| > What is more interesting to me, is what do you think a student misses out on, from a capability point of view, with an applications focused learning as opposed to one focused on reading and writing proofs? The generalizable value is not so much in collecting a bunch of discrete capabilities (they're there, but generally somewhat domain-specific) as it is in developing certain intuitions and habits of thought: what mathematicians call "mathematical maturity". A few examples: - Correcting trivial errors in otherwise correct arguments on the fly instead of getting hung up on them (as demonstrated all over this comment section). - Thinking in terms of your domain rather than however you happen to be choosing to represent it at the moment. This is why math papers can be riddled with "syntax errors" and yet still reach the right conclusions for the right reasons. These sorts of errors don't propagate out of control because they're not propagated at all: line N+1 isn't derived from line N: conceptual step N+1 is derived from conceptual step N, and then they're translated into lines N+1 and N independently. - Tracking, as you reason through something, whether your intuitions and heuristics can be formalized without having to actually do so. - More generally, being able to fluently move between different levels of formality as needed without suffering too much cognitive load at the transitions. - Approaching new topics by looking for structures you already understand, instead of trying to build everything up from primitives every time. Good programmers do the same, but often fail to generalize it beyond code. > Would a student who is not intending to become a mathematician still benefit from this approach? If they intend to go into a technical field, absolutely. > Would a middle aged man who was taught some “Linear Algebra” benefit from picking up a book such as the one referenced here? Depends on what you're looking for. If you want to learn other areas of math, linear algebra is more or less a hard requirement. If you want to be able to semiformally reason about linear algebra faster and more accurately, yes. If you just want better computational tricks, drink deep or not at all: they're out there, but a fair bit further down the road. |