No, really, why are we learning this?

It's a good question, but one that I never feel fully prepared to answer. Much of the math covered in high school algebra classes and onward isn't of daily use to anyone who isn't a scientist, an engineer, or a math teacher. Yes, a parabola predicts the path of a thrown basketball perfectly every time[1], but most of us don't have any use for a method of predicting the movement of the basketball that is slower than both our intuition and the actual basketball. Logistic functions satisfy a simple differential equation that makes them good models for phenomena from population growth to diffusion of infrastructure and innovations[2], but I've never actually used one outside of textbook and classroom examples. And I like using math to describe things!

The truth is, students who aren't planning careers in science (including social sciences) or engineering are learning math for reasons that don't often have a lot to do with what they will need to know later in life. They are learning math because someone hopes it will help them learn to think more clearly, because someone thinks it is beautiful, to keep their options open in case they discover a passion for physics or chemistry, or because the skills they are learning would have been useful when they made their way into the curriculum. In fact, these reasons also play a big role in what particular math is taught even to students who will go on to be scientists and engineers.[3]

This complicated web of reasoning means that the question "Why are we learning this?" isn't a question that can really be answered with mathematics, or even with the applications of mathematics, as tempting as those answers are. The answer to that question, most of the time, is historical, about the growth of mathematics as a discipline, about the development of applications of mathematics over time, and about the influences and opinions of the various people and groups who have shaped educational policy and the idea of the educated citizen.

Right now, I'm investigating the question "Why do so many of us learn calculus?" When I was in high school, calculus had a privileged position in my mind, and in the minds of my family and others around me, as the pinnacle of mathematical achievement. In college, it was the course sequence that came first in the mathematics major, a prerequisite for a wide variety of math courses (few of which explicitly referred to anything learned in calculus classes) as well as for courses in other disciplines which drew directly on the subject matter of calculus. I want to know what gives this particular subject matter its central role in the transition from high school mathematics to college mathematics. Could some other course or sequence of courses do that job as well, or is calculus inevitable?

I'm starting in the 1950s, motivated by this clue

One of the contributions of the New Math movement was the introduction of calculus courses at the high school level.

from David Klein's "A Brief History of American K-12 Mathematics Education in the 20th Century," which also places the New Math movement as lasting from the early 1950s through the late 1960s.[4] The next post in this series will deal with articles published in the American Mathematical Monthly in the early 1950s, in particular those that describe the contemporary math curriculum and those that propose changes to the curriculum in the high schools and the first years of college.

1. http://blog.mrmeyer.com/?p=8483
2. http://en.wikipedia.org/wiki/Logistic_function#In_ecology:_modeling_population_growth
3. At this level, the motivations for learning integral calculus are clear, but the motivations for learning integration by partial fractions (for example) are not! A real function that arose from physics or engineering just might be possible to integrate symbolically using partial fractions, but even if it were the process would likely be messy enough that a reasonable person would hand it off to a computer algebra system to deal with.
4. http://www.csun.edu/~vcmth00m/AHistory.html

On the name of this blog

Before I loved mathematics, I loved language. I still do love language, especially the fragmented patterns of etymology that let us guess at the meanings of some, but not all, unfamiliar words in our own and other languages. It's a fascinating web of connections, and my picture of it grows every time I learn a new word.

In college, I studied ancient Greek, and one of the first words I learned in my first Greek class was the word for student. It is μαθητης; transliterated, it would be something like mathetes. This is only one of the 13 entries in my Greek-English lexicon for words beginning math-, and every one of them has to do with learning.

These are the words that our English word mathematics comes from. In fact two of the entries in my lexicon already refer to mathematics and mathematicians. The Greeks had many words for various arts they studied, and mathematics isn't the only one which has kept its meaning in modern English; rhetoric is still the art of speaking persuasively before a crowd, and philosophy still the pursuit of wisdom through concentrated thought. But mathematics is the only discipline  whose name also means, literally, that which is learned or studied.

This isn't, of course, the only possible etymology for a word which means mathematics. It is, however, a powerful one. For many things we do, we have instincts and history going back even before we were human. We tell and listen to stories so naturally because we are social animals, and stories are part of the bonds that hold us together. Our instincts for mathematics, though, are quite limited. Most people can automatically tell different numbers of objects apart only up to about 5. Everything after that is learned, not in the unconscious way we assimilate new words but through deliberate effort, the way we learn to read. Any mathematical skill, whether counting or a complicated method of proof, has to be learned, practiced, and, often, failed at before it can be used. Even the most accomplished and brilliant mathematicians learn their craft step by step. Their astonishing insights are supported by a background of patient and diligent study-- and that background, at least, we can all imitate if we want to.