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working upon continuous quantity—that is, a whole requiring measurement. Every successive step in the entire course of development should harmonize with this initial stage. To get exact ideas of quantity the mind must follow Nature's established law; must measure quantity; must break it into parts and unify the parts, till it recognises the one as many and the many
There can be no possible numerical abstraction and generalization without a quantity to be measured. Where, then, does the "single closely observed object” come in as material for this parting and wholing?
Beginning with a group is in harmony with Nature's method; promotes the normal action of the mind; gives the craving numerical instinct something to work upon, and wisely guides it to its richest development. This psychological method promotes the natural exercise of mental function ; leads gradually but with ease and certainty to true ideas of number; secures recognition of the unity of the arithmetical operations ; gives clear conceptions of the nature of these operations as successive steps in the process of measurement; minimizes the difficulty with which multiplication and division have hitherto been attended ; and helps the child to recognise in the dreaded terra incognita of fractions a pleasant and familiar land.
Forming the Habit of Parting and Wholing.The teacher should from the first keep in view the importance of forming the habit of parting and wholing. This is the fundamental psychical activity; its goal is to grasp clearly and definitely by one act of mind a whole of many and defined parts. This primary activity working upon quantity in the process of measurement gives
rise to numerical relations; the incoherent whole is made definite and unified—becomes the conception of a unity composed of units. Every right exercise of this activity gives new knowledge and an increase of analytic power. At last the habit of numerical analysis is formed, and when it is found requisite to deal with quantity and quantitative relations, the mind always conceives of quantity as made up of parts—measuring units; not invariable units, but units chosen at pleasure or convenience; parts, given by the necessary activity of analysis, a whole from the parts by the necessary activity of synthesis. This means that always and inevitably from first to last the process of fractioning is present.
A Constructive Process. This wholing and parting, as far as possible, should be a constructive act. The physical acts of separating a whole into parts and reuniting the parts into a whole lead gradually to the corresponding mental process of number : division of a whole into exact parts, and the reconstruction of the parts to form a whole. It can not be said that even the physical acts are wholly mindless, for even in these acts there must be at least a vague mental awareness of the relation of the parts to one another and to the whole. These physical acts of wholing and parting under wise direction lead quickly, and with the least expenditure of energy, to clear and definite percepts of related things, and finally to definite conceptions of number. The child should be required to exercise his activity, to do as much as possible in the process, and to notice and state what he is really doing. He should actually apply, for instance, the measuring unit to the measured quan
tity. If the foot is measured by two 6-inch or three 4 inch or four 3-inch units, let him first apply the number of actual units—two 6-inch, three 4-inch, and four 3-inch units—to make up the foot, and so on. By using the actual number of parts required he will have a more definite idea of the construction of a whole than if he simply applies one of the measuring units the necessary number of times. This operation with the actual units should precede the operation by which the whole is mentally constructed by applying or repeating the sin. gle unit of measurement the required number of times. It is the more concrete process, and is an effective exercise for the gradual growth of the more abstract times or ratio idea.
When the child actually uses the 1-inch or the 3-inch unit to measure the foot, his ideas of these units as well as of the measured whole are enlarged and defined. He applies the inch to measure the foot, and this to measure the yard, and the yard to measure the length of the room and other quantities. Let him freely practise this constructive activity, thus practically applying the psychological law, “Know by doing, and do by knowing." The 2-inch square is separated into four inch squares, or sixteen half-inch squares, and these measuring units are put together again to form the whole. Similarly a rectangle 2 inches by 3 inches, for example, is divided into its constituent inch squares or half-inch squares, and again reconstructed from the parts. A square is divided into four right-angled isosceles triangles, into eight smaller triangles, and the parts rhythmically put together again.
Value of Kindergarten Constructions. In this con
nection it may be noted that most of the exercises of the kindergarten can be effectively used for training in number. The constructive exercises which are so prominent a feature in the kindergarten are admirably adapted to lead gradually to mathematical abstraction and generalization. No doubt much has been done in this direction, but much more could be done were the teacher versed in the psychological method of dealing with number. No one questions the general value of kindergarten training, which on the whole is founded on sound psychological principles ; but, on the other hand, no educational psychologist doubts that its philosophy as commonly understood needs revision, and that its methods are capable of improvement. If its aim is, as it should be, an effective preparation of the child for his subsequent educational course, it is thought that its practical results are far from what they ought to be. It is often maintained with considerable force that kindergarten methods should be introduced into the primary and even higher schools. On the other hand, something might be said with a good show of reason in favour of introducing primary and grammar school methods into the kindergarten. What is radically sound in the kindergarten methods will harmonize with what is radically sound in the methods of the public school. On the other hand, what is psychologically sound in the methods of the public school should at least influence the aims and methods of the kindergarten. Is the present kindergarten training, speaking generally, really the best preparation for the training given in a thoroughly good public school! The function of such a school is to give the best possible prepa
ration for life by means of studies and discipline, which, as far as inevitable limitations permit, secure at the same time the best possible development of character. Among these studies the three R's must always hold a prominent place, in spite of theories which seem to assume that language, the complement of man's reason, and number, the instrument of man's interpretation and mastery of the physical world, are not essential to human advancement, and may therefore be degraded from the central position which they have long occupied to one in which they are the subjects of merely haphazard and disconnected teaching.
The practical methods founded on these theories seem to treat the world of Nature as one whole, which even the child may grasp in its infinite diversity and total unity. The “flower in the crannied wall” is made the central point around which all that is knowable is to be collected. But as the human mind is limited, and must move obedient to the law of its constitution, the theories and methods which overlook these facts are not likely permanently to prevail ; and the old subjects that have stood the test of time will no doubt stand the test of the most searching psychological investigation, and regain their full recognition as the “core” subjects of the school curriculum.
Does the kindergarten, then, accomplish all that may be done as a preparation for such a curriculum ? It is to be feared that with regard to many of them the answer must be in the negative; and this is perhaps especially true concerning the subject of arithmetic. We have known the seven-year-old “head boy” of a kindergarten, conducted by a noted kindergarten teacher,