them face her again.) Now, tell me, how many have I taken away? Child. 3 times 2 and the half of 2 have been taken away. Moth. But how many times 1 has been taken away? Child. 7 times 1. Moth. How many times 2 must be added again, to get the same number as before? Child. 3 times 2 and the half of 2. Moth. And how many times 1? Moth. (Adding one cube only.) many times 2 are now wanting? Child. 3 times 2. How Sometimes the Mother may turn the questions thus: 3 times 2 how many times the half of 2? (Instead of asking 3 times 2 how many times 1;) in order to make it clear, that 1 or the half of 2 means the same thing. Child. 3 times 2 are 6 times the half of 2. Moth. Why? Child. 3 times 2 are 6 times 1; 1 is the half of 2: 6 times 1 are 6 times the half of two, &c. In this manner she may replace by degrees all the cubes, by adding now 1, now 2, sometimes more, to prevent the children from proclaiming empty sounds only, without having a clear intuition of what they say: as is the case with our multiplication tables, by which children soon know mechanically, that after the sounds 3 times 4 follows the sound 12; but are unable to answer when the question is put to them inversely: 4 times 3 how many times 1? because these sounds are unknown to them. Moth. (Having replaced the six pair of cubes.) How many must I add, to restore our former number? Child. None; the former number of 6 times 2 is complete; or, Moth. (Having added one cube to the six pair.) How many have I to add, to complete our former number? Child. much. None; there is already one too In order to fix still more the children's minds, when counting, to the position of objects, she may give the same exercise, in the manner following: Moth. (Placing as before eight pair of cubes, each pair at equal distance, thus :) 8 8 8 8 8 How many times 2 are here? Child. 8 times 2. Moth. How are they ranged? Moth. Are they placed at equal or unequal distances? Child. Each pair is placed at equal dis tances. Moth. Do you observe nothing else? Child. Each pair of cubes forms a rectangle, the long sides of these rectangles run parallel, and their short sides in the same direction. Moth. What position have the long sides of these rectangles, in regard to some part of this room? Child. Their long sides are turned towards the door, and their short sides towards the window. Moth. Very well. Lest we should forget whatever we know of these cubes, let us repeat it (together with the children.) a. 8 times 2 cubes are placed here. b. Each pair is so ranged as to form a rectangle. c. These rectangles are equally distant from each other. d. The long sides of these rectangles are parallel, and their short sides in the same direction. e. They have their long sides turned towards the door, and their short sides towards the window. Now, mark, how many cubes are here, and in what manner they are placed. Turn about. (Meanwhile she may take away some, for instance, five, and the children having again faced the table, she may ask :) What change has taken place? Are the cubes still situated as they were before? Child. Yes; they are exactly situated as before, but some of them are wanting. Moth. How many times 2 are wanting? Moth. Which did I take away? Child. (Describing precisely which were taken away.) Moth. Turn about again. (Replacing the cubes she had taken away) 8 88 88 How many have I taken away? Child. (Having turned about and looked at them.) You have taken away none. Moth. But has any alteration taken place? Child. Yes, the second pair, (from the right) has been moved closer to the first pair, and the fourth nearer to the fifth, and consequently farther from the third. Moth. Right. Has any change taken place with regard to their direction? Child. No, the direction is the same. The Mother repeats with them all the changes which have taken place. Moth. (Separating two pair of cubes, so that one of each pair remained in its original situation, but the two others were moved out of the straight line, their sides remaining unchanged, thus:) ᄆᄆ How many of these cubes have I taken away? Child. None; but they are placed differently from what they were before. Moth. Has the position of all of them been changed? Child. No, the position of two only has been changed. Moth. Exactly! But in what manner is their position altered? Child. They are separated from those |