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students usually are unwilling to accept that $(-a) \times (-b)= a \times b$, or $(-1)\times (-1) =1$ in elementary school. Any good way to teach them such knowledge?

BTW: This $(-1)\times (-1) =1$ can not be proved, it is just extension of ring, so the quesion asking formal proof is in wrong way. My question is not duplicated.

XL _At_Here_There
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  • Comments are not for extended discussion; this conversation has been moved to chat. – Xander Henderson Sep 09 '22 at 12:20
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    As it is currently phrased, this question would be more appropriate to [matheducators.se]. However, the accepted answer, which is nearly identical to those in the linked question, confuses me some, as the answer is simply a proof of the fact, and does not speak to pedagogy at all. – Xander Henderson Sep 09 '22 at 12:22
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    @XanderHenderson I agree about the answer, but I don't see why this is a reason to close as a duplicate. I've seen you do this before, where you judge duplicates based on the answer given rather than the potential for other answers. Anyone who wants to give an answer that does engage with the pedagogy will not be able to add it to the dupe target, so aren't you just preventing the question from being answered properly? Also, please stop closing questions like this; you are breaking your promise when you ran for moderator. – Theo Bendit Sep 09 '22 at 12:27
  • @TheoBendit The question was already closed for lacking context. In its current state, it is closed, but it redirects towards another question with relevant answers. This seems like an improvement, though I would be more than happy to return this question to its previous state, i.e. closed for lacking context (and, frankly, for being off-topic here) – Xander Henderson Sep 09 '22 at 12:32
  • @XanderHenderson How about returning it to its previous state of being already re-opened, and not break your election promise? I was hoping to post a detailed answer, dealing with the pedagogy and the formalism, and their interplay. – Theo Bendit Sep 09 '22 at 12:34
  • @TheoBendit When I encountered this question, it was closed. I had to reopen it in order to close it as a duplicate. – Xander Henderson Sep 09 '22 at 12:38
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    if you really want to teach this to elementary school kids, then obviously formal proofs are out of the question, and they’re not useful. I think more useful is a ‘definition and proof by picture’ in that you tell students about how (positive) numbers are drawn on a number line to the right, and the negative numbers are by ‘definition’ what you get by reflecting to the other side. So the negative of any number is what you get by reflecting; in particular $-(-1)=1$. I vividly recall being taught this way and it made sense (because we all have mirrors at home so reflections make sense). – peek-a-boo Sep 09 '22 at 12:38
  • @Xander My apologies about the re-opened thing. I missed that your name was fifth on the list of re-open votes. The problem is, thanks to your re-closing of the question, my re-open vote (as well as three others) no longer counts. Oh well, there's nothing we can do it about it now. Just please, please, please stop closing questions like this Xander. You are an elected moderator, but you also are just a single voice in this diverse community. You did recognise this once. Please recognise it again. – Theo Bendit Sep 09 '22 at 12:41
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    @peek-a-boo, noone can prove it, it is just extension of ring, I just ask a natural way to teach kids. – XL _At_Here_There Sep 09 '22 at 12:43
  • @TheoBendit And, since you mention a desire to write a detailed answer, why not answer one of the very closely related qeustions [matheductors.se]. For example, https://matheducators.stackexchange.com/q/5794/ or https://matheducators.stackexchange.com/q/13019/ – Xander Henderson Sep 09 '22 at 12:46
  • I might do that @Xander, thanks. The answer I was writing is more suited to elementary school though. I'll think about it. – Theo Bendit Sep 09 '22 at 12:48
  • @TheoBendit This question (linked above) is very much about multiplication in an elementary school setting, and several of the answers there are very much in that context, e.g. this one, which talks about using manipulatives which are commonly seen in the lower grades. – Xander Henderson Sep 09 '22 at 12:50
  • @TheoBendit but the site prohibits you from posting any answer to my question. – XL _At_Here_There Sep 09 '22 at 12:51
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    @XL_At_Here_There a natural way to teach kids? I gave you exactly that. There are two directions kids are familiar with: left and right. Tell them that a minus sign switches the two just like a mirror does. That’s a ‘definition’ and ‘proof’ which elementary school students can understand. Or if you live in a place where houses have several basements, that’s also a reasonable way of talking about negative numbers, and then tell them that multiplying by $-1$ means going that many floors in the opposite direction. There are so many ways of talking about this and relating to everyday life. – peek-a-boo Sep 09 '22 at 13:07
  • Teach multiplication by $-1$ as a reflection across $0$ on the number line. Then demonstrate that $-1\cdot 5=-5$ by showing how it it reflects 5. From there, it should be obvious why the product of two negative numbers are positive because it is just two reflections. – John Douma Sep 09 '22 at 14:25
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    I am not sure why this question and answer attracted this level of debate, and the fact that people have other suggestions for better answers suggests that the question should have been reopened to allow the other suggestions an airing. The reason to vote to close the question as lacking context does not make sense at all, the user was not asking us to do a homework problem after all, since OP is clearly already aware that $(-1)\times(-1)=1$. If the question is to be moved then why not move it already, and if it is a duplicate of the education site, let those people mark it as such? – Suzu Hirose Sep 09 '22 at 23:36
  • @SuzuHirosenIt Elementary school students are between 6 and 12 years old. I doubt they would be able to understand a proof using the axioms of Peano arithmetic. – John Douma Sep 10 '22 at 19:14
  • @JohnDouma, who is using the axioms of Peano arithmetics? of first order or second?order – XL _At_Here_There Sep 11 '22 at 00:28
  • @XL_At_Here_There I have never met an elementary school student that was skeptical about the rules of multiplication and I would be shocked to find one that was convinced upon seeing an algebraic proof. – John Douma Sep 11 '22 at 00:51

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You might like to teach the students that there were hundreds of years of history where people refused to believe in negative numbers, and then after accepting them debated what multiplying one negative number by another would mean.

But the argument is not so hard to make if you have already accepted that $1\times-1=-1$:

$$ \begin{align} 0&=(1+-1)\times(1+-1)\\ &=1\times1+2\times1\times(-1)+(-1)\times(-1)\\ &=1-2+(-1)\times(-1)\\ &=-1+(-1)\times(-1) \end{align} $$

(See "A History of Abstract Algebra" by Israel Kleiner for more on the debates on negative numbers.)

Suzu Hirose
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  • Excellent, But it is question of an extension of ring – XL _At_Here_There Sep 09 '22 at 11:40
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    Unless your elementary school is attended by Ruth Lawrence-type of people you probably do not want to worry about rings, or extensions. I don't think we did any symbolic algebra at my elementary school. – Suzu Hirose Sep 09 '22 at 12:00
  • I don't teach any elementary school, but just asist some kids in doing math. Actually, I will not teach them about Ring, Field, etc, just describe what we are doing to the question. – XL _At_Here_There Sep 09 '22 at 12:04
  • Thank you for your helpful answer. – XL _At_Here_There Sep 09 '22 at 12:12
  • @XL_At_Here_There it is much easier to teach them that multiplication by $-1$ is a reflection across zero on the number line. Then they will easily be able to see why a negative number times a negative number is positive. – John Douma Sep 09 '22 at 14:27
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    If you would like to answer this question, then voting to close the question doesn't seem to be a rational way forward. Answering the question in the comments to my answer also does not seem to be the best possible solution. – Suzu Hirose Sep 09 '22 at 23:38