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As he talked, it was immediately clear to me that he had, indeed, learned this trick without meaning. He didn’t seem to have a sense of how place value played into it, why you could approach the problem one digit at a time, and I also immediately knew the limits of his procedure. So I set a mental agenda. I would do a few more problems with him, and then spring a new problem on him, one where the ones digit in the second number was larger than in the first number.

So we took turns giving each other problems to try out the trick, and then after a couple rounds I said, “Wait, what about this one?” and wrote:

82 – 56 = ___

“Okay,” he said. “8 minus 5 is 3” (he wrote down 3) and “2 minus 6 is …” He paused for a moment, and I waited hopefully for the moment of revelation, and then he wrote down “4” with great confidence.

I really wanted him to be a bit more flummoxed than that. I tried to press his thinking. “2 minus 6 is 4?”

And that was exactly the problem. He had left behind his own thinking and was now taking his cues from us.

“Okay,” I said, wondering if I should have tried easier numbers. “That’s a cool trick you learned, but it looks like maybe it doesn’t work if the ones are bigger in the second number. I wonder what we could do when that happens?” He didn’t seem particularly interested in pursuing that line of thought, and it seemed that our exploration might have come to a clunky end.

But then he said, “Wait. I wonder what you do if both numbers in the second number are bigger than the first number.” And he wrote: 34 – 89 = ___ , except that he accidentally wrote + instead of – , so it read 34 + 89 = ___.

That wasn’t where I was planning to go, but what a great question! “Huh, that’s really interesting!” I said. “But did you mean to write plus instead of minus?” He didn’t, but instead of changing it back to a subtraction sign, seeing the more familiar addition sign seemed to give him some confidence, and he said excitedly, “I wonder if the same trick works for addition!”

“Why don’t you try?”

My agenda was completely toast by this point, but all sorts of interesting mathematical things were happening. “So what’s the answer? Does it look like it worked?” I asked.

“One hundred ... wait, one thousand one hundred thirteen?” He laughed. “Not at all!”

“So you said 3 + 8, but is that really a 3?”

“Yes. I mean no. It’s 30.”

“And the 8?”

And this is where it ended, but I thought a lot about the conversation afterward. On the one hand I was very aware that I hadn’t met my objective of helping my son make meaning of the subtraction method he had learned. But I didn’t feel like the conversation was a failure. That was my objective, not his. I wanted to extend his thinking to a wider range of subtraction problems, but the question that really sparked his interest was his own question of whether he could apply the same type of thinking to a challenging addition problem. In the course of our conversation, he posed a question, evaluated the reasonableness of his answer, rethought his method, and used a really nice, meaningful strategy to arrive at his final answer. I was someone to bounce ideas off of (and yes, I did give him a few nudges along the way), but he was the one doing the real mathematical work, and he came away excited, proud, and with a sense of ownership.