How should mathematics be taught to non-mathematicians?

telescoper:

This post from the estimable “Gowers’s Weblog” passed me by when it was originally published in 2012, but I saw the link on Twitter and decided to repost it here because it’s still topical..

Originally posted on Gowers's Weblog:

Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18 . An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?

A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might think that a course called Use of Mathematics would teach you how to come up with mathematical models for real-life situations, but these questions did the opposite, and still do. They describe a real-life situation, then tell you that it “may be modelled” by some formula, and proceed to ask you questions that are purely mathematical, and extremely easy compared with A-level maths.

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Credentialism and Overexamination

Only time for a quick post this morning as I have to go into the department to get my things ready for tomorrow, when the Autumn Semester starts and I have to begin lecturing (at 9am on a Monday morning). Anyway, the text for today’s sermon is provided by Ed Smith’s Left Field column in the New Statesman, the latest issue of which I read yesterday. His topic is the rise of credentialism and the resulting excessive amount of examination in the British school system:

It is now widely accepted that British pupils are excessively over-examined. Teachers are so busy focussing on examinations that there is little time left for education. Exam-led cramming has become the year-round norm – like an election campaign that consumes the whole political cycle. Exams are obviously necessary. But there is an optimal amount of assessment and it has been far exceeded. Grade inflation – notwithstanding this year’s controversial “crackdown” – is simply accepted as a fact.

It’s well said, and it’s not just the school system that suffers from disproportionate emphasis on assessment over education. It’s rife throughout the university system too, starting with the reliance on A-level grades as criteria for assessing students’ suitability for university study, through the “modular” undergraduate degree programmes with examinations twice a year for three or four years.

We examine far too frequently and the effect of this has been to turn the entire education system into a meaningless exercise in box-ticking.

It is an unfortunate irony: in our age of credentialism, exams have never mattered more. And yet they have never been more unreliable as gauges of academic quality.

I’ve felt for some time that in my discipline, Physics (and Astronomy) A-levels are virtually useless as indicators of the suitability of a student for doing an undergraduate degree. Some of the very best students I’ve ever had the pleasure to teach came into my university with modest A-level scores; and some students who came in with perfect grades at school never adjusted to the different, more independent type of study required of an undergraduate.

As Ed Smith points out, the increased emphasis on examination grades hasn’t expanded opportunity either.  It may appear to be fairer to base university entrance or award jobs on examination results rather than, say, interviews, but this has just led to a system that can be easily gamed – private tutors, cramming, re-sits to improve grades, and so on. He rightly concludes that the “correlation between exam results and intelligence has been steadily weakening”.

So what’s the alternative? Smith mentions the admissions process at Harvard University, which famously ignores high-school grades and relies on its own interview system. Interviews can be very biased if carried out in an inappropriate way. Subjecting a young person to a 30-minute grilling  in a room with a complete stranger can be enormously stressful for applicants who are shy, and would also play into the hands of those whose educational background has involved specific training for such ordeals. But one thing I’ve found by talking to students face-to-face is that it doesn’t take very long to identify precisely those things that the examination system does not: imagination, enthusiasm for the subject,  and a flair for thinking on your feet:

One teacher told me with regret that she had to advise her most academic pupil not to display her full intellectual range: in order to secure all the ticks, first you have to stop thinking freely.

If you don’t believe this, take a look at this GCSE Science Examination. A truly intelligent student would struggle to find any correct answer for many of the questions on that paper!

This is why we still place so much emphasis on interviews in the postgraduate admissions system: we take it for granted that all applicants for PhD places will have good undergraduate degrees. What marks out an excellent candidate for a position as  research student, however, is not the ability to pass exams but a mixture of creative flair and almost obsessive determination to surmount the difficult challenges involved in independent research. The correlation between these characteristics and degree results is by no means strong.

The problem for a UK University in adopting the Harvard approach is that credentialism is now running the system. Students apply to universities largely on the basis of their predicted A-level grades, lowering their sights if their predicted grades would not be expected to get them into a more “presitigious” department. But departments that take in students with low A-level scores also get marked down in the league tables for taking in “weaker” students. We’re all aware that A-levels are basically useless, but both sides are  bound so tightly into the system that there seems to be no escape.

So what’s the answer? I don’t know if there is one, but I’d love to see what would happen if all universities abandoned A-levels and instead set their own entrance examinations and interviews. It would be a huge amount of work, but it would make a refreshing change if universities could gather useful information rather than relying on the uninformative guff produced by the national examination boards.

And here is Smith’s closing remark that rings very true to me for personal reasons,

There is a further dimension to the problem of credentialism. It encourages life’s winners to underestimate their good fortune and to over-rate the extent to which they deserve their success. Far from advancing talent over privilege, credentialism has strengthened the grip of people already at the top.

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English Language O Level, Vintage 1979

Judging by the furore surrounding the last-minute marking down of GCSE English Language examinations this year, I thought it might be interesting to put the old scanner to work and show you the English Language examinations I took at age 16, way back in 1979. In those days the GCSE hadn’t been invented yet, and instead we had two different systems GCE  O Level (which I took) and CSE. Anyway, these be the papers what I sat.

The one thing that surprises me a little in retrospect is the considerable emphasis on poetry in the second paper, which I now think would belong more in an English Literature paper. However, there’s no doubt that my schooldays instilled in me a lifelong love of poetry and for that I won’t complain at all…

I’d be very interested in any comments about the difference in style and content between these and modern-day GCSE English Language.

P.S. If you’re wondering what happened to Page 2 of Paper 1, it’s completely blank so I didn’t scan it.

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Gove Agreement

telescoper:

I’ve had the same worry about finding myself in agreement with Michael Gove, at least on a few things; see here, for example. Anyway, this piece makes some very good points about the corruption of the GCSE system.

Originally posted on Protons for Breakfast Blog:

Michael Gove: Not always wrong.

What do you do when someone with whom you basically disagree, says something sensible? Michael Gove has placed me in this situation three times now.

Firstly he abolished the Qualifications and Curriculum development Authority (QCDA).  Secondly he pointed out at that school IT lessons are at best uninspiring. And now he has gone and acknowledged that our system of competitive exam boards has driven down GCSE standards.

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A Return to O-levels?

I woke up this morning as usual to the 7am news on BBC Radio 3, which included an item about how Education Secretary Michael Gove is planning to scrap the current system of GCSE Examinations and replace them with something more like the old GCE O-levels, which oldies like me took way back in the mists of time.

There is a particular angle to this in Wales, because Michael Gove doesn’t have responsibility for education here. That falls to the devolved Welsh Government, and in particular to Leighton Andrews. He’s made it quite clear on Twitter that he has no intention to take  Wales  back to O-levels. Most UK media sources – predominantly based in London – seem to have forgotten that Gove speaks for England, not for the whole United Kingdom.

This is not the central issue, however. The question is whether GCSEs are, as Michael Gove claims, “so bad that they’re beyond repair”. Politicians, teachers and educationalists are basically saying that students are doing better; others are saying that the exams are easier. It’s a shouting match that has been going for years and which achieves very little. I can’t add much to it either, because I’m too old to have done GCSEs – they hadn’t been invented then. I did O-levels.

It does, however, give me the excuse to show you  the O-level physics paper I took way back in 1979. I’ve actually posted this before, but it seems topical to put it up again:

You might want to compare this with a recent example of an Edexcel GCSE (Multiple-choice) Physics paper, about which I have also posted previously.

I think most of the questions in the GCSE paper are much easier than the O-level paper above. Worse, there are many that are so sloppily put together that they  don’t make any sense at all. Take Question 1:

I suppose the answer is meant to be C, but since it doesn’t say that A is the orbit of a planet, as far as I’m concerned it might just as well be D. Are we meant to eliminate D simply because it doesn’t have another orbit going through it?

On the other hand, the orbit of a moon around the Sun is in fact similar to the orbit of its planet around the Sun, since the orbital speed and radius of the moon around its planet are smaller than those of the planet around the Sun. At a push, therefore you could argue that A is the closest choice to a moon’s orbit around the Sun. The real thing would be something close to a circle with a 4-week wobble variation superposed.

You might say I’m being pedantic, but the whole point of exam questions is that they shouldn’t be open to ambiguities like this, at least if they’re science exams. I can imagine bright and knowledgeable students getting thoroughly confused by this question, and many of the others on the paper.

Here’s a couple more, from the “Advanced” section:

The answer to Q30 is, presumably, A. But do any scientists really think that galaxies are “moving away from the origin of the Big Bang”?  I’m worried that this implies that the Big Bang was located at a specific point. Is that what they’re teaching?

Bearing in mind that only one answer is supposed to be right, the answer to Q31 is presumably D. But is there really no evidence from “nebulae” that supports the Big Bang theory? The expansion of the Universe was discovered by observing things Hubble called “nebulae”..

I’m all in favour of school students being introduced to fundamental things such as cosmology and particle physics, but my deep worry is that this is being done at the expense of learning any real physics at all and is in any case done in a garbled and nonsensical way.

Lest I be accused of an astronomy-related bias, anyone care to try finding a correct answer to this question?

The more of this kind of stuff I see, the more admiration I have for the students coming to study physics and astronomy at University. How they managed to learn anything at all given the dire state of science education represented by this paper is really quite remarkable.

Ultimately, however, the issue is not whether we have GCSEs or O-level examinations. There’s already far too much emphasis in the education system on assessment instead of   learning. That runs all the way through schools and into the university system. The excessive time we spend examining students reduces what we can teach them and turns the students’ learning experience into something resembling a treadmill. I agree that we need better examinations than we have now, but we also need   fewer. And we need to stop being obsessed by them.

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Results and explanation (via Gowers’s Weblog)

I thought I’d reblog this because it pertains to my earlier post from today…

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

via Gowers’s Weblog

Are exams getting easier?

With the publication of this year’s GCSE results there’s been the usual clamour about “dumbing down” of educational standards. So are these examinations getting easier or not? I can’t answer that question because I’m far too old to have done GCSEs. The examinations I took at the equivalent stage of my school career were O-levels. But, being an inveterate hoarder of useless articles, I kept the exam papers that I took, so what I can do is put up and example the O-level papers I took (in 1979) and let you decide. I thought the Mathematics one might be of interest, so here it is or rather here they are, because there were two 2-hour written papers; there was no coursework component, so these counted 100% of the final grade.

If you’ve done GCSE mathematics recently, have a look and see what you think!

(You can click on the images to make them bigger if they’re difficult to read…)

I’d be interested in any comments you might have, especially if you’ve actually done GCSE Mathematics (recently or a long time ago). I suspect the most obvious difference is that in my day we did much more geometry…

I might put up the Physics papers if there’s enough interest!

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Sexism, of course…

I’ve only just recovered from the shock  of seeing the sheer hopelessness of British science education laid bare last week. Indeed, I was so staggered to discover how poorly conceived the current GCSE science examinations are that I forgot that I’d already blogged about the lamentable tendency of the modern education system to concentrate on getting kids to swallow and regurgitate little bite-sized factoids, rather than actually learning to think for themselves.  Leaving aside the issue that quite a few of the things that are being taught seem to be wrong anyway, my point there was that teaching science isn’t about teaching facts at all, it’s about trying to develop critical thinking and problem-solving skills. At least that’s what it should be, if only the dumbers-down would stop meddling.

Well, I’d almost come to terms with my despair when I saw another article (from Friday’s Guardian) which tells a tale that’s not just idiotic, but also sinister and offensive. Here’s the full text

One of the country’s biggest exam boards is developing different GCSE courses for boys and girls, it emerged today.

The Assessment and Qualifications Alliance (AQA) said it was looking into creating a science GCSE with more coursework in it for girls, and one which gave more weighting to exam marks for boys.

Studies have shown that girls perform better in coursework than boys, while boys do better in exams.

AQA said it would not prevent boys from taking the girls’ course and vice versa.

The courses in English, maths and science could be available from September next year.

Bill Alexander, the exam board’s director of curriculum and assessment, told the Times Educational Supplement: “We could offer a route for boys that is very different to a route for girls.

John Bangs, head of education at the National Union of Teachers, said it was “extremely dangerous” to get into gender stereotyping. “There are lots of boys who like the investigative element of coursework as well,” he said.

John Dunford, general secretary of the Association of School and College Leaders, said it was a “wild generalisation” to state that boys did better in exams, while girls performed better in coursework, but that it had “more than a grain of truth” to it.

However, he suggested that as well as sitting the gender-specific exams, pupils’ work should be marked in part by professional assessors.

Experts believe that this year could end a 20-year trend for girls to outperform boys in GCSEs because many new courses have no coursework. Instead, pupils complete work over a prolonged period, but under exam conditions.

There’s also a longer piece on the same topic in the Times Education Supplement.

Different courses for boys and girls? Are they serious? This is gender stereotyping of the worst possible kind. I find it absolutely abhorrent that anyone in any position of authority in the education system could even have contemplated doing something so offensively patronising. What’s next, different courses for different racial groups?

I sincerely hope that the new government intervenes and stops the AQA from going along this road. Better still, it should scrap these worthless examination factories and sack the profiteering dunderheads in charge who are responsible for turning the education system into a national disgrace.

Science Examination Blues

I woke up this morning …

.. to the 7am news on BBC Radio 3, including a story about how GCSE science examinations are not “sufficiently rigorous”. Then, on Twitter, I saw an example of an Edexcel GCSE (Multiple-choice) Physics paper.  It’s enough to make any practising physicist weep.

Most of the questions are very easy, but there’s just as many that are so sloppily put together that they  don’t make any sense at all. Take Question 1:

I suppose the answer is meant to be C, but since it doesn’t say that A is the orbit of a planet, as far as I’m concerned, it might just as well be D. Are we meant to eliminate D simply because it doesn’t have another orbit going through it?

On the other hand, the orbit of a moon around the Sun is in fact similar to the orbit of its planet around the Sun, since the orbital speed and radius of the moon around its planet are smaller than those of the planet around the Sun. At a push, therefore you could argue that A is the closest choice to a moon’s orbit around the Sun. The real thing would be something close to a circle with a 4-week wobble variation superposed.

You might say I’m being pedantic, but the whole point of exam questions is that they shouldn’t be open to ambiguities like this, at least if they’re science exams. I can imagine bright and knowledgeable students getting thoroughly confused by this question, and many of the others on the paper.

Here’s a couple more, from the “Advanced” section:

The answer to Q30 is, presumably, A. But do any scientists really think that galaxies are “moving away from the origin of the Big Bang”?  I’m worried that this implies that the Big Bang was located at a specific point. Is that what they’re teaching?

Bearing in mind that only one answer is supposed to be right, the answer to Q31 is presumably D. But is there really no evidence from “nebulae” that supports the Big Bang theory? The expansion of the Universe was discovered by observing things Hubble called “nebulae”..

I’m all in favour of school students being introduced to fundamental things such as cosmology and particle physics, but my deep worry is that this is being done at the expense of learning any real physics at all and is in any case done in a garbled and nonsensical way.

Lest I be accused of an astronomy-related bias, anyone care to try finding a correct answer to this question?

The more of this kind of stuff I see, the more admiration I have for the students coming to study physics and astronomy at University. How they managed to learn anything at all given the dire state of science education in the UK is really quite remarkable.

Space without Physics…

I’m indebted to a colleague (Annabel Cartwright) for sending me this (coincidentally topical) sample question, illustrating the quality of a modern British school science examination.

Since it’s now clear  that there is no room for science in the new era of the UK Space Agency, I suppose we should get used to the removal of science from other things too. Starting with science exams.

This question is taken from a GCSE Physics examination.

Some people think that governments spend too much money on space research.

Which ONE of the following statements is true?

1. Science can tell us what the planets are made of, and whether they ought to be explored.
2. Science can tell us what the planets are made of, but not whether they ought to be explored.
3. Science cannot tell us what the planets are made of but can tell us whether they ought to be explored.
4. Science cannot tell us what the planets are made of, nor whether they ought to be explored.

Apparently one (and only one) answer is correct. Any offers?