This article presents a case study in which two eighth-grade students developed knowledge for modeling a physical device called a winch. In particular, the students learned (a) to distinguish equations that are true for any value of the independent variable from equations that constrain the independent variable to a unique value and (b) to solve the latter type of equation to determine when specific physical events occur. The analysis of how these understandings emerged led to two results. First, the analysis demonstrated that students have and can use criteria for evaluating algebraic representations. Second, the analysis led to a theoretical frame that explains how students can develop modeling knowledge by coordinating such criteria with knowledge for generating and using algebraic representations. The frame extends research on students' algebraic modeling, cognitive processes and structures for using mathematical representations, and the development of mathematical knowledge.
Andrew Izsák, The University of Georgia, Department of Mathematics Education, 105 Aderhold Hall, Athens, GA 30602-7124; email@example.com