How should one tutor a student in undergraduate real analysis?

Question

I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's Principles of Mathematical Analysis (baby Rudin). We meet once a week for about an hour to discuss concepts and their problem sets.

In the beginning of the semester, I seem to spend about half the time on concepts (mainly repeating definitions and giving examples) and the other half giving hints on the homework problems. Later in the semester (and in the second semester continuation), I spend most of the time giving them considerable help with the homework. I don't have any trouble explaining concepts. However, I have a lot of difficultly with providing the appropriate amount of help with problem sets. The biggest problem I see with my students is a lack of problem solving ability. Part of it is a result of a poor grasp on the material and part of it is laziness. They don't seem to be able to think about a problem constructively for more than 5 or 10 minutes before giving up. Further, many of the problems are relatively easy once you see the essential idea or 'trick'. So either I show them the 'trick' or they just get the answer from another student. I usually end up giving too much assistance.

I have suggested they try easier exercises to build up their problem solving skills. Abbott's Understanding Analysis is a good source of these. However, they are unwilling to do much extra work on their own. Of course, this is ultimately their problem, but as a busy undergraduate myself, I am sympathetic.

Another common concern is preparing for exams. A 1-hour exam is a much different format than an untimed problem set and much less forgiving. My standard advice is to first to memorize all of the definitions, be able to give examples and non-examples (where relevant), and memorize the statements of all of the important theorems. After they have done this, I suggest solving easy exercises from their textbook and trying to prove simple propositions in their book without looking at the book's proof (e.g. a finite intersection of open sets in $\mathbb R^n$ is open).

How should I address these issues? Are there any other techniques I could use to make my tutoring more effective? I am particularly interested in ideas that are specific to real analysis.

Answer 1

memorize all of the definitions, be able to give examples and non-examples (where relevant), and memorize the statements of all of the important theorems

I'd say you are on the right track for definitions, but I'd like to add a tactic for learning theorems. To "learn" a theorem, you should encourage the students to try to break the theorem, as a way to remember how it is constructed. For example, here is the intermediate value theorem:

A continuous function on a closed interval passes through all intermediate values.

Instead of "memorizing" this, students should memorize what the conclusion means, perhaps with a picture -- they will need to know precisely what intermediate values are, otherwise the theorem cannot make sense. But then they should try to break the theorem:

• Seek out an example of a discontinuous function on a closed interval that fails to pass through the intermediate values.
• Seek out an example of a continuous function on an open interval that does not pass through the intermediate values.

This kind of testing-the-limits of a theorem is ultimately what leads to deeper understanding (and as a side effect, better recall).

Answer 2

With lots of pictures/diagrams — giving an intuitive understanding — and then connecting these diagrams to the logic which can be used in proofs. Once students can convert logical sentences to intuitive pictures, and back again they will have "got" (at least the basics of) analysis.

This study guide is also very helpful for students new to analysis: How do undergraduates do mathematics? A guide to studying mathematics at Oxford University. It shows how to read logical sentences in a useful way.

Answer 3

Students reaching for real analysis more often than not do not have a firm grasp of calculus as they should. What ends up happening is that they don't understand how most of that was necessary as a foundation for calculus.

I advocate the following: show constructions where things are working and then exhibit counterexamples as what you get when you push boundaries. They were discovered when people tried (to use a pun) to pass to the limit. Most counterexamples aren't really intuitive: took years of amazingly capable people to realize they were possible and other peers still weren't convinced.

One more tactic: relate what they're doing back to their computations; it's not just about proving how it works but seeing it in action.

Answer 4

You are getting a lot done--feel good. Your approach is very thoughtful and the pedagogy well explained. Can't think of major changes.

You are battling a couple disadvantages (only having an hour, laziness). I was disappointed that so many answers did not really discuss how to help students given the disadvantages, but just opined in a vacuum on how to be more real analysis-y (for ideal students). [But the comment about flaws in calculus was helpful, although would be more so if combined with idea on how it alters the actual tutor session.] I also think your council on exam prep makes sense and that the counters were not realistic (not considering students are not already A students).

One small tip: charge for your time. It makes both you and the student care more.

P.s. General "you", I realize this is old Q. But any Q&A site is meant to help others in same situation.