Is proof-based exercise-oriented math course without solution an effective way to teach pure math?

Question

In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of:

1. notes written during the lecture (with several recommended textbooks that the prof does not follow in detail), and
2. a long list of exercises questions drawn from various sources with no solution provided to the students

The questions for the exams are directly drawn from the list of exercises questions with modifications. There is absolutely no homework, no grades aside from grades of several exams and solution to exercises are never given aside from hints you can obtain by talking to the professors.

From talking to the professors, I get the sense that they are mainly concerned about:

1. cheating, using homework solution from online of standard texts thus hindering mathematical development
2. encouraging students to think more about problems
3. providing ways to incentize students to do exercises by putting modified exercise problems on exams
4. weeding out students who are not committed to learning about math

Has anyone ever went through or conducted such a course? Is this an effective way to deliver a proof based math course? Are there any potential pitfalls and flaws using this approach?

Answer 1

I have taught such courses and I don't think that a long list of exercises without solutions is good idea - at least it didn't work for me. It is because 1) students are not yet able to tell a correct proof from an incorrect proof. 2) often students don't know how to get started, but if you give them a hint, they will be able to solve the problem.

What works for me is that they type their solutions into a shared online document, I write comments, and they modify their solutions according to my feedback. Then, in the end of the semester, they have solutions nicely typed, but the solutions are produced by themselves. Here is a copy of the actual document that my students typed last week (the real document is shared with permission for public to edit)

Answer 2

This sounds vaguely similar to an "inquiry-based learning" course. So although this isn't what you are describing (notably the lectures), you may find resources at the Academy of Inquiry Based Learning helpful. Here is a separate description from a related venture. A most extreme version of this (extreme in a good sense, I think) is the so-called Moore method.

Answer 3

The courses that you mention -- real analysis and topology -- are most often senior-level courses for undergraduate mathematics majors. At this point the program is trying to give you a taste of graduate/professional work, where the overall project is one of generating proofs for as-yet unsolved open problems. It's also pretty universal that textbooks at this level contain no complete solutions at the back (more frequently some rough-sketch hints for select exercises).

Look at this as a good opportunity to decide on whether you want to pursue math further at the next level or not; this is what it's like all the time. As Alfréd Rényi joked, "A mathematician is a device for turning coffee into theorems".