Tim Rowland : Mathematics

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The way we were.

It was very different then. It’s the late 1960s, at Bingley College of Education (aka ‘teachers training College’) in the West Riding of Yorkshire. Some 300 students per year are enrolled in each of the three years of the Certificate in Education course. About 10% of the student body would extend their course by an additional year, and be awarded a BEd degree, which had been introduced as a result of the 1963 Robbins Report on Higher Education. Incidentally, it was Lord Robbins who recommended that the former teacher training Colleges should be renamed Colleges of Education, to signify that ‘training’ should be broadened into the wider concept of ‘education. Some 30 years later, we seem to have turned the full circle.

What was I doing at Bingley? I was just 23, an East London boy, and this was my first real experience of The North. On the long train journey from Southampton, my University town, to my job interview, I discovered that West Yorkshire is some way north of Watford . In time the raw, open landscape utterly seduced me, as it does to this day. I was there because I wanted to study for a PhD in Mathematical Logic but the Science Research Council would not fund me to study full-time. I had imagined that the College of Education Assistant Lecturer was a new kind of Being, since I was only the second at Bingley. I needed a job, and this one had one huge advantage over anything else I had looked at. Only 20 miles from Bingley was the University of Leeds , and several well-known logicians. I was taken on as a part-time research student, and Bingley College readily agreed to pay my fees and allowed me one day a week to do research. I took this to be nothing special at the time, although perhaps I should have. The discipline of mathematics education was in its infancy at that time, a marginal option for doctoral study, but one that I did not consider. Nowadays, no education department would so generously support one of its junior ‘faculty’ to undertake pure subject-based research. As for my work with students, I welcomed the prospect (as I saw it) of being paid to ‘do’ mathematics in public.

As it happened, I did have a very limited kind of experience of school teaching. Following abortive ‘seventh-term’ Oxbridge exams, my own boys’ grammar school had taken me on as an unqualified mathematics teacher for the two terms before I began at University. For £9 a week, I taught across the school age-range. The A-level teaching was a wonderful way of consolidating my own knowledge and, I believe, gave me a flying start as an undergraduate – maybe the supply of graduate mathematics teachers was not so good even then, and even in such ‘easy’ schools. Initially, at Bingley, my job was to teach academic mathematics to the 50-or-so students who had chosen it as their three-year ‘main course’ and to prepare, teach and examine truly undergraduate-level courses to the one or two mathematics students each year who went on to the fourth year. It was not unusual for them to be older than me. For one day a week they came to me for courses in set theory and topology, and sought help with the analysis, group theory and mechanics they had been ‘taught’ by some apparently-reluctant lecturers at the University of Leeds School of Mathematics.

Towards the end of my time, I also taught them ordinary and partial differential equations, when Matthew Linton, the College lecturer who had taught the course, left. He was very bright, a fine musician and mathematician: I was saddened to read of his death in a plane crash 20 years later.

It is salutary to compare the rigorous mathematical demands of that BEd with those of BEd/BA(QTS) courses some thirty years later. . Lynch (1979, p. 24) is surely correct in attributing this to the “paternalistic role [of the validating universities] vis-à-vis the Colleges” and their “exaggerated concern for academic standards” in College students who were deemed to have ‘failed’ to gain a University place. It was only when the Council for National Academic Awards came into being that the Colleges, free of dependence on the Universities for the award of their degrees, were given greater autonomy and the opportunity to consider what the nature of main subject study should be. Before the mid-1980s was there little or no recognition that the nature of the mathematics taught to intending teachers, and indeed the teaching methods they were exposed to, might be different from those where the purpose is purely academic (e.g. Grossman, Wilson and Shulman, 1989). A contemporary account of issues in the design of College BEd mathematics curricula is given in Flisher (1973). This includes a robust defence of the value of in-depth subject study (the ‘main course’ for intending primary teachers, in opposition to “voices” that were calling for its abolition. Flisher wrote as follows:

The student should acquire [from study of a main subject] a special interest in its relevance to education and the teaching … “relevance to teaching” does not mean “directly related to a classroom lesson” I would say that a main subject is not a subject to teach, but rather a subject that gives the student certain attitudes, value and knowledge from which to teach . (Flisher, 1973, p. 322, emphasis in the original)

Jack Ashworth, one of my colleagues, would quip “Tim does the long division with them”. Jack was a Pennine man from Todmorden; he had a lot of ‘street cred’ because he had not been out of school too long. Having hardly been in school, I had no cred to lose. I just taught mathematics to the mathematics students, and nobody seemed much bothered that I knew little about primary schools, such was the divorce between the ‘academic’ and professional aspects of the course. My subsequent experience in undergraduate teacher education leads me to believe that professional/academic integration is a good thing – that much is gained when the same people teach both, and are well-qualified both in mathematics and in mathematics education.

The only aspect of the main subject work that defeated me was a so-called Main Subject Week, set aside in the summer for biology and geography field trips, theatre visits for the drama course, intensive studio and landscape work for the fine artists. The mathematics specialism students were not going to be fobbed off with College-based work, however wacky, yet ‘mathematical trips’ to Jodrell Bank, the Science Museum and the like never quite lived up to expectation, in terms of mathematical outcomes. In time, I became skilled at finding spurious reasons for us to go on expeditions to the Yorkshire Dales and excursions to Bridlington. One such was the empirical verification of Naismith’s rule for the time required to ascend a mountain. It was all very innocent, although the delicious sense of conspiracy compounded the pleasure.

In fact, I also gave a course (about 30 hours, as I recall) on mathematical games and puzzles to all-comers in their first term, when they followed an induction course intended to broaden their academic and cultural horizons in advance of making their main and subsidiary subject choices. I have to thank my Head of Department, David Moore (an ATM man) for assuming that someone with a masters degree in mathematics would know about mathematical games and puzzles. In fact I did not, so the course gave me reason to become acquainted with Ernest Dudeney and Martin Gardner, and was a major turning point in my enjoyment and appreciation of mathematics. I have much more to thank David for. Despite my inexperience and his seniority, he never patronised me. I believe that he developed my confidence in myself as a mathematician. It was David who encouraged me to submit my first article to Mathematics Teaching in 1970. In retrospect, this shows an awareness of mathematical investigation that I had not acquired by study of advanced mathematics. It was, in effect, the seed of a fascination with inductive reasoning that has stayed with me ever since. For this, I owe much to a colleague, Sheila Russell, who came to work at Bingley a little after me. Sheila’s energy and her enthusiasm for doing mathematics powerfully motivated me to make my own contribution to the business of making conjectures and constructing proofs. She had a way of making her students and her colleagues feel that what they were doing was important. I remember once showing her some original (for me, at least) group theory that I had been working on: she immediately proposed my presenting it at a research seminar for our mathematics students. At the time, such events were not a part of the culture.

David also entrusted me – in practice, he had little choice – to set the advanced mathematics papers for the fourth year students. This might entail a fortnight’s work for one paper, in some years for one student! I recall a discussion with a mathematics lecturer at another Leeds Institute College (Bretton Hall, I believe) who confided that “The only creative mathematics that I do these days is setting questions for exam papers”. I understood his remark perfectly, and one such question was the inspiration for another MT article in 1974. The buzz that I got from working on it contrasted so starkly with my depressing experience of ‘proper’ research in axiomatic set theory. I abandoned that particular PhD, not without considerable regret, having invested seven years in its pursuit. Incidentally, on taking up a post at Homerton College some years later, I was delighted to find myself in a team culture of setting exam questions ‘from the bottom up’: very few were merely cribbed from books or from past papers, from Homerton or elsewhere.

My apparent success with the games and puzzles course presumably prompted one of my ‘education’ colleagues, Diane Atkin, to impose on me a set of Nuffield Primary Mathematics Guides, and to ask me what I thought. Diane was already a convert to Nuffield Mathematics, and I felt very unequal to her invitation to comment. My overwhelming impression, as I recall, was that the Guides contained some mathematics on sets, number bases and the like with which I was familiar, but whose relevance to the mathematical needs of young children was not at all obvious.

Nevertheless, the Guides did seem to want to bring into primary mathematics some joy and some breadth, such as I had experienced with the students on the games and puzzles course, and it is for this reason that I regret the loss of most of these topics from school mathematics. I resisted Diane’s suggestion that I should be drawn into teaching a primary mathematics curriculum (quaintly-named ‘Basic Mathematics’) course. Remarkably few limitations were imposed by others on what I might be expected to be able to do!

I recognise how much I learnt from all sorts of people in staff room conversations, especially when everything stopped for mid-morning coffee. It is regrettable that such opportunities are now relatively few and far between. In the absence of any formal training to teach, I learnt a great deal by osmosis – about philosophy of science, about child development, about dance and drama education, about the rationale for the new middle schools. I would say that what I learnt most of all were ideas and principles. I developed a kind of idealism, a philosophy of learning that turned out to be hard, initially, to implement in the ‘real’ world, but which nevertheless remained unshaken in the face of pragmatic constraints. I do not look back on my own experience as an Assistant Lecturer as workplace training for teaching. One could construe it in that way, but that would stretch my own sense of what I was doing somewhat too far.

After three years, there followed a year’s secondment to a secondary comprehensive school, to teach the school’s first-ever A-level mathematics class. I enjoyed the advanced teaching and survived the rest. The fact that the Head only realised that this was my probationary year in the week that I left the school had everything to do with a breakdown of communication and nothing to do with my competence with the 15-year-olds in the seventh mathematics set. On the other hand, I did teach A-level mathematics to Vicki Kellett, who that year became the first pupil from the school to gain a university place – to study textiles, at Leeds University.

I returned to the College as a ‘proper’ lecturer, resigned to teaching how-to-teach, having just been made painfully aware of my own shortcomings when it came to managing 11- to 15-year-olds. David Moore had cornered the secondary curriculum area anyway, so there was no opening for me there. That fact probably determined the direction of much of the rest of my career.

So it happened that I became Jack’s Apprentice in primary mathematics curriculum. Jack Ashworth took retirement when the College closed, and has since passed on. He was a down-to-earth, thoroughly likeable man, paternally kind to me. His previous post had been at Bulmershe College in Reading , where he had been thoroughly infected with Dienes, doubtless due to the influence of his colleague, later HMI, Peter Seaborne. The Basic Mathematics course was based on a strong diet of Logiblocs and Multibase Arithmetic Blocks (Seaborne, 1975). We did all the Nuffield stuff with sets. We made them, we sorted them, notated them, found their union and intersection.

We were amazed to see that common denominators of fractions were merely the intersections of sets of multiples. I may have forgotten, and probably have, but I don’t think we talked too much about children, their learning (they would get that in education ‘theory’) or how one might organise them to do and reflect on these activities. As I remarked earlier, mathematics education was at that time in its infancy as a research domain. Many years later, we realise that mathematics pedagogy is informed by both understanding of mathematics per se and by theories of child development – not to mention theories of knowledge, understanding of social factors, and much more.

When my apprenticeship came to an end, I was set loose with my own Basic Mathematics classes. I tried to do what Jack did, and for good measure threw in some fun with binary arithmetic and punched cards. I had also discovered tessellation, and we did some of that too. I take pleasure in pointing out that I managed to acquire two degrees in mathematics without learning that every quadrilateral tessellates in the plane. About six years ago, I mentioned this to a distinguished Cambridge Professor of Geometry as we climbed the staircase towards his College rooms. He stopped in his tracks and asked me, “Is that really true?” I think our Basic Mathematics courses engaged the students’ interest and raised their awareness that (for better or worse) there was more to primary mathematics than they recalled from their own experience.

I made no visits to students in schools in my first incarnation at Bingley, as an Assistant Lecturer. In my second coming I was, of course, obliged to do so. Time has kindly blotted out all memory of my ‘supervision’ of students in schools, with one exception. I supervised a PGCE student called Alan Barnes at Eccleshill Upper School . Alan had a PhD in relativity, and did far more to further my career than I could possibly have done for his, as I indicated at the time (Rowland, 1994).

On the national scene, there were some tremendously exciting ideas in the air. One of the earliest uses of the term ‘investigations’ is in a report on College of Education mathematics main subject studies (co-authored by Alan Bell) produced for the mathematics section of the Association of Teachers in Colleges and Departments of Education (ATCDE, 1967). In the same year, the ATM published Notes on Mathematics in Primary Schools . It is probably true to say that we did play down the skills of arithmetic in our curriculum courses, at the expense of enquiry-based approaches to learning. No doubt some of it rubbed off, but when our students got into school they, like most of their school mentors, depended heavily on ‘Fletcher Maths’. It was all undeniably exciting, if somewhat chaotic. Perhaps this teacher training environment did for many of these students much the same as it did for me, their ‘teacher’ – the inculcation of some ill-defined but passionately-held ideals and values.

All good things must come to an end and, without doubt, it was a good thing for me that this one did. Seven years after I had arrived, I left Bingley for a school teaching post. Three years later, I read that the College had been closed. The boom years had been followed by a dramatic decline in the allocation of student numbers in response to demographic trends. In common with a great many others, the College became a victim of local government reorganisation and the delightful remoteness of its moorland location.

Tim Rowland : 2008.