The rise of continuous assessment models and the ability to track students’ every interaction with digital learning content may allow for broader, more holistic evaluations of student potential.
This is merely a complex way of saying "metrics". Welcome to the real world Oxford kiddies.
I wonder to what extent there is anyone who could cope with the Oxford course but who could not breeze through A-level Further Maths. I entirely understand that the converse is false: there are people who are extremely good at exams but who simply couldn't cope with the Oxford course.
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.
My Cambridge interview tested for this explicitly. They gave you 45 minutes to prepare answers to 8 'further maths' type questions, then you had to solve some creative problems at the end. These creative problems required knowledge of further maths concepts but were unusual so you needed time to think about them and develop a mathematical framework for the solution, this therefore tested that you could do the further maths questions easily and quickly and that you could solve new problems.
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.
Am I wrong to think that the stretching of the band does not have anything to do with "how far along the band" the mouse will travel? Maybe it's my English as a second language, but my intuition is that the mouse still travels along the band as it expands, and you would probably need a point of reference, e.g. How far along the gap that the elastic band bridges to make the question more challenging.
The mouse is walking along the band, while the end of the band moves away, however the mouse is also being carried along by the expansion of the band. I think the best approach is to think in terms of the fractional coordinate of the mouse along the band, and work out its effective velocity in this coordinate.
Correct, if you structure the problem as the proportion completed at time t relative to the proportion at time t-1, then you can sum in the limit as t goes to infinity to show surprisingly that he gets to the end under both scenarios.
The elastic band could expand in both directions or either. If I assume the band expands so fast that I remain at the same point with regards to a point of reference, have I not still moved by 2m along the band after a minute? I guess it's more about the semantics of how moving along the band applies.
Also how the expansion carries the mouse has to do with its movement. E.g. if it's galloping it would not carry it.
Totally a possibility. I know lots of people who are skilled problem solvers, but sometimes do poorly on exams because they simply aren't motivated enough. That was definitely the case for me entering college. I had work experience that proved I could do complicated computer science work, but I didn't have great exam grades, so I had a hard time getting into the department.
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.
As someone going through this process at the moment, I find it unlikely that a candidate who could naturally cope with the Oxbridge courses would struggle with the further maths content in isolation. For all its flaws, the course is largely a disjointed collection of fundamental problem solving methods and concepts which are reintroduced at University level anyway.
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.
It's important to note the distinction between breezing through course material and breezing through an exam.
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.
My experience is that there are three levels of maths ability among people who breezed A-level, or at least passed it well enough to do university maths:
- 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.
Liked the tone of your post. Did not detect any bitterness. Does your mathematical background serve a use in your current career, which I assume is as a dev of some sort since you are on Hacker News?
One on on interviews don't scale. I feel like it's the biggest problem in any admissions process; the best metrics are one's that don't scale, so you're forced to rely on less optimal ones.
It's not a problem of scale, it is a problem of fairness. I went to one of the two and I can tell you people lose their minds over everything being anonymous and fair even when it is as anonymous as it gets.
When human assessors are involved, it becomes more subjective. Imagine having your assessment with the instructor of your strongest subject vs your weakest subject.
Very fair question - alas, I did not have the authority to change the system (and, for selfish reasons, would not want to as my exam scores flattered my actual ability).
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.
Do you think you're confusing existing "bad exams" with all possible exams?
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?
This is just speculation but I have some ideas of what effects might be in play here.
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.
This article speaks directly to the problem that Art of Problem Solving is trying to address. https://www.artofproblemsolving.com/
Richard Ruszcyk wants to teach kids to think, how to solve novel problems, and hates the normal approach of teaching rote formulas. No student steeped in the AoPS way would crash during the Oxford interviews. An Oxford interview is their "middle school normal".
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.
I didn't do the A-levels but from my understanding of them I have always thought that they don't seem to discriminate between candidates well enough. That is to say that there just are not enough grades. I think the passing grades you can get are A,B,C or D. Most people who get in to the top universities get straight A's. I think if each of these grades was split in 3 it would make the result a lot more meaningful and would go a long way towards ensuring that the best candidates got invited to study at the university (I think the entire interview process is fraught with huge bias problems).
They recently (a few years ago) introduced A* grades for 90+/100 scores, where A was 80/100 before and so, as you note, not very discriminating. I'm not totally sure, but I think a standard offer for one of the elite universities is A*AA these days.
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.
This is spot on. These days the pressure of achieving A* grades in maths and further maths for maths applicants definitely reduces pupils' capacity for taking additional modules 'for fun'. The standard offer for top universities has become A* A* A, often with the proviso of passing an additional entrance exam (STEP / AEA)... I have seen the offer go as high as A* A* A* without the entrance exam, and achieving this is is a feat in itself, requiring both breadth and depth in several subjects, and consistently strong performances in 6 - 10 exams.
[At least when I took them a few years ago] the admissions offices get your raw results anyway - so even if it was just 'A', they could still have their own internal A* for 90+.
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.
From speaking to people that have gone to Oxford and Cambridge, they are definitely not necessarily letting in just the brightest minds. Nor do I think they truly are trying to find them - there is a definite "fit" for these places. If I had to characterise it, I'd say "smart, and very academic/scholarly". They all seem to have a deep appreciation of some subject or topic and it seems like their admissions process is very good at uncovering it.
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.
Ed Frenkel told a great story about how Soviet Russian oral exams were used to discriminate against Jews by giving them harder questions and demanding more sophisticated answers.
So I would argue the easiest way to plot log log x is to parameterize the x domain.
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.
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[ 0.17 ms ] story [ 102 ms ] threadThis is merely a complex way of saying "metrics". Welcome to the real world Oxford kiddies.
Edit: looks like Medium is having issues everywhere.
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.
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?
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 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.
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 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.
http://www.npr.org/2014/03/28/295789948/the-real-problem
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.
https://www.google.ca/search?q=plot%20ln(ln(x))
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.