two . . . six—that is, a first one, a second one six ones. Nor will the most ingenious presentations of sense material free us from this fundamental process. Such presentations help to make the process rational; they do not supersede it. If a good working idea of four has been got, it carries with it the idea of three. But it does not follow that seven (4 + 3) is known without counting. The presentation includes the presenta tion does ; but the mere perception of not give the number seven. We perceive the four and the three, and we know these numbers because we have previously analyzed each of them, and put its units in ordered relation to one another-that is, counted them. We have a perception of the group which represents the union of these two numbers, but we do not really know seven till we put the three additional units in ordered relation to the four; or, in other words, till (starting with four) we count five, six, seven, thus fixing the places of the new units in the sequence of acts by which the whole is measured. We are not to rest satisfied, on the one hand, with mechanical countingmere naming of numbers-nor, on the other hand, with mere perceptions of units unrelated by counting. This consciously relating process gives the ordinal element in number; counting, for example, the units in a sevenunit quantity, when we reach four we must recognize that four is the fourth in the sequence of acts by which the whole is constructed. The following points in the development of number seem, therefore, perfectly clear: 1. The measuring idea should be made prominent by constructive exercises such as have been suggested. 2. There must be rational counting-relating-of the units of measurement. This is addition by ones. It is impossible to know, for example, ten times, without having added ten ones. 3. While use may-rather should-be made of the ratio idea (division and multiplication), the mastery of the combinations to ten should be kept chiefly in view; that is, addition and subtraction, with the emphasis on addition, should be first in attention, but with exercises in the higher processes. This must be the course if the ideas of unit and number are to be rationally evolved. In counting up, for example, the four 3-inch units in the foot measure, the child first feels and at last sees that one unit is 1 out of four; two, 2 out of four; three, 3 out of four; or 1-fourth, 2-fourths, 3-fourths of the whole. 4. To help in this relating as well as in the discriminating process, rhythmic arrangements of the actual units, and of points or other symbols of them, may be used with remarkable effect. The real meaning of five, as denoting related units of measure, will clearly and quickly be seen when the units (or their symbols) are arranged in the form ; and so with other num bers. The results of the entire mental operation of analysis-synthesis, by which the vague whole has been made definite, are given in these perceptive forms. 5. Hence, during the first year at school we need not confine our instruction to 5, or 50, or 500, if we follow rational methods. The first thing, then, is to see that the child gains not a thorough mastery but a good working idea of the number ten.* After exercises in parting and wholing, some notions of unit and number have been gained, and a formal start is made for ten, the instrument of instruments in the development of number. Many a 12-unit quantity has been divided into equal parts. Working ; ideas of the numbers from one to six have been obtained the child works with them to make the 'numbers one to six more definite. If the preliminary constructive operations have been wisely directed the task is now an easy one. Not long-continued and monotonous drill on all that can be done upon the number ten is needed, but a little systematic work, with the ideas which from free and spontaneous use are ready to flash, as it were, into conscious recognition. Using the number forms for six, it will need but few exercises to make perfectly clear the real meaning of six, and then the remaining numbers 7. . . 10 are, as means for further progress, within easy reach. In using these number percepts the "picturing" power should be cultivated. Five, six, seven as five and two, eight as two * For the constructive exercises referred to, various objects and measured things may be used as counters; but since exact ideas of numbers can arise only from exact measurements, and since ten is the base of our system of notation, the metric system can be used with great advantage. The cubic centimetre (block of wood) may be taken as a primary unit of measure; a rectangular prism (a decimetre in length), equal to ten of these units, will be the 10-unit, and ten of these the 100-unit. The units may be of different colours, and the units of the decimetre alternately black and white. A footrule, with one edge graduated according to the English scale, and the other according to the metric scale, is a most useful help. fours, ten as two fives, and five twos, etc., should be instantly recognised. In exercises upon the combinations of six we have the whole and all the related parts distinctly imaged. The analysis of the visual forms has been made 5 + 1 = 6, 4+2 = 6, etc. Now cover the 5; how many are hidden? how many seen? Cover the four; how many are hidden? how many seen? And so with all the combinations, taking care that the related pairs of number are seen as related-for example, 6 = 5 + 1 = 1 + 5. This insures a repetition of the number activity in each case, and the ability to recognise, on the instant, any number; to see not only a whole made up of parts, but also the definite number of parts in the whole. Is Combinations of the 10-Units.—It is hardly necessary to say that, with a fair degree of expertness (not a perfect mastery) in handling these twenty-five primary combinations, rapid progress may be made in the use of the higher numbers. As soon as a good idea of ten is gained, the pupil handles the tens, using as the tenunit the decimetre already described (or other convenient measures), and going on to 10-tens, even more easily and pleasantly than he proceeded to 10 "ones." there any known law, save the law of an utterly irrational method, that confines the child for six months to the number five, for twelve months to the number ten, for another year to the number twenty, actually exacting two whole years of the child's school life to say nothing of the kindergarten period-before letting him attempt anything with the mysterious thirty? If a child has really learned that five and three are eight, does he not know that 5 inches and 3 inches are 8 inches, 5 feet and 3 feet are 8 feet, 5 yards and 3 yards are 8 yards, 5 miles and 3 miles are 8 miles? Do five and three cease to be eight, or ten repudiate 5 times two when the unit of measure is changed? As a matter of fact, when the child, under rational instruction, gets a good grip of three he quickly seizes ten, the master key to number. As soon as he comes to ten let him formally practise with the tens the combinations he has learned with ones: the ten is a unit, because it is to be repeated a number of times to make up another quantity, just as much as "one" is a unit because it is used. a number of times to make up a quantity. When it is known that five units and three units are eight units, it is known for all units of measure whatever. There is absolutely no limitation save this, that the child should have a reasonably good working idea of the unit of measure. If he knows, for instance, that 5 feet and 3 feet are 8 feet, he not only knows that 5 miles and 3 miles are 8 miles, but also has a good idea of the distance 8 miles, provided he has a good idea of the unit mile. So, when he has a working idea of a 10-unit quantity as compared with the 1-unit quantity which measures it, he passes with the greatest ease to a good idea of a magnitude measured by ten such 10-unit quantities. In using the metric units, previously referred to, he has analyzed the decimetre prism, has compared it with the minor unit-the cubic centimetre-has found that it takes ten of these minor units, or one of them taken ten times, to equal it, etc. He goes through the same constructive process with ten of these 10-unit measures; he puts ten of them together to make a square centimetre; he analyzes and compares; he uses |
