Our first real analysis test is coming up next Wednesday, and the other night in class we discussed its design. Dr. P has posted to the course website the first test from the last time he taught this (several years ago), and it was basically structured like this:
Part I - several mandatory questions that are fairly "easy," such as definitions, logical manipulations, and the like, for about 60% of the grade
Part II - several proofs, of which you had to choose a few, for the remainder of the grade
He intends to keep a similar format, but pointed out that, although the point of Part I is to allow you to get the bulk of the points for some fairly easy questions, in practice many students actually more or less bomb Part I.
A small digression, actually, over the topic of "definitions." Some of what we have to know for this test are various math definitions. I wouldn't normally be a fan of needing to memorize definitions for an exam, but that's because in most subjects, it's more important to know what something is than to know its definition. In math, a thing is its definition, and you need to know the definition in order to do proofs or make new theorems about the thing. For instance, if I have to prove that 3/5 is a rational number, I will need to know that a rational number is one that can be written as one integer divided by another.
Anyway, he could expand Part II (the proofs) to be be more of the points, but doing proofs as part of an exam is difficult, and he doesn't think he could realistically ask us to do more than about three in total, even if that was the whole exam. And in that case, if you seriously whiff on one of them, you're already down to a D, which is pretty harsh. He'd rather keep it so that each individual proof is worth no more than about 15% of the total exam score.
He was also talking about comprehensive exams in graduate school, where you might have six proofs to work on. He said that turning in two nearly-perfect proofs might be a pass, but turning in two nearly-perfect proofs and one garbage proof would probably make you fail, because it's important that you know the difference between math and garbage, obviously. Then he reassured us that he won't grade us in such a fashion. (Imagine if particularly stupid answers actually resulted in negative credit. People would flip. It's kind of amusing to think about.)
So he settled on making us do two proofs in Part II, in addition to a certain obligatory proof promised as part of Part I. It was brought up that he could let us turn in more than two proofs (since there will be several to choose from), and then grade them all and give us credit for the two best ones. He doesn't want to do that, though, because he fears that we will then hurry to write several proofs, and they will all be more mediocre as a result. He doesn't want us just "flinging stuff at the wall to see what sticks." I suggested that it is also much more horrible to grade 5 bad proofs than it is to grade 2 better proofs. (5 is more work than 2 anyway, but when you add in the expected difference in quality it gets horrible to contemplate.) He finally compromised and agreed that we could turn in 3 proofs and he would count the best 2. No more than 3, though! If you write more than 3, you have to pick out the best 3!
I'm looking forward to the exam. I'm pretty comfortable with most of this material, though I intend to do some more review, and I like how the test design looks.