Tag Archives: CMP

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.


CMP3 samples are online

If you’ve been following along for a while, you may know that I worked for two years with the Connected Mathematics authors in a revision of the materials responding to the Common Core State Standards.

My work for the project finished this summer. In the meantime the author team has been writing, and Pearson has been continuing development of the commercial product. I understand it may a while yet before we can hold student books in our hands, but electronic samples are now available on the Pearson website.

It was super interesting for me to look carefully through the offerings, as I was present for much of the development of the project. But it’s really hard for me to look at it with fresh eyes.

I’d be curious how y’all think Pearson has managed the tech-development side of things. Be sure to explore the “Online simulation” and report back.

Task design

For the last two years, I have been working with the four surviving original authors of Connected Mathematics on a revision that is responsive to the Common Core State Standards. My task has been top-to-bottom revision of three of the four rational number units, Bits and Pieces I, Bits and Pieces II and Comparing and Scaling.

The process for this work is unusual for commercially available U.S. curriculum materials, so I want to share a few observations from the inside. They will trickle out over the next few months, and they’ll get filed under “Curriculum”.

The task

In CMP2, we focused the initial fractions unit on careful introduction of the number line. The premise was that children had lots of elementary school experience with area models for fractions, and that we wanted to introduce the more sophisticated linear models.

We introduced the licorice lace problem.

In this problem, a group of four kids is going on a hike. They have a 48-inch licorice lace and they want to share it equally among themselves. Sid (the protagonist in our narrative) carefully marks the places where he will make the cuts.

Just before he actually cuts, two more kids show up. Now they need to make new marks on the already marked-up lace. The cycle is repeated a couple of times. At each phase, we ask students to name the part of a licorice lace each hiker receives.

If you try this yourself, you will notice that it’s pretty hard to locate the marks for sixths when there are already fourths marked. Not impossible, but hard.

Ideally, some students in class will try, and some students will go to twelfths; others will go to twenty-fourths. Then when it’s time to name the fractions we have sixths, twelfths and twenty-fourths on the table and we can talk about equivalence and partitioning linear things.

From classroom feedback and my own experience working the problem with adults (both in professional development and college courses), it was clear that the problem needed a redesign. The set up was wordy, using one and a half pages of text to work through a small set of tasks. The marks before cutting were slightly implausible. The sharing and re-sharing was too complex for simple problem-posing.

The redesign

Two years ago, I took on the task of redesigning this problem.

I knew we needed something that was (1) linear, (2) shareable, and (3) already marked.

Linear and shareable are properties of licorice laces (these are not Twizzlers, each of which-while shareable-is not plausibly shareable among four children). That third criterion was new. If I could find something whose pieces were already marked, I could get rid of the complicated storyline and a tremendous amount of text.

“Marked pieces” is important because this is a problem about partitioning and repartitioning. We want kids to have pieces that they need to cut up further, and to have to think about names for these new pieces.

Skittles would not work.

These, while delicious, are mathematically unproductive for our purposes.

A bag of Skittles is composed of the original unit, one Skittle.

I needed something where we partition the original unit. It is perhaps shameful how many hours of thought went into this. But I eventually found it.

It’s perfect. Linear, shareable and already marked. You want to share equally? Each person gets \frac{1}{2}, sure. But can’t you see that each gets \frac{4\frac{1}{2}}{9}? Or that it could also be \frac{9}{18}?

It is helpful that most people don’t know the standard partitioning of a Tootsie Roll. (Did you know it was ninths? Be honest, now. It hasn’t always been; it used to be sevenths.) If you don’t know the standard partitioning of a Tootsie Roll, then we can make as many pieces as we like to start each new task. No more marking and re-marking. We just give a new  Tootsie Roll and a new number of people.

We know from research that sharing is a productive context for understanding fractions. We’re sharing something that is already partitioned, so we need to repartition when the number of sharers is not a factor of the number of pieces.

Feedback from classrooms and my experience working with adults (again-professional development and college courses) suggests that we get more mathematics with a lot less effort setting up than we did with the previous version.


Not every problem in Connected Math has gotten this level of attention, of course. But a lot of them have. This is a curriculum that takes context seriously as a basis for mathematical activity and abstraction.

Once we have committed to a particular mathematical development (e.g. partitioning in linear situations in order to move to the number line), we seek a problem in which the right mathematical activity naturally results. I am proud to have been a part of that.

It’s not just me (Smart Boards)

I’ve been catching up on some podcast listening. Here’s Audrey Watters talking to Steve Hargadon on their weekly podcast back on January 29:

…I think there are lots of systems in play that don’t actually want…I mean the folks who sell you interactive whiteboards don’t really want the content to be accessible elsewhere because then why the hell would you buy an interactive whiteboard?

And here was me recently on the relationship between software and interactive whiteboard hardware:

…here’s something strange about Smart Boards. [The ability to capture student work is] not a feature of the board; it’s a feature of the software. But Smart fancies itself a hardware producer, so it hasn’t designed the software to do much of anything without the board.

If we were designing the ideal piece of software to do what you’re suggesting, I’m not at all convinced that it would require a Smart Board to run on, and I doubt that it would look very much like Smart Notebook at all.

I taught myself to use a Smart Board. I have presented professional development sessions around Smart Boards. I have used Smart Boards in my classes-including College Algebra and math content courses for future elementary teachers.

I’ll say it directly. An interactive whiteboard is a crummy tool, massively overpriced. The software has been designed to sell the hardware, rather than as an excellent interface that stands on its own.

In short, Smart Boards suck.

There is a small amount of meaningful additional functionality that an interactive whiteboard brings to the domain of classroom presentation media. In math, this has mostly to do with manipulating visual images-moving this rectangle onto that one to compare their areas and the like.

Here’s a crummy video of the one lesson I really do want a Smart Board for. It’s from Connected Mathematics: Bits and Pieces II.

[vimeo http://vimeo.com/22362944]

More on fraction division (you know you love it!)

Readers came through for me when I surveyed on ratios. So I come back to the well seeking your insights.

This time on division of fractions.

The Common Core State Standards reserve division of fractions as the only operation on rational numbers at sixth grade. The other three arithmetic operations come in the earlier grades. The Connected Mathematics response to this is going to be to compactly revisit addition and subtraction, linger a bit longer on multiplication and spend some time and depth on division.

This is a topic with which I have considerable experience.

As you surely know, the standard approach is to tell students to invert-and-multiply and then move on.

In previous versions of Connected Math, we have done a single Investigation (4 to 5 days worth of study) on division of fractions, with an emphasis on:

  1. Quotative (or measurement) models, leading to
  2. Common denominator algorithm

For those unfamiliar, there are two standard problem types for division: quotative (measurement) and partitive (sharing), exemplified by the following.

Quotative (measurement): I have 35 apples to package in bags. I can fit 7 apples in each bag. How many bags can I fill?

Partitive (sharing): I have 35 apples to share equally among 7 people. How many apples does each person get?

A typical quotative fraction division problem goes like this:

I have 3/4 of a cup of grated cheese. Each omelette requires 1/4 cup of cheese. How many omelettes can I make?

These types of problems lend themselves to wanting the denominators to be common; then we can divide the numerators and voila! This is justified by thinking of the 3/4 cup as 3 units and the 1/4 cup as 1 unit. The units are the same, so it doesn’t matter that they are fractional units. The basic question is How many of this are in that? Just like with the apples and bags-how many sevens are in 35?

But partitive is trickier. A typical problem goes like this:

I have 3/4 of a cup of grated cheese. This is enough to make 1/4 of an omelette. How much cheese do I need to make a whole omelette?

Now the common denominator doesn’t matter so much. Instead, I want to multiply by 4, based on the reasoning that if 3/4 is 1/4 of the whole thing, then 4 times 3/4 must be the whole thing.

For most people, this has quite a different feel from the apples and people problem that exemplifies whole-number partitive division. Having a fractional number of groups complicates the partitive problems.

What we have decided to do in Connected Math is to keep the quotative investigation more or less intact (tweaking based on our experience in the field), and create from scratch an investigation on partitive division.

But the typical examples of partitive fraction division don’t resonate with me. It’s easy enough to divide a fraction by a whole number this way (1/2 pound of peanuts shared among 3 people), but fraction by a fraction is tougher (1/2 pound of peanuts is 2/3 of a share?)

Notice that partitive division problems-whether whole-number or fractional-give unit rate answers. Apples per person. Cups of cheese per omelette. Pounds of peanuts per person.

But in those contexts, it is implausible that I would know the rate for a fraction of a unit but not the whole unit. How do I know that 3/4 cup of cheese is enough for 1/4 of an omelette without knowing how much is in a whole omelette? There is no plausible scenario under which I would have this information without also knowing the unit rate.

So now we get to the question…

What are some unit rates in which it’s plausible to know the fractional rate without knowing the unit rate?

My man Sean pointed out in a related discussion that unit rates involving time fit into this category. I can plausibly know that I walked 1/2 miles in 15 minutes, notice that this is 1/2 mile per 1/4 hour and wonder what this means for my miles-per-hour unit rate.

But in an article I reread today, the authors suggested “2/3 of a cake exactly fills 1/2 of my container”. I don’t like this one nearly as well. Exactly fills 1/2? I like it better than the omelette problem, but I feel we can do better.

How about it?