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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. –  Qiaochu Yuan Aug 16 '10 at 17:47
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@Qiaochu: I ranted about this some time ago on MO: mathoverflow.net/questions/7584/… –  Pete L. Clark Aug 16 '10 at 18:04
    
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. –  andyvn22 Aug 16 '10 at 18:24
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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! –  Pete L. Clark Aug 20 '10 at 6:54
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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 !!! –  Phira May 9 '11 at 14:07

6 Answers 6

The number line. For whatever elementary school grade it is, it illustrates why you need zero, and why you need negative numbers.

The venerable cake, or probably pizza nowadays, to illustrate fractions.

The Riemann Sphere.

The coin flip as a prototypical random event.

Markov chain as a frog hopping from lily pad to lily pad.

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Can to say more about the Riemann sphere? Sounds interesting! –  vonjd Oct 27 '10 at 6:22
    
See the Wikipedia article on the Riemann Sphere. It summarizes it this way: "In mathematics, the Riemann sphere (or extended complex plane) is the sphere obtained from the complex plane by adding a point at infinity." Riemann's work made a lot of complex analysis easier to visualize, though I'm not sure that the visual models are vital to his theoretical contributions. –  phv3773 Nov 1 '10 at 17:57

I can remember the following at school level -

  1. Cartesian coordinates. Pythagoras theorem. (Metaphor - distance from the wall or from the edge of the floor.)
  2. Variables used in basic algebra. (Metaphor - common nouns, eg. "an animal is green and has two legs...")
  3. Irrationals and the completeness axiom. (Metaphor - can zoom in the real line without limit.)
  4. Complex number and complex plane. ("Add just solution to $x^2+1=0$ to the real line, suddenly every polynomial is solvable!")
  5. Sets, relations. (Venn diagrams are helpful for sets.)
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I don't think the zooming in part is a metaphor for the completeness axiom. You can zoom in indefinitely on the rationals, too, without ever “seeing” a difference. (More formally: for any rational p < q, the rational interval [p, q] is isomorphic to the rational interval [0, 1], exactly as in the reals.) Perhaps I just don't find “doing something infinitely often” as intuitive as you might. –  Christopher Creutzig Aug 20 '10 at 8:29

Using parallel, vertical "gates" to visualize high-dimensional or even infinite cartesian products of intervals. It comes up in topological and measurable spaces (specifically in probability theory), and is nearly essential to intuiting convergence in these spaces.

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My brain only works well in Greekspace (that charming region of mathematics where everything has to correspond to a physical analog), so these are quite familiar to me.

  1. If real numbers represent distance along a number line, imaginary numbers represent rotation. See the incredibly well-constructed explanation here: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
  2. If you want to push a crate, and you don't want it to spin while you're pushing it, the direction you have to push it in is an eigenvector. The amount it moves in proportion to how hard you push it is an eigenvalue.
  3. A matrix is a collection of vectors. If you have a matrix of some dimension less than four, you can visualize the vectors floating in 3D space. If not, you'll have to get more creative.
  4. Speaking of dimensions, a "dimension" is "anything that you can vary continuously," whether it's time, color, temperature, w/e. (I actually set up a 3D grid of different-colored vectors in my head for my linear algebra final. Very "A Beautiful Mind," in retrospect.)
  5. The Heaviside function is the curb that your response-function-car hits.
  6. Vector components/projections are shadows of the vector projected on x, and orthogonal vectors don't cast shadows on each other.
  7. Derivatives and integrals are getting distance/acceleration from the graph of your speed (which is, of course, drawn by a pen attached to your speedometer on a scrolling sheet of paper).
  8. Logarithms & their rules didn't really make sense to me until I saw & learned how to use a slide rule.
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I don't see how the crate metaphor works. If you don't want a rigid body to rotate, you push it along any line that passes through its center of mass. –  Rahul Jan 27 '11 at 19:38
    
I should have defined that you're only pushing normal to one of the faces of the crate. There's only a single vector that won't rotate the system if you push in its direction; it only "scales." Same goes for eigenvectors for matrices. –  user6322 Jan 28 '11 at 16:03
    
That imaginary numbers article made my day! +1 –  ValekHalfHeart Apr 3 at 5:28

A graph (either the drawing of a function or the vertices-and-edges kind).

This might seem like a paltry or too general a metaphor but really, take any kind of algebraic expression (say in arithmetic); yes, one can manipulate it symbolically, but it often doesn't become meaningful until you draw something, something that captures a number of possibilities all at once in a picture (e.g. why is 1/x undefined at x = 0? the graph goes in two different directions there).

The vertices/edges kind is so useful for capturing state/objects and transitions among them either for real world description or for mathematical concepts.

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I find the concept of a digital sundial quite intriguing. From wikipedia:

Roughly speaking, there exists a set with prescribed projections in almost all directions.

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