Perhaps the one question we’re asked most frequently by teachers preparing to integrate mathematical modeling into their classroom is “Is this mathematical modeling?” Usually, the teacher has a specific activity in mind and what they really want to know is “Will doing this activity require my students to engage in the full process of mathematical modeling?” Before I go any further, let me say – not every task that supports the development of modeling skills, needs to engage students in the full process of modeling. We’ll explore that in another post, but just keep in mind that here, I’m talking only about those tasks that are designed to have students engage in the full process.
Unfortunately, I don’t think there is an easy or simple to use “litmus test” to answer the question “Is this modeling?” I do think however, that we can list several tests that together tend to catch most of the key points of confusion. So, here are four things you should think about or ask yourself if you’re wondering if the task in front of you is a mathematical modeling task.
Can you find the “why” or the “how” question?
While mathematical modeling “tasks” are generally not cleanly formulated as one-line questions, if you think about your task, you should be able to see your way through to at least one central question you are aiming at answering. Try to find this central question and then ask yourself if your central question is a “why” or a “how” question. If it is, this is a good sign that your task is mathematical modeling task. If you can envision how a mathematical model will help you answer that question, even better! Let’s think through an example to try and clarify this point. A few days ago we discussed the “Great Lakes Problem.” Recall, that the idea was to investigate pollution in the Great Lakes (or your favorite local body of water). That starts out so “open” that there isn’t even a question there at all! But, we “banged that down” to a toy model where we were asking “If we stopped all pollution into lake X, how long would it take for pollution levels to reduce to some level?” That’s a “how” question and a good indicator that you’ve potentially found a mathematical modeling task. Or, consider the discussion around the “tipping buckets” system. There, we were asking “Why is the period of oscillations X and not Y?” Again, a good indicator of a modeling problem.
Will your model have explanatory or predictive power?
Remember, we build mathematical models for a reason. That reason is generically to be able to explain something we see or to be able to predict (and perhaps control) what will happen next with a given system. If you try and envision the model coming out of your task, ask yourself whether it will have this predictive or explanatory value. Again using the Great Lakes problem as an example, the model there would let us predict how long it would take pollution levels to reach some value under certain conditions. That’s a clear testable prediction. In the tipping buckets problem, a good model would help us explain why the period of oscillation is as observed. It would also let us predict what the period would be if we changed the inflow rate of water or other parameters in the system. If the models you envision don’t have predictive or explanatory power, it’s probably not a modeling problem. Now, note, this does depend on the type of model you are imagining your students will construct. If you are envisioning a purely descriptive model, you of course will not see explanatory value. But, you should see predictive value, albeit limited by the nature of descriptive modeling.
Do you already have in mind one right answer?
If you do, this is probably a sign that this isn’t a mathematical modeling activity. Remember that when constructing a mathematical model, the modeler is making assumptions and decisions along the way. These choices lead down different paths and consequentially result in models that look different from one another. You should be able to imagine students constructing different models while engaged in your activity. You should also be able to imagine that some of these models are better than others and have an idea in mind as to what makes a better model for your situation. Is it more successful in terms of making predictions? Does it explain more of what is observed? If you can only imagine one path to one answer that you have in mind, it’s probably not a modeling problem.
Is it clear what validating your model against the real world would mean or look like?
The ultimate test of any model is whether or not it accurately captures that aspect of the real world it was intended to capture. To test that, at some point, the modeler has to validate their model against real world data. Does the model predict accurately what actually happens in the real world? Is it able to explain and account for the full range of observations about the real world system? When you are thinking through your task, ask yourself “If a student created a model of this situation, how would we know if it is any good?” If there is no way to objectively compare the model to the real world, this is probably not a modeling problem.
Hopefully, asking yourself these four questions will help you determine whether or not you’re on the right track with your modeling tasks. Please feel free to send along any questions you might have or tasks you would like input on. We’re happy to help!