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Part IB Maths for Natural Sciences, from 1984

Posted in Biographical, Education with tags , , , , on October 7, 2012 by telescoper

I’ve been rummaging through my old second-year undergraduate notes and papers trying to compare what I did when I was a student with what we’re asking current second-year undergraduates to do. Since I’m now teaching a fairly mathematical second-year course, it is interesting to look at how the content compares with the Mathematics papers I took way back in 1984.

Looking at these two examinations it’s clear that some of the content is similar (e.g. vector calculus, Fourier transforms) but some big things are entirely missing from our second-year syllabus, specifically Laplace transforms and group theory. The absence of the latter is a matter of particular regret because it’s such a beautiful subject that I think leads onto a deeper understanding of physics than a lot of the other things we make the students learn.

The other striking thing is that the marks for different bits of the questions are not given. That was standard in my day, but nowadays we usually indicate how many marks are available for each part. Moreover, the style of examination is such that even the number of correct answers needed for full marks isn’t given; it just says “You are advised to answer complete questions rather than fragments”.

Anyway, as usual, I’d be interested in comments on the content and difficulty especially from current students in the unlikely event that they have nothing better to do on a Sunday afternoon than have a look at it! For my part, I’ll be in the department getting next week’s lecture materials sorted. Heigh-ho.

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More Maths, or Better Maths?

Posted in Education with tags , , on July 25, 2012 by telescoper

Interesting view from a Biosciences perspective about the recent recommendations to increase the number of students taking Mathematics at A-level.

I’ve always had a problem with the way Statistics is taught at A-level, which is largely as a collection of recipes without much understanding of the underlying principles; would more emphasis on probability theory be a better way to go?

JennyAKoenig's avatarBiomaths Education Network

The introduction of post-16 maths is in the news again with a report from the House of Lords committee on Higher Education in STEM and many of the headlines from the Guardian, Independent and Times Higher  have picked up on the recommendations regarding maths study post-16.

I have written a few thoughts here on my first impressions but would very much welcome comments.

Though I was pleased to see that some of my work showing that only GCSE maths is required for undergraduate biosciences was cited, the conclusion from this was that more students should take maths A level and this is a little worrying.

The lack, or low level, of maths requirements for admission to HEIs, particularly for programmes in STEM subjects, acts as a disincentive for students to take maths and high level maths at A level. We urge HEIs to introduce more demanding maths  requirements…

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Cambridge Entrance Examination – Mathematics for Natural Sciences (1981)

Posted in Biographical, Education with tags , , , on February 7, 2012 by telescoper

I thought I’d take 5 minutes this lunchtime to add another item to the collection of old examination papers I’ve been posting, as someone asked me about this type of examination via a comment recently. This is the Mathematics paper I took way back in November 1981 for entry the following October to do Natural Sciences. I also took papers in Physics and Chemistry, as well as a General paper. Looking at this after a gap of over 30 years it looks pretty tough. One thing I should point out, though, is that the timing of the paper required us to come back after A-levels for an extra term (“the seventh term”)  at my school, the Royal Grammar School in Newcastle,  to form the “Third Year Sixth” who were all Oxbridge candidates. We were then intensively coached for the entrance examination. You will notice, for example, a couple of questions on this paper relating to group theory, which wasn’t on the A-level syllabus but which we were taught specifically for this examination. Some schools couldn’t offer this specialist teaching so pupils from them were significantly disadvantaged by this form of selection. As it happens, I answered both the (relatively easy) questions on group theory and got in to Cambridge…

Comments on the content and/or difficulty are welcome through the box below!

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Closer to Erdös…

Posted in Biographical with tags , , , , on December 18, 2011 by telescoper

After one of my  lectures a year or so ago, a student came up to me and asked whether I had an Erdős number and, if so, what it was.  I didn’t actually know what he was talking about but  tried to find out and eventually posted about it.

In case you didn’t know, Paul Erdős (who died in 1996) was an eccentric Hungarian mathematician who wrote more than 1000 mathematical papers during his life but never settled in one place for any length of time. He travelled between colleagues and conference, mostly living out of a suitcase, and showed no interest at all in property or possessions. His story is a fascinating one, and his contributions to mathematics were immense and wide-ranging.  The Erdős number is a tiny part of his legacy, but one that seems to have taken hold. Some mathematicians appear to take it very seriously, but most treat it with tongue firmly in cheek, as I certainly do.

So what is the Erdős number?

It’s actually quite simple to define. First, Erdős himself is assigned an Erdős number of zero. Anyone who co-authored a paper with Erdős has an Erdős number of 1. Then anyone who wrote a paper with someone who wrote a paper with Erdős has an Erdős number of 2, and so on. The Erdős number is thus a measure of “collaborative distance”, with lower numbers representing closer connections.

I say it’s quite easy to define, but it’s rather harder to calculate. Or it would be were it not for modern bibliographic databases. In fact there’s a website run by the American Mathematical Society which allows you to calculate your Erdős number as well as a similar measure of collaborative distance with respect to any other mathematician.

A list of individuals with very low Erdős numbers (1, 2 or 3) can be found here.

Given that Erdős was basically a pure mathematician, I didn’t expect first to show up as having any Erdős number at all, since I’m not really a mathematician and I’m certainly not very pure. However, his influence is clearly felt very strongly in  physics and a surprisingly large number of physicists (and astronomers) have a surprisingly small Erdős number.

Anyway, my erstwhile PhD supervisor John D. Barrow recently emailed to point out that he had written a paper with Robin Wilson, who once co-authored a paper (on graph theory) with Erdős himself. That means John’s Erdős number is now  2, mine is consequently no higher than 3, and  anyone I’ve ever written a paper with now has an Erdős number no greater than 4.

I’ll be making sure this new information is included in our forthcoming REF submission.

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Serious Brain Teaser

Posted in Cute Problems with tags , , , on October 28, 2011 by telescoper

This one has been doing the rounds this morning, so I couldn’t resist posting it here:

PS. If anyone knows where this originated please let me know and I’ll give proper credit!

PPS. Note that the Mike Disney option (300%) is missing…

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Advanced Level Mathematics Examination, Vintage 1981

Posted in Education with tags , , , , , , on September 26, 2011 by telescoper

It’s been a while since I posted any of my old examination papers, but I wanted to put this one up before term starts in earnest. In the following you can find both papers (Paper I and Paper 2) of the Advanced Level Mathematics Examination that I sat in 1981.

Each paper is divided into two Sections: A covers pure mathematics while B encompasses applied mathematics (i.e. mechanics) and statistics. Students were generally taught only one of the two parts of Section B and in my case it was the mechanics bit that I answered in the examination. Paper I contains slightly shorter questions than Paper 2 and more of them..

Note that slide rules were allowed, but calculators had crept in by then. In fact I used my wonderful HP32-E, complete with Reverse Polish Notation. I loved it, not least because nobody ever asked to borrow it as they didn’t understand how it worked…

I also did Further Mathematics, and will post those papers in due course, but in the meantime I stress that this is just plain Mathematics.

If it looks a bit small you can use the viewer to zoom in.

I’ll be interested in comments from anyone who sat A-Level Mathematics more recently than 1981. Do you think these papers are harder than the ones you took? Is the subject matter significantly different?

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Whatever happened to Euclid?

Posted in Education, The Universe and Stuff with tags , , , , , , on May 24, 2011 by telescoper

An interesting article on the BBC website about the innate nature of our understanding of geometry reminded me that I have been meaning to post something about the importance of geometry in mathematics education – and, more accurately, the damaging consequences of the lack of geometry in the modern curriculum.

When I was a lad – yes, it’s one of those tedious posts about how things were better in the old days – we grammar school kids spent a disproportionate amount of time learning geometry in pretty much the way it has been taught since the days of Euclid. In fact, I still have a copy of the classic Hall & Stevens textbook based on Euclid’s Elements, from which I scanned the proof shown below (after checking that it’s now out of copyright).

This, Proposition 5 of Book I of the Elements, is in fact quite a famous proof known as the Pons Asinorum:

The old-fashioned way we learned geometry required us to prove all kinds of bizarre theorems concerning the shapes and sizes of triangles and parallelograms, properties of chords intersecting circles, angles subtended by various things, tangents to circles, and so on and so forth. Although I still remember various interesting results I had to prove way back then – such as the fact that the angle subtended by a chord at the centre of a circle is twice that subtended at the circumference (Book III, Proposition 20) – I haven’t actually used many of them since. The one notable exception I can think of is Pythagoras’ Theorem (Book I, Proposition 47), which is of course extremely useful in many branches of physics.

The apparent irrelevance of most of the theorems one was required to prove is no doubt the reason why “modern” high school mathematics syllabuses have ditched this formal approach to geometry. I think this was a big mistake. The bottom line in a geometrical proof is not what’s important – it’s how you get there. In particular, it’s learning how to structure a mathematical argument.

That goes not only for proving theorems, but also for solving problems; many of Euclid’s propositions are problems rather than theorems, in fact. I remember well being taught to end the proof of a theorem with QED (Quod Erat Demonstrandum; “which was to be proved”) but end the solution of a problem with QEF (Quod Erat Faciendum; “which was to be done”).

You can see what I mean by looking at the Pons Asinorum, which is a very simple theorem to prove but which illustrates the general structure:

  1. GIVEN
  2. TO PROVE
  3. CONSTRUCTION
  4. PROOF

When you have completed many geometrical proofs this way it becomes second nature to confront any  problem in mathematics (or physics) by first writing down what is given (or can be assumed), often including the drawing of a diagram. These are key ingredients of a successful problem solving strategy. Next you have to understand precisely what you need to prove, so write that down too. It seems trivial, but writing things down on paper really does help. Not all theorems require a “construction”, and that’s usually the bit where ingenuity comes in so is more difficult. However, the “proof” then follows as a series of logical deductions, with reference to earlier (proved) propositions given in the margin.

This structure carries over perfectly well to problems involving algebra or calculus (or even non-Euclidean geometry) but I think classical geometry provides the ideal context to learn it because it involves visual as well as symbolic logic – it’s not just abstract reasoning in that compasses, rulers and protractors can help you!

I don’t think it’s a particular problem for universites that relatively few students know how to prove the perpendicular bisector theorem, but it definitely is a problem that so many have no idea what a mathemetical proof should look like.

Come back Euclid, all is forgiven!

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Ways of Thinking

Posted in Biographical, The Universe and Stuff with tags , , on November 25, 2010 by telescoper

I’m putting one more Richard Feynman clip up. This one struck me as particularly interesting, because it touches on a question I’ve often asked myself: what goes on in your head when do you mathematical calculations? I think I agree with Feynman’s suggestion that different people think in very different ways about the same kind of calculation or other activity.

There’s no doubt in my mind that I’ve become slower and slower at doing mathematics as I’ve got older, and probably less accurate too. I think that’s partly just age – and perhaps the cumulative effect of too much wine! – but it’s partly because I have so many other things to think about these days that it’s hard to spend long hours without interruption thinking about the same problem the way I could when I was a student or a postdoc.

In any case, although much of my research is mathematical, I’ve never really thought of myself as being in any sense a mathematical person. Many of my colleagues have much better technical skills in that regard than I’ve ever had. I was never particularly good at maths at school either. I was sufficiently competent at maths to do physics, of course, but I was much better at other things at that age. My best subject at O-level was Latin, for example, which possibly indicates that my brain prefers to work verbally (or perhaps symbolically) rather than, as no doubt many others’ do, geometrically or in some other abstract way.

Another strange thing is the role of vision in doing mathematics. I can’t do maths at all without writing things down on paper. I have to be able to see the equations to think about solving them. Amongst other things this makes it difficult when you’re working things out on a blackboard (or whiteboard); you have to write symbols so large that your field of view can’t take in a whole equation. I often have to step back up one of the aisles to get a good look at what I’m doing like that. Other physicists – notably Stephen Hawking – obviously manage without writing things down at all. I find it impossible to imagine having that ability.

But I endorse what Richard Feynman says at the beginning of the clip. It’s really all about being interested in the questions, which gives you the motivation to acquire the skills needed to find the answers. I think of it as being like music. If you’re drawn into the world of music, even if you’re talented you have to practice long for long hours before you can really play an instrument. Few can reach the level of Feynman (or a concert pianist) of course – I’m certainly not among either of those categories! – but I think physics is at least as much perspiration as inspiration.

In contrast to many of my colleagues I’m utterly hopeless at chess – and other games that require very sophisticated pattern-reading skills – but good at crosswords and word-puzzles. Maybe I’m in the wrong job?


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Bayes and his Theorem

Posted in Bad Statistics with tags , , , , , , on November 23, 2010 by telescoper

My earlier post on Bayesian probability seems to have generated quite a lot of readers, so this lunchtime I thought I’d add a little bit of background. The previous discussion started from the result

P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)

where

K=P(A|C).

Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down, not by Bayes but by Laplace. What Bayes’ did was derive the special case of this formula for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from the possible n?”, the answer is given by the binomial distribution:

P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}

where

C(n,x)= n!/x!(n-x)!

is the number of distinct combinations of x objects that can be drawn from a pool of n.

You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or px. The probability of (n-x) successive failures is similarly (1-p)n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact that the ordering of successes and failures doesn’t matter.

The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).

So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning. He got the correct answer, eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.

This is not the only example in science where the wrong person’s name is attached to a result or discovery. In fact, it is almost a law of Nature that any theorem that has a name has the wrong name. I propose that this observation should henceforth be known as Coles’ Law.

So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but despite this was elected a Fellow of the Royal Society (FRS) in 1742. Presumably he had Friends of the Right Sort. He did however write a paper on fluxions in 1736, which was published anonymously. This was probably the grounds on which he was elected an FRS.

The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1764.

P.S. I understand that the authenticity of the picture is open to question. Whoever it actually is, he looks  to me a bit like Laurence Olivier…


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Open Admissions

Posted in Education with tags , , , , , , , on August 21, 2010 by telescoper

As I predicted  last week, the A-level results announced on Thursday showed another increase in pass rates and in the number of top grades awarded, although I had forgotten that this year saw the introduction of the new A* grade. Overall, about 27% of students got an A or an A*, although the number getting an A* varied enormously from one course to another. In Further Maths, for example, 30% of the candidates who took the examination achieved an A* grade.

Although I have grave misgivings about the rigour of the assessment used in A-level science subjects, I do nevertheless heartily congratulate all those who have done well. In no way were my criticisms of the examinations system intended to be criticisms of the students who take them and they thoroughly deserve to celebrate their success.

Another interesting fact worth mentioning is that the number of pupils taking A-level physics rose again this year, by just over 5%, to a total of just over 30,000. After many years of decline in the popularity of physics as an A-level choice, it has now grown steadily over the past three years. Of course not everyone who does physics at A-level goes on to do it at university, but this is nevertheless a good sign for the future health of the subject.

There was a whopping 11.5% growth in the number of students taking Further Mathematics too, and this seems to be part of a general trend for more students to be doing science and technology subjects.

The newspapers have also been full of  tales of a frantic rush during the clearing process and the likelihood that many well-qualified aspiring students might miss out on university places altogether. Part of the reason for this is that the government recently put the brake on the expansion of university places, but it’s not all down to government cuts. It’s also at least partly because of the steady increase in the performance of students at A-level. More students are making their offers than before, so the options available for those who did slightly less well than they had hoped very much more limited.

In fact if you analyse the figures from UCAS you will see that as of Thursday 19th August 2010, 383,230 students had been secured a place at university. That’s actually about 10,000 more than at the corresponding stage last year. There were about 50,000 more students eligible to go into clearing this year (183,000 versus 135,000 in 2009), but at least part of this is due to people trying again who didn’t succeed last year. Clearly they won’t all find a place, so there’ll be a number of very disappointed school-leavers around, but they also can try again next year. So although it’s been a tough week for quite a few prospective students, it’s not really the catastrophe that some of the tabloids have been screaming about.

I’m not directly involved in the undergraduate admissions process for the School of Physics & Astronomy at Cardiff University, where I work, but try to keep up with what’s going on. It’s an extremely strange system and I think it’s fair to say that if we could design an admissions process from scratch we wouldn’t end up with the one we have now. Each year our School is given a target number of students to recruit; this year around 85. On the basis of the applications we receive we make a number of offers (e.g.  AAB for three A-levels, including Mathematics and Physics, for the MPhys programme). However, we have to operate a bit like an airline and make more offers than there are places. This is because (a) not all the people we make offers to will take up their offer and (b) not everyone who takes up an offer will make the grades.

In fact students usually apply to 5 universities and are allowed to accept one firm offer (CF) and one insurance choice (CI), in case they missed the grades for their firm choice. If they miss the grades for their CI they go into clearing. This year, as well as a healthy bunch of CFs, we had a huge number of CI acceptances, meaning we were the backup choice for many students whose ideal choice lay elsewhere. We usually don’t end up recruiting all that many students as CIs – most students do make the grades they need for their CF, but if they miss by a whisker the university they put first often takes them anyway. However, this year many of our CIs held CFs with universities we knew were going  to be pretty full, and in England at any rate, institutions are going to be fined if they exceed their quotas. It therefore looked possible that we might go over quota because of an unexpected influx of CIs caused by other universities applying their criteria more rigorously than they had in the past. We are, of course, obliged to honour all offers made as part of this process. Here in Wales we don’t actually get fined for overshooting the quota, but it would have been tough fitting excess numbers into the labs and organizing tutorials for them all.

Fortunately, our admissions team (led by Helen Hunt Carole Tucker) is very experienced at reading the lie of the land. As it turned out, the feared influx of CIs didn’t materialise, and we even had a dip into the clearing system to  recruit one or two good quality applicants who had fallen through the cracks elsewhere.  We seem to have turned out all right again this year, so it’s business as usual in October. In case you’re wondering, Cardiff University is now officially full up for 2010.

There’s a lot of guesswork involved in this system which seems to me to make it unnecessarily fraught for us, and obviously also for the students too! It would make more sense for students to apply after they’ve got their results not before, but this would require wholesale changes to the academic year. It’s been suggested before, but never got anywhere. One thing we do very well in the Higher Education sector is inertia!

I thought I’d end with another “news” item from the Guardian that claims that the Russell Group of universities – to which Cardiff belongs – operates a blacklist of A-level subjects that it considers inappropriate:

The country’s top universities have been called on to come clean about an unofficial list or lists of “banned” A-level subjects that may have prevented tens of thousands of state school pupils getting on to degree courses.

Teachers suspect the Russell Group of universities – which includes Oxford and Cambridge – of rejecting outright pupils who take A-level subjects that appear on the unpublished lists.

The lists are said to contain subjects such as law, art and design, business studies, drama and theatre studies – non-traditional A-level subjects predominantly offered by comprehensives, rather than private schools.

Of course when we’re selecting students for Physics programmes we request Physics and Mathematics A-level rather than Art and Design, simply because the latter do not provide an adequate preparation for what is quite a demanding course.  Other Schools no doubt make offers on a similar basis. It’s got nothing to do with  a bias against state schools, simply an attempt to select students who can cope with the course they have applied to do.

Moreover, speaking as a physicist I’d like to turn this whole thing around. Why is it that so many state schools do teach these subjects instead of  “traditional” subjects, including sciences such as physics?  Why is that so many comprehensive schools are allowed to operate as state-funded schools without offering adequate provision for science education? To my mind that’s a real, and far more insidious, form of blacklisting than what is alleged by the Guardian.

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