school, is considered able to undertake nothing beyond the "number 20." It might reasonably be maintained that, under rational and therefore pleasurable training of the number instinct in the kindergarten, the child ought to be arithmetically strong enough to make immediate acquaintance with the number 20, and rapidly acquire-if he has not already acquired—a working conception of much larger numbers.

Important Points of Rational Method.-In applying the rational method of teaching arithmetic there are important things that the teacher must keep in view if he is to aid the child's mind to work freely and naturally in the evolution of number. The child's mind must be guided along the lines of least resistance to the true idea of number. This movement, in the very nature of things, must be slow as compared with the gathering of sense facts; but under the psychological method it may be sure and pleasurable. The result aimed at can not be reached by banishing the word times from arithmetic; nor by working continually with indefinite units of measure; nor by exclusive attention to manual occupations under the vague idea that physical separation of things is analysis of thought; nor by making counting-emphasizing the vague how many-the single purpose, and unmeasured units the sole matter of the exercises, to the exclusion of the how much, and the measuring idea which is the essence of number; nor by substituting for the rhythmic and spontaneous action of the child's mind in dealing with wholes, both qualitative and quantitative, a minute and formal analysis which properly finds place only in a riper stage of mental growth; nor by any amount of

drill, however industrious and deviceful the drill-master, which substitutes mechanical action and factitious interest for spontaneous action and intrinsic interest, the very life of the self-developing soul.

Number is the measurement of quantity, and therefore the only solid basis of method and sure guide for the teacher is the measuring idea.

1. The Measured Whole.-The factors in number are, as before shown, the unity (the whole of quantity) to be measured, the unit of measurement, and the times of its repetition-the number in the strictly mathematical and psychological sense of the word. The teacher must bear in mind the distinction between unity and unit as fundamental. The entire difference between a good method and a bad method lies here, because the essential principle of number lies here. Vague unity, units, defined unity, is the sequence as determined by psychological law. In the child's first dealing with number there must be the group of things, the whole of quantity to start from; and in every step of the initial stage the idea of a whole to be measured is to be kept prominent. In addition, there is a whole (the sum) to be made more definite by putting together its component parts (addends)—not equal measuring units, but each part defined by a common unit- -so as to completely define the quantity in terms of this specifically defined unit. In subtraction we have a given quantity (minuend) and a component part (subtrahend) of it to find the other component (remainder)—a process which helps to a more definite idea of the given whole, and especially makes explicit the vague idea of the "remainder" with which we began. In multiplication there is

given a quantity (multiplicand) defined by a measuring unit and the times (multiplier) of its repetition; and the process makes the quantity articulately defined (in the product) by substituting a more familiar unit for the derived unit of measurement; in other words, by expressing the quantity in terms of the primary unit by which the derived unit itself is measured. In division we have a whole quantity given (dividend), and one of two related measuring parts (the divisor) to find the other part (quotient), and the operation makes clearer the whole magnitude, and at the same time makes the first vague idea of the other measuring part (quotient) perfectly definite. Briefly, in all numerical operations there is some magnitude to be definitely determined in numerical terms, and the arithmetical operations are simply related steps expressing the corresponding stages of the mental movement by which the vague whole is made definite. Keep clearly in mind, therefore, the inclusive magnitude from which and within which the mental movement takes place-which justifies and gives meaning to both the psychical process and the arithmetical operation.

2. The Unit of Measure-Its True Function.— From the vague unity, through the units, to the definite unity, the sum, is the law of mental movement. The second point of essential importance is to make clear the idea of the unit of measure. More than half the difficulty of the teacher in teaching, and the learner in learning, is due to misconception of what the "unit" really is. It is not a single unmeasured object; it is not even a single defined or measured thing; it is any measuring part by which a quantity is numerically de

fined; it is (in the crude stage of measurement) one of the like things used to measure a collection of the things; it is (in the second or exact stage of measurement) one of the equal parts used to measure an exactly measured quantity. It is one of a necessarily related many constituting a whole.

It is, therefore, an utterly false method to begin with an isolated object-false to the fact of measurement, false to the free activity of the mind in the measuring process. Nor is the defect to be remedied by introducing another isolated object, then another, and so on. The idea of a unit can begin only from analysis of a whole; it is completed only by relating the part to the whole, so that it is finally conceived at once in its isolation and in its unity in the whole. Not only do we not begin with a single object and "develop one," but also even in beginning, as psychology demands, with a group of objects, we are not to begin with the single object to measure the quantity—at least we are not to emphasize the single object as pre-eminently the measuring unit. We separate twelve beans, for example, not into 12 parts, but into 2 parts, then 3 parts, etc.; that is, we measure by 6 beans, by 4 beans, by 3 beans, by 2 beans; and the resulting numbers for the one measured quantity are two, three, four, six; and each of the measuring parts to which the numbers are applied is a unit, is ONE. So in building up the measured foot with 6-inch, 4-inch, 3-inch, 2-inch measures—each in turn measured off in inches-there are two ones, three ones, four ones, six ones. The point to be kept in view is to prevent the mischievous error of regarding the unit as a single OBJECT, a fixed qualitative or an indivisible quantitative

unity, instead of simply a measuring part-a means of measuring a magnitude.

The Unit itself Measured.-As necessary to the growth of the true conception of unit as a measuring part, the idea of the unit as a unity of measured parts must be clearly brought out. The given quantity is measured by a certain unit; this unit itself is a quantity, and so is made up of measuring parts. This idea must be used from the beginning; it is absolutely essential to the clear idea of the unit, and of number as measurement of quantity. Beginning with a group of 12 objects requiring measurement, or with counters representing such objects, we have them counted off into two equal parts, noting the relation of the parts to one another and to the whole; then each of these two units (half of the given whole) is counted off into two equal parts, and the relation of these minor parts to each other and to the whole they compose is noticed; then each of the first units of measurement (halves of the given whole) is counted off into three equal parts, and their relation to one another and to the whole which they make is carefully observed; and so on, with similar exercises in parting and wholing. Such constructive exercises help in the growth of the true idea of the unit as a measuring part, which is or may be itself measured by other units. But the true idea of the essential property of the unit-its measuring function— can be fully developed only by exercises belonging to the second stage of measurement, in which exact and equal units are used for precise measurement. These measurements of groups of like things (apples, oranges, etc.) by groups which are themselves measured by still