What you learn in Mathematics classes...

Bryanston's Head of Mathematics, Alex Hartley, explains why Maths is more relevant outside the classroom than you might think.

As interesting and enjoyable as it is to use a pair of compasses to construct the perpendicular bisector of a straight line, I do confess that pupils have asked me when exactly they will use this skill in real life. Further, I will even concede that my usual response – that it will form part of their iGCSE examinations – is, in the language of youngsters today, a bit lame. So, in this post I will attempt to explain more generally what our pupils learn through studying Mathematics, and whether or not there is more to it than just getting to the right answer.

With a new class, my first lesson of the academic year in September might well involve what I call the Handshakes Investigation. Working in groups, we establish, by getting out of our seats and getting involved, how many handshakes might take place when, say, three, or four, or another number of people meet for the first time. Building up a table of results for various numbers of people, we might seek to explain the patterns that are evident, refining our thoughts to form an algebraic equation and using this to test and predict future results. There’s a lot in there; elements of teamwork, of identifying patterns in numbers, and predicting and testing future results. At a more advanced level, pupils are generalising a particular rule and hence using algebra to construct a simple yet elegant and powerful formula that can be used not just in the cases already discussed, but many more besides. This investigation is a journey for each pupil and there’s a lot more to it than simply getting to the right answer. It also makes for a richer experience than simply working through problems from a textbook, although there is, of course, a time and a place for that and practice does indeed make perfect. Looking towards a real-life application, it is analogous to the problem of asking how many matches take place in a league in which a certain number of teams all play each other*.

Change in the world of mathematics teaching is approaching, with both the British curriculum and the IB offering new courses within the next few years. Following the Chancellor’s encouragement in his Autumn Statement (2017), we’ve offered Core Maths as an accessible course in the sixth form. Core Maths is entirely based around the practical applications of maths – mainly statistical and financial – that our pupils are likely to need both in their personal and professional lives, regardless of the career choices they make. We’re optimistic that our first cohort of pupils, who are just completing A3 (Year 12), will earn a good set of grades which will contribute to their overall accumulation of UCAS points and may enhance their choices at university. The IB curriculum will soon change too, and mathematics will be split into two subjects; Maths Analysis (which will contain a great deal of algebra and calculus, in common with A level courses), and Maths Applications (which will focus upon the uses of maths in real-life).

What would I do differently, if it were up to me to design the curriculum? Three things. First of all, I’d make iGCSE Mathematics, and particularly the foundation tier, focus more upon the applications of mathematics to real life. More questions about, for example, personal finance and perhaps we’d cope without a pair of compasses. Secondly, I’d insist that everybody studied mathematics at some level in the sixth form (just like in the IB) and finally, I’d set more questions that drew in a range of different skills, because that would encourage pupils to better understand the rich connections between each area of mathematics that they study.

So, there’s a lot to enjoy and savour in mathematics lessons; there is of course a range of skills and techniques which do need to be explicitly taught and practised to ensure fluency. But solving problems, talking things through, and creating and testing ideas is all in there too – and this is true for all pupils, no matter what their prior attainment or ability in Mathematics.



* h = ½ x n x (n-1)  is the formula, where h is the number of handshakes, and n the number of people involved. Relating this to sports fixtures, 20 Premier League teams could play each other in ½ of 20 x 19 = 190 matches, but overall 380 matches are played each season because teams meet both home and away.

You can follow Bryanston Mathematics on Twitter here @BryanstonMaths.

Against the clock

It’s a term of exams; one which is more complicated than any of my teaching career, as the latest round of national exam tinkering starts to bite. The country’s Year 12 pupils (we call them A3) are engaging, depending upon their choices, in a mix of legacy and reformed A levels - and it is a cocktail that is producing headaches across the country. Our job, parents and teachers, is to keep our charges sane. And to allow them to do their best.

Exams are one of those unavoidable snapshots of life; not attractive to many of those engaged in them, apart from a few odd souls who relish the adrenalin rush of proving themselves against the clock and without access to any notes that may help them. I remember my tutor at college, the wonderful Stephanie West, asking me as I was getting a little ‘stressed’ preparing for my unseen papers in Greats (which means ‘Finals’ in almost every other part of the world): “If we found a new manuscript of a Greek author, to whom would we give it for the better translation: a nervous finalist student in an exam with no dictionary, or a focused and rested expert in a library surrounded by all the texts one could possibly need?” That helped me a lot to realise what a damn silly idea in lots of ways an unseen translation exam is. It has to be done of course, but the aim of the test is not the production of a perfect translation, but the production of the best you can do in the most unnatural of circumstances. Doing well in exams is all about recognising upon what it is that you are being tested and making your peace with that.

My husband is a big horse-racing fan. He often talks to me about the horses, the trainers and the races and I understand a (very small) percentage of what he says. This is of course the secret of a long and happy marriage. He talked to me recently about ‘breeze up sales’, in which young horses (two-year-olds) are galloped (‘breezed’) for a quarter of a mile or so, the better to inform the on-looking prospective purchasers as to their potential as racehorses. Recently, it has become modish to time this test gallop. And still more modish to pay the highest price for the fastest horse. This has resulted, of course, in young horses being specifically prepared to excel in the test gallop, with less regard for their future health and racing prospects.

I was struck by, and indeed moved by, some of the language of horsemen who see the limits of this exercise. One said: “It’s a fad. But the shrewd guys are seeing through it. They know that if you push these young horses too hard, their heads will be fried, their legs will be fried, and you can forget any idea of them becoming racehorses. Best of luck to these lads with their stopwatches … I know we’re being penalised for not doing fast breezes. But our turn will come. People will come full circle.”

We all have to do ‘breezes’ against the clock. We call them exams. But whilst we accept we have to do them, and that it is generally a good thing to do well in them, I do hope that by next year we shall indeed be seeing a full circle. We will very soon be returning to the days of a two year sixth form (i.e. no AS exams) and, by 2019, we’ll be going back to something like 1926 and the School Certificate, with GCSEs graded 1-9, rather than A* to G. With that might return also the educational memory that not everyone is supposed to get only the highest grade; not everyone is first in a particular race. Or as another racehorse trainer put it: “A lot of the real ‘road runners’ that break the clock at the sales don’t turn out to be ‘race runners’, so to speak … and some of our best horses were bought simply because we liked their make and shape and attitude.”

I have two OB daughters in their early twenties who I very much hope, having been pretty decent ‘road runners’ at Bryanston, are now much more importantly ‘race runners’. All I have ever cared about is their ability to be independent, happy, and fulfilled. I wish them lives of brave choices and paths explored. I wish them loving and loved. These ‘breeze ups’ or exams are, and always have been, just irritating snapshots on the way to these greater goals. I do very much hope that between us we can make sure all our children have the right priorities in life. Some of achieving that is in making sure we don’t buy into any nonsense.

In other words, focus long-term. Or if you prefer, it’s not the snapshot or the stopwatch that matters but how it fits into your plans for a happy, fulfilling, and contributory life.

A level reform: Evolution not revolution

Ian McClary

This week we welcome Ian McClary, Head of Sixth Form, who looks at the changes to A levels that will begin to take effect from September and the opportunities these changes bring.

Next year the reforms to A levels, introduced by Michael Gove as Education Secretary, will begin to take effect. Instead of completing half the A level course in A3 and the other half in A2 (a change introduced by the previous Labour government in 2000), the reformed A levels will, over two years, return to something similar to what we experienced in ‘the olden days’, when we were examined at the end of two years.

In an attempt to halt grade inflation, make A levels more rigorous and ambitious and better prepare young people for the demands of employment and further study, an extensive consultation process, which drew on advice from universities and subject associations, has taken another long, hard look at our educational ‘gold standard’.

Even though it can seem like just another raft of unnecessary and confusing changes, it is important to remember that we have again been presented with a valuable opportunity to reflect on how best to serve our pupils in the sixth form as they prepare for higher education and beyond. Everyone knows that you can’t fatten a pig by weighing it but exams in their right place are an important tool and an important rite of passage.

For pupils currently in B, things won’t look any different. They have chosen their subjects as usual and will sit a mixture of reformed and legacy AS units at the end of A3. Depending on the subjects they have chosen, some of these AS units will count towards their A level grade together with their A2 units; in other subjects they will not and the pupils will be required to be examined on that material again at the end of A2. This will not be much different to what happens already, with some pupils choosing to resit AS exams in the summer of A2.

What we will also be thinking carefully about next year is how we move forward into 2016 when all the subjects we offer will be reformed. There will be advantages and drawbacks to these changes, just as there were to the changes back in 2000. What is important, though, is how we teach children to learn and provide them with a rich experience in the sixth form.

Was it wise, for example, back in 2000, to spend a term of teaching at the end of A3 focusing on preparation for AS exams (knowing full well that trying to get pupils started immediately afterwards on the A2 courses would be a non-starter)? After 15 years of that, why would we continue to offer the new AS qualification when, as a standalone qualification, it is worth less than before (only 40% of an A level) and does not contribute towards the A level grade? And, if the reformed A levels are going to be more rigorous, why would we not jump at the chance to have more time to teach them?

This is something we currently enjoy with the IB Diploma programme, enabling more interesting and diverse assessment methods along the way, as well as terminal exams at the end of a two-year linear course. Given that the IB Extended Essay and Theory of Knowledge qualifications (the core of the IB Diploma) are worth together more than an AS qualification in a fourth subject, which may well not be relevant to what a pupil wants to study at university, how can we instead make those elements of the IB available to a wider range of sixth form pupils studying A levels, together with other supplementary qualifications like the Extended Project Qualification (EPQ) or the Gold Crest Award?

Like an AS qualification these supplementary qualifications will not, in most cases, form part of a university offer (which will continue for the majority of our pupils to be based on three A level predicted grades) but their value as an academically enriching experience goes without saying. I hear time and again from admissions tutors that if a candidate wants to make their application more competitive, apart from demonstrating a genuine interest in their chosen field, the two most important things they need to show are evidence of independent research skills and a commitment to their community through volunteering.

Apart from the shape and pace of the academic year in A3 and A2, I don’t think that much is going to change. Pupils will continue to rise to the challenges placed before them and we will continue to explore ways of providing more opportunities for them to enrich their time in the sixth form so that they are more than just the grades they get. That will be achieved in many different ways, depending on each pupil, so we certainly won’t be seeking to offer a one-size-fits-all programme.