Mathematics

When young children perform long division or multiplication, are they constructing a proof?

Comments

Sure, why not? What they construct---I take it we mean, "write down"---is the very same proof you or I might construct. So they've constructed the same thing we would, so it's a proof. But there is a different question you might ask, namely, do they understand it as a proof? And that, I take it, is an empirical question. Let me ask my brother, who works on mathematics education, and see what he has to say....

So I asked my brother about this, and he tells me that this kind of question is much discussed in the literature on mathematics education. Here's what he had to say:

"Good thing to think about. A related idea I've been considering for some time---and maybe the difference is just a matter of cognitive development---is whether solving an equation algebraically is a proof.

"Another spin on the idea---which is what got me thinking about it in the first place---is whether solving equations ought to be taught as proof, since every step one takes in algebraic solutions can be mathematically/logically justified through some equivalence that leads to the solution set. What most kids end up learning to do is to conduct the procedures of solving without any real understanding (or caring) of why what they are doing is mathematically justified. I have a sense that if solving were taught as proof, then it would be more natural for kids to pay attention to why certain steps that seem OK actually introduce other solutions, or botch everything because they might be dividing by zero, etc."

An important paper here, he says, is Andreas J. Stylianides, "Proof and Proving in School Mathematics", Journal for Research in Mathematics Education 38 (2007), 289-321.

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