Simultaneous Method not Psychological.—It seems clear, therefore, that the fundamental operations as formal processes should not be all taught together; on the other hand, rational use should be made of their logical and psychological correlation. It is one thing to perform arithematical operations in such a way as to involve the use of correlative operations, and it is another thing to force these operations into consciousness, or to make them the express object of attention. The natural psychological law in all cases is first the use of the process in a rational way, and then, after it has become familiar, abstract recognition of it.

The method usually followed violates both sides of the true psychological principle. Because it treats number as so many independent things or unities, it can not mentally or by interpretation bring out how the operations are correlative with one another. It is only when the unit is treated not as one thing, but as a standard of measuring numerical values, that addition and multiplication, division and fractions, are rationally correlative. And it is because this correlation is not brought out and rationally used that-in spite of the teaching of all the operations contemporaneously-division is still a mystery and fractions a dark enigma.

Then, the common method errs in the opposite extreme by attempting to force the recognition of ratio, and fractions, into consciousness before the mind is sufficiently mature, or sufficiently exercised in the use of ratio, to grasp its meaning. The result of this unnatural method is that mechanical drill and memorizing, with the sure effect of waning interest and feeble thought, is forced upon the pupil. To master all the

numerical operations contained in 6, 7, 8, and 9 is a slow and tedious process, and so the method is compelled in self-consistency to limit the range of numbers which are to be mastered in a given time. In reality it is easy for the mind to grasp the fact that $1 is a hundred ones, or fifty twos, or ten tens, or five twenties, long before it has exhausted all possible operations with such numbers as 7 or 11 or 18. It might, indeed, be maintained that a return to the old-fashioned ways of our boyhood, by which we soon became expert in the mechanical processes of addition and subtraction, would be preferable to this monotonous drill on "all that can be done with the numbers" from 1 to 10 and from 10 to 20 in the second year; for this new method is just about as mechanical as the old, and, while leaving the child little if any better prepared for the "analysis" of the higher numbers, leaves him also without the expertness in the operations which is essential to progress in arithmetic.

Division and Subtraction not to precede Multiplication and Addition.-On the ground that the "first procedure of the mind is always analytic," some maintain that division and subtraction (the "analytic processes") should be taught before multiplication and addition.

But just as multiplication, definitely using the idea of ratio, is a more complex process than addition, so division, the inverse of multiplication, is more complex

*It might be asserted with some truth that the first procedure of the mind is synthetic: there must be a "whole "-a synthesishowever vague, for analysis to work upon. Certainly the last procedure of the mind is "synthetic."

than subtraction, the inverse of addition. We may, as we have seen, add a number of threes, for example— giving each addend a momentary attention, and then dropping it utterly from consciousness-without grasping the factor, which, with three as the other factor, will give a product equal to the sum of the addends. So in division, the inverse operation, this factor does not come merely from the successive subtractions of three from the sum until there is no remainder; here, as in addition, a further mental operation is necessary before the factors are discovered that of counting the times of repetition ; i. e., of finding the ratio of the sum (dividend) to the repeated subtrahend.

We are told, too, that when we separate 8 cubes into 4 equal parts it is instantly seen that 8 contains 2 four times, that 2 is one fourth of 8, that 2 may be taken four times from 8, and that these results being obtained independently of addition and multiplication, division and subtraction may be taught first.

There seems to be a fallacy lurking here. We may, indeed, separate 8 cubes into two parts, or four parts, or eight parts; but that is mere physical separation. Granting recognition of the concrete (spatial) element -the measuring units-how does the abstract element -the idea of times-arise? How do we know that there is four times 2 or eight times 1? Only by counting, by relating, by an act of synthesis-the last procedure of the mind in a complete process of thought. Thus, the fallacy referred to ignores one of the two necessary factors (relation) in the psychical process of number. It must presuppose that counting does not imply addition and multiplication. What is counting

but addition by ones? What is five, if not one more than four; and four, if not one less than five? How is four, e. g., defined except as that number which, applied to a unit of measure, denotes a quantity consisting of three such units and one unit more? This counting, which begins with discrete quantity (collection of objects) in the first stage of measurement, is addition (with subtraction implied) by ones, and the idea of multiplication and division involved in it becomes evolved (in counting with an exact unit of measure) with the growth of numerical abstraction and the consequent development of the measuring power of number.

It seems plain, then, that in the development of num、 ber as the instrument of measurement there is first the rational use, leading to conscious recognition, of the aggregation idea that is, addition and subtraction; then the definite use, leading to conscious recognition, of the factor (times) idea—that is, multiplication and division. In other words, the psychological order as determined by the demand on conscious attention is the old-time arrangement-Addition and Subtraction, Multiplication and Division.

It is the order in which numerical ideas and processes appear in the evolution of number as the instrument of measurement; the order in which they appear in the reflective consciousness of the child; the order of increasing growth in psychological complexity. This order may be said to reverse the order of logical dependence, but the psychological order rather than that of logical dependence is to be the guide in teaching.

Not Exclusive Attention to One Rule or Process.But the true method, as based on this psychological

order of instruction, by no means implies that addition and subtraction are to be completely mastered before the introduction of any multiplication or division or fractions. Quite the contrary. On account of their greater complexity the higher processes are not to be taught analytically-made, that is, an object of conscious attention from the first; but they may and should be freely used, and thus relieve the monotony of too much addition and subtraction, and at the same time prepare the way for their conscious (analytic) use.

Because of the rhythmic character of multiplication such forms of it as can be objectively presented in simple constructions—the putting together of triangles, squares, cubes, etc., to make larger or more complex figures, of dimes to make dollars-are much more easily learned than many of the addition and subtraction combinations. The ideas of ratio should be incidentally introduced in connection with certain values (e. g., 9, 12, 6, 16, 100, etc.) practically from the beginning; and consequently the process of fractions in simple forms, and its symbolic statement. Nothing but the demands of a preconceived theory could so nullify ordinary common sense as to suppose that there is no alternative between either exhausting all operations with every number before going on to the next higher, or else mastering all additions and subtractions before going on to ratio-multiplication and division. Practical common sense and sound psychology agree in recommending first the emphasis on addition and subtraction, with incidental introduction of the more rhythmical and obvious forms of ratio, and gradual change of emphasis to the processes of multiplication and division. If the