When I was a kid, I just learned math the way they taught it in school. I didn't spend time figuring out tricks very much. I didn't think in math.
Caleb claims to not love math, but he thinks of tricks all the time.
For example, the other day I was teaching Anda the Saxon Math 54 lesson on "these multiplication facts are the memory group; there are no useful tricks to help you remember them" and Caleb piped up, "Well, there's one for some of those."
"What is it?" I asked.
"When you are multiplying two single-digit numbers that happen to be two apart, like 5 and 7, or 6 and 8, the answer is the number between them squared minus one," he said.
I, of course, had to try a few. It works every time. 6 times 8 does, in fact, equal 7 times 7 minus one.
"How did you figure that out?" I asked.
"Well, I was messing around trying to figure out if I could apply my addition trick to multiplication," he said.
"What's your addition trick?" I asked.
"When two numbers are two apart, and you have to add them, the answer is the number between them doubled," he said.
So I had to try that one, too. It works, of course. I immediately could grasp the logistics of that one--you have two groups of objects, two digits apart, and if you take one from the larger group and stick it in the smaller group, they have equal amounts. You've averaged them and landed at the number between.
That he could apply it to multiplication was cool.
Of course, Anda immediately pointed out that the trick is useless if you haven't yet memorized your square numbers.....