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There are a number of aspects of problem-solving that upper-division students need to be taught explicitly:

  • How to break up a complicated problem into smaller pieces. See the Pieces Link or the Pieces PDF
  • 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.

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