Some Thoughts on the Teaching of Mathematics

I love math. I'm a statistician for a living. And I read books about math. It is my firm conviction that more people can like math; that more people ought to like math. They don't, most of them, not out of any innate problem, but because math is the most mistaught subject in the elementary schools and often in the high schools.

Let me ask you something. When you got out of high school, did you know ANY of the following pieces of beautiful elementary mathematics? Which, if any?

1. Euclid's proof that there is no largest prime
2. Stirling's approximation
3, Any infinite series representation of pi
4. A proof that the square root of 2 is irrational

And yet, all of these things can be taught based on math that is taught in elementary school.

When you got out of high school, had you heard any songs?
Had you seen any paintings?

The fact all (or darn near all) of us have heard many many songs and seen many many paintings is a good thing. The fact that few of us have seen the equivalent in math is a perversion.

First piece of beautiful elementary mathematics:

Euclid's proof that there is no largest prime
A prime number is an integer (whole number) larger than 1, which is evenly divisible only by itself and 1. The first few are 2, 3, 5, 7, 11, 13....

We start by assuming there is a largest prime. Then we will deduce an absurdity from that, thus showing that there can be no largest prime.
Call the largest prime p.
Find all the primes less than p
Multiply them together and add 1, call this Q. Q cannot be divisible by any number less than p (because there will be a remainder of 1). Either Q is a prime number itself (and it is certainly larger than p) or else there is some other number, between p and Q that is prime, but is also larger than p.

Second piece of beautiful elementary mathematics:
Stirling's approximation

A factorial of an integer is all the integers up to and including that integer multiplied together. It is written as an !. For example, the factorial of 5, written 5! = 1*2*3*4*5 = 120.

e, the base of the "natural" logarithms, is approximately 2.718... It is the limit, as n approches infinity, of (1 + 1/n)^n. In other words, about (1 + 1/1000)¹⁰⁰⁰. Even closer to (1 + 1/1,000,000)^1,000,000.

pi is the ratio of the circumference of a circle to its radius. It is approximately 3.14....
Now.....

n! ~ (2*pi*n)^1/2(n/e)ⁿ

just look at it! We approximate a product of integers, and the approximation involves two irrational numbers! WTF?

Third piece of beautiful elementary math:
Any infinite series representation of pi
There are many. Here are some

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 ......

pi^2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 ......

and

pi/2 = 1 + 1/3 + 1*2/3*5 + 1*2*3/3*5*7 + 1*2*3*4/3*5*7*9 + .....

Fourth piece of beautiful elementary math
A proof that the square root of 2 is irrational
As in the first proof, above, we start by assuming that 2^1/2 is rational and then show that that leads to absurdity.
If 2^1/2 is rational, it equals a/b (that's what rational means).
Simplify a/b to lowest terms.
square both sides and get 2 = a²/b²
Then a² = b² *2.
That means a² is even. And that means a must be even (an odd number squared is always odd; see if you can figure out why). So, let a = 2k.
so 2 = (2k)²/b²
or 2 = 4k²/b²
Multiply by b² and get
2b² = 4k²
divide by 2 and get
b² = 2k², so b must be even as well. But, if a is even and b is even, then a/b is not in least terms.