I’m fortunate in that my current position at the University of Delaware gives me the flexibility and opportunity to fairly regularly spend time in Delaware schools talking with and working with Delaware teachers. It’s especially fun to speak with teachers who are just starting to incorporate the practice of mathematical modeling into their classrooms. Today, I’d like to talk about a fairly typical dilemma or stumbling block that these teachers face and share some ideas for how to get past common sticking points.
The common dilemma is this – many teachers find a great real-world problem for their students, find a nice real-world data set for them to work with, and get their students genuinely engaged in investigating this problem. The students then grab the first tool in their toolbox, i.e. curve fitting or regression, fit a nice curve to the data, make a nice plot, and then say “Now what?” That is, the teachers we have worked with are very comfortable carrying out descriptive modeling and very comfortable working in the “compute” and perhaps “interpret” or “validate” boxes of the CCSSM modeling cycle, but then find themselves at a loss for what to do next. These teachers generally sense that there’s something missing, but then can’t quite find their way forward with a next step.
I asked Michelle about this common phenomena and asked for her perspective as a former high school teacher in particular. Here’s what she had to say:
This dilemma resonates with me. Because I did not take any coursework in mathematical modeling as a secondary education major (an issue which is a national problem according to this article by Newton et al. (2014), I did not have a good sense of what mathematical modeling was really about. My notions of mathematical modeling were largely formed by my engagement with curriculum materials. This is problematic because, as pointed out by Meyer (2015), many textbook exercises labeled as “modeling” tasks do not authentically engage students in all aspects of the mathematical modeling cycle. It was not until graphing calculators, and subsequently regression problems, became an important part of school mathematics that I really had any opportunities to engage in any form of modeling. At the time when it was introduced into school mathematics, I either had to teach it to myself or attend professional development workshops to learn about calculating these simple descriptive models. However, unlike the teachers described above, I thought I was done once the line of best fit was calculated, and I verified a “good” r-value. It is promising to see that these teachers are aware that there should be more to modeling than this.
It’s encouraging to know that there has already been a shift in how teachers are thinking about mathematical modeling and that at least in my experience in Delaware, are trying to think beyond curve fitting. But what should a teacher who is faced with this dilemma do? How do you get past that feeling of “Now what?” or take your modeling beyond simple curve fitting? Based on my observations, I want to offer two pieces of advice or two strategies you might follow when faced with this dilemma.
Strategy #1 – Go back to the beginning
It seems to me that a common reason why this dilemma occurs at all is because not enough attention was paid up front to the “Problem” phase of the modeling process. It’s important to remember that what you are doing is trying to answer a question and that question is not a mathematical question! That is, being able to fit a curve to data is not the point of the exercise. Rather, your students should clearly have in mind the real-world problem that you’re trying to solve or the real-world question that you are trying to answer. It’s this question that drives you around the modeling cycle and this question that you should be returning to once you’ve done something like fit a curve to a data set. So, when faced with “Now what?” one answer should always be “What question were we trying to answer to begin with?” Bringing the focus back to your core question is one way to move the conversation forward.
Strategy #2 – Validate and critique the model
What do you do when your students say “we’ve fit this curve to our data, we have a good r-value, and hence we can now predict that in the future there will be X of Y”? That is, what do you do when they’ve done regression and think they now have entirely answered their real-world question? Here’s where you have an opportunity to push the conversation forward by validating and critiquing the model. I’d ask students questions like “how much confidence do you have in your prediction?” or “do you have any reason to believe that the trend you’ve sketched out will continue indefinitely?” Most curves obtained by curve fitting, when pushed beyond the range of data where they were obtained will lead to results that are genuinely open to question. This might be because we have no real reason to expect such trends to continue or it might be because we have no real reason to expect such trends to even exist, as we saw in The Shrinking Mississippi. Opening a conversation like this, allows you to help your students understand the limitations of all models, especially those that are purely descriptive models. This creates an opportunity to return to the start and push your students’ thinking toward analytic models built on a more fundamental understanding of what it is that they are exploring.
It’s important to remember both that mathematical modeling is a cyclic process and that what drives one around the cycle is the attempt to answer a real-world question. When you’ve built a model and are faced with the “Now what?” question, keep the goal in mind, be critical of the models you’ve built, and you’ll find ways to help you and your students be able to say “Oh, that’s what’s next!”