ARITHMETIC
The children possess all the instinctive knowledge necessary as a preparation for clear ideas on numeration. The idea of quantity was inherent in all the material for the education of the senses: longer, shorter, darker, lighter. The conception of identity and of difference formed part of the actual technique of the education of the senses, which began with the recognition of identical objects, and continued with the arrangement in gradation of similar objects. I will make a special illustration of the first exercise with the solid insets, which can be done even by a child of two and a half. When he makes a mistake by putting a cylinder in a hole too large for it, and so leaves one cylinder without a place, he instinctively absorbs the idea of the absence of one from a continuous series. The child’s mind is not prepared 103 for number “by certain preliminary ideas,” given in haste by the teacher, but has been prepared for it by a process of formation, by a slow building up of itself.
To enter directly upon the teaching of arithmetic, we must turn to the same didactic material used for the education of the senses.
Let us look at the three sets of material which are presented after the exercises with the solid insets, i.e., the material for teaching size (the pink cubes), thickness (the brown prisms), and length (the green rods). There is a definite relation between the ten pieces of each series. In the material for length the shortest piece is a unit of measurement for all the rest; the second piece is double the first, the third is three times the first, etc., and, whilst the scale of length increases by ten centimeters for each piece, the other dimensions remain constant (i.e., the rods all have the same section).
The pieces then stand in the same relation to one another as the natural series of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
In the second series, namely, that which shows thickness, whilst the length remains constant, the 104 square section of the prisms varies. The result is that the sides of the square sections vary according to the series of natural numbers, i.e., in the first prism, the square of the section has sides of one centimeter, in the second of two centimeters, in the third of three centimeters, etc., and so on until the tenth, in which the square of the section has sides of ten centimeters. The prisms therefore are in the same proportion to one another as the numbers of the series of squares (1, 4, 9, etc.), for it would take four prisms of the first size to make the second, nine to make the third, etc. The pieces which make up the series for teaching thickness are therefore in the following proportion: 1 : 4 : 9 : 16 : 25 : 36 : 49 : 64 : 81 : 100.
In the case of the pink cubes the edge increases according to the numerical series, i.e., the first cube has an edge of one centimeter, the second of two centimeters, the third of three centimeters, and so on, to the tenth cube, which has an edge of ten centimeters. Hence the relation in volume between them is that of the cubes of the series of numbers from one to ten, i.e., 1 : 8: 27 : 64: 125 : 216 : 343 : 512 : 729 : 1000. In fact, to make 105 up the volume of the second pink cube, eight of the first little cubes would be required; to make up the volume of the third, twenty-seven would be required, and so on.