Section 5.8 Recognizing language differences and representation differences between mathematics and physics.
¶Learning physics is a lot like learning a language, not only because many common everyday words such as “force” have very specific meanings in physics contexts but also because many ideas are expressed mathematically. There are many ways in which physicists and mathematicians differ in how they use words as well as symbolic and visual representations. A companion NSF project, Bridging the Vector Calculus Gap, explored these differences (http://math.oregonstate.edu/bridge). The proposal for that project stated:
A major part of the problem is the traditional mathematics emphasis on Cartesian coordinates to describe vectors as triples of numbers, rather than emphasizing that vectors are arrows in space. This leads to the all-important dot and cross products being memorized as algebraic formulas, rather than statements about projections and areas, respectively. It is hardly surprising that many students are then barely able to compute line and surface integrals, or the divergence and curl of a vector field, let alone understand their geometric interpretation.
The Bridge project resulted in an online book, The Geometry of Vector Calculus http://math.oregonstate.edu/BridgeBook/ (Dray & Manogue, 2009-2015) that emphasizes geometric visualization when evaluating line and surface integrals.
One of the interviewees was a graduate teaching assistant who focused his dissertation research on ways in which the paradigms students used visualizations for problem-solving. In reflecting on his research, he contrasted his own undergraduate experiences elsewhere with what the students were learning to do in the paradigm courses:
It was a lot of stuff in vector fields, trying to get a picture of how students at the junior year were actually using visual representations and connecting those to algebraic representations when they were solving problems. In the symmetry paradigm one of the big themes was how do you use the geometric structure of the physics formalism, how do you use the geometric structure to simplify the problem? which in my experience going through electricity and magnetism and going through similar stuff in classical mechanics as an undergraduate, we didn't think about that at all, we just learned how the teacher solved the problem, you remembered those classic examples when you tried to figure it out.
The paradigms students were learning to think visually as well as algebraically whereas as an undergraduate he had only learned to do the algebraic procedures demonstrated by the professor:
There was a lot more time and effort spent in the symmetries paradigm trying to get you to see a visual picture of what was going on and using that to simplify the algebra; I learned to do the algebra really well as an undergraduate but the visual component of it wasn't there; I just did the things that my professor did because that's what you needed to do.
This approach was important for the students, especially those who were not headed to graduate school:
The thing also that I remember very vividly, the students who did not go on to graduate school, benefited the most really, trying during that junior year to get them to have that deep understanding of the central ideas, rather than just being able to do things, or being able to work through a textbook, because it's so much more like what you do in graduate school, like really trying to have a deep understanding rather than just grinding through the material so that you know stuff.
One of the goals of the paradigms courses was to foster this shift for students from ‘doing what the professor does’ toward a much deeper understanding of mathematical descriptions of various physical systems.