No Limits

After nearly 35 years of teaching, I think I finally get it. Sometimes we have to tell students what we want them to know. I don’t mean math or music theory or macroeconomics. I’m talking about the really big-picture stuff: all the broader skills, life lessons, etc., that we believe undergird a liberal education.

Throughout my teaching career I’ve acknowledged to students the power and effectiveness of studying mathematics as a way of developing thinking skills. I’m very explicit about this at the start of every course: Studying math enhances your ability to think logically and solve complex problems. Math makes you strong, and employers and even law school deans love math majors, etc. I repeat this homily at the end of each term as part of my general review, then send the students on their way.

In recent years, I’ve realized that I miss many golden opportunities to reinforce these lessons in class throughout the term. Smart as our students are, I think they get the message much more clearly if we tell them in the moment, rather than assume they will make the connections themselves or that they will absorb them subconsciously.

Being more explicit about these broader lessons is particularly important in my department, at least on the math side. For instance, many students take calculus, but only a few will use it down the road. Yes, there are benefits to seeing and practicing calculus even if you never use it, for calculus, is one of the great achievements of the human intellect. But what life lessons are imparted by studying calculus?

The fundamental idea of calculus is something called the limit, on which is based the derivative (measuring instantaneous rate of change) and the integral (measuring total or accumulated change). One interpretation of the integral is that it is a measure of the area under a curve. In a classic tradition of teaching mathematics, we consider an example:

In Math 25, one example presents a region bounded by the curve y = 1/x² extending infinitely along the x-axis. Students easily see that the region has 

infinite perimeter. But by careful use of the idea of limit and the definition of the integral, we determine that the area of this unbounded region is actually finite. Wow! This is a great moment in class. An unbounded region with infinite perimeter can have finite area. Calculus is mind-boggling! But what are the life lessons here? 

• Think beyond what seems possible or normative.

• Challenge your intuition.

• Use language carefully and use reasoning consistently to give meaning and clarity to ideas and concepts that might otherwise seem impossible to comprehend.

But does pointing out these broader lessons in class in the moment make a difference? It’s hard to say, since I have no formal data. I have asked students to reflect on takeaway lessons from my courses, which I think helps to reinforce the idea. Here are exit responses from a couple of my Math 25 students:

“Math can be counterintuitive. Life can be counterintuitive. And in that, there lies beauty!”

“Nothing is impossible, especially when something can have finite area but infinite perimeter.” 

These are reassuring sentiments. Perhaps students would come to these conclusions without my deliberate prodding or create lessons that are personally meaningful, but I find that taking moments in class to broaden everyone’s perspective to be both satisfying and liberating. For anyone concerned that I am not tending enough to mathematics itself, be assured that life lessons take only moments to acknowledge. Perhaps for many students those lessons are longer lasting than limits, derivatives, and integrals. 

—Deb Bergstrand is a professor of mathematics and statistics.