Timeline for Virtual algebraic calculation within proofs
Current License: CC BY-SA 3.0
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| Aug 22, 2013 at 22:12 | history | edited | user9072 |
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| Oct 15, 2011 at 22:12 | comment | added | David Feldman | In the classroom, over and over, a lack of intuition about the (big) distribution law seems to stop students in their tracks. Since the geometrical intuition behind the distributive law involves high dimensions when one has many factors, lack of geometric intuition might play the culprit. | |
| Oct 15, 2011 at 22:08 | comment | added | David Feldman | >David: Can you specify what the problem is exactly? Well of course (various) students have many (different) problems digesting theoretical mathematics. I'm looking for examples and then I hope to experiment a bit and formulate precise hypotheses. I suspect that proofs by contradiction exacerbate a more fundamental problem: absent experience doing complex calculations, students have very little intuition about the emergent features of algebraic calculation. Their teachers and books come from a culture that predates technology-reformed-curricula and take such intuition for granted. | |
| Oct 15, 2011 at 16:45 | comment | added | darij grinberg | ... algebraic identities. Of course, this can be somewhat difficult depending on the proof. | |
| Oct 15, 2011 at 16:45 | comment | added | darij grinberg | David: Can you specify what the problem is exactly: (a) They don't understand arguments about calculations when you show them such arguments, or (b) They can't come up with such arguments on their own? And (1) is the problem specific to proofs by contradiction or (2) is it only exacerbated by the fact that they can't get an example to work with? In case (1), I suggest dividing the proofs up in modular pieces in such a way that the reasoning-about-calculations part is contained inside those pieces which are not proofs-by-contradictions but, instead, honest-to-god and constructive ... | |
| Oct 15, 2011 at 9:45 | answer | added | Julien Puydt | timeline score: 5 | |
| Oct 15, 2011 at 7:02 | answer | added | Alexander Woo | timeline score: 5 | |
| Oct 15, 2011 at 6:58 | comment | added | Alexander Woo | I think many mathematicians aren't very good with these either. For the most part, only combinatorialists and certain kinds of analysts really need this skill on a regular basis. Some mathematicians might even argue that part of the progress of mathematics is abstracting away calculations. | |
| Oct 15, 2011 at 6:29 | comment | added | David Feldman | A different sort of example: classification of modules over a PID by means of describing the contingencies in a theoretical sequence of matrix manipulations. What interests me: if our curricula don't emphasize pencil-and-paper computations on examples on substantial size, have we prepared students to understand theoretical narratives that reference such computations? | |
| Oct 15, 2011 at 6:26 | comment | added | David Feldman | Hi Gerhard, So here's another example: the Greene-Nijenhuis-Wilf probabilistic proof of the hook formula. The proof gives a uniform generation algorithm for standard tableaux and succeeds by showing that each one occurs with probability equal to the reciprocal of the desired count. The probability gets factored as a product of conditional probabilities, which, working backwards from the expected answer, get expanded as products with many factors (the big distributive law - this gets my students lost). Finally each of the exponentially many summands gets an interpretation. | |
| Oct 15, 2011 at 4:55 | comment | added | Gerhard Paseman | I would like to see more examples, just so I can be on the same page. Otherwise I might suggest something like constructing the free term algebra for a (universal algebraic) variety as an example, and I do not yet know that that is what you want. If it is, that and similar examples might be found in George Bergman's Math 245 text he uses at UC Berkeley. Gerhard "Ask Me About System Design" Paseman, 2011.10.14 | |
| Oct 15, 2011 at 4:28 | history | asked | David Feldman | CC BY-SA 3.0 |