Mathematics at St Paul’s

Mathematics at St Paul’s in the 1960s

From Climbing Out: the beginning of a life by Bob Phillips (1964-68), Broomfield Press, 2015. Available at lulu.com

There was an outgoing Head of Maths – a small, apparently elderly, balding, rather frail man by the name of AJ Moakes (Maths Department 1931-67). I had very few lessons with him; it was Hugh Neill (Maths Department 1966-72), the man he brought in to become the new Head of Maths who had an impact on me.

It was apparent to me even at 13 that Neill was a very young man; that was, I guess, another indication of St Paul’s standing in the cosmos – confidence in the judgement of one outgoing Head of subject to nominate a successor, even early in his career. A young professional, joining the regimen of St Paul’s teaching, and the regimen of disciplined St Paul’s learning, would get the best possible impetus to success.

One story about Moakes. There was a class that was to St Paul’s standard of discipline, rioting in the next room along the corridor. That is to say, the level of noise was high. Mr Moakes appeared in the doorway, all 5'4" of him, clad in the ubiquitous suit and tie and academic gown. “This noise is intolerable.” Not much response. “If you boys do not maintain silence immediately, I shall be forced to stamp my foot!” Stunned silence. Exit AJ Moakes, triumphant disciplinarian (and famous among the science boys as an inspiring teacher).

I think my year was Hugh Neill’s first in the school. He was good at giving personal guidance in class: brilliant at showing the way through mathematical difficulty. This was a mixed blessing, though – when Neill leaned over your desk to help, one could not help being aware of the time since his jacket had last been fumigated.

Hugh Neill continued the lead of AJ Moakes in taking St Paul’s into the modern mathematics well ahead of the pack. With my very traditional grounding in maths in Africa, it appeared to me not really as maths at all, but a bit of a confidence trick.

Let me recount a very early class with Neill and see if I can explain this apprehension of a sleight of hand, in words. Neill drew a very simple problem on the board. Man at point X trying to get to destination at point Y by the shortest possible route. There is a river between man and destination, represented by two parallel lines a few inches apart. Constraint: when crossing the river, the man can only travel at right angles to the shores. Work out his shortest land path.

“Work out” is what I did – lots of geometry and algebra – good fun, but laborious. Neill at the board: “No, no, no – no hard work required! Suppose we just, as a mental exercise, transposed one bank of the river so that it coincided with the other – it doesn’t make any difference to the man’s path, since he has to cross the two banks at the same point on the river”.

Groans from the class, and a few boys exclaiming, “Of course – it’s just a straight line.” Mr Neill beams. I am consumed with something like anger – that is cheating. You cannot just magic the river out of the way.

Transformation was just one magic trick. Neill taught us vectors and matrices and Venn diagrams as other magic to bring simplicity to situations for which traditional mathematics brought lots of hard work. Probability – a language of beautiful insight into the way in which a portion of the world works. He made of maths a language of illumination rather than a discipline of hard work and rules to be learned. (He was very apologetic later, when we came to the integral calculus, telling us that, sadly, there was no way around simply learning the patterns of different integrals.)

Hugh Neill, making a major innovation in advance of most schools, brought computing to St Paul’s in 1967. He and a couple of other schools – I believe Westminster and Eton – struck up a partnership with BP. They would give schools time on their

 Books by Hugh Neill

Hugh Neill continued the lead of AJ Moakes in taking St Paul’s into the modern mathematics well ahead of the pack.

mainframe computer at night, subject to some heavy constraints. These were the days of huge mainframes – huge in space terms, that is. BP had an ICT 1900. We could program in FORTRAN – the accepted computer language for scientific programming in those days. In order to feed a programme into the computer it had to be embodied in 80 column punch cards.

That was an early hurdle. St. Paul’s invested in a couple of clunky manual Hollerith card punch machines: 16 buttons corresponding to the 16 hexadecimal characters, each of which punched the appropriate coded pattern of holes in the card for that character. We sat with a table that translated alpha-numeric characters into hex (the ASCII table) and pushed combinations of keys on the Hollerith machine. It was very laborious to punch in the codes for each FORTRAN instruction. And then we needed to check that we had coded the instruction correctly. One lucky school had a machine that read punched cards and printed along the top edge the backtranslation of what had been punched.

So, we wrote code on FORTRAN coding sheets. We punched each instruction of the programme onto a card. We mailed the deck of cards to this other school. They put it through their interpreting machine and mailed it back. We read the interpretation and re-punched any erroneous cards. And round again – two mailings, forth and back, each time – until we had a card deck in which all the FORTRAN statements seemed to make sense. This we could send to BP (by mail). They would run the deck in the next available night shift, and send the results back (by mail) in the form of reams of fan-fold printed paper. Only then would we find out what bugs there were in our programming logic. Then – more instructions, more punching, more interpretation and more rounds of mailing.

I seem to remember that in the middle of all this there was a postal strike, which ground the whole process to a halt. None of this dimmed my excitement in understanding the process of turning thought into instructions that could be executed precisely. The word “algorithm” had a fascination: reducing a problem to the essential mechanism by which it would be worked out.

I realised that working my way steadily through the examples of elementary programming that Mr Neill had duplicated for us would require approximately a three week cycle for each exercise. I would have left school before I had got anywhere interesting. So I turned to the back of the book for an interesting challenge – linear programming using the SIMPLEX method. Now that demanded a really interesting grasp of FORTRAN.

So I spent weeks understanding the SIMPLEX algorithm and translating it into loops of sequential steps, and thus into FORTRAN code. That was the fun stuff, in hours snatched from A Level revision and swimming – working out the exact sequence of steps that was implied by the SIMPLEX method; making sure the steps would apply to all variations that I wanted to include; making sure that I had mechanisms to exclude unwanted variations; filling in the coding sheets. Then there were hours and hours hunched over the Hollerith punch machine turning my programme into a card deck, also quite satisfying, in a masochistic way. It was a respectably fat deck of cards that I sent off to be interpreted, and I received it back to make corrections. One more round of interpretation, and the corrected deck was sent off to BP’s computing department.

The conclusion to all this effort was perfectly rewarding. I had got my code to the state in which I knew it would work – every programmer knows that conviction. I never found out to the contrary – the mail was slow; BP was slow; A Levels intervened; I never got the results of the first run back from the computer. »

Bruce Balden (1969-73) provides his memories of Hugh Neill at and after their time at St Paul’s.

Hugh Neill taught me for one term in the Upper Eighth Double Maths set. It was the ‘Pure’ side of the subject. Whenever he set us work it was clear that he examined our mathematical thinking and he would hand it back to the class and show examples of good work which made the individual concerned grow in confidence. I showed the convergence to a limit of a function by reducing it to an obvious case rather than the strict ε method. He commented favourably on this approach.

Later on he played an important role in my professional life. I was by then teaching Mathematics in Division 5 (Tower Hamlets) of the ILEA and Hugh was the Chief Maths Inspector. I was part of a group developing a Maths education resource. We had many working weekends and Hugh would come along to advise and contribute. On one occasion I was part of a small discussion group with Hugh looking at the contrast between mathematical process and content. I did not say a word for two hours but felt I had never worked as hard just by following the argument. The main conclusion was that process is more taken with making decisions on how to tackle the problem rather than applying techniques. This Maths education resource had its own examinations and Hugh oversaw this with rigour. He was always one (quite rightly) to make sure that there were plenty of multi-stage questions i.e. not just applying a formulaic technique such as solving a quadratic equation. He would always prefer a context where the quadratic equation had to be derived first. Again – the emphasis was on mathematical thinking rather than technique. Hugh was an excellent bridge player who took a keen interest in the school team. At that time the school has a real star, Tim Cope (1967-72), who went on to become an international player for South Africa. Under Hugh’s guidance and Tim’s leadership we would qualify for the Daily Mail National Finals but we did not win it due to the rest of the team not being up to Tim’s standards.

Hugh Neill, making a major innovation in advance of most schools, brought computing to St Paul’s in 1967.

Mathematics at St Paul’s in the 1977 – 2019

Owen Toller (Maths Department 1977–88 and 2006–19) describes how Maths has evolved.

With thanks to Crispin Collier (Maths Department 1967-88) and Bob Phillips (1964-68).

Much has changed; much remains the same. When I joined the staff in 1977 a majority of Paulines entered the School in the Fifth Form, took O Levels after two years, took three A Levels in what was then the Middle Eighth, and often stayed on into a seventh, or even a tenth, term to apply to Oxbridge.

In those days A Level failures were far from uncommon, but before league tables there was less pressure on results. In mathematics a crucial difference was that the extended A Level course comprised Pure Mathematics and Applied Mathematics – not the core ‘single’ Mathematics followed by Further Maths as today. Even for single-subject mathematicians there was no AS Level until 2000, so you had to stick to your original choices for two years. Certainly the newer system allows a welcome flexibility and has encouraged many more pupils nationally to take not only Mathematics but also Further Maths at A Level.

However, a pupil of around 1980 visiting a St Paul’s maths class today would probably encounter much of the same material: algebra is largely unchanged (we are still to discover a vaccine against the widespread belief that, and Mathematics A Level has always included coordinate geometry, differentiation, integration, exponentials and logarithms. In 2019 I handed out a sheet entitled Methods of Integration that was barely altered since its first appearance in 1979. The content of trigonometry is also largely unchanged, though the calculations have become much less tedious thanks to the availability of calculators. In 1977 calculations were done using slide rules; I had to teach some clever tricks for using them efficiently. The rather splendid (and very heavy) metal calculating machines that occupied a lot of space in a room on the top corridor were rarely dusted down. To use them you had to turn a handle, just like an old-fashioned bus-conductor’s ticket machine, and sometimes they rang a bell. In about 1980 I bought my first electronic calculator, for £32; it would do only basic calculations.

It may surprise some readers to discover how little impact advances in technology have had on public examinations in mathematics; there is a Further Mathematics option (Discrete Mathematics) which focuses on algorithms, but apart from the ability to study and investigate sequences much more efficiently and enjoyably, there is little in exam syllabuses that involves numerical

In 1977 calculations were done using slide rules; I had to teach some clever tricks for using them efficiently. The rather splendid (and very heavy) metal calculating machines that occupied a lot of space in a room on the top corridor were rarely dusted down.

work. There are two main reasons for this. One is practical: programmable and online technologies pose major problems of security and authenticity in public examinations and may also be considered to benefit the betterendowed schools. But with universal intuitive user interfaces, computing is available to everyone, regardless of mathematical skills, programming has become a specialist area, and mathematics concentrates on the concepts and the techniques – on actually doing the subject – which is and will always be its centre. A few years ago an outstanding pupil, in his last term, was working in class on a problem and suddenly exclaimed aloud, in frustration, “You know what I hate about maths, it’s the numbers.”

Most of the changes affected all good schools, as also would the increase in numbers taking the subject – but probably few to the extent of St Paul’s. In 1977 there was a single specialist Mathematics class in the Middle Eighth (today’s Upper Eighth), known as Middle Maths. Once the restriction of A Level choices to a limited number of groupings of subjects was replaced by a smorgasbord, where you chose any three or (in due course) four subjects, the number taking mathematics grew. It is now some 85% of the year group, with nearly 40% taking Further Mathematics. The size of the mathematics department has grown from 7 in 1977 to 21 today, partly to reflect the greater uptake but also because (as I once complained jokingly to Professor Bailey) “you keep promoting my best teachers out of the classroom”. The idea seems to have taken root that “maths is a good subject to do”. This is fine if you enjoy it and are good at it; but each year I saw boys choosing it without these qualifications. Most would have had a more enjoyable and fulfilling Eighth Form course in other areas, and I tried to persuade them of this, but with mixed success.

I was regularly asked about why relatively few Paulines choose to »

read mathematics at university. There are, I think, two reasons. One is that many Paulines are more concerned about getting to a top university than what they read, and I believe it is true that even a very strong pupil is more at the mercy of luck when applying to either Oxford or Cambridge to read Mathematics than to read Physics, Natural Sciences or Engineering. Meeting the standard Cambridge maths offer, involving STEP 2 and STEP 3, is no joke. Secondly, star mathematicians probably stand out more than in most other subjects – usually everyone knows who the top mathematician in the year is, and many, realising that they are not quite in that class, choose something else. Years without a star often have more applying to read mathematics – though the class of 2016 was an exception, a whole constellation who were not eclipsed by Joe Benton (2011-16).

I think there was a longer “tail” in the 1970s and 1980s than in recent years. I recall introducing one fourth form class to the use of textbooks with answers in the back, stressing of course that they should never just copy out the answers. A few weeks later two boys handed in a prep that consisted only of answers copied out from the back of the book. They were copied from the wrong chapter.

Probably the biggest change in examination syllabuses has been in applied mathematics, which until the 1980s usually meant mechanics. The introduction of modular syllabuses encouraged more people to study

One exercise involved rearranging a group of words into a well-known phrase. One Pauline came up with “To hell with good intentions – the road is paved.”

statistics, and this was especially appropriate for the increased numbers taking mathematics who were not doing Physics. Since 2016 it has been compulsory for A Level students to study both mechanics and statistics. Standards of teaching and learning in Statistics have increased mightily in the intervening years, although there is still a shortage of education in “statistical literacy”, mainly because it is hard to examine.

In 1977 “new maths” was on the way out, though junior classes still used the set of textbooks written by Jack Moakes (Maths Department 1931-67) and Hugh Neill (Maths Department 1966-72), Pattern and Power in Mathematics, which was informed by the concept. (One exercise involved rearranging a group of words into a well-known phrase. One Pauline came up with “To hell with good intentions – the road is paved.”) SMP and MEI projects brought some fresh air, and increasingly textbooks became not repositories of formal mathematics but books that pupils could actually read and learn from. But the one really significant change in the teaching and learning of mathematics over the last forty years was the coursework initiative. The Powers That Be decided that investigative work was The Answer, and from about 1987 all pupils had to submit coursework for assessment. The idea was, I am certain, good, but the implementation doomed it. Assessment schemes were forced to be far too rigid (at first each project had to be assessed on a form with 47 boxes to be ticked, half-ticked or left blank, and each pupil did 6 such projects – later reduced to 2), and the result was that teachers nationwide were pushed into doing stereotyped projects that met the criteria but stifled any initiative. Particularly destructive of enthusiasm were the statistical investigations; nobody foresaw that collection of personalised data by every pupil in the country was a task on a wholly impractical scale, so artificial data sets were produced, and many pupils were completely put off statistics as a direct result. Few were really sorry when coursework in mathematics was abolished. However, the initiative had a positive legacy; it encouraged more informal investigation in the classroom, and naturally this suited the largely Socratic style of teaching at St Paul’s where the default mode is for teachers to ask “what if”?

Another change nationally is that high grades have become the norm. Today exam boards publish detailed mark schemes and examiners’ reports, there is a busy cottage industry in tutoring for examinations, and teachers have become ever more expert at “teaching to the test”. But St Paul’s stands out perhaps even more today than was the case 40 years ago; the Department philosophy is “we teach mathematics, not exams – and exam success is a result”. Inevitably the increase of options and time taken to teach about society and so on make it harder to teach beyond the exam syllabus, but St Paul’s somehow still manages it.

There have always been strong mathematicians at St Paul’s. Perhaps the most famous was J E Littlewood (1900–1903), who went on to form a world-renowned collaboration with the great G H Hardy. In A Mathematical Miscellany, recently republished as Littlewood’s Miscellany, Littlewood describes the extraordinary mathematical education he received at that time; in most schools really outstanding pupils were often simply given advanced books and told to get on with it. The Head of Mathematics in Littlewood’s day was F S Macaulay (Maths Department 1885-1911), a leading mathematician who would later become an FRS (on Littlewood’s proposal) – a distinction not remotely approached by his successors. In the 70s and 80s pupils included Imre Leader (1976-80) and Oliver Riordan (1985-90), today Professors of Mathematics at Cambridge and Oxford Universities respectively. They were given encouragement, but like so much

Now some 85% of the year group take Mathematics at A Level, with nearly 40% taking Further Mathematics. The size of the mathematics department has grown from 7 in 1977 to 21 today.

then the organisation and level of support were more informal than they are today. I did not exactly teach Oliver in the fifth form, but he attended my lessons, and each week I allowed him to do some work of his own choosing instead of the class prep. One week, having observed that if you differentiate the formulae for the area of a circle and the volume of a sphere you get the formulae for the circumference of a circle and surface area of a sphere respectively, he proved that if you differentiate the 4-dimensional volume of a hypersphere, you get its 3-dimensional surface volume. (I told him I did not understand his diagram.)

For most of the period about half of any sixth form cohort did an examined course beyond GCSE (or O Level). This was appropriate enough at the time – better Paulines have never been stretched by GCSE mathematics – but the abolition of the Additional O Level and then of Additional Mathematics GCSE meant that there was no appropriate examination. For some years sixth form pupils took the first module of the A Level course, but this had disadvantages and the practice was discontinued from about 2014. In fact it was increasingly realised that the best pupils were better served by greater depth (harder questions, stronger skills) than by learning more content and taking public examinations early. Putting able pupils in early for routine public examinations such as GCSE is not helpful to their mathematical education but rather the reverse; unfortunately not all parents want to hear this message. St Paul’s has developed its own course for the Fourth, Fifth and Sixth Forms, including problem solving and other appropriate challenges. And today the UK Mathematics Trust runs a world-class series of stretch-andchallenge papers, involving little or no knowledge beyond public exams, from Junior, Intermediate and Senior Mathematics Challenges through to the British Mathematical Olympiad, rounds 1 and 2. The Mathematics department has usually contained some outstanding mathematicians. Paul Woodruff (Maths Department 1978-82) and Gerry Leversha (Maths Department 1986-2011) were probably the brightest stars. Gradually a more formal programme for the highest fliers developed, and it is no coincidence that in recent years St Paul’s has a record of successes in the International Mathematical Olympiad – the world’s most prestigious mathematics competition – which is roughly as good as that of any other two schools combined. Dominic Yeo (2003-08), James Aaronson (2007-12), Sahl Khan (2008-13) and Joe Benton all won medals, Joe representing the UK four times (starting in the Fifth Form) and in his final year coming seventh in the world. Since then St Paul’s has been fortunate enough to appoint several outstanding mathematicians. Outside school Gerry contributes to mathematics nationally by his editorship of The Mathematical Gazette, and no small number of its contributors have Pauline connections.

If you walk through the corridors of today’s St Paul’s when there are boys around, you hear them discussing and arguing about mathematics. If you go into the staff common room, or the mathematics area, you hear teachers talking about mathematics (and not just mark schemes). Teachers continue to investigate unfamiliar details of the subject and to write original papers. In department meetings there is often discussion of mathematics – not admin, not exams, and not just how to teach it, but the subject itself. Staff and boys alike are passionate about the subject, and such huge intellectual curiosity is central to the uniqueness of St Paul’s. »

A roundabout way to teaching Mathematics after being taught by Owen Toller

Matthew Conrad (1979-84)

If my school report from the Summer Term 1980 was to be believed I should never have progressed mathematically beyond O Level. That was the view of my maths teacher, Owen Toller… I am now a Maths teacher at the Weald Community School in West Sussex.

I had never intended to teach, although there were early signs of how my career path might develop when, part way through my A Level studies,

I decided that medicine was not my vocation but that I should study

Economics at Cambridge to prepare me for success as an entrepreneur.

I was one of those pupils that Owen refers to as staying on for the 10th term to take Oxbridge entrance.

My meandering continued when I left Cambridge in the teeth of the late 1980s recession and decided to train as a lawyer – “it’s always good to have a backup plan” my sister told me as we sat waiting for a train at Cambridge station. After brief success as an intellectual property solicitor, I followed what I thought would be my destiny but my entrepreneurial success soon turned sour when my attempt to take on Prêt à Manger in the battle of the sandwich, failed. It was when I was managing the rights for the Japanese cartoon character Hello Kitty that my son (now aged 7) was born that I finally woke up and realised I needed to do something professionally that had some purpose beyond monetary success, that I might enjoy and that

I could explain in a lift what I did for a living. I am not quite sure where the idea to teach came from but it just seemed right and I realised that the subject I had most enjoyed in school, in spite of Owen’s initial misgivings, was Maths.

In 2015 I joined the West London

Free School, the pioneering

Hammersmith state school just across the River from my Alma Mater. I taught there until last year when my family relocated to West Sussex. I then had a term at Durrington High School in Worthing (an Academy for 11-16 year olds with 1,600 pupils) before joining The Weald (another large comprehensive) just in time to go into Lockdown 3.

My experience of teaching at the West London Free School was, in some way, very similar to the teaching I experienced, and which Owen describes, at St. Paul’s in the late 1970s and early 1980s. The ethos of the school was to teach knowledge not for exams. In the maths department I was fortunate to work with an inspirational head of department, Norman Revie, whose philosophy was identical to that which Owen describes at St Paul’s (“we teach mathematics, not exams and exam success is a result”). At Durrington High there was a greater emphasis on tailoring teaching to secure exam success rather than instilling the joy of the subject for its own sake. While it is early days for me at The Weald and currently there are no exams to teach to, it is a school somewhere in the middle of the spectrum.

The fundamental difference between the schools I have experienced as a teacher and the school I experienced as a pupil is the pupils themselves, their level of mathematical attainment and their motivation to improve. There are talented and able mathematicians in these schools who would not be out of place in a maths classroom at St Paul’s. However, I have taught pupils with the numeracy of primary school children, even as they reach the end of their secondary education and who have little or no motivation to improve. Why this is the case is the subject of much debate, and could occupy a whole article, but it is a mix of their personal circumstances and failure of the system.

I had two distinct experiences of teaching in Lockdown – in Lockdown 1 I taught the pupils I had been teaching in the classroom, some for 5 years; at the start of Lockdown 2 I joined a new school and I taught pupils I had never met.

His philosophy was identical to that which Owen describes at St Paul’s, “we teach mathematics, not exams – and exam success is a result”.

 Matthew Conrad’s reports from 1979 and 1980 In spite of these differences, the teaching experience was very similar. It was lonely, in spite of regular contact with colleagues via Zoom, WhatsApp and email. It was frustrating teaching to a blank screen – for safeguarding reasons, pupils’ videos are switched off; and it was extremely challenging to gauge pupils’ understanding during lessons. While you can replicate online some of the tools we use for immediate feedback in the classroom, I found there was no substitute for being able to look into the eyes of your pupils, to measure understanding from the atmosphere in the room (the level of fidgeting, chatting) and to walk around the room looking over the shoulders of pupils as they work.

While I enjoyed experimenting with the technology required to teach remotely, overall it was an unsatisfying experience. It lacked the camaraderie that is such an important feature of classroom teaching. Classroom teaching is a physical profession – you are on your feet and moving around much of the day – teaching remotely is a sedentary experience. Classroom teaching does not require hours glued to a computer screen– something I did not miss from my pre-teaching days. With Lockdown teaching that returned, along with the lower back pain! 