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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
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Teacher Development, Volume 3, Number 2, 1999
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
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)
A CONSTRUCTIVIST APPROACH TO LEARNING MATHEMATICS
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. Focusi