# Spiral learning in real analysis

Has there been any attempts at developing a curriculum for teaching analysis (here let us be narrow and say real analysis in the sense of rigorous integral and differential calculus) in a multipass, spiral approach?

If there is, has there been any studies on its efficacy? Or any guidelines on best practices? Has any textbook been written from this point of view?

To a certain extent that is what we do a bit already: in many institutions the students are expected to first learn calculus and multivariable calculus from the algorithmic, computational point of view, and then re-learn it in an analysis class with more rigorous foundations. But frequently this is a large jump from 0 to 1 in terms of rigor. I am wondering whether someone has developed a textbook or curriculum for a 3 or 4 pass spiral approach to the increase in "rigor".

An example for what I am thinking about is, on the topic of integration, maybe doing a multipass approach would lead to something like

1. The first pass: integration by taking antiderivatives; fundamental theorems of calculus used axiomatically and not justified, ditto mean value theorems.

2. The second pass: Riemann integration of (uniformly) continuous functions, with tagged partitions and so on. The first mean value theorem and the first fundamental theorem proven in detail. Prove that the evaluation of the Riemann integral is linear. Some basic approximation/quadrature theorems and error estimates.

3. The third pass: Darboux version, start with approximations by piecewise linear or piecewise constant functions, take the sup/inf appropriately. Define the class of (Riemann) integrable functions, prove that it forms a real algebra. Reprove in this generality the first mean value theorem and the second (the monotonic one) mean value theorem, as well as the first and second fundamental theorems, and the Lebesgue criterion (in a suitable formulation).

Possibly also talk about directed sets and nets (useful for taking the directed limit over partitions for defining the Darboux integral, and also for topology later) and various other techniques.

4. Fourth pass (optional): Riemann-Stieltjes integration, redo the relevant theorems, integration by parts for RS integration, and expose the second mean value theorem as essentially one integration by parts plus the first mean value theorem.

• I'd want a step 0 for "area under the curve", and a numerical approximation of something like $\int_0^1 \sqrt{1+x^3} dx$, to get away from closed-form integrals quickly.
– user173
Sep 26, 2015 at 21:37
• Maybe abroad. I'd be a bit surprised if this were done in the United States. Oct 8, 2015 at 19:33

I don't know if there is a specific curriculum that adheres to your exact ideas, but there are some historically-oriented ones that could work. In particular, I think David Bressoud's Radical Approach to Real Analysis and Radical ... Lebesgue Integration could conceivably help fill that kind of gap.

(I would imagine there are also combinations of inquiry-based notes that could work, but that wouldn't be a formal curriculum.)

• Interesting suggestions. From a quick glance it looks promising, but starts at a level that is at the second pass or between the second and third passes. It could be conceivably incorporated into a larger curriculum of the type I described, but by itself doesn't seem to fully be what I am looking for. Thanks nonetheless! Oct 19, 2015 at 20:27
• Well, I think that every analysis course ever (possible exception: Spivak) assumes students already know how to do your 1, so that would be tougher. No, I didn't think this would do it by itself. Good luck! Oct 20, 2015 at 1:47
• Ah! I never meant for this to be a single course! I say curriculum because I am thinking of this as a multiyear, comprehensive affair. Roughly I imagine each of the passes taking one semester: the latter three can potentially be condensed into two. Oct 20, 2015 at 2:17