Comments on: A look at a few Tripos questions IV

By: Steve Collyer

Steve Collyer — Tue, 08 May 2012 21:13:23 +0000

There are a couple of things that confuse me about this question of ambiguity in questions.

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.

By: gowers

gowers — Tue, 08 May 2012 12:50:55 +0000

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.

By: Steve Collyer

Steve Collyer — Tue, 08 May 2012 10:57:44 +0000

This is a very nice series of articles. I wish you’d written them thirty years ago, though.

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

By: gowers

gowers — Fri, 04 May 2012 14:58:42 +0000

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.

By: Richard Baron

Richard Baron — Fri, 04 May 2012 14:48:50 +0000

In response to the second part (Suppose that m, M are real numbers …), my first thought, as a non-mathematician, was to sketch some axes and vertical lines at arbitrarily chosen x = a, b, draw any old wiggly line for f(x) between those lines, then draw horizontal lines at y = m, M such that they do not cross the wiggly line, and point out that as you squeeze the horizontal lines inwards without crossing the wiggly line, you must preserve the desired inequalities. But I don’t suppose I would get any marks for that, even if mathematicians have in the past expressed approval of proof by picture. (There is an example in Littlewood’s Miscellany: a secondary source tells me it is on page 55).

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.)

By: gowers

gowers — Fri, 04 May 2012 12:50:49 +0000

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.

By: woutervandoorn

woutervandoorn — Fri, 04 May 2012 11:44:23 +0000

You seem to spend a lot of time thinking about what the examiner thinks about what is and what is not allowed. E.g. can I assume fact X, or use theorem Y, or do I have to be more explicit somewhere? Do you have any ideas on how to make an exam so that students don’t have to spend the majority of their time dealing with these meta-questions?

By: Wednesday Highlights | Pseudo-Polymath

Wednesday Highlights | Pseudo-Polymath — Thu, 03 May 2012 13:21:47 +0000

[...] Tripos (whatever that is) entertainment continues. [...]

By: alja

alja — Wed, 02 May 2012 17:02:40 +0000

I always liked doing the first part of this question by coming up with an alternative definition of the integral of a continuous function in terms of antiderivatives (which is equivalent to the standard definition since all continuous functions have an antiderivative).

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.

By: Vania Mascioni

Vania Mascioni — Wed, 02 May 2012 15:02:27 +0000

Interesting to hear that even in “Cambridge English” there is a potential for confusion when using “any” or “arbitrary” in questions, and your final advice is surely sound! Way too often in my exams I have had students just pick their own example rather than carrying out the general argument I was asking for, and it’s very annoying to have to make the questions longer just to eliminate that possibility.