"If counters are arranged in this way, and impressed upon the child's memory as depicting the relation between the number 3 and 7, it is, in fact, all there is to know about it. Instead of teaching all the variety of possible combinations between 3 and 7, it is sufficient to make the child keep in mind the above picture. The first four rules, as far as 3 and 7 are concerned, are contained in it, and will result from expressing the same thing in different words, or describing the picture in different ways. Looking at the picture, the child can describe it as : 3+3+1=7, or 3 × 2 + 1 = 7, or 7—3—3=1, 7÷3=2 (1). The latter process is to be read: 3 in 7 twice, and I remaining. "Let the number to be measured be 10, and the number by which it is to be measured be 4; then the way to arrange the dots is: "The child will be able to see at once, by reading, as it were, that 4+4+2=10, 4× 2 +2=10, 10—4—4=2, 10÷4=2 (2), and to perceive at a glance a variety of other combinations. The children will, in the course of time, learn how to draw these pictures on their slates in the proper way. Nor will it take long to make them understand that every picture of this kind is to be 'read' in four ways, first using the word and, then times, then less, then in. As soon as the pupils do this, they have mastered the method, and can work independently all the problems, within the given number, which are required in measuring." PREFACE TO REVISED EDITION. Great care has been taken to correct the errors of the first edition and it is hoped that few remain. At the request of several teachers in ungraded schools, the addition, subtraction, multiplication, and division tables have been inserted in the Pupils' Edition, and a number of pages have been prefixed to relieve teachers in such schools of much of the labor of oral instruction. In graded schools the pupils should not have a book until they have been taught orally as far as to the tables of 7's and review, p. 29. The Teachers' Edition contains the answers to the examples in this book, and instruction for oral work with the youngest pupils, together with methods and additional examples. It is, therefore, necessary that the teacher should have it and carry along the work of the two editions together. References at the bottom of the pages in each edition call attention to the pages of the other edition which contain work of the same kind. The Pupils' Edition is bound separately for |