instruction, because the whole sum of external properties of any object is comprised in its outline and its numbers, and is brought home to my consciousness through language." This thesis furnished the basis for his curriculum. The method of instruction was based upon the thesis that sense-impressions are the "absolute foundation of all knowledge."

1

Pestalozzi considered arithmetic the most important means for giving that mental training which would result in the power to form "clear ideas." For this reason he took particular pains to identify the elements of the subject, and to formulate the series of steps in the method of instruction. He considered arithmetic to arise

entirely from simply putting together and separating several units. Its basis is essentially this: One and one are two, and one from two leaves one. Any number, whatever it may be, is only an abbreviation of this natural, original method of counting. But it is important that this consciousness of the origin of relations of numbers should not be weakened in the human mind by the shortening expedients of arithmetic. It should be deeply impressed with great care on all the ways in which this art is taught; and all future steps should be built upon the consciousness, deeply retained in the human mind, of the real relations of things which lie at the bottom of all calculation. If this is not done, this first means of gaining clear ideas will be degraded to a plaything of our memory and imagination, and will be useless for its essential ригрове.2

Both the content of arithmetic and the method of teaching it, as Pestalozzi conceived them, are implied in this statement. The "clear idea" which is represented by a number, e. g., seven, is obtained by counting seven objects. The "clear idea" which is represented by 7 multiplied by 8 is to be obtained by counting the total number of objects in seven groups containing eight objects each. This plan was extended to common fractions and operations with them. The "shortening expedients of arithmetic" were not permitted until "clear ideas" of numbers and number relations had become permanently fixed by having appropriate sense perceptions. In the beginning, the children might use their fingers, peas, stones, or other handy objects for obtaining the necessary sense perceptions. Later a "spelling board" with moveable letters (tablets) was used or the tables which Pestalozzi devised.

The units table consisted of 1003 rectangles arranged in rows of 10. Each rectangle in the first row contained 1 vertical mark. Each rectangle in the second row contained 2 vertical marks, and so on, each rectangle in the tenth row containing 10 vertical marks. The first fraction table was made up of 10 rows of squares, each row con

3 In How Gertrude Teaches Her Children, the translator (p. 216) states that the units table consisted of 12 rows of 12 rectangles each and that each of the fraction tables contained 144 squares. The tables are described by Unger, p. 177, as having only 10 rows, each consisting of 10 rectangles, or squares. The tables were reproduced by Colburn in this form. This form is also given in Pestalozzi's Ausgewählte Werke, by F. Mann.

taining 10 squares. The squares of the first row were undivided. Those in the second were divided into two equal parts by a vertical line; those of the third into three equal parts, and so on. The second fraction table was constructed from the first by dividing the squares in the second column into two equal parts by a horizontal line, those of the third column into three equal parts, and so on.

These tables were used in an elaborate set of exercises which were based upon Pestalozzi's concept of the nature of arithmetic and of the art of instruction. The exercises were prepared by Hermann Krüsi, a teacher of experience and ability who was an assistant to Pestalozzi at Burgdorf and later at Yverdun. They were published in 1803 with the title, Anschauungslehre der Zahlenverhältnisse, in three parts.1

There were eight sets of exercises upon the units table. In the first, the child was to point to the marks in the table and count out the combinations of the multiplication table up to 10 times 10. The second consisted of 540 exercises of the form: "19 times 1 is 9 times 2 and 1 time the half of 2." The object was to express each number as so many twos, threes, fours, etc. In the third, a number expressed in terms of sixes was changed to so many sevens, or if expressed in terms of sevens, it was changed to so many eights, etc. For example, "9 times 9 and 8 times the ninth part of 9 is 89 times 1, 89 times 1 is 8 times 10 and 9 times the tenth part of 10." In the fourth, the tenth parts of numbers are multiplied by the numbers 1 to 10.. The remaining sets of exercises were made increasingly complex, the sixth consisting of 360 exercises of the form, "12 is 2 times 6, 18 is 3 times 6, therefore 2 times 6 is 2 times the third part of 3 times 6." Four of the eight sets of exercises contained a total of more than 2,000 exercises of the formal types illustrated.

The first fraction table was made the basis of 12 sets of exercises and the second of 8 more. One of these contained 17,280 exercises of the form "17 halves are 2 times 7 halves and 3 times the seventh part of 7 halves."

These exercises are purely formal, but Pestalozzi believed that by having a child grind through them laboriously, counting out each on the appropriate table, his mental powers would be developed, because the exercises were based upon his psychological analysis of the process of the development of the human mind. Arithmetic had been reduced to its elements and the instruction psychologized by reduction to an elaborate formula.

1 There is a copy of Anschauungslehre der Zahlenverhältnisse in the Library of Congress. This copy contains only the first part, which was devoted to the exercises on the units table.

The facts of this description are taken from Die Methodik der Praktischen Arithmetik, by Friedrich Unger, p. 177ff.

It should be noted that no practical problems are included in the list, and there is no suggestion that arithmetic has a practical function. In the first stages, the child was expected to count familiar objects. Some years earlier Pestalozzi said in "Leonard and Gertrude":

The instruction she [Gertrude] gave them in the rudiments of arithmetic was intimately connected with the realities of life. She taught them to count the number of steps from one end of the room to the other; and two rows of five panes each, in one of the windows, gave her an opportunity to unfold the decimal relations of numbers. She also made them count their threads while spinning, and the number of turns on the reel when they wound the yarn into skeins. 1

1

No problems are involved here. The children were made to count these objects. But at this time Pestalozzi had not begun to formulate the art of instruction, and when he did, the idea suggested in this quotation was overshadowed by his interest in a psychological method. Since "clear ideas" of numbers and their relations were of the first importance, the symbols of arithmetic were to be delayed until the clear ideas were fixed in the mind of the child. of operations were not included in the published plan. instruction necessarily became oral.

The forms
Thus the

No better summary of Pestalozzi's system of arithmetic can be given than that found in Biber's Henry Pestalozzi and His Plan of Education, which was published in 1831. He says:

In this calculating world shall we be understood if we say that Pestalozzi's arithmetic had no reference to the shop or counting house; that it dealt not in monies, weights, or measures; that its interests consisted entirely in the mental exercise which it involved and its benefit in the increase of strength and acuteness which the mind derived from that exercise?

Again, in this mechanical sign-loving age, shall we be understood if we say that his arithmetic was not the art of handling and pronouncing ciphers, but the power of comprehending and comparing numbers? And, lastly, with a public whose faith is exclusively devoted to what is immediately and palpably "practical and useful,” shall we be believed if we add that, notwithstanding the apparently unpractical tendency of Pestalozzi's arithmetical instruction, he numbered among his pupils the most acute and rapid "practical arithmeticians"?

Such, however, was the case; his course of arithmetic excluded ciphers until numbers were perfectly understood, and the rules of reduction, exchange, and others, in which arithmetic is applied to the common business of life, were superadded at the close of his arithmetical course, as the pupil's future calling might require it. The main object of his instruction in this branch of knowledge was the development of the mental powers; and this he accomplished with so much success that the ability which pupils displayed, especially in mental arithmetic, was the chief means of attracting the public notice to his experiments.

The Pestalozzian movement in America.-William McClure, a Scotch philanthropist, was the first disciple of Pestalozzi in the United States. The earliest presentation of Pestalozzian principles was by him in an article published in the National Intelligencer, June 6, 1806.2

1 Pp. 130-131.

2 Will S. Monroe: History of the Pestalozzian Movement in the United States, p. 44.

This was followed by other articles of a more elaborate nature. In 1806 McClure induced Joseph Neef, who had worked with Pestalozzi, to come to Philadelphia, where he opened a Pestalozzian school in 1809. About three years later Neef removed to Village Green, Pa. From there he removed to Louisville, Ky.; thence to New Harmony, Ind.; and finally to Cincinnati. In 1808 he published a treatise on education, entitled: Sketch of a plan and method of education founded on the analysis of the human faculties and natural reason, fitted for the offspring of a free people and of all rational beings. A chapter was devoted to Pestalozzi's plan of teaching arithmetic.

The work of Neef and the writings of McClure served to advertise the principles of Pestalozzi in the United States, but educational practice was not influenced directly. This was particularly true in New England before 1821. There were leaders in education who were acquainted with the work of Pestalozzi, and a little later there were many enthusiastic disciples of the Swiss schoolmaster. Educational periodicals, beginning with the Academician (1818-19), contained many articles on the work of Pestalozzi. In 1821, when Warren Colburn published his First Lessons in Arithmetic on the Plan of Pestalozzi, the Pestalozzian movement in the United States was beginning to acquire momentum and to influence school practice. Colburn's relation to Pestalozzi.-In the preface to the first editions of the First Lessons, Colburn acknowledges his indebtedness to Pestalozzi as follows:

In forming and arranging the several combinations the author has received considerable assistance from the system of Pestalozzi. He has not, however, had an opportunity of seeing Pestalozzi's own work on this subject, but only a brief outline of it by another. The plates also are from Pestalozzi. In selecting and arranging the examples to illustrate these combinations, and in the manner of solving questions generally, he has received no assistance from Pestalozzi.

The meaning of this statement becomes clear only when we consider the meaning which Colburn attached to the words, "combinations," and "examples to illustrate these combinations." The "several combinations" to which Colburn refers are the number facts such as, "Eight and four are how many?," "Three times seven are how many?," "Fourteen less nine are how many?" "Eight are how many times six?" "6 is one-fifth of what number?" The "examples" are practical problems about things which children can understand. It appears that even in the early editions of the First Lessons, the Pestalozzian tables, or plates as Colburn called them, did not always accompany the text. Colburn, in his address on "The Teaching of Arithmetic," 1830, said: "It has often been asked whether the plates which sometimes accompany Colburn's Intel

1 This acknowledgment does not appear in the first edition, 1821, but does appear in the editions of 1822 and 1826.

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lectual Arithmetic, or anything else of a similar nature, are of any use to the learner."

In addition to this indebtedness to Pestalozzi, which Colburn explicitly acknowledges, a study of his writings shows that some of the underlying principles of his texts are essentially identical with those held by Pestalozzi. This probably means that Colburn was acquainted with Pestalozzi's principles. But to appreciate fully the extent of Colburn's indebtedness to Pestalozzi, it is necessary to consider what he contributed as well as what he borrowed, and how critically he selected what he used.

Soon after Colburn's death, Dr. Edward G. Davis, to whom reference has already been made on page 54, wrote as follows:

His great and most interesting project, that of improving the system of elementary instruction in mathematical science, appears to have occurred to him during the latter part of his college life, and was the subject of painful thought many years before his first work made its appearance. It required, indeed, no small energy of mind thus to break through the trammels of early education, and strike out a new path; for Colburn, like others, had been brought up under a system the reverse of that which he now undertook to mature and introduce.'

Colburn's biographer says:

His First Lessons was, unquestionably, the result of his own teaching. He made the book because he needed it, and because such a book was needed in the community. He had read Pestalozzi, probably, while in college. That which suited his taste, that which he deemed practicable and important, he imbibed and made his own. He has been sometimes represented as owing his fame to Pestalozzi. That in reading the account and writings of the Swiss philosopher he derived aid and confidence in his own investigations of the general principles of education, is true. But, his indebtedness to Pestalozzi is believed to have been misunderstood and overrated.2

After examining carefully all of the evidence which has been obtainable, it is scarcely possible to improve upon the justness of these estimates of the originality of Warren Colburn's work. He died at the age of 40. This, coupled with the fact that he did not begin to prepare for college until he was in his twenty-third year, and that he taught school only two and a half years after graduating from Harvard, indicates the genius of the man. He had the ability and courage "to break through the trammels" of tradition and of his own education. With only slight assistance from the work of Pestalozzi, Colburn produced a text which revolutionized our school practice as no other text has done.

1 Barnard's Amer. Jour. of Educ., 2: 297.

2 Ibid., p. 301.