What is math research?

“This might be a stupid question, but what is math research?” is the most common question I get at social events and family functions, and it’s not a stupid question. Many scientists do their research in a laboratory, or in the field, taking samples or surveys. Humanities researchers read texts or look up primary sources. Mathematics research is unique because it can be done alone in a room with nothing but paper, pencil, and one’s mind.*

(*Math research isn’t usually done this way. Usually mathematicians read many other papers before starting a new research project, reference books as they go, and collaborate with others instead of working alone.)

So what is math research? Math research is the process of discovering truths about numbers, shape, and other mathematical objects. Math research is asking, “Is there a relationship between this and that? Is this always this way? Does that ever exist?” and then trying to answer these questions, for the first time. Let me give an example. Someone once looked at a right triangle (a triangle with one angle measuring 90 degrees) and asked, “Is there a relationship between the side lengths of this triangle?” This person used some of the math they already knew (the area of a triangle, for example) to discover that the sum of the squares of the side was equal to the square of the hypotenuse. Or as you may have heard it: a^2 + b^2 = c^2.

“Well that’s not research. I’ve known that since 6th grade,” you may say. But it was research when the first person discovered it. Now it’s a well-known fact, and most people have applied it many times in school without ever wondering why it’s true. Unfortunately, math is taught in schools as a set of facts, presented to be memorized and applied. Students never have to struggle with discovering the facts; they only struggle with when and how to use them to complete their homework. But this isn’t really math. Math is the process of discovering these equations, these new relationships and patterns, for the first time. Real math is math research!

Math research can involve answering (or at least attempting to answer) all sorts of questions, even ones where the answer is already known. Here are some example questions one might consider:

• How many spheres of radius 1cm can fit into a box that is 1m cube?
• If 100 people are at a conference and all want to shake hands with each other, how many handshakes will occur?
• Are there any whole numbers a, b, and c so that a^3 + b^3 = c^3?
• Does 1 + 1/2 + 1/3 + 1/4 + … equal a finite number, or does it grow to infinite? How about 1 + 1/2 + 1/4 + 1/8 + …?

So when I say that I’m working on math research, I mean that I’m trying to ask questions about numbers and mathematical objects (in my case, about graphs) and answer those questions for the first time. I’m trying to be the first person ever to notice and prove that a certain relationship or pattern exists. Which, when it happens, is pretty cool!