Mathematicians occupy an odd place in the public imagination, as objects of great curiosity and also great misunderstanding. TV and movies portray us as anything from eccentric to insane, though sometimes we get to solve crimes. But there is rather little public understanding of what mathematicians actually do with their time.
Even among mathematicians, applied math has an odd reputation. Many pure mathematicians (those who spend their time working on purely abstract problems) regard applied math as mere "computation", as if we were essentially glorified calculators.
But applied mathematics is not about discovering new numbers, nor solving crimes, nor cranking out long calculations (though some of that is involved). At heart, applied math is about creating, refining, and analyzing models.
The "applied" in applied math means that we work on problems that are in some way relevant to the "real world". However, the real world is a complicated place, and virtually any system you might want to investigate has far too many interactions and unknowns to be understood completely. Imagine, for example, trying to understand the physical properties of a gas by first specifying the mass, volume, and exact location of each of billions of molecules, and then trying to predict where each particle will be in the next instant, and then the instant after that. Even if you were somehow able to do all these calculations, your answer would be valid only for that particular gas in that particular configuration, and would give you little insight into the behavior of gases in general.
So when we want to understand a system, we don't attempt to incorporate every potentially relevant detail. Instead, we model it: we focus on what we believe to be the essential features of the problem and throw out everything else. All models are oversimplifications, but if they are well-constructed, that is, if we have picked the right features to keep and the right ones to discard, they can provide valuable insight into the real-world problem we are studying.
All models incorporate a trade-off, which I've (poorly) illustrated here:
We often hear about models on the right end of this spectrum: models of
of large-scale, complex phenomena such as the global climate or economy. These models incorporate many different variables in order to be as accurate as possible in predicting reality. The trade-off is that there is generally less insight to be gained from such models, because cause and effect relationships can be difficult to untangle with so many variables involved.
Mathematicians are more interested in the simple end. Unlike complex models, which can generally only be analyzed through computer simulation, simple models can often be analyzed using a pencil and paper. Though they do not describe reality as accurately as complex models, they illustrate very clearly how and why certain effects lead to certain outcomes. Simple models also have the advantage of generality: the same set of simple features may be present in a wide variety of systems. The more variables and complications you throw in, the more your model becomes tied to the one specific problem you started with.
I've written a lot in this blog about the Prisoners' Dilemma as a model for cooperation. The essence of the model is this: two players each have a choice whether or not to cooperate with the other. If a player decides to cooperate, they pay some cost, and the other player gains some benefit. Of course, cooperation happens in many different forms in human and animal life, and you could study any particular cooperative behavior by tracing its social and/or cognitive basis, as well as its evolutionary origin. But by studying the particularly abstract, simple model that is the Prisoners' Dilemma, you can gain some insight into the phenomenon of cooperation in general: when and why it evolves, and how it is maintained.
The purpose and method of developing and analyzing models is a strangely absent topic from high school and college math and science classes (a welcome exception is a course I'm currently TAing at Boston University that teaches quantitative reasoning to non-science majors). But given the role that models play in our economy as well as in science, and the catastrophic consequences of their failure, I think that communicating an understanding of the modeling process should be a central goal of science education.