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HARVARD COLLEGE LIBRARY
MAY 21 1936
A new edition, thoroughly revised, and printed
BY WILLIAM E. CHANCELLOR, A.M.,
SUPERINTENDENT OF SCHOOLS, BLOOMFIELD, N.J.
ARITHMETIC is an essential element in every stage of human progress. Though there may be a difference of opinion as to how much of it is essential or desirable, the subject must hold its position as one of the two constants in our fundamental education, of which the other constant is Language. The necessity of Arithmetic in human progress, its value in practical life, and its efficiency as a means of intellectual development demand that it shall be taught in the full light of modern psychology.
Most of the teaching of Arithmetic is determined by the text used. Hence it is evident that the use of a book which is unpedagogical in plan, arrangement, and character gives meagre results, while the use of a book which is in harmony with the laws of mental development and mental growth gives results which are commensurate with the time and energy devoted to it. For various reasons it has lately come to pass that Arithmetic is challenged to defend its position as one of the most prominent subjects in our schools. The author of this series shows a strong conviction that the cause of this widespread dissatisfaction with results now obtained is traceable to the character of the texts in general use.
express the author's confident belief that it is possible to prepare a text which is entirely adapted to the needs and gradually developing powers of the child.
Modern psychology has contributed to our educational practice no truth of greater importance than this, that education is
the process of building up, thought by thought, the structure which is to be the adult mind. From this it follows infallibly that children must be graded not merely hy ages and capacities but by actual attainments, and that the texts provided for their use must also be graded. The educational text-writer must therefore fit his texts to the pupils. Commonplace as this truth nowadays is, the topical manual of Arithmetic, which considers the logic of the subject, not the psychology of the pupils, is still in use in most American schools.
Again, the inductive philosophy, which teaches us that principles are to be derived from the facts, and that facts are not to be selected or made to suit predetermined principles, and the experimental psychology, which demonstrates that nothing is completely learned until it has issued in action, have pointed unmistakably to the principle that Arithmetic should be learned in its examples. But few text-book writers have been willing to abandon the scholastic traditions descended from the centuries before either the inductive philosophy or experimental psychology were even in wise men's minds.
As opposed to the topical plan, what is now known as the spiral plan is rapidly growing in favor among teachers. This plan consists in daily reviewing the important processes so far known, and in frequently adding new processes. It combines the old and the new, the review and advance lesson, systematically. This series illustrates the spiral plan with no concessions to topical traditions. The topical division, if logically followed to all its conclusions, results in teaching addition one term and subtraction the next, fractions one year and denominate numbers the next. Though logical, such a division is neither psychological nor pedagogical. It neglects the mind's need of variety in the daily exercises, and tends to the repetition of mere process work. The topical Arithmetic may be an excellent teacher's manual, but the pupil, through ignorance, is not ready in any early grade for even abridged treatises upon the science and art of Arithmetic.
The alliance of oral, often called mental, Arithmetic with the written lessons illustrates the author's confidence in the principle of grading the topics and developing them through examples. No severer test of a theory can be applied than by putting it into' exclusive practice. Written lessons may be given in which for weeks and months, even for terms and years, the pupils may conceal their failure to understand their work; but oral Arithmetic is the complete revelation of the progress and position of class and pupil in the science. It is impossible for a teacher to conduct an oral recitation which is beyond the pupils' comprehension. Interest at once ceases, and there are no answers to the questions. The successful union of oral and written lessons, as seen in all these pages, is an artistic achievement in pedagogy.
In itself oral work unquestionably deserves the prominence it receives in these pages. Mental Arithmetic promotes attention, concentration, accuracy, ingenuity, skill, familiarity, and confidence. It tests all written work as to the essential point, whether or not the process is understood. It secures the education of the class by the efforts of individuals, who are followed and criticised by their fellows. It is a language training of the highest value.
This series emphasizes to an unusual degree the necessity of employing the imagination in the study of Arithmetic. Most teaching of Arithmetic has appealed to the memory almost to the exclusion of the other faculties. Though the development of the reason is earnestly desired, little or no intelligent effort has been put forth with the certainty of accomplishing this object. The author has made it very clear that thinking or reasoning in mathematics involves the exercise of the imagination. Since faculties grow strong by exercise, work must be provided which will compel the exercise of the imagination.
Here is abundant material in the first three books of this series, of which the direct aim is the training of this faculty. This work not only aims at this power; but if done at all, the