Making Sense of Subtraction

Sarah’s mom asked me to do some math with her to get an idea of what kinds of activities might be helpful and fun. She was 7 at the time, very active, and reportedly interested in math because she liked money a lot. She’d figured out that math was helpful in the financial arena, and wanted to continue to out-save her brother.

When we met I asked her to tell me a little about herself as a math student. She immediately said “I’m good at adding but not “subtracting.”

In general when kids think they can add but can’t subtract, there’s a little something missing from their understanding of adding, or from the connection between adding and subtracting. Once the concept of adding is solid, and the connection between it and subtraction is clear, subtraction itself doesn’t tend to present much additional difficulty.

So Sarah and I did a little adding. I showed her a game I learned from Family Math. The way it works is that you turn over one card – the “target” number – and put it on the table where everyone can see. Then you deal two cards to each player. Each player works with his or her two cards to create an addition or subtraction problem with an answer as close as possible to the target number.

Sarah caught on to the structure of the game right away. In the first few rounds she drew cards that allowed her to use addition to get fairly close to the target number. (I could see that conceptually, she understood what addition was all about, so I was looking for the missing connection with subtraction.) In the fourth round, she drew a pair of 7s and looked up at me uncertainly. The target number was 1.

“So I should subtract?” she asked. “Because 7 plus 7 is way too high.” I nodded, and suggested she give it a try. She set the cards next to each other and said “I don’t know.” I reached for a pile of crayons so I could give her a concrete explanation. “If I start with 7 crayons,” I began, “and then I take away 7, how many do I have?”

“Oh,” she said confidently. “Seven.”

I was, of course, surprised and confused. I was expecting my demonstration to make it clear that 7-7=0. I asked her to explain.

“Well, you said you took 7, so you have 7.”

What I’d said was “I have 7, and I take 7 away.” What I needed to say was “If you have 7, and I take 7 away from you, how many do you have left?”

We as adults do this sort of thing all the time. Things that we have known how to do for a very long time can be incredibly difficult to explain effectively. The only way I have found that really works consistently is twofold: First, conjure up the experience of having someone explain something to you that they understand well and that you have no experience with. Remember what that’s like. Then, imagine yourself with the equivalent life experience of your child. Math and other human-designed pursuits are not natural and logical; they’re languages as any other.* Learners of any new language need lots of exposure to it, and lots of patience as they figure out its nuances and quirks, not to mention the inconsistencies of those who are using it around them!

* The patterns studied in mathematics can occur in nature, but much of what we tend to teach as math is a language used to describe those patterns.