desired result will be inevitable, for the nature of it is such that doing it by rule is precluded, even if the teacher would do So. As is clearly shown, the child can respond to these exercises only by imaging, comparing, and perceiving relations. This is reasoning. Let no one say that this work does not belong to Arithmetic. The purpose of it, and the certainty that this purpose can be accomplished, fully justify its introduction into a text in Arithmetic.

With theory and rules reduced to the smallest amount consistent with a proper presentation of the science, and with examples offered in great variety and number, this series secures the teaching and the learning of Arithmetic inductively. The problems unfold the processes and the forms of reasoning. This feature, together with the adoption of the spiral rather than of the topical plan, not only gives the pupil a graded manual of problems, but also compels thought as to what the problems mean and choice of methods for their solution. On the other hand, the topical plan, in which the process is given and is followed by a score or a hundred problems to be solved by that process, establishes the habit of working by rules without thought or choice. The spiral graded plan, based on problems, as embodied in this series, develops independence and self-reliance.

Drawn, as are the books of this series, upon these lines of modern psychology and of the best pedagogical practice, they deserve educational attention and recognition for the positive and noteworthy quality of being consistent in detail with the principles they profess to represent. These principles are no longer in question. There is nothing experimental here. The series stands to affirm where it may be used, the supreme importance of Arithmetic in education, and the necessity of conforming the methods of instruction to the principles of mental acquisition and mental growth.

SUGGESTIONS TO TEACHERS

PRACTICE in the fundamental processes, with the application of the same to problems, is continued in this book, as in the Third Grade; and the new topics of factoring, cancellation, greatest common divisor, least common multiple, and fractions are carefully treated. The book is made on the same general plan as the two that precede it. The first thirty-eight pages are essentially a review of the processes and types of problems contained in the Fourth Grade book. New work is then gradually introduced, topics previously presented are extended, and a constant review is kept up as the work advances.

In this book, for the first time, the more important definitions are given, principles are developed and formulated, model solu. tions accompany the presentation of new types of examples, and in some topics rules are given. This matter is reduced to the smallest amount consistent with an intelligent presentation of the subject. The definitions and principles pertaining to the fundamental rules will be found from page 277 to page 285. No rules for these processes are given. Pupils have been adding, subtracting, multiplying, and dividing, in the preceding grades. They know the name of each process and they know which one to apply in the solution of a problem, but they have not been required to define them. Reasons have been giver. for not presenting this matter in the Third Grade where each topic is begun. As this is not a topical manual, there is not a definite portion of the book devoted to each subject. It is therefore immaterial where the definitions of the fundamental rules are given, but since the first pages are devoted principally to a review, it has seemed proper to introduce them somewhere

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in this part of the book. As new subjects are presented, the definitions, principles, model solutions, and rules are found on the same pages.

It will appear at first glance that the lessons for written work throughout the book include examples to develop skill in performing arithmetical processes, and problems that require the application of these processes. This union of so-called abstract and concrete work in the daily lesson is intended to produce the best results. It rarely occurs that two consecutive problems involve the same principle or require the same solution. As the pupil approaches each successive problem, he must therefore choose the process that is applicable to the solution. This feature cannot fail in the end to develop greater self-reliance and independence. Pupils who have completed the Fourth Grade book should come to this grade able to add a long column of figures rapidly and correctly. Besides the many examples in addition given in the lessons, teachers may desire to give other drills. If so, they will do well to observe the suggestions for securing rapidity in addition given in the preceding grade.

It is believed that teachers will approve the treatment of fractions contained in this book. Pupils who have used the preceding books approach this work with definite ideas of fractional values. To them fractions are more than symbols. They know fractions as equal parts of things, they know the relative magnitude of fractional parts, and they have the power to image equal parts of things. In comparison with the usual treatment, the work is considerably abridged, and yet from the standpoint of either utility or mental discipline, it is evident that the results will be more satisfactory. One feature of this treatment is the careful avoidance of fractions whose terms are expressed by more than two figures, while most of the work involves fractional parts not smaller than tenths. As the work is limited in the main to fractional parts larger than tenths, the examples, both abstract and concrete, are much less com

SUGGESTIONS TO TEACHERS

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plex, and their solutions are easily performed. While there are enough examples to develop the necessary skill in handling fractions, the constant aim is the development of power, which is of greater value than the mere manipulation of meaningless symbols. Problems are often given employing fractions that are neither seen nor heard of except in arithmetics. The only motive in such problems seems to be to provide opportunity for performing operations with fractions. No such problems are found in this book. The fractions used here are appropriate, and there is always some object besides the mere manipulation of symbols. There are very few examples of so-called complex fractions, and these are simple. Attention is called. to the treatment of multiplication and division of fractions, with confidence that teachers and pupils will appreciate its simplicity.

Cancellation is usually taught with little or no thought that the principle involved is of any value or capable of any practical application. Two or three pages are devoted to the topic, but no reference is made to it thereafter. Cancellation may be employed in the solution of problems in which a division is to be performed, with dividend, or divisor, or both, expressed by their factors, provided both have common factors. It is specially applicable in finding the area in square yards when the dimensions are expressed in feet, in finding the area in acres when the dimensions are expressed in rods, in finding the number of cords when the dimensions are expressed in feet, and in examples similar to the 11th on page 317, the 1st on page 319, and the 5th on page 323. Cancellation materially shortens the solution of such problems, and pupils should be trained to discriminate when it can be used to advantage. To this end many problems, oral and written, are given, in the solution of which cancellation may be applied.

The oral exercises in this book, as in the two that precede it, constitute an important part of the work. The same reasons hold for the prominence and abundance of these exercises, and

the same motives have prompted the preparation of them. All types of examples and problems found in the lessons appear also in the oral work. These include examples similar to those of the preceding grade, also examples in factoring, cancellation, the different operations with fractions, and surface and volume measurements. Cancellation is quite as applicable to oral work as to written. In fact, many problems that could hardly be solved mentally without cancellation are easily solved in that manner by the use of this principle. Many such problems are given on the oral pages; and though they might at first seem too difficult for oral work, in view of the method of solution here suggested the difficulties disappear. The importance of training the imagination is still apparent, and this suggests the necessity of providing work that will compel the exercise of that faculty. Here, as in the preceding grades, such exercises are found. Some lines of work given in the Fourth Grade for this purpose are continued here, and other exercises are introduced in which the pupil is required to determine what form or forms result from uniting squares and rectangles of various dimensions, and in which he is to compare squares and rectangles of different dimensions to determine the relation of one to the other. The first is synthetic, the second analytic, and both appeal strongly to the imagination. The oral work of the grade deserves the most careful attention of teachers and pupils. The thoughtful teacher will see a purpose in it all. Though some of it may at first seem a little difficult, the persistent effort of teacher and pupils will cause the apparent difficulties to disappear.