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(4) Twenty-six standardized Self-Testing Drills for the maintenance of acquired skills and for the systematic self-measurement of pupil progress.
(5) Fourteen standardized Problem Scales, which make a persistent attack on the problem-solving difficulties of children. These "scales," requiring from two to three pages each, clearly make for bulk; but experimentation has proved this use of space well justified.
(6) Ample drill material, particularly on the more difficult combinations, required by the calculated frequency with which the basic combinations must occur in the experience of the child if he is really to learn and to hold them. This book actually contains the drill necessary to accomplish the teaching job, without the customary supplementing that must be made by the teacher from other sources.
II. Special material for recognizing individual differences and provision for optional work. (1) Frequent units of "extra problems" and organized lessons for "good workers." All authorities agree that it is high time for competent writers to begin giving systematic consideration to superior children in the public school.
(2) A carefully planned review of the work taught in the second grade in many schools. The first three chapters (totaling nearly 80 pages), devoted to this review, are of course optional.
(3) Various units of work as alternative to certain other units as a means of meeting the needs of special pupils, groups, or classes. Again, classroom experimentation has shown this aspect of the treatment to be a highly remunerative teaching aid. (4) Self-Help Practice Lessons to aid in training pupils in good habits of using their book as a learning tool under self-guidance.
III. Mechanical demands. (1) Considerable space devoted to making clear the organization of the material, both to teachers and pupils. Chapter headings (showing large units of the work), centered section topics (exhibiting the larger sub-units of each chapter), and numbered article titles (indicating the arithmetical
substance of the lesson under study at the moment) combine to make clear the mathematical "flow" of the text. To these are added parenthetical suggestions for each article, that indicate the authors' advice to teachers concerning the most effective way of handling these units.
(2) Summaries at the ends of chapters, that provide convenient and definite means for a quick test-survey of the progress made by the pupil.
(3) An unusual amount of space devoted to pictorial and diagrammatic aids to understanding.
The foregoing list of space requirements indicates the inherent necessity of more generous texts in arithmetic than have customarily been provided.
Experimental use of this material in a large number of thirdgrade classes widely scattered over the country has demonstrated that expansive treatment takes not more but less time than the customary condensed text material. The average third-grade class, as a matter of fact, normally uses as many pages of material as this book contains, taking about 150 pages from its adopted text and getting the rest as supplementary material from sources that are frequently ill-fitted to the text. This text for class use does not need supplementing. The authors would lay heavy stress on the fact that, while expansive treatment increases bulk, it increases also the rate of progress; and it decreases both the difficulties and the failures.
The authors' acknowledgments are herewith made to the administrators and teachers, unnamed here only because their number runs into hundreds, who so generously aided in passing this material through the refining fires of classroom experimentation, thereby assisting in bringing it to the final form. Special acknowledgment is due Misses Tolosa Cooke, Ruth Pritchard, and Celia Sheldon and Mr. W. C. Findley for valuable assistance in connection with various units of the work.