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.

Ofqual: Recruitment of Key Stage 2 subject experts

A colleague brought to my attention to the following advertisement from Ofqual:

“Recruitment of Key Stage 2 subject experts

[…] We are keen to hear from you if you feel you have a suitable level of experience in Key Stage 2 education and assessment, specifically in reading, writing, mathematics or science. You might be a current or ex-teacher or marker, or have other relevant experience in developing or delivering Key Stage 2 assessments. […]”

He has also raised his concerns:

“I have a gut feeling that any set of minimal requirements for “Key Stage 2 subject experts” invited to  work on “developing Key Stage 2 assessments” should include (at least in case of mathematics) some experience of teaching and assessment not only at KS2, but also at KS3 level. For otherwise how can they ensure the continuity and cohesion of pupils’ study?”

My opinion is that there are few people who will have the necessary experience of both KS2 and KS3 teaching experience and assessment development. They are more likely to find people with the KS2 experience only – of course KS3 experience as well would be a bonus, but most people (I should probably say teachers here) are primary or secondary, not both. People in middle schools (ages 9-13) would have bridged the KS2/3 divide, but perhaps would not have seen the curriculum through to the end of KS3. I think that if they stipulated the KS2/3 requirement then they would be faced with a dearth of applicants!

I was always amazed that pupils who came into secondary education with level 5 mathematics at KS2 never really seemed to have a grasp of the topics at that level – even many of those with level 4 struggled with level 4 in KS3. Now this could have been the 6  week lay-off they had over the summer, or the fact that they had been crammed for the KS2 mathematics SAT tests to get the best results for their primary school data. However there seemed to be little correlation between the algebra at level 5 which was tested at KS2 and the algebra at level 5 for KS3, the latter always seemingly the harder. Unfortunately I do not have any hard data to back up my opinion – it is just a gut reaction. One has to believe in the integrity of the powers that oversee these tests (QCA, QCDA, Ofqual or whatever) and that continuity did and will take place.

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]

Do the math(s)

Gillian Tett in FT Magazine: National identity? Do the math(s). A quote:

What difference does a letter “s” make? When it comes to number crunching and national pride, the answer for some American and British people is “a lot”.

A few days ago I wrote a column in the Financial Times about short-selling bans in which I observed that it was tough for the Federal Reserve – or anyone else – to prove whether bans actually worked on the basis of any “math”.

To be honest, that is not a spelling of the word that I would normally use; the British style is “maths”, while Americans typically say “math”. But I had been chatting with some American academics just before I wrote the piece and was focused on the equity market issues. Thus the word “math” slipped in, and I failed to notice it since that missing “s” seemed such a trivial issue.

Read the whole article.