Theories of Mathematical Learning

Leslie P. Steffe, Pearla Nesher, Paul Cobb, Bharath Sriraman, Brian Greer, Leslie P. Steffe, Pearla Nesher, Paul Cobb, Bharath Sriraman, Brian Greer

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136485541
• Edition: 1
• Topic: Education
• Subtopic: Education General

Chemists, working with only mortars and pestles, could not get very far unless they had mathematical models to explain what was happening "inside" of their elements of experience -- an example of what could be termed mathematical learning. This volume contains the proceedings of Work Group 4: Theories of Mathematics, a subgroup of the Seventh International Congress on Mathematical Education held at Université Laval in Québec. Bringing together multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning worldwide.

SOCIOLOGICAL AND ANTHROPOLOGICAL PERSPECTIVES ON MATHEMATICS LEARNING

Paul Cobb, Editor

Emergent and Sociocultural Views of Mathematical Activity

Paul Cobb
Vanderbilt University

Barbara Jaworski
Oxford University

Norma Presmeg
Florida State University

Taken together, the six plenary chapters presented in this section indicate that a wide range of approaches fall under the rubric of sociological and anthropological perspectives. These contributions all reflect the contention that knowing and doing mathematics is an inherently social and cultural activity. This claim does not merely mean that social interaction serves as a catalyst for otherwise autonomous cognitive development. In the view of each of the contributors, social and cultural influences are not limited to the process of learning but also extend to its products, increasingly sophisticated mathematical ways of knowing. Consequently, in describing their own positions, the contributors each challenge the assumption that students’ mathematical activity can be adequately accounted for solely in terms of individualistic theories such as constructivism or information-processing psychology. Three of the contributors, Voigt, Stigler et al., and Saxe and Bermudez, develop analyses that complement psychological constructivism. In contrast, the perspectives developed by van Oers, Forman, and Crawford are proposed as alternatives to psychological theories that focus on the individual.

In this commentary, we first consider the key assumptions of these two groups of theorists, giving particular attention to the role they attribute to individualistic theories. We then contrast their differing treatments of a variety of key issues and conclude by discussing possible ways in which their perspectives might be complementary.

EMERGENT PERSPECTIVES

Voigt, Stigler et al., and Saxe and Bermudez use differing theoretical constructs to address a diverse range of issues. In our view, this diversity belies several areas of compatibility in their treatment of students’ mathematical activity in classroom situations.

Voigt’s work is premised on the assumption that mathematical learning and teaching are linked via classroom interaction. His work can therefore be viewed as an attempt to transcend individualistic analyses of either students’ or the teacher’s activity (cf. Romberg & Carpenter, 1986). One of his central concerns is to clarify the processes by which the teacher and students develop a basis for mathematical communication. He approaches this issue by analyzing their negotiation of taken-as-shared meanings. It is important to appreciate that Voigt’s use of the term negotiation is derived from symbolic interactionist theory (Blumer, 1969). In this view, social interactions involve subtle shifts and slides of meaning that frequently occur outside the participants’ awareness. Thus, when Voigt speaks of the negotiation of meaning, he declares a theoretical interest—that of analyzing the evolution of mathematical meaning in interaction. His overall goal is to develop a theoretical framework within which to analyze the interrelationships between the teacher’s and students’ activities. The sample episodes he discusses indicate that this framework is not tied to any particular instructional approach. Instead, the theoretical constructs he employed are intended to be applicable to any mathematics classroom, regardless of how desirable or undesirable the instruction might appear.

Voigt’s notion of negotiation can be contrasted with Richards’ use of the term in his contribution. For Richards, negotiation occurs only when the teacher actively listens and consciously adapts to students’ mathematical activity. He contrasts interactions of this type with those that characterize traditional classrooms and argues that, in the latter case, the teacher is a conveyer of information and the students are the recipients. This distinction between desirable and undesirable instructional practices is motivated at least in part by the pragmatic concern to reform traditional American instructional practices. Thus, Richards argues that the teacher should be a trained negotiator whose goal is to initiate students into the consensual domain of mathematically literate adults.

Voigt’s and Richards’ notions of negotiation are clearly in conflict. For example, Richards argues that interactions in which students merely adopt the teacher’s language do not involve negotiation. However, Voigt would contend that even the most draconian teachers necessarily adapt to their students’ activity. The issue for Voigt is not to ascertain whether the teacher and students negotiate meanings, but to understand how they do so by analyzing the microprocesses of their interactions. One way to resolve this conflict is to differentiate between implicit negotiation and explicit negotiation (Cobb & Yackel, 1993). This distinction highlights Voigt’s claim that students’ mathematical learning is frequently indirect and occurs as they participate in implicit negotiations that are outside their own and the teacher’s awareness. For his part, Richards proposes that classroom interactions should involve the explicit negotiation of mathematical meanings in which the teacher encourages students to articulate their mathematical problems, interpretations, and solutions.

Although Voigt focuses on collective meanings, he takes care to emphasize that taken-as-shared meanings do not displace analyses of individual, personal meanings. Indeed, the references he makes to constructivist psychological analyses in his chapter indicate that these and interactionist analyses can be complementary. We can clarify the relationship between the two types of analyses by taking as an example an interaction between a researcher and one student. To the extent that a constructivist psychological analysis takes account of the interaction, the focus is on the student’s interpretations of the teacher’s actions. An analysis of this type is made from the perspective of the researcher, who is inside the interaction and is concerned with the ways in which the student modifies his or her activity. In contrast, an interactional analysis is made from the outside, from the point of view of an observer rather than that of a participant in the interaction. From this perspective, the focus is on the obligations the researcher and the student attempt to fulfill, and on the taken-as-shared meanings that emerge between them, rather than on the student’s personal interpretations. As Voigt makes clear, these taken-as-shared meanings are not cognitive elements that capture the partial match of individual interpretations, but are instead located at the level of interaction. The complementarity between the two theoretical perspectives becomes apparent when it is noted that taken-as-shared meanings emerge as the teacher and students attempt to coordinate their individual activities. Conversely, the teacher’s and student’s participation in the establishment of these taken-as-shared mathematical meanings both supports and constrains their individual interpretations.

This complementarity between interactionist and psychological constructivist perspectives implies that the link between the social and psychological aspects of mathematical activity is indirect. Thus, although it is not possible to deduce individual students’ cognitions from interactional analyses, such analyses can inform cognitive analyses. For example, one can infer from an interactional analysis what, minimally, students need to know and do in order to fulfill their obligations and thus be effective as they participate in classroom interactions. Further, one can infer the conceptual constructions that students might make as they participate in the evolution of such patterns of interaction. Inferences of this type reflect the view that students have to reorganize their own activity in order to learn. As a consequence, Voigt rejects the notion that the social dynamics of the classroom determine students’ mathematical development. He instead analyses how students’ participation in specific patterns of interaction might support their mathematical development by identifying possible learning opportunities.

At first glance, Voigt’s interactionist approach might seem unrelated to the problem of accounting for differences in Japanese and American students’ mathematics achievement. However, the framing of the issue by Stigler et al. focuses attention on the quality of classroom social interactions. Stigler et al. contend that there are cultural differences in what it means to teach and learn mathematics in the two countries, and that these differences constitute two distinct traditions of classroom practice. The goal of their analysis is to identify aspects of these two cultural traditions by comparing representative lessons from each country. Thus, Stigler et al. link the macrosociological and microsociological levels of analysis by treating the microcultures established in particular classrooms as manifestations of broader cultural phenomena. This enables them to address a cultural-difference issue by focusing on teacher–student classroom interactions.

It should be noted that whereas Voigt attempts to tease out hidden interactional regularities, Stigler et al. analyze the teacher’s routines and explore the consequences for students’ mathematical learning. In doing so, Stigler et al. repeatedly hint at the interdependency of the teacher’s and students’ activity, thereby establishing a point of contact with Voigt’s work. For example, on one occasion, they describe an interaction in which an American fifth-grade teacher asked his students for the area of a right triangle that has been divided into square units. The teacher anticipated that the students would find it difficult to count the squares because fractional parts of square units lined the hypotenuse. However, the students persisted in their attempts to count the squares. On three occasions, the teacher attempted to motivate the need for an alternative method by capitalizing on the students’ solution attempts. Thus, when a student called the fractional parts halves, the teacher asked, “Are they halves? This one is not a half, is it? This one is not a half, this one is way more than a half.” The students ignored these interventions and persisted with their counting approach until they jointly developed a viable solution and arrived at the correct answer. It was only at this point that the teacher introduced a planned demonstration that involved placing two triangles together to make a square. Clearly, the teacher felt obliged to follow his scripted lesson plan. This is a key aspect of the American tradition of school mathematics identified by Stigler et al. However, the authoritarian way in which the teacher eventually introduced the demonstration can only be understood by taking account of the students’ unanticipated persistence. The teacher’s role as an authority who controlled the mathematical agenda can therefore be seen to have emerged in the course of the interaction.

Stigler et al. relate their analysis of classroom episodes to students’ mathematical learning by identifying the opportunities that arise for thinking and reflection. Thus, like Voigt, they leave room for psychological analyses of individual students’ cognitions. The characterization of mathematical learning they offer is that of a process of individual construction that occurs in social interaction within a cultural tradition of practice. Given the relationship they propose between classroom processes and broader cultural phenomena, it can in fact be argued that the teacher and students in each country are regenerating culturally specific traditions of practice as they interact in the classroom.

Saxe and Bermudez’s work complements that of Stigler et al. by emphasizing the culturally situated nature of mathematical activity. In their chapter, they illustrate how their Emergent Goals model can be used to analyze the mathematical environments that emerge for children as they participate in classroom activities. Saxe (1991) clarifies that this model was originally developed to account for the mathematical learning that occurs as individuals participate in nonschool cultural practices such as selling candy in the street. The central problem addressed by Saxe (1991) was that of explaining how individuals’ personal goals become interwoven with the socially organized activities in which they participate. The solution he proposed involves the explicit coordination of two theoretical perspectives—a constructivist treatment of children’s mathematical activity and a sociocultural treatment of cognition. Thus, like Voigt and Stigler et al., Saxe’s theoretical work complements psychological constructivist analyses of individual children’s mathematical activity.

The four parameters incorporated into the Emergent Goals model deal with the cultural aspects of mathematical activity (activity structures and artifacts), the interactional aspects (social interactions), and the individual psychological aspects (prior understandings). The view of mathematical learning offered by Saxe and Bermudez is therefore similar to that implicit in the Stigler et al. analysis: a constructive process that occurs while participating in a cultural practice, frequently while interacting with others. It can also be noted that Saxe and Bermudez follow Voigt and Stigler et al. in using the notion of learning opportunities to link the cultural and social aspects of mathematical activity to conceptual development. In effect they propose that learning opportunities and thus the mathematical knowledge constructed are relative to the socially and culturally constrained goals that an individual attempts to achieve.

The sociocultural treatment of cognition that Saxe and Bermudez incorporate into the Emergent Goals model draws heavily on Leont’ev’s (1981) activity theory. For example, Saxe (1991) follows Leont’ev in arguing that an individual’s actions are necessarily elements of a broader sociocultural system and cannot be adequately understood unless this relation is explicated. Further, he notes that the term activity in activity theory refers to this broader system of practice and to its associated motive (cf. Crawford, Chapter 9, this volume). In the case of Saxe’s prior work with Brazilian street vendors, the activity was that of candy selling and the motive was that of economic survival. In line with Leont’ev’s theory, Saxe analyzed the structure of the candy-selling activity by decomposing it into a cycle of goal-directed actions (i.e., purchase boxes of candy, price candy for sale, sell candy, and select new boxes for purchase). The Treasure Hunt game that Saxe and Bermudez describe was developed to provide a classroom simulation of mathematically productive out-of-school activities such as candy selling. In this case, the activity is that of playing the game, and the associated motive is to win the game by accumulating treasure. As before, the structure of the activity is delineated by identifying a sequence of goal-directed actions (i.e., challenge, rent, purchase, region, and check).

Despite these commonalities in Saxe’s analyses of mathematical a...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Preface
• Part I: Sociological and Anthropological Perspectives on Mathematics Learning
• Part II: Cognitive Science Theories and their Contributions to the Learning of Mathematics
• Part III: The Contribution of Constructivism to the Learning of Mathematics
• Part IV: Perspectives on the Nature of Mathematical Learning
• Author Index
• Subject Index