Math as Incantation

Our department recently hosted a series of talks by graduate and undergraduate students. I managed to participate1 in the graduate student portion. I gave a mini-rant on the overuse of the phrase 'big data' (even though I included it in my title), and then presented a gentle introduction into computational mechanics. Overall, presenting was a very positive experience. I tried to follow some best practices, despite my crippling monotone.

The undergraduates spoke as part of the Math departments Directed Reading Program. We had a good mix of topics: variational calculus, compressive sensing, the algebra of the Rubik's cube, the list goes on. The beauty of studying mathematics is that everyone can do something completely different. And they all did an amazing job. They would have blow undergraduate-me away.

It was interesting viewing undergraduate presentations from the perspective of a graduate student. I've traced my first undergraduate presentation to a course titled, appropriately, Effective Communication for Chemists. I talked about evolutionary psychology and theories positing that the human mind evolved to its current greatness so that guys and gals could get laid. As you do when you're a sophomore in college.

Thinking back on my undergraduate career, I gave very few talks. At least, in comparison to the three years I've now spent in graduate school. Part of that is because of the Scientific Computing project course sequence. But also just the new desire to share cool new ideas with others.

While watching the undergraduate talks, I noticed something weird. A lot of the students, especially the freshman, would take to reading equations off their slides, almost as an incantation. For example, in a talk about Black-Scholes, they might have a slide with the following expression:

\[ \frac{1}{\sigma \sqrt{T - t}} \left[ \log \left( \frac{S}{K} \right) + \left( r + \frac{\sigma^{2}}{2} \right) (T - t) \right]\]

Never mind what any of this means2. In a graduate talk, we would speak around this expression. We might point out some key terms, to give a flavor, but we would never just read out the expression.

But a few of the undergraduates did this. Which, in retrospect, seems only natural. After a decent amount of training, you start to see a mathematical expression as a whole. But until then, you have to look at it a piece at a time, so it only seems natural that this is how you should present it to your audience.

The students who took this approach clearly knew what they were talking about. But they hadn't developed the fluency, or trust in our fluency, to skip past the phonics approach and embrace a whole language approach.

I wonder at what point in ones education that leap occurs? I certainly don't recall when it happened for me, but I know it must have.

Fortunately, this is entirely different from the cargo cult mathematics that I often see practiced. That's an entirely different 'mathematics as incantation.' None of the talks this year suffered from that. Which I take a very good sign of health for our department.

  1. This presentation will not stand alone. But I take that as a point of pride with my talks. So... Good?

  2. If you want to learn how to ruin the economy, you can read more here.