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Closing up the blog

With deepest apologies to all who had hoped for more frequent updates as we worked on CMP3, this will be our final post. The amount of work required to get the basic job done left us without time to keep you all updated as we had planned.

Here are a few short updates and then we’ll redirect all inquiries through the CMP website.

Reader Jenny asks in the comments:

What’s new in the development of CMP3? On target for Spring 2013?

Would it be possible to post an outline or Table of Contents so we may begin planning for the transition to the Common Core? My district would like to move a unit or two this coming school year followed by full implementation the next year. Any information you could provide would be greatly appreciated!

The target is still to have the curriculum ready for classroom use in Fall 2013. Business negotiations between Michigan State (which owns the copyright) and Pearson (which edits and produces the physical product) are in their final phases. As soon as that wraps up, the first set of final drafts of units will go to Pearson for production. At that point, all timelines will be out of the hands of the Connected Math crew at MSU.

But the goal is certainly to have samples of at least some units available in the coming school year. And Pearson will definitely be producing Common Core alignment materials ASAP, including scope and sequence information for the units. Please keep an eye on the CMP website and stay in touch with your Pearson rep.

We have plans to make 2013’s CMP Users’ Conference in February be an official rollout of CMP3. Detailed information about that should be available mid-to-late fall. Keep an eye on the CMP website for dates.

Compacted Trajectories and Other Common Core Concerns

Barb asked in the comments recently:

What are the authors thinking about how to organize CMP2 for the Compacted Trajectory courses in the Common Core?  Which units would go in the 7th/8th grade course, and what would be left for the 8th/9th grade course?  Any suggestions?

This is honestly not a conversation that we have had. It is possible that Pearson has designed something that nominally fits the request, and that will likely suffice until CMP3 is in print. In the meantime, we would be happy to look over anything you’ve put together and give some quick, critical feedback.

If it’s on the web, post a link in the comments and we’ll get to it as soon as we can. Otherwise, we can catch up by email; we’ve got yours since you left it when you commented.

The issue of what is going to happen in eighth grade in the Common Core era concerns us tremendously. We greatly welcome your feedback on this question:

How will districts handle the fact that the eighth grade CCSS have minimal algebra when the recent trends have been towards more algebra in eighth grade, not less? In particular, will you teach to both the eighth grade CCSS standards AND the Algebra standards for high school?

 

Building an online community

Twice now in the comments on this blog, we have gotten requests for a place where CMP teachers can share online. For example:

Where is there access to teachers talking to others implementing CMP? I do a reasonably good job searching the web for resources, but find nothing much about CMP. Why? I would like to participate in an active discussion about the investigations in various units, but none seem to exist? Are they not allowed? I am curious. I have taught CMP for a long time, currently for 7th grade and such conversations would be helpful.

We agree.

The true answer to the question, Why is there no online CMP community?, I’m afraid, is the simplest one. No one has done it.

Back in the mid 90’s, when CMP was in the original piloting phase, there was a bulletin board (you dialed in directly to it), and that didn’t get much traffic (for obvious reasons). In the late 90’s, there was a listserv for CMP leaders that did fairly well. But in the age of Web 2.0? No one has put it together.

Twitter would be one way to go. The disadvantage there is that Twitter is a synchronous communication medium. If you teach Prime Time at the beginning of 6th grade and someone else does it at the end of 5th, you’re unlikely to find each other.

(UPDATE: In the week between writing this piece and posting it, CMP has gone live on Twitter: find us as @connectedmath)

A blog requires curating. This blog is focused on keeping the field informed about the revision for CMP3—and allowing for input as the revision proceeds—rather than on supporting dialogue about the current practice of CMP2 teachers.

We have looked into a few other platforms and haven’t been able to craft the vision.

So, again, please let us know if you have any examples of the sort of thing you’d like to see exist. We’d be delighted to give something a try. Through Facebook and the CMP3 blog (and our known network of leaders, and the publisher, etc…) it would be easy to spread the word widely that we’re trying to build something. We just need to have a vision for what to build.

Division algorithms

Julie asked in the comments recently (slightly edited here):

I see that the CCSS has 5th grade teachers teaching the partial [quotients] method of division and then 6th grade teachers are to link this understanding to the standard division algorithm. The kids don’t understand it nor do they want to. Nowhere in the 6th grade CMP2 does it show teachers how to make this transition. It seems like a small detail but it causes major problems for our students, teachers and parents at grades 5 and 6. I have in the past asked the teachers to continue w/ partial [quotients] but now w/ the new CCSS it clearly says to fluently divide multi-digit numbers using the standard algorithm. Do you have suggestions?

As you can imagine, the issue you point to caused some consternation at CMP. Common Core calls for sixth-grade students to:

Fluently divide multi-digit numbers using the standard algorithm.

This occurs frequently in CCSS, I’m afraid. The standards seem unusually concerned with which algorithms students are using at which grade levels, and there is little justification for why a particular algorithm might be the right one. (For further extended discussion of related matters, see Christopher’s blog, Overthinking My Teaching).

We are of two minds on the particular standard Julie points to, and on wide range of other similar issues.

1. Spirit of the law

Feedback from the field has been quite strong indicating that high-stakes assessment is what is going to drive instructional priorities in the roll out of Common Core.

We have tried to imagine a principled test that would be able to sort out whether a student knows the standard algorithm. This test would have to have one of two types of questions: (1) an extended response scored by a human being who had been trained to look for the standard algorithm, or (2) a multiple choice question in which a student is supposed to fill in a missing digit somewhere in the algorithm but not in the final quotient. The first seems really unlikely and the second is unconscionable.

So in a version of Bits and Pieces III (with a tentative title of Decimal Ops) that we are field testing this year, we have split the difference. We are developing sense for the standard division algorithm by connecting to division of fractions, as we did in CMP2. Then we have this:

And we have some text and problems around this that support connecting to the thinking behind the partial quotients algorithm. But in the student edition, we don’t go so far as getting to the digit-by-digit standard algorithm.

There is a principle at play here. The standard algorithm is efficient (although not a lot more efficient than a well-executed partial quotients algorithm, and certainly not much more efficient than the algorithm depicted above). But the efficiency of the algorithm comes from ignoring place value and operating on each digit, or subset of digits, as though they represented units. The standard algorithm does not develop children’s number sense.

Furthermore, a common justification for teaching the standard division algorithm is that it is necessary for algebra-it generalizes to polynomial long division. But this argument is not quite right. Polynomial long division-to the extent that it can be justified as a necessary algebra topic-is more closely tied to partial quotients than it is to the standard algorithm.

2. Letter of the law

So even if sixth-grade CMP3 doesn’t go all the way to digit-by-digit standard long division algorithm in the student materials, we want to be supportive of teachers who need to meet every detail of the standards. We will have support for that in the teacher materials. The student book gets us awfully close to the standard algorithm. We will write teacher materials to help teachers build to the standard algorithm by going further with the ideas in the student edition.

Will CMP3 just mean more topics?

Reader Mark Schommer asked in the comments recently:

I fear time. It appears to me that CMP has added time to a schedule that is very full already. My teachers have difficulty finishing the requirements and to add additional (even though it is necessary) makes me much more concerned.

Will CMP3 be removing current CMP2 topics or re-aligning?

This is an important question.

At present, there exist materials produced by Pearson (our publisher) that are supplements for aligning CMP2 with Common Core. These materials are what Mark fears-putting more stuff into an already overloaded year of instruction. They are a stopgap measure-something to help schools and teachers get by during this time of transition to CCSS.

But they are not a satisfactory long term solution.

The long term solution is a full revision of the materials. That’s what CMP3 will be.

We do not have the National Science Foundation grant money that we had in CMP1 and CMP2, and we do not have the kind of time we did in those versions either. As a result, we are trying to use a light touch on the materials. But if you have tried to do an alignment with CCSS, you know that a light touch won’t cover everything. And this is especially true at sixth grade.

Here are a few of the more substantive changes at sixth grade that we are field testing this school year. This list is not comprehensive; it is intended to give you the flavor of what we are working and how CCSS will impact Connected Math classrooms over the next few years.

  1. Number properties (especially the distributive property) is going into Prime Time so that we can refer to them throughout sixth grade.
  2. Shapes and Designs will become a seventh-grade unit.
  3. Variables and Patterns will become a sixth-grade unit, with some adjustments and the inclusion of some beginning equation solving.
  4. Ratios will go into Bits and Pieces I.

This last one may have been the most challenging feat of all to pull off, and it is one of the units on which we are most looking forward to feedback from the field. Our first try at this had us tacking on an extra investigation to Bits and Pieces I. Of course, this is the exact fear Mark expressed in his comment. Our latest version weaves ratios throughout the unit-asking students to make comparisons among the fundraising goals and progress of the various classes, for instance. We have decreased attention to things that come earlier in Common Core in order to make space for ratios.

Oops!

After a summer hiatus, we’ll get back to weightier matters shortly.

In the meantime, you will be pleased to learn that we are setting up in-house procedures to make sure we don’t misspell this word again:

From the notepads we distribute at conferences.

 

A Brief History of Algebra in Connected Math

Does 8th grade CMP count as Algebra 1?

Short answer: Yes.

But this is a question with a history. Let’s begin in 1989.

1989 NCTM Standards and CMP 1

In 1989, the National Council of Teachers of Mathematics issued Curriculum and Evaluation Standards for School Mathematics. These standards formed the basis for the National Science Foundation funding of the writing of Connected Mathematics and other elementary, middle and high school curricula.

In particular, the Standards called for increased emphasis on algebraic ideas in the middle grades.

One critical transition [between the elementary and the high school curriculum] is that between arithmetic and algebra. It is thus essential that in grades 5—8, students explore algebraic concepts in an informal way to build a foundation for the subsequent formal study of algebra. Such informal exploration should emphasize physical models, data, graphs, and other mathematical representations rather than facility with formal algebraic manipulation. (p. 102)

…by the end of the eighth grade, students should be able to solve linear equations by formal methods and some nonlinear equations by informal means (ibid).

In fact, the authors created algebra units in CMP 1 that addressed all of this and went above and beyond these standards.

Variables and Patterns introduced tables, graphs—and to a lesser extent equations—and relationships between variables in seventh grade. Moving Straight Ahead, also seventh grade, had students working extensively with linear relationships-slope, y-intercept, tables, graphs and equations. At eighth grade, there were four algebra units. Thinking with Mathematical Models revisited linear relationships and introduced (1) modeling and (2) the idea that not all predictable relationships are linear. Growing, Growing, Growing used exponential relationships to strengthen students’ understanding of linearity. Frogs, Fleas and Painted Cubes introduced quadratics and Say It with Symbols summarized the symbolic algebra work in the curriculum.

Taken together, these units created a much more challenging and much richer eighth-grade mathematics curriculum than was typical at the time.

It didn’t take long before the question was being asked.

Does 8th grade CMP count as Algebra 1?

In writing CMP 1, it had not been the intention of the authors to answer this question one way or the other. In the early 1990’s, there was no large scale emphasis on Algebra 1 for eighth graders. Instead, the authors had set out to write a challenging curriculum rich in algebraic ideas.

Principles and Standards 2000, and CMP 2

In 2000, NCTM issued a revised set of standards-Principles and Standards for School Mathematics. These standards also stuck to algebraic ideas and an emphasis on linearity in the middle school curriculum.

Students in the middle grades should learn algebra both as a set of concepts and competencies tied to the representation of quantitative relationships and as a style of mathematical thinking for formalizing patterns, functions, and generalizations … They should connect their experiences with linear functions to their developing understandings of proportionality, and they should learn to distinguish linear relationships from nonlinear ones. In the middle grades, students should also learn to recognize and generate equivalent expressions, solve linear equations, and use simple formulas.

The authors received funding from the National Science Foundation in 2000 to revise the materials from beginning to end, with Principles and Standards as one of the major driving forces. On their own, these standards would not have required a fundamental change in CMP’s algebra strand.

But this time around, they couldn’t ignore the important question of whether eighth-grade CMP is equivalent to Algebra 1. Teachers and district decision-makers had been asking the question, and there had developed a policy environment in which Algebra 1 in eighth grade was being seen as essential for all students.

(For more about this, see a Brookings Institution report, a Washington Post op-ed and a District Administration document. Also, discussions of algebra for eighth graders invariably cite The Algebra Project)

So the authors made a commitment at the beginning of their work creating CMP 2 that eighth-grade CMP 2 would be equivalent to Algebra 1. Now…that said, there is no universal agreement on the definition of Algebra 1. But it was clear that a commitment to aligning CMP2 with Algebra 1 would mean several changes, including (1) more formal symbolic work, especially with respect to the distributive property and quadratic functions, (2) increased attention to algebraic inequalities and (3) some formal work with systems of equations.

In order to address these needs, the existing algebra units were revised and a new unit, Shapes of Algebra was written. Some teachers may have to tweak or supplement in order to meet their idiosyncratic state or district guidelines for Algebra 1, but the eighth-grade CMP 2 algebra units now comprise a principled Algebra 1 course.

Now what? Common Core and Algebra in CMP 3

The present driving force for curriculum and testing is the Common Core State Standards. One of the puzzling things for us is illustrated by this image from the CCSS webpage:

Algebra and Functions are two different standards, while CMP has taken a functions approach to algebra. These different orientations are sure to make alignment challenging in schools.

Eighth-grade Common Core does not comprise a full Algebra 1 course in the standard American curricular sense. We are curious to know how the standing pressures to offer Algebra 1 to all eighth graders will interact with the new pressures of testing in the Common Core era.

We’ll share on this blog the decisions we are making as we work on the algebra strand. But please check in and share your stories from the field. What are the pressures you are working under as you rethink eighth grade in your districts and buildings in light of Common Core?

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):

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.