Assoc. Prof. Dr. Suzanne White Brahmia (UW Seattle): Calculus reasoning in physics contexts: Beyond the first year

Most physics majors enroll in calculus courses and calculus-based introductory physics courses in their first year of study. One of the goals of this curriculum is that these students will be able to use calculus in the situations when physical quantities are no longer held constant, and other assumptions are relaxed.
While many students master procedures in their calculus courses, and possibly develop skills using the logic structures of mathematical proof, research shows that they might not develop a skill of using calculus reasoning in the situations that require physics modeling. They view the mathematics in mathematics courses as distinct from what they are doing in physics.
In this talk I will build on prior evidence of this calculus-physics mismatch to show possible ways for closing this gap. For example, a known difficulty is visualizing what happens to the physical quantity represented by the infinitesimal dr in a definite integral, like \int_0^R F(r) dr, in the limit that dr goes to zero. What are you summing up if you are multiplying by zero? Where do the units go? Physics quantities can help students grasp challenging mathematical ideas, like limits and products, with a careful weaving of activities that develop sensemaking with both mathematical and physical quantities in a situation that blends the two.
Our current research focuses on activities that can be used with small groups both in lectures and smaller group meetings, as well as exam settings, designed to help close this gap in student reasoning. These materials are useful both as applications in vector calculus and differential equations courses, and as transition materials to advanced physics courses in mechanics and electricity and magnetism---where the students need a foundation from these mathematics courses to succeed.
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