ODEs and the meaning of life

What life lessons can you take away from an introductory ODE class? Does mathematical content implicitly promote a worldview? As a TA or instructor, to what extent is it appropriate to wax philosophical during class?

Recently, a friend sent me this adaptation of a Zen koan, originally about a rhinoceros fan, now about differential equations:

One day, Yanguan called to his assistant, “Bring me the ordinary differential equation.”
The assistant said, “It is broken.”
Yanguan said, “In that case, bring me the ordinary.”

I love teaching linear algebra, because I can occasionally drop hints about how out-there, imaginative, and self-transcendent modern algebra is. The content, I think, suggests a wonderful way of looking at the world. We study structure for its own sake. We explicitly establish assumptions, then rigorously break free from those assumptions. I think there are some hidden life lessons in a linear algebra class – about the value of playfulness, imagination, and mistakes. I’m not afraid to mention these during class.

Maybe these are messages you could take away from any math class, or really any college class. But something about linear algebra, for me, suggests these values naturally.

On the other hand, when I taught multivariable calculus, I found myself contemplating – alone and sometimes out loud in class – the nature of our spatial intuition and visual perception, our notions of space and dimension, and the power and utility of generalization in mathematics and in life.

But now I’m teaching introductory differential equations, and I’m struggling to figure out satisfying big-picture themes or values. I see a strong metaphor with cooking: in class we’re building a collection of recipes or routines to follow, when presented with various types of ODEs. That’s fine, but the students did the same thing when they learned about different integration techniques in Calc II. Someone suggested: the value of thinking both quantitatively and qualitatively – we try to arrive at specific solutions, but also classify families of general solutions and their global behaviors.

Please, if you can think of any others, I’d love to hear them.

Along these lines, I’ve been asking myself two questions: Does mathematical content implicitly promote a worldview? As a TA or instructor, to what extent is it appropriate to wax philosophical during class?

I think the answer to the first one is yes. I’ve ended up an algebraist and category theorist for a reason. In my opinion, those fields are founded on abstraction and imagination, and promote large-scale vision and curiosity – and this appeals to me. On the other hand, an introductory ODE class necessarily promotes pragmatism, over-simplification, and determinism, by relying so much on “real world” applications like idealized springs and wolf populations.

It’s the last one, determinism, that bugs me the most. Maybe I’m being to sensitive, but I cringe inside every time we arrive at a specific solution, describing exactly and perfectly the position of our mass on our string, for all time. I really feel impelled to talk about all the non-linear systems out there, and about relativity and quantum mechanics. But I can’t present these topics thoroughly, and fear that my students will walk away from the class with a 300 year-old Newtonian worldview. For me, determinism is the wrong way to think of the world.

Which brings up the second question: As a TA or instructor, to what extent is it appropriate to wax philosophical during class?

For example, the other day, in the few minutes before the bell rang and class started, I read a quote by Gandhi, about the importance of making mistakes and allowing each other to make mistakes. It took a minute, and didn’t take up class time.

Compared to humanities classes, in a math class there are perhaps fewer opportunities to express something big-picture. But there are opportunities. When presenting non-commutative matrix multiplication, you can mention ring theory and the value in isolating and extracting the essence of a concept or question. When presenting Stokes’, Green’s, and the divergence theorems in multivariable calculus, you can point out the power and utility of generalization. Or, when talking about springs and masses, you can go on a rant about determinism…

Sure, it’s probably not a good idea to spend too much class time on vague, imprecise opinions, right? But the larger themes are always the hardest to glimpse, and probably the most useful, interesting, and important things for the students to take away from the class.

Most importantly, I would argue that a teacher cannot avoid teaching about life. If you don’t do it explicitly and actively, then you are doing it implicitly and passively. So rather than ask if you should, ask what deeper lessons you could impart, and how. How would you like to affect your students?

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