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by benrbray
1734 days ago
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As I mentioned elsewhere, my biggest complaint about my undergrad math education is that the topics felt entirely disjoint. We could take courses in any order, and it effectively felt like we started over from scratch each semester, rather than allowing the material to compound over four years. If courses are offered in a more rigid order, I think that the gradual introduction of categorical language could go a long way. It does not need to be emphasized, but it should not be neglected. Textbook authors let some categorical language slip through so the time, and I think eductors are doing students a disservice my neglecting it. For instance, I was utterly confused about the "natural" isomorphism between a vector space and its double dual, as well as the "universal property" of tensor products, free modules, etc.. I also saw commutative diagrams in my complex analysis and topology books, but no one bothered to explain where they come from or what the rules are. Much later I finally took the time to study category theory on my own, and I can't help but think the whole experience would have been much more efficient had my professors exposed us to the idea of working with abstract morphisms much earlier. |
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I agree. When teaching, I try to choose my terms carefully so someone would not go astray from the subject matter. It is remarkable how easy it is to get stuck on a side-quest with looking up abstract terms, while-so missing the main topic. imo the graceful way to handle it is to drop references (and disclaimers in case they are time-consuming) when introducing new things.
Somehow math learning has the property that 'struggle with x, finish x. learn y. if I knew y before x it would've been so much smoother!' sounds correct, but in practice it often does not go so well.