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Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)?

For example, I think the comparison of modular arithmetic to arithmetic with times, on a clock face, is fantastic, because it corresponds so perfectly and because clocks are so widely known.

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    The comparison of modular arithmetic to clock times is quite lacking. It explains addition but not multiplication.2010-08-16
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    @Qiaochu: I ranted about this some time ago on MO: http://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/26154#261542010-08-16
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    But once the clock face is introduced and students are comfortable with addition, doesn't multiplication make very good sense? It's not as if the clock pushes your multiplication intuition the wrong way; it just doesn't help.2010-08-16
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    @Pete: yes, I was trying to remember where I had read that criticism before...2010-08-16
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    I thought this was asked on MO, but cannot find the link. Anybody remember something similar over there?2010-08-16
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    Fine: clock times vs. modular arithmetic sucks. But there are good analogies out there. I for one think this question deserves more upvotes than it's getting.2010-08-16
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    I think apples in buckets is a pretty good metaphor for addition and subtraction (of natural numbers) :)2010-08-19
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    @andyvn22: in my opinion, the intuition which is damaged is not that of modular arithmetic but rather the (much more fundamental and important, for most private citizens) intuition for dimensional analysis. There is no doubt that K-12 education is failing students on this: e.g. many freshman calculus students will suggest formulas like $V = \pi r h$ for the volume of a cylinder. They don't see that this is not only incorrect, but ridiculous -- the dimensions of volume need to be length cubed! So yes, I do think suggesting that time * time = time is bad for intuition!2010-08-20
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    Function Monkey! http://math.stackexchange.com/questions/1730/how-do-you-define-functions-for-non-mathematicians/1744#17442011-01-27
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    @Pete I know that this is an old question, but I have to say that I come from a school system where we were docked points in primary school if we did not write the unit next to each number during the calculations. I was very angry that I was not allowed to write $3+5=8$ and the rationale was like yours. I totally disagree that a motivation has to be all-encompassing to be useful and I wonder why you did not protest the "apples in buckets" post above, because: You cannot multiply apples !!!2011-05-09

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