Tag Archives: CMP

A really big experiment…really big

It is just beginning to dawn on me how big an experiment the Common Core State Standards really is.

Ladies and gentlemen, this is huge.

Huge like No Child Left Behind was huge. Huge like charter schools and vouchers. Much bigger than Teach for America.

Allow me to elaborate.

I work part time for the Connected Mathematics Project (CMP). We are in the early phases of a revision in light of the Common Core State Standards (CCSS or Common Core). An important feature of the development of CMP has been extensive piloting and field-testing, as in the diagram below.

As we have been working on revisions of the materials, a major goal has been to use a light touch. We have tried to add content where necessary to meet CCSS without disrupting what we know from research and field testing is working well. Because this time we don’t have National Science Foundation funding for the kind of field-testing we did in the past, and we don’t have the time we did in previous versions because Common Core is coming very, very quickly (see, e.g., the timeline for the state of New York.)

And if we aren’t able to get the level of feedback that we did for previous versions, I can guarantee that no one else is either. The development of CMP (and, to an extent, the other NSF-funded curricula of the 1990’s and early 2000’s) has been unique in American math curriculum.

But a light touch won’t do it.

Not only do we have to add ratio to sixth grade (formerly in seventh), we have to solve equations there too. And we have to do volume and surface area of a rectangular prism there, but volumes and surface area of other prisms in seventh grade, where formerly they were all together in seventh grade.

And of course there’s more. Operations on fractions, one of the most carefully developed ideas in CMP? Those are done by fifth grade (not in the purview of CMP), with the exception of division of fractions, which is at sixth grade. And on and on…

The more we ask around, the more evidence we have that schools will adopt curriculum and will structure their students’ pathways through the curriculum based on the letter of the Common Core standards, not on the spirit. So if we write a unit that does all operations on fractions, only the part that does division will get taught (or supplementary material will be used instead, or the curriculum won’t get adopted).

And so in a very real sense (protestations to the contrary on the CCSS website notwithstanding), Common Core is developing a curriculum.

Recall the CMP development diagram above. Has Common Core been subjected to the same development process? Consider the following from the Common Core FAQs:

  • Aligned with expectations for college and career success
  • Clear, so that educators and parents know what they need to do to help students learn
  • Consistent across all states, so that students are not taught to a lower standard just because of where they live
  • Include both content and the application of knowledge through high-order skills
  • Build upon strengths and lessons of current state standards and standards of top-performing nations
  • Realistic, for effective use in the classroom
  • Informed by other top performing countries, so that all students are prepared to succeed in our global economy and society
  • Evidence and research-based criteria have been set by states, through their national organizations CCSSO and the NGA Center.

Do you see where evidence and research fit in this scheme? Last. Not only are they last in the list, there isn’t even a claim that decisions about scope and sequence in the standards are based on research or evidence of any kind.

Why do we teach rectangular prisms at a different grade level from other prisms? Unclear. It’s certainly not based on research evidence of how students learn about volume and surface area. It’s unlikely even that it’s based on sound learning theory. And it’s certainly not based on meaningful mathematical connections.

So it’s an experiment. And 44 states have signed on.

I previously thought of the experiment as a benign one. I thought that for CMP it would mean a few tweaks around the edges, maybe some gentle and healthy nudges to increase the level of mathematics students are expected to do at sixth and seventh grade.

But I now understand that it’s pernicious. The standards are badly designed. They have nowhere near the research and field-testing base of existing curriculum. The impending assessment components of CCSS seem likely to dictate an even tighter quarterly adherence to curricular sequencing than the present annual testing that has had such a profound impact in the wake of NCLB.

And Connected Math gets derided as experimental; as the new new math?

Please.

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Data kids might find relevant

I know this is a non-starter. But I hope it sparks some others to think about something important with me.

In the Connected Math unit Data about Us at sixth grade, students collect information about themselves as a class, they represent the data in a variety of ways and they draw some rudimentary inferences.

Two subsequent units in the curriculum draw on these ideas. In Bits and Pieces I, students use percents to summarize survey data. In How Likely Is It? students use data analysis and proportional/fraction reasoning to study probability.

In HLII, there is an Investigation involving inherited traits, such as the ability to curl one’s tongue, attached earlobes, curly v. straight hair, etc. Many of these are standard chestnuts of Mendelian genetics; nearly all have been debunked.

So future versions of the curriculum will not use the Punnett-square for theoretical analysis of trait inheritance. But they remain (I think) reasonable areas for data collection. They are age-appropriate and interesting to middle school kids.

So we get rid of the genetics lesson and focus on descriptions of populations instead. Fair enough.

But frankly, how interesting is the following task?

table of data from a survey of genetic traits in the US

The original task from Connected Mathematics 2: How Likely Is It?

Answer: Not very.

It’s fun to think about our own attached/detached earlobes. But not so fun to look at survey data on the matter.

So I started thinking about what kinds of rudimentary data inference kids could do instead. And what if we took Dan Meyer’s challenge seriously and applied it to this problem?

I will reiterate that my first idea is a non-starter. No way is this the right problem. But consider what an interesting question can result.

The task

Here is a fifth-grade class, circa 1984.

photograph of a fifth-grade classWhere in the United States is the school located?

I would love help thinking about this problem on two levels:

  1. This problem. What do you notice in the picture that might help answer the question?
    What data do you attend to?
    How do you find yourself wanting to answer? Regionally? By state? Rural/suburban/urban?
    How sure are you of your answer? It’s probably…? It might be…? It’s got to be…?
  2. This kernel of an idea. My hunch is that this task as stated now is too sensitive for sixth grade math classrooms. But do you agree that it’s a more compelling question than the original? If so, what might the mashup of this task with the original look like? Is there a version of this idea that poses as intriguing a question, without setting off political-correctness alarm bells?

Resources

Some potentially useful census data.

The answer.

Acknowledgments

I Googled “Class pictures” and this site was the first one that had pictures of classrooms full of kids. I didn’t set out to find a classroom with any particular characteristics (other than being in 5th-8th grade).

Math 2.0: cont.

Having argued that Dan Meyer is using technology in ways that are novel in American mathematics classrooms, I want to turn to the problems he is using technology to solve (I refer to problems of teaching, not math problems).

This is the area in which Meyer is most explicit about his work. He gave an online seminar (and while we’re on the topic, can we please agree never to use the term webinar again?) recently in which he described the genesis of the escalator problem. Some of my observations will surely match his.

In the Connected Mathematics Project (CMP), which I have worked with for quite some time, we talk with teachers about a teaching model-Launch, Explore, Summarize. CMP is based on problems which form the basis of most daily lessons. Teachers engage students with the context and the mathematical challenge in the Launch, give them time to work on the problem in the Explore phase and then uses students’ ideas and solution methods to Summarize and help students to meet the lesson’s goals.

Meyer is working hard on engaging students in mathematics lessons. He is developing excellent launches.

When I work with CMP teachers, I emphasize two key aspects of launching problems. (1) Students need to understand the context, and (2) Students need to understand what the mathematical challenge is within that context.

Not every student is going to have experience with even the best chosen contexts. That’s OK, but it means teachers need to pay attention to setting contexts up for students, and in helping students to pay attention to important features of the context.

You don’t need to live on the coast to solve a problem involving the ocean, but the teacher has a responsibility to bring important aspects of the ocean to the students’ attention.

But it’s not enough to get students engaged with the context, teachers also need to make sure students understand the mathematical task embedded in the context. Everyone needs to agree what the question is.

Setting up both of these in a finite amount of time is challenging, and Meyer is upping the ante.

The opening shot in the escalator video (below) establishes the context instantly-escalators at the mall. Is there a teen in America for whom this is not a meaningful context? Love it or hate it; having few or many opportunities to visit it, the mall is part of teen culture.

Opening shot of the video-Dan Meyer in the mutliplex.

The opening shot: a scene familiar to high school students in this country.

The next 20 seconds suggest the mathematics embedded in this context. We are going to be looking at rates-how fast Meyer (and by extension the students) can go up and down the stairs and escalators.

And here, in my opinion, is the one weakness in the Launch (and it’s a minor one). The video ends with Meyer getting to the top/bottom of the stairs. I want the video to hammer home the implicit question, How long does it take to go up the down escalator? I want him to turn from the bottom of the stairs, go to the bottom of the down escalator and begin to take his first step, then have the video freeze.

But that’s a mere quibble with a masterfully designed launch. So let’s dig a bit deeper.

If teachers want to engage students, they need to know the target audience. Meyer is a high school teacher and he knows his students well. Consider the following elements of the escalator video:

  1. He smiles slightly and slyly at the camera in the opening close up. Dig this, he seems to be saying to the viewer. While most high school students won’t know who this guy is, he is no longer some random guy; he is a sympathetic accomplice.
  2. He puts in his earbuds. Adults may not notice this as significant, but high school students will pick up on it right away. It builds their identification with the context.
  3. The question. I cannot say enough about the question. How long to go up the down escalator? is brilliant. It’s just transgressive enough to be interesting to high schoolers, and nowhere near the border of inappropriate for school-endorsed investigation. Compare to the original-What is the speed of the canoe in still water?-and it’s no contest.

So Meyer has some novel uses of technology, including to launch problems in high school classrooms. For him, the problems of teaching include, (1) How to engage high school students with meaningful problem situations, and (2) How to focus their work on a common question.

But to what end? What happens once the video is finished? Next post.

Cubeyness

I have been interested recently in the mathematics lying just under the surface of middle school math problems (see also “How many ways to roll a 10?” ). I enlisted a former student, Ben Harste, to help me write an article on such an investigation. The premise is below; until and unless it is accepted for publication, you can read the full article by downloading it from my Papers page.

In the Connected Mathematics unit Filling and Wrapping, students find all of the rectangular prisms that can be made using whole number side lengths and 24 unit cubes. Then they find the surface areas of these prisms and notice that surface area can vary greatly for prisms with fixed volumes.

Commonly, teachers have their students conjecture how to tell which of two prisms will have a smaller surface area based on appearances alone, and a frequent observation is that the closer a prism is to being a cube, the smaller its surface area will be (in comparison to other prisms of the same surface area). This property could be called cubeyness.

It is easy to judge the relative cubeyness of a 1 by 1 by 24 prism against a 2 by 3 by 4 prism. But what about when two prisms appear similarly cubey? How can we measure cubeyness without calculating the surface area? And furthermore, how can we compare the cubeyness of two prisms that have different volumes (so that, for example, a 2 by 3 by 4 prism would be just as cubey as a 4 by 6 by 8 prism, or so that the cubeyness of a prism does not depend on the units we use to measure it?)

These turn out to be much harder questions than they appear to be on the surface, and answering them calls into question a lot of what many teachers know about measurement, statistics and geometry.

A visual launch-probably interesting only to CMP teachers

Connected Mathematicsa 6th through 8th grade mathematics curriculum-is designed with a teaching model in mind: Launch-Explore-Summary. In my professional development work, I help teachers to understand the teaching model in general, and I demonstrate effective techniques for particular problems.

For a few summers now, I have been demonstrating a Launch for a problem in the 8th grade unit Frogs, Fleas and Painted Cubes. I came up with the idea after leaving the middle school classroom, so I have not used it with students. But every summer, teachers ask where they can find written instructions for it and I have had to say that they do not exist.

My summer teaching partner has reported to me that it has been effective in her classroom, so I am now motivated to write it up and share it widely. In what follows, I will assume that the reader has access to the Student and Teacher Editions of the unit. The problem in question is Problem 2.1 of Frogs, Fleas and Painted Cubes.

The original launch

In the original problem, students consider the effects of a land swap in which a square piece of land is swapped for a rectangular one with the same perimeter. So a 5 unit by 5 unit square, might be swapped for a 3 unit by 7 unit one. Students record their information in a table and look for a pattern in the relationship between the areas of the two pieces of land.

In designing my alternate launch, I wanted to achieve two ends: (1) increasing access for visual learners, and (2) increasing the generality of the results.

(1) was important to me because I have been working in my own teaching on visual representation in mathematics, and because the problem asks us to represent something geometric in a table. It seemed like a natural place to increase the visual components of the problem.

(2) is important as a teaching principle. The original problem asks students to consider a variety of squares, but to always change each dimension of the square by 2 units. Thus a 5 by 5 square becomes 3 by 7, and a 6 by 6 square becomes 4 by 8. But there is nothing special about 2 units. A more general pattern emerges if we consider different numbers of units.

The new launch

For this problem, students will work in groups of 3 or 4. Each group receives a bunch of colored (say pink) inch grid paper and a bunch of white inch grid paper. After setting up the context as in the Student and Teacher Editions, have each student draw a square (with whole number side lengths) on a sheet of the white grid paper. Students should coordinate to make sure that each student has a square of a different size from their groupmates.

Each group is assigned a number from 1 to 6, repeating numbers if there are more than 6 groups.

Using the pink grid paper and the group’s assigned number, each student draws a new rectangle. Say my group’s assigned number is 3. Then I will transform my original square (say it was a 5 by 5 square) into a new rectangle by increasing one dimension by and decreasing the other by 3. My new rectangle would be 2 by 8. This preserves the perimeter. What does it do to the area?

To investigate this question, each student tries to cover his/her original white square with his/her pink rectangle. Students will need to cut these pink rectangles apart in order to cover as much as possible of the white square.

Then each group puts their covered squares onto a large poster paper and these are displayed around the room, in order of the assigned numbers (see below).

Now the class is ready to do the mathematical work of investigating the relationship between the assigned number, the size of the original square, and the area of the pink rectangles. Some possible observations that students might make and questions they might pose include:

  • It does not matter what size square we start with; the difference between the area of the original square and the area of the pink rectangle is the same within each group.
  • The group’s assigned number matters-as the assigned number increases, so does the difference between the areas of the original square and the pink rectangle.
  • The area of the pink rectangle is always less than the area of the original square. We can see this because we never quite cover the white with the pink.
  • Several of these images show a white square peeking out from behind the pink. Is it always possible to rearrange the pink so that a white square peeks out from behind?
  • When will it be possible to arrange the pink so that it is in the shape of a square (with whole-number side lengths)?
  • etc.

After some work observing patterns and asking questions, now students should be ready to make the table in the text. However, students should alter the table to match their group’s assigned number, and they should conjecture how the tables of groups with other assigned numbers should look.

How many degrees in a polygon?

I nearly always learn new mathematics when I work with teachers. Recently I led a daylong session on the Connected Mathematics unit Shapes and Designs with a group of sixth-grade teachers. In that unit, there is a problem that asks students to figure out the total number of degrees in the interior angles of an n-sided polygon.

Two strategies are presented in the text by fictional students. Tia’s strategy begins with this diagram:

Tia’s strategy

Cody’s strategy begins with a different diagram:

Cody's strategy

Most of the teachers in my session knew the formula T=(n-2)*180 where T is the total number of degrees in all interior angles of the polygon and n is the number of sides of the polygon. But it quickly became clear that they were struggling to use the diagrams to justify the formula. As a group, they liked Tia’s diagram because it shows (n-2) triangles, but they could not answer my questions about why Tia gets (n-2) triangles when she cuts up an n-sided polygon. Furthermore, they tended to see Cody’s strategy as incorrect because he gets n triangles, not (n-2).
We spent 45 minutes sorting this out as a large group. I pushed them to explain to me why Tia will get (n-2) triangles. And at the end I had four ways of understanding this. Three of them were new to me.

(1) Sides

This is the way I have always thought about the formula. In this way of thinking, each side of the polygon corresponds to one triangle in the dissected polygon, as in the shaded triangle below. The remaining sides of the shaded triangle are formed by the diagonals that Tia draws.

Relating sides of the polygon to the number of triangles.

But there are two sides that do not get used this way. The sides adjacent to the vertex from which Tia draws her diagonals are incorporated into the first and last triangles. Therefore, we always get two triangles fewer than the number of sides of the polygon. So (n-2) refers to the number of sides of the polygon.

(2) Vertices I

The remaining strategies were new to me. I have fleshed out the details of the arguments the teachers made.
Tia draws diagonals from one vertex (call it A) in order to form her triangles. In doing so, she draws (n-3) diagonals. Those diagonals cut the angle at A into (n-2) parts. At the other end of each diagonal (at C, D and E in the diagram below) the angles of the polygon are cut into 2 parts. There are two angles that remain intact. In forming her triangles, Tia get (n-2)+2(n-3)+2 angles in all of the triangles combined. This expression simplifies to 3n-6. Each triangle has three angles so the polygon has been cut into (3n-6)/3=(n-2) triangles.

Counting vertices in Tia's triangles

In this argument, n stands for the number of vertices in the original polygon. Of course this is the same as the number of sides, but the argument focuses on the vertices rather than on the sides.

(3) Vertices II

A simpler way to use vertices is to notice that from vertex A we can draw (n-3) diagonals. This is because we draw a diagonal to each vertex except A and the two vertices adjacent to A. Just as cutting a cake 1 gives 2 pieces, cutting our polygon n-3 times gives one piece more: (n-3)+1=n-2. So (n-3) diagonals make (n-2) triangles.

(4) Connecting Cody’s strategy to Tia’s

The extra 360 degrees in Cody's diagram

These vertex arguments were interesting for me to consider. I had always thought about n in the formula as the number of sides. More profound for me was a teacher’s observation that Cody’s strategy can be connected to Tia’s. In this case, n refers to triangles, not to sides or vertices.

Cody’s strategy involves putting a new point in the interior of the polygon. Connecting that point to each of the vertices gives n triangles. But when we total the measures of the angles of these triangles, we are including the angles in the center of the polygon and these angles really have nothing to do with the angles of the polygon. So we need to subtract their combined measure from the total. These extra angles completely surround the center point so they total 360 degrees. Therefore Cody’s formula is n*180-360, which is equivalent to (n-2)*180.

A teacher in my session observed that this new point could be anywhere in the polygon. Furthermore, if we pull that point towards a vertex of the polygon, two of Cody’s triangles get smaller and smaller in area. When the point is pulled all the way to a vertex, those triangles collapse and disappear, and the extra 360 degrees around the point disappears at the same time. So Tia’s strategy is the same as Cody’s strategy except that she has chosen her point to coincide with a vertex of the polygon. (n-2) refers to n triangles that Cody would draw minus the collapsed triangles. Click here to view a brief animation demonstrating this strategy.

(5) A new strategy

Following this teacher’s logic, I wondered what would happen if we dragged the point to the side of the polygon instead of to a vertex. I turns out that only one of the triangles collapses, and the extra point now has 180 degrees surrounding it. This leads to a new formula: (n-1)*180-180. Click here to view a brief animation demonstrating this strategy.

Others?

Readers are invited to submit further ways of thinking about the number of degrees in the interior angles of a polygon. Don’t worry about the art-describe your thinking in detail and I can draw the picture for you!

Addendum 4/20/2011

I have written a Sophia packet on angle measures in polygons-considering the general case of the argument involving exterior angles of a polygon. If you’re looking for more information on this topic, the packet is worth checking out.

Wump hats, part II

In the previous post, I described a workshop I was leading with seventh grade teachers recently. We were working on a problem in which the Wumps are given hats. The first hat is described by points in the coordinate plane. The other hats are transformations of the original hat and the problem has students investigate the effects on the image of multiplying and of adding and subtracting.

I had 6 small groups of teachers and there are 5 image hats. I had each small group use gridded chart paper and markers to draw the original hat and one assigned image hat on the same coordinate axes, as below.

Wump hat 1

Wump hat 1

I asked the sixth group to make up their own rule and to draw the original hat and the image, but not to show the rule on their chart paper.

We put the drawings on the classroom wall and gathered to talk about them. I asked the teachers to match the hats with the known rules, which went well. The group came to consensus about which was which and observed informally that:

  1. adding and subtracting moves the original figure around, and
  2. multiplying changes the size of the figure

I then asked teachers individually to think about what rule might have created the sixth image (see below).

What rule generated this image?

What rule generated this image?

Based on the rules above, I expected that most of the teachers would create a rule such as (2x-1, y+3). This is based on the two observations above, but applied in the opposite order. The most obvious change from the original to the image is that the image is twice as wide, hence the “2x”. The other change is that the hat moved. The left corner of the image is 1 unit to the left of the original’s corner, and the image is 3 units higher. This accounts for the “-1” and the “+3”.

I then asked the group that made the hat to reveal their rule. Much to my teachers’ surprise, the rule was (2x-2,y+3).

2x-2, not 2x-1

This is what my friend Jim had been asking me about over the years. The reason 2x-2 is correct is that observation number 2 above is not complete. Multiplying changes the size of the original, and by necessity it moves the points around. Consider this: if two points define the left and right bottom corners of the Wump hat, and if the Wump hat doubles in size, then those endpoints cannot possibly both be in the same place. For then they would be the same distance apart and the hat could not have changed size.

So multiplying changes the size, but it also moves the points around. The hat moves to the right as a result of the “2x” part of the rule. So we need to subtract 2 (not 1) in order to shift the hat to its final resting place.

I felt that I had created a wonderful moment where teachers were ready to learn some mathematics. They had made a prediction and they had been presented with evidence that these predictions were incorrect. In resolving such conflicts, we have opportunities to learn.

A close look at the hat in question will reveal that there is something unexpected about the diagram. The scales on the two axes are not equal. This became an important point in the ensuing conversation and I’ll examine that in the next post.