The contention that knowing and doing mathematics is an inherently social and cultural activity has gained increasing acceptance in recent years. At least in the United States, this attempt to go beyond purely cognitive analyses reflects a growing disillusionment with mainstream psychology. The sociocultural perspective treats mathematical learning as primarily a process of enculturation wherein students appropriate their intellectual inheritance; that is, the mathematical ways of knowing institutionalized by wider society. The motivation for developing the interpretive framework described was primarily theoretical. Such work does have practical relevance in that analyses of classroom events or of individual students' mathematical activity typically lead to suggestions for educational improvement. A primary reason for conducting the analysis of the first-grade classroom was to explore ways of accounting for learning as it occurs in the social context of the classroom. This chapter outlines Vygotsky's key insights, although leaving room for the conception of children as active constructors of their ways of knowing.

In this chapter, the theoretical construct of guided reinvention is extended to include desirable pedagogical practices for teachers implementing RME sequences. First, we explain what a guided reinvention teaching approach looks like and how it evolved out of over 25 years of research. We then articulate the planning and teaching practices of guided reinvention teachers and describe how those practices move beyond what many call "inquiry approaches" to mathematics teaching. We end the chapter by offering a set of learning goals that professional developers might use when mentoring aspiring guided reinvention teachers.

MFOI/PCO2 Plus Postage. *Constructivism (Learning); Elementary Secondary Education; *Learning Theories; *Mathematics Instruction; Models; *Sociocultural Patterns; *Socioeconomic Influences The overall intent is to clarify relationships between psychological constructivist, sociocultural, and emergent perspectives by grounding them in attempts to understand what might be happening in a variety of teaching and learning situations. The first part of the paper outlines an interpretive framework developed in the course of a classroom-based research project. At the level of classroom processes, the framework involves an emergent approach in which psychological constructivist analyses of individual activity are coordinated with interactionist analyses of classroom intercctions and discourse. At the level of school and societal processes, the perspective taken is broadly sociocultural and focuses on the influence of individuals' participation in culturally-organized practices. In the second part of the paper, the franework is taken as background against which to compare and contrast the three theoretical perspectives. The emergent approach augments the psychological constructivist perspective by making it possible to locate analyses of individual students' constructive activities in social context. In addition, the purposes for which emergent and sociocultural perspectives might be appropriate are considered and observed to span the individual students' activities, the classroom community, and broader communities of practice. Contains 75 references. (Author/MKR) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. Constructivist, Emergent, and Sociocultural Perspectives in the Context of Developmental Research Paul Cobb Erna Yackel A Paper Presented at the Seventeenth Annual Meeting for the Psychology of Mathematics Education (North American Chapter)

The analysis reported in this chapter focuses on a second-grade classroom in which the children typically displayed positive emotional acts as they attempted to solve personally challenging mathematics problems. We argue that, within the microculture established in a particular classroom, certain emotional acts but not others are appropriate in situations such as solving challenging mathematical tasks. The emotional acts that are warranted in particular situations can differ significantly from one classroom to another depending on the nature of the social norms that have been established. We illustrate how the negotiation of social norms that contrasted sharply with those established in most US mathematics classroom made possible the children’s generally positive responses to mathematical problem solving. In the final section of the chapter, we discuss the implications for the development of productive classroom learning environments in which debilitating emotions such as frustration while solving mathematical problems are not warranted.

We discuss mathematical tasks used in a first mathematics content course for elementary teachers at our university to foster a deep conceptual understanding of early arithmetic, including basic concepts of number, number relationships and strategies, and coordinating units of different rank. Our approach is to immerse our students in a base 8 world for up to six weeks. A key aspect is that we develop base 8 vocabulary. We use base 8 analogs of instructional sequences developed in classroom teaching experiments in the elementary grades that have been proven successful to promote deep conceptual understandings of basic arithmetic and place-value numeration in young children. As a result, our students have unique opportunities to develop a reconceptualized view of early arithmetic and learn how it can be advanced.

Current interest in mathematics learning that focuses on understanding, mathematical reasoning and meaning making underscores the need to develop ways of analyzing classrooms that foster these types of learning. In this paper, the author show that the constructs of social and socio-mathematical norms, which grew out of taking a symbolic interactionist perspective, and Toulmins scheme for argumentation, as elaborated for mathematics education by Krummheuer [The ethnology of argumentation. In: The emergence of mathematical meaning: Interaction in classroom cultures (1995, pp. 229– 269). Hillsdale, NJ: Erlbaum], provide us with means to analyze aspects of explanation, justification and argumentation in mathematics classrooms, including means through which they can be fostered. Examples from a variety of classrooms are used to clarify how these notions can inform instruction at all levels, from the elementary grades through university-level mathematics.

In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by elaborating how the constructs chains of signification and models were used to design an instructional sequence involving linear growth. The second and third cases illustrate vertical didactising by developing argumentation analyses and generative listening, respectively, as instructional design tools. In the second case, argumentation analyses emerge as a tool that other designers can use to anticipate the quality of conversations that can occur as students engage in tasks prior to implementing the instructional sequence. The third case develops the notion of generative listening as a conceptual tool within the context of designing differential equations instruction to gain insights into what are, for students, experientially-real starting points that are mathematical in nature and to provide inspirations for revisions to instructional sequences.

The central purpose of this chapter is to demonstrate that by coordinating sociological and psychological perspectives we can explain how changes in beliefs might be initiated and fostered in mathematics classrooms. In particular, we examine: 1) the coordination of students' beliefs about mathematical activity and classroom social norms and 2) the coordination of specifically mathematical beliefs and classroom sociomathematical norms. Examples from a university level differential equations class are used for purposes of clarification and illustration.

Increasing emphasis on “Algebra for all” (NCTM 1997a, 1997b) compels educators to identify and address fundamental ideas that build the foundations for algebraic thinking and reasoning. Identifying these foundational concepts and developing appropriate instructional approaches are the focuses of our work. One area in which students often experience difficulty is adding and subtracting algebraic expressions. Although students may be able to memorize a procedure, such as “distribute the negative” when subtracting algebraic expressions, they are often unable to make sense of this procedure. Our work suggests that part of students' difficulty in this area is that they do not conceptualize an algebraic expression as a composite unit. In the paragraphs below, we explain what is meant by composite units and how this construct helped frame our development of an instructional sequence to help students make sense of, and find meaning in, algebraic expressions and operations on algebraic expressions.

Students in Ms. Jones's second-grade class have just reconvened from working on several problems. Once the children are seated in the front of the room, Ms. Jones asks Casey to explain his answer of 16 for the following problem: “Aunt Mary has 31 pieces of candy on the counter and Uncle Johnny eats 15 pieces of candy.

In this paper, I use analyses of collective argumentation in a variety of classroom settings, from elementary school to a university-level differential equations class to illustrate various roles the teacher plays. These include initiating the negotiation of classroom norms that foster argumentation as the core of students’ mathematical activity, providing support for students as they interact with each other to develop arguments, and supplying argumentative supports (data, warrants, and backing) that are either omitted or left implicit. We gain two important insights from these analyses. First, an emphasis on argumentation can be used productively to provide openings in mathematical discussions for new mathematical concepts and tools to emerge. Second, the analyses demonstrate that teachers need to have both an in-depth understanding of students’ mathematical conceptual development and a sophisticated understanding of the mathematical concepts that underlie the instructional activities being used.

Contents: E. Yackel, P. Cobb, K. McClain, Preface. E. Yackel, Introduction: Perspectives on Semiotics and Instructional Design. Part I:Theoretical Considerations. P. Cobb, From Representations to Symoblizing: Introductory Comments on Semiotics and Mathematical Learning. A. Sfard, Symbolizing Mathematical Reality Into Being--Or How Mathematical Discourse and Mathematical Objects Create Each Other. W. Dorfler, Means for Meaning. B. van Oers, The Appropriation of Mathematical Symbols: A Psychosemiotic Approach to Mathematics Learning. R. Nemirovsky, S. Monk, "If You Look at It the Other Way...": An Exploration Into the Nature of Symbolizing. Part II:Instructional Design Issues Related to Symbolizing, Communicating, and Mathematizing. K. Gravemeijer, P. Cobb, J. Bowers, J. Whitenack, Symbolizing, Modeling, and Instructional Design. J. Bransford, L. Zech, D. Schwarts, B. Barron, N. Vye, The Cognition and Technology Group at Vanderbilt, Designs for Environments That Invite and Sustain Mathematical Thinking. R. Lehrer, L. Schauble, S. Carpenter, D. Penner, The Innerrelated Development of Inscriptions and Conceptual Understanding. R. Lesh, H.M. Doerr, Symbolizing, Communicating, and Mathematizing: Key Components of Models and Modeling. J. Bowers, Postscript: Integrating Themes on Discourse and Design.

This paper extends analyses of social interaction patterns that have been successful at characterizing elementary and secondary school classrooms to the learning and teaching of undergraduate mathematics. Using data from a classroom teaching experiment in differential equations as an example, we document the social and sociomathematical norms regarding explanation and discuss how these norms were constituted in this specific case. We focus on the social norms that students explain their thinking and try to make sense of other students' thinking and, for first-order differential equations, we document the sociomathematical norm that explanations be grounded in an interpretation of the rates of change. The analysis makes explicit certain social aspects of classroom environments that contribute to the conditions that make meaningful learning of mathematics possible.

The manner in which a horizontal addition and subtraction number sentence activity was constituted in one second grade classroom is analyzed for the purpose of discussing and illustrating how mathematical meaning is interactively constituted in the classroom. In particular, the teacher's emphasis on different solutions contributed to students' development of increasingly sophisticated concepts of ten. In turn, students' solutions contributed to the teacher's development of an increasingly sophisticated understanding of the children's mathematical activity and their concepts of ten.