Generally speaking, it would be nice to have a foundations class at the initiation of the Math major. Some of my colleagues envision this course centered around teaching college algebra. Well, to be more precise, algebra done with care, algebra beyond symbol pushing and problem solving.
The Foundations course we currently run in our major requires Calculus II as a prerequisite, but, this is mostly for maturity. Logically, it ought to go at the start. If the proofs course was first then more students would have a better view to what major course work looks like in Math. Calculus is still very much a general education requirement, so, the inculcation of students into native math culture really only begins in the proofs course at our institution.
But, it's not really this simple. In fact, we do have epsilon delta proofs in our first semester calculus. Indeed, we also study convergence and divergence of sequences, sometimes the least upper bound property and other bits a pieces of analysis in the second semester. However, these are not the true flavor of the calculus sequence. They're deviations from the main story of calculation and application. In my personal opinion, the real analysis needed for second semester calculus is far more sophisticated than mere epsilon delta proofs. For example, the proof that the derivative of a power series is the derivative of a power series is quite beyond the students. I'm certain of this as none of them ever complained the many years I failed to appreciate the need of the proof.
We introduce integrals in calculus I in terms of limits of sequences! Many texts do this. So, this is after we cover sequential limits, right? Wrong. The limit appearing in the Riemann integral is merely intuitively framed by justifying approximations. Now, don't misunderstand, I understand the place of providing an intuitive version of calculus before rigor sets in. But, I am aware of the hypocrisy of carefully covering epsilon deltas in the same semester I fail to properly define limits of sequences, much less the integral itself.
So, why study epsilon deltas in calculus I? It doesn't seem practical to maintain the level of detail throughout the calculus sequence. We must admit there are tools we wish to use, but, not construct; how many major theorems do we have time to present the proof? I have a lot of time here, but, even so, it's not possible to cover everything. I can tell you from experience, the students only appreciate so much honesty in a given semester. So, why epsilon deltas?
- it gives students a much needed chance to study how inequalities work.
- it can be a great chance to better understand how graphs can organize analytical thought
- the logic isn't that complicated if you do a half-dozen examples. Keep it simple, basic functions like linear, quadratic, reciprocal all are helpful.
- it allows you to be precise in the meaning of a limit not existing, what we mean when we say the limit point need not be considered in the evaluation of a limit
- it does make introduction of sequential limits a natural continuation of an existing discussion
- the students who can think can do it. It's not that big a deal.
- for students who are math majors at heart, it's a breath of fresh air in an otherwise logically questionable development. I have had the topic gain the major new students. In fact, my best student to date came to Math from a different major in large part because I was overly formal in Calculus I. It can make a difference.
Ok, so, how many students don't get it? Lots. Just like lots of students don't get a number of topics in calculus. Convergence, divergence, the chain rule, the idea that you plug the number into the derivative to find the equation of a tangent line, this list is endless really. There is a risk and a possible reward, you have to decide on the basis of your audience whether it's worth the risk.
That said, do you need a logic class beforehand? No, not in my experience.