I often teach a course with the enigmatic title “Fundamentals of Mathematics I”, intended for liberal arts majors. This is usually the last encounter with math for students in non-scientific disciplines. The syllabus contains a decent amount of optional topics so it is quite possible to tailor the material to one’s taste and professional interests. As many of us are well aware, a course like this poses its unique challenges. Unlike calculus, where the topics are fairly standard, a “fundamentals” course must, in my view, depend much more on mathematical ideas and much less on drudgery computations. Of course, students need to compute, but the result of their computational efforts should be exciting and fun, rather than a more or less meaningless answer that matches the solutions page at the end of the chapter.

These are the topics I usually teach: set theory, logic, group theory, and combinatorial methods. All lend themselves to a great deal of enjoyment, where students are confronted with deep ideas (e.g. what is truth? what is counting? does infinity come in different “sizes”?). Some students feel bewildered when they discover how difficult it can be to “count”.

It is a time-tested favorite of mine to teach them about groups. But unlike a formal course in abstract algebra, I tell them briefly what a group is, how abstract notions can be fun and useful, and after showing them the cyclic groups, both infinite and finite, I proceed fairly quickly to the dihedral groups D_{3} and D_{4} (of orders 6 and 8 respectively). I construct them as the groups of rigid symmetries (rotations and reflections) of the triangle and square. D_{3} is such a revelation since it is the smallest group that happens to be non-commutative; and I usually spend several lectures drawing pictures and discussing composition of symmetries. It is really exciting to reveal to them how “multiplication” is not a universal idea and it can be defined as a non-commutative binary operation. After producing the multiplication tables of both D_{3} and D_{4} we set out to explore the orders of individual elements as well as the subgroup structures of each one of these groups. We emphasize the subgroups of rotations and reflections.

One of my favorite results in elementary group theory is Lagrange’s theorem: If G is a group of order n and H is a subgroup of order m, them m divides n. I take advantage of the sheer simplicity of this result and “test” it for D_{3} and D_{4}. It is not magic, I tell them, it’s a theorem!… I also exploit Lagrange’s result by bringing the (finite) cyclic groups back and sharing the pleasant fact that the converse of Lagrange is true in that context: if C_{n} is the finite cyclic group of order n and m is any divisor of n, there exists a subgroup (necessarily cyclic) of C_{n} whose order is m. And yes, we “test” the truth of this and explore the consequences when n is prime.

Many topics presented in more advanced courses in combinatorics, abstract algebra, logic, etc. can certainly be made accessible to a liberal arts audience. The trick, of course, is to explain the ideas in layman’s terms, progress to some level of formalization, and work out many enlightening examples. Teaching this course has been very satisfactory indeed. My hope is and has always been to leave my students with some long-lasting interest in the ideas behind mathematics, as well as a taste of what mathematicians do.

Marcos! It seems you have a certain amount of freedom in choosing the content in your “fundamentals”-type courses, which I envy – ours are more-or-less proscribed by the department, via a detailed syllabus. But I still like the variety that these courses provide, and the there is great opportunity to do math you can “get your hands on” (like messing with triangles and squares!). I think the focus on concrete examples is likely to have the most lasting effect at this level – these are the things that might just stick around in someones head beyond the completion of their degree. I want to take your class – it sounds fun!