We have seen how in the first half of the nineteenth century there were rapid and considerable developments in the teaching of mathematics at Cambridge University and within what we should nowadays term the secondary sector of education. There was also considerable mathematical activity outside schools, colleges and other such educational institutions. In a wider educational context, the period saw the State's first involvement in elementary education and teacher-training. The need for a national educational system was gradually being realised, as was the desirability of having teachers who had not only achieved mastery of academic material but had also received instruction in the art of teaching. Thomas Tate's interest in mathematics was aroused in an unconventional way and he himself was to carry on the ‘popular tradition’ in mathematics education before being appointed as effectively the country's first mathematics teacher-trainer, and becoming renowned as an educator and textbook author.

THOMAS TATE AND THE POPULAR TRADITION

Thomas Tate was born in Alnwick, Northumberland, in February 1807, the second son of a builder. Both he and his elder brother, George, attended the Borough School in Alnwick, but only George appears to have become a pupil at the Dukes (Grammar) School (1). It may well be that the elementary school catered more for Thomas' scientific tastes, since its master, James Ferguson, was to show his interest in, and ability for, mathematics by producing a new edition of Hutton's Arithmetic (2).

In the preceding chapters we have seen how, in the absence of any Stateor Church-coordinated system, the provision of mathematics education at a pre-university level often came to depend upon private enterprise. Frequently, in order to attract financial support, this led to an emphasis being placed on utilitarian ends. In this chapter we consider a mathematician who was largely self-taught and whose contribution to education was for many years a ‘private’ one. Later in life he was to head the mathematics department of England's leading military academy where once again emphasis had to be laid on the subject's practical aspects. Hutton, then, was an educator biased towards a utilitarian view of mathematics. Through his books and other writings he was to have a considerable influence on English mathematics, particularly outside the two ancient universities and the established grammar schools.

The most obvious difference between the subject of this final biographical essay and those of previous chapters is doubtless that of gender. By the twentieth century it had become possible for a woman to be ranked alongside her male colleagues as both a mathematician and a mathematics educator. There are, however, other significant differences: for the first time the study is of someone who trained professionally as a teacher, who studied at a university other than Oxford or Cambridge, and who undertook postgraduate work in mathematics education. Most importantly of all, Elizabeth Williams' interests were to span all sectors of school education. For, during her lifetime, the concept of two independent elementary and secondary systems existing side-by-side was first weakened and later replaced by a unified scheme in which all children received first primary and then secondary education.

PRIMARY SCHOOL EDUCATION

Elizabeth May Larby was born in January 1895, the second in the family of four children. Her childhood was to be divided between the country where her forbears had farmed for generations, and the suburbs of London in which the family lived during term-time until the children's education had been secured [1].

So it was that at the age of four Elizabeth began school at St Michael's Church of England Infants' School, Chelsea. She attended that school for two years – each Monday paying her ‘fees’ of 3d (slightly over 1 p) – before transferring to Shaftesbury Road Elementary School in Forest Gate.

That a highly-paid civil servant in the Admiralty (earning £350 p.a.), who was educated at St Paul's School and Cambridge University, should have to employ a tutor to teach him the multiplication table is an indication of the status of mathematics education in the seventeeth century.

1802: CAMBRIDGE TRIPOS PAPERS

[Students took up to four papers. The seventh and eighth classes (containing the ordinary degree students) were set only bookwork questions.]

Monday morning problems—Mr Palmer

First and second classes (i.e. the expectant wranglers)

1. Given the three angles of a plane triangle, and the radius of its inscribed circle, to determine its sides.

2. The specific gravities of two fluids, which will not mix, are to each other as n : 1, compare the quantities which must be poured into a cylindrical tube, whose length is (a) inches, that the pressures on the concave surfaces of the tube, which are in contact with the fluids, may be equal.

3. Determine that point in the arc of a quadrant from which two lines being drawn, one to the centre and the other bisecting the radius, the included angle shall be the greatest possible.

4. Required the linear aperture of a concave spherical reflector of glass, that the brightness of the sun's image may be the same when viewed in the reflector and in a given glass lens of the same radius.

5. Determine the evolute to the logarithmic spiral.

6. Prove that the periodic times in all ellipses about the same centre are equal.

7. The distance of a small rectilinear object from the eye being given, compare its apparent magnitude when viewed through a cylindrical body of water with that perceived by the naked eye.

It is possibly still too early to attempt to chronicle and disentangle the various movements that have taken place in mathematics education since i960. Even if the time were ripe, then it would be impossible to present a detailed, critical account without effectively embarking on the writing of a separate book. Yet to end our story in i960 would be unfortunate in two respects. First it would leave many important changes unreported and unreferenced. Secondly, and perhaps more importantly, it would not illuminate the way in which so many of the changes of the last two decades grew out of, and were consequences of, beliefs, actions and decisions which have been alluded to in the story of mathematics education prior to 1960. For that reason, I have chosen to end this book with a brief overview of trends in the post-1960 era.

What were the most significant events of these times? If the question had been asked in 1965, then the reply would almost certainly have been the formation of the large ‘modern mathematics’ projects such as the School Mathematics Project (SMP) and that of the Nuffield Foundation concerned with the age range five to thirteen. These projects brought a new, more open approach to curriculum development and the stress they laid on corporate involvement was an important factor leading to the establishment of a more widely-based ‘mathematics teaching community’.

The first half of the nineteenth century witnessed many significant changes in mathematics education. Mathematics teaching at Cambridge was revivified; the hold of Oxford and Cambridge on university education was broken; a new type of secondary school arose in which mathematics was awarded a place of honour in the curriculum; and the particular problems of teaching and learning mathematics gradually became more explicitly and professionally scrutinised. The case for a state system of education was argued as were the objectives which such a system should seek. Closely concerned with many of these changes and developments, as well as with the general diffusion of mathematical ideas, was Augustus De Morgan.

EARLY YEARS AND SCHOOL

De Morgan, the son of an army colonel, was born in Madurai, southern India, in 1806. He spent only seven months in India, but the stay was to prove of great significance for he contracted an eye infection and lost the sight of his right eye. His father brought the family back to England, but he himself soon returned to India. Indeed, Augustus was to see little of his father. They had the years 1810–12 together, during which time his father taught Augustus ‘reading and writing’, but then Colonel De Morgan set out for India once more. There he became ill with a liver complaint and died on the voyage back to England.

By that time Augustus had begun to attend a series of private schools run by single ladies or clergymen.

To my knowledge this is the first book to be published which attempts to tell the story of the development of mathematics education in England. That this should be so is rather surprising; for one can turn to histories of the teaching of science and to a history of mathematics teaching in Scotland. Any attempt to fill such a gap is, therefore, fraught with difficulties, for the ‘only book’ is likely to be invested with an authority it may not deserve. Extra problems may also arise as a result of my having chosen to present the material through the medium of biographies. Emerson's claim that ‘there is properly no history; only biography’ could be used to justify this decision. The truth is, however, more mundane; for I abandoned a ‘chronological’ account thinking it would have little appeal for the general reader as opposed to the serious student. I believe also that a biographical account, even though it requires frequent scene-settings and ‘flashbacks’, better demonstrates the great part which individuals have always played in the advancement of mathematics education in England. The book, however, is not to be compared with Macfarlane's Ten British Mathematicians of the Nineteenth Century, for that author set out to identify the ten ‘greatest’ mathematicians of that age. My basis for selection has been different, for I have chosen subjects from various periods and traditions whose own mathematical education and whose contributions to mathematics teaching provided a framework around which I could construct a representative story.

The early English schools were the offspring of the Church and their curricula were designed with one principal aim in view, the more widespread dissemination of Christianity. If Christianity and civilisation were to be brought to the Anglo-Saxons, it was essential to have priests who were trained to read the Bible – a literate priesthood was a prerequisite for evangelisation.

After the coming of St Augustine in 597 schools were established by the great monastic churches, for example at Canterbury, Westminster and York, in an attempt to supply this basic, vocational training. These schools provided instruction in the Latin language, the Latin scriptures and in the Church's liturgy and music. Gradually, at the more important centres, the curriculum widened to include astronomy and arithmetic {1}. There was a utilitarian need for the latter – if only to calculate the dates of movable feasts and to keep rudimentary accounts.

We cannot be certain exactly when mathematics first was taught in England, but Bede (674–735) in his Ecclesiastical History {2} tells how Theodore, Archbishop of Canterbury in the late seventh century, and his deacon, Hadrian, ‘gathered a crowd of disciples … and … taught them the art of ecclesiastical poetry, astronomy and arithmetic’. Bede, himself, in his De Temporurm Ratione (3) demonstrated the arithmetic needed to make the necessary calculations concerning Easter. Of course, at that time the West had no knowledge of the Hindu–Arabic notation and so Bede made use of ‘finger-reckoning’, a form of arithmetic taught in monastic schools for many centuries (cf. p. 15 below).

As the writer of the first series of mathematics texts in English, Robert Recorde clearly holds a special place in the history of mathematics education in England. Yet it can be argued that Recorde should be recognised for yet another reason: that he was the first mathematics educator. Not only did Recorde teach mathematics, but his writings show clearly – both implicitly and explicitly – that he had also given serious consideration to the problems of learning and teaching mathematics.

THE MAN AND HIS TIMES

Robert Recorde [1] was born in Tenby, Wales, about 1512. That his exact date of birth is not known is regrettably typical, for there are many details of Recorde's life and career that are lacking. It is also significant that the sole piece of information we have concerning Recorde's age should be the result of a political trial [2]. The times in which Recorde lived were indeed turbulent ones, and the various political and religious struggles {3} affected both his personal well-being and his plans for publishing mathematical texts.

The first stages of the Reformation took place whilst Recorde was a student at Oxford. He graduated BA there in 1531 and was the same year elected a Fellow of All Souls' College, a graduate foundation for the study of theology, law and medicine. Clearly, Recorde must have devoted considerable time to the study of the last named, for, according to the Cambridge records, he had been licensed in medicine at Oxford about 1533.

It is not perhaps surprising that James Maurice Wilson became a ‘schoolmaster’ and ‘divine’, for that was a family tradition; that the Dictionary of National Biography should describe him also as scholar, mathematician, astronomer and antiquary gives an even greater indication of his remarkable all-round ability. As he himself put it, ‘I had no special aim or bent in life; but if circumstances showed me that something ought to be done, and no one else came forward to do it, I thought that I ought to try’ (1).

SCHOOLING

Wilson was born, one of twins, at King William's College in the Isle of Man in November 1836. His father, the first principal of the College, an Anglican institution established in 1833 {2}, was soon to leave teaching to become a parish priest, but it was to King William's that Wilson returned to begin his formal education in 1848. They were not happy times; the school was cheap and attracted a mixed clientèle. Those of ‘good birth’ were ‘swamped in a very rough lot’ and there was a considerable amount of bullying and cruelty {3}. Neither could young Wilson find much consolation in his lessons, for ‘no one on the staff was a scholar, and no one even a tolerable mathematician’. When he, with the aid of a textbook, learned some algebra, his solutions of ordinary quadratics ‘were kept and copied by the Principal, in case anyone in the future should reach such a high-water mark’.

The Origins of the Theory

1. As soon as man began to think of abstract problems at all, it was only natural that speculations as to the nature and ultimate structure of the material world should figure largely in his writings and philosophies.

Among the earliest speculations which have survived are those of Thales of Miletus (about 640–547 B. C.), many of whose ideas may well have been derived from still earlier legends of Egyptian origin. He conjectured that the whole material universe consisted only of water and of substances derived from water by physical transformation. Earth was produced by the condensation of water, and air by its rarefaction, while air when heated became fire. About 500 B. C. Heraclitus advanced the alternative view that earth, air, fire and water were not transformable one into the other, but constituted four distinct unalterable “elements”, and that all material substances were composed of these four elements mixed in varying proportions—a sort of dim anticipation of modern chemical theory. At a somewhat later date, Leucippus and Democritus maintained that matter consisted of minute hard particles moving as separate units in empty space, and that there were as many kinds of particles as there are different substances.

Unhappily nothing now remains of the writings of either Democritus or Leucippus; their opinions are known to us only through second-hand accounts. From these we learn that they imagined their particles to be eternal and invisible, and so small that their size could not be diminished; hence the name ἄτομος—indivisible.