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ARITHMETIC AS A MENTAL PLINEy of California.

Another writer said:

Having such prominence, the subject came to be taken as the basis of gradation and of promoting pupils.'

It is difficult for the teachers of to-day to realize that arithmetic has not always been one of the fundamentals of the school curriculum. There is the general impression that the curriculum consisted of the three R's until it was enriched by the addition of the more modern subjects. Hence we fail to appreciate that it was not until the second quarter of the nineteenth century that arithmetic was accorded its place in our schools as one of the traditional educational trinity.

Inductive method. The complete title to Colburn's First Lessons? contained the phrase, "on the inductive method of instruction," and this method was a conspicuous feature of his texts. During the active period from 1821 to 1857, authors frequently included some reference to the inductive method in the title of their texts. In the construction of their texts many followed closely Colburn's plan. Some authors adhered to the deductive plan, and after 1857 the texts, even those which had previously embodied the "inductive method," were generally organized deductively.

Skill and thoroughness.-Increasing emphasis was placed upon skill in performing the operations of arithmetic. This is testified to by the increased space given to drill exercises and the publication of "Lightning Calculators," which were numerous in the last half of the century. In the preface to the New Intermediate Arithmetic, Felter

says:

This book is designed to make the pupil quick and accurate in calculation, and to give him a knowledge of those principles and processes of arithmetic which are needed in the ordinary transactions of life, together with skill in their application.

To accomplish this, the drill card exercises are arranged to furnish any desired amount of practice in computation; while the processes and analyses leading directly to the rule, together with the number, gradation, and character of the practical examples, give the knowledge of necessary principles and skill in their use.

Felter says in the preface to An Introduction to Arithmetical Analysis: "The importance of being thorough in the elements of arithmetic can not be too often impressed upon the teacher."

In brief these are the significant features of arithmetic as a school subject in this period. In each of them there are evidences of Colburn's influence. In the next chapter the important texts of the period are described, and in them we shall see more clearly the influence of Colburn upon the arithmetic of this period.

1 J. M. Greenwood: Principles of Education Practically Applied, 1887, p. 154.

2 Edition of 1826.

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Chapter VIII.

FORMALIZED PESTALOZZIAN ARITHMETICS.

In this period, arithmetic was usually presented in a series of three books for the common schools and a higher arithmetic which was primarily for academies and colleges. While there was no absolute uniformity in the planes of division of arithmetic in the common schools, yet in general there was, first, a primary arithmetic which covered the work pursued in the primary school, which varied from two and one-half years to four years; second, an intellectual, or mental arithmetic; and third, a text often designated as "practical," or "common school," and which was a complete text in itself. In case a text became at all popular, it was provided with a key for the use of the teacher. The originally distinct line of cleavage between mental arithmetic and written arithmetic became less and less distinct by reason of combining the two, which was especially popular in the latter portion of this period.

Two classes of texts of this period will be described; first, those which were widely used and hence exemplify the practice of the times, and second, other important texts. This second class contains texts which were not used as extensively as the first class, but which indicate the course of development. The extensiveness of the use of a text has been determined by considering the number of editions through which the text passed and the length of time it was before the public.

For description the texts have been grouped under authors. The order is determined by the date of the first text by an author.

ARITHMETICS WHICH were widely used.

Warren Colburn's First Lessons, 1821, which we have described, was the most extensively used mental arithmetic during the active portion of this period and must be counted among important texts of the entire period.

Frederick Emerson (1788-1857), who wrote the North American Arithmetics, was for a number of years a teacher in the Boston public schools, later principal in the department of arithmetic at Boylston School, and finally superintendent of schools. The Part First appeared in 1827, Part Second in 1832, and Part Third in 1834. Part First is distinctly an elementary book, and the author states, "The

slate and pencil are not required in the performance of the lessons contained in Part First." The first part of the Part Second consists of oral arithmetic, and the second of written arithmetic. Part Third is designed for advanced scholars, and as such is a scholarly presentation of the subject from a mature point of view.

As soon as the series was complete, it displaced Colburn's texts in the Boston schools, and the North American Arithmetics, Part First, was an alternative text as late as 1866-67. The series had been used in Chicago preceding 1866. In an edition of Part First, it is stated that it has been adopted in Boston, Salem, Portland, Providence, New York, Philadelphia, and Louisville. I have examined copies of Part Second dated 1832, 1839, 1848, 1854, and of Part Third dated 1834, 1844, 1850. Part Third appeared in two forms, both copyrighted in 1834. One of these is announced as revised and enlarged. The enlargement is a list of questions for examination. Otherwise the series does not appear to have been revised.

Charles Davies (1798-1876) graduated from the Military Academy at West Point in 1815. He was professor of mathematics and natural philosophy in that institution from 1823 to 1837, and professor of mathematics in Trinity College, Hartford, 1839 to 1841. Later he taught mathematics in the normal school at Albany, N. Y., and was professor of higher mathematics in Columbia College, New York City from 1857 to 1867, when he was made emeritus professor. Davies's primary arithmetic was published under the title of First Lessons in Arithmetic in 1840. There is also a Primary Table Book, which appears to have been published separately at first. In 1856 the primary book is advertised as Davies Primary Arithmetic and Table Book. Davies Intellectual Arithmetic was first copyrighted in 1838 and recopyrighted in 1854, 1862, 1881. The practical arithmetic was first published in 1833 under the title Common School Arithmetic. In 1838 it was "enlarged and improved" and called Arithmetic Designed for Academies and Schools. In the preface of this edition Davies describes the book as an "elementary treatise." In 1848 this was revised and called Davies' School Arithmetic, and in 1855 another revision changed the title to Davies' New School Arithmetic. Later a "New Series of Arithmetics" was prepared, and the School Arithmetic became Practical Arithmetic and a new work, Elements of Written Arithmetic, was added to the series. The first edition of the University Arithmetic was published in 1846. It passed through many editions and was often revised. Greenwood says: "Whenever the discovery of new methods of presentation demanded a revision, the publishers and authors at once complied."

In 1912, the following arithmetics by Charles Davies were listed by the American Book Company: Primary Arithmetic, Practical Arithmetic, Elements of Written Arithmetic, and University Arithmetic.

Davies prepared "a full analysis of the science of mathematics," and explained "in connection the most improved methods of teaching." This was published in 1850 under the title, The Logic and Utility of Mathematics. This was based upon the system of mathematical instruction which had been "steadily pursued at the Military Academy (West Point) for over a quarter of a century." In describing this "system of mathematical instruction," Davies says:

It is the essence of that system that a principle be taught before it is applied to practice; that general principles and general laws be taught, for their contemplation is far more improving to the mind than the examination of isolated propositions; and that when such principles and such laws are fully comprehended, their applications be then taught as consequences or practical results.

This view of education led, at an early day, to the union of the French and English systems of mathematics. By this union the exact and beautiful methods of generalization, which distinguish the French school, were blended with the practical methods of the English system.1

And he sums it up by saying:

And in that system (at the Military Academy) Mathematics is the basis; Science precedes Art; Theory goes before Practice; the general formula embraces all the particulars.

This system was the basis of Davies's arithmetics. In them, arithmetic is first a science.

In estimating the work of Charles Davies, Greenwood says:

The influence of Dr. Davies's writings on subsequent authors in this country can hardly be overestimated. It may be very properly regarded as the beginning of a revolution in schoolbook making. Simplicity and extreme clearness became the leading ideas in the minds of authors, who studied how to be understood by children and young people.2

Joseph Ray (1807-1855) entered the Ohio Medical College in Cincinnati in 1828, graduated, and became a surgeon. In 1831 he became a teacher in Woodward College and professor of mathematics in 1834. This position he held until the institution was changed to Woodward High School in 1851, when he became president.

Ray's primary book was first published in 1834 with the title, Ray's Tables and Rules in Arithmetic and sold for 6 cents. In 1844 it was remodeled and became Part First, of Ray's Arithmetical Course. Since then it has been revised in 1853, 1857, 1877, 1903, and has appeared under several titles. In 1857, it was called Primary Lessons, in 1877, Ray's New Primary Arithmetic. The intellectual arithmetic was first published in 1834 under the title, The Little Arithmetic; Elementary Lessons in Intellectual Arithmetic, on the Analytic and Inductive Method of Instruction. In 1844, it was enlarged and called Ray's Arithmetic, Part Second; in 1857 it was

1 Preface.

2 James M. Greenwood and Artemas Martin, "Notes on the History of American Textbooks on Arithmetic." Rept. U. S. Commis, of Ed., 1897-98, p. 839.

known as Intellectual Arithmetic, by Induction and Analysis; and in 1877 as Ray's New Intellectual Arithmetic. Under this last title it was copyrighted in 1905. Ray's Eclectic Arithmetic on the Inductive and Analytic Methods of Instruction was first published in 1837. In 1844 it was "carefully revised" and called Ray's Arithmetic, Part Third, and in 1857 it was again revised and called Practical Arithmetic by Induction and Analysis. In 1877 it became Ray's New Practical Artithmetic. In 1879, a two-book series was issued, Ray's New Elementary Arithmetic and Ray's New Practical Arithmetic. This series was revised in 1903 and the word "new" changed to "modern." Ray's Higher Arithmetic was published in 1856, the year after Dr. Ray's death. The text was completed and edited by Prof. Charles A. Mathews. It was revised and called Ray's New Higher Arithmetic in 1880.

Of all the texts of this period, the series by Joseph Ray has enjoyed the most extended and continued use. Ray's arithmetics became popular soon after their first publication in 1834, and it seems that their popularity increased rapidly for a number of years. Until within the last quarter of a century, no arithmetics were published which supplanted them except locally. Even now (1913), after more than a decade which has been characterized by texts of another type, they are still a widely used series of arithmetics. The average yearly sale for the last ten years has been approximately 250,000 copies.

J. M. Greenwood sums up his estimate of Joseph Ray and his work in these words:

To many it has appeared strange why Ray's arithmetics have such a hold on the popular mind. The reason is, I think, obvious. Dr. Ray was, in a large sense, a self-made mathematician and a self-made teacher. He had learned well the lesson of self-help, and in the preparation of his books he always kept before himself all the difficulties he had experienced in mastering each topic. No one knew better just when and where and how to bear down on certain points. In an eminent degree he possessed that rare combination of assimilation and clear presentation. He knew how to make the subjects stick.1

Benjamin Greenleaf's (1786-1864) first book, the National Arithmetic, Combining the Analytic and Synthetic Methods, was published in 1835. Greenwood says: "It soon became a favorite treatise with teachers who preferred sound attainments in this science. The first edition was exhausted within a year." It was revised in 1836, 1847, and 1857, but the "general plan of the work was never changed." In 1836 it was announced as "Forming a complete mercantile arithmetic, designed for schools and academies." The edition of 1857 has the title, The National Arithmetic, on the Inductive System, combining the Analytic and Synthetic Methods; Forming a Complete Course of

1 Op. cit., p. 840.