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”.
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.
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 , sure. But can’t you see that each gets ? Or that it could also be ?
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.