The funny thing is that, if you (wikipedia’s version of) the definition of multiplication in terms of the Peano axioms, the kid is right and the teacher is wrong.
I recall proving that a × 1 = a on an assignment in first year algebra.
I also recall watching my professor spend an entire class proving there were a finite number of even primes.
Those were the days.
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I almost completely agree, but I’ve seen too many “college” algebra students who get a correct numerical answer after a series of true mistakes in their process (like 1-2=3 or sin(x)/x=sin) which miraculously cancel each other out. They do need to use some correct method.
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I would have thought the very definitions of primes and even numbers would have made that a no-brainer. An entire class?
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Okay, sin(x)/x = sin is a big problem. What I should have said is that if everything that is written down is true and derivable from truth then you get full marks, if you write down your method and the method is just plain wrong, that’s something else. For most of school I would just write down the answer and then I’d get part marks and my teachers would say “show your work” and I’d politely decline to do so.
It was a bizarre experience, made more bizarre by the fact that he didn’t really tell us what the end goal was when he started out. He just kept proving one thing after another until he finally said, “Therefore there are a finite number of even primes.” I honestly have no idea if he was just having fun with us or if he felt it was important.
I see. Well, 1 is certainly finite, and I want whatever he had that day.
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Here’s a cogent response, from a math teacher (basically, there’s more going on here conceptually then most folks see at first look): Why Would a Math Teacher Punish a Child for Saying 5 x 3 = 15?
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I’m guessing that he was into Model Theory
and wanted to prove things, not just for the usual underlying set and successor function: (N,+1) but also non-standard models.
These include weird things like non-archimedian elements (infinite elements basically, in the sense that you will never hit it by doing successive +1’s to a natural number.) Such a system might have lots of pathologies. I’m told that some of them have “numbers” which are divisible by every standard prime.
“Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad; but chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom.” --Chesterton
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Oh. That’s what school is for? Job preparation? I thought it was to create informed citizens, not slavish peons. And while we’re at it, very few “real jobs” will require students to factor quadratics. Those that might tend to value ingenuity… you know, the word that shares a common root with the term, “engineer.”
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Also primes devisable by 3 5 and 7 are also finite in number.
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Assuming you can enumerate these corollaries, it therefore follows that there are a finite number of primes. 
I have to correct lots of stuff (grad school) and I freely admit to my students that I sometimes make boneheaded mistakes because I’m grading so many assignments back to back and after a while doing it I get all turned around. I tell them to speak up if they notice something is amiss. I marked a point off a guy’s assignment because my rounded answer key said 0.38 and he wrote 0.375! He was all butthurt, and I said look man, here’s the point back; I really don’t care and I’m not punitive. Sometimes I am tired of grading all these things and forget to think…
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That was my thought also. Five times three is five threes added together. Picky, but correct.
But five times three is also sixty divided by four, and -15 * e^(pi*i).
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Depends on what the meaning of “is” is. (Can’t believe I actually typed that.) But seriously, I would accept saying that five times three is equal to sixty divided by four, which is equal to fifteen, etc., but I wouldn’t say that five times three is sixty divided by four. To me, what five times three is, is five times three. YMMV! (And my own mileage might be different on a different day under different conditions )
[Edited for spelling.]
It does, but again, we’re in math class doing arithmetic, what “is” means is clearly defined.
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five threes or three fives… it’s both reflexive and symmetric.
I like (2^4) - 1, since I’m a nerd.
Also, F in hex
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That’s a very specific sort of nerd.
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