Is it a good idea to use past questions in examinations?

Question

When it comes to writing examinations, there are two options I am considering.

1. Write the entire exam originally (which will of course mean some well-known proofs, thereby technically not making it completely original)

2. Use a ratio of 'original' questions and actual past paper questions.

My students have access to perhaps 50 past exams to practice from. Not necessarily exams that I have written, but indeed examinations that students have been able to keep and were provided with worked solutions sets.

The theory I had behind using actual past paper questions was to encourage them to do more past papers. Perhaps not for the best of reasons (as I presume they'll be hoping that they'll get lucky and get a question in my exam that they've seen already) but motives aside, they may complete a larger volume of past papers.

I'd like to hear some thoughts regarding this. Is this a good idea, is this a terrible idea? I can imagine that a few students may get lucky but in terms of the majority of the cohort (around 60), it may boost overall past paper attempts.

Answer 1

It depends on your goals for the examination. As you are mentioning proof, and using some well-known ones, it seems that you are aiming for more of a "show me that you've thought about all of these types of proofs" style.

As they have access to a large collection of examinations, I would, perhaps, pick a few exercises from those of the most important ideas and run with those with a few 'newer' style problems. As they have a large collection, you can perhaps place emphasis more "write down the core idea behind the proof of 'Big Theorem X'", or knowing how to prove a key lemma that is critical to a 'Big Theorem'. Other options are just having them prove an inductive step, if that is a tricky part, or a special case of some Big Theorem (or Large Lemma) or even a homework problem. The key point is that they can show you that they are familiar to the key proof strategies for the particular course you are teaching. While it sometimes boils down to "memorize this list of 100 proofs", hopefully, along the way, they pick up the proof strategies to simplify their list.

Answer 2

It depends on your goals for the examination. Typically the goal is to assess students' understanding, or measure them in some similar way. Will having some prompts on your exam that are identical to prompts some students have answered before throw off the accuracy of this measurement? I don't think so. Especially since there's a positive correlation between how well a student does on an exam and how much time they spend studying (looking over past materials) anyways.

A relevant anecdote: a past instructor of mine gave us access to every midterm and final exam he wrote in his decades-long career. In that week before each exam I learned so much working through those past exams. Granted, it was a CS course and the exam was assessing "can I do this" more than "do I know this."

Answer 3

I think it's a great idea. The point is to drive learning, not trickiness. If they've mastered the typical problem set (including the workings), that's fine. You can't know how to work 50 tests without having learned the topic.

I would try to avoid tests that are all or nothing (single question). And avoid infrequent exams. I.e. biweekly is better than "final/midterm only". This will reduce the luck component sufficiently. (Also, many other reasons why you should not give infrequent super high stakes problems, regardless.)

Answer 4

In general there is no good reason to repeat questions as such (even if they are not made available, one should assume they are available, being circulated online or via messaging). If students know that questions might be repeated and solutions to past exams are available, many students will simply memorize the solutions.

It's reasonable to put similar questions from year to year, as one generally wants to test similar competencies from year to year. However, excessive similarity from year to year generates a situation where students learn how to solve particular problem typologies rather than how to do mathematics. Being able to solve problems of a particular form and structure is a particularly useless skill, and it is a bad idea to encourage students in this direction because they already receive a lot of push in this direction, from family, friends, themselves, and paid tutors.

It will work if there are variations in the problem statements (what appear to an instructor minor variations can be far less minor from a student's point of view - I give an example below) and if the subset of problem typologies used on any given exam is a small subset of all those available (that is to say, over the years there are 15-20 different kinds of problems, in any given year maybe 4 or 5 appear). This makes the exam less predictable but retains the desired effect of guiding students in their preparation.

What I mean by minor variation is the following. I can ask

1. Find all $r$ such that $Ax = rx$ has a nontrivial solution.
2. Find all $r$ such that $(A - rI)x = 0$ has a nontrivial solution.
3. Find the eigenvalues of $A$.

These are mathematically equivalent problems, but their statements are not equivalent in difficulty (they are listed in order of decreasing difficulty). In particular, many students are unable to solve 1 because they can't see that it is the same as 2; many students learn a procedure for finding eigenvalues (calculate some determinant) and don't relate eigenvalues to the equation appearing in 2, so 3 is easier than 2. One might put some variant of this question some years (varying A, and not every year), varying the difficulty, by changing the phrasing as above, depending on where one wants to center the exam's difficulty and what other problems appear on the exam (maybe form 3 is ok this year because other questions are more conceptual).