My new course “Mathematical Structures” begins next week. (This is my slightly perverse way of celebrating my retirement.)
Every week, after the lectures, I will post the lecture notes and supplementary material on the course web page. This will be done in the by-now old-fashioned way, on a public web page rather than on a closed ILE page or whatever the fashionable acronym is.
I hope that this experiment will be of some interest to anyone who is involved with teaching or learning mathematics at first-year undergraduate level. We have found that there is an increasing disconnection between what our students learn in the exam-centred environment at school and what we would like them to know to tackle the study of university mathematics. So this course is at least as much about raising the students’ game as it is about actual mathematical content.
There are some other new features about the course (at least as far as Queen Mary is concerned): the tutorials will be in small groups, with the academic adviser as tutor; and the students will be giving presentations to their tutorial groups.
In an informal document to the course tutors, I described the course as having three main aims:
- to introduce some of the basic objects of mathematics (sets, functions and numbers);
- to get across to the students some good mathematical habits, such as using definitions and how to construct proofs;
- to make the students enthusiastic about our beautiful subject.
As the course progresses, we will see to what extent these aims have been achieved!
The notes will be in ten chapters, and will be posted week by week on the course page, except for week 7 when we have no lectures. I will also try to post a brief discussion here about what that section of the notes is trying to achieve and why I made some of the decisions I did. In my Gresham College lecture next year on 14 May, I want to look back over the process and provide some kind of summing-up. The week-by-week plan goes as follows:
- Functions and relations
- Natural numbers
- Integers and rational numbers
- Real numbers
- Complex numbers
- Constructing and debugging proofs
Each chapter of the notes will also include a section on study skills, and
will be accompanied by supplementary material.
I would be very interested in your comments on the course material, though I don’t promise to act on them! Please bear in mind, as you read what I have written, that the slogan of the course (taken from T. S. Eliot) is “precise but not pedantic”. So, for example, real numbers will not be defined by Dedekind cuts or by Cauchy sequences. This doesn’t mean that I think there is anything wrong with these constructions.
Naturally, if you are a student taking the course, and find your way here, your opinions are especially welcome and valuable!