Timeline for Short Course Suggestions For High School Students
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 22, 2020 at 23:49 | comment | added | Joel Dodge | I'd hate to be judged by my worst take from 10 years ago so here's hoping darji has improved their bad opinions since then. I'll just answer the feeling that calculus is for lesser creatures by recalling something that efim zelmanov said once in a seminar: "I love teaching calculus because I get to stand in front of the class and talk about the most beautiful mathematics ever created as if I thought of it yesterday." | |
| Nov 22, 2020 at 23:17 | comment | added | William Bell | I think basic set theory, analysis, and elementary number theory are all great ideas for a high school student wishing to get used to what real mathematics is like. In particular, I think analysis is probably the best preparation for higher mathematics. | |
| Dec 22, 2011 at 16:17 | comment | added | darij grinberg | My feeling about set theorists is that many of them have been brought to cardinal arithmetic etc. through its applications (forcing, decidability and computability questions etc.), rather than doing it for its own sake. A course taking these real applications (instead of the faux naturality of studying infinite sets) could be very interesting. | |
| Dec 22, 2011 at 8:12 | comment | added | Lovre | @Richard: I must say I had similar feelings about what I wrote, but I feel that [12 workings days * 2 hours] might be enough, if everyone could stay focused, all examples nontrivial and carefully chosen, and little time spent on computational aspects (that's how I envisioned the courses). | |
| Dec 22, 2011 at 8:06 | comment | added | Lovre | .. resonate as something fundamental and deep about natural numbers, while modular arithmetic and quadratic residues would probably seem as something interesting, but not very deep or general. | |
| Dec 22, 2011 at 7:56 | comment | added | Lovre | @darij: Thanks for your detailed feedback. I personally don't consider set theory to be devoid of intuition and understanding, and it certainly doesn't lose everyone, since some people go and become set theorists. For analysis, I can see where you're coming from; my feelings for analysis are probably influenced by the fact that I have taught myself everything I know about it, and that I don't see the technical work as automatically dry and boring. Concerning the elementary number theory, I somewhat agree, but you could prove other facts about the primes.. at that level, primes would.. | |
| Dec 22, 2011 at 2:23 | comment | added | Richard Rast | While they might be interesting topics, all three seem to be a huge overreach for a two week course. | |
| Dec 22, 2011 at 1:48 | comment | added | darij grinberg | As for elementary number theory, it is a good proposal, but the Prime Number Theorem is a problematic matter: just stating it doesn't help a lot; proving it might be too much for the course. I'd better go with quadratic residues and some of the more interesting modular arithmetic. | |
| Dec 22, 2011 at 1:44 | comment | added | darij grinberg | ... If you want to change something about this, I think the better idea would be to introduce some of the more surprising and fresh material (complex analysis, $p$-adic analysis, nonstandard analysis - handle with care -, even some of the nicer numerical analysis), rather than just to do the standard 1-dimensional real calculus with more attention to proofs and details. | |
| Dec 22, 2011 at 1:42 | comment | added | darij grinberg | ... already overstressed ad nauseam in school, such an analysis. In my circles (German IMO training 2004-2006) the predominant feeling towards (single-variable) calculus was annoyed contempt; differentiating and integrating was something to be left to lesser beings (applied mathematicians, schoolteachers), whereas topological stuff like continuity and the construction of the reals was considered reasonable but boring and technical. Only complex analysis evoked better feelings. This probably has to do with the fact that in school, the only analysis taught is trivial and boring analysis. ... | |
| Dec 22, 2011 at 1:39 | comment | added | darij grinberg | I can't help stating the fact that neither of the three suggested courses would improve my interest in mathematics in the long run. Set theory fascinates people when they first hear of it (I too had such a phase during my school time), but at some point tends to lose them and leave nothing behind except for an impression that mathematics is about playing around with virtual infinities beyound any hope of real understanding or intuition. What, for instance, does Cantor-Schröder-Bernstein mean? I also don't consider it a good idea to give a more technical and axiomatic take on a subject ... | |
| Dec 21, 2011 at 20:41 | comment | added | Alan Haynes | The fact that you are a high school student can not be ignored here... it is good to hear what you think, and I agree that those are interesting and natural topics. | |
| Dec 21, 2011 at 20:18 | history | edited | Lovre | CC BY-SA 3.0 |
edited body; deleted 43 characters in body
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| Dec 21, 2011 at 20:13 | history | answered | Lovre | CC BY-SA 3.0 |