understandings are ultimately based upon individual constructions whether gained through individual or group experiences, and that teaching should encourage appropriate interactions that allow a sharing of ideas and further constructions to be made. So both share some ideas about how knowledge is generated and both question the view that knowledge is built up through dispassionate observation:

    [constructivism] ... starts form the assumption that knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her experience ... The experience and interpretation of language are no exception. (von Glasersfeld, 1995, p. 1)

    On the most abstract level, we might say that what we count as knowledge are temporary locations in dialogic space samples of discourse that are accorded status as knowledgeable tellings on given occasions. More concretely, knowledge is in continuous production as dialogue ensues. To be knowledgeable is to occupy a given position at a given time within an ongoing relationship. (Gergen, 1995, p. 30)

    Radical constructivism and social constructionism also question the view that individual minds can accurately reflect the true nature of an independent world but in so doing acknowledge a belief about how such a world is assembled:

    If ... people look through distorting lenses and agree on what they see, this does not make what they see any more real it merely means that on the basis of such agreements they can build up a consensus ... (a consensual domain) ... in certain areas of their subjective experiential worlds. (von Glasersfeld, 1991, p. xv)

    The personal success of generating accepted contributions or, as von Glasersfeld has coined it, to arrive at viable constructions rests on both the participants active engagement in the social practice of mathematizing as well as on the qualities of the social interaction. It is only through becoming actively involved that the experienced resistance from the social group can develop selective power and force for the subject. (Bauersfeld, 1995, p. 152)

    Dow

    nloa

    ded

    by [

    UT

    SA L

    ibra

    ries

    ] at

    09:

    22 0

    5 O

    ctob

    er 2

    014

  • A CONSTRUCTIVIST APPROACH TO LEARNING MATHEMATICS

    175

    For the purposes of this article a consensual domain is taken to be an area of agreement derived from reflexive activity and giving rise to meaningful communication (Richards, 1991). Richards points out that a consensual domain is a necessary but not sufficient condition for successful communication and in a broader sense the consensual domain of school mathematics is put forward as an example about which there is general agreement but in which meaningful communication does not take place. Despite the many initiatives in numerous countries worldwide over the last 15 years this is sadly still true.

    As long ago as 1982, Mathematics Counts (the Cockcroft Report) drew attention to the need for discussion between teacher and pupils and between pupil and pupil. The Department of Education and Science (DES) said in 1985, The quality of pupils mathematical thinking as well as their ability to express themselves are considerably enhanced by discussion. Of course, research into the beneficial effects, or otherwise, of grouping has gone on over the years as a means of developing meaningful dialogue between pupils and has not been wholly part of the constructivist school. For example, Barnes (1976), with my own brief interpretation added Simmons (1993), lists five important factors for effective group organisation:

    1. Feeling of competence by pupils: this is an outcome of the composition of groups and the pupils perception of the teachers role in the whole process. 2. Common ground: pupils abilities to restate and reorganise presented situations into presentations they can all share (a consensual domain); the use of language at the pupils level is all-important at this stage. 3. Focusing: the teachers ability to present a problem framework that provides the kind of help needed to start pupils thinking and to start asking their own relevant questions. 4. Pace: basically allowing time for exploratory talk to take place; again a balance is required between allowing time for learning to take place and time to show that the job has been done. 5. Making public: the need to refine language appropriate to a wider audience. For example, groups reporting back to the whole class would require a spokesperson to collect and present findings of the group using more explicit language so that other groups less familiar with the situation can appreciate the points being made.

    These five points to my mind sit easily with social constructionism that In its radical form, ... does not commence with the external world as its fundamental concern or with the individual mind but with language (Gergen, 1995, p. 23) and where, unlike the radical constructivists, importance is given to other thinkers beyond that given to other viable constructs. The difference between others and objects as offered by the social constructionists is that we construct others out of ourselves and others contribute to the image of ourselves:

    Students have to learn mathematics as social knowledge: they are not free to choose the meanings they construct. These meanings must not only be efficient in

    Dow

    nloa

    ded

    by [

    UT

    SA L

    ibra

    ries

    ] at

    09:

    22 0

    5 O

    ctob

    er 2

    014

  • M. Simmons

    176

    solving problems, but they must be coherent with those socially recognized. (Balacheff, 1991, p. 89)

    Clearly, how we conceptualise knowledge affects our choice of educational process. If we take the view that mathematics is ... both a cognitive activity constrained by social and cultural processes and a sociocultural phenomenon that is constituted by a community of actively cognizing individuals (Wood et al, 1995, p. 402), then we must adopt teaching strategies that fully recognise the opportunities and constraints afforded by such processes. It effectively means that if teachers are to make appropriate choices about the strategies they use then they have to be aware of the intended outcomes as they relate to their particular discipline. As Bereiter (1994) points out, there is nothing incompatible about the ideas put forward by constructivism and socioculturalism except for the practical difficulty of doing both at once. Cognitive and social process are then seen as complementary and necessary, but according to Bereiter not sufficient; a theme I shall return to in a moment.

    When students attempt to solve problems in mathematics, it is clear that there are periods when they need to interact with other students, often in language that indicates uncertainty and is therefore imprecise, but which seeks a forum in which ideas may be validated. Such interaction may be supported to a lesser or greater degree by objects or symbols, the use of objects often giving way to symbolic representation and manipulation as a more refined view of the problem emerges. It is also clear that there are periods when students need time to think and contemplate (interact with?) objects or symbolic representations in order to deal with internally perceived discrepancies or to move forward towards a particular solution. In mathematics, like other subjects, this often means making connections through the development and interpretation of symbols that represent the problem domain. However, the frequency and style with which these circumstances prevail is, I believe, unique to mathematics.

    For example, during childrens investigation into Pythagorass Theorem by a variety of methods, an aim of the lesson will be to encapsulate the findings in some general way, which might be through the use of language or through the use mathematical notation, viz. c2 = a2 + b2. However, in order to develop a wider concept of the theorem, this representation can be manipulated in a simple way to read kc2 = ka2 + kb2, which when translated back into the geometric problem domain gives rise to variants of the more familiar theorem. However, where is such knowledge located?

    The debate between radical and social constructivists does not appear to provide a clear answer, locating it in the head or in the social interaction that takes place. Yet it would be difficult to deny the existence of the Pythagorean Theorem if it were not located in either of these two locations. One could argue that such a mathematical relationship exists whether or not we are capable of understanding or expressing it. Knowledge in this objective sense is totally independent of anybodys claim to know; it is also independent of anybodys belief, or disposition to assent; or to assert, or to act. Knowledge in

    Dow

    nloa

    ded

    by [

    UT

    SA L

    ibra

    ries

    ] at

    09:

    22 0

    5 O

    ctob

    er 2

    014

  • A CONSTRUCTIVIST APPROACH TO LEARNING MATHEMATICS

    177

    the objective sense is knowledge without a knower: it is knowledge without a knowing subject (Popper, 1972, p. 109). Perhaps this additional dimension, based on theories of knowledge long debated, in which theoretical systems, problem situations and critical arguments reside, which appears at odds with constructivism but which does not deny their value, is one that is required to help bring about balance in the redirection in school mathematics and to support the natural application of mathematics to real w...

Recommended

View more >