"A mathematical model is a mathematical representation of a particular real-world process or phenomenon that is under examination, in an attempt to describe, explore, or understand it. When students engage in mathematical modeling, they often have the opportunity to leverage mathematics to understand and critique the world. Mathematical modeling is the creative, often collaborative, process of developing these representations. Modeling always requires decision making that involves determining which aspects of the phenomenon to include in the model and which to suppress or ignore and what kind of mathematical representation to use. As noted by the mathematician Henry Pollak (2012), throughout the modeling process, both the real-world situation and the mathematics must be taken seriously."

I rewrote my content standards for the course trying keep this in mind. I was recently getting observed for a personnel thing, and it was a day we were thinking about modeling. The observer was David Austin, who's teaching a new modeling course that we're offering as a part of developing an mathematics emphasis for the math major. I asked him to share the idea of the course and we tried to connect it to them to encourage future teachers to take it. Part of how we addressed that was a classic trigonometry problem of fitting a cosine model to the height of a seat on a ferris wheel. But they weren't using video of a real Ferris wheel. Why would we need a trig function for a real Ferris wheel? So I'm pretty sure that this was not real world modeling.

Then Dan Meyer took on modeling last week. He was at a panel with the folks who produced THE report about this, GAIMME. (People just say 'game'.) They identify four aspects to mathematical modeling. It's real world that bothers Dan. He said/writes:

If your definition of “real world” labels the US tax code asrealand polygons asnon-real, your definition is not useful. To most US K-12 students, the US tax code is verynon-realand polygons are veryreal.

If you define “real-world” as a property that is binary rather than continuous, that is fixed across all cultures and time rather than relative and mutable, if your definition doesn’t account for the ways (per Freudenthal) that contextsbecomereal in someone’s mind, it isn’t useful.

And if your distinction between “mathematical modeling” and “learning” depends on “real world,” a descriptor without a definition, it isn’t a meaningful distinction.I love Dan's willingness to wade into a fight, and to constantly rethink and refine his ideas about teaching and learning. But I don't see how he's helping here. Real world may not have a sharp mathematical definition, but the idea that we can bring things from the real world into the math class is helpful. Many of Dan's first big ideas were about this, and I don't think he's denying it now. He also wrote a lot about pseudo context, which was helpful in getting at the idea that real world is not the be all end all.

Dog Eat Doug by Brian Anderson |

I don't care about labeling or not labeling tasks as real world or not. I do care about the process the learners are engaged in with these problems.

GAIMME outlines this as the process:

- Formulating the problem or question
- Stating assumptions (often requiring simplifications of the real situation) and defining variables
- Restating the problem or question mathematically
- Solving the problem in the mathematical model
- Analyzing and assessing the solution and the mathematical model • Refining the model, going back to the first steps if necessary
- Reporting the results

Paula Beardell Krieg was sharing some of her usual awesome mathart, and it led to a quick conversation. She made an image that was interesting to her with trigonometric functions then did her arty magic to make it into paper that she then origamied into boxes. I commented that I'd been wondering if the modeling aspect was part of what I found so engaging about engaging in art in math class. (I was working on a David Mrugala idea, and, though not real world, I tried a lot of different functions trying to get different effects.) Getting the math to make what you want is messy, open and genuine. She said:

This idea of starting with something fuzzy then refining resonates with me in two ways. One, mathematically, is that more and more it seems like (from the glimpses I've had of higher level math) is that what mathematicians are doing is always trying to get more and more accurate models. ... What the artist does is similar in that we start with a sketch then keep refining until we decided we've gotten close enough. When I am playing with these functions to get images that I like, what's really awesome is that I get to use the math that I know as well as learn new, or refresh myself, with new stuff. It's an amazing process for me, as I have to push my math envelope to push my art envelope. It is so satisfying, but it does require so much mucking around. That's the mess that you talk about. It's also the play that I think about for my own work and for the work I do with kids. So much discovery in mucking about. I know you know that this is where so much learning happens. I am convinced that imposing too much structure (which is different than NO structure) sucks the potential out of learning. No structure can do the same. Finding that sweet spot, which is actually quite large, is a good place to aim for.

For Dan this leads into a big idea "Teachers need fewer ideas about teaching." Whoa! I'm definitely not ready to get into that. But I want one of those ideas for math teachers to be "learners need to be doing mathematics" and some of what doing math is is encapsulated in this process.

The end of the conversation (for now) with Paula was a direction for me to pursue these ideas.

So it comes back to play! I think this is inherently tied up in the first step: the mathematician poses the problem.

I was interested in Dan's points too.

ReplyDeleteI do (-!-) think there's a real world, irrespective of whether students are interested in it or not. It's a wonderful, complex, important real world that we've got a lot of experience of and also a lot more to discover in.

But I'm with Dan in that:

- what feels real to students may not coincide with this;

- problem solving with abstract ideas - numbers, shapes, pattern - can follow very similar steps to real world modeling;

- it's the steps that are the thing more than the context.

I love Paula and your emphasis on problem posing, which of course is so closely linked to play.

It would be nice to see interest to learners be greater than real world or not.

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