Timeline for Thinking and Explaining
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2010 at 12:26 | comment | added | Bill Thurston | @Deane Yang: Have you read the classic Proofs and Refutations: the Logic of Mathematical Discovery, by Imre Lakatos? It makes a compelling case for the dialectical method, the value of mistakes and corrections. @Ryan Budney: It's very tricky to figure out what will convey better to students. Often students have complex thought processes we're unaware of, struggling to fit into simplified explanations we give. It might well work better if you explained your way of thinking. I'm not yet convinced, though, that going to 2 dimensions is a clearer mental image than scaling sin(x) by factor x. | |
| Oct 6, 2010 at 12:05 | comment | added | Bill Thurston | I agree, beginning math is a very rich and intriguing area. I've discussed arithmetic questions with many young children, and they are often very creative in strategies to think their way to an answer. It really requires being on your toes to discern their thought processes, because the words do not match adult expectations; they often take phrases with logical meanings that I've suppressed because of convention. To teach math to kids, I think it's paramount to encourage them to think, rather than teach conventional "borrowing 1" type stories. Early math teaching usually suppresses thinking. | |
| Oct 4, 2010 at 0:18 | comment | added | JBorger | I also have the same divided brain when doing arithmetic, especially when adding up 8 or so 1-digit numbers, when there are often several good orders to add them. (This mostly comes up around exam time.) The attitudes of the two halves to each other are also the same. Amazing. | |
| Oct 3, 2010 at 22:08 | comment | added | Deane Yang | Actually, I like explaining ways of solving problems that involve making mistakes, intentional or not, and then figuring out how to correct them. I like this better than trying to teach error-free algorithms, because it incorporates the error-checking as a natural part of the process. I believe we should be teaching more systematic methods for finding and correcting errors. Students should learn when guessing, checking, and correcting is faster and easier than a more direct algorithm. Integrals that require more than one integration by parts is an obvious example of this. | |
| Oct 3, 2010 at 21:13 | comment | added | Ryan Budney | I have similar problems when explaining graphing to calculus students. When I see a functional expression like $x\sin(x)$ my mind jumps to decomposing it as a composite $x \to (x,\sin(x))$ with with $(x,y)\longmapsto xy$, and since I know the graphs of these two functions, assembling the graph of the composite is quick, like pasting two images together. But when teaching calculus we're deliberately not thinking in a multi-variable context, and in particular trying to deduce everything via a linear outlook: differentiate, find zeros, check 2nd derivative there, find where increasing... etc | |
| Oct 3, 2010 at 21:05 | history | answered | gowers | CC BY-SA 2.5 |