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

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.

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).

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.

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 …

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) )

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

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!]

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

lots of love,

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

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.