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Why math? JHU mathematician on teaching, theory, and the value of math in a modern world

Q+A with Richard Brown, director of undergraduate studies in Johns Hopkins University's Department of Mathematics

Math as both profession and course of study can be a hard sell, something even Don Draper might have trouble pitching. The field unites numbers, theories, and ideas that, yes, can be physically represented but remain intangible. Math is a language unto itself that for some might as well be Latin or Klingon. Even its rare turns in popular culture—A Beautiful Mind, Proof, and The Big Bang Theory come to mind—typically depict brilliant but troubled and/or socially handicapped thinkers more absorbed by theory than reality.

To Richard Brown, however, math can be as beautiful as a ray of morning sunlight cast upon an orchid's petals, as lyrical as a Beethoven symphony. "Math is not about the numbers," says Brown, director of undergraduate studies in Department of Mathematics at Johns Hopkins University. "It's the ideas behind the numbers. Yes, you can say it's built on a rigid set of rules, but out of that comes an infinite amount of creativity." And, he adds, beauty.

In recent years, Brown has served as math spokesman of sorts. Last year, he presented a talk at the inaugural TedxJohnsHopkinsUniversity titled "Why Mathematics?". He opened up with a question: "Why on earth would anyone choose math as a field of discipline to study, or construct a career in the field?" To answer the query, Brown dissected his own career path: He began his undergraduate studies at Temple University as an architecture major, then switched to engineering. Neither felt quite right. He would find his true calling in math, which he describes as "the thoughtful making of logical structure," and earned a master's degree in applied mathematics from Temple and a doctorate in math from the University of Maryland, College Park. Prior to coming to Johns Hopkins in 2005, he worked at MIT's Lincoln Laboratory and NASA's Goddard Space Flight Center on the Hubble Space Telescope Project. He also spent four years as a mathematician in a private research engineering firm before returning to academia at American University in Washington, D.C.

He edited the book 30-Second Maths (Icon Book, 2012), which took the 50 most engaging mathematical theories and succinctly explained them to the general reader. Brown also contributed to a new book titled Mathematics for the Curious: Why Study Mathematics? (Curious Academic Publishing, 2015). He wrote the book's first chapter, "What Actually Is Mathematics?" in which he talks about math as more art than science.

The Hub sat down with Brown to talk about a field that is both misunderstood and underappreciated.

You keep asking the question: why math? Are you basically advocating for math as a field of study, or is there a deeper concern here, such as not enough people entering the field, or that too many Americans exit the school system with deficiencies in their math abilities?

I wouldn't say there are major issues. But one of the concerns I do have is this transition from middle school/high school mathematics to university mathematics. Sure, a lot of students come in very prepared for the university experience, but a lot of students come in not prepared at all. So, yes, there is some concern, but I wouldn't say there's a worry about not enough people going into mathematics. We tend to have a pretty flush crew coming in every year.

When you get beyond the basics of addition and subtraction, math seems to suffer from a dissociation with real world application. I've heard elementary and high school math teachers tell students such things as, and I'm quoting here, "Most, if not everything, I'm going to teach you this year you're never going to use in your life." What is your reaction to such statements?

That gets down to one of the root concerns we do have. At the primary and secondary level, some of the people who teach mathematics don't get it. They don't get math, and they don't really understand it. They don't see math's purpose and understand its bigger structure. And they immediately convey that to their students.

When we see students come into the university level not prepared for this experience—or they don't like mathematics, or they're afraid or anxious about it—that typically means they've had a teacher like the one you mention, someone who has no full understanding of what mathematics truly is or how to teach it to a younger audience.

I think the problem is that [the math] is difficult for the instructor. The only reason they would come across with an attitude like that is that they themselves don't understand the math or are anxious and fearful. They think math is only useful in balancing a checkbook, or torture (laughs). They don't see math as the process of thinking logically.

I think there's a perception that some forms of math are problems just for the sport of it, like a crossword puzzle or Sudoku. They exist to challenge your brain, but they won't help cure cancer or put a man on Mars.

Without any context of why you're learning something, yes, it becomes just a bunch of problems that you need to solve. There are certain concepts, constructions, or equations that you're never really sure if they are going to show up anywhere or not. But presented in context, math has a number of real-world applications. But the problems themselves have value. Even at the primary school level, you can play Sudoku games and teach the logical structure of things. If you give people a sense that there's a creative nature to mathematics, then at some point, even at the primary school level, they can learn and wonder at some of the logical structures they're building. That is something they take with them the rest of their life, a curiosity about how things are structured logically.

Can you give me an example of what you mean?

When my son was 9 years old I decided to play a game with him and asked, 'What is 1 + 1?' And he said, "Dad, 2.' And I said, well, sometimes the answer is 10. He laughed at me, but I said, if you only have two numbers to play with, 0 and 1, and not 10 numbers, then what is your number system going to look like? It will be 0 and 1, and you run out of numbers. So you put a 1 as the next digit and you get 10. It's the binary system basically. 0, 1, 10, 100. So, in this system, 1 + 1 = 10. 1 + 10 = 11. Then I asked him what was 11 + 1? And, he said 100. And then we started to play with three-digit numbers. Suddenly he was interested, because it was playful. It was just a game. But, it turns out, this is how computers work.

In both your essay and TedX talk you mention the "beauty and the art of math." But surely math is just a series of symbols and numbers; where is the beauty in that? Or are you not referring to the physical representations, but more the logic behind an equation?

It's actually the logic and the ideas that all that junk on a blackboard might represent. There's a lot of ways to address this question. I compare it to music. I'm not the first to make this connection. There is a famous essay out there called "A Mathematician's Lament" by Paul Lockhart. He compares the teaching of math to the teaching of music. He uses that to say that the way we teach mathematics at the primary school level is just abysmal. If we taught music like that, students would never listen to music until they got to college. They would just be working on scales all day on a piece of paper. They would never understand what the subject is all about.

The comparison is nice. Music and math both have this nice logical base set of rules from which all the complexities of their structures are built upon. There are not that many rules in music, and yet we can compose something as beautiful as a Bach fugue. That fact seems obvious to people. Mathematics is the same way. We build our number systems over a base set of rules, and yet there is this an infinite amount of creativity and complexity to the stuff that we do. As mathematicians we see this, and we live for it. But to people at grammar school and high school who are just learning the quadratic formula because they're going to need it some day, they don't see that. They don't get a chance to see the beautiful part of mathematics that we see.

Can you explain this beauty?

One other nice aspect that music and math share is that in music, when you put together a logical construction, you attach a value to it. People get an emotional response from it. You attach a value to it in the form of how nice it sounds, how harmonious it is. When we create a new mathematical structure, we create a new fact. In and of itself, it's not really interesting. What's interesting is how it fits into the whole of mathematics, whether it addresses a long-standing question or is so counterintuitive a result that it's just stunning to look at. Or maybe it establishes a relationship between two disparate fields of mathematics that nobody ever dreamed existed. The aesthetic value lies in how interesting it is, and how it fits into a greater whole.

What pushes math forward? It's not so much lab-based, where we are trying to unravel something like how a cancer cell moves and multiplies. What are the catalysts that make someone create a new formula or structure?

Every once in a while, someone comes up with a very clever idea to establish a new result, like something as mind-boggling as Fermat's Last Theorem. The first thing you consider when you have a new result is, why does the logic exist the way it does? Why does it work out this way? Does it always work this way, or do I need restrictions to make it work? Then you start to ask, where can I go from here? I have a new perch to stand on, what else can I reach for? Really, that is what research is all about. Every question that is answered raises 10 to 12 new ones. Math works this way.

OK, I'm making you chair of a national committee to change how math is taught in the United States. What are some things you're changing?

Actually, there was an advisory committee to President Obama formed some years back. One of its conclusions was that kids need to understand why math is important to them, and one way to do that would be to bring more scientists and engineers into the classroom to study mathematics so they could see how it's useful. My first reaction to that was that might not be correct. I think what would be better would be to bring more mathematicians into the classroom to teach math. I think what happens at the primary and secondary level is that people teaching math only have rudimentary understanding of math, they don't understand it at the research and application level. The experts in that field should be the ones teaching the subject. I shouldn't be teaching physics, for example, I should be teaching mathematics.

I'm not so sure that showing why math is useful, like balancing a checkbook or calculating a distance, would really help mathematic education. I think treating it more like an artistic endeavor would be useful, where you're teaching mathematics in a much more creative sense. You're teaching kids how to think analytically and reason deductively, as you teach the structure. The attitude at how math is taught is the tough part.

I'm wondering how math education in the United States differs from how it's taught in Japan, China, Germany, and other nations.

I think the students from the countries you mention are typically ahead of us in terms of math understanding by the time they get to the university level. I've seen the prior work of students from these countries who come here to Hopkins, and it seems to be much more focused on the pure mathematics than on the reasons why it's applicable to the world. For example, they are more advanced in calculus, which tends to be more theoretical in nature than practical in nature.

Some find math daunting and think: my brain is just not wired to understand these principles. Do you give any credence to that line of thinking, that a person is just not mathematically inclined, so to speak?

I don't think that is true. I think anyone can learn mathematics. Now, maybe it's not the case that anyone can learn advanced mathematics if you start them right here and now, after they've gone through their entire childhood running away from the basic concepts of mathematics. But anyone can learn.

So you would tell a student, I don't want to hear the words 'I can't do this,' or 'it's too hard.'

And I do (laughs.) I tell some students, perhaps your background is insufficient to grasp what we're doing at the moment, but anyone can learn math at the higher level. Certainly at a level of calculus and above.

Do you think learning math might actually be good for us, in terms of brain development and sharpness?

I don't know how much research has been done on this, but my gut tells me it's very good for you. I do know that when I read the newspaper and listen to people speaking, one thing as a mathematician you learn to see quite clearly are the logical flaws in arguments. That way of thinking is also useful when you read the legal part of a contract you're about to sign. Why is that? Maybe in a legal endeavor, or when there is a politician speaking, they are trying to use the logic to deceive. But I think that a lot of people are just not aware of the logical flaws in the reasoning. Learning mathematics is learning how to think analytically. Everyone, I think, should learn math so they know how to reason deductively and logically, and put together a complicated idea that makes sense.

Do you watch The Big Bang Theory, and, if so, you might know where I'm going with this…

I don't. But apparently the math on the dry erase board in the background is all correct.

That was my question!

Yes, apparently the math on the board is all checked by mathematicians and physicists, because sometimes they put physics on there, too.

Math was not your first passion or interest. What about the field won you over?

Wow, it was such an evolution. Hard to tell if there was one moment. But when I was studying architecture, there was this sense that I wanted to build something useful that could generate an emotional response. When I got to engineering, the idea that I could find creative ways to solve problems sounded very cool to me, but it didn't quite work for me. When I got to mathematics, I began to understand that there's so much more to math than just playing with equations. There is an actual logical structure to everything that we do. Being able to see and understand that structure really started to speak to me. Here, I thought, I could play in a world that is artistic and I get to do useful things. Either as a teacher or like the work I did for NASA. I felt like I was home.

Speaking of NASA, would aliens be able to understand our math? Would we be able to understand theirs?

The answer is yes, because there is nothing earthly or tangible about mathematics. Everything that mathematicians do, create, or think of is imaginary. It's made up. It's just logic. So it wouldn't make sense that the logic wouldn't hold up in an alien world. The physics might not hold up, but the logical structure of thought would still be the same. It's a universal thing.

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