making of their own examples. Such pages as these for drill in various topics are merely illustrations of exercises which may be set by the pupils for themselves. Every week there should be a period in the program for invention of problems by the pupils. One caution is to be observed. It is best for the teacher to inspect the problems to see that they are capable of solution. This invention of problems makes the pupils independent of printed answers and stimulates them to try proofs for themselves. 6. It is undesirable to pay much attention to the words of arithmetical rules and definitions at this stage lest the forms conceal the principles. Reasoning is the soul of arithmetic. 7. Incidentally such facts of the business world as the children see it for themselves may well be used. What facts will depend upon the neighborhood of the school and of the children's homes. These facts may be used as the bases of their invention of problems. This book uses standard facts such as may be seen almost anywhere. Continued observation of local facts will result in the accumulation of great stores of knowledge for use from time to time in the instruction and drill. Among the facts that ought to be known by children and are valuable as material for making problems are such as these; viz.: (a) Railroad fares are usually three cents per mile. Children pay half fare. What are the costs for familiar distances? to great centers of population? for "excursion" and "return" tickets? دو Children pay (b) Street-car fares are usually five cents. three cents. How many miles may one ride for one fare? Are several tickets sold at lower rates? Are special school children's tickets sold? (c) Make market lists of prices for provisions, meats, groceries, shoes, dry goods, clothes, carpets, furniture, etc. Place these lists on the blackboard, and use them both in written and in oral work. (d) Take imaginary trips, singly or in parties, paying all expenses,fares, food, purchases, admissions to places of amusement or recreation, etc. (e) As far as possible, get the material from the children, but have it accurate. 8. It is unquestionable that a great variety of problems. and exercises tends to make the principles themselves very clear. But avoid too much consecutive repetition of processes. We can secure the memory of processes best by variety of associations. Hence, though we ought to make each topic clear at its first presentation, we ought not to dwell too long upon it, but to go forward to the next, returning occasionally to refresh and to enrich our former knowledge. 9. Neatness in writing tends to accuracy in all exercises. Let us encourage excellent work with pen, pencil, or chalk crayon by commending it. Poor work may be due to defective eyesight, which causes increasing trouble with every added year of life. The figures written by most school children are too small and cramped. 10. In working through such a book as this it is well for a teacher to remember that in the early part of a term or year some problems may prove too hard for the pupils which they will be able to solve easily later in the term or in the year. If the problems are too easy, it will be found a very simple matter to increase the difficulties by adding another step or by dealing with larger figures. While the class is studying the topics, the teacher is studying the class; and while the class improves in industry and ability and technical knowledge, the teacher improves in instruction and in assignment of work through greater familiarity with the needs and powers of the individuals in the class. 11. In the practical use of this book a greater or a less proportion of the problems will be found too difficult for even the best pupils, not to say the average pupils, upon the first consideration of them. (a) Sometimes it will be found advisable to give oral instruction regarding a problem. (b) Many of the class will be able to return a few weeks later to a hitherto unsolved problem with power to solve it unaided by the teacher. (c) Where pupils fail upon many problems, there may be need of a systematic review, for the purpose of developing in the imagination the concrete bases of arithmetic which are given by ratio and counting. But more often it is the better plan to go forward, using oral problems that involve the processes suitable to the ages and grade of the pupils. By mere change of numbers almost any problem may be made oral. Many classes are weakened in activity, power, and memory by too much oral instruction. 12. A book in which every problem is easy for nearly every child is too easy to give proper inducement for earnest, steady, upbuilding effort on the part of its students. The plan here has been to make a systematic outline, leaving to the teacher's judgment of the daily needs of the class exactly the number of problems and upon exactly what pages they should be for the ever insistent "next lesson." Something new for curiosity and a good deal that is old for power and for skill make the right combination usually for lessons in any subject. From a half page to a page will be usually a reasonable lesson, including both class and home work. |