Timeline for Teaching undergraduate students to write proofs
Current License: CC BY-SA 2.5
16 events
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| Jun 22, 2022 at 8:14 | history | edited | CommunityBot |
replaced http://www.math.uga.edu/~pete with http://alpha.math.uga.edu/~pete
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| Feb 17, 2019 at 18:08 | comment | added | WestCoastProjects | I was a top engineering student taking "math" courses with the good math majors (tested out of first 2 calc courses which at the time was unusual). There were no proofs. I will eternally feel robbed. | |
| Jan 27, 2011 at 19:28 | comment | added | Pete L. Clark | @Mark Adams: of course you can ask! However, I have almost nothing by way of an answer, because I know very little about software to check proofs and I have absolutely never used such things in a course setting. It's certainly an interesting idea. Could you say a little more about it? For instance -- and forgive me if this is a painfully naive question -- what are the fundamental differences between a computer proof assistant and a human grader? (Not philosophically, I mean, but in terms of the effect on the students' learning.) | |
| Jan 27, 2011 at 16:40 | comment | added | Mark Adams | Pete, can I ask whether you think it would be instructive for the students to use supporting software in such a course? What I mean is having something like a theorem prover (also called a "proof assistant" or "proof checker") checking the steps that students do in exercises. The student would perhaps first do a proof on paper, and then check it step-for-step using the software. (Current theorem provers would be of little use for this, by the way, because they are incapable of supporting a one-to-one correspondence between pen-and-paper steps and theorem prover steps.) | |
| Jan 14, 2011 at 15:38 | comment | added | Harry Gindi | Hey PLC, I learned Algebra II in that same self-paced course at CTY the summer before my freshman year in HS as well! Comrade! | |
| Jan 14, 2011 at 11:02 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Jan 14, 2011 at 10:39 | comment | added | Pete L. Clark | @Zen: there is more to the chapter on logic than just truth tables. But this is part of what is done, and it is (I have found) a useful part. For instance, we want students to understand the notion of vacuously true implication. Part of this is to show them the definition of $A \implies B$: in particular, that it is true except when $A$ is true and $B$ is false. Certainly most of the exercises in the chapter on logic do not involve "meaningless algebraic manipulation". | |
| Jan 14, 2011 at 9:45 | comment | added | Zen Harper | ...to illustrate what I'm talking about: why is it that students have such difficulties solving easy things like, say, $|x+1| + |2x-3| < 4 + |x-2|$ ?? Because they cannot think logically. Almost none of them split it up into different cases for $x$; they try to do horrible stuff with squaring and $|y|^2 = y^2$ and, of course, eventually give up. They're not thinking about what they're doing; they think Mathematics is about meaningless algebraic manipulation. And truth tables and formal logic will only reinforce this, I think. | |
| Jan 14, 2011 at 9:22 | comment | added | Zen Harper | I don't claim to have any useful teaching experience related to this, but anyway: I think spending two weeks on truth tables is a really terrible idea. If students don't already intuitively understand basic logic, then they need lots of practice in simple "naive logic" exercises; formal manipulation of (to them) meaningless symbols is not the way to go (surely only logicians and computer scientists actually use that stuff?) (I also think that students who are not intuitively logical are probably hopeless cases, as far as proper mathematics is concerned; but we'd better not go there!) | |
| Jan 14, 2011 at 5:17 | comment | added | Pete L. Clark | @LSpice: when I covered the unit on logic, I believe I was (for the most part, on average: there are few "flawless victories" in teaching) successful in conveying the relevance it had to actual mathematical statements. This is not to say that students never make the converse fallacy again, but rather that I can expect them not to in the sense that when they do make it I can point to it, say exactly what they did wrong, and they understand and agree. It's definitely something to keep emphasizing throughout the course. | |
| Jan 14, 2011 at 5:16 | comment | added | Pete L. Clark | @Amit: the course consists almost entirely of math and math education majors (including double majors). Regarding your second question: I think so, yes, but I can't offer you anything quantitative. | |
| Jan 13, 2011 at 20:57 | comment | added | LSpice | “spending say, two weeks setting up logic is a small price to pay for being able to expect that students will not confuse the converse with the contrapositive for the rest of their careers.” I agree whole-heartedly with this, but is it what actually happens? In my experience, unless the audience is quite sophisticated, what actually happens in many cases is that they can fill in truth tables, but attach no meaning to them and so go right back to making the confusion it is intended to avoid. | |
| Jan 13, 2011 at 19:08 | comment | added | Amit Kumar Gupta | Pete, thanks a lot for your response. What is the composition of the student body in your 3200 class, mostly math majors, or is there a mix of math, computer science, and other majors? I TA'd a course that sounds very similar to your 3200, except there were 100-200 students in the course and a minority were math majors, so perhaps that's why I didn't notice the marked improvement in proof-writing you mention seeing in your class. Also, could you tell when looking at your students' solutions to your second midterm that what they had learned for the first midterm was paying dividends? | |
| Jan 13, 2011 at 16:32 | comment | added | Pete L. Clark | If you're asking me personally: the UGA math department works hard to keep all class sizes below 40. Beyond the freshman calculus level, most classes have 20-30 students. We have 3 hours of class time per week over a very long (even by American standards) semester: something like 15 weeks. | |
| Jan 13, 2011 at 8:27 | comment | added | Matthew Daws | This is a very nice, long description. However, as someone based in the UK, two facts I'd love to know (and this applies to other answers as well) are: What is the class SIZE? And how much contact TIME did you have? (For comparison, I teach a similar course, with size=150 and time=3 hours a week over 9 weeks). | |
| Jan 13, 2011 at 6:43 | history | answered | Pete L. Clark | CC BY-SA 2.5 |