Consider these two division problems:
Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?
Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?
These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)
The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.
Do you know which is which?
We talked about this on Twitter today. (Click through for some really outstanding discussion….seriously.)
I have written about the two major types of division problems before, and they are relevant here.
Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.
So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.
Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.
When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.
What’s the moral of the story? Let me know your thoughts in the comments.
It took a bit of thinking but there is a similar situation with “subtraction” :
I have a box which will hold 24 cans.
I have 18 cans, and I put them in the box.
How many more cans will fit in the box?
You have 24 cans, and you put them in your box. Only 18 fit in your box.
How many left over are there.
I would have guessed wrong. My thinking is that the first problem is quotative (measurement model): fix the size of a share, fix the total, determine how many portions can be allotted. The second is partitive (fair-sharing model): fix the number of shares, fix the total, determine how big a fair share is.
Most kids have experience before school with fair sharing, not with the measurement situation.
Your explanation goes to something that usually only happens from schooling: an understanding of place value and division in the context of an algorithm.
I’m not claiming you’re wrong (after all, you collected the data). But can you see why from my perspective the result is surprising? It makes me think that she’s old enough to have had her more intuitive understanding of division as sharing overridden by something to do with formal operations combined with understanding of place value.
I suspect that slightly younger students would be able to do the second problem more easily, though they would for the most part do it the way you described: doling out “cookies” or gummy bears or erasers into 10 piles to determine equal shares. And I think they’d be a little more confused by the first problem, though of course they could certainly count out piles of 10 assuming they grasped what was being asked.
Where I’ve seen preservice elementary teachers struggle is to come up with problems to model division by (and/or of) fractions using these two models, particularly with creating a quotative problem like “I have a recipe that calls for 2/5ths of a cup of flour and I have 8 cups of flour. How many batches can I make?” to model 8 / (2/5ths).
While I think you are correct in assuming that kids have more experience with situations similar to problem B, just having an algorithm for dividing objects evenly doesn’t equate to being able to solve the problem directly. If anything, the experience with problem B works counter to being able to solve it using division – the student already has an algorithm that works (handing out the 22 cookies to the 10 kids, one at a time) It takes conscious effort in *thinking* about the problem to recognize that in each “pass”, 10 cookies are distributed, that the total number of passes is how many 10s are in 22, and that the total number of passes is equal to the number of cookies. It’s easy to forget that humans are creatures of habit – once we know how to do a task, we don’t often think about better ways of doing that task, even after we learn new things.
Moral of the story: What we think may be easy or difficult for a child is not at all what they are thinking. Every child is unique in their thinking and brings different intuitive feelings to problems that unless we talk and listen to them, we could make poor assumptions about their understandings. (that is the second, equally important, moral of the story;).
I was thinking the 2nd problem would be more difficult for a different reason. It may be too early for a 3rd grader to know about fractions like 2/10 (the last 2 cookies being shared by 10 kids) but I’d say that 10 kids could share 22 cookies equally without having any left over.
I love the differences in these two problems, simultaneously the same and different. No wonder division is so hard for kids to wrap their heads around. After all, sharing cookies with two people is not the same as sharing with 10. BTW, since there are really only two kids involved in problem A, I’m pretty sure they’d come up with an obvious solution for the two left over!