## UMD
PERG PhD Dissertations: |

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 deal- ing 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 intro- ductory physics for life sciences course at the University of Maryland. There are both qualitative and quantitative threads to this work. The qualitative work an- alyzes 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 signif- icantly more often in playing certain e-games, and describe the signicance of this nding 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 dierent 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.

- Introduction
- Blending physical knowledge with mathematical form in physics problem solving
- How do students' ontological metaphors for equations change depending on what epistemic game they play?
- The Math Epistemic Games Survey (MEGS)
- Comparing factor analysis and network-based clustering methods in analyzing multiple-choice tests
- Conclusions

Appendices

- Interview protocols
- List of epistemic games
- Codebook for ontological metaphors for equations
- Math Epistemic Games survey
- MEGS test administrations

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