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As Robert Davis has stated, anyone "who observes mathematics education has to be impressed by the quite sudden eruption of `constructivism' as a central concern of so many researchers" (Davis, 1990, p. 114). While constructivism has probably failed to have more than a peripheral impact on the thinking and practice of most mathematics teachers and curriculum developers, there can be no doubt that over the past decade mathematics education researchers have been challenged to reflect on what a relativist, constructivist mathematics education research agenda might look like (Ellerton & Clements, 1991).

Each of the keynote addresses at the 1987 annual International Conference of the Group for the Psychology of Mathematics Education, for example, was dedicated to the theme, and since then three major edited collections have been prepared in which the theoretical bases and implications of constructivism for mathematics education have been explored. In 1990 a collection of 12 papers was published by the National Council of Teachers of Mathematics in the United States (Davis, Maher, & Noddings, 1990), and in 1991 a book edited by von Glasersfeld carrying chapters written by leading American and European writers, appeared, under the title Radical Constructivism in Mathematics Education. The Special Interest Group on constructivism which met at the seventh International Congress on Mathematical Education in 1992 generated much lively debate on the pedagogical implications of constructivism, and papers presented to this Special Interest Group have subsequently been published (Malone & Taylor, 1993).

Radical Constructivism

The radical constructivist movement takes on the relativist position that mathematics is not an "out-there" pre-existing body of knowledge waiting to be discovered, but rather is something which is personally constructed by individuals in an active way, inwardly and idiosyncratically, as they seek to give meaning to socially accepted notions of what can be regarded as "taken-to-be shared mathematical knowledge." As Ernst von Glasersfeld (1990), possibly the best known advocate of the radical constructivist position in mathematics education, has stated:

. . . knowledge is the result of an individual subject's constructive activity, not a commodity that somehow resides outside the knower and can be conveyed or instilled by diligent perception or linguistic communication. (p. 37)

According to von Glasersfeld (1990, p. 37), all good teachers know that guidance which they give to students "necessarily remains tentative and cannot ever approach absolute determination," because, from the constructivist point of view, there is always more than one solution to a problem, and problem solvers must approach problem situations from different perspectives.

In this acceptance of the relativist position, radical constructivism is in harmony with the major relativist trends of twentieth century philosophers and mathematicians. However, a major distinguishing cornerstone of radical constructivist theory is its acceptance of Piaget's emphasis on action (that is to say, all behaviour that changes the knower-known relationship) as the basis of all knowledge. An individual gets to know the real world only through action (Sinclair, 1990; von Glasersfeld, in press). Hermine Sinclair, Piaget's successor in Geneva, described von Glasersfeld as perhaps "an even more radical constructivist than Piaget" (Sinclair, 1987, p. 29).

Implications of Radical Constructivism for Mathematics Learning Environments

For radical constructivists working in the field of mathematics education, the crucial issue is not whether mathematics teachers should allow students to construct their own mathematical knowledge, "for the simple reason that to learn is to actively construct" (Cobb, 1990a). "Rather," Cobb (1990a) says, "the issue concerns the social and physical characteristics of settings in which students can productively construct mathematical knowledge." (For further commentary on constructivism in general, and on radical constructivism in particular, see for example, Cobb, 1986; Confrey, 1987; Dorfler, 1987; Kilpatrick, 1987; Labinowicz, 1985; von Glasersfeld, 1983.)

Cobb (1990b, pp. 209-210) called for constructivist mathematics educators to develop a new context - a "mathematico-anthropological context" - that will assist coherent discussion on the specifics of learning and teaching mathematics. According to Cobb there is research support for moving to establish mathematics classroom environments that incorporate the following qualities:

1. Learning should be an interactive as well as a constructive activity - that is to say, there should always be ample opportunity for creative discussion, in which each learner has a genuine voice;

2. Presentation and discussion of conflicting points of view should be encouraged;

3. Reconstructions and verbalisation of mathematical ideas and solutions should be commonplace;

4. Students and teachers should learn to distance themselves from ongoing activities in order to understand alternative interpretations or solutions;

5. The need to work towards consensus in which various mathematical ideas are co-ordinated is recognised.

Although many teachers of mathematics would accept all five of these points, all too often the rhetoric of mathematics teachers and the realities of what transpires in their mathematics classrooms do not bear much resemblance to each other (Desforges, 1989). Nonetheless, radical constructivists are determined to refine and apply their ideas to mathematics classrooms however difficult and time-consuming this process might prove to be (Ackermann, in press; Steffe, 1990).

Notwithstanding certain tensions between theory and practice and lack of clarity in the definition of radical constructivism (as opposed to other forms of constructivism - see Ellerton & Clements, 1992), the effects of radical constructivism on mathematics education research and on school mathematics are being strongly debated by many mathematics education researchers around the world. In the second half of the 1980s, there was a wave of research (see, for example, Wood and Yackel, 1990), built on earlier theoretical works (for example, Bauersfeld, 1980; Chomsky, 1957; Mehan & Wood, 1975), aimed at identifying the roles of teachers of mathematics who wish to adopt radical constructivist approaches. Cobb (1990a), in a paper entitled "Reconstructing elementary school mathematics," summarised research which attempted to assess the effectiveness of the application of radical constructivist ideas to mathematics teaching and learning. He made five main points:

1. To claim that students can discover mathematics on their own is an absurdity.

2. Students do not learn mathematics by internalising it from objects, pictures, or the like. Mathematics is not a property of learning materials, structured or otherwise.

3. The pedagogical wisdom of the traditional pattern of first teaching mathematical rules and skills, and then providing opportunities to apply these in real life situations, is questionable. An alternative approach takes seriously the observation that from a historical perspective, pragmatic informal mathematical problem solving constituted the basis from which formal, codified mathematics evolved.

4. The teacher should not legitimise just any conceptual action that a student might construct to resolve a personal mathematical problem. This is because mathematics is, from an anthropological perspective, a normative conceptual activity (see Shweder, 1983), and learning mathematics can be seen as a process of acculturation into that practice. This is evident from the fact that certain other societies and social groups have developed routine arithmetical practices that differ from those taught in Western schools.

5. Mathematical thought is a process by which we act on conceptual objects that are themselves a product of our prior conceptual actions, and from the very beginning of primary schooling, students should participate in and contribute to a communal mathematical practice that has as its focus the existence, nature of, and relationships between mathematical objects. From this perspective, understanding mathematics is constructing and acting on what might be called "taken-to-be-shared" mathematical objects.

Other writers have drawn up lists such as this summarising characteristics that should apply to radical constructivist teaching and learning in mathematics. Steffe (1990), for example, has elaborated ten principles for mathematics curriculum design that are in keeping with the main radical constructivist thrusts.

Pateman and Johnson (1990) have claimed that it has been "constructivist" teachers of mathematics who have led the recent important movement towards establishing mathematics learning environments that nurture interest and understanding through co-operation and high quality social interaction. Pateman and Johnson maintained, as did Steffe (1990), that the belief that children construct their own mathematics out of their own actions and their reflections on those actions (in social settings) provides a new framework for those responsible for devising mathematics curriculum and school mathematics programs. According to Pateman and Johnson (1990), three aspects of curriculum need to be considered - content, methodology and assessment - if environments are to be created which foster socio-cognitive conflict and challenge. For Pateman and Johnson (1990), constructivist educators can hardly rigidly prescribe content in advance; the leaning environments they create need to be idiosyncratic to children and context: and the assessment methods they employ should foster growth and cooperation (which is particularly difficult for those so used to competitive ratings). Constructivist teachers need to be opportunists, "willing to continue to learn both about mathematics and children in the attempt to develop them as autonomous creators of their own mathematics" (Pateman & Johnson, 1990, p. 351).

While radical constructivist mathematics educators tend to emphasise the socio-cognitive importance of student-to-student and student-to-teacher interactions, there is no suggestion that whole-class environments cannot facilitate rich mathematical learning. Cobb (1990b, p. 208) has identified the following social norms for worthwhile whole-class discussion in mathematics classrooms:

1. Explaining how an instructional activity that a small group has completed was interpreted and solved;

2. listening and trying to make sense of explanations given by others;

3. indicating agreement, disagreement, or failure to understand the interpretations and solutions of others;

4. attempting to justify a solution and questioning alternatives in situations where a conflict between interpretations or solutions has become apparent.

Although most experienced teachers would like to think that these norms already apply in whole-class discussions that occur in their own classrooms, mathematics classroom discourse analyses indicate that this is not the case. Nevertheless, studies of interaction patterns in mathematics classrooms in elementary schools in Japan (see, for example Easley and Easley, 1992) suggest that quality mathematics learning has occurred in many whole-class teaching and learning environments in classrooms where teachers have never heard of the constructivist movement. If this is indeed the case, then it makes little sense to claim that the radical constructivist movement has been primarily responsible for the development of whole-class teaching methods that are likely to generate high quality learning.

We have argued elsewhere (Ellerton & Clements, 1992), in fact, that it is a moot point, whether developments which have produced such lists are a direct outcome of the radical constructivist movement. As Kilpatrick (1987) has pointed out, the principle that teachers should attempt to create classroom environments in which learners regularly engage in mathematically rich, social interactions has been advocated by many who would not regard themselves (and would not be regarded) as "radical constructivist." Furthermore, although radical constructivist educators such as Paul Cobb are strongly in favour of the development of richer social interaction patterns in mathematics classrooms, it is not altogether clear whether this has its origins in their identification with the radical constructivist movement or has merely arisen because they wish to improve the quality of mathematics teaching and learning.

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