In this thesis, I study some aspects of how students learn to use math to

make sense of physical phenomena. Solving physics problems usually requires dealing

with algebraic expressions. That can take the form of reading equations you’re

given, manipulating them, or creating them. It’s possible to use equations simply

according to formal rules of algebra, but most students also learn to interpret the

equations and use the equations as ways to bolster their physical understanding.

Here, I report on three years of studying this mathematical sensemaking an introductory

physics for life sciences course at the University of Maryland. There are

both qualitative and quantitative threads to this work. The qualitative work analyzes

a series of problem-solving interviews. First, I use case studies from these

interviews to survey the variety of rich cognitive tools students bring to bear on

problems around use of algebraic expressions and equations and make observations

on potential applications to instruction. Next, I draw a connection between the

ontological metaphors students use for equations and the epistemic games they play

while solving problems. I show that certain ontological metaphors are used significantly

more often in playing certain e-games, and describe the significance of this

finding for problem solving. The quantitative thread of this thesis describes how my

collaborators and I created and analyzed the Math Epistemic Games Survey, a math

concept inventory that studies how students’ uptake of problem-solving strategies

such as “check the extreme cases” progressed over the year-long physics course. I

show that students on average make little progress on the MEGS over a semester,

which suggests that curriculum development in this area has great potential upside.

Finally, I test several different methods of analyzing the multiple-choice test data

that go beyond counting correct and incorrect answers to extract lessons from the

distractors students choose. Using these methods on computer-simulated data and

real data from the MEGS, I caution against drawing too-strong conclusions from

their results.