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ARITHMETIC AS A SCHOOL SUBJECT.

Two more problems are similarly explained, though somewhat more briefly. He then draws a conclusion as follows:

In the last three problems the division is performed by multiplying the denominator. In general, if the denominator of a fraction be multiplied by 2, the unit will be divided into twice as many parts, consequently the parts will be only one-half as large as before, and the same number of the small parts be taken, as was taken of the large, the value of the fraction will be one-half as much. If the denominator be multiplied by three, each part will be divided into three parts, and the same number of parts be taken, the fraction will be one-third of the value of the first. Finally, if the denominator be multiplied by any number, the parts will be so many times smaller. Therefore, to divide a fraction, if the numerator can not be divided exactly by the divisor, multiply the denominator by the divisor.

PART III. THE INFLUENCE OF WARREN COLBURN IN DIRECTING THE DEVELOPMENT OF ARITHMETIC AS A SCHOOL SUBJECT. ACTIVE PERIOD, 1821-1857; STATIC PERIOD, 1857-1892.

Chapter VII.

ARITHMETIC AS A MENTAL DISCIPLINE.

During the first half of the nineteenth century the growth of cities, the rise of manufacturing, the invention of machines, new modes of travel and transportation, and other factors combined to produce a demand for a higher degree of education than had been necessary when life was more simple. At the same time, the home began to contribute less to the child's education. As a consequence there came to be a new concept of the purpose and scope of the education provided by the schools and an awakened interest in public schools. This movement which has been known as "the common-school revival" was most prominent between 1835 and 1850. The interest in the work of Pestalozzi, which we have noted in Chapter IV, the production of texts by American authors,' and the extension of the public-school system to include primary schools and high schools were phases of the larger movement.

The production of arithmetic texts by American authors, the modification of the content of arithmetic, the extension of the instruction in the subject, and the attempts to provide texts for young children were elements in the general development of arithmetic as a school subject in the United States. This movement had been growing since the close of the Revolutionary War, and the adoption of a Federal money was a phase of the "great awakening." In the three preceding chapters we have told of Colburn's contribution. It is the problem of this chapter and the two following to show in what ways and to what extent Warren Colburn augmented and directed this development.

The limits of the period. The importance of Colburn's First Lessons justifies the selection of 1821 as marking the beginning of this period in the development of arithmetic as a school subject. Following this date there was a period of very rapid development. New types of texts appeared. Some of these were revised frequently to keep pace with the growing ideas of the time. But, beginning about 1860, these revisions ceased, and after this date it is seldom that we find a new type of text which attained any importance.

1 Arithmetic texts by American authors have been mentioned on page 14.

There was no great event, such as the appearance of Colburn's First Lessons, to mark the close of this period. At times from 1821 to 1892 innovations were attempted, some acquiring a considerable following. However, after about 1860, there was no essential change in the aim or content nor modification in the method of teaching which was not local or merely temporary until well toward the close of the century. Then new types of texts became popular and replaced those which had been used for over a quarter of a century. Also radical changes in the method of teaching were urged. Several events indicate that the date of this transition was about 1890. We have chosen 1892, the date of the Report of the Committee of Ten. Although this report dealt only incidentially with arithmetic, it was the official declaration of the teachers of the United States and marked the alignment of a number of our greatest educators on the side of arithmetical reform.

The date marking the end of the process of formalization and the beginning of a stationary period is likewise difficult to determine with exactness. We have chosen 1857, the date of the last of a series of revisions of Ray's arithmetics. Just prior to this date, revisions of Ray's arithmetics were frequent, but in 1857 a form was attained which was not altered until 1877 and only slightly then. Other texts and events do not, in general, specify the date 1857, but they agree in indicating the beginning of a relatively static period about 1860. In view of the popularity and the widespread and continued use of Ray's arithmetics it is appropriate that we select the date marking their maturity.

Mental arithmetic.'-The arithmetic of the preceding period was confined to calculations with written symbols. There were no examples or problems in which the quantities were small to be solved without the use of pencil or pen. In fact, the subject was frequently spoken of as "ciphering." Colburn intended that the problems of his First Lessons should be solved without the aid of written symbols, and he constructed the book is such a way that this was made necessary unless the teacher supplemented the text by instructions in "written arithmetic."

After 1821 the more popular arithmetics were issued in the form of a series. Usually one book of such a series was devoted to mental arithmetic. A few authors united the two types of arithmetic in the same text. Mental arithmetic was universally taught, frequently in a course paralleling the one in written arithmetic.

1 The term "mental arithmetic" became quite generally used to designate that arithmetic which did not involve computations with written symbols. Colburn and some other authors used the term, "intellectual" instead of "mental," and still others called this type of arithmetic, "oral." The use of the term, "mental arithmetic," has been criticized on the ground that arithmetic which involves calculations with written symbols is just as truly mental as that which does not, but the term has been and is still so generally used that its use here is justified and will serve to avoid confusion.

Texts for young children. The texts of the preceding period were not suitable for young children. Thus when arithmetic was taught to them no text was used in the hands of the pupils. It was only a few years prior to 1821 that there was an attempt to provide a text for young children.' But soon after 1821 many primary books appeared and a series of arithmetics was not complete unless it contained a text specifically intended for young children. There were texts prepared to precede Colburn's First Lessons, which Colburn claimed was simple enough for children 5 or 6 years of age.

Most of the primary texts embodied the use of objects. In many of them there were pictures in which the pupil was to count the number of objects. In some texts examples were represented graphically by means of marks, dots, etc., or by actual pictures of the objects mentioned in the exercise.

Arithmetic as a mental discipline.-Throughout the preceding period, as we have shown, arithmetic was taught because of its practical value in certain trades and commerce. A disciplinary function of arithmetic was emphasized by Pestalozzi, who believed that it was to be attained by drill upon a set of abstract exercises which were to be solved by the use of his tables or other sensible objects. Colburn recognized mental discipline as one of the important functions to be realized from the study of arithmetic. The recognition of the disciplinary function, particularly as attached to mental arithmetic, grew after the appearance of Colburn's First Lessons until it overshadowed the other functions. Davies says in the preface to his School Arithmetic, 1855: "In the preparation of this work, two objects have been kept constantly in view: First, to make it educational; second, to make it practical." The educational value which Davies has in mind here is mental discipline. Joseph Ray says in the preface of his Intellectual Arithmetic, one-thousandth edition, 1860:

By its (mental arithmetic) study, learners are taught to reason, to analyze, to think for themselves; while it imparts confidence in their reasoning powers and strengthens the mental faculties.

Davies puts it somewhat more forcibly in his Intellectual Arithmetic:

It is the object of this book to train and develop the mind by means of the science of numbers. Numbers are the instruments here employed to strengthen the memory, to cultivate the faculty of abstraction, and to sharpen and develop the reasoning powers.

In the New Normal Mental Arithmetic, by Edward Brooks, 1873, the author says:

The science of arithmetic, until somewhat recently, was much less useful as an educational agency than it should have been. Consisting mainly of rules and methods

1 See p. 39.

of operations, without presenting the reasons for them, it failed to give that high degree of mental discipline which, when properly taught, it is so well calculated to afford. But a great change has been wrought in this respect; a new area has dawned upon the science of numbers; a "royal road" to mathematics has been discovered, so graded and strewn with the flowers of reason and philosophy that the youthful learner can follow it with interest and pleasure; and one of the most influential agents in this work has been the system of mental arithmetic.

The importance of this change can hardly be overestimated. The study of mental arithmetic, introduced by Warren Colburn, to whom teachers and pupils owe a debt of gratitude which can never be paid, affords the finest mental discipline of any study in the public schools. When properly taught, it gives quickness of perception, keenness of insight, toughness of mental fiber, and an intellectual power and grasp that can be acquired by no other elementary branch of study. An old writer on arithmetic quaintly called his work "The Whetstone of Wit." Mental arithmetic is, in my opinion, truly a whetstone of wit. It is a mental grindstone; it sharpens the mind and gives it the power of concentration and penetration. To omit a thorough course of mental arithmetic in the common school is to deprive the pupil of one of the principal sources of mental power.

Arithmetic as a science. Since the time of the Greek philosophers arithmetic has been conceived of both as an art and as a science, or as some authors put it, as practical arithmetic and theoretical arithmetic. The writers of the texts which were used during the ciphering-book period usually recognized both of these aspects of arithmetic, but they seem to have done so mainly for traditional reasons. In the schools arithmetic was an art. But in this period a number of texts became colored with a philosophic point of view. The theoretical part of arithmetic was given more emphasis. The principles were more carefully formulated, and special attention was given to their interrelation and organization into a logical system. Greenleaf in the National Arithmetic (first published 1835, revised 1847, 1857) gives elaborate lists of definitions, axioms, and principles, and a chapter on properties of numbers. By some writers the "science of numbers" is used synonymously with arithmetic.1

Arithmetic, the important school subject.-By reason of more simple texts and by reason of the emphasis upon the disciplinary function of arithmetic, its relative importance as a school subject grew during this period. It became the custom for pupils to receive instruction in arithmetic when they began to attend school, which in some cases was before their fourth birthday."

Frequently, mental arithmetic was recognized as a separate subject, and two periods a day were given to arithmetic in several of the grades, in some schools from the third or fouth grade to the eighth, inclusive. William B. Fowle said in 1866:

Arithmetic is the all-absorbing study in the public schools of Massachusetts, and, probably, in those of every other State. As far as my observation goes, it occupies more of the time of our children than all other branches united.3

1 See the citations from Davies and Brooks, quoted above.

2 Clifton Johnson: Old Time Schools and School Books, p. 37.

3 The Teacher's Institute; or Familiar Hints to Young Teachers, p. 45.