Table of Contents
Reasoning With Tools and Inscriptions
Paul Cobb
The unit of analysis that I use when discussing the 2 sample episodes is that of a classroom mathematical practice together with the students' diverse ways of contributing to its continual regeneration. Analyses cast in terms of this unit account for the mathematical learning of the classroom community. As I clarify, a classroom mathematical practice is itself composed of 3 interrelated types of norms: a normative purpose, normative standards of argumentation, and normative ways of reasoning with tools and inscriptions. In keeping with the theme of this special issue, I step back from the sample analysis by focusing on the last of these 3 aspects. In doing so, I introduce the notion of a chain of signification to illustrate a way of accounting for mathematical learning in semiotic terms.
Teachers' and Students' Understanding: The Role of Tools and Inscriptions in Supporting Effective Communication
Kay McClain
The 2 episodes featured in this issue provide a rich setting in which to investigate not only the influence but also the confluence of students' and teacher's understandings on the quality of whole class discussions. In particular, I focus on the communication between the students and myself as the teacher in the classroom. This unique perspective allows me to offer insights into the teacher's decision-making process and how those decisions influence the opportunities for learning. As part of the analysis, I consider the role of tools in both supporting and constraining communication in the classroom. The analysis therefore makes explicit the tensions in teaching by highlighting the importance of the teacher's understanding of the students' offered explanations and justifications and the mathematics that is to be taught.
Orchestrating the Multiple Voices and Inscriptions of a Mathematics Classroom
Ellice Ann Forman and Ellen Ansell
The purpose of this article is to explore how inscriptions are used to create argumentative positions in the 2 classroom episodes, Batteries and AIDS. This activity is similar to a prominent practice of scientific communities: the use of inscriptions to advance knowledge claims. We begin by discussing the meaning and function of inscriptions in scientific communities and then apply some notions from the history and sociology of science to our analyses of the case study data in the 2 episodes. In addition, we address some of the challenges for teachers in incorporating scientific practices into their classroom activities. Among these challenges is the need to alter the nature of the problems and inscriptions used as well as change classroom discourse structures. In our analyses of the 2 episodes, we found that the classroom activities resembled those of scientific communities in several ways: Real-world, dilemma-driven problems were presented; students evaluated the inscriptions offered in terms of their adequacy for advancing particular knowledge claims; and the teachers helped their students reflect on, clarify, expand, evaluate, oppose, or integrate each other's explanations into their argumentative positions. We also found that the teachers legitimated student contributions to the discussion by revoicing their arguments. Despite the positive findings from our analyses, we were able to identify and illustrate one additional challenge for future design experiments: How do we orchestrate the multiple voices of the classroom in an equitable fashion while pursuing our instructional goals?
Children's Developing Mathematics in Collective Practices: A Framework for Analysis
Geoffrey B. Saxe
This article presents a cultural-developmental framework for the analysis of children's mathematics in collective practices and illustrates the heuristic value of the framework through the analysis of videotaped episodes drawn from a middle-school classroom. The framework is presented in 2 related parts. The first targets the children's emerging mathematical goals in collective practices, with a particular focus on the complex role that artifacts play in children's emerging goals. The second part focuses on children's developing mathematics that takes form in their goal-directed activities: (a) Microgenetic analyses concern the process whereby children structure cultural forms like artifacts to serve particular functions as they accomplish emerging mathematical goals; (b) sociogenetic analyses concern the spread or travel of mathematical forms and associated functions within a community of individuals; and (c) the functions that they serve over the course of children's development. The analyses of the classroom episodes points to the promise (and limitations) of the framework as a method for furthering our understanding of the interplay between social and developmental processes in children's mathematics.
Representational Tools and Mathematical Understanding
Analucia D. Schliemann
Understanding how cultural mediators and social interaction promote meaningful learning requires that each student's perspective, reasoning, and construction processes be taken into account. In my analysis of the classroom episodes, I consider individual students' progress as they use tools, discuss data distributions, and interact with teachers and their peers. I argue that data display tools provide a partial context for discussions but do not constrain the students' interpretations or the way they reason about the data. Students' approaches to the mathematical relations discussed in the classroom result rather from the meaning they attribute to the different features of the displays, the teachers' questions, and the evolving interaction with their peers.
The Interplay of Intimations and Implementations: Generating New Discourse with New Symbolic Tools
Anna Sfard
The analysis of the Batteries and AIDS episodes presented in this article is guided by the assumption that thinking can be conceptualized as an activity of communication, and learning can be regarded as modifying and extending one's discursive ways. Within this framework, 1 of the aims of mathematical learning is to become skillful in the discursive use of designated symbolic artifacts supposed to mediate solving certain types of problems. My analysis of the learning episodes is aimed at uncovering the ways in which the discursive uses of such new symbolic tools were interactively constructed by the students. I argue that the construction process is extremely complex because of a certain inherent circularity of this process. The analysis reveals that this difficulty may be overcome by the gradual dialectic adjustment of former discursive habits to new contexts. The adjustment happens gradually, through cycles of intimations about the applicability of the old habits followed by implementations in which this applicability is examined. This intricate interplay of intimations and implementations is found in both the Batteries and the AIDS episodes and is presented in detail in the article. I also show that, with time, students significantly increased their mastery of this particular discursive mechanism.
Commentaries
ANNOUNCEMENT