THE

TEACHING OF ELEMENTARY

MATHEMATICS

CHAPTER I

HISTORICAL REASONS FOR TEACHING ARITHMETIC

Importance of the question-For one who is preparing to teach any particular branch, and who hopes for success, the most important question is this: Why is the subject taught? More important than all methods, more important than all devices or questions of text-books, or advice of the masters, is this farreaching inquiry. Upon the answer depends the solution of the problems relating to the presentation of the subject, the grade in which it should be begun, the time it should consume, the text-books, the methods, the devices, in fine, the general treatment of the whole matter in hand. It is the old, old cry, "We know not whither Thou goest, and how can we know the way?" Unless the goal is known, what hope has one to find the path?

Of course the inquiry is of no interest to the machine teacher, the teacher who is content to follow

the book unthinkingly, to see the old curriculum remain forever unchanged, and to follow the path his teacher trod, even though it be rough to the foot and without interest to the eye. But in England and America to-day we have a host of young and enthusiastic teachers who are anxious to make the AngloSaxon educational system the best, and who are willing to inquire and to experiment.

teachers this question is vital.

For such

The evolution of reasons - This search after reasons may be pursued either from the standpoint of a mere inquirer into the conditions of to-day, or from that of one who is interested in the evolution of the ideas which are now in favor. While it is not possible in a work of this nature to enter into the details of the development of the reason for the presence of arithmetic in the curriculum to-day, some slight reference to this development may be of interest, and should be of value.

The beginning utilitarian-In the far East, and in the far past, the reason for teaching arithmetic to children was almost always purely utilitarian. To the philosopher it was more than this, but in the early Chinese curricula it was given place merely that the boy might have sufficient knowledge of the four fundamental processes for the common vocations of life.1

1 Schmid, K. A., Geschichte der Erziehung vom Anfang an bis auf unsere Zeit, Stuttgart, 1884-98, Vol. I, p. 78. Hereafter referred to as Schmid.

This was done in the common schools almost from the first, but in the middle ages1 the subject so increased in importance that special schools were established for the study of arithmetic. A little later2 it was taught as a special course in the high schools, open to those who had a taste in this direction, although even then children must have continued to learn common reckoning in the earlier years. In general, however, it has been taught in the far East for two thousand years, because of the utilities which it' possesses, or merely for the purposes of examination, or because it correlated with a study of the sacred books.3

Early correlation — In India little could be expected for arithmetic in the schools. The aim of education, as summarized in the first book of Manu, was to bring man to lead a religious life. The reading of the Veda, the giving of alms, these were fundamental features of education. Even to-day is this the case. For more than two thousand years the curriculum and the methods have remained quite unchanged, and even in our day, in the native schools, the boy's work is largely that of memorizing the Hindu scriptures and

1 Under the Sung dynasty, 961-1280. Schmid, I, p. 80.

2 Under the Ming dynasty, 1368-1644.

3 Laurie, S. S., Historical Survey of Pre-Christian Education, London, 1895, p. 128, 141, 148. Hereafter referred to as Laurie.

4 Schmid, I, p. 105-107.

picking up other knowledge incidentally, a classical example of extreme correlation. For such people, arithmetic, beyond the mere rudiments, is of value only as it throws light upon the central subject, and hence it has little place in the curriculum.1

The same idea characterized the early Mohammedan schools, where the Koran furnished the core of instruction, a plan of education still obtaining, on a slightly more liberal scale, in the present schools of Islam.2 It also held quite general sway in the monastic schools of the middle ages, where arithmetic, like everything else, was either warped to correlate with theology, or confined to the simplest calculations. That arithmetic was popularly considered merely as having some slight value in trade is shown by a familiar bit of monkish doggerel, as old at least as the beginning of the fifteenth century. It thus sets forth the values of the seven liberal arts, grammar, dialectic, rhetoric, music, arithmetic, geometry, and astronomy:

"Gramm. loquitur, Dia. vera docet, Rhe. verba colorat ;

Mus. canit, Ar. numerat, Ge. ponderat, As. colit astra."

1 For a description of the arithmetic in the native Hindu schools of the present consult Delbos, L., Les Mathématiques aux Indes Orientales, Paris, 1892, - pamphlet.

2 Schmid, II (1), p. 599.

3 Ib., II (1), p. 86. In this line is the rule attributed to Pachomius, "Omnino nullus erit in monasterio, qui non discat literas et de scripturis aliquid teneat."

4 Ib., II (1), p. 114.

For the medieval cloister schools the computation of Easter day was the one great problem. On this depended the other movable feasts, and every monastery was under the necessity of having someone who knew enough of calculating to determine this date.1

Utilitarian among trading peoples Among the Semitic peoples we find arithmetic more extensively taught. The Semite has generally interested himself not in the thing for its own sake, but for what it contained for him in a practical way. Hence the Assyrians and Arabs and related peoples have no national epos and no enduring art.2 But they found in arithmetic a subject usable in trade, and hence it was extensively taught in their schools. Among the ruins in and about ancient Babylon it is not uncommon to find tablets containing extensive bank accounts, and lately some interesting specimens of pupils' work in arithmetic have come to light.3

Among the Jews, after elementary instruction was made obligatory, arithmetic formed, with writing and the study of the Pentateuch, the sole work from the sixth to the tenth year of the child's school life.

1 Rashdall, H., Universities of Europe in the Middle Ages, I, p. 35. Schmid, II (1), p. 117.

2 Schmid, I, p. 142.

8 Ib., I, p. 152, 153. The firm of Egibi and Sons is often mentioned in these tablets; it was long famous in banking business from Nebuchadnezzar's time on.

4 A.D. 64. Laurie, p. 97.