What is Conceptual Understanding?

This article deals with the issue of conceptual understanding in mathematics education. Still, it is a fascinating article that may even apply to how we understand things in physics.

One interesting aspect to this essay that deals with physics and engineering is this:

One of the principal reason why mathematics majors students progress far, far more slowly in learning new mathematical techniques at university than do their colleagues in physics and engineering, is that the mathematics faculty seek to achieve full conceptual understanding in mathematics majors, whereas what future physicists and engineers need is (at most) functional understanding. (Arguably most of them don't really need that either; rather what they require is another of the five strands of mathematical proficiency, procedural fluency.) I have taught at universities where the engineering faculty insisted on teaching their own mathematics, precisely because they wanted their students to progress much faster (and more superficially) through the material than the mathematicians were prepared to do.

I actually rather agree with that. In my "So You Want To Be A Physicist" essay, I've stated that some time, physics majors need more mathematics than mathematics major. However, they don't need to know it that deeply. They only need to know how to use it correctly. That is why many physics departments teach their students their own brand of "mathematical physics", mainly to arm their students with the necessary "tools" that they would need to tackle advanced physics courses without having to take all the equivalent mathematics classes that deal with these topics with rigor.

So yes, I'd say "procedural fluency" would be an apt description of what most physics majors need as far as mathematics is concerned.

Zz.