USING THE INTERNET FOR TEACHING MATHEMATICS

Email lists

Excerpts from recent email discussion conducted on both lists at:
ucam-nrich-support@lists.cam.ac.uk,
maths-gcse-a@felsted.rmplc.co.uk

The original message
From Dave Harrington, Walton High School, Stafford
office@walton.staffs.sch.uk
22.10.98
One of my bright year 13 students has applied to one of the Cambridge colleges to study maths. They have asked him for a list of topics that he is interested in so that they can discuss them with him. He has asked me for advice on what sorts of topics they are looking for. Bearing in mind that he has currently studied all of the A Level syllabus, but as yet very little of the Further Mathematics syllabus, does anyone have ideas of the sort of things they are looking for?
In particular:
Should I suggest topics that he currently has little knowledge of, but then give him a crash course before the interview (Complex Numbers, Matrices, Vectors spring to mind)
Should he stick to easier, and possibly less inspiring topics that he already knows?
What sort of depth are the questions likely to be ?
Are they knowledge based or will they deliberately ask about ideas beyond those he is likely to have met? If anyone has any experience or ideas that they are willing to share, we would be very grateful.

from Anthony Back, 26.10.98:

I think they are more interested in potential rather than present knowledge.
Questions such as:
If two random chords are drawn in a circle what is the probability they intersect.
What is the probability that there will be an adjacent pair of numbers in the national lottery draw.
What is the expected number of tosses to get 3 consecutive heads with a fair coin.
How many different necklaces could be made from 4 red and 4 blue beads.
How many different rectangles could be drawn on an 8 x 8 chess board.
Probability and problems on permutations and combinations give ample scope for 'thinking' as opposed to knowing. These problems can quickly test for originality and imagination.

From David Sanders. (Trinity College, Cambridge), 26.10.98:

This is exactly right. The interviewer is not interested in what he has been *taught*, he is interested in what he has *found out*, by reading, or trying to solve questions that have taken his fancy or whatever. I would almost go so far as to say that if he hasn't done any background reading outside the course (of the Ian Stewart or Martin Gardner kind, or whatever -- 'popular science' and 'popular maths' books) then he won't get in to Cambridge. Obviously this is an overstatement, but if he hasn't then the chances are he's not sufficiently motivated. If he has, then great -- that's exactly the thing they want to know about, judging by your report of what they've asked for.
These are good examples, but I'm not sure that anybody would really find them intrinsically *interesting* (although maybe I am exhibiting some bias here...). When I was at the same stage, it was books like 'Does God Play Dice', 'Chaos', 'The Selfish Gene' etc., that were interesting, not so much what we were doing in class. Cramming is not the way to go.

From Gareth McCaughan Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@dpmms.cam.ac.uk Cambridge University, England, 26.10.98:

WARNING: frank answer follows.
They're probably looking for the truth: he should tell them what areas of mathematics he's actually interested in, rather than picking topics he expects them to like.
If there *aren't* any areas of mathematics he's interested in, then I suggest he probably won't be all that happy studying mathematics at university, particularly such a demanding course as the one at Cambridge: there's no point putting three years of intensive study into something that doesn't interest you much.
I suppose he might be interested in *everything*. In that case, I suggest some such form of words as "I have found most of the mathematics I've looked at interesting. The things I've learned about most recently are X, Y and Z; I'd be happy to talk about those".
But, I repeat: if he isn't really interested in anything much then he shouldn't be doing mathematics at university; and if he is interested in some particular things then he should be honest and say what they are, rather than trying to pick topics that will impress.
One thing that is likely *not* to impress the interviewers is any evidence that instead of answering the question "What are you interested in?" he has answered the question "What do you think we might be impressed by?".

… a brief look at such things now suggests to him that he might find them interesting, there's no reason why he shouldn't tell them "I'm just starting to look at complex numbers, and they seem fascinating. I might be in a position to discuss them by the time I come to interview" or something. In that case, a crash course might be very sensible.
They will be much more interested in how he thinks than in what he knows.
The precise nature of the questions varies from college to college, and from interviewer to interviewer. My admissions interview at one college asked me some quite demanding questions, but they were demanding because they required good thinking, not because they required knowledge I didn't have. My wife's admissions interview at another college involved only very simple questions about one or two topics.
I would expect most interviewers to try to elicit thinking rather than recitation of knowledge; that might well involve "ideas beyond those he is likely to have met", but that isn't the same thing as "facts beyond those he is likely to have met".
I should mention that despite being a fellow at a Cambridge college doing mathematics, I am not myself involved in the admissions process; all the above is my opinion only, and I don't guarantee that it reflects reality accurately.