Universities and A-levels

In the two weeks since I posted my election statement blog, A-level reform has featured in the news more than once. The TES article below shows that how universities should be involved with A-levels, already unclear, is becoming contentious.

Some structure is required to allow HE mathematics to play a part in the review of A-level. To me this is part of a wider problem of curriculum design and maintenance which we currently have in England. I am not an expert on educational systems worldwide, but I believe that many countries have a standing committee for the development and review of the school curriculum. I believe that it would be valuable to have some such structure in England, and that the LMS, working with others, should be able to contribute to the design of a body with the working title ‘Standing Committee for Mathematics’. Continue reading

The ownership of A-levels

From TES, from an article Gove under fire, by William Stewart, 19 October 2012:

“By far the most important thing we are doing on A levels

is getting university academics back in the driving seat instead of the Department for Education,” the source [very close to education secretary Michael Gove] said. […]

[Mr Gove] wanted government to “step back”, allowing universities to take “real and committed” ownership of new A levels, giving the qualifications their endorsement so that they, rather than exam boards, “drive the system”.

But in an official response to the plans, seen by TES, Universities UK states: “We do not think it would be advisable or operationally feasible for the sector to take on the ‘ownership of the exams’, particularly in terms of formally endorsing all A levels as currently  proposed.”

[UUK] argues that because A levels are a national qualification, “ultimate responsibility and accountability” for them should remain with the government. […]

Ministers believe there is a split over A levels between academics and the universities they work for, which represents a “huge problem”.

“Almost all academics want linear A levels, but universities are not run by academics and admin offices have totally different views, partly because of the cursed focus on ‘access’ which has poisoned intelligent discussion of (the) real problem, which is too many rubbish schools,” the source close to Mr Gove said.

Read the whole article.

A Bacc is coming?

From Mail Online, By Ben Spencer:

Education secretary Michael Gove is […] said to be developing an Advanced Baccalaureate which would see students studying a mixture of A-level subjects, writing a 5,000-word essay and undertaking voluntary work. […]

If his proposals are enacted it would mean the entire exam system for secondary schools will have been replaced in the space of thee years.

The new baccalaureate system would require A-level students to study ‘contrasting’ subjects to give them a broad education, The Times reported last night.

A candidate who chose A levels in maths, further maths and physics, for example, would be expected to pick a humanity, such as history or French, as a fourth subject. 

Mr Gove also wants shorter and more open-ended questions in exams.

One option he is apparently considering is to limit the A-Bacc to teenagers who choose at least two A levels from a list of subjects specified by Russell Group universities – maths, further maths, English literature, physics, biology, chemistry, geography, history, and modern and classical

languages.

Read more: http://www.dailymail.co.uk/news/article-2218880/Gove-transform-A-Levels-stage-examination-overhaul.html#ixzz29YA9h5eN

In Praise of Pick’s Theorem

Whose theorem?” you may be thinking. That was certainly the question I was asked by several of my colleagues when I mentioned that I was giving a talk on this subject. The slides from that talk are here [pdf], and this post contains some meta-mathematical thoughts I had while planning it. My main conclusion is that Georg’s Pick’s theorem is a truly wondrous thing, deserving of a much higher level of celebrity than it currently enjoys. In fact, in this post I’m going to go further than that, and argue that PT merits a place on the maths A-level syllabus. I should quickly say that I’m only thinking out loud rather than making a considered policy proposal (so I’m not addressing obvious next questions such as what should be cut from the curriculum to make the necessary space). All the same, I’d be interested in any reaction.

Before I go on I had better tell you what the theorem says: the action takes place on a square grid (or “lattice”) comprising those points on the plane whose x & y coordinates are both whole numbers. Against this background we can draw all manner of geometrical objects simply by connecting dots with straight lines. Any non-self-intersecting loop built in this way will carve out a shape (known as a “lattice polygon”). Of course, this figure might be horribly jagged and irregular, with thousands of edges. Nevertheless, Pick’s theorem will tell us its area in a single, simple formula. All you need to do is count the number of grid points which lie on the shape’s boundary (call that B) and the number which lie fully inside the shape (C). Then the area is A=½B+C-1.

Here are some observations which I’d say make this a great piece of maths:

  1. It is easy to state.
  2. It is easy to apply: all you have to do is count dots.
  3. It is very general, valid not just for triangles and quadrilaterals but highly irregular shapes too.

And yet…

  1. It is by no means obviously true.

Together (1)-(4) add up to…

  1. It is genuinely useful: it will very quickly tell you the area of shapes which would be horrible to calculate from first principles.

So far this could be an argument for including Pick’s theorem at GCSE or even primary school level…. but I don’t think that would be a good idea. As we all know, mathematicians deal above all in proofs. So if Pick’s theorem is to be on the syllabus, then its proof had better be too. And I think there is a lot to recommend this as well.

So, before I go further, here’s an rough outline of how a typical proof goes (see my slides for a more detailed sketch, or Cut the Knot for an alternative approach). First step: establish that the result holds for triangles. Second: prove (by induction) that every lattice polygon can be constructed by gluing triangles together. Third and final step: show that when you glue two shapes together, if PT holds for each separately, then it holds for the amalgam. Here are some remarks in praise of this proof:

  1. It is a good level of difficulty. It is certainly not trivial, at the same time there are no major technical obstacles to overcome.
  2. Taken as a whole, the proof is reasonably lengthy – I’d argue this is a good thing, as there is real satisfaction in proving something meaty, rigorously and from first principles. At the same time, the summary is short, and the overarching strategy fairly easy to grasp.
  3. What’s more, it comes naturally in three pieces, each of which is of a manageable size, any one of which could make a reasonable bookwork-type exam question.
  4. It is a good illustration of an important philosophy: to address a complicated problem (an arbitrary irregular shape) we break it down into simpler things we know how to deal with (triangles).

Here are a couple of other miscellaneous things in PT’s favour:

  1. It is a comparatively recent discovery. With much of school-level geometry dating back to Euclid, Pick’s theorem (proved in 1899) would be the most modern thing on the maths A-level syllabus. (I’m open to correction here!)
  2. It is always good to place science in its human context, and PT offers several possibilities for worthwhile cross-disciplinary research. Georg Pick was an Austrian Jew who lived most of his life in Prague, and was eventually murdered by the Nazis. He was also a friend of Albert Einstein, and played an interesting indirect role in the development of General Relativity.

Back with the maths, PT naturally opens up several further lines of enquiries – these are outlined in more detail in my slides. I don’t suggest these should be on the syllabus, but their proximity is certainly a bonus, and they would make excellent topics for project-work or extracurricular reading:

  1. What happens if we make the grid finer? If we make it fine enough, can any shape with straight edges be turned into a lattice polygon? (No! This leads to topics like constructible numbers, squaring the circle, and transcendental numbers.)
  2. Does PT generalise to shapes with holes in? (Yes! This leads directly into discussion of topics like simple-connectedness and Euler characteristic.)
  3. Does it generalise to 3-dimensions? (No! Or not immediately, anyway. The basic counterexamples are Reeve tetrahedra, which can be grasped without too much difficulty. It is illuminating how these shapes eliminate the possibility of any version of Pick’s theorem in 3d: the basic idea being that two Reeve tetrahedra can have the same number of boundary and internal points, but different volumes.)
  4. Beyond this, the more enthusiastic student can delve as deeply as they fancy into the beautiful theory of Ehrhart polynomials, which will lead them to further elegant theorems and very quickly to open problems. This is great for showing that maths is not all finished, and might perhaps inspire them to have a go at tackling these questions themselves.

What should be the priorities of Ofqual A Level Reform Consultation?

Having read the Ofqual A Level Reform Consultation I suggest that DfE:

  1. Be wary of changes which may lead to a reduction in numbers taking Mathematics and Further Mathematics.
  2. Accept that there MUST be a common core in at least the pure parts of Maths and Further Maths.
  3. Accept that if the country/government is serious about wanting a more numerate population then the maths curriculum must be drawn up to do the job, not fitted into an unsuitable mould for the sake of ‘consistency’ across subjects.
  4. Accept that, at A level, no one exam can test satisfactorily the whole ability range in mathematics.
  5. For minimum disruption, redesign something like AEA or STEP to stretch the top ability range with problems (where students are not led through to the solution) and where rigour and good style are recognised and rewarded. Fund it and make it more accessible than the present AEA/STEP, with on-line support as for Further Maths. (I imagine HE are not so unhappy with the content of Maths and Further Maths but with the lack both of rigour and of problem solving.)

[Related posts: Universities to set A-levelsA Level Reform ConsultationCommons Select Committee on Education: Introduce National Syllabuses]

A Level Reform Consultation

Ofqual:

This consultation considers proposals for the reform of A levels in England. We are seeking views from higher education, employers, learned societies, colleges, schools and others so that A levels are the best that they can be.

You can download a copy in PDF format, A Level Reform Consultation [PDF, 557KB].

The deadline for responses to this survey is 11th September 2012.

Find out how to respond. To register for one of the events, or to request more details, please email  reform@ofqual.gov.uk:

  • 18th July in Birmingham and 5th September in London for schools and colleges 
  • 25th July in Birmingham and 24th August in London for the Higher Education community  
  • 27th July in Manchester, 10th August in Birmingham and 29th August in London for all interested stakeholders including those who are unable to make the other dates above. 

Universities to set A-levels in new qualifications overhaul

 in The Telegraph:

Examiners will be expected to enlist the help of at least 20 British universities when drafting exam syllabuses and test questions as part of a major drive to raise standards, the Telegraph has learned.

All new qualifications will require a formal “sign-off” from universities – particularly leading research institutions – before being sat by sixth-formers.

The reforms, to be outlined on Tuesday by Ofqual, the qualifications regulator, are intended to ensure teenagers have the appropriate levels of subject knowledge and study skills required to get the most out of a degree course.

Read the whole article.

 

Universities to set A-levels

Later addition: see the letter from Michael Gove  and response from Glenys Stacey. A key phrase in Gove’s letter:

Different subjects have different requirements; I am interested in your views as to how the system should develop to allow for approaches to — for example — mathematics that provide for differential level of challenge.

From the Glenys Stacey’s response:

Making sure that A levels are fit for purpose means getting four things right: subject content (curriculum), teaching, assessment and level of demand. We would look to universities, working with learned societies and awarding organisations, to agree the subject content of A levels. Ofqual would be happy to work with whatever arrangements are put in place to do this, provided that they enable universities to develop high quality content. We will want to be sure that respected university departments and learned societies support the content defined for each new A level. Content will vary to some extent between different A levels in the same subject, but we would want to see all A levels being widely accepted. So even if a particular A level is developed by a small group of universities, we would want to see a significant number of key universities signed up to it. We will look to the university sector to put in place sensible arrangements for this as soon as possible.

From BBC:

The Russell Group of leading universities said they were “certainly willing to give as much time as we can into giving advice to the exam boards”.

But Wendy Piatt, the group’s director general, cautioned: “We don’t actually have much time and resource spare to spend a lot of time in reforming A levels.”

From The Guardian:

Mark Fuller, director of communications of the 1994 Group, which represents small, research-intensive universities, said it was “absolutely right that leading universities and academics have an influence on A-level qualifications alongside others, including employers”.

He said: “This influence must not be restricted to any single group of institutions which, by definition provide higher education only for a minority of 18-year-olds. Universities and employers need A-levels which are robust, fit for purpose and which recognise academic excellence. This excellence is widely distributed across the UK’s higher education sector.”

 

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“University dons to vet A-level exam papers”

Julie Henry in  The Telegraph, 19 Feb 2012:

School exam papers are to be vetted by university academics in a radical overhaul of the A-levels set by one of the country’s biggest exam board.
Mathematicians, historians and scientists from leading universities are working with the OCR board on the design of new “gold standard” syllabuses and assessments.
Universities including Cambridge, Oxford, Birmingham, Manchester, Nottingham, University College London, Sheffield, Leeds, Liverpool, Durham, Surrey, Warwick and York are involved in the programme which covers nine subject areas.

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