# “Modeling” – Part 1

In what I’m pretty sure was the first-ever instance of a live-tweeting of our weekly math department meeting, Paul sent out a tweet about a presentation I was giving in which he called me out to blog about its contents.  So here it goes.

First, a backdrop: this year for the first time, our department is reserving a chunk of our weekly meeting to share curriculum.   Nice.   I signed up to be the sharer for the second week.  I wasn’t sure what I was going to present.  Many of the things that I do in my classes I’ve shared with my colleagues over the course of time in less formal ways, and I didn’t want to be redundant.  In the end, I decided to share two different takes on mathematical “modeling” that I’ve done with my middle schoolers.  This post is about the first one.

Here is a first worksheet of an investigation of something that’s called a “measure task”:

The idea is to take some mathematical property that has some intuitive meaning and to attempt to formalize it—in particular, to try to come up with a way of measuring it numerically.  I think this “squareness” measure task is an awesome one because it has a very low threshold and there are many ways that students can approach it, regardless of their prior knowledge.

For example, last year I did this “squareness” worksheet with a faster-paced seventh grade class and a slower-paced sixth grade class.  The seventh graders had a fair amount of experience with expressing quantities algebraically, both from prior years and from our previous units.  They went straight for comparisons of length and width, coming up with formulas like $\frac{l}{w}$ and $\frac{l+w}{w}$.  This brought up interesting questions—like whether these different expressions would give the same ordering of the rectangles, and whether they were really capturing our intuitions.  On the latter point, notice how the formula $\frac{l}{w}$ results in widely different measures for the squareness of 1×5 and 5×1 rectangles.

My sixth graders approached this problem through the lens of fractional parts.  They decided to measure squareness by the (smallest) fraction of the rectangle that needs to be chopped off in order to make a square.  Lacking algebraic sophistication, they landed on a simpler, “uncoordinatized” method for measuring squareness that was visceral and re-upped a core concept that we had been discussing.

I’m pretty sure that I first heard about this kind of task from Avery.  He’s even blogged about it here.  Some fabulous resources for finding out more—both in terms of theory and practice—are these two articles:

‘Creating Measures’ Tasks by Swan and Ridgway

Formulating Measures by Schwartz

In my presentation, I said that the measure task paradigm is both flexible and mathematically potent–in terms of content, lesson structure, and aims.  Measure tasks can be used to tackle mathematical concepts from the most elementary to the most advanced.  Like I said, I’ve used the “squareness” task with 6th and 7th graders, and I’m sure that this and similar tasks could be done with elementary schoolers.  On the other hand, some of the tasks that my colleagues brainstormed—like measuring how curvy a curve is—would be great tasks for a calculus class.  Once you get rolling, there are so many mathematical concepts that can be approached in this manner.

The lesson structure of measure tasks is also flexible.  They can serve as brief intros to a new concept at the beginning of a period that then swiftly moves on to other pedagogical modes.  They can be one-off lessons.  They can be used as review or extension activities, or even as luxurious several-day explorations.  As with content, the time-frame and commitment needed to bring measure tasks into a classroom are both as flexible as could be.

Finally, the goals that one might have for introducing measure tasks to students are varied.  The focus could be on particular content (rectangles or squares, in my example), applying concepts in new contexts (fractions or algebraic translation), or just on the mathematical process of creating, testing, modifying, and comparing.

I found talking (and now writing) about measure tasks to be exciting–I can’t wait to use them more often in my classes!  My colleagues really got into the idea as well, making great connections and brainstorming task ideas of their own.