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IV. The course must continue the process of rationalizing mathematics. As stated in the Preface of Book I, mechanization is entirely justifiable in the earlier grades. In the junior high school, mechanization is still necessary, but must be based upon rationalization.
Rationalization by induction (generalization) is continued in Book II, especially in teaching geometrical facts (See Chapters IX and XIV), but also in teaching algebraic principles (See Exercise 76, Exercise 77, and the instruction on pages 186-188, etc.). However, the spirit of deduction is introduced informally in the manner of solving equations (See pages 18, 21, 197, 203). Mechanical operations like transposition (See pages 202, 203), clearing of fractions (See § 92, page 175), and cancellation (See Example 2, page 175) are shunned. Only by so doing can algebraic processes avoid becoming meaningless manipulation of abstract symbols.
V. The course must prepare for subsequent courses in mathematics. While preparation for the next course is by no means a major aim of any course, failure to so prepare is certainly a major fault.
The gap between the eighth and ninth grades is indeed bridged by the material of this text, even if the formal algebra of Chapters XII and XIII is not studied. The consistent use of the formula and the equation and the avoidance of mechanical processes throughout the course gives the pupils a foundation of knowledge and of habits which will enable them to pursue successfully a high type of ninth grade mathematics.
VI. The course must be flexible. possible is it to give in one single course material suitable for all schools. Especially true is this for a subject in the teaching of which there is the experimentation which characterizes the teaching of junior high school mathematics. The text may be expected to furnish timely material for such experimentation. On the other hand, the author is indeed remiss who fails to offer some guidance in the selection of material for a minimum course.
In this text, the minimum modern course is comprised in Chapters I-XI, excluding the supplementary topics (See pages 29, 53, 106, 131, etc.) Teachers are urged to teach Chapters II and III as given in the text, for, as already said, they are the backbone of the course. Most schools can do more than this minimum course. Whether to spend the time on some or all of the supplementary topics, or, omitting some of them, to complete Chapters XII-XIV, must be decided by local conditions. Unquestionably many schools will be able to do some of the supplementary material and Chapters XII and XIII.
Problems Solved by Proportion
ADVICE TO THE PUPIL
Accuracy in arithmetic. Careful writing of figures and neat orderly arrangement of the solution of an example will help you solve your example accurately. Avoid doing "scratch work." Do upon your final paper any computing which you find necessary. Place such computation at the right side of your paper, arranging the main steps of your solution at the left side of the paper.
Checking solutions. Even expert computers check the solution of a problem in some way to make certain that the result is correct. You must form the habit of checking your solutions.
A first check is to examine the result to determine whether it appears reasonable.
Thus if the rent of a house turns out to be $7550 per month, it is likely that a mistake has been made in locating the decimal point, and that the result should be $75.50. The solution should be examined carefully to determine what is the correct result.
Other checks will appear as you study the book.
Speed in computing is chiefly a result of long practice, and partly a result of the use of short methods of computing. Some of these short methods you may know; others are new to you. In this grade you should seek such short cuts and use them whenever possible.