### Math Is Hard

I learned addition and subtraction in a very mechanical way: here are sheets of problems, now do them! And I wasWell, division can be thought of as--in fact, is--repeated subtraction. That's one way of defining what division is, just you can define multiplication to be repeated addition and exponentiation to be repeated multiplication (and taking roots to be repeated division; the cube root is the answer to: "what number can I divide this by three times to get one?").

You can see that division can be thought of repeated subtraction most clearly in long division: 50,008/14, say. First we subtract 3,000 fourteens from 50,008, and get a remainder of 8,008. Then we subtract 500 fourteens from 8,008, and get a remainder of 1,008. Then we subtract 70 fourteens from 1000, and get a remainder of 28. And then we subtract two fourteens from 28, and get zero. Voila: we have subtracted 3,000 + 500 + 70 + 2 = 3,572 fourteens from 50,008--we have divided 50,008 by 14 and gotten 3,572.

The idea that "division is repeated subtraction" is much better when a student is first confronted by division by a fraction--3/4 divided by 1/4, say--than is the alternative of "division is dividing into piles." You divide 50,008 into piles of 14 and you have 3,572 piles. But you divide 3/4 into 1/4 of a pile and... a student who thinks "division is dividing into piles" is immediately lost. By contrast, if the student starts out thinking that "division is repeated subtraction," it is easy for him or her to see what 3/4 divided by 1/4 is: how many times can you subtract 1/4 from 3/4 before you get zero? And the answer is three.

**spectacular**at those sheets. So all of this "understanding" sort of escapes me. I could prove the equivalence of those understandings, but I'll stay with my trusty method of taking the reciprocal of the denominator and multiplying. But then, there is a reason I have no intention of ever teaching math. It's hard to make math make sense to others.

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