Monday, March 31, 2014

Mathematical Modelling and Adventures in Blogging

Back in February, I blogged about iTAGs, the meet-up groups that aim to help teachers "build community, develop as activists, and link social justice issues with classroom practice." I've joined two of them: "Social Justice Educators on the Path to Cultural Relevancy" and "Locally Relevant Mathematics with the Community Based Mathematics Project."

For the latter, I was invited to blog about one of our recent sessions, where we discussed practices for "modelling" in mathematics classrooms. I am re-posting my entry below, or you can view it in its original post on the Community Based Mathematics Project website.

[Note, since I'm a proofreader by trade and a bit of a perfectionist in my own work: I stuck with the double-L spelling of "modelling" rather than "modeling," even though I like the latter more, because the former was used more consistently in the literature we were drawing upon. However, whoever posted the entry used the single-L version in the title. Please forgive the inconsistency. And please forgive this ridiculous apology.]

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The Locally Relevant Mathematics iTAG is a place where we discuss ways to make mathematics lessons personally meaningful and culturally responsive -- but we also keep a focus on the way that these responsive practices can help promote particular forms of inquiry and mathematical understanding. For the latest session, we looked at modelling as a great classroom practice to support all of these goals.

But what actually is modelling?

Admittedly, it's not something I could clearly explain before, and even now it's still a bit tricky to define. As we discussed it, modelling is different from traditional problem-solving in that the "problem" is inextricably tied to the context, and students have to engage with a given scenario in order to figure out best how to approach it with mathematics.

Because that seems abstract and not very descriptive (it did to me!), we spent much of our meeting doing an example of a modelling task. We were given a scenario: given a particular set of Philadelphia neighborhoods, we were tasked with determining how many basketball courts would be an ideal number to build in each. Rather than provide us with a particular approach to take, or data to support us, our instructor instead asked us to brainstorm what kinds of data would be most important to help us make that decision. With a whiteboard full of potential data sources, we then discussed and narrowed down the list of desired data. Our instructor then let us know which data he had available for us. Working in small groups, we were free to request data and develop our own approaches and answers. Finally, we shared out with the class, explaining the various methods and responses we had developed.

While modelling tasks don't have to be this abstract, what mattered about this task was that there was no way for us to strip away the context and say, "oh, this is the formula the teacher wants us to use." We had to deal with the scenario as a real-world challenge, grapple with the data, and determine the approach that would most allow us to be successful.

While it might be hard to explain exactly what modelling is, it's easier to see how this kind of exercise fits the mission of our iTAG. First, the scenarios and contexts can be carefully chosen to be engagingly relevant to students' lives and experiences. Second, it uses that connection to challenge students to apply their understandings of mathematics in a way that can promote higher-level thinking and deeper understandings. This sort of task is more of a challenge to create, and it requires a lot of trust in one's students to provide them this sort of relatively open exploration; however, I remain convinced that the potential rewards make it entirely worthwhile.

After our modelling activity, one of the ITAG members shared valuable ideas from an article on "launching complex tasks" in the classroom. The article spelled out ways to get a complex problem started, like 1) Discuss key features of the context, 2) Discuss key mathematical ideas, 3) Use students' ideas to develop common language for the features of the problem, and 4) Avoid giving students a particular solution method. Using these strategies when launching a modelling problem, she said, made the process go smoother.

The article is "Launching Complex Tasks" by Kara Jackson, Emily Shahan, Lynsey Gibbons, and Paul Cobb. It was published in Mathematics Teaching in the Middle School (Volume 18, No. 1, August 2012).

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