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Bulletin No. 1 of the San Francisco State Normal School*

BY FRANK F. BUNKER

Supervisor of Arithmetic in the San Francisco State Normal School

PREFACE.

Until education becomes less an art and more a science the thing which will prove most helpful to the teacher down in the heat and dust of the schoolroom' will be the experiences of her fellow-teachers. From the philosopher and psychologist she generally gets nothing but "inspiration" plus an overpowering desire to bolster up her schoolroom practices with some kind of a theory. From our theoretical men who learnedly discuss general questions at teacher's institutes she too frequently gets only momentary entertainment. The educational literature which has recently emanated from these savants is voluminous, but the chronicle of the successes and failures of the teacher who triumphs in the end, through the grace of her own common sense, has yet to be written.

We purpose in this monograph to have our student teachers present an account of their work in teaching arithmetic to classes in our training school. Each of these teachers has had entire charge of a class in this subject for one term. At the end of the term each made a written report of her work. It is from these reports that this monograph has been written.

This record by no means exhausts the methodology of the subject. It is to be taken rather as a tentative effort on our part. We have in our work in the practice of teaching arithmetic consciously held to what we feel is a conservative and safe policy. We are exceedingly careful that our teachers do not go out into the state with some one or more of the many arithmetic experiments elevated to the dignity of serious trial. This will, in a measure, explain the lack of any direct reference to some of the recent departures in arithmetic which, though perhaps sound in general principle, have not been sufficiently trimmed by experience, and which in practice so far, have served mainly to unsettle and stampede many otherwise sane teachers.

No claim is made for originality of method or device. Each student has

* Single copies of this Bulletin, in monograph form, will be sent to any teacher in the public school service of California upon receipt of four cents postage. Orders from others will be filled at twenty-five cents per copy. Address, State Normal School, Powell Street, near Clay, San Francisco.

read widely, and with enthusiasm. Successful teachers have been interviewed and observed. Each day the gleanings from these sources have been brought to the conference room and there subjected to discussion by supervisor and class. Whatever seems of value is tried in the schoolroom by our students and the results carefully observed. It is the story of these trials and their results which we have here recorded, and which we hope may be read with profit by our comrades in the work.

The work which we have here presented is limited to the first three years, in our school, taken in the first three grades. We intend to complete this methodology by publishing soon a similar report on our work in the intermediate grades, and at a later time still, a third on that done in the higher grades. For the purpose of clearness, the reports which follow are broken into topics. The grade in which the work was done, and the student-teacher doing it, are indicated in brackets.

San Francisco, Cal., Jan., 1902.

INTRODUCTION.

FRANK F. BUNKER.

Much of the bad teaching in primary grade arithmetic is due to the confusion which has arisen in the minds of teachers over the relation of objective to abstract number work. The idea is prevalent that in some way, not clearly understood, accuracy and facility in handling figures will grow out of facility in handling objects. A knowledge of the theory of the growth of fruit trees will not give any facility in the mechanical work of hoeing out the weeds or cultivating the soil about them. Neither will a series of exper iences in handling, feeling, and seeing blocks give ability to add, subtract, multiply or divide figures. On the other hand, facility in operating with the four fundamental processes on figures will give no one ability to apply these figures to objects or to visualize the terms of a concrete problem. Ability to use figures comes only through using figures. The foundation of success here rests in memorizing the combinations and the tables. Ability to apply intelligently figures to things comes by giving suitable exercises in which this association of objects and figures is required.

The study of every topic, generally included in a course in arithmetic, can be begun either from the objective side or from the side of the purely formal. For instance, in beginning the study of fractions, the teacher can give a more or less extended course wholly within the field of the concrete, or she may choose to begin with the formal and mechanical side; the side which is concerned alone with the various manipulations of fraction symbols. Just so with square root, with division, with multiplication, or in fact with almost any phase of arithmetic; on the one hand there is the field of the objective, the concrete; on the other, the field of the formal. Careful observation of practice work will show, as we have just said, that facility in one field will by no means give facility in the other. A child by careful teaching in the field of the concrete will soon acquire great skill in adding

simple fractions, and yet he may never have seen those same fractions expressed by figure symbols. He does this by reason of the fact that to him fraction is as much a concrete thing as is his dog or his horse. To him adding fractions is nothing more than calling up and counting mental images of familiar things. On the other hand the mind is never more devoid of mental images than when engaged in formal calculation. To have images of things floating around at such a time means that attention is diverted with ineffectiveness as a consequence. Obviously the child needs training in both these fields. He needs to be accurate and tolerably rapid in the mechanical work of fractions and at the same time he needs the power to see visually the relation between and of a foot.

If it be true then, that there is no transference of power between these two fields of work, it follows, we think, that teachers are wrong in declaring that objective work should always precede the formal. In some cases it seems to us the formal may well come first with an application to objects at a later time. Most children find it easier to learn the names in the number series than to apply them to things. Again we fancy it is much less trouble to find mechanically the product of 33 and 2/3 than by the use of objects. On the other hand, in combination work, in the multiplication and division tables, in approaching the study of fractions, as well as beginning the study of many other phases of arithmetic work the objective would better be taken first. However, all that we want to point out here is merely this: no general statement as regards the order of the objective and formal can be made. The question must be decided wholly with respect to the particular topic under consideration.

Counting and Writing Numbers.

In this work we begin with the formal side. Children like the rhythm of the number series. We make use of this impulse and have our children when they first enter our receiving classes begin running up the number series. Not until after they have learned the names to 25 or 30 and can write the symbols do we begin objective work. The details of this work are discussed in the following reports:

[Receiving class.] The first thing which I took up in this grade was the number series. There is a certain rhythm about this which the child enjoys. Running up the number series was at first done by the children individually; when confident that each knew the series correctly, I asked the class to give them in unison, and while giving them in this way I frequently introduced simple calisthenic exercises, allowing the children to count the motions. In this way the pupils had a brief rest and by combining the motions of the body with counting impressed the rhythm more firmly on their minds.

The above work was done entirely with a view to memorizing the number series. I now began systematic work in teaching the children to apply this series, spoken and written, to familiar objects about them in the schoolroom. However, in applying this series I at first did so in connection with counters in order that the children could arrange them in groups as they counted them out as, one two three four etc. It was only by thus

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exercising care that I kept the children from thinking that the second object counted out