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There are a number of aspects of problem-solving that upper-division students need to be taught explicitly:
- Using geometric and harmonic reasoning. Students who go fluidly back and forth between algebraic symbols and a problem's geometry are referred to as thinking “harmonically”.
- Recognizing what is variable vs. constant throughout a problem, including during integration.
- Engaging in reflection such as doing dimensional analysis, thinking about the meaning of their answer, and thinking about limiting cases.