By Dan Teague, posted November 9, 2015 —

It was early September 1963. At John Hanson Junior High, I was part of a new program in which a small group of eighth graders were taking Algebra 1. Mr. Green was my teacher, explaining the difference between a number and the numeral representing the number and why x = 3 wasn’t the solution to the equation 2x = 6; rather, it should be {x ∈ ℜ | x = 3}. (New Math—those were the days). As far as we knew, we were the first kids in the history of the world allowed to take Algebra 1 in eighth grade. We thought we were hot stuff.

Then Mr. Green, in an effort to contain our youthful arrogance, said something equivalent to, “You guys are good, and math is easy for you. But always remember this: Everyone has their Green’s theorem.” Probably this stuck with me because of Mr. Green talking about Green’s theorem (no relation), but I’d like to think I would have remembered it anyway. He went on to explain what he meant. When he was a student, mathematics was easy. Algebra, Geometry, Trigonometry, and Calculus—all were all very easy for him. But then he got to Green’s theorem in Multivariable Calculus. From then on, in every mathematics course, although he was just as good as he had always been, he had to work really hard to master the content. His meaning for us was that, at some point, mathematics becomes difficult—for everyone. And when we hit our personal Green’s theorem, as we all eventually will, we need to learn to work at it like everyone else.

Fast forward fifty-two years, and I’m talking with a student about courses she wants to take next term. She tells me, “I really like mathematics, but I’m not very good at it.” Her proof? “I have to work really hard to be successful.” In the world of students, if you can do something easily, you are good at it, and if something is not easy for you (no matter how successful you are and no matter how much you enjoy it), you are not very good at it. Moreover (and this is the dangerous part of this thought), if a subject isn’t easy for you, then you just aren’t cut out for it. My student could not imagine mathematics as a major and certainly not as a career option.

But Mr. Green would tell her that she is using the wrong metric. For everyone, mathematics becomes challenging. Mathematics is one of the greatest intellectual achievements of humankind. Of course, it will be challenging. Everyone has to work at it—some earlier than others and some later than others. But everyone has a personal Green’s theorem.

It is important for students to understand that, whatever their area of interest, the subject will eventually become challenging. So what? Hard isn’t bad. The real question isn’t whether a student finds a subject easy or hard; all subjects will all be hard eventually. The real question is, Do you enjoy that challenge? That’s how to decide what career path to pursue. You actually don’t want an “easy” job. You want to be challenged by your work and to draw joy and contentment from accepting the challenges it offers and working hard to meet them.

Mathematics is the hardest thing I know, with the possible exception of teaching mathematics. I do them both not because they are easy. Like you, I do them both because I love the challenges they offer and because each day I have to work really hard to be successful. Thank you, Mr. Green.

DAN TEAGUE, teague@ncssm.edu, teaches at the North Carolina School of Science and Mathematics in Durham. He is interested in mathematical modeling and finding problems that connect concepts from different areas of mathematics.