Open Set

I was explaining long division to my son last weekend. (Don’t ask, long story.) As I was going through the steps, I found myself doing something I have done many times in my teaching career: I asked for help with the arithmetic facts. “So now we write the 8 up here and multiply. What is 8 x 7? Great, I’ll write the 56 here. Now, we need to take away this 56 from the 58?. What’s 58 minus 56?” 

Then I thought about it. What was going on? My goal was for him to learn the steps and theory of long division. If I wanted him to remember the steps (divide, multiply, subtract, bring down, repeat!) then I should be asking him questions like, “What step comes next?” Or maybe, “Where should I write the 8?” Things that get him to recall parts of the algorithm, and make use of it in the moment. 

If my goal was to teach the theory – why this was a valid method – I should ask him, “When I wrote this 8, it represented a bigger number. Is it really an 80 or an 800?” Or maybe, “Why do you think we need to subtract over here?” 

But no. I asked “What is 8 x 7?” This leads to neither algorithmic processing nor conceptual understanding. Why would I do that? 

Maybe I thought that he needed some practice with multiplication and subtraction skills, and here was an interesting context to engage him in otherwise rote practice. But he gets that at school on IXL, I’m not concerned about him memorizing tables. 

Maybe it was about attention. Was I trying to keep him interested by giving him parts to solve on his own? Let him feel like he was part of creating the answer? Or perhaps I was checking to see if he was dutifully following along, by asking a question to see if he could answer it. Totally reasonable, things that I do a lot in the classroom, but not really what was going on here. He was interested in the process! I didn’t need to test that or maintain that, or if I did, not through arithmetic. 

So, why was I doing this? My best answer is that it’s a habit. This is what it sounds like to teach math. And I know it’s not just me. It’s something I see all the time when I watch other math teachers at work. The teacher is at the board, working through a long and complex problem. Students are dutifully taking notes. The teacher tries their best to make each step clear. They explain the flow of the algorithm, the series of steps. Or, they explain the heuristics that they used when selecting which step to use, how they thought about this particular problem. The teacher points out common pitfalls and makes sure they repeat the most important phrases over and over, so students remember them. And, they periodically pause to ask students to answer simple arithmetic questions, as they come up. 

And when they do, they pause for a moment, and raise their voice just a little, and then restate the arithmetic question.  “So now we divide by 3 to get rid of the leading coefficient, and whatever I do to one side I always have to do to the other side, so I’m going to divide this 12 by 3 as well. … What is 12 divided by 3?” Student who were only half paying attention tune in, hear the question, and produce the 4 they need to move on. No connection to the larger problem, the algorithm, the problem solving. Students are only practicing material they learned years ago. Not a bad question, just a meaningless one, in that context. 

And yet, we keep doing it. But hey, my son did learn long division that day, and I learned that he knows 7 x 8 as well.