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Apr 18, 2014 · That said, the main reason for proving obvious things is that proofs are the fundamental building blocks of mathematics. If something is true, a mathematician should be able to prove it. If something cannot be proven, that will (or should) stick in the mathematician's craw.
9. As Arturo and Qiaochu have pointed out you get better at proving things by practicing a lot, solving exercises and by seeing how other people have proved things. There's a big part in learning how to prove statements by repeating certain strategies that may have worked for particular types of problems.
96. There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would naturally expect a statement in the latter category to be easy to prove -- and they usually are. I'm interested in examples of theorems that are ...
- Imported vs. Enculturated
- The Things in Proofs Are Weird
- Examples
- Conclusion
- Notes and References
To frame the inquiry, I posit that there are imported and enculturatedcapacities involved in reading and writing proofs. Teachers face corresponding challenges when teaching students about proof. Capacities that are imported into the domain of proof-writing are those that students can access independently of whether they have any mathematics traini...
I recently encountered an article by Kristen Lew and Juan Pablo Mejía-Ramos, in which they compared undergraduate students’ and mathematicians’ judgements regarding unconventional language used by students in written proofs.One of their findings was that, in their words, “… students did not fully understand the nuances involved in how mathematician...
Here are some specific instances of student struggle that seem to me to be illuminated by the ideas above. 1. In the paper of Lew and Mejía-Ramos mentioned above, eight mathematicians and fifteen undergraduates (all having taken at least one proof-oriented mathematics course) were asked to assess student-produced proofs for unconventional linguisti...
My claim is that the mathematician’s skill of mentally capturing classes of things by positing “arbitary, but fixed” universal members of those classes, and then proceeding to work with these universal members as though they are actual objects that exist, is an enculturated capacity.I think it’s a little bit invisible to us—at least, it was so to m...
I trust that any reader of this blog who has ever taught a course, at any level, that serves as its students’ introduction to proof, has some sense of what I am referring to. Additionally, the research literature is dizzyingly vast and there is no hope to do it any justice in this blog post, let alone this footnote. But here are some places for an...
- Ben Blum-Smith
The obvious solution is to stop trying to put proofs on dead trees. Displaying proofs on a computer gives you the ability to present them in a way that shows the important steps and overall strategy by default, but allow the user to expand it to show the intermediate steps when needed.
Apr 28, 2022 · In 1993, Wiles announced the final proof of Fermat's Last Theorem, which was originally stated by Fermat in 1637 and was "considered inaccessible to prove by contemporaneous mathematicians" (Wikipedia article on the proof). So clearly, in many cases, mathematicians have to bend over backwards to prove certain logical conclusions.
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Since j, k, and l are all 1, the three rational equations above give us n = m, p = n, and m = p and so all three are equal. Proof 2: We can ignore some of the information in our givens and just draw from divisibility a simple inequality. Namely, if m divides n, then m n. Likewise, n p and p m.
