My son (6 yo) was eating his breakfast before school today, and started asking me questions about math. It started by asking about groups of 1 – is 4 groups of one 4? Is 8 groups of one 8?

I was trying to get everything done to get us out the door, and not paying much attention, except to reply distractedly.

Then he dropped this bombshell.

6yo: If 0 + 0 is 1, then would 8 times 0 be 4?

Me: Uh… I need to think about that.

This is the kind of question that feels near gibberish. 0 + 0 is not 1, so what is he even talking about?

But as the truism goes (and I can’t remember who said this), children’s wrong answers are usually right answers to a different question. And in this case, these “What if” questions are a sign that a child is taking a sort of radical ownership over their mathematical thinking. I advocate for precisely this kind of mathematical play, based in what if questions, even though it’s very hard to find sensible examples of them.

In a way, it’s a Big If question. He wants something to be true to explore an idea, so he’s just doing it. And on reflection, what he’s saying makes sense.

If 0 + 0 = 1, then 8 x 0 = (0 + 0) + (0 + 0) + (0 + 0) + (0 + 0) = 1 + 1 + 1 + 1 = 4.

Right?

He’s sort of saying he needs a number that has the property that 2 of them makes 1 (we’d call this number a half, but he’s not making that connection yet, and so he just grabs 0 as a number less than 1 that he wants to imagine as having this property. In a way, he’s using 0 as a variable. From one perspective, his knowledge is incomplete, but from another he’s dealing with that incompleteness with an extraordinary feat of abstraction.) And if you have a number that has a property that two of them make 1, then eight of them would indeed be 4.

And that’s how, quietly at breakfast with his dad distracted and trying to get us all out the door, a 6-year-old is connecting mathematical dots and taking control of his understanding.