The Oxford maths interview, shaking up admissions
higheredrevolution.com
higheredrevolution.com
Almost all of those on the Cambridge maths course (which I'm taking to be similar) found A-level maths/further maths very easy. Of course, the admissions process selected for such people, but I think "finds A-level maths content very easy" is a prerequisite for the course.
The 'creative' problem I had was this: A mouse walks along an elastic band at 2 meters per minute. The elastic band stretches by 1 kilometer every hour. How far along the band does the mouse get? If you got that right, then they gave you a stretch question where the band doubles in length every hour.
Also how the expansion carries the mouse has to do with its movement. E.g. if it's galloping it would not carry it.
That's not to say I couldn't have done well on the tests, I just didn't care enough to study hard for them because I had more interesting things taking up my time.
Now, that's not to say there aren't problems with that philosophy too - but it's certainly one reason.
The exams themselves however are an entirely different matter, and I know of several capable students who have had their university prospects diminished by poor performance in the exam room - whether due to anxiety, poor 'exam technique' (less relevant for maths of course), or other factors.
Notably, Cambridge interviews a high proportion of candidates who apply for maths, which presumably enables them to take some of the emphasis away from the exam results on paper, and give much more consideration towards the logical thinking and problem solving skills useful for the Tripos.
I have taught plenty of students who possessed a deep understanding of mathematics but could not perform within the confines of a structured exam. They may be berated for performance or even lack of preparation, but given that problem-solving is the main criteria for admissions this surely reflects the limitation of the measuring tool.
Similarly, many students who breeze through A Level exams may have thought they'd mastered topics like calculus, but it turns out the exams only called on their computational fluency. The moment they are pushed conceptually and asked to formulate their thinking with the rigour that undergraduate mathematics, they come unstuck.
- fundamentally does not grok University-level maths at all;
- gets "ordinary" university maths (ie redbrick and "below"), but would not be able to cope with Oxbridge level;
- Oxbridge level
There are stark step-jumps between these three, and exam results are not a good indicator of which one an individual actually falls into. In particular, I believe you can be a good jobbing middle-level mathematician by learning the relevant stuff, providing you have the basic aptitude for it, but you need a real gift to be able to cope with the top level, and that's something that an exam format cannot elicit.
Speaking personally, I did an interview at Cambridge, which served the excellent function - for both parties - of determining that I was not in the top group.
First the exams test knowledge, not problem solving ability, which is what he wants. It may be possible to design a test for problem solving ability. That's basically what IQ tests are designed to do. As I understand it, the SAT test used to be like an IQ test and highly correlated with IQ.
Second exams probably do correlate with IQ (it correlates with literally everything), but correlation isn't enough. You can't just take the top n results from a test. You can use a test as a filter, e.g. sampling the top 5% of results or whatever. But when you set the filter too high, you just get outliers.
E.g. the top 100 people who are freakishly good at taking tests, and nothing else. Maybe they have abnormally good memories and just remember everything they read in the textbook. Or maybe they have parents that pushed them to study excessively, or just did so on their own.
You only should use tests as a filter, a minimum standard. Not optimize for it directly.
The other thing everyone from there talks (whines) about is the sheer volume of work they are expected to complete, and for that, exams seem like a good (albeit imperfect) proxy for gauging this.
I'm not sure I like the idea. Only having to hit 80 reduces stress quite a bit, and it gave me time to do extracurricular things like programming, and way more maths modules than anyone should :) If I had to hit 90+ in everything, I would have probably been risk adverse and cut down. It also feels like it selects a bit too much for people who over perform in exams. That said, I did have to send in my raw module marks anyway when I applied ~10 years ago, and did hit 90+ anyway.
Surprised nobody's mentioned STEP, three exams at least two of which are required for the mathematics tripos (and I think engineering at some colleges) and is significantly more likely to require preparation.
I don't think any applicant would be worried about FM - because if they were worried about anything it would (should) be STEP.
But then, uh, I may be biased...
But as I understand it, they can't make an offer based on that, so they're already obliged to accept you by the time they get those results.
The AoPS text books are the best math textbooks I've seen anywhere. The online classes are great, but the pace is blistering. But most kids should use the AoPS books, just at a more suitable pace. (And by blistering, I mean my daughter was very seriously challenged to keep up with her peers all the way through single variable calculus, where AoPS tops out. She then enrolled in multi-variable calculus at a local engineering university at age 13 and blew away the curve.)
AoPS is an example of where learning can go. AoPS teaches how to think, not how to takes tests.
When human assessors are involved, it becomes more subjective. Imagine having your assessment with the instructor of your strongest subject vs your weakest subject.
Worth noting that written exams do not feature at all in research. There's a recognition at this level that mathematical understanding can not be captured through blunt testing instruments.
I don't recall ever sitting an exam with 10 questions where if you got 2 of them out you'd come first. You'd mark such a "hard" paper based on "this is a reasonable avenue of exploration", "this is a good idea that won't work", "this approach will prove ultimately futile but given the student hasn't seen anything like it before it's definitely worth some points."
Why are you trying to achieve just that in an interview?
Research isn't done in interviews any more than research is done by sitting exams. However sitting down and exploring ideas with a pen and paper by yourself is likely closer than "creating an impression" in an interview. An interview where you, the interviewers, are saddled with the irksomeness of having to roughly account and adjust for your bias as much as you can be aware of it (does she really look like a mathematician? I didn't like sportsman Ben's approach as much as bespectacled whoever - is the impression based on the maths alone or is the form of it's presentation fairly important).
The huge advantage of grading an exam is you can exclude all of that gumf we all carry (with differing expressions of it) by simply not knowing who wrote it. Not their name or anything.
The other big win is that if you publish the blessed thing you can make such exams more normal. Budding mathematicians might practise such skills and get better at them, younger.
As it is you're selecting for the children of mathematicians who get such practise at home and excluding the very smart who honed their exam skills alone because that's all they encountered.
I also got very flattering scores on mathematics exams but I wouldn't mistake that for anything beyond my ability to act the student performing seal at that particular game. Any exam where it is remotely possible to get 100% does not test anything like creativity - creativity is the thing that is most important, no?
It's weird. Fortunately we have Wolfram Alpha so it was trivial to quickly see it: http://www.wolframalpha.com/input/?i=plot+log(log+x)
For f x = log x where log has base b, we know that f b^0 = 0, f b^1 = 1, etc. what does g x = log log x look like? Well, we know log x is undefined for x < 0, which implies g is undefined for x < b^0 = 1, or b^b^0. More generally, you can say when f x = y, g b^x = y. So we know that g intercepts 0 at b^b^0 = b. You could do this for several more points for y = 1, y = 2, and so on, or you could draw a graph of log x and just relabel the x-axis to be 1) shifted by 1 with the x-intercept at b and 2) scaled logarithmically.
The important parts to include on a sketch are that the function is monotonically increasing, has an asymptote at x=1 [0], intercepts the x-axis at x=e (approx 2.7) and quickly becomes almost flat. Both graphs agree on these points.
Wolfram Alpha is using the complex logarithm (https://en.wikipedia.org/wiki/Complex_logarithm) to extend to x < 1; Google isn't. I'm fairly sure an interviewer would expect a sketch that looks like Google's, perhaps with a comment that achieving ln(ln(x)) < 1 is possible (only) if the ln function is allowed to take complex values.
[0]: For values of x < 1, ln(x) < 0, so ln(ln(x)) is not a real number.
Edit: looks like Medium is having issues everywhere.
This is merely a complex way of saying "metrics". Welcome to the real world Oxford kiddies.