Today, I met with a student at a local library for our first session. She is a college student taking Calc II at a local JC over the summer, and had run into a bit of a snag with it. It seems that she hasn't taken many math courses recently, so she was a bit rusty, and she has a test next week. Well, I was running a bit late, as I had not factored quite enough time for the rush hour traffic and an accident which blocked half of the freeway, but called ahead to let her know I would be there about 10 minute late. I finally got to the library, which was nice and big. After I found where she was, we found a study room where we could get down to work.
First of all, I really like tutoring math, and anything from calc II down I feel like I have a pretty good handle on, and can usually get through most problems no problem without a book handy, though I always take one with me just in case I forget something or to check myself. Today, we were going over integration by parts, trig integrals, and trig substitution. Integration by parts is something that always seemed to give students trouble, because they either forget the formula or they don't really understand what it does or why it works. I started off by going through a pretty basic derivation of the formula using the product rule, but it seems that she is allowed to use all of their formula sheets for their tests, and that their final grade is based solely off their test grades! Now I know why all my friends want to take math at the CC! Well, I think I was able to make some headway once I determined that the formulas and the understanding of the calculus wasn't her problem as much as the algebra. I think she understood better once I moved on to help her with her algebra.
Next, we got to trig integration, and while we didn't have a really hard problem, it sure felt like it was. It was just a run-of-the-mill sine function raised to an even power with a cosine function to an odd power, but her formula sheet she was trying to use had something about "if the power m-1, where m is even....use sin^2 + cos ^2 = 1", and another one about using the double-angle formulas. Well, it confused me, because I just learned it the easy way. Whichever function has an odd numbered power, save one of the them, and use the trig identity to substitute for the rest. If they are both even, then use the double-angle formula. If they are both odd, pick one and substitute. Well, I don't think she understood how the substitution was really working until I forced her to look at the integral and said, "If I had [sin(4x)]^4* cos(4x), I could just pick u = sin (4x), du = 4cos(4x), and were good. But we have [sin(4x)]^4*[cos(4x)]^3, so we have to substitute for two of the cosines so we are left with just one, then we can do the u substitution." It never ceases to amaze me how much easier math is when you know why you are doing something as opposed to just knowing how to do something.
Well, after that, we did another problem with trig substitution, which for most people, even me, is about as much fun as a root canal. This one wasn't too bad though, but the problem required that we do a trig substitution, then a u substitution, and then re-substitute to get our final answer in terms of x. Lots of algebra, and lots of places to make mistakes. I even goofed on the coefficient, leaving in one too many 3's in the denominator. It was more difficult, I think, because I was late and felt like I needed to rush to get through the material. Also, as she was a college senior, I might have slightly overestimated how quickly she would follow the material as I tried to explain it to her.
I guess in the end, we may not have been quite the right fit for tutor and student. I think she got some benefit from the lesson, but I could have done a better job of relating the material to her. Of course, the first day is always tough, since you are kind of flying blind, not really sure how someone's mind works, and only by hunting and pecking away at spots can you get an idea of how their comprehension is different than yours, so that you can help to correct any misconceptions that are keeping the student from reconciling their old learned material with the new, unlearned ideas. I won't be surprised if I don't get a second chance to help her before her test, but I hope that the next time I am able to get her to do more work, and prove to herself that she gets it before she goes on to the next problem.