Math, in the questions

Oh, math.  It can be so trying.  I was reminded yesterday, though, that the real access to having a peaceful relationship to numbers and the language we speak around them is inquiry.  If we pay as much attention to the questions kids are asking about numbers as we do to the answers they’re coming up with (or aren’t), we’ll see their actual understanding developing, and we’ll also send the message that math is an investigative process, not a dull answer-finding mission.  We might even reveal its creative element.

Yesterday a 9 year-old asked me how it’s possible that 3 + 7 x 4 can equal 31 sometimes and 40 sometimes.  She was looking at an exercise in parentheses placement, so 3 + (7×4) = 31 and then (3+7) x 4 = 40. How, indeed.  This scenario seems to violate what she’s been taught: that when you add or multiply things, you get an answer.  We forget to tell kids that math is just a language. As you learn more of it, the “rules” bend and snap to accommodate what the language is being used to convey.  (The same is of course true of English, which is why it’s dangerous to teach rules about how things are going to go with endings and vowel combinations without mentioning that they’ll only be true sometimes, or for a while.)

But thank goodness she asked me the question.  The moment kids stop asking us to account for the inconsistencies and the paradoxes in our language of numbers, they’ve given up.  They may continue to excel at getting answers right, but they’ve stopped doing math. They’ve decided it’s a mystery that belongs to someone else and they’ll just do enough in their books or classes to give us what we ask for.

The good news is that they don’t actually need quick comprehensive answers to their questions.  They’re actually asking for us to ponder the questions with them.  I wasn’t quite sure what to say when I got the question about the parentheses.  The answer is probably something like “the parentheses are part of the language we have that helps us describe different situations with numbers.”  What I said was “I haven’t thought about that in a long time, but it’s weird, isn’t it?  How could you do the same thing to three numbers and end up with a different answer?  The parentheses are sort of telling us to do the things in a different order.  It seems like maybe the order changes what happens.”  Then we tried moving the numbers around and doing the operations in different orders to see what happened.  We played with the question a little bit, and when we stopped, we didn’t really have an answer but I’m sure that the next time she has a question like that one, she’ll ask me.  And she’ll keep her mathematical self alive that much longer.