An account of the first decade of AT(A)M
“The mathematics and the metaphysics,
Fall to them as you find your stomach serves you.
No profit grows where is no pleasure ta’en.
In brief, sir, study what you most affect.”
The Association - which to start with was called the Association for Teaching Aids in Mathematics (AT AM) - was founded in 1952 by a group of enthusiasts led by Caleb Gattegno and Roland Collins. Gattegno was then a lecturer at the London University Institute of Education and already involved in promulgating materials like films and geoboards; Collins was a teacher at a Doncaster school and producer of Mathematical Pie, a termly magazine for students.
At the time, mathematics teaching was fixed in traditional ways. In the grammar schools, the syllabus was dominated by algebraic exercises and formal euclidean geometry. And mathematics in the relatively new secondary modern schools was usually seen in terms of' practical' activities deemed more suitable for the supposedly less intelligent. One of the aims of the Association in its early days was the notion that all children could and should be offered real mathematics in a lively and interesting way. This engaged the interest of a few people who were already involved in producing models, films and other teaching aids to use in their classrooms. More generally, the idea of ‘mathematics for all’ was to inspire an increasing number of teachers over the last half of the twentieth century. (Whether - and how - it will survive in the twenty-first remains to be seen.)
There were twelve people - teachers from all over England - at the first meeting of a steering committee meeting held on June 28th, 1952. These founding members had responded to a brief notice in the Mathematical Gazette, the journal of the Mathematical Association, inviting the formation of a teachers' cooperative to produce teaching aids. Such a venture would not have been considered at the time as the province of the older established Association. There was perhaps also a sense that a new Association would attract younger teachers in the recently expanded secondary sector.
After a preliminary discussion on the notion of multi-sensorial aids as opposed to purely visual aids, the committee elected its first officers: Gattegno as Director of Studies and temporary Chairman; Collins as Secretary, to be assisted by Yvonne Guiseppi (a London teacher, later the head of Abbey Wood, and then Sydenham, comprehensive schools); and G T Sare as Treasurer. It was then agreed that three members would prepare for the next meeting accounts of the work each had originated in his own school.
By the next year, the Association had about 150 members and the steering committee had co-opted further people, including Claude Birtwistle (a teacher from Nelson, later an editor of MT) and Trevor Fletcher (a lecturer at St John Cass College, a pioneer maker of mathematical films, the instigator of the first book by members of the Association, and later to become senior HMI for mathematics).
A residential week-end conference, attended by twenty-seven teachers, was held at Brazier’s Park, a conference centre in Berkshire, in October, 1953. Discussions ranged over various teaching aids, and included a viewing of some Nicolet films and Fletcher’s new film on the Simson line. Similar week-end conferences for a small number of people were to become a regular feature of the Association’s activities over the next few years. A report of the next such meeting in May, 1954, observed that everybody had arrived tired on the Friday evening, but had left happy and stimulated on Sunday. It concluded, “Naturally, we cannot expect a week-end like this to change the whole situation of the theory and practice of teaching aids, but we are sure that it changed the attitudes of many participants and made them much more aware of the problems involved”.
The first annual general meeting of the Association was held in February, 1954, at the London Institute, when the steering committee was replaced by an elected committee, with the same officers as before. This was also the occasion for a very comprehensive exhibition that included models (mainly supplied and demonstrated by Jack Peskett, an original committee member who taught at the Sandhurst military college), an ‘awe-inspiring’ differential analyser (made by R 0 Knight, a Worcester teacher), and the showing of a number of films from various sources. Some demonstration lessons were given by Roland Collins and Ron Fielding (a London teacher who was very active in these early meetings) who both showed films and filmstrips to grammar-school children, and by Gattegno using his recently encountered Cuisenaire rods with junior-school children. The rods were referred to as ‘briquettes’ in a later report, which also described the lesson given by Gattegno as having “aroused admiration for his own quiet persistence and for the plump young individualist who preferred his own name for fractions to the usually accepted ones”.
Similar meetings were held in Manchester, Exeter and Birmingham over the next year, and shared observation of lessons followed by small-group discussions was to become a standard feature of meetings of the Association.
A Bulletin (with stapled, cyclostyled pages) was first circulated by post to members in 1953 under the editorship of Collins, but pressure of work led to his being replaced for the second issue later that year by A C Hepburn. This had an article by Fletcher on films, and a report by Gattegno of his first contact that year with the Belgian teacher, Georges Cuisenaire, and his coloured rods.
Continuing teething problems meant a further change of editorship and Bulletin 3, a twelve page issue in October 1954, was edited by Fletcher (who recalls that he had wanted to devote his time to film¬making, but that Gattegno had pressed him very hard to take over the editorship).
This issue had reports of some of the above meetings, and articles on various teaching aids, including a note on geoboards from Gattegno (which also advertised their availability from his newly founded Cuisenaire Company), and a review of some Nicolet films by Owen Storer (a teacher who had been at the 1950 inaugural meeting of the International Commission founded by Gattegno “for the study and improvement of mathematics”, and who was later a lecturer at Birmingham University). Fletcher also wrote in this issue about the formation of a film production unit, and reported that Ian Harris (a Dartford teacher, who became a treasurer of the Association, and later a lecturer at King’s College, London) was working on a film on tangency, originally scripted by Collins and financed by his own publication, Mathematical Pie. There was also an announcement of an evening showing of films at the London Institute to be held in December, 1954 - the first of a number of such meetings.
The 1955 annual general meeting of the Association, in Birmingham, saw the election of a new treasurer, Brenda Briggs (a London teacher, later a lecturer at Southampton University). The meeting was part of a weekend seminar, which also included a public exhibition on the Saturday afternoon, when various demonstration lessons were given: Storer taught eight-year-olds with Cuisenaire rods; Gattegno showed secondary school girls a Nicolet film; Collins (with the same girls) used film strips; and Fletcher showed his film on the cardioid to college students. The report of this meeting, by Gattegno in Bulletin 4, claimed that such lessons were an important part of the work of the Association and should be continued and expanded: "Teachers would rather see how aids are used than hear about their uses by word of mouth alone."
The Bulletin was replaced later in 1955 by a twice-yearly printed publication, Mathematics Teaching. The first issue had a review of progress to date by Gattegno, as Director of Studies:
“The mainspring has been the maintenance throughout of relations with schools and contact with the problems in the reality of a self-education process. All those who joined the Association did so either because they had already worked with teaching aids on one or other of the problems in question, or were in need of help in acquiring more technical knowledge for producing or using various aids, or because they wished to support the activity of a group engaged in discovering whatever was needed for the improvement of the teaching of mathematics. The Association was therefore a somewhat heterogeneous body, needing to educate itself to a better understanding of how to develop its own work successfully...”
Other articles in this issue included an account of a meeting in Doncaster which reported that a demonstration lesson given by Gattegno had provoked strong reactions, but that the children had been given “joyous glimpse of uncluttered truths”. A weekend seminar in Staffordshire was thoughtfully reported by Cyril Hope (a training college lecturer, later to lead the Midlands Mathematics Project). Some films were reviewed by Ian Harris and Bill Brookes (a Liverpool teacher, later a lecturer at Southampton University).
By 1956, the AT AM committee included Cyril Hope, Margot Fyfe (a London teacher, who became secretary of the Association the following year, and was later to work in Nigeria), and John Trivett (a Bristol teacher, later a lecturer at Simon Fraser University, British Columbia, where he inspired a generation of Canadian teachers). At a weekend committee meeting in June 1957, it was resolved that all present should write 300 words on ‘how I teach’ within the following three weeks (this idea was to be taken up many years later when a whole issue of Mathematics Teaching 139 was devoted to the theme). Such self-examination was held to be a way of conducting research on teaching. The committee was urged not to rest content with the arranging of conferences, but should develop a ‘corporate mind’ as a way of revolutionising the teaching of mathematics.
Gattegno left the London Institute in 1957 to take up a UNESCO language teaching consultancy in Ethiopia. He also resigned from the AT AM committee and was subsequently elected to a newly¬created post of President of the Association. An awkward situation then arose about a proposed nomination of Collins to the now absent chair. Gattegno objected to this, ostensibly because of some financial problems with the bulletin that Collins had incurred and had not told anyone but Gattegno about. He even suggested that Collins be made a joint President - as an honoured but inactive adviser - leaving the committee free to elect someone else to the chair. In the following year, Gattegno wrote a letter from Ethiopia to each committee member, referring to all this. His influence on the committee was stronger - Collins was eventually elected Vice¬President (and remained active in the Association for some years). Whatever the real reasons that lay behind this issue, it is clear that Collins had never really got on with Gattegno, and had his own quite different aspirations for the Association. According to Trevor Retcher, it was perhaps surprising that they worked together long enough to achieve what they did.
By 1958, there were 900 members of the Association, and there were a few thriving local branches. David Wheeler (a London teacher, later a lecturer at Leicester University, an influential A TM secretary and editor, and who then had a distinguished career in Canada) became an assistant editor for Mathematics Teaching 7. This was a particularly lively issue which included an article on the theory of games by Fletcher and some thoughts on mathematical problems from Madeleine Goutard (a French teacher, who later collaborated with Gattegno in New York), who emphasised that" i n all fields of education, and especially with young children, we must start with indefinite situations (for such is the reality in which they live)".
There was also a long account by Birtwistle of the annual conference at Blackpool. This conference had covered a number of the issues that preoccupied members of the Association that year. Some of these issues could be summarised briefly as follows.
- The need for a re-organisation of the mathematics syllabus - in his lecture, Fielding had urged that mathematics teachers should write their own syllabus “which must be constantly modified”.
- Criticism of the utilitarian mathematics taught in secondary modern schools - it was suggested that investigation might reveal that “all children think mathematically far more than we believe”.
- Continued interest in the use of materials such as films, cuisenaire rods, geoboards and various home-made aids - with distinctions drawn between the notion of a teaching aid and a learning aid.
- New approaches to teaching - in his lecture, Hope had suggested that teaching started with pupils “who know what they know, which we must find out experimentally”; he recommended that lessons began with" a simple but pregnant situation which the children can comprehend". The notion that a lesson should start from a ‘situation’ (as described by Hope and Goutard) was taken up and developed over the next few years.
By now, branch meetings were becoming a thriving and important part of the Association’s activities. A typical branch meeting at this time would be that of the Middlesex branch held on a Saturday in October 1958 at Mountgrace Comprehensive School. This included a demonstration lesson given by Fielding, who showed a sixth-form class the Nicolet film about an angle-bisector property of a triangle. After the lesson the observers were split into four groups and asked to discuss four questions posed by Wheeler:
- Why are so many of us here?
- Do we learn anything at these meetings that we could not find out just by reading Mathematics Teaching?
- Is there anything of value that we can give to our colleagues that we could not give to an even wider public by writing articles?
- Is there anything of value that a group of teachers can do as an organised group that individuals could not do separately?
These issues also reflected some of the concerns of the Association in the sixties. Another concern was to continue to offer demonstration lessons at meetings of teachers. The journal for November, 1958, reported that during the previous twelve months there had been ten day-conferences, each of which had included demonstration lessons. Margot Fyfe offered some reasons for this deliberate policy, and specifically addressed the often encountered request for a mathematical aim of the lesson. For her, the demonstration lesson was an opportunity for teachers to study how children learn:
“To ask what is the purpose of giving a demonstration lesson is at the same time to ask what genuine meeting is possible between man and man. If it is I who give the lesson, can I receive not only the response of my class, but also the reaction of the onlooker? Can the onlookers, individually, experience the unity of what is taking place, entering into it, and then communicating what they met? ... My aim is that the pupils move forward from where they are. [...] It is impossible that I should know beforehand what will happen. My intention is to free myself entirely from any such preconception, so that my energy will be available to meet what I find before me and to play an active part: looking and listening.”
The editorship of Mathematics Teaching passed in 1959 to Claude' Birtwistle, who redesigned and enlarged the tenth issue; the journal was now printed in Nelson - with a print run of 1800 copies. Officers of the Association at this time were Harris, (Chairman), Wheeler (Secretary), Briggs (Treasurer) and Hope (Director of Studies). The tenth issue contained articles by S Vajda (a mathematician who wrote on operational research), by Geoff Sillitto (a lecturer at Jordanhill College of Education, who became an influential member of the team that wrote the Association’s first publication) and an obituary of Jack Peskett, a founder member who had been an enthusiastic and inventive designer of teaching aids. Notices of future meetings included five scheduled for October (1959), spread over the country from Yeovil to Newcastle-on-Tyne.
Gattegno contributed a ‘message from the President’ to Mathematics Teaching 12, in which he wrote of the need to become aware of the ‘structures’ now being emphasised by contemporary mathematicians. He also urged what he called ‘experimental teaching’, and suggested that “teaching questions are complex and we have to learn to think in a complex way about complex questions...the other way is illusion again”. This also reflected the belief that teaching also involved a life-long learning.
In 1960, Fletcher was elected President of the Association, and the committee now included Bill Brookes, Geoff Beaumont (a London teacher, later a statistics lecturer at Exeter, and then London, Universities) and Alan Bell (a London teacher, later a leading member of the Nottingham Shell Centre). The journal now listed seven pamphlets published by the Association, and priced at sixpence or one shilling (postage 2d) - these included a list of films and filmstrips, and three on the use of Cuisenaire rods. In a brief presidential message, Fletcher hoped that the Association would become more involved in research - not studying teaching in terms of ‘I always do it like this’, but by searching more deeply into the links between mathematics and psychology and philosophy. He suggested that:
“The difficulties which many pupils have with our subject are emotional, and they cannot be overcome by changing syllabuses or writing textbooks; they may be overcome by understanding the mainsprings of human action and mobilising pupils' entire energies for the task.”
Blackpool was again the venue for the annual conference in 1960. According to the report in Mathematics Teaching 13:
“...the producer was the talented Mr C B of Nelson lie Birtwistle] and the cast of stars included (no seniority in the billing) Mr R H Collins of Doncaster, Mr J V Trivett of Bristol, and Mr D H Wheeler of Leicester. With acknowledgements to Continuity by Clutten, Harmony by Harris, Balances by Briggs, and refreshments by Redmans (hotel caterers), the stage was set for yet another production by the AT AM travelling players.”
The final lecture was given by Collins on the evaluation of effectiveness. He emphasised the virtue of self-criticism and listed some questions which succinctly reflect some of the preoccupations among members of the Association at the time; for example: Are we prepared to accept the child’s proof? [...] Do we find time to study all the alleged incorrect answers? [...] Can we pose situations and let the children do the talking?
The next issue of the journal had an article on “real ‘real-life’ mathematics” by David Fielker (a London teacher, later the head of an innovative mathematics department at Abbey Wood School, the director of a mathematics centre and an editor of the journal), and an article on a logical computer (made by pupils) from Dick Tahta (a St Albans teacher, later a lecturer at Exeter University and a co¬editor of M7). There was also - significantly in terms of later developments - an article on modern mathematics and teaching by Gustave Choquet (a Bourbaki author and a personal friend of Gattegno).
By 1961 there were 1 500 members. The growth in membership reflected the enthusiastic recruitment of new members by the first founders of the Association. This also involved an important element of personal support for teachers who often felt isolated in their own schools. As someone [Alan Bell?1 once said at a residential weekend meeting, “I teach with the strength of all of you”.
Mathematics Teaching was enlarged to 72 pages for the 15th issue. The following issue appeared with a different coloured front cover, and the annual membership fee was raised from ten shillings to one pound. The journal became a quarterly with the 17th issue at the end of 1961, and appeared in an enlarged format in the following year.
Articles now began to reflect a widespread interest in so-called ‘modern mathematics’ stimulated by a series of conferences, on the one hand sponsored by industry concerned at the overly abstract nature of newer university courses, and on the other hand initiated by mathematicians interested in bringing the school syllabus up to date. The Association became involved partly because it was felt that when teachers had to reconsider what was to be taught they would then automatically have to reconsider how they taught. (This was an argument that was also to be deployed many years later - for example when there was an interest in ‘unstreaming’ mathematics classes; it is still often invoked for any new issue in mathematics teaching.)
A demonstration lesson, based on the use of arrows to represent binary relations, was given at the 1961 annual conference by a Belgian mathematician, Georges Papy, who was later to write a series of colourful school textbooks which remain unsurpassed for their original and stimulating approach to some quite ‘abstract’ mathematics. The lesson stimulated a lot of discussion in the Association, as did the report of the Royaumont Seminar sponsored by the Organisation for European Economic Co-operation (OEEC). The British representatives at this seminar had included Cyril Hope, who in his lecture challenged the conference to develop a new structure for school mathematics; he himself was eventually to lead the Midland Mathematics Project (which emphasised vectors and was a state-school contrast to the at first exclusively private-school based School Mathematics Project - SMP).
There was a sparkling account of this conference in Mathematics Teaching 16, by David Fielker. The ‘meat’ of the conference, he suggested, was that of modern mathematics, this being “surrounded by the vegetables of Piaget’s educational psychology”. This neatly encapsulated the developing interests of the Association at the time.
“Rather than two complementary themes one felt that Dr Beard’s lecture on Piaget’s work was a supplement to those of Mr Hope and especially of Prof Papy, in the sense that Piaget’s experiments showed that modern mathematics was, to put it simply, the correct and more natural thing to teach. In this almost revolutionary atmosphere the seminar groups with their more familiar topics - aids, first year in the secondary modern school, the sixth form, film-making, methods of teaching this and that, syllabus construction - seemed to hark back to another era.”
It was also significant - in view of later developments - that the exhibition of aids at the conference had included hand-calculating machines (with a manufacturer’s two-page advertisement in the journal) as well as an electronic computer made at a Croydon school under the direction of Maurice Meredith (later a lecturer at Southampton University).
As well as the demonstration lesson, Papy had given a lecture on teaching modern mathematics, moving from his classroom representation of relations by arrows drawn on the blackboard to more sophisticated detail:
“Prof. Papy swiftly passed on to Stone’s theorem on lattices and Pasch’s theorem about the preservation of order in parallel projection, saying as he did so that one must go slowly for pupils - and for teachers! - for in presenting mathematics we knew, the others had to think.”
Asked at the end of the lecture how people could find all this out, Papy replied: "Study!".
The conference certainly stimulated a number of study groups that were set up in various branches in order to work on some of the issues that had been raised. This was not, reported David Wheeler in the journal, “a disinterested pursuit of knowledge for its own sake, but a small attempt to help us all to teach better”. The committee was itself also a study group and continued to meet as such, as well as to conduct the business of the Association.
In the sixties, leading members of the Association were very committed to the notion of group writing. In 1962, Trevor Fletcher (back from Canada, where he had been making mathematical films - notably Dance Squared - with the National Film Board of Canada) convened a group of twenty A TM members who collaborated in the writing of the book, Some Lessons in Mathematics. This book was drafted in a residential writing week, held at Leicester, where each individual contribution was carefully worked over by others so that the result was very much a joint responsibility. The book was published (by CUP in 1964) - the Introduction began and ended as follows:
“If the teaching of mathematics is to remain healthy it must be continually refreshed at no time in history did the teaching of mathematics present more problems, make more demands, or offer the promise of richer satisfaction.”
SLM (as the title became abbreviated) and its aftermaths could be seen as a sort of turning point for the Association, which soon marked its now wider scope and interests by changing its name to the Association of Teachers of Mathematics (ATM).
This also marked the beginning of a slow change in the national standing of the Association. In general, the first members had been young teachers, who were working in some isolation in their schools and who were stirred by what seemed at the time fairly radical ideas. Some, no doubt for a mixture of reasons, were disinclined to join the existing older association of mathematics teachers; others became members of both. As the Association gained influence - through the energy and commitment of its first members, through its published books and pamphlets, through the activity of its local branches, through its influence in teacher training (in which a number of the early members became involved) - it became respectable and earned representation on national committees of various sorts.
The Association is now a thriving organisation with many diverse interests and publications, and an international reputation. It would be a mammoth task to prepare a comprehensive account of its activities since the early years. So the switch from ATAM to ATM marks a suitable time to end this account of the first decade.