Task design

For the last two years, I have been working with the four surviving original authors of Connected Mathematics on a revision that is responsive to the Common Core State Standards. My task has been top-to-bottom revision of three of the four rational number units, Bits and Pieces I, Bits and Pieces II and Comparing and Scaling.

The process for this work is unusual for commercially available U.S. curriculum materials, so I want to share a few observations from the inside. They will trickle out over the next few months, and they’ll get filed under “Curriculum”.

The task

In CMP2, we focused the initial fractions unit on careful introduction of the number line. The premise was that children had lots of elementary school experience with area models for fractions, and that we wanted to introduce the more sophisticated linear models.

We introduced the licorice lace problem.

In this problem, a group of four kids is going on a hike. They have a 48-inch licorice lace and they want to share it equally among themselves. Sid (the protagonist in our narrative) carefully marks the places where he will make the cuts.

Just before he actually cuts, two more kids show up. Now they need to make new marks on the already marked-up lace. The cycle is repeated a couple of times. At each phase, we ask students to name the part of a licorice lace each hiker receives.

If you try this yourself, you will notice that it’s pretty hard to locate the marks for sixths when there are already fourths marked. Not impossible, but hard.

Ideally, some students in class will try, and some students will go to twelfths; others will go to twenty-fourths. Then when it’s time to name the fractions we have sixths, twelfths and twenty-fourths on the table and we can talk about equivalence and partitioning linear things.

From classroom feedback and my own experience working the problem with adults (both in professional development and college courses), it was clear that the problem needed a redesign. The set up was wordy, using one and a half pages of text to work through a small set of tasks. The marks before cutting were slightly implausible. The sharing and re-sharing was too complex for simple problem-posing.

The redesign

Two years ago, I took on the task of redesigning this problem.

I knew we needed something that was (1) linear, (2) shareable, and (3) already marked.

Linear and shareable are properties of licorice laces (these are not Twizzlers, each of which-while shareable-is not plausibly shareable among four children). That third criterion was new. If I could find something whose pieces were already marked, I could get rid of the complicated storyline and a tremendous amount of text.

“Marked pieces” is important because this is a problem about partitioning and repartitioning. We want kids to have pieces that they need to cut up further, and to have to think about names for these new pieces.

Skittles would not work.

These, while delicious, are mathematically unproductive for our purposes.

A bag of Skittles is composed of the original unit, one Skittle.

I needed something where we partition the original unit. It is perhaps shameful how many hours of thought went into this. But I eventually found it.

It’s perfect. Linear, shareable and already marked. You want to share equally? Each person gets \frac{1}{2}, sure. But can’t you see that each gets \frac{4\frac{1}{2}}{9}? Or that it could also be \frac{9}{18}?

It is helpful that most people don’t know the standard partitioning of a Tootsie Roll. (Did you know it was ninths? Be honest, now. It hasn’t always been; it used to be sevenths.) If you don’t know the standard partitioning of a Tootsie Roll, then we can make as many pieces as we like to start each new task. No more marking and re-marking. We just give a new  Tootsie Roll and a new number of people.

We know from research that sharing is a productive context for understanding fractions. We’re sharing something that is already partitioned, so we need to repartition when the number of sharers is not a factor of the number of pieces.

Feedback from classrooms and my experience working with adults (again-professional development and college courses) suggests that we get more mathematics with a lot less effort setting up than we did with the previous version.


Not every problem in Connected Math has gotten this level of attention, of course. But a lot of them have. This is a curriculum that takes context seriously as a basis for mathematical activity and abstraction.

Once we have committed to a particular mathematical development (e.g. partitioning in linear situations in order to move to the number line), we seek a problem in which the right mathematical activity naturally results. I am proud to have been a part of that.


11 responses to “Task design

  1. >If you don’t know the standard partitioning of a Tootsie Roll, then we can make as many pieces as we like to start each new task.

    Do you tell the kids it’s a Tootsie Roll? (Even if you don’t , but do show a photo, some will figure it out.) Do you give them problems that give the wrong number of segments to start out? That seems problematic to me. There’s the issue of people feeling like “this isn’t real, even though the book pretends it is” aka pseudo-context. One solution for this that I liked, which you may not have seen is at: http://blog.mathpl.us/?p=101. Basically, using out-of-this-world contexts is helpful.

  2. Sue: Due to publisher constraints, we will not call them “Tootsie Rolls” but instead “chewy chocolate rolls”.

    I accept your critique, but I’ll push back. The major consideration here is whether sharing a (say) five-segment Tootsie Roll is pseudocontext. Presumably we’re working off Dan Meyer’s definitions namely:

    1. Context that is flatly untrue, and
    2. Operations that have nothing to do with the given context.

    And I further presume that 1 is the basis of the critique.

    While it is untrue that there are five-segment Tootsie Rolls (there are not), there are or have been one-segment, seven-segment, nine-segment and fourteen-segment Tootsie Rolls. Is it such an unreasonable stretch to ask kids to imagine and operate on a five-segment Tootsie Roll?

    Must a context be only about the world as it is while completely avoiding the world as it might just as well be? There’s something interesting and debatable in that question.

  3. I’m not sure of the answer to that. But I wasn’t suggesting that you use the world as-it-is, I was suggesting you depart further from reality, so there’s less clash.

    I linked to someone who had a great idea related to this question. You run less risk of kids turning off because of a (small, perhaps) betrayal of trust when the allegedly true context is discovered to be not really true, if you use obviously made up contexts, like potions, and the candy eaten by wizards, trolls, vampires, hobbits, etc. I’d really like to know what you think of the post I linked to.

    It sounds to me like you’re doing great work. But any time you’re working on curriculum that so many thousands of kids will be forced to use, you have big obligations to do it as right as possible. I’m not sure what that means exactly, but I thought this was an interesting issue to explore.

    It’s probably no fun to think about how it could have been done better, when you’ve put so much thought into getting it right already.

  4. Maybe I shouldn’t have used the term pseudocontext, because your operations do seem to work well within your context. And your part (1) of the definition isn’t how I would state it. I’d put it that the book pretends something is true or matters that isn’t or doesn’t.

  5. Oh, it’s definitely an interesting issue to explore, Sue.

    If I read you right, you’re a bit happier not having these be “Tootsie Rolls” since they don’t actually come in a whole mess of different numbers of segments. Having them be “chewy chocolate rolls” removes the condition of “pretending something it true…that isn’t”. That’s an interesting perspective.

    But the piece you linked to is advocating an even stronger move. This should be about some fictional/fantastic situation. Like a world where the good people of Naraboo eat dragon turds, which are conveniently marked in segments according to the dragon’s age. I kid, but that’s the direction, right? Make up something silly to avoid clashing with kids’ reality?

    My feeling on that is that it needs to be done with a really light touch. I have used Sybilla Beckmann’s book for my math for elementary teachers course. I like a lot of what’s there, but there is a surplus of Harry Potter-esque problems that always rubbed me the wrong way. Perhaps it was the context of working with future teachers in a college course, but there’s more to it than that.

    Considering the example in the article you linked to, I agree wholeheartedly that a problem about buying CDs at Tower Records is troublesome. But I’m not sure Harry Potter is the way out of the trouble. Isn’t there something in kids’ actual lives that costs about 12 dollars and they might want to buy in multiples? Movie tickets? T-shirts? Museum passes? Bus passes? Something?

    And when we bump up against something that is realistic, but outside our students’ experience (as in earning money for babysitting-some people do earn money for this), isn’t it an opportunity to enlarge our students’ experience vicariously?

    And we also need to discuss (I think) the simplifications we make in order to think in particular ways. Yes, if I buy multiple boxes of cereal, I might get a discount. Yes, we are ignoring sales tax. Yes, I trimmed a bit of extra Tootsie Roll from the ends before I took that photograph. Let’s solve the simplified version first and then ask how reality impacts our idealized solution. Because that’s what math is.

    Or so I would argue.

  6. Sounds like we’re on the same page, or close to it.

  7. (Somehow Sue keeps beating me to the comments of every blog that I want to leave a comment for this afternoon! 🙂

    I’m a little bit familiar with Connect Math via Bits and Pieces. I’m just happy to learn that you do take the time and care in “the redesign” of the curriculum because as Sue said, “so many thousands of kids will be forced to use, you have big obligations to do it as right as possible.” Not that the authors of our current adoption are mean spirited or are numbnuts, but they seem to have collaborated with all the wrong people for the wrong reasons.

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