I am in Austria right now; having spent a week in Innsbruck visiting Hans-Peter Schroeker and Manfred Husty, I am now in Graz at the Technical University, and I am talking with Anton Gfrerrer, Sybille Mick, Johannes Wallner and Johann Lang of the Institute for Geometry, and giving some talks. Central Europe has a long and distinguished history of excellence in research and teaching of geometry. Here in Austria students learn Descriptive Geometry, Hyperbolic Geometry, Projective Geometry and CAD systems for visualization: if only Australian students were exposed to half as much!!
Its a pleasure to be in this part of the world, both personally and academically. My father is from Austria, born in Linz, and my mother from Liechtenstein, and so I have been travelling to this part of the world on and off for many years visiting relatives. Here is a picture from Triesenberg in Liechtenstein, where I stayed for a few days visiting my aunt Maria, who lives a few kilometers down the mountainside.
Here is a shot from a town in Austria called Landeck where I overnighted on the way to Innsbruck. Some lovely walks around the countryside there. And I suppose I better show you a mountain, of which there are many around, but sadly the weather has not been entirely cooperative for sunny photos. This taken from the train on the way to Graz from Innsbruck.
Today I want to tackle the challenge of succinctly summarizing some essential features, in my opinion, of effective tertiary level mathematics education. This follows naturally from my last blog on MOOCS. Of course there is much to be said here, but suppose we had to just write a paragraph or two: an excellent exercise for focusing one’s thoughts.
Putting together an effective university level mathematics course requires:
1. A prior solid understanding of the mathematical content of the subject, and its connections and application to other areas within and outside of mathematics.
2. A carefully chosen syllabus that lays out a logical sequence of topics: not too many, not too few, pitched at the right level for the intended audience.
3. A written text, either notes or a book, which covers in detail the syllabus of the course, including a wide variety of examples. This can of course be online.
4. A series of lectures, given either live or via video, which explain the course content but perhaps do not go into quite as much detail as the written notes. These lectures should be obviously accessible, interesting and useful to the students for learning the material.
5. A complete and comprehensive collection of exercises for students to attempt. These should help students gain familiarity and mastery of the course content, to develop problem solving ability, and to spark further interest in other aspects of the subject. The exercises may possibly be organized into various levels of difficulty if appropriate.
6. A carefully prepared set of worked solutions to many of the exercises, and summary answers to the rest. This could be either in written or video form, or both.
7. A mechanism for grading student work at solving exercises and writing up solutions, and providing a reasonable level of feedback on their written work.
8. Effective and fair tests and final exam, that motivate students to study, review and ultimately absorb the material.
So if you can manage to incorporate all these aspects in a coherent way, you will for sure have an effective mathematics course.
And how much of this could be done on-line? Almost all of it, with the important exceptions of 7, and possibly 8. This is the key challenge in setting up online courses in mathematics—(in fact also of regular courses at university level!)—how to provide good feedback to students on the exercises that they ought to tackle.