# How to assess students in real analysis?

Terence Tao says the following in the preface to his book Analysis I:

With regard to examinations for a course based on this text, I would recommend either an open-book, open-notes examination with problems similar to the exercises given in the text (but perhaps shorter, with no unusual trickery involved), or else a take home examination that involves problems comparable to the more intricate exercises in the text. The subject matter is too vast to force the students to memorize the definitions and theorems, so I would not recommend a closed-book examination, or an examination based on regurgitating extracts from the book. (Indeed, in my own examinations I gave a supplemental sheet listing the key definitions and theorems which were relevant to the examination problems.)

Based on Tao's comments, we can separate the most common types of examination into the following categories:

1. Closed-book/closed-notes examination without supplemental sheet.
2. Closed-book/closed-notes examination with supplemental sheet.
3. Open-book/open-notes examination.
4. Home examination.

Based on your experience (or on sources on the subject that you know), in the context of real analysis examination:
(a) What is the best category?
(b) What are the pros and cons?
(c) How similar should the exam questions be to the homework questions?

If you use a different category from those listed above, feel free to comment.

Addendum: I'm talking about an introductory course that covers the basics of natural and real numbers, convergence of sequences and series, limits of functions, continuity, derivatives, integrals, sequences and series of functions.

• It really depends on the abilities of your students. If I gave an open-book/open-notes examination of the type Professor Tao imagines, more than half of my students (even those who study quite hard) would be able to answer zero questions - they have no ability to apply what they have learned to situations that are not exactly the same as those they have been explicitly taught to work in. Commented Mar 10 at 18:37
• Adding to what @AlexanderWoo said, remember that Tao is at UCLA, which has a world class mathematics department, and is incredibly selective about the students it takes on. Tao's students are, on average, the best of the best of the best. What works at UCLA may or may not work elsewhere. Commented Mar 10 at 22:11
• My point was not that an open-notes approach is either a good idea or a bad idea. My point is that Tao likely suffers a bit from the curse of knowledge, and that his recommendations might not work for everyone. In particular, UCLA students tend to already understand that they have to memorize a great deal of information, thus basic "regurgitation" questions should be gimmes for those students. My students (calculus, not analysis---we don't go that far here) need exam problems which do require some rote work, since no one has ever made them memorize anything before. Commented Mar 10 at 23:18
• @shoover: "we had a well defined and strictly enforced honor code." I assume that this honor code said that cheating is forbidden. How can this be strictly enforced when the exams are unproctored? (By the way I'm also wondering what's the point of an "honor code" that only works because it is strictly enforced.) Commented Mar 22 at 20:23
• I primarily tutor. I think anyone offering take home exams underestimates the number of students who are just desperate to pass and have given up on actually learning the exercises. I regularly get requests, "Will you do this exam for me?" Many or most exercises, can be googled. Even if it's a totally unique question, the student can simply post it on MSE and likely get an excellent answer. An honor code in this situation only put the honest students at a disadvantage. Commented Mar 26 at 6:58

I suspect that the rules of assessment are the same for every discipline, calculus not excluded, and can be just derived from first principles and your particular version of common sense without any help from Terence Tao. Mine are the following:

1. Decide on a few (3-5) topics you want to test and create one problem for each. Don't give more than 5 problems for a one hour test or include problems that require long solutions with extensive computations.

2. Tell the students that the exam will be concentrated on these topics and give them sufficient number of practice problems well before the test.

3. Out of 5 problems make 3 completely routine (for the students who know the subject, of course), 1 with a minor twist and 1 really hard (though requiring nothing beyond the covered material).

4. Invent your problems from scratch and tune them up exactly to your satisfaction yourself.

5. Conduct the exam in a fully controlled environment (ideally in a classroom with your own presence).

6. Grade on some simple scale with minimal partial credit (mine is 6 - full solution, 5-a solution with minor error, 1-just a good idea, 0-everything else; occasionally I do give the scores of 4 and 2, but those are exceptions).

Whether it is open or closed book/notes doesn't matter too much as long as you understand that each of those kinds of exams just requires a different tune-up of the problems. I never give take home exams to undergraduates though: tuning those up properly would require creating open problems based on the elementary calculus material. That is possible, but the most likely outcome will be that nobody will solve anything. Everything short of that will result in shameless cheating and total mayhem with grading, IMHO.

Needless to say, this is just my approach based on my common sense. Your common sense may be totally different. Just follow it and don't be afraid to experiment with various formats. Whatever works to your satisfaction is the right way, whatever doesn't is the wrong one. It is always as simple as that.

Just my 2 cents :-)