Background: I am fairly new at teaching, and in every subject I have taught, I have had difficulty estimating the length and difficulty of an exam. I need to write an exam for a university senior-level knot theory course. The exam must last one hour due to university constraints.
What is the most effective way to write an exam that must fit a given time constraint?
My background is in high school teaching, so my experience may not directly transfer, since the types of exams are different.
However, I have found a very useful rule of thumb to be this:
After writing the exam, I sit for it myself, i.e. I sit down to write down full answers in one sitting. Most students will take six to eight times as long as I did.
Here's a few useful strategies:
1) Once you've written the exam, time how long it takes you just to write down the answers. That gives you a baseline on how much time someone (you!) who already knows all the questions would take to answer the exam. This is particularly useful for exams that are writing-heavy (like proofs and "explain this" questions).
2) If you have TAs, ask them to take the exam. Give them some fraction of the time you intend to give the students. Depending on the topic/course, this could be 1/5, 1/3, 1/2. Your TAs (arguably) should get near perfect scores on your exams (unless you want a very difficult question). Make sure you tell them to actually time themselves and actually write the answers down. Also, make sure you actually review what they've written. Once, one of my TAs finished in the time he was given, and I hadn't reviewed his answers. It turned out (after the students took it) that he'd accidentally answered something incorrectly, and the exam was harder than I had intended. Ideally, you'd want to use undergraduate TAs for this, if at all possible, because they're "closer" in experience to the students in the course.
3) Try to mark your idea of a "difficulty level" for each question you write on the exam, as you add them. Generally, the best exams have a combination of "easy", "medium", and "difficult" questions. Certainly, if you don't balance this for your idea of what this means, the exam might end up skewed to one side. Then, if you have your TAs take it, ask them to rate the questions too. If their perspective is totally different from yours, that might indicate a problem.
I think there is no general solution, but here are some ideas which could help estimating the time:
A useful rule of thumb I got when I started up here (which has turned out surprisingly accurate) is "Solve the exam yourself, multiply by four." In any case, when possible I schedule twice that (some time is lost until everybody is seated, exams are handed out, etc.; and there is always some straggler arriving half an hour late...). But we don't have regulations on exam lengths
Once I've written an exam, I take it to determine my rubric or how I'm going to grade each question. If a question has $n$ parts/steps to its solution, then that question will take about $n$ minutes. This is a rough estimate and some steps follow quicker than others, but it seems to average out and has worked well for me for the last handful of years. This works best in non-proof courses, but can also be used for very formulaic proofs. One good/bad aspect of this approach is that it gives students a hint as to how difficult/long the solution should be.
Srsly, the expansion factor should not be 4 or 10 but something like 40. If you are a quick, fluent, experienced person, and if we're talking about mathematics at any level beyond cook-book/filter calculus, and if we're talking about situations with kids who're still "not dry behind the ears" mathematically, a huger expansion factor is appropriate.
Indeed, as in other answers, one should be wary of self-deceit about the mere time to write out an appropriate response.
But, more insidiously, further, even for kids who do approximately know "the right answer", there will be a huge amount of dithering. And, there will be a huge amount of no-op self-doubt, etc.
In summary, as the level of the mathematics goes up, the expansion factor goes up, hugely. It is admittedly subtle in undergrad math, depending on circumstances, prior experience, etc. But, e.g., for PhD work, I allow an expansion factor of about 1,000. To explain why this is not a joke, and why this gives perspective on undergrad exams, I seriously comment that it is my (very considered) opinion that in starter-thesis situations, "the advisor" should "see" how to do any thesis problem suggested. And for the student this should involve much confusion, much reading, etc. So, if we ask about expansion factors, ... which, !!!, include self-education, etc., _of_course_ it'll take a novice 1,000 times longer than an expert. Hopefully one's advisor is sufficiently expert so that they can do things that much faster... as dis-heartening as this might be to a naive beginner.
When we dial-back this deconstruction to undergrad stuff, the lesson still has weight...
The dumbed-down summary's rhetorical version is "whatever factor you think is good, you should multiply by 5".
(But/and think how to avoid "time constraints"... which test things that are irrelevant...)
