At the start of every high school calculus class, Mr. Platt would ask which problems people wanted to see explained. Then he'd ask for volunteers to demonstrate their solutions to each of these problems. If a single requested problem went without a volunteer, he'd walk from desk to desk to check everyone's homework. As a class, we sought to avoid this outcome whenever possible. One day, all the problems save the last one were accounted for. I had a solution for it, and I was pretty sure it was right, but I wasn't confident in the technique I'd used. Nobody else was volunteering. Mr. Platt made ready to stand up. My classmates and I glanced nervously at each other. If I had anything at all worth explaining for that problem, it was my responsibility to keep him in that chair! So I raised my hand and wrote my solution on the board. When my turn came, I walked the class through my reasoning. Mr. Platt's annoyed response: “Your answer's correct, but we haven't done integration by parts yet.” (True enough, I had no idea what that meant.) And then he got up and checked all our homework assignments.
Moral of the story? Beats me. Suggestions welcome.
The results of my first Calc II midterm are in. I scored a solid B, which is better than I expected, and not shabby for my first math exam in 10 years, but nonetheless not to my satisfaction. There are only a few techniques of integration. To apply them intelligently is mostly a simple matter of pattern recognition. To recognize the patterns is a simple matter of practice, practice, practice. I've got my work cut out for me. If I want to be a physicist, math needs to flow in and out of my brain with facility, creativity, and correctness.
Whether this is even possible is something of a gamble. The last time it was true, I was a junior in high school. Such a different Amitai. The areas in which I've since grown are unquestionably valuable. I wouldn't go back and spend my time any differently. But my life as a mathophile (and polymath) suffered — irrevocably, in the sense that I can neither instantaneously summon my skills of yore nor make up for the lost time. In the sense that I might yet relearn what I've forgotten, including the love I once had for the discipline of mathematics, and make something of it? We'll see.
With or without the midterm score, early results are encouraging. There are typically two assignments per week, each of which has been taking me about 8 hours to think my way through: a traditional handwritten assignment due every Friday, and a web-based one every Monday or Tuesday. I figure out as much as I can on my own, then take the rest to the Barnard Math Help Room and pick their brains. It puts a grin on my face to be the needy beneficiary of a math Help Room. It'll amuse me even more if in a year or two I'm still hanging out there, doing the explaining.
A couple weeks ago, Prof. Ross was elucidating by an example the technique of integration by partial fractions. He started doing the obvious transformations toward a system of equations that'd give us the numerators of some simpler, more easily integrated fractions. I raised my hand and pointed out a simpler way to get there. The professor liked the idea, and since it took only a moment to do it my way, verified that the two methods were equivalent. Maybe I've still got the knack for this stuff.
I appreciate the design of his homework assignments. No matter what day of the week it is, there's always math to be doing; since I need the practice, this is great. The web assignments give limited but instant feedback: is my solution right or wrong? Many a problem I've retried and retried. A memorably simple one I managed to get wrong in 10 novel ways before finally solving it cleanly. (Which illustrates another nice property of the web assignments: to get full credit, all one has to do is work hard enough.) It's a bit of a high to chug through a couple new problems, feel good about them, dash over to the computer and type the solutions in, and be right. This is happening with increasing frequency.
Calculus is by far my most challenging course. Czech is no joke either, though. I'll probably have some doozies in the next 7 semesters, too.
Can I do the work, math and otherwise? Yes, I've been doing it. At the end of every day I feel I've done what was necessary and sufficient to stay on top of things. At the beginning of every day that feeling is gone, waiting to be re-earned.
Can I keep doing the work? We'll see. Ask again next year, and the year after that. But there is a key difference now vs. my last go-round with college: when studying, I haven't felt like I'd rather be doing something else. On the contrary, when I'm not studying, it feels wrong enough to get on my nerves. I've been regularly skipping extracurriculars in order to ensure sufficient quality study time.
Eventually math assignments might take less than 8 hours! That'll help too.
It's early yet, but thus far, Columbia's living up to the hype. It's a pleasure to be here, living up to more of mine.