alt Hacker NewsI think for many people (myself included) understanding mathematics is rooted in application because it helps bridge the divide between intuition and rote memorization. Without the application, IMO instructors are doing a disservice to their students and pedagogy of mathematics itself. They’re intentionally ignoring a significant fraction of the class, unless they’re teaching some esoteric grad level pure math.
Yeah this was my position. I don't mean "Sally has six apples" necessarily, but my brain needs some concrete handles to grab onto so it can actually manipulate this stuff intuitively.
See also: Using spaced repetition to see through a piece of mathematics
https://news.ycombinator.com/item?id=18895613
Where the author describes his mind eventually turning the mathematical objects (symbols) into some kind of mental objects he was able to actually manipulate "directly". As a result of this facility, his mind spontaneously produced many new insights into the matter.
That level of fluency obviously requires a great deal of work, but in my experience it can be approximated and accelerated with analogies. (e.g. see Feynman's teaching.)