Research on the Teaching of Math: Formal Arithmetic at Age Ten, Hurried or Delayed?

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The Historical Perspective

Strange though it may seem at first, it is nevertheless quite clear that addition, subtraction, multiplication and division – comparatively simple operations, which we inflict on our children while they are still quite young – were, in antiquity, far beyond the horizon of any primary school. The widespread use of calculating-tables and counting-machines [abacus] shows that not many people could add up – and this goes on being true to a much later date, even in educated circles. ­ A History of Education in Antiquity, by Henri I. Marrou, translated by George Lamb, Sheed and Ward, London, 1956, page 158.

[page 204] Before the Reformation there was little or nothing accomplished in the way of public education in England. In the monasteries some instruction was given by monks, but we have no evidence that any branch of mathematics was taught to the youth (some idea of the state of arithmetical knowledge may be gathered from an ancient custom at Shrewsbury, where a person was deemed of age when he knew how to count up to twelve pence. See Tylor’s Primitive Culture, New York, 1889, Vol. I., p. 242.) . . . In the sixteenth century, on the suppression of the monasteries, schools were founded in considerable numbers . . . and for centuries have served for the education of the sons mainly of the nobility and gentry. In these schools the ancient classics were the almost exclusive subjects of study; mathematical teaching was unknown there. Perhaps the demands of every-day life forced upon the boys a knowledge of counting and of the very simplest computations, but we are safe in saying that, before the close of the last century, the ordinary boy of England’s famous public schools [page 205] could not divide 2021 by 43, though such problems had been performed centuries before according to the teaching of Brahmagupta and Bhaskara by boys brought up on the far-off banks of the Ganges . . . . All the information we could find respecting the education of the upper classes points to the conclusion that arithmetic was neglected, and that De Morgan was right in his statement that as late as the 18th century there could have been no such thing as a teacher of arithmetic in schools like Eton. In 1750, Warren Hastings, [page 206] who had been attending Westminster, was put into a commercial school, that he might study arithmetic and book-keeping before sailing for Bengal.

At the universities little was done in mathematics before the middle of the 17th century . . . . During the reign of Queen Elizabeth, fresh statutes were given, excluding all mathematics from the course of undergraduates, presumably because this study pertained to practical life, and could, therefore, have no claim to attention in a university . . . .

[page 207] This scorn and ignorance of the art of computation by all but commercial classes is seen in Germany as well as England.

It was not before the present century that arithmetic and other branches of mathematics found admission into England’s public schools. At Harrow “vulgar fractions, Euclid, geography, and modern history were first studied” in 1829. At the Merchant Taylors’ School “mathematics, writing, and arithmetic were added in 1829.” At Eton mathematics was not compulsory till 1851.

Since the art of calculation was no more considered a part of a liberal education than was the art of shoe-making, it is natural to find the study of arithmetic relegated to the commercial schools. The poor boy sometimes studied it; the rich boy did not need it. In the Latin schools it was unknown, but in schools for the poor it was sometimes taught . . .

[page 217] The first arithmetics used in the American colonies were English works . . . . The earliest arithmetic written and printed in America appeared anonymously in Boston in 1729. Though a work of considerable merit, it seems to have been used very little . . . . In 1788 appeared at Newburyport the New and Complete System of Arithmetic by Nicholas Pike . . . . It was intended for advanced schools . . . . [page 218] Reform in arithmetical teaching in the United States did not begin until the publication of Warren Colburn, in 1821, [page 219] of the Intellectual Arithmetic. This was the first fruit of Pestalozzian ideas on American soil . . . . The success of this little book was extraordinary. But American teachers in Colburn’s time, and long after, never quite succeeded in successfully engrafting Pestalozzian principles on written arithmetic . . . .
­ A History of Elementary Mathematics with Hints on Methods of Teaching, by Florian Cajori, Ph. D., London: Macmillan Company, 1917, pages 204-207, 217-219.

. . . The American Calculator, first published in 1828 . . . is reasonably typical of the colonial period. This text was used with older students (beginning at about the age of eleven). It was complete in itself, not one book of a series such as the texts that evolved in later years. ­ Readings in the History of Mathematics Education, edited by James K. Bidwell and Robert G. Clason, National Council of Teachers of Mathematics, Washington, D. C., n. d., page 2.

. . . The study of it [arithmetic] used to be put off to a very late period. Scholars under twelve or thirteen years of age were not considered capable of learning it, and generally they were not capable. Many persons were obliged to leave school before they were old enough to commence the study of it. ­ Readings in the History of Mathematics Education, page 25, taken from “Teaching of Arithmetic,” text of an address delivered by Warren Colburn before the American Institute of Instruction in Boston, August, 1830, reprinted from the Elementary School Teacher 12 (June 1912): 463-480.

[In the early seventeenth century, the grammar school curriculum was] almost certainly confined to Latin grammar, the Catechism and Bible study . . . . [P]upils arriving at Oxford and Cambridge frequently did not have any knowledge of Arabic numerals, to say nothing of the elementary arithmetical operations. ­ A History of Mathematics Education in England, by Goeffrey Howson, Cambridge University Press, Cambridge, 1981, page 30.

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