Supervised study in mathematics and science

Front matter

EDITED BY

ALFRED L. HALL-QUEST

MACMILLAN & CO., Limited

Preface

In attempting a book on supervised study which will cover even approximately the subjects of mathematics and science, it is impossible to do more than give suggestive lessons. This, therefore, has been my plan — to give only one or two typical outlines of a topic or subject, but to leave an intimation of its application whenever or wherever the teacher may elect. Thus, only one Red Letter Day lesson is presented in Algebra, but the teacher will undoubtedly desire to use many such plans during the year. The material in the lessons mentioned may be suggestive for the planning of others.

I have not tried to add another learned book in pedagogy to the many already on the market. It has rather been my aim to write a book that may be of explicit and direct value to the teacher or principal who is daily striving to teach his children how to study and how to learn, I have tried to write it in simple language, so that the reader may get the meat, if there be any, without too much stuffing.

It is needless to say that I am a firm believer in supervised study. It has done much for our children ; I am sure it will do more as we progress in the proficiency of its administration. It is not a panacea for all pedagogical ills, but it is valuable for what it claims to be, and it holds great promise for the future.

I am greatly indebted to Professor Alfred L. Hall-Quest of the University of Cincinnati, who, as editor of this series, has not only made it possible for this volume to be, but who, viii Preface

through the reading and criticism of the manuscript and through innumerable other suggestions, has been of inestimable help to me.

Deep appreciation is also here expressed to Professor Charles M. Rebert of St. Lawrence University, for valuable suggestions and advice ; to Mr. A. E. Breece of the Hughes High School, Cincinnati, Ohio, who made a very careful and valuable critical review of the manuscript as relating to mathematics ; to my teachers at Canton, N. Y., who made it possible to actually try out many of the lessons ; and to my wife, for her constant counsel and encouragement.

In addition, I wish to acknowledge my thanks for the courtesy of The Macmillan Company, the American Book Company and the Charles E. Merrill Company for permission to quote more or less extensively from their publications.

S. Clayton Sumner.

Palmyra, N. Y.

January 31, 1922.

Table Of Contents

PAGE

Introduction. Supervised Study a Moral Imperative . . The Editor xiii

Chapter One. Management of the Supervised Study Period in Mathematics 3

First Section. Algebra (Elementary)

Chapter Two. Divisions of Elementary Algebra; Units of Instruction and Units of Recitation. A Time Table . . 20

ILLUSTRATIVE LESSONS

I. The Inspirational Preview 26

II. Introduction. Unit of Instruction I. A Lesson in Correlation 34

EEL Introduction {Continued). A How to Study Lesson . . 43

IV. Introduction [Continued). An Inductive and How to Study

V. Addition. Unit of Instruction IH. An Inductive Lesson:

Addition of Monomials 59

VI. Addition (Continued). An Inductive Lesson: Addition of

Polynomials 67

VQ. Simple Equations. Unit of Instruction X. An Expository

and How to Study Lesson : The Equation and Problems . 73

VHI. Factoring. Unit of Instruction VII. A Socialized Lesson . 80 x Table of Contents

LESSON PAGE

IX. Fractions. Unit of Instruction IX. A Deductive and How

to Study Lesson 82

X. A Red Letter Day Lesson 85

XI. Radicals. Unit of Instruction XIV. A Socialized Lesson . 88

XII. Quadratic Equations. Unit of Instruction XV. An Expository and How to Study Lesson . . . . . .92

XIII. An Examination 95

Second Section. Plane Geometry

Chapter Three. Divisions of Plane Geometry; Units of Instruction

and Units of Recitation 105

I. The Inspirational Preview 106

II. Rectilinear Figures. Unit of Instruction II. A Deductive

and How to Study Lesson : Vertical Angles . . .110

III. Rectilinear Figures {Continued). A Deductive Lesson: Tri-

IV. Rectilinear Figures {Continued). A How to Study Lesson:

Originals 123

V. Rectilinear Figures {Continued). A Deductive Lesson : Originals 129

VI. A Socialized Review : Book I 134

VII. An Exhibition or a Red Letter Day Lesson . . . .135

Third Section. Advanced Mathematics

Chapter Six. Divisions of Biology ; Units of Instruction and Units of

Recitation. A Time Table 155

A. — BOTANY

I. The Inspirational Preview 156

II. Introductory Topics. Unit of Instruction I. A How to Study

Lesson: Preliminary Experiments 161

III. Introductory Topics {Continued). An Inductive Lesson.

Problem: No Two Plants Are Alike 165

IV. Introductory Topics {Continued). An Inductive Lesson.

Problem : Struggle for Existence 169

V. Seeds and Seedlings. Unit of Instruction II. A How to Study

Lesson: Seeds and Their Germination . . . .171

VI. Seeds and Seedlings {Continued). A Deductive Lesson: Laboratory Experiments . . . . . . . .175

VII. Seeds and Seedlings {Continued). A Socialized Lesson: A

Field Trip 179

B. — ZOOLOGY

VIII. Insects. Unit of Instruction IX. A How to Study Lesson :

The Grasshopper 181

IX. Insects {Continued). A Laboratory Lesson: The Grasshopper 187

X. Insects {Continued). A Correlation and Research Lesson . 189

XI. Insects {Continued). A Socialized Lesson . . . .192

XII. A Red Letter Day Lesson 195 xii Table of Contents

C. — PHYSIOLOGY

LESSON PAGE

XIII. Bones and Muscles. Unit of Instruction XVI. A How to

Study Lesson : Muscles 197

XIV. Muscles {Continued). A Laboratory Lesson Using Micro-

scopic Slides 200

XV. Muscles {Continued). A Deductive Lesson. Problem: How

Muscular Activity Is an Aid to Good Health . . . 202

XVI. Muscles {Continued). A Lesson in Correlation . . . 204

XVII. An Examination Lesson 206

Fifth Section. Physics Chapter Seven. Further Lessons in Science 213

I. Fluids. Unit of Instruction. An Expository and How to

Study Lesson 214

II. Fluids {Continued). A Laboratory Lesson . . . ,219

III. Fluids {Continued). A How to Study Lesson : Problems . 223

IV. Red Letter Day Lessons 225

Bibliography 229

Index 233

INTRODUCTION BY THE EDITOR

SUPERVISED STUDY A MORAL IMPERATIVE

Millions of words have been written about education. Theories have abounded and still are fertile. The visitor at educational conventions, especially in the department of school superintendents, is impressed, however, with the rapid multiplication of devices for visualizing educational practice. A rich variety of moving picture machines and already voluminous catalogues of educational films witness to the dawn of a new era in the technic of teaching. Later we shall no doubt find boards of censors passing upon these films — boards composed of theorists, critic teachers, educational scientists, et al., — but at present the field is open for all. Doubtless many teachers will find in this form of visual education an opportunity for enlargement of income as well as for the demonstration of teaching skill.

Increasing Emphasis on Demonstration. Demonstration and description are rapidly coming to the front in discussions of methods of teaching. One carefully prepared and successfully performed demonstration is of more value than many verbal descriptions, however clear these may be. A series of vivid verbal descriptions makes definite and concrete a volume of abstractions and theorizings on educational practice. Theory is important ; it must not be discounted ; but here, as elsewhere, an illustration turns on light and makes

xiii xiv Introduction by the Editor

objective and easily understood the necessarily vaguer discussions of abstract theory.

This series of volumes on Supervised Study attempts to visualize one form of study supervision. Each book is written by a teacher who has had considerable experience in this type of work. The emphasis in each discussion is to make concrete in as detailed description as possible, what the author has actually done in his own classroom. Very briefly each author states the theory underlying his practice, but beyond this brief statement he refrains from a discussion of principles. Teachers desire to see how theory is applied. One cannot be too clear and too definite in describing the mode of procedure in supervision.

At Present No Generally Accepted Meaning of Supervised Study. In answer to those who believe that Supervised Study as described in this and other volumes of the series is different from the general understanding of the term, it should be emphasized that at present there is no generally accepted form of Supervised Study. It is the conviction of the editor of this series that a standardized form is undesirable. The main objective is teaching children — all children — how to study and guiding them while they apply the principles of correct studying. It is of comparatively little importance how this is done, providing it is done effectively. If the teacher makes this type of teaching superlatively significant, it follows that the management of the class and the method of presenting subject matter will change accordingly. But each teacher must be the final judge of how to adapt this new point of view to local needs.

The Imperative Need of Preventing Failures in School. It should be said, however, that any plan which seeks to prevent

Introduction by the Editor xv

failures and which aims to train all pupils to study as effectively as native ability permits is superior to plans that simply correct improper methods of work and that are concerned only with the retarded pupils. If school work is limited to the assigning and hearing of lessons, only a few — the highly endowed — will permanently profit by such experience. There are well-meaning people who sincerely believe that the school is the place for eHminating society's mentally unfit, and that the surest way of such elimination is to assign lessons, long and hard. Those who can will ; those who cannot will not. Those who will and can are the fit !

Some there are who learn to swim by the "sink or swim" method ; they are destined, forsooth, to be swimmers if they do not sink. But how many of you who read these pages learned to swim by this fatalistic method? There are children who early judge themselves incapable of school work. Nobody cares ! They either can or cannot study. By means of the hard, soulless machinery of assigning and hearing lessons they are cast out. We call this a safe test and out they go labeled mentally weak, unfit to partake in a world of thrilling knowledge, unfit to climb to altitudes of self-revelation and social worth. If, however, they could have been taught how to use their minds, how to partake in the feast of knowledge, who knows but that many of them would have found a new meaning of their destiny !

Supervised Study Is Not Only an Intellectual Necessity; It Is a Moral Imperative. As teachers it is our plain duty to teach children how to study. The whole class period must be conducted in this spirit. The specific aim of every class period must be to so direct the pupils that their grasp of the new work is adequate for independent application. The teacher xvi Introduction by the Editor

is preeminently a director of study and not primarily a dispenser of subject matter.

The Point of View in This Volume. The author of this volume is convinced of the effectiveness of Supervised Study. He and his teachers have tried it long enough to know its advantages. The subjects of mathematics and science are especially favorable to this method. A comprehensive view of the courses in the high school is given in a series of typical lessons describing in great detail how children may be directed in beginning, continuing, and reviewing their study of particular units of subject matter. The author is well aware of the movement for reorganization of courses especially in ninth grade mathematics, but inasmuch as such revision is not likely to be possible in all schools for some time to come the usual division of courses is considered in this volume. It is believed also that even where general mathematics is taught, not a few pupils will elect additional special courses in the field of mathematics. Inasmuch as general science is at present little more than a combination of various special sciences the separate treatment used in this volume seems preferable. It is hoped that general science will evolve increasingly along the lines of natural correlations through which the pupil will be able to understand the intimate relationships that exist among the phenomena of nature.

Chapter One The Management Of The Supervised Study Period In Mathematics

Causes of Failures in Mathematics. — There are a number of contributory causes which, together or separately, might account for the high mortality in mathematics classes. That it is high is so commonly accepted among the profession, that a large percentage of failures has almost come to be an established expectation. In eleven high schools near Chicago, the percentage of failures in algebra and geometry was found to be greater than in any other subject. 1 In the report of the New York State Education Department on statistics for Regents Academic Examinations, the failures in mathematics for the past five years have been between thirtythree per cent and forty per cent. 2 The nearest competitors for scholastic dishonors are the commercial subjects which are largely mathematical in content.

The causes of these failures are psychological, pedagogical, and physical. Psychologically, mathematics has been by

1 School Review, June, 1913, p. 415.

2 Annual Report of the State Department of Education (10th to 14th inclusive), New York State.

almost common opinion accorded the position of being the hardest subject in the school curriculum. This estimate of the subject, persisted in by pupils, teachers, and the laity, has inevitably resulted in a state of mind that predetermines a large percentage of failures. Until we teachers succeed in dispelling this opinion, pupils in many instances will expect to fail, and they will fail. There is no sane reason why mathematics should be so considered, and with the new vision of teaching the subject and with the readjustment of the course of study, combined with its scientific treatment (which will emphasize the functional and practical side instead of the formal aspect), this view of the severity of mathematics doubtless will gradually disappear.

Mathematics Taught with Deliberate Unattractiveness. — It is repeating a platitude to refer to the fact that mathematics has been very poorly taught in the public school. There has been no serious lack of scholarship and of emphasis on the acquirement of knowledge of subject matter, but this very emphasis has tended toward the serious neglect of training pupils to apply mathematical rules and formulas to practical reasoning. Too much emphasis has been laid on the formal examination, the " spectacular " effects according to Schultze. 1 Too 4 much is attempted in the time allotted, with insufficient assimilation of the matter studied. Pupils are not taught how to study mathematics. They are only drilled on abstract formulas. The result is an overdeveloped memory and undeveloped powers of reasoning.

Because of the above noted unsound pedagogical methods, with the resulting formal examinations, and because the

1 Arthur Schultze, "The Teaching of Mathematics in Secondary Schools"; The Macmillan Company, 191 2.

pupils are graded chiefly on mechanical ability, their progress in mathematics can be determined to a highly refined nicety. They have failed to " do " a certain number of problems. Ergo, they are just that much deficient in ability and improvement. There is no leeway for difference of opinion, for the exercising of the reasoning faculty, for the training of individual characteristics and differences. Being largely a fact subject, as now taught, it resolves itself mainly into a question of " yes " or " no," and this accentuates the probability of failure. Individuals differ vastly in their ability to memorize, and therefore the poor memorizer is placed at a disadvantage. The pupil who can reason out a new demonstration in geometry knows infinitely more geometry than he who can transcribe on paper every one of the prescribed demonstrations in a book on this subject.

The Value and Place of Supervised Study. — This leads us logically to a discussion of the value of supervised study in mathematics. Unsupervised study is inefficient study because much time and energy are lost in misdirected effort. Pupils do not know how to attack a lesson any more than they know how to perform the mechanical processes, until they are carefully taught. Class exercises avail little for the majority of the pupils because no two minds react in the same way. To clinch the class exercise individual guidance is required. The unsupervised recitation as a rule does not provide for this. Problems in algebra and originals in geometry are entirely dependent upon the characteristics of the individual mind, which can be developed and trained only through the individual himself. To quote from an article by the author, 1 " the school must teach its pupils not to be perfect

1 Journal of the New York State Teachers 1 Association, November, 19 18.

automatons, responding with machine-like accuracy to the whim of the examiner, but to become thinkers, with power and knowledge of how to attack and study out a problem, how to form personal opinions, how to get results, by themselves. This, then, is the function of supervised study: to properly direct the pupil in his work so that he may develop the best methods of attacking problems ; that he may avoid wrong methods of reasoning; that he may most efficiently employ his time ; and that he may eventually acquire a power of skill that will classify him as a finished thinker, an educated man."

Supervised study is only one of the several methods that need to be employed in bringing about a closer relationship between teacher and pupil and in the development of the pupil's native endowment in the field of mathematics. Such relationship might be illustrated as spokes of a wheel. Just as every spoke (Figure I) is necessary in the connection between the rim which represents the pupil and the hub which represents the teacher, so supervised study should be given its proper position in the devices of the schoolroom. The other spokes, each with its peculiar evaluation, might be the recitation, the assignment, the equipment, tests and quizzes, standard tests and measurements, inspiration and sympathy.

Division of the Course into Units of Instruction, Recitation, and Study. — In our discussion of the technic of the supervised study of mathematics, let us first agree on our use of terms, as formulated by Professor Hall-Quest in his pioneer book, " Supervised Study " * and followed out in the other books in this series. In program of studies, let us include all the work offered in a school ; by curriculum, let us understand a 1 Hall-Quest, "Supervised Study"; The Macmillan Company, 1916.

group of subjects leading to a special end, as college preparatory curriculum, domestic science curriculum, etc. ; and by course, any single subject as algebra, civics, etc. Then, as a means of evaluating the course and giving it a definite and

Figure I comprehensive development, we shall separate the subject matter into various units. By units of instruction, we shall mean the large topics around which the material revolves. In many cases these are the divisions noted in the table of contents ; they are the divisions of the subject into " type lessons." 1 Such general topics as percentage, banking, factoring, graphs, would thus become units of instruction. Then the subdivisions of these larger units into smaller ones, around which one or more recitations would revolve, may be called units of recitation. These units may again be subdivided into the work planned for a single day, or units of study.

Types of Recitation. — In addition to this analysis of any course of study into its various units, the careful teacher will further decide on the technical form of presentation of each unit of study, or the type of lesson. Following the treatment of this phase as detailed by Strayer 2 and Earhart, 3 we may employ, as occasion prompts, the deductive, inductive, habituation, expository, how to study, socialized, or review lessons. Since each type, however, has its peculiar aim and technic, the teacher will do well to make a careful study of them and of their application to the subject in hand.

In general terms, the deductive lesson aims to draw forth new conceptions from our present knowledge. It is based on the process by which we think. The inductive lesson, on the other hand, leads up to new concepts by a series of successive steps, each definite and complete in itself. It is the process by which we accumulate knowledge. In the drill or habituation lesson, the mechanical side of learning is stressed. By drill, needful automatic reactions are established. The expository lesson seeks to make the new assignment as clear as demonstration and analysis make it possible. It is usually employed as a connecting link between the old and new mate-

1 McMurry, "How to Study" ; Houghton Mifflin Co., 1909.

2 Strayer, "Brief Course in the Teaching Process" ; The Macmillan Company, 1912.

3 Earhart, "Types of Teaching"; Houghton Mifflin Co., 1915.

rial of a unit of study. The how-to-study lesson is self-explanatory. Under review are usually included written or oral examinations. In a larger sense the review should cover the bringing together for periodic consideration, the clinching of a unit of recitation or unit of study. The socialized lesson may be poorly named from a standpoint of nomenclature ; possibly a better term at the present time would be the democratized lesson. At any rate, it is that type of lesson which introduces the human element into the school work. By it the work outside and that inside of the schoolroom are correlated. Although this type of lesson may be used sparingly, it is none the less important, and the teacher should feel that his greatest opportunity for reaching the consciousness of the child is presented through this form of lesson organization. Group assignments, dramatic productions, class programs, dual projects, mathematical clubs, and like devices for impressing the interdependence of individuals in solving social or economic problems, will tend to vitalize and democratize the subject matter. A more elaborate classification of exercise types as applied to the teaching of high school mathematics has been evolved in an illuminating article by Professor G. W. Myers, of the College of Education of the University of Chicago, in High School Mathematics and Science, June, 1921. In this article, the following types of mathematical class exercises are discussed, and specifications or norms are given for judging each :

I. The conceptual type. VII. The problem type.

II. The expressional type. VIII. The topic type.

III. The associational type. IX. The applicational type.

IV. The assimilational type. X. The test type.

V. The review type. XI. The research type.

VI. The drill type. XII. The appreciational type.

io Supervised Study in Mathematics and Science

While space does not permit a detailed review of these, the article in question is commended to the reader for careful study. To treat it adequately here would be to quote it as a whole.

The Time Schedule. — In order that the plan of supervised study may be carried out in its finest application, the period should be long enough for the pupil to do most if not all of his studying in school. This would mean a period of from ninety minutes to two hours in length and would also involve an extension of the school day in most cases. Superintendent L. M. Allen x remarks that a shorter time than the above is " neither hay nor grass," and that less than forty-five minutes for the study period itself will not suffice. The author of this book will grant that the longer the period the better the results, but from the experience in his own school for the past five years, he is constrained to disagree with the above conclusion of Superintendent Allen. In the school at Canton the periods are all one hour in length, the first thirty-five minutes being devoted to the recitation and the assignment, and the last twenty-five minutes being given up to the study of the lesson for the next day. Realizing that a longer study period would be very desirable, the author knows from experience that even this length of time will justify itself by increased and better work, as shown from the statistics as applied to the Canton school.

It seems, therefore, only fair to conclude that, when it is impossible to increase the length of the periods to the limits suggested above, the installation of supervised study is still feasible and good results may be secured from the shorter period. In any case, the period will be divided into three School Review, June, 191 7.

Supervised Study Period in Mathematics n

parts: the review, the assignment, and the silent study. When the periods are sixty minutes in length, the approximate division of time among these three parts should be as follows :

The Silent Study 25 minutes

With a longer period, the study section will be increased more than the others. A sixty minute period means, of course, that only part of the study will be done in class ; a ninety or one hundred twenty minute period should make home work unnecessary.

The Review. — The review will take the place of the old recitation and, while its length has been decreased, by intensive and well-applied questions the work ought to be thoroughly covered in this time. The class review should essentially be a re-view of the difficult parts of the day's lesson ; and, while the weaker pupils will get the most attention, it will be a sort of summing up for all. It is unnecessary to review every minute step ; half of the usual recitation is spent in reciting on perfectly well-known and understood things. The review is not the time to show of what we know but to clear up the things we do not know or know only indistinctly. It should always be a real step forward, a sort of clearing house for the previous day's assignments. Again, the usual type of recitation is apt to be a kind of monologue with the teacher taking the leading part. As a matter of fact, the teacher should remain in the background. In war, the generals give orders but the rank and file does the fighting. The review, then, should be incisive, intensive, and conclusive. The entire class should not be held back by a few backward pupils. On the other hand, the bright pupils should not be conciliated by insipid questions. The teacher should address the review to the weaker ones but by methods that will appeal to all.

The Assignment. — The assignment is always the most important part of the recitation period. It should include, in addition to a definite allotment of new work, very clear explanations. The advance lesson should be carefully planned beforehand, so that there will be a definite amount of ground to be covered, a definite objective gained, and a definite advance made. If this is slurred over, the pupil will have no clear idea of what the lesson is about or of what he is to do. The assignment to be prepared should also be made with due consideration of its difficulty, the varying abilities of the members of the class, and the often overlooked fact that the pupils have lessons also in other departments. If all these elements have been taken into consideration and given thoughtful planning, the time allotted to this particular section of the period ought to be sufficient. But there must be plenty of time to cover the assignment fully and thoroughly; therefore, it is better to assign too little than too much. In any quota of problems or examples, special difficulties likely to be encountered should be pointed out and possible methods of attack suggested. The assignment is perfect only when every pupil knows exactly what is the aim of the new work, what is the best method for its solution, and just what ground he is expected to cover before the next recitation.

The Study Period. — The study section is the part devoted by the pupil to the study of the advance lesson, under the direct and sympathetic supervision of the teacher. In the sixty minute period used in the author's school, the pupils are not all expected to complete the assignment during the period. Some will, however, and it will be an incentive to all to strive to complete the work during the time allotted to study. At any rate, all will have been able to get a start and a start in the right direction.

The teacher will have two classes of pupils to look after during the silent study period. One will be those who have some little technical difficulty with the new lesson. A welldirected question will usually set them on the right path. The other division will be those who are commonly considered failures, but who in many instances are simply pupils with some individual characteristics which react unfavorably for maximum efficiency. These pupils should be carefully studied by the teacher and their cases diagnosed. The problems thus presented to the teacher should awaken all of his determination to solve them. The silent study period thus gives the teacher an opportunity for a study of the individual personnel of his class. The elimination of pupils from the list of failures should become the predominant effort rather than the elimination of pupils from class and school. Often a pupil, who would ordinarily have failed, has found himself through a little attention and study on the teacher's part during the study period. When the teacher can sit down with him, note his manner of work, detect his deficiences and weaknesses, and by a little tactful and sympathetic guidance, lead him into the paths of success, such a pupil will gain confidence and later economical independence. He must be taught how to walk — how to study.