Discussion meeting: Are we maintaining standards?

Tuesday June 23, 10.00-11.30, Room B3.02

Speakers (briefly): Derek Holt, Mario Micallef, Claude Baesens

Our MA3 and MA4 modules cater to a wide range of students: 3rd year BSc students, 3rd and 4th year MMath students, and MSc students in the case of the MA4 modules. Does this work? Do we succeed both in challenging the MMath students and simultaneously giving the 3rd year BSc students something they can cope with? Or do we achieve an apparently satisfactory average only by giving the MMath students material that is too easy and the 3rd year BSc students material that is too hard?

Organisers: Derek Holt   David Mond

Before the meeting, two histograms, prepared by Annette Anderson, were circulated. They show the performance of the different groups of students (3rd year BSc, 3rd year MMath, 4th year MMath, MSc) on 3rd year modules and 4th year modules in 2007-8.

Here also are the corresponding histograms for 3rd year and 4th year modules in 2008-9.

Report:

After some discussion of the shortcomings of the current arrangement, Colin Sparrow made a concrete proposal: we should eliminate the requirement that MMath students must take two MA4 modules in their third year. At this point a show of hands gave strong support to the proposal, with (roughly) twelve in favour and four against. Some of the arguments for and against were then listed on the blackboard, and are reproduced here.

Arguments for:

1. Simplicity and clarity.

2. The straw poll!

3. It would enable MA4 modules to aim more clearly, both as regards their level and their sequence.

4. Some second years are currently scared into leaving the MMath by the prospect of having to take two hard MA4 modules in their third year.

5. MA4 modules aimed at third years have to take place in term 2, in order to allow students some experience of their third year before plunging into the difficult modules. This is a problem for fourth year and MSc students, since it means they cannot use these particular MA4 modules as preparation for other term two modules. [Of course, some MA4 modules will have to be taught in term 2, but they should not be the ones most evidently useful as preparation for later modules, which is what the ones specially recommended to third years are, by and large.]

Arguments against

1. There would be a logistical nightmare at the end of the second year for the students on the borderline between continuing on the MMath and reverting to the BSc.

2. The current system obliges MMath students to do some hard modules early on, and this is a good thing.

3. The lack of distinction between BSc and MMath stream in the third year would make it easier for MMath students to decide to graduate at the end of their third year. Without the hard MA4 modules to bring their third year marks down, MMath students might see an advantage to graduating at the end of their third year. The incentive to do so might even be increased by the fact that in their fourth year they would have to take all five of the MA4-level modules that (national) regulations of the MMath degree require. Once such a trend is established, it would be very difficult to reverse.

4. The problem is not that the MMath students find the MA4 modules they take in their third year too easy; it is that there are not enough accessible third year modules for the BSc students. If there were more `capstone' modules providing a suitable finale to the BSc, then the current problems, such as they are, would be solved.

5. [added by Dave Wood, who was unable to attend] If there is a student who has ill-advisedly stayed on the MMath and they totally bomb in their fourth year, there is not the option for them to graduate with a BSc at that point. Currently if a 3rd year MMath student does terribly in MA4 modules in their 3rd year they are given the option to graduate with a BSc. and get out before it's too late. If any changes such as this are made, we would have to place a lot stricter controls on who is allowed to stay on the MMath.

Conclusion John Jones, Derek Holt and Colm Connaughton volunteered to think through the arguments and draft a proposal for change. Dmitry Rumynin, John Cremona and Diane Maclagan agreed to work on an alternative proposal, along the lines of counter-argument number 4, above.

1. (Dmitry Rumynin) One particularly useful proposal (not reflected on the summary above) was by Mario "to introduce more fun MA3*** modules".

I'd like to stimulate a discussion within various groups and solicit people's opinion on the matter. Here I just repeat the main points of the proposal.

PROBLEM: Think of a weaker 3rd B.Sc. Mathematics student in Warwick. His/er typical 3rd year curriculum consists of Introduction to Topology, Knot Theory and History of Mathematics padded with LL133, IB215 and several MA2***. Is it possible to offer more 3rd year maths options which would be deemed easy by students and at the same time useful and enjoyable for future high-school teacher and accountants?

PROPOSED SOLUTION: Offer "fun" modules in other areas which will be roughly in line in the level of difficulty with the named three modules. There should be a critical mass of people willing to teach a module so that it runs each year and deserves a reputation of "fun" and "easy" among the students.

PROPOSED FUN MODULES: Some ideas for modules were proposed during the discussion but I see some problems with them.

1) Fractals. I am not an expert here but we do have MA3D4 already. Is the proposal to dumb it down?

2) Galois Theory. I am not sure why this one has been aired but we do have MA3D5 already. Is the proposal to dumb this one down as well?

3) Problem Solving in Euclidean Geometry. This was aired by me with future teachers in mind. The idea is to go over solutions to Geometry problems from International Mathematical Olympiads. It is not clear to me whether we should comprise a 3rd year module from essentially high school maths but if people think that it is a good idea I can give it another look.

3 years BSc (including the modules up to MA3)

2 years MSc (including MA4, MA5 and TCC modules)

3 years PhD

Something like this has been proposed by Robert in a previous meeting (but still trying to keep the 4 year MMath alive in parallel). I did not want to open that box in the discussion, but something became clear also as a result of our discussion: The students who have lined up a job leave after 3 years. The students who want to stay in academia and/or got for PhD will be very happy to do a 2 year Masters with a good variety of challenging modules, an international crowd of graduate students etc. So in my view a 4 year degree does not make any sense. I know that British academics might feel somehow emotionally attached to it, but I think the only reasonable way for the future is 3+2+3. It will solve all the problems of different levels in 3rd and 4th year (simply by eliminating them), and this is what we should be talking about and invest our time in, in my opinion (which I realize might be rather controversial).

3rd year choices for BSc and MMaths students. By deleting this distinction,

we will encourage the following 3rd year strategy:

1) register for MMaths by any means necessary;

2) if conditions are good (job offer or marks are good) graduate at the eleventh hour.

In particular, I am worried about lower 2.1-students overplaying this strategy. I personally have one student one average per year with about 62 average who wants to stay on MMaths. At present, I dissuade most of them for putting forward a special case to stay on MMaths. In fact, over my years here only 2 students acted against this advice and it worked for one of them but went disastrously wrong for another.

If the proposal goes through, my advice to such student will be to use the above strategy. As a result we may have up to 60 students staying on MMaths for completely wrong reasons and considering graduation after the exam results are out. Are we ready to handle 60 students who suddenly change their minds and graduate?

5. (Bill Hart) The course (in Dmitry's list above) Problem Solving in Euclidean Geometry was aired by me with future teachers in mind. The idea is to go over solutions to Geometry problems from International Mathematical Olympiads. It is not clear to me whether we should comprise a 3rd year module from essentially high school maths but if people think that it is a good idea I can give it another look.

I am sceptical that our students could manage this. Here is a problem which was shortlisted for the 1988 Olympiad, but I think rejected:

Q. The triangle ABC is inscribed in a circle. The interior bisectors of the angles A, B and C meet the circle again at A', B', C' respectively. Prove that the area of triangle A'B'C' is greater than or equal to the area of triangle ABC.

I would wager that few if any Warwick students without quite some prior training in solving IMO style problems could solve this within a week of seeing it. (I tend not to wager ever unless I am absolutely confident of winning the wager - I am a mathematician after all.)

I think such a course would be doomed to failure, and certainly would not be seen by the students as either "fun" or "easy".

(As an aside, at the institution where I did my undergraduate degree, the vast majority of the professional mathematicians there were unable to solve such problems without a full day's work - olympiad participants are given about 1.5 hours per problem. :-) )

Usually such a course covers:

Archimedes

Euclid's Elements

The Arab mathematicians

Napier

Various other topics, e.g. solution of quadratics, various methods for estimating areas and volumes, etc

Newton and Leibniz

There are three problems with such a course which make it less fun to teach.

1. If a textbook is prescribed for the course students wonder why there are lectures for the course - why can't they just read the textbook.

2. If you improve on the textbook in the lectures (easy to do, many of the textbooks on the subject contain horrible mistakes, such as teaching trigonometric techniques for solving problems which the ancient Greeks solved), then the students become very upset that such a terrible textbook was prescribed.

3. Students *cannot resist* plagiarising Wikipedia, even if you tell them over an over again not to do it. So if you set problems, they probably can't solve them using the techniques of the time. If you set them reading exercises, they do not use the library.

However, the course *can* be made fun with sufficient, careful preparation.

I conducted only a cursory search of online resources to see if we already offer such a course, so my apologies in advance if we do this already.

Let a,b,c be the angles in the triangle ABC, R the radius of the circle. Easy calculation with angles overlooking similar arcs show that the angles in the triangle A'B'C' are (a+b)/2, (a+c)/2, (b+c)/2. Now expressing the areas and remembering a+b+c=\pi

Area of A'B'C' = 2R^2 sin((a+b)/2)sin((b+c)/2)sin((a+c)/2) = 2R^2 sin((a+b)/2)cos(a/2)cos(b/2) while Area of ABC = 2R^2 sin(a)sin(b)sin(c) = 2R^2 sin(a)sin(b)sin(a+b) = (Area of A'B'C')\times 8 cos((a+b)/2)sin(a/2)sin(b/2) Hence, it suffices to establish that cos((a+b)/2)sin(a/2)sin(b/2) \leq 1/8 for a,b \in (0,\pi) with a+b <\pi. This is routine, for instance, rewriting it as (1-cos(a)-cos(b)-cos(c))/4 will do the job but there may be a more elegant way through trigonometry... Notwithstanding all this, I may be the last person that students will regard as "FUN". When I was suggesting this, I was thinking of going over solutions of about 60 problems over the course (2 per lecture). Assessment could indeed be a problem as giving unseen problems like this on the exam is not fair... but there is nothing wrong with giving seen ones...

Thanks for the criticism anyway.

I'd be keen to be involved in something `fun' and `easy', because in my view our whole task is to produce people who are friends of mathematics, be they on their way to being brilliant mathematicians or simply people who can help their children not to be afraid of it. But we already have a history of mathematics course, and two seems confusing when there are so many other wonderful things about mathematics.

Moreover, it seems to me that we don't do badly: there are a number of second-level courses that strike us as `fun' and `easy', and which our supposedly weaker students do. Number Theory is one, our differential equations courses are surely others. So I'm guessing that the problem is that these courses are seen as easy by students who also feel a bit of a failure for choosing them, who may think that mathematics was not, after all, the right choice for them. They want fun with dignity – an elusive commodity everywhere in life. And if we provide dignity by making it challenging we risk simply making it hard.

By the way, I also doubt that the students we are talking about are exactly the same as those who choose to become teachers, although I’d guess there is a considerable overlap. So making it another course for teachers doesn't quite get it right, and it panders to the feeling we all want to combat that if you can't do mathematics you teach it instead.

Do I have a solution? No, but I have a crazy idea. At the risk of saying the obvious a popular course is a lot of work unless you are willing to let too many students to do very little. So it should have lots of exercises and yet be thought-provoking. Make it Web-oriented: build your own showcase of things in an area of mathematics that you like! Marks for range, marks for depth, marks for saying why it's interesting, a half-day of presentations halfway through and another at the end (good for their c.v.s) , a final exam to test understanding as opposed to skills at downloading from websites. Work in small teams might be allowable, given some agreement about how the final marks were allocated, topics might be restricted to a small subset of those the staff know about and can police (cheating being what it is). The topic could be genuinely historical (Newton, anyone?), given a historical flavour, be entirely history-free (ditto with applications, pure theory, etc.). Like the current system of third-year essays people should be steered away from reproducing material in existing courses, but this shouldn't rule out topics being such things as My favourite finite groups, My favourite rings, My favourite differential equations, My favourite Fourier transforms, . . . . And if you don't already think this is crazy, you will also find the more enthusiastic students link their work on this course to their Facebook pages.

The subtext here is: if a student is really not sure they like, or are willing to do, mathematics any more give them a chance to find a bit of it interesting and find they enjoy exploring it.

Bill and David: if you think this is too crazy, or, if we already do something like it (Trevor Hawkes used to run team projects and a Wiki-course, I recall) then just bury it. If you think it has any merit - over to you.

1) The Open University offer 3rd level courses which are accessible without 'dumbing down'; I taught on the Number Theory and Logic, and Geometry and Groups courses (topics included Turing and Abacus machines, decidability etc, and tilings and wallpaper patterns, also projective and inversive geometries).

2) Look at the chosen essay titles in recent years to see what captures the imagination of students; I have had essays on Mathematics and Music, Mathematics and Art, Graphs and Networks and other Discrete Mathematics, Population Dynamics and Cryptography all of which would lend themselves to interesting courses.

3) Consider offering 2-stage courses where 3rd level course could be converted to 4th level with additional reading material and a suitable piece of assessed work.