If there were a top-ten list of “things that make math teachers cringe,” the question “When will we ever use this?” would surely be at the top. That’s pretty independent of whether you teach at the elementary grades, middle school, high school, or college. Quite rationally I suppose, most students want to know there is some *utility* in what they’re learning, that this lesson is not just another “eat your spinach, it’s good for you” type of lesson, but is something they’ll be able to see as relevant to their own lives and their own careers.

One of the nice things about teaching mathematical modeling is that it’s incredibly relevant in a wide variety of contexts and to people working in a tremendous variety of fields. As I read the news each day, I keep an eye out for neat places where mathematical modeling shows up and today, I thought I’d share a few recent ones with you.

One of the coolest is the recent discovery of evidence for the existence of a ninth planet (poor Pluto!) in our solar system. This discovery, announced by Caltech researchers in January of this year relies entirely on indirect evidence provided by a mathematical model. In this case, no one has actually seen the ninth planet, all of the evidence comes from observations of objects in what is known as the Kupier Belt. These objects are moving in ways that just don’t make sense…unless there is some other very large mass out there as well. By constructing a mathematical model of how these objects should move and inserting an unknown large mass into the model, the Caltech team has shown that the most likely explanation for the motions that are observed is the presence of something that isn’t observed, i.e. a ninth planet. How cool is that? Note that the reasoning of the Caltech team is exactly the same as the reasoning we’ve been discussing here. They *observed* a pattern, they sought to *explain* that pattern, they made a *hypothesis* about what could be causing that pattern, they built a *mathematical model* incorporating that hypothesis and showed that the model predicted the observed pattern, and hence can claim that the probability that their hypothesis is true is now very high. In this case their hypothesis happens to be the very exciting one that a previously undiscovered planet exists!

In an entirely different direction, a team from the University of Aberdeen recently built a mathematical model that explains how things go viral. In this case, the team wanted to understand how things like the Macarena could suddenly become wildly popular, or how “Numa Numa” could garner more than two million views on YouTube in just three months, or more importantly how social movements, ideas, or products could catch on or fail to do so. Here, the team borrowed from mathematical models used in epidemiology, similar to those we explored in “Pictures and Stories,” and added in the effects of acquaintances, such as those we maintain through social media, to construct a new model that could examine the spread of ideas. The Aberdeen team showed that while an individual’s resistance to the spread of a “contagion” might be high, when bombarded by that contagion from many directions, such as happens through FaceBook or Twitter, transmission occurs, i.e. you go view Numa Numa as well. That synergy leads to explosive transmission and we say that something has gone “viral.” This is not only a wonderful example of the use of mathematical modeling to explain a real-world phenomenon, but also a wonderful example of the generalizability of mathematics and mathematical models. The same mathematics and the same types of mathematical models that can be used to study the spread of Ebola here have been used to study the spread of ideas.

Another example that caught my eye recently was work that appeared in PLOS One, where researchers investigated the impact of deploying a test for a type of drug-resistant tuberculosis. The question here was whether or not having a test that detected the particular drug-resistant strain, in addition to existing tests for TB and another type of drug-resistant strain, would impact the spread of TB throughout a population. Knowing the answer to this question allows researchers to effectively direct their time and resources. If this third test would help contain the spread of TB, it would be worthwhile, but if it didn’t, that time and money could be more usefully directed toward something that would actually save lives. The answer here, arrived at through extensive mathematical modeling, was surprising. The additional test did nothing to impact the spread and hence is not worth developing or deploying. As you might imagine, this is the type of question that not *only can* be answered by a mathematical model, but *can only* be answered by a mathematical model.

Discovering new planets, explaining the spread of viral videos, and determining where to invest time and money in medicine, are just three very recent examples of mathematical models and mathematical modeling impacting lives and people around the world. I encourage you to keep an eye on the news; I’m certain you’ll quickly collect your own stable of such stories to share with your students when they ask you “When will we ever use this?”

John