Modeling: Looking for definitions

Modeling is at the center of what I love about math and math education, so I’m thrilled that the Core Standards highlight modeling and that it figures in our latest drafty California framework.

But I’m worried about definition creep. I’m worried that, in two years, when they’re trying to come up with modeling curriculum, people in schools doing the hard day-to-day work will be tempted to say that practically anything is modeling and come up with plausible rationalizations. That, in turn, will dilute the importance of including modeling in policy documents, and result in students who can’t model.

To forestall this, it’s important to know what modeling is and isn’t. So it’s with some embarrassment that I, modeling maven and aficionado, have trouble drawing the lines. So consider this a first step in clarifying these questions for myself:

• What is modeling?
• What isn’t modeling?
• How much do we care whether we can come up with a definition?

Let’s start with the Framework (Modeling appendix, April 2013 review draft, lines 13–14):

Put simply, mathematical modeling is the process of using mathematical tools and methods to ask and answer questions about real world situations (Abrams, 2012).

Of course they go on at length, but the key is a connection to the real world. Here is another definition that I have used recently:

A model is an abstract, simplified, and idealized representation of a real object, a system of relations, or an evolutionary process, within a description of reality. (Henry, 2001, p. 151; quoted in Chaput et al., 2008)

Here, the key ingredients are abstraction and simplification.

Another distinction worth noting is that the first is a definition of modeling—a process—and the second defines a model—a representation. I have no clue whether that matters much.

What do the Core Standards themselves say? First of all, the document identifies modeling as one of eight Mathematical Practices, a great list I have mentioned before. Here is the one called Model with Mathematics, and it’s worth quoting in its entirety:

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

This is great, but it’s easy to imagine this lofty overarching idea getting lost when you’re designing a curriculum—or an assessment—and you have a chart of content to fill in. Fortunately, the Core Standards promote Modeling to the level of a content standard at high school. What do they say? Here’s a quote I find chilling:

Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol.

That is, you should find it everywhere, so we won’t list very many actual skills and goals. One could see this as a good thing: we’re celebrating the ubiquity of modeling. But I’m less sanguine about our ability to keep “overarching ideas” in mind, especially as we design assessments. For example, Number and Quantity at the high-school level has this (N-RN 3):

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

That’s pretty specific. If we pick one of the standards with a star (F-TF 5), we see:

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (*)

Which is great, too, until you imagine the item-writer. If you’re thinking formally—formally enough to write an item about the product of a rational and irrational number—and now you see this standard about trig functions, you might be tempted to read it as being about (for example) transformations of functions. Which it is, sort of, but consider this imaginary item:

Transform the function f(x) = –cos x so that it has an amplitude of 18, a midline of y = +20, and a period of 45.

Do students need to be able to do this? Yes. Does the item fit the standard? I don’t think so, because there’s no phenomenon being modeled. So suppose they did this:

A Ferris wheel’s hub is 20 m off the ground. It has a radius of 18 meters, and goes around once every 45 seconds. You start at the bottom at time t = 0. Find the function h(t) for your height at time t.

I still don’t think this is modeling. Problem-solving, yes; modeling, no. But I’d bet a dozen donuts that a problem like this one will be passed off as modeling someday soon in an assessment near you. On the other hand, if I showed you the graph from the previous post, one of actual data:

And said, model these data with a periodic function, and use your model to estimate the period, that would be modeling. It’s not the only kind of modeling there is, but it’s one I can recognize for sure.

My biggest anxiety about modeling in the Core Standards really comes down to assessment. So what are the assessment folks doing? The “Smarter Balanced” people have what they call “claims,” of which the fourth of four is:

Modeling and Data Analysis: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.

I’m sure we can all get behind that. The other claims are about Concepts and Procedure, Problem Solving, and Communicating Reasoning. This is a terrific list, and makes us optimistic. I mean, look: these are four really big ideas we care about. And modeling is one of them!

Then you look at the actual assessment plan, and you see that they will report three scores. Why three? Because problem-solving and modeling get lumped into one.

Immediately I get defensive. Will modeling get submerged under problem-solving?

And that is enough for this post. What’s next? I have many thoughts about the differences between problem solving and modeling, and why they are important. I also want to write about what we might call “genres” of modeling—for example, modeling data with functions; creating probability models; using geometrical models—but am mightily confused about what to call them.

Stay tuned.