The child says “six”

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.

  1. To get the child to say, “six”.
  2. 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.

No.

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.

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17 responses to “The child says “six”

  1. Thx. This draws a fantastic contrast between a traditional mindset in math with the growth mindset of the future!

  2. Christopher.
    To me, this is a hugely important piece. Thank you. You’ve taken one of the biggest (if not the biggest) issue in elementary math (and beyond) and laid it out beautifully. I will be sending this to my District Math Department leads and asking them to include it in back-to-school professional development.

    Thank you.

    Seth

    PS Louie Armstrong said, “If you have to ask, you’ll never know.” You didn’t have to tell us that the video was supposed to be funny.

  3. This was on the back on my Cheetos bag. https://twitter.com/LukeSelfwalker/status/468778228480765953 It provided a good talk with my seven year old. Maybe you and Tabitha could spark up a conversation about it.

  4. I’m having issues with an online group of adult ed folks … we were presented a “challenge problem” supposed to be typical of the “new GED.” (http://resourceroomblog.wordpress.com/2014/05/29/challenge-problem/ )
    It was poorly written… and the “correct” answer assumed that two variables were equal, when there was no indication in the problem that they were equal; they were even given letters.
    However, if you sort of ignored that and put some stuff in the table they provided, you’d get the answer they wanted.
    When I described my difficulty with the problem, the person who posted the problem replied with all kinds of assurance that I shouldn’t feel frustrated; she tried to explain why the answer was right, and encouraged others to “help me understand.”
    I’ve said about five times that I understand the problem… and that it’s wrong… but that isn’t even considered.
    It seems so painfully clear that The Object Is To Get THe Right Answer — even if it isn’t. It’s so Orwellian to be encouraged to use “better” mathematical reasoning…

  5. I feel your pain, xiousgeonz. Believe me that I feel your pain. Thanks for backing me up on the This is not a straw man bit.

    When in the situation you have encountered, I find myself trying to state my case twice, and then acknowledging that the folks who argue back are not ready to hear what I have to say. I find that I need to tell myself this and move on. Otherwise my head becomes bruised from too much banging against the wall.

  6. It’s taken me four (second time I made the mistake of saying I would cease and desist, which of course got the reply that no, no, this should be a *safe* place for us to try to grow in our mathematical understanding…) Repeating my responses … is time better spent … on just about anything.

  7. There was nothing about that video that didn’t bother me, from the refusal to understand number sense to the blatant racism.

  8. @JustonAion – me, too.
    Unlike Seth Levett, I’m glad you said the video was supposed to be funny because otherwise I would have thought it was just sad and intended to flame the anti-Common Core sentiment.
    Is anyone helping parents understand how Common Core math is different and why? (And by the way, to just ask their child “how does your teacher want you to do this?” and then listen! Grrr.) I don’t know how to get this done, but it seems to me if we don’t get parents on our side we won’t succeed and will increase the anti-math sentiment and incidence of math anxiety, rather than reduce it.

  9. Agreed. If the schools are going to ask kids to explain their strategies on problems their parents think don’t have strategies, then there had better be clear messages home to the parents.

  10. Hello again.

    I wish Christopher had not included the video in this post. It’s distracting from what I think is the larger point. He states,

    “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.”

    To me, he zeroes in on a major issue in early Math Ed (and later Math Ed). Are we seeking answers or are we seeking understanding? Put bluntly like that, the issue seems simple. But it’s not so simple. Christopher’s interaction with Tabitha and his reflections on that interaction demonstrates that.

    It doesn’t have anything to do with Common Core.

    Seth

    PS I thank those of you who pointed out the racism in the video.
    PPS I still think the interaction between mother and child are funny.

    • Sorry. Knee-jerk reaction to the video. I concur 100% on Christopher’s main point: answers versus understanding. Although, the implementation of Common Core is what’s driving the wider push for understanding as opposed to rote memorization and regurgitation.

  11. I think that the question is not the right one. It leans desperately towards the symbolic side of math, something which is extremely overemphasized in the early stages. The child level question is “The man needed 14 more cents. I gave him 20. How much too much did I give him?” (I really wanted to write “many”, you try it!!). or variations on this question.
    (or: How much more did I give him?).

    Whatever happened to “add” and “take away” ? Perhaps they died.

  12. The particular question at hand, howardat58 is not really germane to the argument here. The argument is that when we find ourselves in a situation where a child doesn’t know how to do something, we ought to lean on support strategies that help to improve the child’s thinking, not on support strategies that help the child to utter the answer.

    A more nuanced discussion of the question itself and how we settled on it in this case will be up on our sister blog, Talking Math with Your Kids in the coming days.

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  15. The video is hilarious, I was in splits and peals of laughter by the end :D (sorry, I know this is serious stuff, but the video is just too funny)…

  16. But, on more reflection, and on a more serious note: In my experience with my 6-yr-old, the best understanding has come from doing the same problem multiple times in multiple ways. So,
    1. 14 plus something is 20.
    2. 20 minus something is 14. (through different play examples, getting him to see that addition and subtraction can be different faces of the same idea: putting together two smaller parts to make a bigger part).
    3. Friends of 20: this is something their school introduced as songs: friends of 10, friends of 20. (to again see different partitions of the same number).
    4. and the list goes on. So, like turning friends of 20 into a game that we play with buttons, or hot wheels cars, or marbles, and form two groups, and keep shifting objects from one group to the other to see that the big number stays the same, but many small numbers can come together in different ways to form it…etc.
    I don’t know about the curriculum issues, since I am not from the US, (education systems anywhere in the world are very interesting though), but I do see value in doing the same thing in multiple ways to support deep learning and understanding.

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