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The role of student work in CMP

May 24, 2011

There are a few places in CMP where we use the following structure:

  1. Pose a rich and interesting problem in a fairly open way
  2. Follow up with a more directed version of the exact same problem, usually incorporating hypothetical student solutions.

Moving Straight Ahead

We do this, for instance, in Moving Straight Ahead. Problem 2.1 poses a relatively open task:

  1. How long should the race be so that Henri will win in a close race?
  2. Describe your strategy for finding your answer to Question A. Give evidence to support your answer.

Then in Problem 2.2, we have students make tables and graphs, and write equations with given independent and dependent variables. We ask points questions such as:

After 20 seconds, how far apart are the brothers? How is this distance represented in the table and on the graph?

Say It with Symbols

We do it again in Say It with Symbols (grade 8). In Problem 1.1, we pose the fairly open task, based on this picture:

  1. Write an expression for the number of border tiles N based on the side length s of a square pool.
  2. Write a different but equivalent expression for the number of tiles N needed to surround such a square pool.
  3. Explain why your two expressions for the number of border tiles are equivalent.

Then in the next problem, we present students with several expressions/equations that other students created and we ask about the equivalence of these.

I remember being a participant in a workshop with our very own Glenda Lappan in the summer after my fourth year of teaching (this would be 1998). Glenda told us, with respect to the second Say It with Symbols problem, “I hope you never have to teach this problem.”

The idea was that the rich variety of equations/expressions and the classroom summary focused on equivalence should come from students’ work on the first problem. But if it doesn’t, you teach the second one.

Comparing and Scaling

The orange juice problem from Comparing and Scaling is a classic CMP problem, and versions of it are to be found in many different places. The basic set up is this: You are making orange juice from concentrate. You have several different recipes, each involving a different number of cups of concentrate and cups of water. Your task is to decide which is the most orangey.

In the original CMP, we posed a relatively open version of this question.

Which mix will make juice that is the most “orangey”? Explain.

In CMP2, we retain this question and append the following.

Which comparison statement is correct? Explain.

  1. 5/9 of Mix B is concentrate
  2. 5/14 of Mix B is concentrate

We are trying to think about whether this problem is a place to apply the technique described above. Do we want a second problem in which we pose student solution strategies for whole-class consideration? There are really two questions implicit in this-(1) Should we do this more often? and (2) Should we do it here?

We would be interested to answers to both of these, but especially 1.


Arguments in favor include the following.

  1. There are two important categories of solution: part-part and part-whole. These need to come out. The second problem ensures that they do. This is the insurance principle.
  2. Doing so can convince skeptics flipping through the materials that important concepts are embedded in the curriculum. This is the math wars principle.
  3. A problem focusing on student solutions encourages a key mathematical practice, critiquing the reasoning of others. This is the common core principle.
  4. Incorporating entire problems focused on student work communicates the importance of student thinking to teachers. It helps to reinforce that CMP is not a curriculum focused on efficient teacher telling; it’s one focused on teachers helping students make their own ideas better and more clear. This is the professional development principle.


The main one here is time. We are acutely aware of the time-based pressures under which schools operate. We are trying to trim unnecessary problems. We can’t really afford to make existing content longer.


As we move forward, we need to be considerate of the time it takes to teach a unit. We also need to keep some principles in mind that may argue in favor of introducing a second problem that focuses student thinking after a more open exploration of the original context.

In this particular case, we would get to show this lovely example of student thinking (as well as others):


From → Grade 7, Grade 8

One Comment
  1. Dana Cox permalink

    I am in favor of including more of these types of problems. I certainly understand the time argument against. Consider this, however.

    1. You are not always doubling the time it takes to wring as much mathematics as possible out of the lesson. As Glenda says, it is the curriculum designers hope that these second stage problems would not need to be assigned. The work generated by the first stage problem should take care of that.
    2. The second stage supports teachers. In the event that the student work is not as varied as we would have liked, teachers need insurance. The insurance principle is a good one, in my opinion, as tasks do not always give us that fertile soil to grow rich classroom discussions. I like that you are considering what a second run at a concept might look like. The work of reteaching is difficult work.
    3. You will inspire teachers to look for new mathematics. Once you have solved a problem yourself, the task of going back and trying it with fresh eyes is a very difficult one for everyone. In my work with in/preservice teachers it is rare that a task will lead to the varied solutions that are possible with students. I find it only demeaning to continue to ask the question, “can you think of another way to solve it?” While it is true that a classroom of students will provide reasoning that a teacher has not previously considered, the second stage problems (even when omitted) give teachers insights into particular strategies of interest and highlights potential mathematics that they may not previously have thought could drive classroom discussion.

    I’m in favor, but I also think there may be a few more arguments against worth discussing.

    1. Providing second stage lessons may influence teachers in the opposite direction. They may tailor the first stage lesson to fit the mold provided by the second. In that way, it may curtail the open math discussion.
    2. Another con I might add would be the potential reluctance of teachers to teach the first stage. Why spend the time on an open exploration when we can get to the heart of the matter?

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