The week passed roughly according to my plan. The guiding problem, which is central to my approach, worked almost as intended. The students worked on it in groups on Monday and Tuesday. On Wednesday I had planned an extra algebra session. Since there were so many good questions on Tuesday about power functions, I had to use some of the Wednesday time for polynomials to keep up the pace. Teaching in the way I do invites questions and these questions reveal the depths of possible misunderstandings. Not tackling them is no way forward.
I also talked (on Wednesday as planned) about the axioms for the real numbers as the basis for algebra. And I had time to talk about irrational numbers and the fact that you need an algorithm in order to calculate the square-root of 2, because √2 is just a symbol satisfying √2·√2=2. I used an algorithm to compute successive approximations. So it all became very concrete. I think it worked well.
On Thursday, I decided to use all the time for a thorough discussion about exponential functions and logarithms, rather than having group time on the guiding problem.
A difference between Swedish university and American college is the fact that it is not compulsory to attend class (except laboratory classes and such). The students may come and go as they please (which they do). This means that if you start out with say 40 students in a class, the number actually attending drops down to say 25 after about two weeks. The half-life for group work of the kind I’m trying to have is also on the order of a few days. This is something you have to live with in Sweden. The teaching environment of college is more like the Swedish gymnasium. But I will continue with the guiding problems and try to convince them about to usefulness of them.
On Friday I ended with the trigonometric functions and we discussed the guiding problem. So we had a detailed run through of the elementary functions. And we had an outlook towards the next week which will be about the derivative.