“Of course we should,” I hear you say. “What good is math if it doesn’t have any read-world applications?”
Let me unpack your reply. In one short paragraph it brings up at least half a dozen responses. We’ll discuss a couple.
First of all, we need a context. I’ll give you two: one from this morning, one from 54 years ago. We’ll start with the earlier one. As an undergraduate, one of the courses I took was in the Applied Math Department; it was titled something like “Data Storage and Retrieval” (there was no such department as Computer Science in those days). In the last week of the semester I had the effrontery to ask the professor why the course was called “applied” math when we had learned exactly zero applications.
“Well,” he replied, “the department name may be a bit misleading. It doesn’t really mean ‘math that has been applied’; it’s more like ‘math that can be applied.’ ”
This, of course, is nonsense. All math can be applied. Sometimes it takes hundreds of years—I’m fond of pointing out prime numbers as my favorite example, which took over 2000 years before we found a use for them, and now they are completely fundamental to internet security. And what, then, is pure math? Is it math that cannot be applied? No such thing; there’s just math that hasn’t yet been applied.
My example from this morning is cryptography, which includes the aforementioned internet security but we haven’t gotten there in the course (called “Quantitative Reasoning”). The students are learning to complete a few traditional exercises and even fewer genuine problems, all of which either have or could have real applications. On the second day of the course I asked the students to come up with ideas of where crypto would be used in real life, and they had quite a lot of legitimate answers. During the unit they will learn not only about prime numbers but also about modular arithmetic, different alphabets, and a variety of cryptosystems that have been used throughout the past couple of millennia. It has always been a popular topic, and it mixes pure and genuinely applied math.
So, how is this different from what’s usually taught in high school? The answer comes back to my old professor. Textbooks for high-school math and teachers who use them almost always focus on fake applications, not real ones: totally made-up problems that would never come up in realistic contexts. There are, of course, some notable exceptions, but that’s the usual situation. Now I have to admit that I love pure math myself, as do a handful of high-school students, but mostly we get the “When am I ever going to use this?” question that I’ve written about several times before. Both of the two most common ways of teaching what purport to be real-life applications of math in high-school are unsatisfactory, either because they involve fake uses (“given this formula, how high does the baseball rise?”) or else they are boring regurgitations of what the student already knows. So yes, we should teach real applied math in high school, but keep it real!