Physics, Math and Mental Gymnastics

While we enter physics to study the fascinating world of black holes, quarks and the quantum, the brutal truth is that mathematics is the central tool of the physicist. Gauss called mathematics the "Queen of the Sciences", and with good reason. If you don't have a solid grasp of mathematics, you aren't going to get very far.

One thing I noticed when getting my degrees in physics was that many of the students found math to be a painful "aside". In one case that really stands out in my memory, I was in a mechancs course and one of the homeworks required the calculation of a brutal integral. I worked very hard by myself over the weekend and managed to get the calculation out with a couple of pages of work. When I returned to class, I was surprised to find that the vast majority of the students had not even attempted to work out the integral. One student had obtained the answer-so he thought-using Mathematica. I looked at it carefully and saw that he had gotten the wrong answer. He argued with me-asserting that the computer cannot make a mistake-but we brought the TA over and it turned out he had entered the integral incorrectly. I had obtained the right answer by working it out by hand.

The student in question had thought he was interested in physics but didn't want to bother with the work of physics-which involves diving into the mathematics. But to become a good physicist-or a solid engineer-you need to bite the bullet and become a master of mathematics. It doesn't matter if you're going to be an astronomer, experimentalist, or engineer-in my view if you want to be the best at what you do in these fields, you should have a solid command of math. So if you are interested in physics but aren't a mathematical hot shot, how can you pull yourself to the top of the field? In my view, the answer is to view mathematics the way you would athletics. A friend of mine who shared this view coined the term "mental gymnastics" to characterize his outlook and study habits.

We all aren't Math Genuises
While for some students thinking mathematically comes natural, most of us aren't ready to master the intricacies of studying proofs when we're college freshmen. This article is written for those of us who aren't automatic math whiz kids. If you are a mere mortal who finds math a bit of work, don't be discouraged. It's my belief that average people can raise themselves up to become very good mathematicians with a little bit of hard work. What we need is some training--we need to train our minds to think mathematically. The best way to think about how you can get this done is to draw an analogy between math and athletics.

To master a sport you have to build your muscles and train your body to react in certain ways. For example, if you want to become a great basketball player, you could be lucky enough to be born Michael Jordan. But more likely, you'll have to work at building a basic skill set, and the truth is even players like Michael Jordan put extra work into their craft. Some activiities you might consider that could make you a better basketball player are

• Lifting weights to build muscle mass

• Run sprints to improve your ability to run up and down the court without getting tired

• Spend a large amount of time shooting free throws, doing layups and practicing basic skills like passing

It turns out that becoming a successful physicist or engineer is in many ways similar to athletics. OK, so suppose you want to study Hawking radiation and string theory, but you are not a hot shot mathematician and weren't the best student. Instead of just reading a bunch of books or lamenting the fact we aren't an Einsteinian genius, what are the mathematical equivalents to lifting weights or running sprints we can do to improve our mathematical ability? In my view, we can begin by following two steps

• Learn the basic rules first-and don't focus on trying to learn proofs or do the hardest problems.
  

• Repeat, repeat, repeat. Do similar types of problems over and over until they are second nature. Only after a topic becomes second nature calculationally do we consider reading the proofs or theorems in detail.
  

That is do tons of problems. In my view a student should start off simple. Don't try to understand the proofs. For example, in my recent book, "Calculus In Focus", I take the perspective that students need to learn math by following the formula: show, repeat, try it yourself. That is

• Show the student a given rule, like the product rule for derivatives

• Focus on mastering calculational skills first. Do this by showing the student how to apply the rule with multiple examples.

• Repeat, repeat, repeat. Do a given type of problem multiple times so that it becomes second nature.

Once the "how" to solve problems is second nature, then go back for a deeper look at the material. Then learn the "why" and start learning the formality of mathematics through proofs and theorems