When I tell people that I’m a math major, I am always met with a comment along the lines of, “Oh, you study math? I don’t know how you do it.”

I used to think this was funny, but the longer that I settle into my field, the more I realize how tragic this actually is. Comments like this remind me how math education is failing in our country. Comments like this remind me how no one really knows how math works and what it is all about until they are a couple years into being a math major.

After some investigation, learning a few definitions, and I came to the conclusion that most people could see that mathematics is, at its core, focused on making arguments and exploring structure. I would like to shed some light now on the true nature of mathematics.

For most people, mathematics has been a story of problem solving. The average person with a high school education ends their study of mathematics at some level of pre-calculus, and in college, most will end their education with some sort of minimum math requirement.

The typical experience in these courses is one of problem solving. First, you learn about some mathematical theorem or object, and then you learn a method to solve for some number. Once the method is down, you are asked to repeat this process on homework problems and on exams.

This method is used from the start of your education with doing doing multiplication tables and continues until you are doing partial derivatives in calculus III. But, in fact, the onslaught of problem solving does end. There comes a time when it is no longer of interest to “solve” for a number.

Mathematics is not the process of solving for a number. Using the quadratic formula to find a root of a function is not mathematics. Mathematics is about making arguments. Everyone has heard of the Pythagorean Theorem. Even as you read the words “Pythagorean Theorem,” I predict you can recall the equation a^2+b^2=c^2. But what does this equation mean? Some reading this may remember that the full statement of the Pythagorean Theorem is this:

“Given a right triangle with leg lengths a, b and hypotenuse length c, it must be true that a^2+b^2=c^2”

In the past many of you have used the Pythagorean Theorem in homework problems to solve for the side lengths of various triangles. But, I press you, do you believe that the Pythagorean Theorem is true? Why should that equation above hold for every single right triangle that you can dream up?

The answer is this: there is an incontrovertible argument. The Pythagorean Theorem isn’t true because we want it to be true, or because it would be convenient if it were true; the Pythagorean Theorem is true because it has been demonstrated to be true by a sequence of incontrovertible statements which are collectively called a proof. While I would love to expose the details here, as they are not complicated, this isn’t the best place to give a proof.

The point of this is that mathematics cares about proving that statements are true and making arguments. Doing exercises, solving word problems and applying mathematics to physics does not constitute the whole of mathematics; these actions merely reap the fruit of mathematics. Someone proved that the quadratic formula works. From there, math students are told that it works, and then they are told to use it to find roots of quadratic polynomials.

The tragedy is that learning to use the quadratic formula is far less engaging than learning why the quadratic formula actually works. I sympathize greatly with people who lose hope and do not want to carry on learning. Why should anyone be motivated to learn mathematics as it is taught?

To keep myself honest, I should not fail to mention that there is clearly value to knowing how to solve equations and apply results like the Pythagorean theorem and the quadratic formula. In fact, our civilization would be centuries behind if it weren’t for the results offered by calculus.

However, I think that learning to apply these results would have much more motivation after learning how and why the results are, in fact, true. Instead of the instructor saying, “The Pythagorean is true all the time, believe me. Now solve these problems,” the instructor should instead say, “Want to know something neat? The Pythagorean Theorem is true! Here is why. [insert proof] Now, let’s see some examples of how it works.”

I think this new format would not only provide students with better argumentative skills, but also it would give students more motivation to try their hand at solving the problems. Also, this educational method would remove the notion that mathematics is dogma and would improve the intellectual development of the student as a whole.

I invite you all to recall the various math theorems that you have learned and to now think about them skeptically. Why should you believe that they are true? If any of you are more curious about how mathematics is truly done, and maybe want to see a proof of some of these theorems, feel free to email me (it is indicated under my name above).