L D Adams Presidential Address
Louise Doris Adams was President of the Mathematical Association 1959-60, only the second woman to be elected to this role, the first being Mary Cartwright. She delivered the Retiring Presidential Address Full Cycle to the Mathematical Association on 7 January 1960 in King's College, London. A version of her address was published in The Mathematical Gazette 44 (349) (1960), 161-172. We give a version of her address below.
Full Cycle, by Louise Doris Adams.
The year 1959 saw the completion by the Association of a first comprehensive survey of mathematical education in schools. Up to 1955 the reports of committees of the Association had dealt with various aspects of selective education, but with the publication of Mathematics in Primary Schools and four years later Mathematics in Secondary Modern Schools the cycle of school reporting was completed. These two reports reviewed what I shall call 'popular education' in order to distinguish it from the selective education of pupils who are chosen by ability, or in independent schools to some extent by circumstances. The modern school report also refers to pupils on a mathematical borderline who find themselves in selective schools without being as mathematically mature as their fellows and of course to a large number of pupils at comprehensive schools.
The publication of two reports on popular education does mark an epoch in the history of the Association. Our membership includes a relatively small number of primary and secondary modern school teachers, since the number of persons specially qualified in Mathematics in those schools, though increasing, is still small. We have therefore offered advice in two fields in which as actual teachers most of us are not concerned. There are however two other groups of members who have direct contact with these schools. It is significant that there were four members (including myself) of both primary and secondary modern school sub-committees who belonged to these groups, namely two training college lecturers (or ex-lecturers) and two inspectors (or ex-inspectors) of schools. Lecturers in training colleges and the inspectorates, local and national, do indeed form vital links between persons who are generally of specialist education and those who are concerned with education as a whole, either because their pupils are young or because they are still so intellectually immature that they are best served by modes of education which are not too highly specialised.
May I be unashamedly personal for a few minutes? My own experience of education divides itself neatly into two halves. In the first half as pupil and as teacher in various schools and colleges I experienced almost every form (other than technical) of selective education. During the second half by my own choice I have been immersed in popular education carrying this interest on through contacts with training colleges during ten years of retirement. For much of this later half of my life I functioned as jack-of-all-subjects and I have even tried to conceal the fact that I viewed education with the eyes of a teacher trained as a mathematical specialist.
I am grateful for this non-specialist experience which cured me of many of the mistakes with young children to which those versed in teaching older pupils are liable. It gave me a lot to learn in middle life, in fact a second cycle of personal experience. I can recommend observation of young children as a refresher course, with the warning that to observe as an aunt or a parent does not give a sufficiently random sample of children to serve as substitute for a wider field of observation.
Such mathematical comment as has survived this non-specialist experience has been put at the disposal of the two sub-committees on which I have served. What then can I say to you now? After searching the annals of the Association I can find no more than passing reference to the influences which have been shaping popular education. Past Presidents, notably the late Mr Siddons whose account of the early work of the Association reported in the Gazette of February 1936 is a classic, have dealt fairly fully with influences on selective education and with the history of various aspects of Mathematics.
I am no historian, but I had the good fortune to begin my observation of elementary education, as it was then called, at a critical period, 1925, when powerful new influences were at work and receiving administrative recognition but when many schools still retained clear evidences of their Victorian traditions. This period and ensuing years are well-documented for education as a whole, but I shall try to pick out some mathematical aspects of changes and add to them my personal impressions as an ordinary grammar school teacher of Mathematics plunged into what was in 1925 a startlingly different educational world.
It is convenient to consider influences on popular education in two categories; firstly the purely educational developments arising from increasing knowledge of young children and how the immature learn, and secondly administrative changes, some favourable to educational development, for instance reduction in the size of classes and others only measures of expediency which may even be adverse to it.
It is natural that educational research should be rich in its study of the youngest children especially in the mathematical field, for even at infant level thought processes are immensely complex. Fortunately infant schools have been comparatively free from administrative interference though sharing in administrative advances and teachers in them have been well enough informed and supported to make good use of educational theory. In the 1920's infant schools already had an established reputation and mathematically the best of them had learned many of the lessons of Froebel and Montessori. Moreover these intellectual advances had gone on side by side with advances in the treatment of young children as whole persons, supported by knowledgeable or at least sentimentally indulgent public opinion. This outside support had had administrative backing. The establishing of School Boards in 1870 had brought into contact with the schools persons of goodwill and good education. As an example my own grandmother served on the School Board of a Midland city from 1871 and it was written of her at her death in 1894, 'It was to her very largely that thousands of young children ... in the infant schools ... were indebted for having their infantile years made happy by the introduction of kindergarten methods. Being a woman and a highly educated one (she left school at fifteen), with deep and keen insight into child nature she was not trammelled by preconceived notions ... she brought to bear on the schools an insight and intelligence that was extremely valuable because it was not merely official ...'. This is a favourable picture of outside influence, but I could name from another personal source a teacher who was dismissed in 1888 for refusing to keep in after school five year olds who did not know the multiplication table expected of them. Her reason for refusing was that in her class of between 70 and 80 infants these children were already dropping off to sleep and falling from the classroom's galleries.
I think that when I first visited infant schools they had already passed one of their climaxes of development mathematically. The teachers' adult need of a systematic framework was imposing itself too rigidly on what children were given to do, there was a multiplicity of formal apparatus and a tendency to assume that if a little child had been through progressive stages of teaching he had learned what he had been taught. Influences from upper departments were strong and I well remember one whole area in which infant schools were acquiring merit with them by teaching the children to do addition of hundreds, tens and units before promotion. They did this by the simple devices of adding numbers in the columns by making and counting strokes and then using a remembered routine for passing from column to column. Unfortunately junior teachers did not realise that ability to get answers right argued no knowledge of addition tables or understanding of how we write numbers. The lack of experience of using small numbers was clearly traceable at fourteen years of age in the slowness of using addition and subtraction bonds and there must have been all sorts of difficulties for the less able arising from the lack of groundwork in decimal notation. On the other hand I remember an infant school so obsessed by the power of decimal notation that it was imposing on children methods of dealing with small numbers which were logical but questionable as mental habits and emphasising this by elaborate apparatus which must have impeded automatic response. I was at the time quite taken in by the beautiful sequence of the teaching.
Just before and during the second world war the expansion and progress of nursery school work again threw emphasis on balanced child development. The good sense of teachers and the support of such organisations as the Froebel Foundation carried work with younger children again into the advance guard of primary education. Mathematics teaching tended to lag behind other aspects of advance. The subject was still referred to as 'one of the skills', but it was beginning to be thought of as something much more than this with skill in reckoning to be encouraged, but as a bi-product of use and the servant of need.
The work of Piaget and others into the nature and timing of mathematical concepts is thus being welcomed as support for teachers' own growing realisation that maturation must have its way and its time. I think that at present the idea that the maturation of the individual sets limits to fully effective teaching is more prominent than the more constructive aspect that it makes demands on ways of working. We await this interpretative stage in which we find out in terms of children as we know them how best to help mathematical ideas to take healthy root. At present there is perhaps some danger that interested teachers may mistake instruments of research, necessarily somewhat artificial and limited, for instruments of teaching. With the youngest children all that may be needed is the appropriate intervention of the adult in everyday happenings, but as the curriculum becomes more systematised more and more does the teacher have to create suitable opportunities for new mathematical learning.
The findings of contemporary research of course extend far beyond the infant school, though we may as yet be unready to heed more than the quite general implications about stages of learning and rates of maturation.
The influence of administrative changes is not a dominant theme in infant education but it is the best thread on which to hang the story of change in popular education for pupils of seven years of age and upwards. When I first visited elementary schools in 1925 they were on the eve of a great re-organisation which was both administrative and educational in intention, but schools above the infant stage were still organised in 'standards', a term which for 60 years had implied standards of formal attainment and which had been extended in number to cover the raising of the school age. In many schools rigid views about attainments led to wide ranges of age in some of the classes, as much as five or more of the seven years of school life from seven to fourteen.
It was a shock to me to find that what I had been taking for granted as a normal syllabus of arithmetic for pupils up to eleven years of age was so patently unsuitable for many boys and girls who were grovelling in lower standards at a late age. I think for some years I accepted wide differences of ability as the only explanation and expected the cure to lie solely in changes of organisation rather than in changes of approach. Constant, contact with infant classes, however, helped me to see how the teaching of Mathematics might fit in to a more general theory of education for primary school stages, and some acquaintance with the history of 'standards' made me sceptical of the theories by which organisation by attainment without regard for age and interest was often bolstered up.
The name 'standard' goes back to before the days of universal popular education. In the Revised Code of 1862 the payment of grant to schools qualifying for it was based on the individual examination of pupils in the 3 R's at various levels called standards. Syllabuses were quite cut and dried: thus in arithmetic, Standard 3 worked a sum in any simple rule as far as short division (inclusive). Standard 4 a sum in compound rules (money). Standard 5 a sum in compound rules (common weights and measures). This pattern, with its emphasis on the dry bones of applied arithmetic, was designed when school life was very short. It is still visible in many primary schools today.
The examination for grant was administered by inspectors and their assistants. The effects on arithmetic teaching are summed up by Matthew Arnold, [Reports on Elementary Schools 1852-1882. Matthew Arnold. H.M. Stationery Office (Eyre and Spottiswoode) 1910, page 128] who from the first disapproved of this Revised Code. In his report of 1868 he writes '. . . To teach children to bring right two sums out of three without really knowing arithmetic seems hard. Yet even here, what can be done to effect this (and it is not so very little) is done, and our examination in view of payment by results cannot but encourage its being done. The object being to ensure that on a given day a child shall be able to turn out, worked right, two out of three sums of a certain sort, he is taught the mechanical rule by which sums of this sort are worked, and sedulously practised all the year round in working them; arithmetical principles he is not taught, or introduced to the science of arithmetic .... The most notable result obtained will be that which has been most happily described by my colleague Mr Alderson. "Unless a vigorous effort is made to infuse more intelligence into its teaching Government Arithmetic will soon be known as a modification of the science peculiar to inspected schools, and remarkable for its sterility".'
Matthew Arnold indeed saw the teaching of arithmetic as a means of education and deplored the extinction of a like vision in a section of the teaching profession. We have not yet learned our lesson. If we are to go on examining the mathematically immature whether in primary or modern schools, we must learn how to test what matters in learning and not solely the mechanical end-points of it, which may or may not have been reached by desirable means.
The code of 1862 continued in force for some thirty years. I suspect that under it began the habitual use of incentives to correct computation such as the tawse and the cane. Its effects were perpetuated by the examination which accompanied the great increase in secondary school places at the beginning of the century. For a grant-paying machine was substituted a step-ladder for the ambitious. Able junior pupils improved their chances in the 'scholarship examination' by promotion into higher standards at the expense of the less successful, and generally tended to set the pace for these standards and for the examination. Also a vicious circle to which examinations are liable had begun. Questions were set because such work was being done in the schools and the schools went on doing the work because of the examination. I do not wish to harp on examinations, but these have been one dominant influence on primary teachers especially in Mathematics which appears so deceptively easy to test. I was told only the other day of a primary school master who was exceptionally knowledgeable about the possibilities of primary school work in Mathematics but who was afraid, perhaps mistakenly, to make changes, because of the selection test in his area which put a high premium on speed in mechanical work.
It took the re-organisation of schools with the break at eleven following the Hadow Report of 1926 to focus attention on the needs of primary pupils of all types. Primary schools were to be stabilised by a top age group which passed to a Senior or to a Secondary School leaving none behind, however low their attainment. The many unreorganised schools remaining (and some remain to this day) gradually accepted a policy of keeping their top classes for older
Although this break at eleven is not universally agreed as the right age for change of school it has some great advantages. If a later age were chosen to end the comprehensive primary stage, small schools would have great difficulty in coping with individual differences within age groups in Mathematics. The range at eleven is already too great for some small or weakly-staffed primary schools to make good enough provision for the ablest pupils. The really precocious are seldom catered for.
The educational theory of primary education, which was elaborated in a second Consultative Committee report in 1931, is often stated shortly as learning from activity and experience. It was least well understood in its application to Mathematics. Attempts to give juniors things to do, often at too low a level of experience, and to let them 'progress at their own rate' in arithmetic (a slogan of the 1930's) were so often time-wasting that they brought the theories themselves undeservedly into disrepute. I was told only the other day of a girl just transferred to a grammar school who was surprised at the fact that she had mathematics lessons. Her progress in arithmetic in a small primary school had been dominated by a text book through which she worked at her own rate with occasional help. Small wonder arithmetic was not her favourite subject. Arithmetic text books even when garnished with a few pictures are amongst the dreariest of school books. They deal only with limited aspects of learning the subject; even the best of them cannot provide a substitute for discussion with the teacher and fellow pupils or deal adequately with the need for experiment and experience.
However, after re-organisation, ways of dealing with differing needs within classes gained ground; group work in arithmetic (more easily organised in schools where there is no homework) became quite usual even though the size of classes was reduced but slowly. (In 1926 the legal limit for a class was sixty but I actually came across a class with seventy-three on its books as late as October 1928.) Again many small infants' schools were, on re-organisation, absorbed into junior and infant departments and, although this was perhaps a temporary set back to infant education, there was an over-all gain in understanding the change from infant to junior classes with, as I see it, a gradual and encouraging improvement in the mathematical aspects of school work in the vital period six to nine years of age. There is still far to go; this period is a great time for exploration and we are still in too great a hurry to tell children how to work things in our way. Time does not allow me to give you examples of the wonderful original thinking which may occur when a quite ordinary child has had the luck to meet a situation before being taught the adult rule for dealing with it.
All types of school except nurseries suffered great setbacks during the years of the second world war, but in primary education there was progress as well as setback. Examples spring to mind both of outstanding success in bringing the learning of Mathematics into the full stream of experience and understanding, and also of its antithesis, a horribly mechanised approach with its accompaniment of unbelievable howlers when memory failed.
From my now limited field of observation I should say that primary education in Mathematics is steadily gaining both in mathematical scope and in the degree of understanding behind work in number. But, the art of reckoning is still in popular estimation the only evidence of progress to be sought, and soundness in this art is impeded by obscuring fundamentals and usefulness under pile upon pile of rules.
As teachers we surely ought to try to bring up our pupils to form a public opinion which is better educated about Mathematics, especially about these early stages which are within the understanding of any person of good general education (even if they are slow computers).
The war years bore more heavily on the mathematical education of older pupils which had been making great progress during this century up to 1939. There had been in addition to the increased selection for grammar schools much concern for older scholars in the elementary schools particularly for those abler pupils that we might now call 'A-stream modern'. The curriculum had been widened for all, and even before Hadow re-organisation Local Education Authorities had been encouraged to try out various ways, including selective central schools, of extending the work for the ablest elementary pupils. Mathematical advances had been mainly towards teaching at a slightly later age Mathematics as taught in secondary schools with perhaps, for boys, developments of a practical kind in geometry and mensuration. Girls often missed geometrical experience and therefore suffered even more than boys from that most virulent disease, still I fear prevalent today, the learning of techniques of algebra by rote without either clear purpose or adequate background knowledge of number. This sort of superficial teaching also arose in the elementary schools when teachers used with pupils still at school the syllabuses they would meet again in the evening institutes. Where separate schools with some technical, trade or commercial bias existed Mathematics was treated as a purposeful tool, but bias was sometimes carried too far and spoiled the balance of the subject as an instrument of education.
The Senior Schools which followed Hadow re-organisation and which flourished in the 1930's had to face a wider problem, the teaching of all grades of ability remaining under the elementary code. They were indeed Secondary Modern Schools in all but name, privilege and the one year of age, fourteen to fifteen. They were for the most part staffed by teachers whose experience was with the children of the elementary school. Many teachers had themselves been so educated up to fourteen, so that they understood their pupils' background as though they were fellow scholars. Many of them had a real interest in Mathematics even though not highly qualified; some were such as might today have entered teaching through a University. To these old hands were added young teachers interested in a new venture and perhaps better qualified as subject specialists. The combination was often a strong one.
I could illustrate, in approach if not in detail, from what I saw in these schools most of the recommendations contained in our recent modern school report; in particular the best of these school tried to suit their curriculum to their pupils' world: to the rural community, to the life of an industrial town or to the kind of folk living in a dormitory area of a city. One girls' school in a vicious quarter of a great sea-port became famous for the way it enlisted the co-operation of girls who certainly had no interest in learning for its own sake and who, whatever their ability or the scholarliness of their teachers, would have spurned a merely conventional approach to Mathematics. Such an extreme example casts a spotlight on an essential difference between compulsory education and a chosen school career.
The number of pupils who regard Mathematics as of importance to their future, or enjoyable in itself, is increasing, but there still remains a majority who, in the words of the modern school report, must be helped 'to see Mathematics as a subject that touches their lives and is worth their attention.' Recent discussions in the press about further raising of the compulsory school age make me feel more strongly than ever that for ordinary children at least, we need to take so-called applied mathematics out of cold storage, to include in it mathematical usages of any and every sort and to re-think how we treat such usages as vehicles both of general and of mathematical education.
I think that scant justice has been done in our thoughts to the pioneer efforts of Senior Schools and indeed of the top classes of many unreorganised schools of the pre-war period. The best of them took advantage of their freedom and showed flexibility and powers of adaptation controlled by a knowledge of circumstances which was the fruit of long experience.
Unhappily Mathematics and Science in Senior Schools were among the earliest war casualties in education, especially in boys' schools. Almost every teacher with any pretensions to good qualifications in Mathematics who survived a call to the services was transferred to teaching in a Grammar School. The generation of wise rather than highly qualified men and women who had been the cream of the old elementary school were nearing the end of their service and the burden which fell on them was very heavy. New teachers coming into the service were those who had passed to grammar schools at eleven and who, for all their advantages, had less in common with their pupils.
I consider it a tragedy that the natural evolution of mathematics teaching through the development of Senior Schools was almost obliterated by the war. It cannot be revived in spirit without enormous effort, if only because of the conservatism of our profession. The 'set' of our minds is inevitably to teach how and what we were ourselves taught. Our idea of elementary mathematics is coloured by what we learned at eleven, twelve and thirteen, but our mode of learning is not necessarily the best mode for our pupils who are in a different generation and may be in very different categories as thinkers.
The Modern School, cut off from its mathematical history by the war, is thus extraordinarily susceptible to the influences of today some of which are far from educational. Questions of prestige and expediency may lead schools to try to acquire a veneer of success. To me the most hopeful tendency is the upsurge of primary education in Mathematics. It seems to me most appropriate that our report Mathematics in Secondary Modern Schools should be so definitely a sequel to the 1955 report Mathematics in Primary Schools.
When I drafted this address I tried to avoid except for incidental references one of the main means of influencing popular education, one which completes the cycle of influence, namely the education and training of teachers. The Association has not yet reported on this subject, though there has been a sub-committee discussing it. Very great changes arising from the extension of the two-year course in colleges to three years are imminent and this is not a moment to make detailed comment; but what I have said about the mathematical staffing of schools makes some comment almost necessary as a conclusion.
Interesting changes in the organisation of the education and training of teachers have gone on side by side with the changes in popular education and I can at least speak of these as one who has had the good fortune to be linked with the work of the colleges in one capacity or another continuously since 1926.
In 1926 only a few teachers in elementary schools for pupils from five to fourteen years were subject specialists, and few of these had any mathematical qualification other than perhaps Mathematics as one of several subjects in the Teachers' Certificate. Almost all teachers taught arithmetic and a problem of colleges was, as indeed it still is in dealing with primary school work, that of coping with a wide range of attainment, interest and aptitude amongst students. Wide differences existed (as they do today ) even amongst those who elected Mathematics as a special subject.
In 1926 the certificate examination was still conducted by the Board of Education. As a junior member of a panel of inspectors entrusted with the examination of Mathematics and coming to this task from teaching in a grammar school, I was immediately struck by the difficulty of setting papers which were neither soft options for the apt well-prepared candidates nor dreary drudgery for the less able but not unpromising future teachers. Inspectors were very interested in this work and in trying to keep some balance between the needs of the schools they visited and the personal education of the teacher, but I do not think that at the time of which I speak the examination was keeping pace with changing conditions in the colleges. The pre-college education of students was changing very rapidly and the gap between student and student, not least in Mathematics, was widening, in spite of, or even because of, the fact that the 1920's saw the last of the pupil teacher centres devised to deal with the fourteen-year-old school leaver.
Finally for the record, let me note what is familiar to so many of you. The certificate examination by the Board of Education was discontinued in favour of a decentralised system administered by Joint Boards under the auspices of the Universities and later by the Institutes of Education now in operation.
It would be impertinent for me to comment on all that the colleges have done for the schools under these systems and on the courageous experiments tried, including such temporary measures as the Emergency Training Scheme and the longer term but now inevitably temporary measure, the third year Supplementary Courses for two year trained and emergency trained teachers. I hope, however, that those of you who are grappling with the problems of Mathematics in the colleges, together with the many who are teaching their future students in the grammar schools, will not mind my emphasising what I think to be important however it may be accomplished.
The problem of making the most of the talents of pupils and students who, as things are, may have little interest in Mathematics but will have to teach it in its fundamental stages, is as important for the health of the subject as turning out more specialist teachers; indeed in the educational cycle it is an important step towards the making of more mathematicians. Moreover amongst those taking Mathematics as a special subject in college there is great scope for dealing with differences of interest and aptitude. Popular education in Mathematics at its various levels can absorb without detriment a very wide range of talent in its teachers, provided that such mathematical knowledge as they have, however elementary, is happily based on genuine learning and is fundamentally sound.
The Small House, Oxford Street, Eddington, Berks.
L D Adams.
Last Updated January 2021