This is not a text book for advance Maths, but I do claim that most of the content of the syllabus can be taught in a more imaginative and investigative way than the textbooks let on. These are just a few examples which I expect will grow.
An investigation to explore scalar products intuitively. It should lead to understanding of the mail properties, and may lead to a proof of those properties.
Logs tend to confuse students for a variety if reasons. The difference in the notation we use between two inverse functions -exponential and log- makes the connection difficult to grasp. I have found that this is a topic that is susceptible to the Tutor Problem, on which there will appear a post soon. Teaching logs without calling them logs enables us to reintroduce the discovery and investigation elements that enable understanding and which the tutors often remove. This activity is fun, but allows all the key rules of logs to come out in natural way.
How to introduce integration in a way that students will understand it as a summation process with a limit, rather than a trick of anti-differentiation. The Fundamental Theorem should come out as a property rather than a definition. It was, after all, thought of first – see Archimedes “The Method”.
This way of teaching also subverts the negative effects of the tutor. This is a personal crusade!
A colleague, Jennie Wathall, asked me how I explained the Fundamental Theorem to students. This provoked a lot of thought, as I realised that I didn’t really explain it at all. It comes out of the way I teach calculus as an observation, but we do not really get into the Why? It seemed to me that an item, in a paper I once read, about the number of Elvis Impersonators could serve as an inspiration to understand the relationship between rate of change and accumulated values. This short discussion has an activity at the end that explores the notation and language we use in calculus.