I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other management or business sciences etc., the course has to be a generic one. By this I mean that we teach Calculus almost exclusively. Sure, there are topics like the Binomial Theorem and general remarks of proving theorems, but students that are interested in mathematics don't find the material particularly interesting. What is more, I can relate to them since I found the first two years of university mathematics somewhat boring. This included calculus, linear algebra, convergent/divergent sequences, multiple integrals etc. Real analysis (except for sequences), complex analysis, abstract algebra, topology and even elementary number theory do not appear until the third year.
The sad thing is that many students take mathematics only up to second year and do not get to see any of the "cool"/"interesting" mathematics even though they might be interested in mathematics. So I wondered: Is there any way of introducing "interesting" mathematics to them? ("Them" – in particular first years, but this question is also relevant for second years, who might have a higher degree of maturity.)
Some things that I (and the other lecturers of this course) have thought of are:
Adding challenging questions to tutorials (e.g. IMC or Putnam, though these are harder than we would like)
Writing short introductory "articles" about fields or groups or perhaps the Euler characteristic (as an introduction to topology) etc. Of course, this is very idealistic, since one often doesn't have time or energy to do this.
Referring them to library books where some of these things are explained. Also, quite idealistic, but how many will actually go to the library.
The best solution is probably to combine the three. Have a question which has a strange answer or solution, which can be explained by some interesting mathematics. Shortly explain how this is done, and have a reference where the student can go if he is interested enough to pursue it further.
Are there any other ways of achieving this goal? Do you know of any questions to which this (combined) procedure can be applied?
Best Answer
I taught a class for advanced first year students at my university some time ago with the aim of showing them interesting aspects of university mathematics. It basically turned into a baby manifold course - with the goal of understanding the concept of de Rham cohomology. The point is that using just a little more machinery than the theory they already knew from calculus (e.g., div, curl, derviatives), it was possible to still prove some quite interesting results like the Brouwer fixed point theorem.
I felt having this 'goal' in the course to be quite effective: the students always had some kind of idea where the course was going and most importantly, they knew why new concepts were introduced. There are a lot of motivating examples and questions to give them, like, 'How can one detect the shape of a space'? This made it easier to motivate concepts, like tangent spaces, differentials,etc.
Even though your students would probably benifit from learning the standard material in calculus (you say they are not math majors), I think you should be able to incorporate some interesting examples like the above into your course. For example, when talking about Taylor series, you could give the nice proof of the irrationality of $e$ or $\pi$. It would certainly make it more fun for you to teach, and probably not take too much of your time.
For reference, here are some of the topics we covered in our course:
There are plenty of good books you could take a look at for finding interesting examples, e.g., I found the following books helpful.
Proofs from the Book by M. Aigner, G. Ziegler
From calculus to cohomology by Ib Madsen, Jørgen Tornehave
The fundamental theorem of algebra by B. Fine, G. Rosenberger.