A Theoretical Framework for Research in Algebra: Modification of Janvier's "Star" Model of Function Understanding.

A pentagonal model, based on the star model of function understanding of C. Janvier (1987), is presented as a framework for the design and interpretation of research in the area of learning the concept of mathematical function. The five vertices of the pentagon correspond to five common representations of mathematical function: (1) graph; (2) table; (3) algebraic formula; (4) verbal description; and (5) situation. The 10 line segments forming the sides and diagonals of the pentagon represent translations between representations. As an example of the usefulness of the model as a framework for designing research and interpreting research data, a study of the understandings of 52 preservice elementary teachers concerning tasks involved in building and using linear mathematical models is presented. Objectives of the study were to determine if the concrete activity of data collection has a measurable effect on student performance on the model building and using tasks, and to test the usefulness of the model for framing research. Data collection activities appear to interfere with, rather than enhance, performance on these tasks. The model was found useful for framing and interpreting the research. It can be expected to be a useful aid in designing instruction. Five figures and two tables present study data. (SLD)