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This evening, we are working on long division with decimal divisors, and comparing the sizes of two fractions. We are working merely on getting these skills down: nothing too deep. When I first showed Ben the cross-multiplication algorithm for comparing two fractions, I showed him why it works the way it works. "It's easy to compare two fractions when they have the same denominator, right?" I said. "Well, it's easy to get two different fractions to be over the same denominator. Just multiply on each side by 1, written as the other fraction's denominator over itself. Then you notice what you get on the left side is the numerator times the right side's denominator, and vice versa on the other side. All you do is compare those numbers. That's called cross-multiplication because it makes a cross. Now you show me." He tried to follow the steps in my first demonstration, and didn't get it right. "It's like this. The numbers move in an x when you do cross-multiplication, like this. They just go swoop, and swoop, like this": And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly. But that doesn't mean that nothing is going on in the kid's head. If a kid really has those moves down, it frees his mind to think about doing the next thing, and he becomes more receptive to learning why the moves need to be what they are, because the anxiety of not being able to handle the calculation is gone. Learning the piano or the violin involves a lot of repetition, while your eyes and your mind and your fingers make the connections that allow you, eventually, to experience the music you're playing on a higher level, without calculating where your fingers need to go next. Math is just like that. Math is something you learn to do, like playing an instrument or riding a bike, not something you learn about remotely, like Magellan's circumnavigation. It has a huge kinesthetic component.
swoop and swoop
the craft of math
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math
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