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TO THE TEACHER

It is the business of schools to give children during the first six years of school life that kind of instruction in mathematics which will lead them to a quick recognition and a ready knowledge of number combinations and number operations, and then enable them to apply these number combinations and number operations to the solution of simple problems. The instruction for this period, if it is to serve its purpose, must be definite and specific.

At the end of the sixth year the pupil should have a thorough mastery of the number combinations in integers; of the four fundamental operations with integers, common fractions, and decimal fractions; of the common measurements; and of the use of all of these in the solution of problems. Since eternal review is the price of excellence in all mathematical work, especially in computations, this book contains a complete but not lengthy review of the work of the first six years. The reviews are arranged elastically, however, so that the time devoted to them can be determined by the needs of the class. Monotony, the one great drawback of reviews, has been removed by connecting the matter reviewed by historical references of interest, by looking at it from a standpoint of business, by number contests, by using the matter to be reviewed as a background for new work.

Teachers will find the study of mathematics developed here so that every principle is studied at sufficient length to make a lasting impression upon the child and to show at the same time its significant meaning when applied to the activities of life. The principle is not, however, dropped when it has been thus studied but is brought continually into play in the problems following. A careful scrutiny of the exercise material will show clearly this feature of the book. Much spiralization has been found to lead to loose, disconnected ideas of the whole subject, one of the things for which mathematics and mathematics teaching is now very justly criticised. On the other hand, it is equally bad to hermetically seal each topic in a separate compartment and place this on the shelf to gather dust after it “has been completed." But extremes have been avoided.

During the transition period of the Junior High School the pupil should be led to rationalize and to generalize. He should now be asked to discover and observe for himself the interrelationships found in his school activities and in the life about him. His mathematics from this point on should train him to study every question that arises from its quantitative side. He should be led to solve the various interrelated problems involving quantity as he sees them around him. Thrift and economy and social relationships are especially stressed in the problems and under suitable topics. Such a treatment lends itself to a continuous and progressive appeal to the child to know and to get the qualities that make for good citizenship. Therefore the author treats mathematics throughout as an instrument for use, and to that end problems and projects to which each mathematical principle is applied have been graded to conform to the successive stages of development of the child mind and of child activity.

The mode of instruction is less and less specific and more and more suggestive. To accomplish this, simple language has been used, and the subject expanded by easy stages, so that the pupil may be given self-reliance and at the same time constant encouragement to be ever on the alert for new discoveries. His senses are employed to introduce him to rationalization and to generalization according to the best-established psychology. Geometrical construction work accordingly receives special attention in this the first book of the series. Literal numbers are studied for their applied value in stating geometrical formulas. The pupil meets here for the first time a new form of mathematical expression, a shorthand symbolical language.

By way of emphasizing some of the more practical features of the book, the author begs to call attention to the following facts: In connection with computations, checks are given a prominent place. Estimates and approximations as well as other forms of simplifying number work are used continually. The three forms of fractions--common fractions, decimal fractions, and percentage-while placed in separate chapters for convenience, are still treated as fractions; merely different forms of expression. The relations between the three forms are clearly brought out. Weights and measures are placed in the book only for those needing review in that work. The metric system is taken up to show its simplicity and the close approximation of the metric units to our English units. Literal numbers are confined to monomials, which are used principally in stating laws and geometric formulas. A new generalization arises in the extension of the number system to negatives. In the geometric work the appeal is made particularly to the constructive and discovering nature of the child. The work requires much drawing and presents many opportunities for sense judgment and will continually raise questions for investigation.

The author desires here to acknowledge the very valuable aid rendered by Mr. W. H. Keller, of the Kansas State

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