Twenty-five years after I took Calculus II, I am enrolled in Calculus III at City College. I've heard from many students that the instructor is one of the best in the program — he is leading one of ten sections this semester and I drew him by chance. Below are some observations recently sent to a few correspondents.
The instructor has said firmly:
The more problems you do, the more shortcuts you will figure out for yourself. The only way you can do this is to do a ridiculous number of problems. You must work almost every day on this stuff, for a couple of hours a day.
Homework is not graded, but we are supposed to keep a special notebook in which "assignments", meaning hard problems apart from the homework proper, are to be done — and those he will examine on occasion. If we've done them. On Thursday morning we were supposed to visit his office before class to show him that we do, indeed, possess such a notebook. The point was to encourage students to make the effort to identify a notebook for those problems early in the term. I was the only one who came, though, and I'm going to be using LaTeX, with his blessing.
The assignment sheet for the semester went up just before class on Tuesday and he announced it in class and told us to do the first three sections, 23 problems in all, mostly vectors, which are not well handled in LaTeX. I spent six hours doing the first assignment and got through 19 of the 23; the last four remain undone several days later, despite my good intentions. On the day they were official due, several of the students were still asking him when he would post the problem sets.
This instructor says he is good at solving problems but his memory is terrible, so he has never learned LaTeX. Instead, he uses MathType, which provides a GUI. It's a pity, because a math course is really the ideal place to introduce LaTeX and guide students in elementary use of it.
The instructor made a pitch to interest me in abstract algebra. I must admit, what he showed me seemed quite interesting and intuitively clear. I have yet to understand what the place of math will ultimately be in my life — I only know I am not yet done with this question.
I am the only student in the class of around 30 who is taking notes on a computer. Some students do open up a computer briefly, but it seems to have something to do with messaging. I ran into a little trouble today with LaTeX because I had anticipated that we was going to introduce determinants, which I haven't learned how to handle yet (actually it's not hard, I now see — the amsmath package has everything I'm likely to need).
A correspondent, seeing the comment about doing "a ridiculous number of problems," replied:
That sounds reasonable to me. The "problem" is that the advent of software like Wolfram|Alpha removes any real usefulness from this kind of skill ... it is now a purely aesthetic amusement.
But I disagree. Skill brings understanding, and understanding leads to insight into other things whose existence you can predict but whose content and requirements you can't easily anticipate. Much of the mathematical and theoretical component of the computer science education at City College consists of exposure to proofs, or to things like proofs such as building a linked list in C++, and so on. It is not as though we will ever need to build our own linked lists or derive Chebyshev's inequality. But struggling to produce them myself helps me to understand and retain them, and there is considerable value in that.