An even number plus an even number makes an even number.
An odd number plus an odd number makes an odd number.
An odd plus an even makes an odd.
You were probably taught that simple rule in elementary school. I was. And it seems to be true. Try it a few times, with a few different numbers, and it always works. (If it doesn’t, check your work. If it still doesn’t work, publish.)
But does it work for all numbers? No matter how large?
The difference between the math we are usually taught in school, and the mathematics that mathematicians do, is this:
- In school, we are taught these kinds of rules so that we can use them when ‘doing math’.
- Mathematicians try to discover what the rules are, and come up with the most concise, elegant arguments possible to show why those rules are (or are not) true.
The problem with school math is that it is, for many or most people, indescribably boring. And it’s also just hard to do math…or, really, arithmetic…if we are just memorizing a bunch of rules, without seeing the beauty and symmetry and patterns in shapes and numbers.
As Paul Lockhart persuasively (and humorously) describes in his essay A Mathematician’s Lament, the art of finding truth is both true mathematics, and a lot of fun. And it doesn’t have to be the formal, rigid proofs that are sometimes taught in school. It’s just about looking for patterns and making an elegant argument.
Instead of telling young learners rules about sums of odd and even numbers, what if we first asked them to figure out what the rules might be, and then asked them to come up with an explanation for why it’s a rule?
Here’s an example of the kind of thinking that might go into a ‘proof’, which is just one of many possible solutions:
First, let’s count not with abstract digits but with tangible objects, in this case squares. Here are five squares:
[picture of several arbitrarily placed squares]
Since an even number means that it can be divided by two, we know that we can arrange an even number of squares into two rows of the same length, and the ends will be ‘square’:
An odd number, on the other hand, will always have a ‘ragged’ end where the rows don’t line up:
Rearranging these pictures, we can now see that our rules seem to be true. Two even numbers, laid end to end, have even ends.
By flipping one odd number over and sticking the two ragged ends together, two odd numbers also have even ends.
But one odd and one even, no matter how we flip and rotate, never gives us even ends.
This will be true no matter how long our numbers are, because all that matters are whether the ends are ragged or square. (Those lightning bolt things are meant to suggest an arbitrary distance…imagine there are thousands of squares in there.)
Q.E.D.
Is this a valid mathematical proof? Does it matter? A child, or a group of children, who have spent the time to think up these sorts of ‘proofs’ will be developing an understanding about and maybe an enthusiasm for mathematics that no amount of rote drills will give them. More importantly, they will begin to learn “what to do when you don’t know what to do.” That is, the confidence to solve problems you haven’t seen before, instead of just following the steps of problems you have.
