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
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
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
Pòlya, G. (1957). How to Solve It (2nd ed.). Princeton, N.J.: Princeton University Press.
I would like to thank Ms Teo Siok Tuan for suggestions that
improved the presentation of this article.