Trivium Pursuit

Delaying Formal Math: History, Part I


Here is a link to an old encyclopedia called The Cyclopedia of Education. Although it was published over 100 years ago, there are a number of articles worth reading — they offer a different perspective from some of the modern encyclopedias. In particular, read the articles “Arithmetic” and “Arithmetic, Hygiene of.”

Below is a quote from the article “Arithmetic”:

Arithmetic, The History of Teaching. — As stated in the section on the history of arithmetic, the subject was taught in early times under two distinct aspects. In the preparation of the merchant and those who had to do with land measure and building, arithmetic was taught purely from the side of calculation, the abacus playing an important part in most countries. This phase of the subject was often left to slaves, to those who prepared apprentices, and later to schools of arithmetic (Rechenschulen). The science of numbers was taught in such schools of philosophy as those of Pythagoras and Plato, in such early universities as that at Alexandria, and in the medieval universities of Europe.

It was not until printed books appeared that an effort was made to unite these two phases. One result of printing was to spread abroad a knowledge of the superiority of the Arabic numerals over the Roman, and this led to the abandoning of the abacus. The effect on the teaching of arithmetic was not fortunate in one respect, since the giving up of the counters led from the concrete, visual, palpable arithmetic to the abstract arithmetic of figures. Counting and reckoning came to be more matters of words and abstract rules than before, and arithmetic was probably more poorly taught than it was under the abacus system.

To break away from this method of the mere rule, efforts began to be made in the seventeenth century, and, in a more pronounced way, early in the eighteenth century, notably in the Francke institute at Halle. These efforts were in the direction of making the number concept more real and the applications of number more genuine. Thus in the Braunschweig-Luneburg school decree of 1737 there are directions for beginning number work by counting the fingers, apples, and other objects, and for basing the multiplication table upon addition. Christian Wolf (1728) insisted upon getting at the foundation principles of number by questions as to the child’s reasons for proceeding; Christian von Clausberg (1732) urged that a clear understanding should accompany each rule; and Basedow (1763) emphasized the danger of having the child feel that he proceeded merely on the authority of his teacher. It was in the Philanthropin at Dessau, however, that serious attempts at reform were first made in any noteworthy fashion. The leader in this phase of the work of the Philanthropinists was Christian Trapp (1777). He advocated teaching the fundamental operations with nuts and other objects before the figures were learned, a feature emphasized by Pestalozzi a little later. Trapp was succeeded by Gottlieb Busse (1779), who wrote extensively upon primary arithmetic, and who introduced the so-called number pictures by which he brought to the child’s senses a group of dots along with the figure, a scheme that is still found helpful in all schools. Much credit is also due to Freiherr von Rochow of Rekahn near Brandenburg, and his associates, who about this time sought to free elementary arithmetic from the mere demands of mechanical business and raise it to the plane of a culture subject in the best sense. Mention should also be made of the work of Peter Villaume of Halberstadt (1779), who insisted that arithmetic “is a practical logic,” and based all of his work with numbers upon perception, and brought oral arithmetic forward as more worthy of attention. In this movement there also joined Bernard Overberg of Minister (1793), A. H. Niemeyer (1802), and G. F. Dinter (1806), and each was influential in preparing the way for Pestalozzi. It is to the latter (1803) that we owe the greatest impetus in the rational teaching of arithmetic to young children. The essential features of his reform are as follows: (1) He taught arithmetic to children as soon as they entered school, basing his work on perception. (2) He insisted upon a knowledge of number and the simplest operations, using objects, before the figures were taught. (3) He approached the subject of fractions in the same way. (4) He made arithmetic the most prominent subject in the school, and it is to his influence that its present prominence is due. (5) He emphasized oral arithmetic, a movement that led to the great success of Warren Colburn in the United States. The next noteworthy name is that of Tillich (1806), who attempted to improve upon Pestalozzi by a systematic use of material, inventing’ for this purpose a set of prismatic blocks of different lengths so arranged as to make prominent the decimal feature of our number system. The plan failed, as all such narrow plans do, and from the time of Tillich to the present there have been innumerable illustrations of this law. An enthusiastic teacher finds some device; he exalts it to a “method”; it succeeds because of his enthusiasm; it proceeds fairly well in the hands of his pupils, and then it is forgotten save for some little feature that becomes impressed upon the permanent educational body. Of these semi-forgotten methods a few may be mentioned. Turk (1816) did not wish arithmetic in what we call the first grade, nor before the child reached the age of 10 years, and of this idea there is just at present a temporary revival as if it were a new discovery, although it was practically universal before Pestalozzi. Kawerau (1816) made formal culture the great aim in teaching, and his extreme views provoked a reaction that is perhaps reaching its climax at the present time. Krancke (1819) suggested the concentric circle plan, keeping the child in the number space 1-10 until that was mastered, in spite of the fact that a child’s interest in and need for counting runs far beyond his work in the operations at every step in his progress. Grube (1842) carried Krancke’s plan to an extreme, but had at least the merit of thoroughness. He used objective illustration more extensively than any one would advocate at present, and attempted the impossible task of having a pupil master each number before proceeding to the next. Knilling (1897) and Tanck (1884) carry to an extreme the plan of building all number work upon counting.

The arrangement of matter in the textbook has occupied the attention of writers for the past half century. There are two plans, more or less connected. The first and oldest is the topical arrangement, the one that was followed for thousands of years in the teaching of arithmetic. This has the advantage of a thorough treatment of each topic as it is met, thereafter not returning to it. The other is the “spiral method,” first brought into prominence by Ruhsam in 1866. It has the advantage of returning to each topic, each time with a higher treatment. Neither plan has proved successful in its extreme form, as might have been expected. The spiral plan has had to give way to a topical plan within a period of one year or two years, thereafter again reviewing some or all topics. This has given rise to the favorite three-book series in the American schools of to-day. The extreme spiral plan resulted in scraps of information and lack of continuity and a falling off in interest from want of a feeling of mastery. At present, in the United States, the tendency is to merge the two plans, preserving the strength of the topical 2arrangement within each half year, each year, or within some longer period, reviewing systematically and on a higher plan, one or more times according to the subject, all of the important topics of arithmetic.

Present Status in the Curriculum. — Placed in a position of preeminence by Pestalozzi, arithmetic has for a century been looked upon as the most important subject in the curriculum. In spite of all the attacks made by modern subjects that call for their share in the course, it has maintained its position even to the present time. One reason for this is its definiteness; it has an exact, well-arranged body of material, and it has proved its usability. Other subjects, like handwork and nature study, however, have not yet developed in such way as to present definite reasons for being, or a definite, well-arranged sequence of topics. For this reason arithmetic will probably maintain a prominent place for a long time to come.

It is not, however, to be expected that it should keep the position it once held. There is no reason why it should take the time that it does, if other well-defined subjects appear that can justify their inclusion in the curriculum. The actual business part of arithmetic can easily be taught in less time than is now given to the subject, and arithmetic must surrender a portion of the time it occupies as soon as these other topics are sufficiently ordered and thought out to justly claim a share.

At present arithmetic is commonly taught in the schools of the United States during the first eight school years. There is some attempt to return to the pre-Pestalozzian view, advocated also by Turk, of omitting it from the first school year, but it is a doctrinaire idea that is not taken very seriously. There is also an attempt to merge algebra and arithmetic in the eighth, or the seventh and eighth, school years, and this seems likely to succeed, in spite of the efforts of some writers to make the algebra rather useless. There is no reason why algebraic notation should not come to the help of arithmetic and mensuration in the seventh school year, but there is a good reason why algebra should not replace arithmetic at this time. The pupil in his seventh school year begins to appreciate the larger problems of business, and they must be taught here if ever. Some arithmetic must therefore be retained in these years, and particularly the commercial part and mensuration, and so far as algebra can throw light upon these subjects it should be employed….

References: —

Ball. History of Mathematics. (London, various editions.)

Cajohi. History of Elementary Mathematics. (New York, 1896.)

Cantor. Geschichte der Mathematik. (Leipzig, various editions.)

Fink. History of Mathematics. (Chicago, 1900.)

Sterner. Geschichte der Rechenkunst. (Munich, 1891.)

Unger. Die Methodik der praktischen AriUimetik in historischer Entwickelung. (Leipzig, 1888.)

Jackson. The Educational Significance of Sixteenth Century Arithmetic. (New York, 1906.)

Kuckuck. Die Rechenkunst im scchzehnten Jahrhundert. (Berlin, 1874.)

Smith. Rara Arithmetica. (Boston, 1909.)

Janicke. GrundzOge der Geschichte und Methodik des Rechenunterrichts. (Gotha, 1879.)

Smith. Teaching of Elementary Mathematics. (New York, 1900.)

Branford. A Study of Mathematical Education. (Oxford, 1908.)

Mclellan And Dewey. Psychology of Number. (New York, 1898.)

Mcmurry. Special Method in Arithmetic. (New York, 1905.)

Smith. The Teaching of Arithmetic. With bibliography. (New York, 1909.)

Stone. Arithmetical Abilities. (New York, 1908.)

Young. The Teaching of Mathematics. (New York, 1907.)

One Response to “Delaying Formal Math: History, Part I”

  1. helene Says:

    facebooking this…spreading the (documented) word!