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The role of student work in CMP

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.

Pro

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.

Con

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.

Summary

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):

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A fundamental dilemma: the case of circles

The Connected Mathematics Project has always stood for coherence in units, and for telling a complete mathematical story. So we don’t do a little bit with fractions now and then a little bit three weeks from now. Instead we spend 4-6 weeks on fractions, decimals and percents-exploring equivalence, representations and models.

The Common Core State Standards presents a challenge to some of the story lines of our units. Consider area. In CMP, area and perimeter are dealt with in the sixth-grade unit Covering and Surrounding. We begin with irregular figures that are composed of whole numbers of square units, move to formalizing area of a rectangle and then use rectangles to develop area formulas for triangles, parallelograms and circles (as well as circumference of circles). All in one unit; all in sixth grade.

How many radius squares (shaded) fill the circle? More importantly, are students better served learning this in sixth grade with other area concepts, or in seventh as Common Core mandates?

But now…

Common Core, grade 6 geometry

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.

Common Core, grade 7 geometry

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

There are two schools of thought on the problem this presents.

1: Don’t worry about it. The argument goes like this: Do circles well at sixth grade. You’ll use area of circles at seventh grade, which will remind students of everything they need to know before it’s tested there. Don’t worry about it. Keep circles where they belong in the curriculum-with the area and perimeter of other plane figures.

2: Do as you’re told. The argument goes like this: Wait! Schools and districts will make decisions not just about which units to teach, but which parts of units to teach based on the Common Core Standards. Schools will skip circles at sixth grade in order to focus on what’s being tested there. If you don’t write a development of area of a circle into seventh grade, teachers will supplement and tell; this important topic won’t get done well.

Versions of this dilemma arise frequently in looking at CMP through the lens of Common Core. Volume of a cylinder is separated from volume of prisms. Volume of rectangular prisms with fractional side lengths is separated from volume of rectangular prisms with integer side lengths. Fractions and ratios are at one grade level; rates involving fractional units are at another. And on and on.

It would be helpful to hear from the field on this.

CMP and Common Core

By now you have heard of the Common Core State Standards. If you teach any subject at any grade level in almost any state in the US, you know that they will greatly affect what you teach in your classroom in the next few years. And you are probably working hard to get ready.

So are we at Connected Mathematics.

The first step was to produce supplementary materials for CMP2 so that classrooms using the curriculum as it is right now can teach all Common Core standards at each grade level without shuffling books between grade levels and having to write the occasional fill-in lesson. Those materials are available now from Pearson.

And for the past year, we have been working on a revision of the materials that will be fully aligned with Common Core (natively compliant, as the techies would say). We have been writing CMP3.

This blog is an effort to communicate with and gather information from the field as we work-teachers primarily, but we also hope to hear from researchers, curriculum directors, principals, parents and students.

In the coming weeks and months we’ll write about the opportunities and challenges involved in the revision work and we’ll seek your feedback.

We hope to encourage thoughtful conversation about teaching and learning mathematics in the age of Common Core State Standards.

Use the comments on our blog to tell us your hopes for the revision, to help us understand the concerns of teachers, schools and districts as you gear up for Common Core. Talk to us.