Mathematical actions as procedural resources: An example from the separation of variables

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in separation include division or multiplication after properly grouping terms in an equation, moving terms (again, at times grouped) from one location on the page to another, or simply carrying out separation as a single act without showing any steps. We describe student use of these procedures in terms of Hammer’s resources, showing that each of the previously listed procedures is its own “piece” of a larger problem solving activity. Our data come from group examinations of students separating variables while solving an air resistance problem in an intermediate mechanics class. Through detailed analysis of four groups of students, we motivate that the mathematical procedures are resources and show the issues that students must resolve in order to successfully separate variables. We use this analysis to suggest ways in which new resources (such as separation) come to be.

Figure 1

When solving a typical air resistance problem, all student groups sought to separate variables in the last equation shown.

Figure 2

Sarah’s procedure of separation contains multiple procedures. These include formal mathematical operations and explicit activity to group terms in the equation appropriately. She also indicates an equivalence between a description of moving and the mathematics of multiplication.

Figure 3

Dan’s resource of separation contains the procedures of grouping and moving.

Figure 4

Jared’s procedure of separation included grouping and division and an explicit statement that subtraction was an inappropriate procedure.

Figure 5

Bill’s two uses of moving without grouping are in contrast to Derrick’s use of multiplication and division after grouping.