Tabitha (7 years old), Griffin (9 years old) and I walked to the local convenience store tonight. We had a math talk that I will describe in more detail on Talking Math with Your Kids soon.
In the meantime, I will excerpt a piece of that conversation here. It will give us some useful language and ideas.
Tabitha was using her own money to buy some hot Cheetos. She was under the impression that they would cost $1.35. While she waited in line, she had me verify that her 5 quarters and 1 dime matched this sum. I assured her that it did.
The Cheetos turned out to cost $1.49.
There were people in line behind her. This was a time to grease the wheels, not to slow down everybody else’s Saturday evening. So I told her to give the cashier 2 more dimes.
As she did so, I told her that she had given the man 20 more cents when he only needed 14 more cents, and asked her how much change she should get.
The cashier finished off the transaction. I stuck out my hand to grab her change (so as not to give away the answer to the question I was about to ask, and she was way more interested in the Cheetos anyway). We turned to leave.
I asked how much change she should get back. She seemed confused by the question. After going back and forth a couple of times, we settled on this question:
14 plus something is 20; what is the something?
Now we get to the question I pose to you, Dear Reader.
What is the goal of asking a child this question?
There are many possible goals, of course. I want to highlight two of these. I think that they stand in stark opposition to each other.
- To get the child to say, “six”.
- To get the child to think about number relationships.
Six is the right answer. I would like for her to be able to get there. But getting her to say, “six” is not the goal of the question for me.
Before I elaborate, I want to make clear that this is not a straw man argument.
Griffin piped up while Tabitha was thinking and asked, “How old were you last year?” The only thing that question had in common with mine was the answer. I have been in math classrooms where teachers offered these kinds of hints.
So not a straw man at all.
While the video below is supposed to be funny, it draws on this idea that the goal is to get the child to say (write) the answer.
My goal in asking this question is to get the child to think about number relationships. I want Tabitha to think her way through to an answer. I want her to be able to say, “six,” yes. But I will be happy with a few productive wrong answers along the way because that will be an indication that she is thinking.
You see, options 1 and 2 above speak to very different ideas about how people get better at mathematics.
Option 1 speaks to the idea that fourteen plus something is twenty is a problem that has the same structure as many other problems (this plus that is something else) but that bears no other relationship to them.
Option 1 is related to a behaviorist view of mathematics learning—that we create associations between stimulus and response, and that learning is the formation and strengthening of these associations. With this view, fourteen plus something is twenty is a unique stimulus that requires a unique response: “six”. The strong version of this view would require me to tell her the answer, have her repeat the answer, and to make sure I ask her about fourteen plus something is twenty again in the near future in order to strengthen the bond.
Option 2, by contrast, speaks to the idea that learning arithmetic is about becoming familiar with number relationships. Option 2 suggests that fourteen plus something is twenty is not an especially important problem on its own, but that it provides us with a place to practice noticing and using relationships in order to strengthen our familiarity with these relationships.
The thing I need to do if Tabitha is struggling with fourteen plus something is twenty is very different if I choose option 2. I need to think about what related problem is likely to be easier for her than this one. I need to think about how to help her make progress.
Here, the most likely productive direction (based on what I know about her, and about her mathematics learning experiences) is to ask:
Do you know this one? Fifteen plus something is twenty.
She probably knows that five is correct here. This is because she has counted by fives many times. Once she establishes that fifteen plus five is twenty, she will likely be able to reason that fourteen plus six is twenty. Fourteen is one less than fifteen, so the other addend must be bigger to get the same sum. She wouldn’t say it that way, of course, but she can think that way.
She can think that way for two reasons: (1) it is natural for children to think this way, and (2) this sort of thinking has been modeled, supported and encouraged.
In short, I and her teachers have taught her in ways that support powerful mathematical thinking.
What we see in the video above does not support that. While I (mostly) get the joke, it is not so far from the truth. This is precisely what goes on in many classrooms and homes. The parent does not ask the child what he is thinking. The child has gotten the message that there is a right way to perform the computation, and that it involves the 4 turning into something else. The whole thing is a mess and it is very very true.
It is too true.
Everything about that interaction needs to change. Everything.
But really, if we change one thing we’ll be on our way to changing everything.
It is a big change, of course.
We need to stop worrying about the child says, “six”. We need to start worrying about how (and whether) the child is thinking.