more on math teaching

[acroyear]

Good Math, Bad Math : Teaching Multiplication: Is it repeated addition?:

All over subjects in early education, we don't teach children the whole truth about things. In fact, we often teach them things which aren't even incomplete truths, but are utter falsehoods. We teach reading by phonics - when phonics doesn't work a lot of the time. We teach them in the US that our government is a democracy, when in fact it's a republic, which isn't the same thing. In physics, we teach them that mass is a constant, and that energy and matter are different things. We teach them Newton's laws. In chemistry, we teach them that electrons orbit the nucleus of an atom. All of these things, we teach at best incomplete truths at the start, because the whole truth is too complicated to be understood when you're just starting to learn.

I don't see why math should be any different than all of those other topics. We start teaching math using the natural or whole numbers. We don't start off with the full real number system - we start with non-negative integers. We start by saying "you can't subtract a larger number from a smaller one". Then later, we teach them fractions, negative numbers, irrational numbers, complex numbers.

As I keep saying, I like the idea of teaching with intuition. I don't really want to lie to kids - I think that the best way to teach is to be honest about being incomplete. Don't say "You can't subtract a larger number from a smaller one"; say "We haven't learned to do that yet.". Teach the intuition, but show a couple of different ones, to make it clear that the intuition isn't the whole story; but don't spend time explaining the whole story to second-graders - they're not ready to understand it. Tell them what they can understand, and let them know that there's more to learn about it.