Results and explanation

I’ve had a healthy number of responses to my question from the previous post. In case you are reading this post without having read the previous one, I shall continue after the fold, because if you read on it will render you ineligible to participate in the little experiment I am conducting.

Every year in Britain, at round about this time of the year, we have the same debate. The GCSE and A-level results come out (these are taken at the ages of 16 and 18, respectively) and they show a modest improvement on the results from the previous year. Some people can be relied upon to seize on this as evidence that exams are getting easier. Others can be relied upon just as strongly to leap to the defence of the latest generation of schoolchildren, praising their hard work, and that of their teachers, and condemning those who, they claim, are trying to belittle it.

So are the exams getting easier? I certainly think that today’s A-level papers in mathematics are much easier than the ones I took, and am planning to write a post in which I shall attempt a detailed demonstration of this, though actually finding the old question papers (which I think I have somewhere) may be more of a challenge than comparing them with today’s papers.

But I thought I’d prepare the ground by taking the questions from two A-level papers and seeing whether people could tell which paper was easier if they did not know anything more than what the questions were. I then had to decide which two papers to choose. What I’d have liked to do is choose two papers from about 15 years apart, but then I had the problem that from time to time the syllabus and exam format gets changed. I wanted to avoid any possibility of somebody’s arguing that an apparently easier syllabus was in fact just as hard because it “developed a different set of skills”. So I needed two papers that were more directly comparable, and I also wanted one of them to be very recent. Unfortunately, this forced me to take the other one to be fairly recent too.

The two I ended up choosing were Further Pure 1 from January 2005 and from June 2010. The board was MEI, which stands for Mathematics in Education and Industry. (Links to these and all the intermediate papers can be found on this page on MEI’s website.) I would have liked to have separate polls for every question, since there was almost a natural bijection between the two papers, but the marks on offer weren’t quite the same so in the end it seemed safer to split the papers up by section and no further.

Here, a few days later, are the results of the polls as they currently stand.

Section A.

205 people have voted, and their votes are as follows.

A1 is probably easier than A2 … 49 votes
The two sections seem to be of equal difficulty … 49 votes
A2 is probably easier than A1 … 43 votes
I am certain that A2 is easier than A1 … 35 votes
I am certain that A1 is easier than A2 … 29 votes

A1 was Section A from January 2005, and A2 was section A from June 2010.

Section B

130 people have voted, and their votes are as follows.

The two sections seem to be of equal difficulty … 33 votes
B1 is probably easier than B2 … 32 votes
I am certain that B1 is easier than B2 … 29 votes
B2 is probably easier than B1 … 26 votes
I am certain that B2 is easier than B1 … 10 votes

B1 was Section B from June 2010 and B2 was Section B from January 2005.


Obviously, these results should be treated with extreme caution, so I’ll just make a few remarks (as uncontentious as possible) and see whether anyone else has anything to say.

1. One would not expect two papers that are just five years apart to be radically different in their level of difficulty.

2. Even if one of these two papers is found to be clearly easier than the other, these are just two papers from one examination board in one subject, so the result could be put down to random variation. (I should say that I chose those two papers because they were at the two ends of the range that was available to me, and not because I made some prior judgment about whether one of them looked easier than the other.)

3. One might argue that even if the two papers were of identical difficulty, they are so similar, as are the ones in between, that candidates taking the exam in 2010 were at a significant advantage over candidates taking it when the format and syllabus had just changed. (I haven’t checked what the papers just before the change were like, so this point isn’t necessarily a valid one.)

4. It is amusing that several people can be certain of a proposition, and several others can be certain of the opposite of that proposition. This phenomenon does have possible explanations: for example, maybe one person finds one kind of question difficult and another finds another kind difficult, so their perceptions of difficulty are genuinely different.

5. The number of people who found 2005A easier than 2010A is exactly equal to the number of people who found 2010A easier than 2005A. However, those who found 2010A easier were a little bit more sure of their judgments.

6. The number of people who found 2010B easier than 2005B was 61, compared with 36 who found 2005B easier than 2010B. Again, those who found 2010B easier were more confident of their judgments: 29 people were certain that 2010B was easier, compared with 10 people who were certain that 2005B was easier.


Based on those results, I find it tempting to conclude that the 2010 paper was genuinely a little bit easier than the 2005 paper. However, I do not claim that this experiment establishes that beyond all reasonable doubt. I also have arguments that are based on the questions themselves, but I would prefer to wait to hear what others think before giving my own judgments. It would also be interesting to conduct further experiments along these lines. I suppose I could just repeat this one: people would know what I was doing, but perhaps that wouldn’t matter too much. If you have suggestions, I’m ready to listen to them.

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49 Responses to “Results and explanation”

  1. plm Says:

    I think your point 3 is the most probable main factor in results improvement.

    Perhaps this can be checked in other series of exams.

    Also you could check exam results from years where major program changes were introduced, for instance compare them to the previous year. (As you perhaps parenthetically mention.)

    If you have graphs, a bit more data for us we may understand better.

    • gowers Says:

      That reminds me of a point that I forgot to make, which is that I do not know whether the results improved in mathematics specifically. Still less do I know whether the results for the MEI board in particular improved between 2005 and 2010. And even less than that do I know about paper FP1 (if zero could be said to be even less than zero).

    • Victor Says:

      I agree with plm. In India, we used to solve problems from many years past, and knowing the 2005 questions would definitely have made the 2010 questions easier (and more predictable). So it would be interesting to see the results of the 2005 and 2010 exams. This would also tell us if students spend many hours working over previous exams.

  2. Results and explanation (via Gowers’s Weblog) « In the Dark Says:

    [...] I've had a healthy number of responses to my question from the previous post. In case you are reading this post without having read the previous one, I shall continue after the fold, because if you read on it will render you ineligible to participate in the little experiment I am conducting. Every year in Britain, at round about this time of the year, we have the same debate. The GCSE and A-level results come out (these are taken at the ages of 1 … Read More [...]

    • gowers Says:

      It’s worth following the link above, as it will enable you to see what an O’level maths paper was like in 1979. (Before GCSEs were introduced, there was a two-tier system, with brighter pupils taking O’levels and weaker ones taking exams called CSEs. But it was felt that splitting people into sheep and goats like that was a bad idea, so GCSEs came in in 1986 and unified the two previous exams.)

      If anyone has some A-level papers from the 70s, 80s or 90s and is prepared to scan them and provide a link, it would be great. (Almost as good would be if you were prepared to send me a photocopy.)

    • Mark Bennet Says:

      Tim

      I have just moved, and have found a wealth of London Board A-level material from about 1978 – 1982 (example papers etc). I haven’t got time to scan it, but could drop it in to Trinity about 24 Sept, when I’ll next be in Cambridge – interested? markbennet@btinternet.com.

      Mark

    • gowers Says:

      That would be wonderful! Thanks also to others who have already provided me with material.

  3. Qiaochu Yuan Says:

    It would be sensible to expect that the exams are getting easier if whatever organization writes the exams derives some kind of benefit from increasing test scores (e.g. funding). Is that the case here?

    • gowers Says:

      Well, a fairly obvious benefit is that the boards make money from examination fees. Since there are competing boards setting equivalent qualifications, and since schools want their results to look good (this sometimes seems more important than the education that they actually impart), there is plenty of pressure in this direction. Of course, if a paper is made too ridiculously easy too quickly, then the exam board loses credibility, so one would also expect the pressure to result in gradual change.

    • plm Says:

      If the boards are smart they can lure schools without making exams “obviously easier”: simply providing alot of past exams (like MEI does) and practice material makes results improvement probable and it is hard to blame for.

      Then the objective is to be the board with more practice material than the others. They can even make exams very slightly harder than competing boards to prove that their exams are “at least as hard as others’” -but actually not for the students who practiced their tests. This can lead to a sort of monopoly where the most popular board can keep exams as hard as the other boards’ with better marks, because students feel more confident with their tests, even if other boards copy them.

      The justifiable similarity from year to year makes accurate data easy to analyze and the design of tests much less “risky”.

      It is not so easy to analyze confidently though, knowing the boards’ executives in person would probably make clearer their incentives, or other similar information, and comparing different boards’ exams, etc.

      In any case, it may be interesting to know the reasoning/theory underlying this UK institution.

      Actually I am coming to think that this policy may be a little UKish, privatization perhaps without the required game theory/mechanism design theory analysis that it is more efficient than tax-funded government work. But I know little UK politics -much of it thanks to Tim’s posts- and I’d be glad to read knowledgeable opinions.

  4. JuanPi Says:

    Hi,
    I find your discussion really interesting.
    However, I think that the result not only should be taken carefully, but they do not help to drive any conclusion.
    “Difficulty” is something that highly depends on many subjective properties like culture, background, etc… Since there was no effort to make sure everybody is evaluating the papers with the same criteria, the results are not surprising. In particular the apparent contradictions could be just the evidence of the issue I am rising.
    Also, the way a question is described can affect a lot how people will solve it, and therefore the “difficulty” in this sense. A question can be obscure, e.g. using technical jargon and words subjects to definitions that are not given. Advances in pedagogy tend to make questions less obscure, since we want to evaluate what the subject really is able to do and not how tricky we can be in questioning. If results are getting better, it can be because questions are easier to understand, and therefore students can use their knowledge better, e.g. they can establish links faster and easier.
    No selection method should be based only on written problem solving skills. A corresponding interview where the student is ask to explain his methods and way of thinking improves a lot the quality of the evaluation.

    You have taken a hard problem into your hands. Be careful what you conclude. In human situations reductionism is almost always the wrong way to go.

    Cheers,

  5. girlsangle Says:

    This reminds me of something called the “Flynn Effect” which I learned about in the book, The Mathematics of Sex, by Ceci and Williams, on pages153-154: “The Flynn effect refers to the steady upward creep in intellectual performance over time that comes about as a consequence of better environments…Flynn has shown that every year is associated with a 0.3-point gain on IQ-tests, although this gain is masked when the test gets renormed every 15 years or so.” I haven’t read the original papers so I don’t know what “better environments” mean and there are controversial interpretations and explanations of the effect, but I thought I’d mention it because it might be relevant.

    • Sean Barrett Says:

      The Flynn effect should certainly factor into this debate. One important difference with the way the Flynn effect is measured is that subjects from different points in time take the *same* IQ tests. I imagine this is harder to do with A-levels for reasons other people have mentioned: the syllabus changes over time, and also it’s pretty standard to practice on old papers so it’s not so easy to find a cohort of people who haven’t already seen the older papers before.

      I’ve never seen an IQ test, so I don’t really know what I’m talking about, but one reasonable alternative hypothesis for the improvement in A-level grades is that the mathematical ability of 18 year olds is somehow correlated with whatever it is that IQ tests are measuring (e.g. “problem solving skills”). In which case one might conclude that the improvement in A-level grades represents an improvement in actual ability.

  6. wrongrook Says:

    I needed 9 minutes for A1 and B1, but 12 minutes for both A2 and B2 so was one of those who voted those ones certainly harder. This was partly because the arithmetic seemed slightly harder in each case, but also because I wasted quite a lot of time on A2:6 and B2:9 because of the sign errors in the questions. (In A2 u_(n-1) should be u_(n+1), and in B2 matrix M should have a -0.8 instead of 0.8) Perhaps the poll would be fairer if you could correct these typos?

    • gowers Says:

      Thanks for pointing that out. I’ve corrected them now.

    • Bryan Blunt Says:

      There is also a typo in:

      The cubic equation 2z^3=z^2+4z+k=0, where k is real, has a root z=1+2j.

      I spent a couple of wasted minutes having assumed the = should be a +.

      This and the other typo in A2 made me vote “A2 certainly harder”.

      Having then moved on to B2 a seen that M clearly wasn’t a reflection (even after all this time from A levels I can still remember a few things!), I concluded that this was a blog post about errors in exam questions.

    • gowers Says:

      That’s unfortunate. I suppose it means that the results have to be taken with even more caution than they did anyway. I’ve corrected this error too. (The problem was that I had to copy out the questions, and that was a tedious thing to do so I did it too quickly.)

  7. Anonymous Says:

    I still remember back in the late 70′s I had to worked on the chapter exercises in “Techniques of Mathematical Analysis” by C.J. Tranter to avoid being bulldozed in the exams. Most of these exercises are questions from the General Certificate Examinations and Examinations for Scholarships in Oxford and Cambridge.
    There is absolutely no doubt in my mind that there is a huge gap in difficulty between then and now.
    Students today, at least in the West, are simply not asked to perform at a higher level.

  8. Mark Bennet Says:

    The biggest difference I saw with the questions was that some of them led you into the subject by breaking the question up into smaller parts. Maybe the difference is something like the difference between “Show me you know what you are talking about, and can do the calculations” (for the broken up ones) and “show me you are confident in using your knowledge and technique in this area”. So the broken up questions seem as though they might require less mathematical maturity and confidence, which might not be quite the same thing as difficulty.

  9. Spencer Says:

    This was fun and I have a few comments (mostly conjectural and inconsequentia!):

    1. Asking “Which is easier?” might be genuinely different from asking “Which is harder?”. I somehow had “which is harder?” in my mind when I got to the voting button and had to look twice for the button I actually wanted to click (occasionally seemingly minor differences like that in an experiment like this can actually affect outcomes).

    2. It would be interesting to see what would happen if you had asked that we may only vote if we actually work through the questions.

    3. Perhaps some people worked out what the experiement was about. This could have affected voting. I guessed what it was about and, with hindsight, because I believe that exams have got easier, I think it made me want to say that one set was indeed easier rather than admit I couldn’t tell and say that I thought they were equal.

    4.Personally, I was more certain about the B’s and less sure about the A’s. I said B2 was easier and this seemed clear to me: You weren’t given any of the sketch in question 7, you had to sketch loci in question 8 (absent in B1) and experience seems to suggest to me that candidates would find the non-integer entries in the matrices more difficult. With hingsight, I think I wanted to believe that the B’s were more obviously different in difficulty because I thought the B’s were more advanced questions in the first place than the A’s and I lean towards believing that there is more serious decline in difficulty at the `top end’ of A-level questions, i.e. there is more significant difference between the hardest questions from older papers and the hardest questions now than there is between the easier questions.

    5. I should say I am pretty familiar with the system, having taken GCSE’s and A-Levels and also tutored A-level maths students.

  10. John Peacock Says:

    Interesting exercise, although not sure the time baseline is large enough to give a really clear effect. Like Mark Bennet, the differences I saw corresponded to a little more hand-holding and splitting questions into bite-sized chunks. On those grounds, I rated the second of the pair as being at a slightly less sophisticated intellectual level in both the A and B cases.

    Over a longer timespan, larger changes in the treatment of maths in schools are readily apparent. This isn’t just in the style of exam question, but in available options. I took maths O-level in 1972. I also took “additional maths” O-level at the same time, which was largely introductory calculus, and was encouraged if you were planning to take A-level maths. So there was a way then to stretch the kids who showed some capability at the subject, but I don’t believe this option exists in GCSE today.

    I now work in the university system, and I lecture mathematics for physics to first years. Several things are depressing about this. First of all, the material I cover is in many cases not much beyond the 1972 additional maths O level – but the students are three years older, and it’s so much easier to soak this stuff up when you are younger. But of course it’s not the first time they have seen calculus etc. – and if you look at the course syllabuses and exams they passed in the final years at school (with A’s in most cases), you would conclude our first-year courses could be made much more challenging. But we have learned by bitter experience that this is not the case: the students have very weak skills in the material allegedly already mastered.

    I don’t feel confident that I know why we have this “maths problem” (which I know is experienced by colleagues in universities around the country). My guess is that there are two main ingredients. One is the trend towards handholding in questions mentioned above: we trust the students less to show initiative and wrestle with a problem where the approach isn’t clear (this would be seen as “unfair”). The other likely problem is modularisation: learning topics for a test and forgetting them (the “oh that was last term” syndrome). As a result, the students lack confidence that they are on top of a coherent body of knowledge.

    But things have been going to the dogs for a long time. I have a nice calculus textbook (written in 1914 by G.W Caunt) which seems to have been intended for students at the end of public school who were hoping to win a scholarship to Oxbridge. I couldn’t think for a second of using it in my current university teaching: it’s far too advanced, detailed and serious in its approach. I wonder if it was thought to be dumbed-down compared to the standards of the 1800s?

    • Sean Barrett Says:

      The change mathematical ability of 1st year undergrads in maths/science/engineering over time isn’t necessarily a good proxy for the quality of school level teaching (or the difficulty of the final exams). The university sector has grown considerably since 1914 – I don’t know the numbers but it’s clear just from wandering around Cambridge that there’s a bunch of colleges that didn’t exist before the second world war.

      The likely conclusion is that even the top universities are taking on students with a broader spectrum of abilities than they were 50 or 100 years ago, so even if there were no change in the ability of final year school students, one would expect the ability of 1st year undergrads to decline.

    • John Peacock Says:

      Sean: agreed, the comparison with 1914 isn’t quite fair (although still striking to see what the best schoolkids were once considered capable of tackling). But university numbers at the top end haven’t changed so much over recent years. I graduated in physics from Cambridge in 1977, and I believe the numbers taking that course today are within 10-20% of what they were in my day. But colleagues there tell me that 1st-year mathematical methods teaching is now much more remedial: competence that would have been taken for granted in my cohort tends to be lacking today.

      Now in the UK as a whole, your argument does apply: a broader range of people are tackling maths, and going to university to study physics – so it is inevitable that the mean technical standard of undergraduates will decline. But I think it’s clear we don’t face just a growing tail with unchanged achievements at the top.

      Growth in numbers may be part of the problem: maths is now considered important for society, as witnessed by recent government statements that children should all study the subject to age 18. If this happens, then there is a danger that the best students will settle for a lower standard, because they will be in the top x% of their class with less and less effort. Possibly what is needed is for further maths A level to become the minimum standard for entry into undergraduate physics.

  11. Emilio 'Mnemonic' Pierro Says:

    Just another consideration: bear in mind one way students prepare for exams these days is to look at past papers (at least that’s one thing we did). And so a student taking a 2010 paper from the same (or equivalent) course that has been running for 5 years will have access to the intermediary papers.

    Perhaps one fair way of testing people is to create a test which isn’t intended to be completed in the allotted time (with the students having that knowledge).

    (My tutor at University was Dr. Tony Gardiner and I have a great interest in education in general so I’ve often wondered about what would constitute a better educational/examination system…)

  12. Anonymous Says:

    “Perhaps one fair way of testing people is to create a test which isn’t intended to be completed in the allotted time (with the students having that knowledge).”

    I strongly disagree, because these kind of tests are highly sensitive to different strategies which in turn are much more sensitive to background, stereotype threats, etc.

  13. Erik Says:

    I voted A2 (certainly) and B2 (probably). I went back and looked through the problems again and, even after reading your explanation, I think I would still vote the same way (or maybe change A2 to probably). The reason is that although, for example, B2 may take longer to solve, the difficulty of each problem seems more of uniform.

  14. Micrographia Says:

    The marks in these two papers are not, strictly speaking, comparable. The raw mark achieved in a module is converted to a UMS score, and these form the basis of A-level results. The conversion is different for each paper, the idea being to account for slight differences in difficulty across the years. The grade-equivalent thresholds for June 2010 can be found here: http://www.mei.org.uk/files/papers/June10Marks_Grades.pdf Unfortunately, the MEI website only contains these documents for as far back as June 2005, not January 2005, so we can’t tell which of the two papers had marks which counted for more.

  15. Uwe Stroinski Says:

    Not giving any information on your goals has some influence on the voting. The two parts are so comparable in difficulty that I thought they were from the same exam maybe given to different groups of students who now complain. Because of how the questions where formulated I had the impression that you have swapped the parts. Since on that educational level the ‘how ‘ is very important I have slightly favored one of the combinations A1/B2 or A2/B1 (I do not remember which one though).

  16. telescoper Says:

    I have A-level papers from Oxford & Cambridge in the following subjects: Mathematics, Further Mathematics, Physics & Chemistry. I also have the special papers (Paper 3) for the last two of these. I’d be happy to scan them in and put them up on slideshare if anyone is interested….

  17. Which way do you carry the one Says:

    How can you be certain that the papers were harder if you don’t have any old papers?

    lots of love,

    Idiot who’s 2011 Part III is only worth tupence ha’penny in old money

    • gowers Says:

      My exact words were “I certainly think that” which doesn’t quite mean “I am certain that”. The reason I think it is simply that the modern A-level papers I have looked at look much easier than I remember them being. I admit that that reasoning is fallible: I have had much more mathematical experience in the interim, and my memory of what the papers used to be like is rather vague. That is why I plan to get hold of some papers and look into the matter properly. FWIW I think Part III standards have probably held up much better than A-level standards …

    • Which way do you carry the one Says:

      Apologies I was being a bit glib. I have become rather cynical about how every generation seems to think that the standard of education in the world steadily rose throughout human history right up to the point that they were 18 when it went in a nosedive along with the quality of popular music and people’s manners.

      Most metrics of human achievement advance steadily (say, the speed of the UK under-18 400m finalists to pick something completely random), not because of any real innate change in the little fellas but because they are trained, ie. taught better.

      There is blatantly grade inflation, because exam boards are ridiculously “marketised” and tempt schools to buy papers by giving out A’s – I know this for a fact because my dad is a headmaster at a private school which switched to Welsh Board for modern languages because its sister school recommended doing it to boosts results (it worked! Yay!).

      However, this has nothing to do with the actual difficulty of the exam or more importantly, with the quality of the education. Prices are universally higher than the 70s but some things (like computers) have got cheaper and some things (like houses) have got more expensive.

      Why don’t every year we see reams of articles about how, oh my god, university entrance requirements have gone way up again for the 30th year and now Cambridge demand much more than three As! Why in my day you could get in with two Bs and a C!

      Well, because that doesn’t allow the generation currently in power to feel smug and superior.

      [ I'm not accusing you of most of the above by the way, just general annoyance with British press, and should probably say a big thanks for your own contribution to my wonderful education : ) ]

      [P.S. I think that looking back at papers from a subject you went on get a FM in does skew your perspective a bit - certainly when I was helping someone prepare for STEP I thought, wow these have got easier and in fact it was the very paper I took! Try looking at modern papers from the A Levels you *didn't* carry on with!]

      ——

      tl;dr – It’s artifically easier to get an A but educationally standard are higher and school is still challenging. Haters gonna hate.

  18. An A-level Physics Examination Paper, Vintage 1981 « In the Dark Says:

    [...] A-level examination paper to see what people think about it. It might add to the discussion over on another blog I read [...]

  19. Danny Black Says:

    I remember doing Further Maths 20 years ago. The level of toughness for the questions was the same for the year I did it and the year before. It is just they cut out about half the topics you would be examined on. I know people who claimed the new exam was “just as hard”.

  20. Mr. B Says:

    gowers – “Well, a fairly obvious benefit is that the boards make money from examination fees. Since there are competing boards setting equivalent qualifications, and since schools want their results to look good ***(this sometimes seems more important than the education that they actually impart)***, there is plenty of pressure in this direction. Of course, if a paper is made too ridiculously easy too quickly, then the exam board loses credibility, so one would also expect the pressure to result in gradual change.”

    maybe i’m the a-level maths teacher that bites on this one, but I can’t let this go. unfortunately this is all schools nowadays, but it is not the fault of schools or teachers that this is the case.

    what you are saying is like saying a football club finds winning matches more important than playing attractive football, for example. essentially all the football club is assessed on is whether it wins football matches or not, yes it may have a lovely half time pie, or may bring through youngsters with excellent passing skills, or have the biggest attendances for example, but at the end of the day the only output that they are assessed upon is how many football matches they win.

    the only output schools are essentially assessed upon is exam results. yes other factors come into play, but if exam results are seen as poor then it cannot be considered a good school. exam results provide the headline figures for a school. it is not the school that decides what outputs it is assessed upon, it is the culture that we live in.

    so that a school chooses an exam board that they perceive would give them enhanced gcse or a-level results should not be surprising, but it feels hard to critisise schools for wanting to preserve their position in a market the government seems intent upon making more and more competitive.

    and i’m only speaking about the state sector here.

    by the way:
    thanks for the excellent blog (speaking as someone who sat in your lectures after my 2004 further maths A grade ;o) )

    • gowers Says:

      I completely agree that it’s not the schools’ fault. In my view a large part of the fault lies with those who take the data and reduce them to a single number so that they can produce a total ordering. If they do that, and if there are real consequences of the ordering, then schools are acting rationally if they choose easy exam boards or push their pupils into subjects where it is easier to get good grades. I think what I would advocate is making available a much more complex set of data. (The practicalities I’m not sure about, I’ll admit.) Amongst other things I’d like to see an anonymized list of pupils, the subjects they took, and the grades (and module scores if available) they got in each subject. I’d also like to know what those same pupils got in GCSE. That way, if some newspaper decided to convert the A-level scores into points and rank schools by the average point score, one could point to the data and make more refined observations, such as that people took more traditional subjects in a particular school, or that the improvement since GCSE was greater, or that there were lots of people getting almost full marks in their mathematics modules, etc. etc.

      In fact, I’d like to be a bit fascist about it. I think that as much data as possible about schools should be available, for the benefit of parents of the pupils at those schools and for the purposes of identifying genuinely failing schools, but league tables should be banned.

      Having said all that, I also think that some exams, such as GCSE maths, are very very easy for some people, such as anybody who ends up reading mathematics at Cambridge (but not just those people by any means). I therefore think that the way to teach people in top sets at schools is not to work towards those exams but just to teach them maths at the pace they can manage. They will then discover that they can do GCSE when it comes round. (I should make clear that, like Tony Gardiner, I don’t advocate acceleration so much as a broadening and deepening. So I’m not suggesting teaching integration to 13-year-olds, even if they can manage it. But making sure they really understand logarithms, or can derive the formula for a quadratic, or can do proofs in Euclidean geometry — all those would be wonderful.)

      Finally, let me try to preempt any criticism by admitting that I have no experience of teaching in schools. Undoubtedly, implementing what I would like to see implemented would not be easy. And probably there are many teachers who are, against all the odds, trying to teach in the way I would like to see. I’m very pleased to hear that you (whoever you are) are in the teaching profession, transmitting some of the ethos of real mathematics to the next generation.

      PS I support Arsenal …

  21. Anonymous Says:

    Improvement on the results from previous years could also be due to lower grade passmarks.

  22. student Says:

    Students can often do better on an exam than their depth of understanding might reflect. A curriculum is always finite and specified prior to the exam. Students can therefore learn how to answer the range of questions they are expected to know without necessarily having a `deep’ understanding.

    That is, there is some distinction between student understanding and the difficulty of question they can answer, when they have been told beforehand what they need to be able to do. E.g. “Prove: difficult theorem” is a lot easier when they have memorised the proof already, than when a theorem to prove is chosen that has not been seen before.

    Basically: testing never consists of a random selection of questions of objective difficulty because students know what they will be asked beforehand. This relates to the point on having seen past exams.

    In the book on algorithms and problem solving – How to Solve it, Modern Heuristics (Not the Polya book) – the authors give an example of a high school problem requiring the use of the triangle inequality that they gave to a range of subjects, including math professors etc. They note that the problem is a lot easier when it occurs in the context of triangle inequality problems (e.g. as an exercise at the end of a chapter on the topic) than when presented in a vacuum. Many of the subjects struggled to first identify the correct technique, but if told the problem is simple.

    So while it is interesting whether the `vacuum difficulty’ of questions has changed, it is also relevant to what extent material requires students to recognise problem types, generalise results etc, i.e. use problem solving techniques. There seems to be more discussion of problem solving as an emphasis these days in education circles, though I don’t claim reliability for that observation.

    Having said all that, I do think that a lot of material is `dumbed down’ these days. To me it has something to do with us becoming less `harsh’ on our students – more afraid to fail them, whether or not that has anything to do with external pressure or instinctive feelings. To use a crass analogy, professors may be like parents who grew up poor themselves but made it to a cushy middle class lifestyle and now moddy-coddle their children, leaving them `soft’. A lack of tough love, say.

  23. Dominic Yeo Says:

    A remark that might also be made is that a reasonably solid student (ie one who doesn’t get fazed by slightly different manifestations of the same general idea) would probably do reasonably well in the 2010 paper even if his entire study consisted of reading the 2005 paper and a model solution or markscheme. I think that, while mathematics A-level contains far more interesting material than say physics, it is unfortunate that the papers tend to be so formulaic.

  24. Costermonger Says:

    I graduated in Mathematics more than fifty years ago, being one of the first school pupils to take the new A-levels (replacing Higher School Cert). I still have a copy of Barnard and Child (‘A New Algebra’, 1946) that I used for part of my A-level. If anyone wants to see what was expected of school pupils moving to serious mathematical studies they should find a copy of this book. Twenty years later my eldest son also took A-level mathematics. I can tell you it was much much easier than what I had to face. Now my grandsons are contemplating A-level choices and saying ‘mathematics is too hard’ – what a laugh! I tackled all the questions in the other post, and struggled with some – the memory is not what it was – but I am in no doubt the A2 and B2 sections are easier.

  25. Costermonger Says:

    When I took A-levels, all the papers were set by examination boards ‘owned’ by universities. In my own case, my papers were Joint Matriculation Board maths papers. I later found out that the questions were mainly set by academics and senior sixth form teachers. In those days, the Government kept its nose out of the examining process and the various bodies valued their standards and reputations. Standards were high. I am not sure when governments began to take an interest, nor when examining boards began to become divorced from universities, but both of these moves were malign as far as standards are concerned.

  26. Mark Bennet Says:

    I once (over 10 years ago) did some remedial sessions for an A-level physics student whose only knowledge of quadratic equations was how to solve them my putting the coefficients into a calculator. (A in Maths GCSE).

    Since the properties of second-order differential equations are rather fundamental to physics, even in one dimension, I was shocked at how little insight she had into what was going on (which I was taught when I was 14).

  27. Emilio 'Mnemonic' Pierro Says:

    Has there been any response from anyone who didn’t take Mathematics beyond A-level (or equivalent)? Bear in mind anyone who has, which goes for most if not all people in this thread, will make up a very specific demographic of the population. The purpose of a National Curriculum is to cater for the whole spectrum, surely?

    Another thing to bear in mind is “National Curriculum” itself is a very idealistic concept. My classmates from elsewhere in the country and different types of schools had studied pretty different things to me. We never did any Geometry or Logic or Set Theory at my school (of whom Eric Temple Bell is an alumnus…) and even the Combinatorics we studied was only as a circumstance of probability theory. No graph theory! Shocking!

    I definitely think Logic and Set Theory would go a long way if introduced pre-University, not just for those wanting a career in research, but some understanding of formal logic would be a great addition to the Curriculum. Obviously I’m not expecting a proof of somethin by Gödel, but just an understanding of how language and Logic differ.

    The sad thing is, many people (my family included) genuinely thought I spent my time at Uni doing long division, which is a great shame.

  28. Costermonger Says:

    In response to Chris above, I should have made clear that it was Vol 2 to which I was referring. The page illustrated is from Vol 1. The other point I should make is that it is not just content but the sparseness and rigour of approach that is notable.

  29. James Heather Says:

    I’m pretty sure that if you ask people on here you’ll get photocopies of quite a few old exam papers. I’m pretty sure I’ve got my A-level Maths and Further Maths papers (1993).

  30. telescoper Says:

    I’ve now posted the 1981 A-level Mathematics Examination that I took (both papers).

    http://telescoper.wordpress.com/2011/09/26/advanced-level-mathematics-examination-vintage-1981/

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