Modern policy discussions in America almost always leave out the biggest question – why are we doing what we are doing in the first place? Leaving out first principles always leaves people trying to find the most practical way to accomplish nothing in particular. We have become accustomed to not asking questions about first principles because they always sound too doctrinaire, but then we wind up, at best, making the misplaced assumption that everyone is reaching for the same goal, or, far worse, viewing the activities themselves as the goals.
One place where this problem repeatedly rears its ugly head is education, and especially mathematics education. Why are we teaching math? What do we want people to get out of a math education? Humans have survived eons knowing nothing more than the basic arithmetic they pick up naturally as part of the world. Therefore, if we’re going to dedicate such a large portion of our children’s lives to learning mathematics, we had better know why we are doing it.
Knowing the reason(s) for teaching mathematics informs what mathematics is taught, how it is taught, and what level of understanding we should require of students. Some have suggested that learning mathematics is equivalent to learning the other great feats of skill in human technical history. With this understanding, the goal in math would be to simply know what the mathematics developments are and why they are important. That is, it is more of a history of mathematics than mathematics itself. For instance, in art class I learned that Leonardo da Vinci developed rules for making perspective images. However, I did not learn to paint such images myself. For some, mathematics should be an intellectual history lesson, not an interactive sport. If there is a practical use here, it is for us to be “art critics” of the use of numbers in society. Perhaps we will never be able to create equations, but we may be able to recognize the artistry of the mathematicians who do (and hopefully also recognize the shortcomings of the imposters).
For others, we need to learn mathematics because we live in a highly numbers-oriented society, and we need to learn which numbers get put in the right places, what those numbers mean, and some basic tips and tricks for dealing with them. The is the basic mindset behind the “practical math” courses. Do this set of operations to calculate taxes. Do this other one to see how much interest you will be paying. You can express things as a decimal or as a fraction, and here’s how to convert between the two. In such courses, the focus is on practical manipulation for daily interactions, not deep meaning.
Another viewpoint is that mathematics is a preparation for STEM (science, technology, engineering and math) subjects that the student might encounter in the future. Engineering, physics, electronics, and some computer science take a lot of mathematical understanding, so the point of mathematics courses is to prepare students for these future studies. They need thorough training in various mathematical ideas because they will be directly useful in later studies. Additionally, here, mathematics can be used to “weed out” students early on who are not likely to succeed in higher-level courses. Of course, one of the issues here is that the necessary mathematics is highly dependent on the future courses. For example, calculus for mathematics students actually tends to look very different than calculus for engineering students. Additionally, zoologists rarely need the intense mathematics of the biophysicists, but may need more advanced numerical training for graphing and data entry than music majors.
Still another view of mathematics is that math is mind training. While the choice of the specific mathematics used may have some importance, the main goal of mathematics is not the specific content being relayed. Think about sports. Why do we do sports in school? Because we need to exercise our body. I can’t play “sport,” I have to choose a specific sport to play. Let’s say I play basketball. Even if I’m only playing basketball to exercise, I need to learn the rules of the game. Learning basketball entails all sorts of benefits – improved speed and strength, better hand-eye coordination, learning to play as a team, and making quick decisions. It’s hard to construct a means of getting those benefits in a generic fashion – one has to actually pick a particular sport with particular rules and learn them in order to enjoy the benefits. However, those benefits stay with you, even if you never play the game again. Likewise, for mathematics. It’s not so much that students need to learn trigonometry specifically, it’s that they are learning a mental game with specific rules that can be applied and checked. The rules lay foundations for practice and for judging correctness. The goal is being able to identify, set up, and use the proper tools for the job from the current selection. The fact that it is trigonometry that is used is incidental to the end goals.
The exemplar class where this makes the most impact (and is the most generally damaged course) is the typical college statistics class. Nobody is sure why this class is in the curriculum, but almost everyone is required to take it. Actually, the truth is worse than that – there are so many differing reasons for this class that it fails to do hardly anything worthwhile. Is this a STEM class where students learn all they need in order to properly prepare statistics for usage in journal articles? Is this an intellectual history class where students learn to see how statistics has been used, and how to spot bad uses of statistics? Is this a practical math class so that students can more correctly make plans based on weather forecasts? Is it a fundamental mathematics class, where the foundations of statistics are taught? For the most part, the statistics courses I have seen don’t have a distinct understanding of themselves or why they are there, and if you have taken one of these courses you probably know what I mean.
The point is that to make an effective curriculum we need to know what we want to gain out of the mathematics courses. Only by answering that question first will we know how to structure math in the future. Which of these goals are being sought? Or, more likely, to what degree is each of these goals being sought? Until we know why we are doing mathematics, the technical details of what should be fixed will never be agreed upon.