1. Given that you (i.e. the Cambridge lecturers in general) are setting the questions (and not some 3rd party), surely you can a) define standard wording for common question types, that question writers should adhere to and b) vet all questions so that they are checked for possible ambiguity and that they adhere to a set of standards?

2. You refer to the problem of guessing the question writer’s mind; can you not fix this problem (to a large extent) by having each question more explicitly state what assumptions the student may make, and perhaps a general statement at the top of the exam paper of what kind of results may be used without proof? (One potential problem that occurs to me here though, is that a “statement of usable results” is also a fairly nice “statement of hints about what the professor wants you to do” which may make the question trivial, so maybe I’m underestimating the subtlety of the problem)

Thirdly, in general, a mathematician has better tools for removing ambiguity than most other professionals; they can narrow down the type of objects that may be considered by specifying a tight enough set, for example. E.g. your “ambiguous function” problem could be written along the lines of “Let A be the set of all continuous functions on [0,1]. Prove that if f is an element of A, then X is true”. That, I guess, would remove ambiguity at the expense of sounding robotic.

The question of how a good question should be written is a timely one though: I’ve seen some pretty dismal efforts in GCSE and A level exams, and those affect far more people than the Cambridge Tripos, of course.

]]>I agree: wording that is open to misinterpretation, even if it is an unreasonable misinterpretation, is not as good as wording that is completely unambiguous. But it can be hard to predict when an unreasonable misinterpretation will occur, so it’s just an unfortunate fact that wording of Tripos questions is not always optimized.

]]>Re: the question of ambiguity: why not simply choose a wording like: “Prove that for all continuous functions defined on blah, X is true” ?

]]>In response to the last point, there is scope for doubt, but not scope for reasonable doubt. Some people sometimes think as follows: the question says, “Let be an arbitrary function”; the function is pretty arbitrary; so I’ll go for that.

]]>So let me say something a bit more formal that is related to the pictorial approach, and that argues directly from the definition that is given of an integral of a continuous function.

1. Given any dissection, its lower sum will not be less than m(b – a) and will not exceed M(b – a). (I was going to rely on pictorial imagination to justify this, but I don’t need to because I am about to place no reliance on the claim.)

2. That is not enough, because the supremum of the lower sums won’t usually be any particular lower sum. You usually have to carry on dissecting more and more finely, without any end to that process.

3. But what we can say is that for any dissection of f(x) over [a, b], if we had matching dissections of g(x) = M and of h(x) = m, over [a, b], where “matching” means that the xi are the same, then the lower sums would comply with the inequalities we wish to prove. (I think I can have this on the basis of elementary arithmetic, without relying on pictorial imagination.)

4. So for any sequence of finer and finer dissections, we can have corresponding sequences of matching dissections of g(x) and of h(x), and at any point in the sequence, the inequalities would hold.

5. Am I now allowed to say that the inequalities would still hold when we moved on from particular dissections to the supremum? Intuition says yes, but rigour would doubtless demand a better justification, presumably based on the formal properties of the reals.

On the terminological point at the end, I do not see that there is any scope for doubt. Either of “Let f be an arbitrary function” and “Let f be any function” must require a proof of the result for all functions, not for a particular function that the candidate happens to like. (I conclude that on the basis of my grasp of English, not on the basis of any supposition as to the cruelty of examiners.)

]]>I’ve certainly written a lot about that, but that’s because it is something that has often worried people I’ve supervised. It certainly doesn’t reflect the amount of time I would recommend thinking about it in an exam: by and large I would hope that a bit of common sense and some sensibly worded questions would be enough.

So how does one word questions sensibly? I think there are certain conventions (that I have mentioned in the posts — things like that it’s OK to assume something from earlier in the course, or that there is a presumption that if you can use an earlier part to solve a later part then that is what the examiner wants), and that if there is any reasonable doubt, even taking into account those conventions, then the setter of the question should be explicit about what can and can’t be assumed — as Tripos examiners often are.

]]>The next two parts are then immediate from the Mean-Value Theorem and the Cauchy Mean-Value Theorem.

I don’t think I got any marks, but it’s the most use I’ve gotten from the Cauchy Mean-Value Theorem so far.

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