Student reasoning with series expansions in introductory mathematical methods

As part of a project to investigate student use of mathematics in upper-division physics courses and develop instructional materials, we have examined how students use series expansions as a means of approximating and simplifying expressions in theory-oriented physics courses. Student responses were collected on pre- and post-instruction written tasks in which students were prompted to use a series to approximate an expression arising from a problem in electricity and magnetism. Students were required to identify an appropriate variable in which to expand and use the expansion to generate an approximation. Despite prior experience with series in calculus and physics, students struggled to determine appropriate quantities in which to expand and most did not attend to the units of the terms or the convergence of their series. While student success rates improved after targeted instruction, many students continued to expand in quantities that were neither dimensionless nor small, and thus unproductive for modeling.