Some undergraduates may share our love of the beauty of mathematics and be motivated by their interest in the subject matter and the style alone. However, I would suggest that there are many other students who could benefit from knowing more about the motivation for and applications of the subjects they are studying.
I was talking to the mathematicians in UCC (Cork), and I heard of a very good system they have there for motivating statistics for their service teaching. They found out who was actually teaching the students some of their other subjects (e.g. biology) and asked them to provide some examples of actual data that they had done some recent statistical analysis on. They were then able to say to the students “Here is some data that Dr X had to analyse last week”, with the result that the students actually sat up and said “Hey, Dr X is the one teaching me biology, and he really had to do some statistics!” This system has proved very effective in increasing motivation and engagement for those students.
So, can we transfer this approach to motivate (for example) the study of mathematical analysis for mathematical physicists? This is relevant to me, because my second-year module G12MAN is compulsory for students on our Mathematical Physics course, but it is NOT a prerequisite for any of the “applied” mathematics modules that they will take as undergraduates, except for those who study one or both of the modules Differential Geometry and Black Holes in fourth year. Nevertheless we know that mathematical analysis is important in more advanced “applied” mathematics: quantum mechanics implicitly works in Hilbert space, and so deeper study here often requires the theory mathematical analysis. Advanced numerical analysis often requires results from functional analysis (e.g. Sobolev spaces). Partial differential equations appear throughout mathematics, and functional analysis is again needed here when investigating the existence and uniqueness of solutions. Probability theory, as axiomatized by Kolmogorov, requires an understanding of the Lebesgue theory of measure and integration. Of course this is far from a comprehensive list, and I apologize if I have missed out your favourite application. (… and I haven’t even started on applications of algebra, number theory, etc. Comments welcome!)
My plan is to ask the applied mathematicians here to let me know which specific problems they (implicitly or explicitly) used mathematical analysis for last week (or “recently”, or possibly “ever” ). I can then follow the Cork Dr X approach, though probably not in as concrete and detailed a way as in the Cork statistics approach.
Of course, knowing more about this should also prove helpful to us as the financial cuts bite and we need to explain just how important it is to maintain our strength in all areas of mathematical study.
The concrete probably wins over the abstract here: it would not be as strong to tell students in the abstract that some mathematical physics requires results from mathematical analysis, as it would be to give specific examples. And I think that the specific examples are particularly strong if they involve the students’ current teachers actually using mathematics for specific problems that we can tell the students something about.
Does anyone else have any success stories (or other!) based on the Cork Dr X approach?