IV. THE BEGINNINGS OF THE MIND AND LANGUAGE

were hampered in our arithmetic throughout life by this defect. Counting should be taught be means of small cubes, which the child can arrange and rearrange in groups. It should have at least over a hundred of these cubes—if possible a thousand; they will be useful as toy bricks, and for innumerable purposes. Our civilization is now wedded to a decimal system of counting, and, to begin with, it will be well to teach the child to count up to ten and to stop there for a time. It is suggested by Mrs. Mary Everest Boole that it is very confusing to have distinctive names for eleven and twelve, which the child is apt to class with the single numbers and contrast with the teens, and she proposes at the beginning ( The Cultivation of the Mathematical Imagination , Colchester: Benham & Co.) to use the words “one-ten,” “two-ten,” thirteen, fourteen, etc., for the second decade in counting. Her proposal is entirely in harmony with the general drift of the admirably suggestive diagrams of number order collected by Mr. Francis Gallon. Diagram after diagram displays the same hitch at twelve, the predominance in the mind of an individualized series over quantitatively equal spaces until the twenties are attained. Many diagrams also display the mental scar of the clock face, the early counting is overmuch associated with a dial. One might perhaps head off the establishment of that image, and supply a more serviceable foundation for memories by equipping the nursery with a vertical scale of numbers divided into equal parts up to two or three hundred, with each decade tinted. When the child has learnt to count up to a hundred with cubes, it should be given an abacus, and it should also be encouraged to count and check quantities with all sorts of things, marbles, apples, bricks in a wall, pebbles, spots on dominoes, and so on; taught to play guessing games with marbles in a hand, and the like. The abacus, the hundred square and the thousand cube, will then in all probability become its cardinal numerical memories. Playing cards (without corner indices) and dominoes supply good recognizable arrangements of numbers, and train a child to grasp a number at a glance. The child should not be taught the Arabic numerals until it has counted for a year or more. Experience