Tag Archives: CMP

Connected Mathematics posts

Posted on June 21, 2011 | Leave a comment

This week is our annual Getting to Know Connected Mathematics workshop on the campus of Michigan State University.

I’m meeting a large number of teachers this week and I expect more than a few of them will find their way here.

In that spirit, the following is a CMP teachers’ guide to the past couple of years on this blog.

Curriculum and standards

Commentary on Common Core

Commentary on one particular Common Core standard

How does CMP compare to some newer ideas in math education?

Algebra strand

The world’s most complicated launch (For Frogs, Fleas and Painted Cubes Problem 2.1)

Data strand

A risky context that we rejected (but interesting nonetheless)

Geometry and measurement strand

Cubeyness (or Overthinking Filling and Wrapping)

Wump hats I (Stretching and Shrinking)

Wump hats II (Stretching and Shrinking)

How many degrees in a polygon? (Shapes and Designs)

Number strand

Division of fractions (Bits and Pieces II)

Questions from middle schoolers

This has been an ongoing series of questions and answers from a CMP class in Alaska. This link takes you to the full collection.

Juvenile middle school humor

Come on. You know you couldn’t last a year in middle school if you didn’t find this stuff funny yourself.

postscript

CMP teachers…Got anything else you want addressed here? Let me know in the comments or by email; I’ll be glad to get to it.

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Huh? Making sense of Common Core

Posted on May 20, 2011 | 12 comments

OK, I get it. The Common Core State Standards are about large-scale coherence. Stay focused on the big picture of getting everybody going in the same direction, then tweak things later, blah, blah, blah… I get it.

And yet kids’ education is at stake. And teachers’ jobs in the era of No Child Left Behind and Race to the Top. And the quality of curriculum that has to bend over backwards to align with these standards.

So when I dig into the details in my capacity with Connected Math, I get indignant about places where things don’t make sense. Consider the case of ratios at sixth grade:

6.RP.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

I’m OK with this. I’m not thrilled with the “unit rate a/b” part, but it’s not a train wreck. Let’s look ahead to seventh grade, shall we?

7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

Huh?

The complex fraction (1/2)/(1/4)? Are you kidding me? Just try verbalizing this:

I walked one-half-over-one-fourth miles per hour.

Does anyone ever talk about rates this way? Ever?

No way! The only way to even come close would be to say,

I walked a half a mile in a quarter of an hour.

But then that’s not a unit rate. For some reason Common Core is obsessed with unit rates-strictly defined. If I thought this were a throwaway line, I wouldn’t be worried. But it’s not a throwaway. That sixth grade standard above? It had a footnote:

Expectations for unit rates in this grade are limited to non-complex fractions.

So the Common Core writers didn’t just make this (1/2)/(1/4) unit rate nonsense up on their first pass through seventh grade. Oh no-it was important enough to go back and exclude it from sixth grade. And important enough to use up one of only three footnotes in the entire 6-8 math standards. The other two? Here’s the next one:

Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

Are you sensing a theme here?

Right, it’s this other odd obsession with complex fractions (i.e. fractions in the numerator and/or denominator).

And the third footnote:

Function notation is not required in Grade 8.

Phew.


UPDATE: Reader Sean steps up to defend this standard in the comments below. I highlight his objections in a later post, and then respond in yet one more post on this topic.

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Posted in Curriculum

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A really big experiment…really big

Posted on May 4, 2011 | 8 comments

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

Posted on March 15, 2011 | 2 comments

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).

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Posted in Launch, Problems (math), Problems (teaching)

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Math 2.0: cont.

Posted on March 11, 2011 | 9 comments

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.

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Posted in Technology

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