but one of the chief reasons is that the primary teacher knows why she is teaching arithmetic, while often the one in the higher grades does not. In the first grade the subject is being taught largely for its utilities, and induction plays the important part; this the teacher knows and hence she succeeds. In the seventh grade the teacher is apt to think that induction still plays the leading rôle, an error which gives rise to much poor teaching.

Recognition of the culture value- This culture value is brought out first by letting the amount taken on authority of the book or the teacher be a minimum. "In education the process of self-development should be encouraged to the uttermost. Children should be

led to make their own investigations and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible. Any piece of knowledge which the pupil has himself acquired, any problem which he has himself solved, becomes by virtue of the conquest much more thoroughly his than it could else be."1

This is not to be construed to mean that nothing is to be taken for granted. We must assume, for example, that equals result from adding equals to equals. But when Euclid was criticised for proving that one side of a triangle is less than the sum of the other two, as having proved what even the beasts 1 Spencer, Education.

know, his disciples were entirely right in saying that they were not merely teaching facts, but were engaged in the far more important work of giving the power to prove the facts. As Bain puts it, referring to the higher grades, "The pupil should be made to feel that he has accepted nothing without a clear and demonstrative reason, to the entire exclusion of authority, tradition, prejudice, or self-interest." 1

What, then, shall be said of text-books which give long lists of "Principles Principles" as a kind of inspired revelation to pupils? So far as these are statements of business customs they have place; but they are generally theorems, capable of easy proof, and of no great value without this proof.

Furthermore, if we would make a clear thinker of the pupil, he should not be compelled to learn, verbatim, all or even a majority of the definitions of the text-book. This does not exclude those which are true and understandable and valuable in subsequent work; but it refers to those which are false, unintelligible, and not usable, and to partial definitions in all cases where the memorizing of the same hinders the comprehension of the complete definition subsequently. For example, what teacher of arithmetic can define number in such way as to have the definition both true and intelligible to young pupils, those below the high school? And if he could do so, of what

1 Education, p. 149.

value would it be? Or who would care to undertake the definition of quantity?1 The fact is that the simpler the term the more difficult the definition. Since a definition must explain terms by the use of terms more simple, it follows that one must sometime come to terms incapable of definition.2 In daily life. we do not learn definitions verbatim; if asked to define horse, the definition would probably include the mule and zebra and numerous others of the equine family. The usual definition of multiplication has hindered the work of many a child in fractions, and yet, even in the first grade he multiplies by the fraction. While it is true that partial truths precede complete ones, it is poor teaching to impress this partial truth on the mind so indelibly, by a memorized statement, as to make the complete truth difficult of assimilation. For example, a teacher drills a class to memorize the fiction that if the second term of a proportion is less than the first, the fourth must be less than third, a statement entirely unnecessary in the logical treatment of proportion, and then, when the pupils come to meet 1 : 2:4, they are lost. To test the matter a little further, let any reader

2=

1 Those who may be ambitious to make the attempt might first read Laisant, La Mathématique, Paris, 1898, p. 14, hereafter referred to as Laisant, or the simple definition of number in the Encyklopädie der mathematischen Wissenschaften, I. Heft, Leipzig, 1898, now in process of publication.

2 Duhamel, J.-M.-C., Des Méthodes dans les Sciences de Raisonnement. Ţière partie, zième éd., Paris, 1885, p. 16.

repeat the definition of number, as it was once burnt into his memory, and see if (= 3.14159) is a number according to this definition,—or √2, or √— I. Or try the definition of arithmetic and see if, by this statement, the table of avoirdupois weight is any part of the subject. Does the definition of multiplication, as usually memorized, cover even the simple case of , to say nothing of √2 × √3 or ? 3 By the common definition of factor is a factor of ? By the definition of square root, as usually learned, have we any right to speak of the square root of 3, since 3 has not two equal factors? Are our arithmetics clear enough in statement so that the memorizing of their definitions will tell a pupil whether the simple series 2, 2, 2, 2, is an arithmetical or a geometric progression, or neither?

...

The old argument that learning definitions strengthens the memory and gives a good vocabulary, has too few advocates now to make it worth consideration. "The rôle of the memory, certainly necessary in matters mathematical as elsewhere, should be reduced in a general way to very limited proportions in rational teaching. It is not the images, the figures, or the formulae which must be impressed upon the mind, so much as it is the power of reasoning."1

1 "Ce ne sont pas les images, figures ou formules, dont il faut surtout laisser l'empreinte dans le cerveau; c'est la faculté du raisonnement." Laisant, p. 191.

This opposition, on the part of leaders in education, to the burdening of children's memories, is not new. Locke voiced the same sentiment: "And here give me leave to take notice of one thing I think a fault in the ordinary method of education; and that is, the charging of children's memories, upon all occasions, with rules and precepts, which they often do not understand, and constantly as soon forget as given." 1 "Teachers at one time believed that the first object of primary instruction is to cultivate the verbal memory of their pupils, when, in fact, the verbal memory is one of the few faculties of our nature which need no cultivation." 2 Of the two, to learn all of the definitions of a text-book or none, the latter plan is unquestionably the better.

But while memorized definitions may not unfrequently be justified, this is rarely true of the memorized rule. The glib recitation of rules for long division, greatest common divisor, etc., which one hears in some schools —what is all this but a pretence of knowledge? "If learning is a process of gaining knowledge, that is, a true apprehension of realities, it excludes verbal memorizing, cramming, and everything that resolves itself on close scrutiny into a pretence of knowledge getting.” 3

But not only is this old-fashioned rule-learning (unhappily not yet extinct) a sham; it is wholly unscientific. Tillich, one of the best teachers of arithmetic of the

1 On Education, Daniel's edn., p. 126.

2 Tate.

8 Dr. James Sully, in the Educational Times, December, 1890.