One of the most commonly asked questions in math classes is, "When are we going to use this?" Granted, sometimes this is just a form of whining, but often the question is sincere.

The reasoning behind the question may be, "Please help me to put this in some sort of context I can relate to and assure me that learning this is worth the effort." The professor, however, may hear, "I think this is useless—justify yourself!" You can prevent a defensive retort by rewording the question. For example, it sounds more sincere to ask, "Could you give us an example?"

When I started teaching, I dreaded the inevitable "When are we going to use this?" question. Later on I came to appreciate it as a great opportunity. I'd give a sincere answer, even when I suspected the question wasn't sincere. The best answers are specific to the given context, such as “Here's how you might apply this bit of math in your major.” Here I can give some general answers to how you might find math useful.

We've all heard that math "helps you think clearly." I believe that's true but so vague as to be unhelpful. Just how does math help you think clearly?

One way is that it helps you recognize patterns and think abstractly. For an elementary example, consider circles. There are no circles in the real world, not in Euclid's definition of a circle, but there are lots of round things that lead to the idealized Euclidean definition of a circle. For a more sophisticated example, sometimes a whole is the sum of its parts—and sometimes it isn't. Mathematicians call these situations "linear" and "non-linear," respectively.

You can predict the value of a stock portfolio if you can predict the value of each individual stock. In this sense stock portfolios are linear. A portfolio is the sum of its parts. Ecosystems, on the other hand, are non-linear. They are not the sum of their parts. You might know what will happen if you introduce rabbits to an island, and you might know what will happen if you introduce wolves. But you can't just add these two pictures together to see what will happen if you introduce rabbits and wolves. Drug combinations may be linear or non-linear. Suppose you take aspirin for a headache and a decongestant for runny nose. If you have both a headache and a runny nose, you might take an aspirin and a decongestant and get relief from both. In that case the drugs are linear. But some drugs that are beneficial separately are harmful together. These combinations are non-linear.

Another way that math helps you think clearly is by making you express your assumptions explicitly. For example, can you divide 3 by 2? The answer is "no" in the context of the integers (whole numbers) but "yes" in the context of the rationals (fractions). Do the numbers 5 and -3 have square roots? In the context of integers, no. In the context of real numbers, yes for 5 but no for -3. In complex numbers, yes to both. Introductory math classes may be informal about assumptions, but more advanced classes are very careful about definitions and assumptions. You learn that stating a problem correctly may be half the effort of solving it.

After I left teaching, I worked as a software developer. A math major sent me an e-mail asking what math classes I found most useful in "the real world." My response was topology, a course that had no direct application whatsoever to my work. I said this because my topology course made me think precisely about definitions. It also taught me how to use doodles to guide my thinking without using them as a substitute for rigorous logical reasoning.

Math can be useful in direct application. I've found a use for most of the math I've learned, though in some cases there were a couple decades between first exposure and application. Indirect applications are far more common. Pattern recognition, abstraction, and precise thinking are immediately useful in any career.