Tuesday, 12 August 2014

GCSE Mathematics: the 'new' curriculum

I have been trying and failing to get The Guardian interested in issues around mathematics. Following Jennie Golding's wonderful presentation at the IMA Festival of Mathematics in Manchester last month the plight of teachers and pupils seemed worth highlighting. Given that I have failed to attract national interest, let me at least post my first draft here.


 
GCSE Mathematics Exam Madness

The last overhaul of the school mathematics curriculum had a catastrophic effect: the number of students taking A level maths plummeted by 20% and it took ten years to bring the numbers back to historic levels. A subject that had always been seen as ‘hard’ was up-graded to ‘well-nigh impossible’ in the eyes of pupils. Given the centrality of mathematics to a wide range of disciplines and careers, the long-term effects of Curriculum 2000s introduction have yet to be understood.

Teachers returning to schools in England this September will be teaching the ‘new’ GCSE curriculum for the first time. This is designed to provide pupils with a deeper understanding of material and to enhance the UKs standing in international education league tables. These are laudable aims, but the experience of Curriculum 2000 shows that great care needs to be taken in a subject where answers require precision, and where even small changes in the difficulty of examination questions can lead to large differences in performance.  There is very little evidence that such care is being taken.
Last week Ofqual, the examiners’ watchdog, indicated that that there are likely to be significant and unexpected changes in this years’ GCSE results, and that grade boundaries might have to be lowered. This should be yet another warning to those introducing the new curriculum which will be examined for the first time in 2017. This year’s GCSE examinations were made ‘tougher’ on the orders of then Education secretary Michael Gove. But the warnings emanating from Ofqual suggest that toughening has had unpredictable effects and may yet lead to paradoxical outcomes.  For example, if, as Ofqual seems to be suggesting, grade boundaries have to be adjusted, candidates will do as well as previous years by showing less understanding of the material (because they are not being given the opportunity to show what they know, but rather what they do not know). In these circumstances, it is not at all clear what, if anything, will have been achieved by ‘toughening’ up.

The new GCSE curriculum presents schools with even greater challenges. Not only are teachers expected to provide pupils with a deeper understanding, but they are to do this based on a syllabus with 30% more material in it. It is important to recognise that the deeper level will be new to many of the teachers as well. They will require training so that they have the knowledge and confidence to deliver the lessons effectively.
Independent schools, with their greater resources, will be more able to negotiate these challenges than state schools, and this will exacerbate the inequalities in the education system. A recent report from the Sutton Trust shows that the proportion of state school educated students going to the country’s top universities is already falling. Therefore the pool of people from which we draw our best mathematicians and scientists is shrinking. Changes which have not been properly though through risk making things even worse. Indeed, one of the criticisms of the Tomlinson Report of the roll-out of Curriculum 2000 was that there had been insufficient work on the effect of the proposals.
In June, Liz Truss, then Education Minister, announced the creation of 32 Maths Hubs nationally to act as ‘centres of excellence’. This seems a woefully inadequate response. How will 32 hubs cover the country as a whole? How will other schools interact with these hubs? How and to whom will they be accountable?

The experience of Curriculum 2000, which was designed to broaden the mathematical experience of pupils and also modularised the examination system, suggests we need to be careful when we tamper with the mathematics curriculum – and perhaps other curricula as well. Even during periods of stability there can be variations of 10 marks (out of 75) in some grade boundaries of A level mathematics papers, so the system is very sensitive. There is still time to act, but it is running out, and my fear is that the consequences of getting the new GCSE curriculum wrong will be a massive fall in the level of mathematical skills for most pupils. This is, of course, the opposite of what the new curriculum was designed to achieve, but it will also have potentially severe consequences far beyond secondary education.

This is because mathematics matters, not just in our schools and universities but across our economy. Last year a report from Deloittes on the economic impact of mathematics estimates that mathematics research is responsible for some 16% of the country’s economy; thus anything that reduces our capacity to produce good mathematicians risks seriously damaging our economy.

At a less advanced but no less important level, people in a range of industries need to be able to use mathematics accurately and with confidence. Nurses need to calculate drug dosages, tellers count change, bank employees and their customers need to understand the effect of changes to the interest rate, and we all need to be able to complete our tax returns. The analysis of risk in making life choices – what foods to eat or avoid, which surgical procedures to consent to – involves understanding basic probability, and much of the information we are given is statistical.

Nearly all technological innovations -- GPS, mobile phones, medical scans, financial markets, computer programmes -- have mathematics at their heart. Whilst the user does not need to know the mathematics in detail, it is hard to appreciate this increasingly digital world without a basic sense of mathematics and what it can do. This extra level of appreciation and understanding will be lost if pupils have a bad experience of mathematics. Fewer pupils will go on to do more mathematics with resulting skills shortages – not least in the teaching of mathematics.
And from a purely intellectual point of view, mathematical ideas can be beautiful, and that beauty should be available to everyone.
So mathematics does matter. It matters for the individual in terms of earnings, confidence of being in the world and intellectual stimulation. It matters for our country in that we need a workforce able to do crucial jobs with confidence. And it is crucial for an economy which is increasingly dependent on the innovations that mathematics supports.
Pre-16 education is the beginning of a process which produces the next generation of mathematicians at all levels. My plea to the new Secretary of State is to fix the obvious problems (educationalists have been pointing them out for months) so that it will not also be the end.    

Paul Glendinning

Paul Glendinning is Professor of Applied Mathematics at the University of Manchester and Vice-President of the Institute for Mathematics and Its Applications. The opinions expressed in this article are his own.

Monday, 28 July 2014

Dave Broomhead

Since Dave died last week a poem I read a while ago but had not then connected with Dave keeps going around my head. Dave inhabits both the magician (able to do spectacular deeds but aware of the limitations of what he can do) and the student (able to ask the simple but searing question). It is by Miroslav Holub, translated by George Theiner.

                    Zito the magician

To amuse His Royal Majesty he will change water into wine.
Frogs into footmen. Beetles into bailiffs. And make a Minister
out of a rat. He bows, and daisies grow from his finger-tips.
And a talking bird sits on his shoulder.

There.

Think up something else, demands His Royal Majesty.
Think up a black star. So he thinks up a black star.
Think up dry water. So he thinks up dry water.
Think up a river bound with straw-bands. So he does.

There.

Then along comes a student and asks: Think up sine alpha
greater than one.

And Zito grows pale and sad: Terribly sorry. Sine is
between plus one and minus one. Nothing you can do about that.
And he leaves the great royal empire, quietly weaves his way
through the throng of courtiers, to his home
                                                                         in a nutshell.


Of course, Dave would have smiled and talked of an imaginary alpha.

And now I think of Dave having retired into his nutshell.

Goodbye Dave.

Friday, 5 October 2012

Toral Communities


In dynamical systems, a wild torus is an invariant torus whose cross-section is a nowhere differentiable closed curve. The picture above, taken earlier this summer, is rather different: it is a torus in the wild. I noticed some last year, but wasn't able to get a good photo. They returned this year and the light was better, but even so I have had to enhance the contrast to make the torus clearer.

Tori in nature are usually either somewhat contrived (humans, earthworms, ...) or the result of a complicated extrusion process (smoke rings). This, I believe, is a community of simple cellular organisms that has formed a natural torus. For a while I thought it must be an egg-containing jelly (extrusions again) but a bit of research on the web makes me think it is much more likely to be a jelly-like community that can create columns, and the ends of the cylinder have joined to form a torus.

Why would this be beneficial? Perhaps it is a mechanism to limit further growth or recruitment. The active area of a cylinder is likely to be the ends, so closing in on itself like this could be a way of preventing further organisms from joining. A gated toral community! If the column becomes too large it might be more easily torn or snagged on other objects, so some limitation on size might be necessary.

Fairy rings (holes in the growth of a clump of plants) may also be toral in nature, extending below the surface. But here I think the mechanism is different -- either nutrients in the central region have already been used up, or dying plants leave behind a chemical residue that prevents new growth in that region. Of course the reason the latter mechanism is beneficial might be due to the former suggestion.
 
On the subject of natural tori, I seem to remember Bilbo and Gandalf engaging in a smoke ring competition in Lord of the Rings (The Fellowship of the Ring). Ah yes, here's a CGI version!



Tuesday, 25 September 2012

Tribute to a quiet wonderwoman


Anne Bennett died on 6 September 2012. She was not old and not young (she left two teenage children -- she was my sort of age), and her name probably doesn't mean much to many mathematicians. It should, and it does to those who worked with her.

Anne was part of the secretariat of the London Mathematical Society; she was also seconded to the Council of Mathematical Sciences, which is how I knew her. During the past year the profile of mathematics within Parliament has increased enormously. We have had several articles on mathematics in the magazine of the Parliamentary and Science Committee (PSC), the first seminar on mathematics in the Houses of Parliament that anyone can remember, and mathematicians were invited to the 60-year celebration of science in the UK (jubilee related) seminar and dinner of the PSC.

This unprecedented coverage of mathematics was orchestrated, quietly and effectively, by Anne. It was Anne who knew who to talk to, Anne who knew when to press home an advantage (and when not!) and Anne who knew what was possible.

After her death, two emails (alarmingly alive) came to mind. The first arrived just after the first parliamentary seminar on maths, when I had tried to follow up the success of the occasion by encouraging my co-chair,  Andrew Miller (MP and Chair of the PSC), to  move forward on some of the ideas we had discussed about  the engagement of the mathematics community with Parliament. Anne wrote (and I should say that she also sent me a word of congratulations on my performance, but later!)

Sorry Paul but this is what I did not want you to do right now.
The inside information I have received is NOT to do this.  I need to talk further with my internal contacts first.
Anne


The second, on the day before she died, was a simple word of thanks:

Thank you to Paul and Ken for your considerable efforts in producing the copy for the Science in Parliament magazine and to everyone for their comments.
Best wishes, Anne



These two emails sum up her contribution perfectly: she understood what we were trying to do and was not afraid to tell us what was needed if she felt that we could do with a bit of direction; but she also recognised when people had put effort into projects she valued, and had the grace and confidence to tell people when she thought they they had done something well.

Isn't that what leadership is about?

You can find a more complete obituary on the LMS web pages.


Wednesday, 19 September 2012

EBacc to the future



Michael Gove, the Secretary of State for Education, has just announced plans for a radical change in school exams for those aged 16. On the face of it this tougher, more rigorous system looks good for mathematics. I am concerned that in fact it will reduce mathematical activity in the UK.

I do think that the examinations have become easier over the years; this is an inevitable consequence of the incentives on student success for both teachers and the examination boards.

I do think that there is a consequent danger that the brightest students will become bored; though there is nothing to stop teachers from introducing more demanding and varied material (why is non-examinable a dirty word in education?).

However, this does not mean that I support the proposals outlines by Gove and Clegg a couple of days ago. Some of my reasons are fairly standard:

  1.   Weaker students  (and students with less academically supportive family backgrounds) are likely to struggle  to achieve anything – particularly in all or nothing subjects like mathematics
  2.  Girls are likely to fare worse in the more traditional exam-only culture.
  3. Clegg’s intervention to ensure that there is only one examination may backfire: many people will leave school with only a transcript of attainment showing very little achievement.

An inevitable consequence of (3) is that a second exam system will eventually be brought in (sound familiar?). The system is to be brought in in two stages – English, Maths and Science starting in 2015 (examined in 2017) and the rest the following year. This creates a strange hybrid generation in the middle!

But there are other knock on effects of the current plans.

     4.  Wales. My understanding is that Wales is not forced to follow England’s lead here. Will it go it  alone, keeping GCSEs going, or will it join Scotland’s system? Or move with England? 
     5.  A levels: a harder exam system at aged 16 is likely to discourage students from continuing to A level, leading to a less well-educated work force, precisely what this initiative is trying to address!
     6. In particular, the numbers taking maths, which is seen as hard, at A level is likely to fall and hence… 
     7. …. the numbers taking maths at University will also decrease.

Teachers and Universities have been working hard to overcome the sudden drop in numbers taking mathematics at A level following the 2000 A level exam fiasco. Those in favour of this policy argue that those coming to university will be better prepared. I won’t hold my breath, and anyway, we need good mathematicians getting good degrees to be the next teachers, industrial innovators and financial marketeers! Maths education is not just about the elite next generation of academics.

So my concern is that whilst the best young mathematicians will thrive in the new environment, most of them would  have succeeded anyway, and there is a danger that this policy will actually reduce the supply of good mathematicians!    

Be careful what you wish for.


Thursday, 13 September 2012

Wither Hefce (Hefce withers?)?

The International Review of Mathematics (see earlier blogs) praised the breadth and depth of UK research, commenting on how research excellence is distributed across the country. This excellence is possible because of the current dual funding system: researchers can apply for support for research assistants on specific projects through the Research Councils (RCUK). There is not enough money to support all mathematics research through this mechanism, so the majority of research is supported through Hefce's research contribution to the universities, which makes it possible for people to go to conferences and keep in touch with the international community.

Hefce, in case you didn't know, is the Higher Educational Funding Council for England (other UK countries have their own versions). As the name suggests they are a funding body, and have been awarding money to universities for teaching (the Hefce T) based on agreed numbers of students, and to support basic research activity (the Hefce R contribution) based on the RAE and now REF assessment exercises. Here's a quote from their webpage:


The Higher Education Funding Council for England promotes and funds high quality, cost-effective teaching and research, meeting the diverse needs of students, the economy and society.

However, the change in the student fee system means that effectively all the Hefce T support will be withdrawn over the next few years. This is, of course, the majority of the Hefce funds, and Hefce is by definition a funding body. So will Hefce continue to act as a funding body, disbursing the Hefce R? If it does we will be in the curious state of having two research funding mechanisms, and this is bound to create tensions as to the different roles, and attract attention as a target for efficiency.

My crystal ball suggests that Hefce's days are numbered. Universities will be expected to use student fees to support the basic (non-RCUK funded) research of their staff.

But will they? Can they? And if they do, will it be with strings (preference to those who already have RCUK grants or matching funds from other sources). If the system evolves in this direction we may well see the UK maths research output drastically cut.

Two codas.
1. It could be argued that the RCUK supports the best UK research and this is all we need. Wrong on two counts! Fields medallist and blogger extraordinaire Tim Gowers has never held an RCUK grant. Also, the EPSRC and other research councils are becoming every more involved in choosing research directions, so different sub-disciplines are not treated equally. It is the dual funding mechanism that ensures that as fashions change we have the expertise to adapt and lead.
2. Research informs our teaching in many ways. Losing that research base will impact on the quality of teaching in universities. 

Tuesday, 28 February 2012

The Oak's Shilling


My new book, Maths in Minutes, is formally published on 1 March. It's a great format due to Quercus: 200 short articles on areas of mathematics, intended to introduce the non-mathematician to the subject. Even with 200 sets of 200 words it is impossible to cover the whole of mathematics, so I have concentrated on core mathematics rather than applications.

It looks good, and some of the entries are (I hope) very good, though there are inevitably some weaker areas! I am particularly proud of the introduction of the ideas of proof, and how Holmes' deductive method fits into this (the role of the contrapositive). On the other hand, the short topic-led format makes it harder to bring out the deep inter-connectedness of mathematics. You win some, you lose some.

You can buy it at all good bookstores, but I get no royalties (boo hoo).