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

by | Math | 2 comments

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.

You can read the entire article here.


  1. Jonathan Vos Post

    “Formal Arithmetic at Age Ten, Hurried or Delayed?”

    Delayed. My son, age 10, could take the derivative or integral of a polynomial if you spoke it to him. He’d visualize the polynomial, calculate the the derivative or integral in his head, and dictate it back to you.

    How old were Ramanujan and Gauss when they could do figurate numbers in their heads?

    Public schools in the USA have essentially given up, as badly as the prestigious schools in England a couple of centuries ago.

    Fortunately, there ARE some good students, good media (the CBS-TV show NUMB3RS for example), many good books, and the Web.

    There is no meaningful limit to what a child can learn, given a good teacher and a motivation for the student, and if you start early enough (neural plasticity).

    Self esteem? I suggest that it be deserved, by allowing children to succeed with actual content and understanding. Not “teaching to the test.” Not “No child left behind.” Those are red herrings. I mean real education.

    For the children.

  2. Perla Sarmiento de Adams


    It is true that children can learn many math concepts very early, but also it is true that in ‎the last century the advance of the Math has being very poor, I think the only interesting ‎fact was The Demonstration of the Theorem of Fermat, that is.‎

    The math we have comes from those who spend the first years of life learning grammar ‎and logic and then, when they were well prepare they develop the calculus and geometry ‎which our little children play today, but this children miss the enough good foundation to ‎develop something better from there.‎

    So, we as civilization, are not winning teaching complex math concepts to the babies, and ‎little children, we are lose the possibility to develop great minds to solve the science’s ‎problems, the math is a tool of the science, it is the language of the science.‎



  1. Michi’s blog » Blog Archive » Carnival of Mathematics #3 - [...] the mix. Following that, Rebecca Newburn discusses equation solving strategies and Laurie Bluedorn takes a historical view on the…
  2. » When should one learn formal mathematics? « e pur si muove - 智に働けば角が立つ;情に棹させば流される - [...] of mathematics back in the 16th century to the esoterica of modern mathematics2 is mindboggling. « Trivium Pursuit «…

Submit a Comment

Your email address will not be published. Required fields are marked *