Pestalozzian primary arithmetic. Arithmetic generally neglected before 1800.-The third subject which exhibits the profound influence of the Pestalozzian methods of objective and oral instruction is the teaching of primary arithmetic. By primary is meant the work of the earlier years of the elementary schools. In Chapter IV it was shown that arithmetic was often omitted from elementary schools in the United States down to 1800; that it was sometimes specifically prohibited by the public-school committees; and that in most places where it was taught, it was not begun until about the fourth year and was then studied by the most mechanical ruleof-thumb methods. The work of arithmetic was commonly known as "ciphering," indicating that the "figuring" on paper (usually according to fixed rule) was the fundamental process. Pestalozzi largely the founder of primary arithmetic. The general significance of Pestalozzi in the development during the nineteenth century is shown in the following quotation from our foremost American authority on the history of the teaching of mathematics, Professor D. E. Smith: The evolution of the teaching of primary arithmetic extends over a period of about two hundred years, although numerous sporadic efforts at teaching the science of numbers to young children had been made long before the founding of the Francke Institute at Halle [1694]. During the eighteenth century not much progress was made until there was established the Philanthropin at Dessau [1774], and perhaps it would be more just to speak of primary arithmetic as having its real beginning in this institution at about the time that our country was establishing its independent existence. It is, however, to Pestalozzi, at the beginning of the nineteenth century, that we usually and rightly assign the first sympathetic movement in this direction, and it is the period from that time to the present that has seen the real evolution of the teaching of arithmetic to children in the first school years. (21: 67.) Pestalozzi's fundamental principles. Mental operations to replace ciphering; objective beginnings.- Pestalozzi's fundamental contention was that the arithmetical mental processes of the pupil are the most important factors in arithmetic study. This may be interpreted to mean either the special mental operations which go forward in working a particular problem, or the general mental processes of judging and reasoning, which Pestalozzi believed could be trained. Two consequences in practice followed from this emphasis on the child's thinking (1) In order to eliminate the old emphasis on "ciphering" according to rule, all written arithmetic was postponed until the child had made considerable arithmetical progress; I II II II II II I II A PESTALOZZIAN NUMBER CHART An objective aid in teaching number combinations. Concerning it Kruesi says, "The table which was intended to bring these facts before the perception of the pupil formed, for a long time, a prominent feature in all Pestalozzian Schools. It even appeared in the first edition of Warren Colburn's Mental Arithmetic " thus originated the "mental," "intellectual," or oral arithmetic of the nineteenth century. (2) In order to assure that the pupil got real number ideas instead of mere words, all the elementary number combinations were learned in connection with the arranging, grouping, and using of material objects, lines, charts, etc., instead of simply being memorized. Incidental language instruction overemphasized. — Thus we find the two main principles of instruction which we are considering in this chapter, namely oral instruction and objective methods, at the basis of Pestalozzian primary arithmetic. As in all other subjects, Pestalozzi used the arithmetic lessons also for purposes of language instruction, according to the principles stated on page 330. In fact, this incidental language training was given so much time, that the method has sometimes been criticized as being too much language and too little arithmetic. This mixture of training in general observation, number combinations, and oral expression is shown in the following lesson from Pestalozzi's school. The teacher rearranged some beans on the desk while the children looked away; then the following conversation ensued: "What change do you see in the position of the beans?" "What then do you say when two beans are taken from eight beans? "Two beans taken from eight beans leaves six beans." Examples of a Pestalozzian chart. The number work with actual objects was followed by work with charts in which appeared various combinations or groupings of straight lines or dots, each line or dot being considered as a unit. The chart shown on page 351 is an example. Pestalozzian primary arithmetic adopted in Germany and England. - The influence of Pestalozzi's methods in primary arithmetic on European practice was of the same profound character as in the other subjects. They soon dominated the elementary schools of Germany and Holland. In England the Pestalozzian arithmetic was developed in the schools of the Home and Colonial Infant School Society by Professor Reiner, who had been teacher of mathematics at Yverdon until the school there was closed in 1826. In America: Warren Colburn's "First Lessons," 1821.— We have such a striking example of the early adoption of Pestalozzi's methods in the United States, however, that we will confine the further discussion to it, namely, Warren Colburn's "First Lessons in Arithmetic on the Plan of Pestalozzi," published in Boston in 1821. This book represents the only phase of the Pestalozzian methods which secured widespread adoption in this country before the Oswego movement of 1860. Pestalozzian origin of Colburn's book. - Colburn's book would be significant as an example of the movement for intellectual arithmetic even if its conception were independent of any Pestalozzian influence. It is interesting to note, however, that there was probably a direct connection with the Pestalozzian movement, although Colburn may have been largely independent in his conception of the method. Colburn states his debt to Pestalozzi in his rather long preface, which is phrased in typical Pestalozzian language; and the full title of the second edition (1822), which is given above, is clear evidence of an acquaintance with “the plan of Pestalozzi." Warren Colburn (1793-1833) graduated from Harvard in 1820 and spent several years teaching school before and after graduation. He was noted in college as a mathematician, and he developed the material of his arithmetic while teaching in the elementary schools. He prepared other textbooks in mathematics, but spent most of his life as superintendent of manufacturing plants. Colburn's book ranks with Webster's speller in impor tance.—The adoption of Colburn's arithmetic was so general, and it continued in use so long, that it is scarcely an exaggeration to rank it with "The New England Primer" and Webster's speller in historical importance. It was translated into foreign languages, was widely used in England, and millions of copies were sold in the United States. Forty years after it was first issued it was still generally eulogized as the best and the only almost perfect book that had appeared in that line. Pestalozzian principles in Colburn's preface. Criticizing the ordinary method of teaching arithmetic by "ciphering" according to rule, Colburn wrote: The pupil, therefore, when he commences arithmetic, is presented [ordinarily] with a set of abstract numbers, written with figures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures signify and what is meant by addition, subtraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of these things, and that is, how to perform these operations on figures. He can perhaps translate the figures into words, but this is useless, since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has not the least conception. Emphasized acquiring correct "number ideas." - After stating that very young children in their ordinary play show an understanding of many elementary number facts, Colburn's preface contains the following characteristic Pestalozzian statement in which the "idea of number" is emphasized: The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to all things with which we are acquainted, we obtain an abstract idea of number. We first make calculations about sensible objects; and we soon observe that the same calculations will apply to things very dissimilar; and finally that they may be made without reference to any particular things. Hence, from particulars, we establish general principles, which serve as the basis of our reasonings, and enable us to proceed, step by step, from the most simple to the more complex operations. It appears, therefore, that mathematical reasoning proceeds as much upon the principle of analytic induction, as that of any other science. |