Modeling the co-development of strategic and conceptual knowledge in mathematical problem solving

Congralations to Mariana Levin who successfully completed her dissertation entitled: Modeling the co-development of strategic and conceptual knowledge in mathematical problem solving

Abstract: This dissertation elaborates the process by which a program of research around studying the co-development of strategic and conceptual knowledge in mathematical problem solving emerged from a microgenetic learning analysis done in dialogue with a heuristic epistemological framework, Knowledge in Pieces (diSessa, 1993). The core analytic work in developing a theoretical model of the co-development of strategic and conceptual knowledge was based on an in-depth microgenetic analysis of a single pre-algebra student who largely independently constructed a deterministic and essentially algebraic algorithm for solving algebra word problems of an underlying linear structure. This strategy emerged over the course of the students’ work on several problems in the context of semi-structured sessions. The student’s later strategy is seen to emerge as a series of gradual refinements to an earlier strategy used in solving problems of a similar underlying form. To an observer with no knowledge of the intervening episodes, this shift in strategies would look quite dramatic and discontinuous. However, by conducting a close analysis of intermediate episodes, one can see significant continuity. The data for the analysis involved a complete record of videotaped interactions between the researcher and the student, as well as all student work.

While this is only a particular case of strategy emergence, the pursuit of a principled explanation of such a learning event implicates a number of foundational issues. In addition to characterizing the evolution of the student’s approaches to solving problems, we might ask what conceptual knowledge the student drew upon. We might wonder if the student only learned a new way to accomplish the local goal of solving problems that he recognized as having a similar structure to problems he had solved previously, or whether there was some deeper conceptual residue of the experience. We might wonder if strategies themselves should be considered as a “type” of knowledge and if so, how might we characterize strategic knowledge.

The existing strategy change literature (see Siegler, 2006 for a review) uses microgenetic methods to track observable changes in strategy usage. In my analysis, I share a similar analytic grain size and top-level approach. However, through the process of negotiating with a heuristic framework, Knowledge in Pieces (diSessa, 1993), I came to characterize both strategic knowledge and the particular conceptual knowledge that undergirded the implementation of strategies as complex knowledge systems. The development of a new knowledge ontology — a strategy system — is a central innovation of the analysis and was instrumental in modeling the interplay between strategic and conceptual knowledge in problem solving.

Advisors: Alan H. Schoenfeld and Andrea A. diSessa

Currently, Mari is a Postdoctoral Scholar in the Program in Mathematics Education (PriME) at Michigan State University.


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