Practical mathematical ability, like other parts of the body (since it is encoded somewhere in the brain), atrophies over time. By the end of high school, I was able to score a 5 on the AP calculus BC exam, which allowed me to skip Calc I & II at Tulane. Freshman year, I learned more math, although I didn’t really like it and was going through a major adjustment to college life, and I didn’t pick it up very well. By sophomore year, I’d forgotten most of the integration tricks I knew, and some of the derivatives.
From time to time, I’ve re-learned this stuff. Experience (and scientific studies) have shown that it’s normal to re-learn things much more quickly that the initial learning time, which is nice. My research places me in kind of an odd spot though.
If I were doing purely theoretical work, I’d have pencil to paper all of the time, integrating, derivatizing (as we said in HS), solving differential equations, substituting variables, etc, etc. But I don’t do purely theoretical work.
On the other hand, were I an experimentalist or a clinician, my work would be much more hands-on, so my math skill would atrophy a great deal, but it wouldn’t matter as much.
I work somewhere in the middle. Our software is sufficiently advanced that my work is a lot like hands-on experimental work. It’s important for me to understand the underlying theory, and I do, but understanding it and sitting down to work out math problems (especially if you’ve gained the understanding by doing so already in the past) is a different story. I go through long periods without serious math work, punctuated by short bursts where I have to dig all the way back in. Lately it’s been for a class.
I used to feel guilty about this — I probably wouldn’t have even posted it on this blog a couple of years ago. However, after hearing many other academics — several of whom are much more advanced in their careers than me, and well respected — lament a similar problem, I no longer feel guilt. It’s a common problem. For professors that teach university classes, the problem is somewhat alleviated, as they have to teach classes on this stuff and it keeps them fresh.
Not to say that I want to have to teach any classes right now, mind you.
I’d be interested to hear if you have any suggestions for staying sharp when I’m in a mathematical lull.