This paper reports research into how mathematical explanations are constructed during conversation based on videotapes of pairs of student math teachers collaboratively writing explanations in geometry. In particular, we analyzed how disagreements about parts of their explanations were managed in these conversations. In contrast to research on disagreement in everyday conversation, explanation disagreements were more likely to overlap with preceding turns and to be stated baldly without prefaces, token agreements or qualifications. However, the observed frequencies of different kinds of disagreements were not consistent with a model favoring explicit substantive disgreement either. Instead, it is proposed that both the interpersonal concerns that would motivate a preference for agreement and the conceptual concerns for a quality explanation that would motivate a preference for substantive disagreement are being managed by participants. Disagreements are co-constructed, and conversants are seen to jointly employ complex devices for introducing and managing disagreement across turns that can satisfy both kinds of concerns with much less conflict betweeen them than might have been expected.

We propose three analytic representations of collaborative problem solving. Activity nests, a generalization of goal-subgoal trees, represent functional decompositions of task activity into components, using nesting to indicate operations that satisfy task functions. Semiotic networks, an extension of semantic networks, represent meanings as refers-to relations between symbolic expressions and other signifiers, and relations in situations and situation types, along with general relations between these meanings. Contribution Vagrants, an adaptation of contribution trees (Clark & Schaefer, 1989), represent how turn sequences collectively achieve task components. W e developed these representations to analyze how pairs of middle-school students constructed tables to represent quantitative properties of a simple physical device that models linear functions. Variations between activity nests of dififerent pairs support an explanation of activity in terms of attimement to constraints and to affordances and abilities, rather than following procedures. The semiotic networks support a hypothesis that task components are completed through accomplishing alignments of refers-to relations.which is a generalization of goal satisfaction. Similarities between the contribution diagrams support a general pattern that we call the turn structure of collaborative operations, in which task information is recognized and task operations are initiated, performed, and accepted. Interaction is organized into this structure in order to support mutually aligned intentions, understandings, actions, and agreements.

We present a theoretical analysis of students' implicit understanding of the concept of variables and functions, and present a cognitive model of this understanding based on the idea that reasoning involves a successful interaction between psychological agents and the things and other people in a situation. In the first part of the paper, we provide evidence that middle-and high-school students demonstrate implicit understanding of functional relations among quantities when they reason about a physical model of linear functions. Implicit understanding is knowledge of concepts or principles that enables and constrains performance, but is not articulate. In the second part of the pa^r, we describe several theoretical properties of our computational model: a.) activities are modeled as interactions between a parson and a situation; b.) reasoning is modeled as a form of activity that produces new information; and c.) understanding is modeled as attunement to the constraints of conceptual activities.

We discuss the process of mathematization in science, focusing on uses that theorists make of physical representations that w e refer to as abstracted models. W e review abstracted models constructed by Faraday and Maxwell in the mathematization of electromagnetic phenomena, including Maxwell's use of an analogy between continuum dynamics and electromagnetism. W e discuss ways in which this example requires major modifications of current cognitive theories of analogical reasoning and scientific induction, especially in the need to understand the use of abstracted models containing theoretically meaningful objects that can be manipulated and modified in the development of new concepts and mathematized representations.

We report analyses of the construction and interpretation of mathematical symbols that refer to quantitative properties and relations of a physical system. Middle-school students solved problems that involved contracting tables, equations and graphs to represent linear functions of a device where blocks are moved varying distances by turning a handle that winds string around spools of different sizes. Previous research analyzed activities of reasoning about quantities of this system as attunement to constraints and affordances, a characterization of students' imphcit understanding of concepts of variable and linear functions. This report concerns activities of representing quantitative properties and relations using mathematical notations. W e are developing analyses of constructing and interpreting tables, equations, and graphs in terms of attunement to constraints and affordances of the represented system, the system of notations, and relations between the constraints of the notations and the represented domain. W e present examples that illustrate concepts of semantic clumps, groups, and morphisms; descriptive and demonstrative representations; multiple referent domains; and constructions of meaning in contributions to conversational discourse.