Resources for Refining Mathematical Conceptions: Case Studies in Learning About Linear Functions

In this article, I describe the refinement of a conception in the domain of linear functions, the use of the x-intercept in equations of the form y = mx + b, to illustrate the resources used by students to refine a conception. The examples presented show how students refined their use of the x-intercept by (a) narrowing the problem contexts in which they applied this conception, (b) making connections between conceptions, (c) using a mathematical procedure, and (d) refining their verbal descriptions of how the graph of a line changes as a result of a change in the equation. The analysis draws on diSessa's (1993) theory of science learning and the recommendations made by Smith, diSessa, and Roschelle (1993) for describing conceptual change. The results extend this prior work from science to mathematics and identify resources used by students that teachers can draw on to support the learning of mathematical concepts.