I've posted a number of diaries about math, statistics, and math education. Today, I'd like to distinguish two subjects that get taught as if they were the same thing, when they are not. It's as if we thought literature and spelling were the same subject.

We call both these things math. But one of them is what mathematicians do, and hte other is not. And, while many people who are good at one are good at the other, it's not always the case.

If you'd like to read more, join me below the fold.

One subject is mathematics. This is almost never taught before high school, and often not until college. That's a shame. It could be taught much earlier. Mathematicians do lots of things, but perhaps the one central idea is PROOF. Paul Erdos, one of the best and most prolific mathematicians of the 20th century, said

A mathematician is a device for turning coffee into theorems

Bertrand Russell, another great mathematician, said

Mathematics is the only subject in which we never know what we are talking about, nor whether what we are saying is true

Doesn't sound much like math, does it?

That's because what is taught in school isn't really math. It's quantitative reasoning. It's about numbers. In high school, you might get a little abstract with algebra and geometry and maybe even trigonometry, but you probably won't be introduced to formal proofs.

Here are two simple bits of real math, that don't require more than 2nd or 3rd grade 'math'....

- You have a stadium full of people, and a bunch of sacks of bags of candy. How do you tell if you have more candy or people?

The quantitative way to do this is to count each. A royal pain. You'll probably count wrong. Then you'll have to check. Oy.

The math way uses one-to-one correspondence: Give each person a bag. If you run out of bags, you have more people than bags. If you have more bags left at the end, you have more bags than people. If they match, the numbers are equal.

Note that no counting is involved. No calculation of any kind.

preliminary info for problem 2: A prime number is one that is divisible only by itself and 1; all other numbers are composite. e.g 5 is prime. 6 is composite, because 2*3 = 6

- Are there an infinite number of prime numbers, or do they stop?

A. Assume there are a finitie number of primes. If so, there must be a biggest one. Call it p.

B. Find all the primes smaller than P. (They knew how to do this even when Euclid came up with this proof).

C. Multiply all the numbers in B together.

D. Add 1; call this number Q. Clearly Q is bigger than P

now, one of two things must be true:

a) Q is prime, in which case P is not the largest prime

b) Q is composite. But, if it is, then it must have a factor larger than P, because, if you divide by P or any of the primes that are less than P, you will have a remainder of 1.

Note, again, that no calculations need be done.

Quanttative reasoning is different. That's a 'number sense'. A simple example is this:

Suppose you are helping your son with his homework. He has the following problem:

54 + 32 + 20 + 98

and he has given an answer of 752. If you have good number sense, you know that this is wrong before you even add anything. All the numbers are less than 100, there are 4 of them. The total must be less than 400.

Of course, both math and QR get more complex than this; and, of course, there are SOME connections. But by teaching these two subjects as if they were one, we do a disservice to everyone. Math, as I've written before, should be taught because it is beautiful. QR should be taught because it is useful.

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