G1M91 Ideas in Mathematics 1995/6

This is an unusual course which has been developed and is still being developed as a result of experimentation and discussion over many years. Because of its unusual nature, the basic philosophy and evaluation is here presented in fuller terms than would be sensible for the majority of courses.

Overall aims of the course

This course is clearly different from others at this stage, and it has different aims. To what extent these should, can, and are achieved remains a subject of debate.

The course attempts to begin to tackle problems found by most students of mathematics. These can be summarised as follows.

It has been agreed for a long time in the School that these are important issues. It is not expected that students will suddenly be totally changed by such discussions, but it is expected that it will help them over the years to consider these ideas.

It is also agreed that such discussions are especially important for the wide range of ability that we take: many students do badly because they are confused about what is going on. Clarity on this can help to compensate for a slower speed of learning or slower degree of retention.

It is also important for the more able students to appreciate the role of mathematics, and to study its methodology. This view may not be popular, but we have the support of Einstein for it!

It has been agreed that these issues are important enough for them to be addressed at the start of the  degree course, so that the awareness of them can benefit students throughout the whole course.

On the other hand, it is not expected that we have the final solution to addressing these problems, and so the course is subject to regular evaluation, review and revision.

Training in writing good mathematics

This problem was tackled by assigning to each student a question from Part A of a previous WJEC Single Subject Mathematics A-level paper. It was originally intended to use a Part B question, but it was decided, fortunately, to start with a Part A question, and no Part B questions were ever used. Each student was handed a different question, and they were entitled to turn down a question if they felt they were not familiar with the subject matter. This recognises that not all A-level syllabuses are identical, and not all material in the A-level syllabus is covered at school.

Students were asked to hand in questions, faults in these were marked, and the solutions went to and fro till I was satisfied with the result. The assessment was originally that the assessed work on the other assignments would be increased by 5%, and this was later increased to 10% when the work involved was realised.

One surprise to me was the amount of work that was often expected at A-level for the 5 to 7 marks for this type of question. On the other hand, a few of the questions did not give much opportunity to display good exposition, and this just had to be accepted, with the matter being discussed with the student.

I was very pleased with the way the exposition improved, and I welcomed the opportunity to discuss these points of exposition, and to explain what I was looking for.

This was a new experiment. I would rate it as successful, and definitely worth repeating.

Training in logic

In order to write good mathematics, the basic principles of logic have to be understood, and these are not part of the A-level syllabus. So the course started with an exposition and exercises on the meanings and use of the usual logical terms, and the use of definitions. These ideas were illustrated with the notion of neighbourhood of a point on the real line.

This area is necessary to be treated in the first year courses, but was probably not particularly exciting. It is conveniently placed here, particularly as ideas of logic are related to the "inner issues in mathematics", as discussed in Davis and Hersh.

Problem solving

Students tend to come up to University thinking of mathematics as a series of standard techniques to be learned and used, and they have little knowledge of the strategy for dealing with a new situation. They often have little awareness of how to use the knowledge that has been presented to them in lectures, unless they can actually remember it at the time a problem is presented. They can be unfamiliar with the idea that writing out the question, copying out possibly related ideas, jotting down ideas you might have, are all part of the basic procedures of a mathematician, and indeed are useful in a variety of situations. The aim of this part of the course was to discuss these approaches to problem solving skills, and to illustrate them.

This part of the course could well be doubled in time. A previous year had five weeks on problem solving. There is a problem on assessment, which should for this be based on more on a problem solving diary type of approach. This has been used for a third year course on problem solving at Southampton.

Discussion of these issues would seem to be essential for first year students.

Wider contexts

This was tackled by setting essay type questions which could be better done if students looked up some books. Students are often criticised for not reading, yet they often know that what is required to be known will be written up on the board at one time or another, so that obtaining more from the library is not necessarily a sensible procedure. Our experience is that if reading is clearly required, and contributes to the assessment, then it will be done, with the usual variation from student to student.

Books referred to in student assignments for this course included the set texts

The notion of space

The aim of this section of the course was to show mathematics involved in the extension of the imagination through the development of new concepts, and that the notion of mathematical space as the repository of motion, or change of data, is important.

Two relevant graphics were shown: the Banchoff/Strauss film of the hypercube, and the Brown/Sen video on "Pivoted lines and the Mobius band".

Departing from previous years, the hypercube film was shown after students had been introduced to and worked through some of the mathematics. The film is known to be quick, and previous familiarity with the concepts helped students to understand and learn from the graphics.

They found it less easy to appreciate the video and the accompanying Dirac string trick. I suspect there are two aspects:

  1. Though it may be technically convenient, it is probably not a good idea to mix the two graphics, which involve quite different mathematical ideas, though still with the underlying theme of representing motion.
  2. The accompanying mathematics of the video was not developed (rotations, axis of rotations, representation by projective space, and so on, ..) so that they found it more difficult to relate to. None the less, the fact that it is not appreciated now does not mean it, and the accompanying trick, will not be remembered. I would conclude that if it is to be used, then the accompanying mathematics needs to be developed before a showing of the graphics. Graphical demonstrations need to be more extensive and better developed if they are to be understood by themselves. In any case, it is important to develop the relation of the graphics with the accompanying mathematics.

Judging from the assignment, this part of the course was appreciated, but the use of polynomial generating functions for determining the general number of r-faces of an n-cube was too difficult, as was the accompanying exercise. (One student got it all out, but most did nothing.)

This part of the course continued with a too brief discussion of the chapters in Davis and Hersh on "Inner issues". The success of this might be rated at 50% (it got over to some but not to all). Nonetheless, this topic is the core of the course, and really needs more time, more discussion by and with students, and more assessment. The aim of the course is also to make students aware of these issues, as a preparation for the rest of the degree course, and for future employment.

Assignment marking

Students do need training in tackling questions outside their apparent ability, and in learning methods for coping with this situation. This is related to problem solving techniques.

The setting of more difficult problems has however a danger if the average continuous assessment mark for courses in the department is fairly high, and indeed for most of the courses rightly set high. Students need assessment of basic routine skills, and do need their confidence to be developed. On the other hand, they also need to be prepared for the more testing work in the following year, and this can be done only by giving them more difficult work which they are likely to fail to complete. This is the basic dilemma.

The method adopted to deal with this problem is to assess assignments with both a completion factor and a difficulty factor. This allows for a more experimental use of questions, and allows for difficulties students find with a new type of question.

This also accords with the teaching philosophy of high expectations combined with an acceptance of slow or small movement towards this expectation.

Grading of essay type questions

The problems this present are:

On the other hand the aims of these essay questions are fairly modest, partly because of the problems mentioned above:

For all these reasons the grading is largely judgmental, and represents mainly an assessment of work done and maturity of views.

It is also a general rule that the more "accurate" the assessment, then the less interesting are the qualities being assessed. It is, for example, easy to assess students' ability to add and multiply natural numbers, but such an ability is only a minor part of an undergraduate education.

We come back to the old distinction between art and craft, which in music comes to the distinction between technique and musicality. The aim of this course is to pay some attention to the development of the mathematical equivalent of musicality, and to link it with technique. There is a lot of work still to be done to decide how best to achieve these aims, and to decide how much this kind of teaching can and should contribute to students' mathematical development.

Changes in later years

Various experiments were tried. In one year I tried out with them presentations from the MasterClass for 13 year olds we have been running in the area since 1985. On the whole the students did not do as well as the 13 year olds! One lesson was a lead up to counting the number of 2-dimensional faces of a 5 dimensional cube: one student loved the idea of higher dimensions and she went on to do research in higher dimensional algebra. This suggests the importance of having some first year course which aim at engaging the imagination and showing that new ideas are going on in mathematics. In another year, a topic covered was some Euclidean Geometry. There were several reasons for this.
  1. Euclidean Geometry is part of the heritage of mathematics, and students should be aware of some basic results, such as why the angle in a semicircle is a right angle.
  2. In this area, one can prove easily some surprising facts, and this justifies the notion of proof. For example let ABC be a triangle, and choose points D,E,F on BC, CA, AB repectively. Then the circumcircles of AEF, BFD, CDE meet at a point.
  3. Situations can be developed to more complicated ones, for example in the last situation one can prove that the triangle formed by the centres of these three circles is similat to ABC.
All this is part of the aim of Popularising Mathematics to students: see the article Promoting Mathematics

See also the article `The methodology of mathematics', R. Brown and T. Porter, Math. Gazette, July 1995.

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February 11, 2012