Comparisons of a Physical Model and Computer Representationsin Reasoning and Learning About Linear Functions

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The proposed research will investigate students' reasoning about quantitative variables and functions in different situations that support understanding: a physical device in which quantitities are related by linear functions, a computer simulation of the device that includes numerical information, and a computer-based "functions machine" that presents only numerical information. The research will focus on the different ways that these models and representations present resources for reasoning and learning about functions. Students will be interviewed in pairs, with parallel series of problems and questions for the three systems. Students who will be interviewed will include middle-school students who have not studied algebra and high-school students who have studied algebra for one year. Teachers will be asked to recommend students whose performance in mathematics is about at the average of national norms. In one study, students will answer questions and solve problems with the systems with whatever additional representations they choose to use. In two subsequent studies, students will learn to construct (l) formulas and (2) graphs to represent functional relations among quantitities in the situations. A fourth study, to be begun in this grant period, will combine learning of formulas and graphs as representations of the systems. Conclusions about the students' reasoning and learning will be based on their success in solving problems and answering questions, the explanations they give of their solutions and answers, and the linguistic terms that they use to refer to quantitative properties and relations. Finally, the work will focus on specific aspects of students' reasoning and understanding that are similar and different situations, relating differences to the different kinds of information and activity that the situations provide.