I met this week with nine undergraduates, all of whom have elected to take the DAMTP’s course in Mathematical Biology. They came in pairs (except one), and with them a deluge of problem sets to mark. On the whole, their math was good, but one bit was largely missing: interpretation. The problems had to do with real-world systems, such as competing insect populations and over-fishing in lakes, but their answers looked more like “period = 4 +/- 2n” than “Stop fishing now!!!”. I assumed, initially, that it must have been a simple lack of time that prevented them from writing the implications of their answers on paper. These students are busy, after all. When I met my supervisees in person, however, I realized that interpretation had taken such a mental back seat that it somehow missed the bus entirely.
One student made a comment that particularly struck me. A self-proclaimed "pure-mo" (pure mathematician), his mathematics was pristine, but I was forced to leave a glaring “Interpretation?” with my green pen beneath each of his answers. During his supervision, it took real effort to coax real-world ramifications from him ("how would this affect a strategy to re-stock a lake? to treat a blood disorder? When can we guarantee the rabbits will survive?” - silence), not because he was unwilling to work with me, but because he missed some crucial link between the figures on his page and the outside world. “I’m afraid the course might get too applied,” he confessed, clearly referring to his difficulty in interpreting answers. As someone who has spent most of his mathematical career with a fear of the opposite sort, that gave me pause. What could make this student so averse to interpretation?
I think the answer lies in a particular - and problematic - understanding of mathematics as a whole. To many of the students with whom I’ve worked, solving a problem is fundamentally a sort of ‘closing.’ Mathematics to them is a sport, a set of puzzles to be worked out. The fact that we might derive some significance from those results is a passing curiosity, accidental to the game itself. When the problem is solved and the answer boxed, they are done, and can move on to the next contest. I argue that the match hasn’t even begun - that the real mathematics happens when they sit back, take a moment to let the result sink in, and realize what their boxed answer means. The best mathematicians I’ve known, both pure and applied, are masters of interpretation. True, the pure mathematician prefers to answer ‘how does this reshape my understanding of mathematics?” while the applied mathematician tends to answer ‘how does this reshape my understanding of the world?” but the distinction, I argue, is of secondary importance, and indeed, all mathematicians should hold both questions in mind. Interpretation, of any sort, inverts the problem-solving act from a closure to an opening. By interpreting a result, we not only strengthen our foundation of understanding, but we also open ourselves up to the possibility of being surprised - surprised by, say, an inconsistent result, or an unexpected similarity to another field. This opens up new questions, new possibilities, new modes of thought, and ultimately breathes life into a field.
The beauty of Mathematical Biology, and of applied courses in general, is that physical explanations are often the easiest ways to entice students to begin the practice of interpretation. They care about the world in which they live, so when they realize that an esoteric math problem can help them better understand their surroundings, they get hooked. I’ve seen the fire light in their eyes, and it won’t be going out soon. We who have taken a responsibility to educate students need to bear in mind that our role is not only to help them master theory, but to assist them in drawing ever-deeper meaning from the questions they answer, and in that way, to fan the flames.