When learning math, it's imperative to not just read passively, but to actively play with the material: Why is it defined this way? What would happen if we tweaked the definition? What's a counterexample of something not claimed by this theorem?
This frequently leads to excursions: Questions related to, but not explored in or needed for, the material being studied. Examples:
It's easy to prove that $x \sin \frac 1 x$ is uniformly continuous, but hard to find a specific $\delta$ for a given $\varepsilon$. That leads to excursions like: How can we find a $\delta$? (A. Use the Prosthaphaeresis identities). Is there an optimal $\delta$? (A. Yes, and it's well defined, although surprisingly discontinuous and difficult to compute.)
A proof that if $f(x+y) = f(x) + f(y)$ and $f$ is continuous at $0$, then $f$ is continuous everywhere leads to the excursion: Are there additive nowhere continuous functions? (A. Yes! They're surprisingly wild, being dense in the plane, yet extremely regular, being invariant under a family of translations.) Exploring this lead to a better understanding of infinite dimension vector spaces.
Clarifying the difference between countable sets and orderable sets. This originated in a failed proof attempt, which yielded some discouraging comments (ibid.), finally yielding a clear explanation of why order isn't generally important for metric spaces.
These excursions tend to a reoccurring pattern:
- First, I get intrigued by the question, and begin exploring it, with pencil and paper
- I quickly exhaust the limits of what I can do on my own, but, in the process, develop some leads that I can search for
- Searching usually leads to some authoritative references discussing the idea in a way that is insightful but largely beyond mean; I scan the source, glean some insight, but don't attempt to follow each detail and certainly don't attempt to work things out on paper
Clearly, these excursions have what to offer, but also have costs. On the one hand, it's great to stimulate independent exploration. On the other, mathematics is learned by doing, not searching and scanning: With limited time available, is it worth taking away from the matter at hand, which is designed at the right difficulty for me to actually do, to explore tangents which I can only read about? And is it worth taking away from curated material selected for its importance and replace it with excursions that may, even if fruitful, be random?
- Given the above, and limited time, what role should excursions have?
- How can a student, especially a self-studying student, maximize excursions value?
- In your experience, have they benefited students (or yourself), hurt them, or both?
Update: Starbird and Su (p.9) acknowledge some of these issues in a university setting:
Individual research projects for undergraduates present many challenges... and the resulting experiences for the students are variable. Sometimes they are the... intellectually satisfying experience we seek; and sometimes the project turns out to have some unexpected defect, such as not yielding sufficiently many results or a sufficient variety of challenges to be of great interest... An alternative is... a quasi-research experience by asking students to tackle questions that are new to them, while not being new to the world. In many ways this experience is often superior to individual research projects because the careful control of the challenge questions gives students a predictable range of research experiences.
I'd argue that exercises from a good text can do exactly that: carefully control the challenge questions to give a predictable research experience. Is that an argument to avoid excursions entirely, and simply find a text with good exercises?