Max discovers a theorem

The deep mathematics of elementary school

What does it look like when a child does mathematics like a mathematician? I’ve been thinking about this question for a long time. Here’s a story that captures it better than anything I could say in the abstract.

In the 2000s I was an advisor for the curriculum Investigations in Number, Data, and Space. At one meeting we were studying professional learning materials that described a group of fourth graders discussing a subtraction problem:

\(145-98=47.\)

They have understood 145 – 100 = 45; some understand that to go from there they have to add 2 to the 45, some think they should subtract it, others would prefer to ignore it. Max, listening to all this, says excitedly:

Yeah, the less you subtract, the more you end up with. AND . . . in fact the thing you end up with is exactly as much larger as the amount less that you subtracted.

I’ve always wished I could have been there. At that moment Max discovered what it feels like to be a mathematician, and it reminded me of the pleasure I had when I discovered the same thing (much later than Max).

Let me try to explain exactly what it is about Max that so interests me. It is not simply that he has understood the problem. Many students have gotten to this point earlier than Max: Lorenzo has explained how to get 145 from 98 by successively adding four 10s and a 7; Brian has drawn a diagram showing blobs of area designed to illustrate the fact that when you subtract a smaller number you get a larger number; inspired by this, Rebecca has written

\(\begin{align*} 145&-100 = 45\\ 145&-\ \ 98 =47, \end{align*}\)

drawn an arrow from one equation to the next, and said

If you have a big hunk of bread and you take a big bite you’ll get something small. If you take a small bite, you’ll get something big.

Lorenzo clearly knew how to transform 145 – 98 = 47 into an addition problem that he could solve, and Brian and Rebecca clearly understood why it followed from 145 - 100 = 45 before Max did.

In fact it was Brian who formulated a rough statement of the general principle at work here, and it was Rebecca who provided Max with the key insight; the first part of his outburst is a rephrasing of her statement. Both Brian and Rebecca exemplify important stages in the process of mathematical discovery. So what is different about what Max did?

It’s what he said next: “AND . . . in fact the thing you end up with is exactly as much larger as the amount less that you subtracted.” Max has gone beyond understanding the problem: he has discovered a way of capturing this understanding in clear, public language. He has stated a theorem that works in general, not just in the particular example of this problem. Moreover, he has discovered just the right way of formulating his statement: compact, general, crystallizing fragmentary connections into solid hard truth that others can see.

Formulating the right theorem is an important part of mathematical discovery, and is perhaps the most creative, mysterious, and difficult to explain. A popular view of mathematics is that mathematicians start with what they know and then work out the consequences using logic and proof. It’s not really like that most of the time. Doing mathematics is like poking around in a mental junkyard of facts, examples, misleading statements, and possible theorems.¹ Lorenzo, in the story, has unearthed the fact that 98 + 47 = 145 and is looking at it from one angle. Every now and then you look at something from just the right angle and discover a true theorem. Rebecca provides Max with the angle he needs. Sometimes the proof of a theorem and the right way of stating it arrive hand in hand, not always fully formed, perhaps in a battered shape needing much nurturing, but together. It may be that the proof isn’t really there, just the ghost of an idea; but for the thing to be there at all the statement has to arrive, and the creative act in formulating the statement is one of the most exciting moments in mathematical discovery. It is, perhaps, what a poet feels on penning exactly the right line, or what a painter feels on executing the right brush stroke. It is the excitement, not only of discovery, but of communication.

The job of the mathematician is not only to figure things out, but to figure out how to say them so that others can see them. Sometimes a mathematician who has discovered a new theorem needs only to state the theorem for others to be able to reproduce the work. Max was lucky enough to be able to stand on the shoulders of Lorenzo, Brian, and Rebecca, and see the theorem they were all groping towards. The class was doing a mathematician’s work that day in the classroom.

1. As Andrew Wiles said of proving Fermat’s Last Theorem, “Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room...”