Tutoring Physics

Just fine as long as it is not advanced

Curiously enough, these days I find myself tutoring Physics sometimes.
Three students I previously tutored in Calculus, as well as one I am currently tutoring for the GRE, all of them have recently asked me to help them with Physics.
The first time it was a surprise for me because my student called me over the phone and all she said was she wanted to schedule an appointment. I just assumed she was taking another calculus class. When we started the session I looked at her book and said, “This is Physics”! Her only reply was a monosyllable: “Yeah,” along with a completely natural, matter-of-fact look on her face. I realized she expected me to know Physics so I went ahead and helped her with the problems she had to study for her exam, and we got all of them right! It was a very nice surprise for me. I thought it funny that it was not a surprise for her because all along she simply had assumed I knew Physics. Fortunately I did not disappoint her.
So far I have been able to successfully help these four students except for the last session with one of them, who is taking a Static Mechanics class at UCSD for his engineering major. Most college courses are packed full with topics to cover, and professors typically move very fast through the material. I had no problem helping this student with the first few topics, including up to finding momentums of forces in three dimensions but when we got to the chapter on force couples and reactions in systems at equilibrium I realized it is going to take me a while to figure it out.
The reason is at first any new subject seems esoteric to me until I find the meaning of its concepts, and the reasons for each step in its particular problem-solving processes.
Most likely I wont be able to go any further right now with this student but chances are I will soon find time to do some research, and I will be better prepared on this Static Mechanics Physics subject the next time around.

Integration by Parts in Calculus 2

It is really neat when students make their own discoveries.

This morning I had a Calculus 2 tutoring session about Integration by Parts.
My student was having some trouble with those integrals that are calculated by applying the integration-by-parts formula multiple times.
In this type of problem you get somewhat complicated expressions with parenthesis nested inside parenthesis, often with multiplying coefficients and/or negative signs in front of each parenthesis.
Students usually get confused when doing these integrals for the first time. The main reason is because they do not expect so much complexity. They believe the problems are going to be shorter than they actually turn out. So one key for getting these integrals right is to keep track of every step separately, identifying each new integral, and labeling it with a new variable of its own (using I, I1, I2, I3, ... works well). Then you keep working all the way down until you finally reach an integral you can actually solve, without any new “left-over” integrals. Then you re-trace your steps back one at a time, substituting each (ever longer) expression into the corresponding place for the previous partially solved integral, until you get to the original one.
We were working with the function (x^3)(e^x), where you start by deriving x^3, and integrating e^x.
My student understood everything but he wanted to make sure he would remember it later, so he started all over again. I suggested this time to start from the bottom up, so he integrated x(e^x), then (x^2)(e^x), and then (x^3)(e^x). I suggested for him to keep going so he integrated (x^4)(e^x).
At this point I said he could even memorize all the resulting formulas just for the test but it did not seem a good idea to him. Then I asked: “Well, maybe there is a pattern here. Do you see after factoring out e^x the last number (the constant term in the polynomial factor) is a factorial? And the polynomial always starts with the power of x we have in the original integral, does it not?”
I started looking for a way to factor out the polynomials. It turned out to be not always possible (the second-degree one already had complex roots) but all of a sudden my student exclaimed: “They are all derivatives!” So the pattern was not apparently multiplicative, but one more closely related to calculus, because it involves successive derivatives.
The best part here is my student saw it himself without my help. He realized what the pattern was before me.
That really made my day. I enjoy watching students making their own discoveries. This was not a textbook exercise, but a question I came up with after we went beyond the examples in the book, and my student was able to come up with the answer faster than me, showing he really understood the question. And now he has a very effective reference point to remember all those integrals if they show up in his exam! He won’t forget it because he discovered it on his own.