I found myself yesterday in a small-town bookshop, kicking my heels for half an hour. Prompted by recent press discussion of the standard of A-levels (the UK 18+ end-of-high-school examination), I browsed through some books intended for A-level further maths students. I must say that they did seem really rather noddy to me, though of course it is only too easy to be seduced into the thought that things are going to the dogs!

Still, that prompted me, just for fun, to look out the papers I sat aged seventeen and a bit, to get into Cambridge, back when the world was young. So here's a small selection of some of the shorter questions:^{1} click to enlarge. (There were four three-hour papers with ten questions apiece: as I recall you aimed to get out at least half-a-dozen a time).

The questions do seem tougher than anything I saw in the contemporary text for further maths. But it would be interesting to know from anyone with their finger more on the pulse how many reasonably bright school kids are in a position to tackle this sort of thing these days. Or indeed -- though the answer could be depressing -- how many of their teachers.^{1} I don't guarantee my proof reading in copying the questions!

## Wednesday, September 02, 2009

### School maths, from the distant past

Posted by Peter Smith at 12:39 PM

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## 18 comments:

Only to(o) easy to be seduced by the thought that things are going into the dogs. Well, not a very seducing thought, but it does seem true that, at least in some respects, young generations are asked to know less than the previous ones. On the other hand, I do recall you pointing out that Cambridge University is now requiring from philosophy students much more than it used to 40 years ago (but, of course, a few things have also happened in analytical philosophy during the last 40 years).

I don't know what a radiusis or a harmonic conjugate are, so I wouldn't be able to do all of them.

Oops, that should be "radius is" in Qn.8.

Blogs are so informative where we get lots of information on any topic. Nice job keep it up!!

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The comparison should be with STEP papers, which are the contemporary equivalent of Cambridge entrance exams. Have you looked at these, and if so have they got easier?

" ... it does seem true that, at least in some respects, young generations are asked to know less than the previous ones.Then, perhaps in other respects they are asked to know more.

In any case, I don't think comparing a Cambridge entrance exam with some further maths texts shows anything very definite about whether maths has gotten easier.

Yep, STEP would be a good comparison. But I'm afraid that having spent too much time too far away from maths, I don't have a sense of what is or isn't equally difficult. Which is why I'd be interested to hear from those who are closer to the business!

From the American side, I'd say that the fraction of our students capable of solving the stated number of these in that time is low --- perhaps not terribly far off of the fraction admitted to Princeton or MIT, which is not a bad comparison for a Cambridge entrance exam. (Of course, I reckon only with the fraction of the students, not the efficacy of any one admission algorithm in selecting them.)

In my opinion:

That papers is significantly easier than current STEP, but significantly harder than current A-Level further mathematics.

I think it is hard to argue that the older A-Levels papers don't require more knowledge, but on the other hand it does require people to do more things to do with translating word problems into mathematical problems, which might be a more relevant skill 'for the real world'. It's certainly easier to pass the exam, but remember an E is a pass...

I got very, very lost when they started talking about monkeys!!!

NS

Dear Prof. Smith,

Allow me add a few similar comparisons (I'm not in the UK):

(1) I taught mathematics for nearly 15 years (at university level); when I left, almost a year ago, the level of the exams (where I taught) was below the ones I took *before* entering university myself.

(2) I compared the exams I drafted during these years and the trend is unmistakable: downhill; as is the other, more objective, one: my marks, despite the lowering of the bar, were dropping even faster.

(3) I used, as a pastime, to answer a few interesting mathematical questions in sites such as Yahoo. I stopped for almost a year and return to take a peek a few weeks ago; someone might have warned me about the step.

The fact is, as you say, that is all too easy to be seduced with the impression that you're going to the dogs but, as far as mathematical and scientific knowledge is concerned, I don't have any doubts that things are going to the dogs.

Best Regards

J. Soares

What, if anything, is the solid evidence that standards have declined? Subjective impressions are not enough, in my opinion.

This is supposed to the evidence with respect to A-Levels. See page 99ish.

Personally, I don't feel this proves anything as the tests they have done aren't really that likely to work, especially across different subjects.

For instance, many peoples who do mathematics can't write and spell. Many people who do history can't do even simple mathematics, and they haven't seem to have taken that into account.

(Perhaps for each subject the tests should have been combined in a sort of ax + by way and the constants a and b found by statistical magic?)

A.C., while I agree that subjective impressions ought not to weigh heavily as evidence, I wonder what would constitute solid evidence as far as you're concerned? Also, it isn't quite fair to dismiss Joao's impressions as merely subjective - surely they fall into the category of expert testimony, which is perhaps the best we are able to expect on matters where questions of value are concerned.

I get the impression that, even over the relatively short time since I did them, maths A-levels have become, if not exactly easier, certainly less wide-ranging.

I don't think that you really need rigorous statistical analyses to have a, at least, justified and true belief that rains more in winter and here, as a teacher, you have a similar situation, particularly if you are not in an elite institutions and even more if you are in a state controlled one.

Anyway, what is "subjective" in the observation that, in mathematics courses for engineering students, the percentage of failed exams tripled over (roughly) the last five years, while both the scope and depth of the said exams narrowed to the incredible point that, in probability and statistics courses you are instructed (meaning: you don't comply, your contract won't be renewed) to avoid question with percentages, because they confuse the students?

Where is the subjectivity when, having taught discrete mathematics (for computer science students) for more than five years, I ended up with a syllabus with lenght less than a third of the original and with students that are unable to understand integer division? (because they are inured to pocket calculators since their first school years).

Besides, upon showing Prof. Smiths' questions to a former colleague today, I got the atonished reply: "they asked students to prove stuff?"

João, to have a justified true belief that it rains more in winter, it actually has to rain more in winter. That is, it has to be true.

There is pleanty of solid evidence about rainfall levels, so that we're not stuck at impressions, anecdotal evidence, and the like.

By the way, Question 3 really does need that it is *not* differentiable at x=0. (As otherwise f[x]=0 is possible)

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