We were emptying the dishwasher in the morning, and my younger son’s job was putting away the silverware. He brought the silverware basket over to the silverware drawer, and said:
You know what I’m going to do? First I’m going to collect together all the spoons, and put them in the spoon bin. Then I’ll take all the knives and put them in the knife bin. Then I’ll take the forks…
Any idea that a five year old can come up with must be really simple, right? But simple ideas can still be deep and powerful. Among other things, this one is at the heart of an important mathematical technique called Lebesgue integration, which one of my favorite math teachers once explained to me like this:
Say you’re trying to count a really big pile of money. You can stack it really high and count it a bill at a time. Or you could separate it into piles of ones, fives, tens, and twenties, count how many bills are in each pile, multiply the count in each pile by the denomination, and add the results. Lebesgue integration is when you break the bills into separate piles first.
What my son figured out was that if you focus on one kind of utensil at a time, you can work faster because everything you pick up goes in the same place, so you don’t need to think about switching from one bin to another all the time. From a computer science perspective, you’re doing fewer operations. From a math perspective, you’re representing a single function that appears complex (because it jumps around all the time, from knife to fork to spoon to knife or from $1 to $10 to $5 to $1) in terms of a few simple (constant) functions defined on different domains. I’ve written before about how math is about finding, creating, and making use of order, and this is a great example.
You can apply this idea to the problem of finding the area under a really jumpy curve. Henri Lebesgue is famous, with an integration technique named after him, because he worked out the details, about 100 years ago. But the underlying idea truly is accessible to a five year old. At least, as long as that five year old pays attention to his chores.