learning/teaching approach to rigorous math with the goal of improving

Question

I will state this now: yes, this is a subjective question. But I feel the answers people give may benefit students.

I want to get better at doing non trivial proofs. Real analysis is standard material some students have to learn. I have, broadly speaking, two choices for a textbook: 1) something like "baby Rudin", where I may struggle for long periods of time on problems, often having little direction and no progress, or 2) something easier, where smaller theorems are developed in the book and where there are less sophisticated jumps of reasoning (demonstrated, required) in the proofs.

What are the preferences of educators/students for using one over the other? Do you have reason to believe one is more successful at actually helping students improve? How do you determine success?

My beliefs: I tend to think the easier book may help me develop my own techniques (I could be wrong). I see more examples and I get more confidence in doing basic things. But more advanced material will probably be less accessible, so I know that at some point I will have to just deal with the sort of terseness that occurs in Rudin. Anyway, I can keep going back and forth about this.

Answer 1

Both have their place. It depends in part on the taste of the reader. I personally would go for the "harder" text, complementing it with Wikipedia, random lecture notes, and sites like this one. It the "harder" is too hard for you, start with the "easy" one but if you are really interested you will have to tackle the "hard" one (and much, much harder stuff) pretty soon.

Answer 2

If you haven't had much experience with writing proofs, I would suggest starting with something like Spivak's Calculus or, in fact, some abstract algebra, which most students find more accessible than analysis.

Answer 3

"Mathematics is no more computation than typing is literature." - John Allen Paulos

There are two things that alienate people from math: 1. mindless computation. 2. obscure proofs. While both are important, the emphasis must be on discovery, exploration, and understanding. Oliver Heavenside once said, "I don't reject my food because I don't understand digestion," when defending his use of certain mathematical ideas he didn't fully understand. Exploration is a prerequisite to proofs. To extend the analogy; How can we determine what is happening in digestion if we are unaware the stomach exists? Few things help people understand better than Geometry, Symmetry, and Patterns. Rigorous math requires motivation which for me occurs when the context and importance are apparent. I agree with your assessment that the "easier" book is better since it will likely provide more opportunity to contextualize and thereby motivate you in some direction that may be far removed from the proofs you would learn in the "more advanced" book.