Over the last several months, we’ve explored mathematical modeling from a variety of perspectives here in this space. We’ve talked about the CCSSM, the modeling cycle, thought tools used by modelers, and we’ve explored a variety of examples, all with the goal of trying to understand how mathematical modelers think and how we might best train our students in the art of mathematical modeling.
One thing we have not done is follow the path of a typical course in mathematical modeling. Today, I want to talk a bit about what such courses look like, and where we might benefit from typical paths and why we might want to deviate from typical paths. Until recently, at least in the United States, mathematical modeling courses have largely been restricted to higher education. They’ve been taught primarily at the advanced undergraduate level and the graduate level. There are exceptions and there have been efforts to teach such courses at the introductory undergraduate level, but it’s reasonable to claim that the majority of the effort around teaching mathematical modeling has primarily occurred at the upper undergraduate and graduate levels.
This means that the majority of textbooks focusing on mathematical modeling are also aimed at the upper undergraduate and graduate levels. If you search for “mathematical modeling” on Amazon, you’ll, in fact, find several thousand books along these lines. Some are more specialized, some are less specialized, but generically, they are aimed at this advanced audience. If you examine the most popular of these, the ones that focus on “mathematical modeling” rather than “mathematical modeling in X,” you’ll find they all follow a typical pattern and that this pattern is reflected in course syllabi at institutions across the country. Roughly speaking the pattern is one where the texts are organized by mathematical topic first, and application areas within those topics. That is, you’ll see them divided into chapters with titles like “Modeling with difference equations,” “Modeling with ordinary differential equations,” “Modeling with partial differential equations,” and so on. Within each chapter, the authors will introduce an application area, and then show the reader a bunch of models developed in that application area using the mathematics of the chapter heading.
The philosophy behind this approach is one of “modeling can’t be taught, but it can be caught.” That is, authors and instructors largely assume that if they show students enough examples of mathematical models, they’ll eventually catch on and know how to model. I freely admit, that I too was a member of this camp for a long time. After all, it had worked for me! But, in teaching mathematical modeling over the last twenty some-odd years, and especially as I’ve had the opportunity to work more closely with the K-12 community, I found myself growing increasingly frustrated and disillusioned with this approach. In the large majority of cases, it became apparent to me that while a handful of students did “catch it,” the vast majority did not. They could perhaps use models they’d seen before, and engage in somewhat trivial extensions of such models, but when faced with a new situation, they were lost. They hadn’t, by and large, learned how to actually model. They hadn’t become modelers.
I gradually realized that I don’t agree with the “can’t be taught, but can be caught” philosophy and many of the blog posts here have been inspired by my personal change in perspective on this issue. I believe that we can, in fact, deconstruct and understand the process that constitutes mathematical modeling, articulate this process more clearly for our students, design activities to engage students in essential competencies that comprise this process, and end up teaching a lot more students how to actually model. My colleague, Rachel Levy at Harvey Mudd, says that what we’ve been doing for a very long time is teaching “model appreciation” rather than teaching the art of mathematical modeling. I think there is a lot of wisdom in that statement and that we can make this shift, at all levels, from teaching model appreciation to teaching the art of mathematical modeling. As I’ve had the chance to explore the mathematics education literature on this topic, especially the international literature, I’ve learned that there has been progress made in this regard and I’ve worked to incorporate the best of these ideas into my own teaching and try and share some of them here.
But, in the back of my head, there is also this whole “baby with the bathwater” thought that’s been nagging me for a long time. In an earlier blog post we explored the notion of “thought tools.” These are those ways of thinking that are often widely applicable and are characteristically wielded by those who engage at a high level in a particular practice. We talked a bit about some of the thought tools wielded by mathematical modelers. This thought tool perspective pushed me into doing a lot of meta-thinking – trying to observe my own thought processes as a mathematical modeler and compare them with those of students just learning the art.
One thought tool that I find myself wielding frequently and have observed other mathematical modelers utilize as well is “analogical thinking.” That is, when faced with some unfamiliar modeling situation, a modeler will often start by arguing “well, this situation is kind of like situation X, so perhaps we can proceed as follows…” They have, in their head, a repository of modeling approaches for a wide variety of problems. They rely on the miraculous fact that mathematics and mathematical approaches to applied problems are often wonderfully generalizable, and so develop solutions to new problems relying at first on what they’ve seen work before. The uniqueness of each situation requires adaptation and creativity, but they can often grasp a ready starting point from which progress can be made.
That brings us right back to model appreciation, because after all, where did they get this broad experience with various modeling approaches? Well, they’ve looked at and worked through lots of models built by others to tackle lots of different situations. Seems they caught something useful somewhere along the line. The seemingly presents us with a dilemma. The vast majority of students don’t catch modeling by the model appreciation approach, but the experience gained via model appreciation seems essential to be a good modeler. I don’t think this actually a dilemma. I think that we should conclude that we need to engage students in both some model appreciation and some deconstruction and practice of the art and the process of thinking like a mathematical modeler. I believe, and can claim from my own personal experience, that this dual-approach, is a more successful approach.
So, as you think through and work to incorporate mathematical modeling into your classroom, I suggest you keep this dual approach in mind. Spend time giving students the opportunity to practice modeling on new and fresh situations, help them understand the process and the practice, pay attention to developing the thought tools they’ll need and the competencies they must master, but, from time to time, don’t be afraid to engage in a little model appreciation and help your students start to build their own “model repositories” in their heads.