The three papers on early number in the May issue of this journal (Baroody, 1984; Carpenter & Moser, 1984; Fuson, 1984) focus attention on the crucial role that child-generated or invented methods play in the development of arithmetical knowledge. I will consider just two of the many issues raised in these closely related papers. The first concerns the difficulties involved in identifying developmental sequences of methods that children construct to solve specific tasks such as subtraction problems. The second issue relates to the instructional implications of recent research on early number.
Three models each posit a sequence of developmental levels that culminate with the child's construction of the part-whole concept. Two are information-processing models of arithmetical word problem-solving performance, and the third is a model of children's counting types and thinking strategies. The three research teams make several of the same basic psychological assumptions. Nevertheless, the models offer qualitatively different explanations of similar behaviors. The models are compared in terms of detail, scope, attention to individual differences, and how they deal with qualitative change and semantic processing.
In this response to Robert Orton, I address each of the major points he raises and attempt to clarify the discrepancies in our positions. I give particular attention to his approach of reframing issues in terms of the categories of traditional academic philosophy. In adopting this stance, Orton (a) presents a highly idiosyncratic interpretation of the Cartesian dualism, (b) creates a gulf between theory and practice, and (c) implies that the social and cultural aspects of mathematical activity can be dismissed. I discuss each of these points and further develop my position by outlining Putnam's (1987) pragmatic realism, clarifying why Rorty's (1979) work might be of interest to mathematics educators, and revisiting Bereiter's (1985) learning paradox. I then conclude by exploring the relationship between the resulting nondualist approach and John Dewey's philosophy and pedagogy.
This study investigates the role that four second graders' use of the hundreds board played in supporting their conceptual development over a 10-week period. Particular attention is given to the transition from counting by ones to counting by tens and ones. The analysis indicates that the children's use of the hundreds board did not support the construction of increasingly sophisticated concepts of ten. However, children's use of the hundreds board did appear to support their ability to reflect on their mathematical activity once they had made this conceptual advance. The constructivist perspective exemplified in the analysis is contrasted with a sociocultural perspective on mathematical development. The differing roles attributed to cultural tools are clarified, and potentially complementary aspects of the two perspectives are discussed.
The notion of intuition frequently crops up in accounts of mathematical experiences (e.g., Davis & Hersh, 1981), and we have an intuitive idea of what is meant. As Fischbein notes, “intuition is generally seen as a primary phenomenon which may be described but which is not reducible to more elementary components” (p. ix). To rectify this situation, Fischbein presents a theory of mathematical and scientific intuition. In doing so, he synthesizes empirical research on problem solving, images and models, beliefs, and developmental stages of intelligence, drawing on examples from the history of science and mathematics. The book is marked by a masterly display of scholarship and makes a fundamental contribution to the analysis of mathematical cognition.
Leslie P. Steffe and Paul Cobb
One primary reason for writing this critique of the report by Hiebert, Carpenter, and Moser (1982) is to emphasize the intricate relationship between theory and data. First, we question the classical empiricist assumption that observation is free from the prejudices of theory. We then consider the role that explanatory theoretical constructs played in the study. Finally, we discuss briefly a claim that number conservation is not a readiness variable for acquiring counting strategies.
Paul Cobb and Leslie P. Steffe
The constructivist teaching experiment is used in formulating explanations of children's mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations of theoretical constructs--that represent our understanding of children's mathematical realities. However, the models must be distinguished from what might go on in children's heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children's construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher's understanding of the children.
Kay McClain and Paul Cobb
The analysis reported in this paper contributes to the effort to understand how mathematics teachers might proactively support their students' mathematical learning by documenting one first-grade teacher's role in guiding the development of sociomathematical norms in her classroom. We highlight the learning opportunities that arose for both the teacher and her students during this process. The analysis therefore serves to clarify what teachers might actually do to support the emergence of the type of mathematical disposition advocated in reform documents. In addition, the analysis describes the decision-making processes that were involved in setting the teacher's agenda. This aspect of the paper focuses on the teacher's interactions as part of a research team whose dual focus was to support both the students' mathematical learning and their development of mathematical autonomy. The analysis was informed by and builds on Yackel and Cobb's (1996) discussion of sociomathematical norms.
Erna Yackel and Paul Cobb
This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.
Melissa Sommerfeld Gresalfi and Paul Cobb
This article presents an analytical approach for documenting the identities for teaching that mathematics teachers negotiate as they participate in 2 or more communities that define high-quality teaching differently. Drawing on data from the first 2 years of a collaboration with a group of middle school mathematics teachers, the article focuses on a critical initial condition for teachers to improve their practice—determining that the effort required is worthwhile. The results speak directly to a central issue that arises when supporting teachers' efforts to improve their instructional practices: their motivation for affiliating with a vision of teaching that involves centering instruction on student thinking.