Teaching

You do not really understand something unless you can explain it to your grandmother
---Albert Einstein

Today, just as when I started my teaching career in grad school in 1978, as soon as I understand something that inspires me, I want to teach it, or write about it. Since mathematics has always been one of my inspirations, and I consume lots of it, it follows that on an almost daily basis I have the urge to communicate to others what I have encountered in the mathematical realm. Traditionally, in my profession, that's been accomplished in the classroom Monday to Friday, between August and May. Nowadays, however, for academics at supportive liberal arts colleges, there are additional opportunities to explore all year round, such as expository writing, giving public lectures, conducting workshops for college level mathematics faculty and high school teachers, writing online columns, and other outreach activities. Most recently, for me, this has extended to professional blogging and tweeting. One consequence of all this is that the divide between teaching and scholarly endeavours-and sometimes service too-gets blurred. I enjoy equally teaching our three constituent audiences: mathematics majors, other service science/ engineering/econ majors, and those students only taking math as a general education requirement.

My unwavering goal every time I enter the classroom is to stimulate students intellectually, no matter what their major or what level they are at, with new material relevant to the progress of their syllabus. At the core of all my courses is content-intensive delivery. Early on, I set high standards and expectations. I hold students accountable every step of the way. Deadlines are not lightly disregarded, but compassion can also be shown when warranted. I also set high standards for myself, and hold myself fully accountable, not only for clear, direct instruction with adequate examples, but also responsiveness to student questions, concerns and needs, in and out of the classroom.

The computational and utilitarian aspects of mathematics, for which the subject is most well-known, are only part of the story. It's just as important to share the beauty and elegance of mathematical thought, and our majors need to be exposed to some of the truth about the creative process of mathematical discovery. Contrary to how we generally present them in class-both out of habit, and due to time constraints-new mathematical ideas don't arrive fully formed. Rather, they are the end result of hard work, and a sometimes hit-and-miss process which takes time, in which meaningful failures, when revisited, can sometimes spawn success. Students need some experience of that. Their first tentative proof attempts, under our guidance, as they move towards our core theory classes (Abstract Algebra and Real Variables), can be frustrating and painful for them-and for us, watching nervously from the sidelines-but are pivotal stages in their mathematical maturing. Mathematics, like art, takes time to learn to appreciate and create.

Early on, I strive to get across two complementary messages to all of my students about college level mathematics: (1) It's not a spectator sport; don't be fooled by how the teacher can make something look easy, and (2) In many ways, it's like a language, which can only be mastered with a lot of practice, a modicum of repetitive immersion, and a significant investment of time.

It's also important to know what nobody knows, and I always seek opportunities to alert students to famous and easy to explain open questions. Are there infinitely many twin primes? (That one is back in the news in May 2013!) Is every even number the sum of two primes (the Goldbach Conjecture)? What are the seven “million dollar math problems"? How many of them have been solved since being posed in 2004? Nationally, it's debatable if our curricula are set up so that all mathematics graduates are familiar with those questions, and in addition can name (1) A mathematician of note alive today and (2) a mathematical result of note proved in the past 50 years. Would colleges be happy graduating music, art or literature majors who weren't familiar with any work in their field from the past half century, and couldn't name a living composer, painter or writer of note? How can we make sure that math grads do better in this regard?

Albert Einstein is alleged to have said, “You do not really understand something unless you can explain it to your grandmother." I'm a firm believer that if one really understands something one should be able to explain it at three levels. In the case of mathematics, as I tell my students, that translates into: being able to explain it to a classmate, to a roommate (in a different discipline) and to your grandmother or grandfather. That's no indictment of anybody's grandparent.

Good teaching is in essence effective communication, about our chosen subject, carefully targetted to the perceived audience. Correctly divining one's audience is one challenge; in education today the bigger challenge is lifting our student audience from where we find them to where they need to be. It's a continual battle, but one worth fighting.

— Colm Mulcahy