THE TEACHING OF NUMERATION 329 separate units, and initiates the child's mind into the idea of numerical groups, at the same time fixing before his eyes the suc- cession of the signs 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This material, called the tray of spindles, is divided into compartments each named with one of the ten figures, arranged in sequence. Within these com- partments, the child accumulates in group separate objects corresponding to the figures; that is, he groups the units. In our case these objects are represented by long, spindle-shaped sticks. The other material which has been mentioned consists of a group of cards in a box containing certain objects (coloured markers); the cards are separate (mixed) and on them are written the ten numbers from zero to nine. The child must first of all arrange the cards in a row, by himself, showing by this that he has learned the numerical series, and recognizes the figures which represent the numbers. Under each figure he then places an appro- priate quantity of markers, arranging them two by two, that is one pair under the other; in this way there is made plain the difference between odd and even numbers. This is all the material which we have considered necessary for laying the foundations of number-teaching and some arith- metical, operations. What follows here is a more detailed description which may be of practical assistance to the teacher. The rods having been arranged together in juxtaposition ac- cording to length, the red and blue marks have to be counted, beginning with the smallest piece, that is: one; one and two; one, two and three, etc. always beginning again with the one for every piece, commencing from side A. After that, the separate rods have to be named from the shortest to the longest, according to the number of sections which they contain, touching the extremities of them with the finger, starting from the side £ where they go up in staircase style; there results from this the numbering of the longest piece: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. This correspondence of the 10 on the three sides is verified by the child, who repeats the exercise many times of his own accord because it interests him.