A blank slate

When the idea of a freshman seminar module (FSM) was first introduced in AY 2006/07, it was novel to NUS. To me, it was not just about teaching a new field which we can readily find books and material to lecture on. FSM does not seem to be well-defined. It has no specific syllabus and no prerequisites. It practically starts with a blank slate. The freedom from the constraints of a highly structured regular curriculum gives the lecturer a sense of naïve optimism that he can now try out some of his pet ideas about teaching that could not possibly be implemented in a regular curriculum.

I started by asking myself what would be my primary objective in teaching an FSM? Clearly, it is not a module meant for students to attain a certain level of competence in a specialised area. Rather, I thought that it should give students a foretaste of learning mathematics, if only at a superficial level. In addition, it should foster ‘out-of-the-box’ thinking among students.

So, what are students’ expectations? I suspect that most students perceive FSM as a ‘soft’ module, but there could be some who are expecting something novel and flexible, or even something exciting and fun. Whatever their expectations, students would be naively enthusiastic.

Making learning fun

Formal mathematics has always been perceived as dull and demanding. A distinguished research mathematician and educator Pòlya (1957) said, “Mathematics, besides being a necessary avenue to engineering jobs and scientific knowledge, may be fun and may also open up a vista of mental activity on the highest level” (p. ix). Pòlya underscores the role of heuristics in mathematical problem-solving. Mathematical discoveries are often made by analogy with other known results in mathematics and physics. The necessary rigour is often established later.

Another powerful driving force behind mathematical creativity is intuition. Starting as a leap of faith, intuition is often pursued to its logical conclusion to produce breakthroughs. When intuition based on conventional viewpoints leads to a blind alley, its antithetical companion, counter-intuition is often invoked to break out of the impasse.

It occurred to me that these two powerful approaches could be used to learn about the development of mathematics in an interesting way. A seminar module can be spawned by searching for examples of analogy and intuition (and counterintuition) in modern solutions of classical problems dating back a couple of thousand years ago and in the creation of new fields with surprising subsequent consequences for other fields.

I also thought that freshmen should soak in the flavours of analogy, intuition and counterintuition in as many areas of mathematics as possible before preconceived ideas and prejudices begin to seep in. Depth will have to be sacrificed in favour of breadth. With sufficient breadth, however, students will be able to trace some of the common strands linking different areas. In the process, I hope they will catch a glimpse of the history of ideas and of the faces behind these ideas. Mathematics will then come alive as a historical and social development.

Thus, the philosophy and approach of the module is to look at the big picture, not the devilish details, and to experience the spirit of enquiry of a universal mental enterprise. One learning outcome of an FSM would be the realisation that ideas are not static and that no idea is an island.

Thinking out of the box

Though the mandatory class presentations trained students to become independent in sourcing and organising information, there was little participation from the audience during the presentations even with my encouragement. However, there was one semester where some student presenters took the initiative to engage the audience by giving out simple problems and getting the audience to participate in the exercises.

I realised that I had to find a way of getting students to participate actively in class and, more importantly, to think critically. With a fixed syllabus, students’ thoughts tend to revolve around solving tutorial problems. Without a fixed syllabus, any questions or problems for class discussion will have to be general and not confined to anything too specific.

I found that solving puzzles requires an unconventional approach akin to thinking out of the box. Solving suitably chosen puzzles not only injects fun into the lessons but also gives one a sense of satisfaction. Even if one is unable to solve it, one will be amazed at and sometimes amused by the simplicity of the solution. In a way, puzzles encapsulate research in miniature.

In order to foster ‘out-of-the-box’ thinking among students, each seminar session would constitute student presentations (with most of the discussions initiated by me) and solving puzzles which I selected. These included logical, geometric and number puzzles that can be solved with elementary mathematics. The class is given some time to think about the puzzle, and the solution is presented to the class by a student. For harder problems, I would give hints to stimulate discussion. If no student comes up with any solution, I would provide the solution and explain the problem-solving heuristics involved.

What about the future?

FSM has only just taken off at NUS, but it is never too early to talk about its future. For my own FSM, the avenues for hands-on learning experiments, simulations, games, tricks and so on in mathematics are waiting to be explored. We are limited only by time and our own ignorance. In my view, FSM should be more informative than didactic, more informal than formal, convey more fun than pedagogy, generate more inspiration than ‘perspiration’, promote more awareness than expertise, and foster more curiosity than rote learning.

Not all students will pursue research careers; many of them will become entrepreneurs, managers and policy makers. Long after they have forgotten the esoteric details of the great theories, I hope that this FSM will leave them with an unforgettable impression of the indomitable spirit of enquiry of the human mind and the relentless energy that drives research. Who knows what positive effect this may have on some of the crucial decisions they will be making in the course of their future careers?

Reference

Pòlya, G. (1957). How to Solve It (2nd ed.). Princeton, N.J.: Princeton University Press.

Acknowledgement

I would like to thank Ms Teo Siok Tuan for suggestions that improved the presentation of this article.