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### Geometric and Harmonic Reasoning

We have begun to study how students use their understanding of the geometry of a problem to make progress in problem solving. We are also looking at the extent to which students go from geometric understanding to algebraic symbols or go from a mathematical expression to geometric understanding. Students who go fluidly back and forth between the geometry and the symbols are referred to as thinking “harmonically”.

#### PhD Dissertation in Progress regarding Geometric and Harmonic Reasoning in E&M Problems

A Science Education PhD student is working on a dissertation related to geometric and harmonic reasoning in the context of solving E&M problems in groups. This work will include written materials, as well as video clips of students using different types of reasoning. Link to Dissertation

#### Video of a group of students translating algebraic symbols to a geometric picture

In this one-minute video clip, students have already been working in a group of three for about 15 minutes trying to find the magnetic vector potential in all space for a spinning ring of charge. This activity is described in MagneticVectorPotential Activity. They have already determined the current and other aspects of the problem and are now trying to work on the denominator of the integral which is | r - r' |. They are creating a geometric understanding of what this means.