QM\ V/ffH»WCnC6UBRAF«Y Jfout $ork Hate (^allege of Agriculture 2U QJorncll Mniucrattj) SItbrarti Date Due Jwzjssto I ^ kApr9 ! 57f Library Burea Cat. No. 1137 Cornell University Library QA 11.S85 Supervised study in mathematics and scie 3 1924 002 953 655 r— i Egg \ t\S% Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002953655 ^uptttusea £>tuD2 petite EDITED BY ALFRED L. HALL-QUEST SUPERVISED STUDY IN MATHEMATICS AND SCIENCE THE MACMILLAN COMPANY NEW YORK • BOSTON • CHICAGO - DALLAS ATLANTA - SAN FRANCISCO MACMILLAN & CO., Limited LONDON * BOMBAY ■ CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO SUPERVISED STUDY IN MATHEMATICS AND SCIENCE BY S. CLAYTON SUMNER, M.A. SUPERVISING PRINCIPAL, PALMYRA, N. Y. (Formerly at Canton, N. Y.) Nrfn fforft THE MACMILLAN COMPANY 1922 All rights reserved PRINTED IN THE UNITED STATES OF AMERICA S'oS Copyright, ig22, By THE MACMILLAN COMPANY. Set up and electrotyped. Published November, 1922. @ 3 ? G (, NotfaooS ffitss J. S. Cualiing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. Go tbe rtBemorg of MY FATHER WHO BELIEVED AN EDUCATION WAS THE RICHEST HERITAGE A PARENT COULD BEQUEATH TO HIS CHILDREN PREFACE Ix 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 typi- cal 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 men- tioned 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 administra- tion. 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 ines- timable 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 mathe- matics ; to my teachers at Canton, N. Y., who made it possi- ble 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 cour- tesy of The Macmillan Company, the American Book Com- pany 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, 192a. TABLE OF CONTENTS PAGE Introduction. Supervised Study a Moral Imperative . . The Editor xiii PART ONE. MATHEMATICS Chapter One. Management of the Supervised Study Period in Math- ematics 3 First Section. Algebra (Elementary) Chapter Two. Divisions of Elementary Algebra; Units of Instruc- tion and Units of Recitation. A Time Table . . 20 ILLUSTRATIVE LESSONS LESSON I. The Inspirational Preview 26 II. Introduction. Unit of Instruction I. A Lesson in Correlation 34 HI. Introduction (Continued). A How to Study Lesson . . 43 IV. Introduction (Continued). An Inductive and How to Study Lesson 50 V. Addition. Unit of Instruction HI. An Inductive Lesson: Addition of Monomials 59 VI. Addition (Continued). An Inductive Lesson: Addition of Polynomials 67 VH. Simple Equations. Unit of Instruction X. An Expository and How to Study Lesson : The Equation and Problems . 73 VIH. Factoring. Unit of Instruction VII. A Socialized Lesson . 80 ix x Table of Contents LESSOH PAGE DC. Fractions. Unit of Instruction EX. 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 Exposi- tory and How to Study Lesson 92 XHI. An Examination 9S Second Section. Plane Geometry Chapter Three. Divisions of Plane Geometry; Units of Instruction and Units of Recitation 105 LESSON I. The Inspirational Preview 106 H. Rectilinear Figures. Unit of Instruction n. A Deductive and How to Study Lesson : Vertical Angles . . .no HI. Rectilinear Figures (Continued). A Deductive Lesson: Tri- angles 117 IV. Rectilinear Figures (Continued). A How to Study Lesson: Originals 123 V. Rectilinear Figures (Continued). A Deductive Lesson : Orig- inals 129 VI. A Socialized Review : Book I 134 VH. An Exhibition or a Red Letter Day Lesson .... 135 Third Section. Advanced Mathematics Chapter Four. Special Methods of Supervised Study in Higher Math- ematics 141 Table of Contents xi PART TWO. SCIENCE PAGE Chapter Five. The Management of the Supervised Study Period in Science . , , 149 Fourth Section. Biology Chapter Srx. Divisions of Biology ; Units of Instruction and Units of Recitation. A Time Table 155 A. — BOTANY LESSON 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 VT. Seeds and Seedlings (Continued). A Deductive Lesson : Lab- oratory Experiments 175 Vn. Seeds and Seedlings (Continued). A Socialized Lesson: A Field Trip 179 B. — ZOOLOGY VHI. Insects. Unit of Instruction DC. A How to Study Lesson : The Grasshopper - 181 IX. Insects (Continued). A Laboratory Lesson: The Grasshop- per 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 LESSON I. Fluids. Unit of Instruction. An Expository and How to Study Lesson 214 H. Fluids {Continued). A Laboratory Lesson . . . .219 HI. 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 prac- tice. 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 edu- cation 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 suc- cessfully performed demonstration is of more value than many verbal descriptions, however clear these may be. A series of vivid verbal descriptions makes definite and con- crete 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 xiv Introduction by the Editor objective and easily understood the necessarily vaguer dis- cussions of abstract theory. This series of volumes on Supervised Study attempts to visualize one form of study supervision. Each book is writ- ten 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 em- phasized 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 effec- tively 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 eliminating society's mentally unfit, and that the surest way of such elimination is to assign les- sons, 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 f atalistic method ? There are children who early judge themselves incapable of school work. No- body cares ! They either can or cannot study. By means of the hard, soulless machinery of assigning and hearing les- sons they are cast out. We call this a safe test and out they go labeled mentally weak, unfit to partake in a world of thrill- ing 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 knowl- edge, 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 con- ducted 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 dis- penser 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 par- ticular 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 in- creasingly 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. PART ONE MATHEMATICS SUPERVISED STUDY IN MATHEMATICS AND SCIENCE 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 es- tablished 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 mathe- matics for the past five years have been between thirty- three 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 in- clusive), New York State. 3 4 Supervised Study in Mathematics and Science 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 mathe- matics 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 em- phasis has tended toward the serious neglect of training pupils to apply mathematical rules and formulas to practical reason- ing. Too much emphasis has been laid on the formal examina- tion, the " spectacular " effects according to Schultze. 1 Too much is attempted in the time allotted, with insufficient as- similation 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, 1912. Supervised Study Period in Mathematics 5 pupils are graded chiefly on mechanical ability, their prog- ress in mathematics can be determined to a highly refined nicety. They have failed to " do " a certain number of prob- lems. 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 prob- ability 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 demon- stration 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 be- cause 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 major- ity 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 individ- ual 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' Association, November, 1918. 6 Supervised Study in Mathematics and Science 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 them- selves. This, then, is the function of supervised study: to properly direct the pupil in bis 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 be- tween 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 super- vised 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 'Hall-Quest, "Supervised Study"; The Macmillan Company, 1916. Supervised Study Period in Mathematics 7 group of subjects leading to a special end, as college prepara- tory 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 FlGUKE 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 8 Supervised Study in Mathematics and Science lessons." x 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 sub- divided 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, ha- bituation, 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 habitua- tion lesson, the mechanical side of learning is stressed. By drill, needful automatic reactions are established. The ex- pository 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- ' 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. Supervised Study Period in Mathematics 9 rial of a unit of study. The how-to-study lesson is self-explan- atory. 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 proj- ects, mathematical clubs, and like devices for impressing the in- terdependence of individuals in solving social or economic prob- lems, 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 dis- cussed, 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 care- ful 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 1 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 Review 15 minutes The Assignment 20 minutes 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 inten- sive and well-applied questions the work ought to be thor- oughly covered in this time. The class review should es- sentially 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 re- view 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 off 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 12 Supervised Study in Mathematics and Science 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 thought- ful 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; there- fore, 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 Supervised Study Period in Mathematics 13 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 well- directed 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 deter- mination 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. The Assignment Sheet. — The assignment sheet used in Canton, which is similar to the one described by Miss Simpson in her companion book in this series, 1 is reproduced at the end 1 " Supervised Study in History" ; The Macmillan Company, 1918. 14 Supervised Study in Mathematics and Science of this chapter. The object of the sheet is to induce the teacher to have a definite plan for each day's work. Work not carefully planned is apt to be poorly done. Nothing begets carelessness and indifference on the part of the class so much as a lack of purpose and plan on the part of the teacher. These sheets need take scarcely any time ; in fact, they will save time, because the teacher will know exactly what he is going to do, what material he is going to use, and where it is to be found. In addition thereto, he will have at the close of the term a complete record of work accomplished. How to Make a Lesson Plan Sheet. — Under Review note exactly the things that need to be reemphasized. The ref- erences to other books for supplementary material may be noted under Memoranda. A good scheme is to write on the back of the assignment sheet the names of the pupils who should receive especial attention during this review. The Threefold Assignment. — As Professor Hall-Quest explains in his book, the assignment should be in three parts : one to take care of the inferior pupil, one to take care of the average pupil, and one to take care of the superior pupil. Hence we have the three part assignments, or the minimum- average-maximum plan. The minimum assignment should cover the minimum es- sentials, i.e. the work that all must do at the very least, and that the majority can easily do within the twenty-five minutes allotted to study. It should be so planned that pupils who never do any more than this amount will be able to pass the final examination, which is or should be the minimum require- ment, but too often becomes the only aim of the teacher. These pupils will not obtain a high mark, but they will have mastered enough of the subject to get a passing grade. The Supervised Study Period in Mathematics 15 object of the average and maximum assignments is to produce pupils who will not only pass but pass high. The average assignment should include more examples and different kinds of examples but not necessarily of a much harder nature. However, the more problems a pupil can solve correctly in a given time, the more skilled he will be- come, up to a certain limit. The maximum assignment is to take care of the brighter pupils, — those who are able to do more than the average pupil and who should have some incentive to do advanced work. Usually this assignment will be given from other texts and will consist of more difficult material. This section should be so limited in amount as not to discourage but to incite to a desire to cover it. It will incidentally keep a disturbing pupil out of mischief. But the tasks should be constructive and not simply the old fashioned " busy " work which employs but does not develop. Under Study note any points to be kept in mind while the pupils are working, especially in regard to mistakes they are likely to make. This section of the sheet may be made very effective, if the teacher is fully alive to the function of the supervised study idea. In the course of a year this plan will have become a real series of methods. The loose-leaf notebook makes an excellent method of preserving the sheets. They will be found of inestimable value another semester. How to Use the Assignment and Study Sheet. — Do not become a slave to the sheet; make it your servant. If it means omitting your daily recreation, eliminate it, not your recreation. It ought not to take much time and, as has been suggested, it will eventually be found to save time. 1 6 Supervised Study in Mathematics and Science Strive always to inspire the class to reach the maximum assignment, to raise the maximum and lower the minimum end. Determine the proper proportions of pupils who should get the various sections. The following is a normal distribution : 80 per cent should complete the average assignment, 10 per cent the minimum only, and 10 per cent the maximum. When these percentages vary greatly from the established norm, the nature and length of the assignments should be modified proportionately. That is, in a class of 30 pupils, 24 should do all the average assignment before the end of the period. Three will be behind and will need special attention next day ; three will be working some of the examples in the maximum assignment. This of course is subject to varia- tion from day to day, but may serve as a guide. At the close of the period, the teacher should ascertain how many of the class have completed the minimum, the average, and the maximum assignments. This may be done by having all hand in their papers a few seconds before the period ends. The pupil may note on his paper which assignments he has completed. What remains of the assignment may be required of the pupils the next day. Various schemes may be evolved ; some will be touched upon in connection with the illustrative lessons in Part One. The assignment should be written each day upon the board and numbered I, II, III to correspond with the different assignment groups. The pupils need not be told the reason for this as it will be better if they do not understand the reason for the differentiation. The pupils should form the habit of noting down the assignment, thus getting practice in keeping a written record of important facts and engage- ments. Supervised Study Period in Mathematics 17 The Management of the Supervised Study Period. — As soon as the class meets, take the roll call and at once start the review. The time spent in taking the roll should be reduced to a minimum ; certainly not over two minutes at the most. An excellent method is to have on a sheet of paper a diagram representing the seats and in each space, which represents a seat, put the name of the pupil who occupies it each day. Then those absent can be quickly noted through the vacant spaces, and their names checked. The time allotted to the review and the assignment during the first thirty-five minutes should be held to as closely as possible ; a schedule or time table is absurd unless adhered to conscientiously. At the end of the thirty-five minutes, a bell may be rung simultaneously in all rooms from the study hall or some other central place, and the next two minutes devoted to physical drills, setting-up exercises, etc. Then, for the next twenty-five minutes or fraction thereof, the pupils should be required to work on their new lesson. The teacher should insist that no other work be done during the study period, until the lesson in hand is finished. At first, some may try to study the succeeding lesson but a little firmness will soon bring desired results. As soon as possible after the class has been organized, it may be expedient to reseat the pupils according to their abili- ties ; those of minimum ability on one side of the room, those of maximum ability on the opposite side, and the average pupils between. This allows the teacher to come into closer contact with the weaker pupils, and he can give them addi- tional attention. This classification must be done with tact so as not to hurt any child's feelings ; therefore, the earlier in the term it is done the better. It is not necessary to name 18 Supervised Sttidy in Mathematics and Science the groups. Calling them A, B, C without any further char- acterization will suffice. Some method of checking the results of the work accom- plished during the period should be worked out. Various schemes will be mentioned in connection with the illustrative lessons ; but a few general remarks may be made here. Any plan which would depend entirely on the amount of work done in class must take into consideration the fact that some children are accurate but slow ; these must not be discouraged by low marks. While rapidity is desired, accuracy is more important, and these pupils should be encouraged to make an effort to maintain accuracy and secure greater rapidity. On the other hand, it is very important that as much of the work be done in class as practicable so that the teacher may know that the work is the pupil's own. Thus there is presented to the teacher a fine question for analysis : to discover why a pupil does not accomplish so much as is expected. The teacher should be untiring in his effort to solve such a problem. Theoretically the pupils who habitually solve only the first or minimum assignment should receive a mere passing grade, those doing the average assignment should receive marks between 80 and 90, and those doing the maximum amount should receive honors. But this must eventually be deter- mined by the ability of the pupil to work similar exercises in the review, by his lack of dependence upon the teacher during the study period, and to a lesser extent by the periodic test. Supervised Study Period in Mathematics ASSIGNMENT AND STUDY SHEET Subject .Period .Date Unit of Instruction Unit of Recitation Unit or Study Lesson Type 19 Review: Memoranda Assignment : 1. Minimum 2. Average 3. Maximum Study: Number of pupils solving minimum assignment... Number of pupils solving average assignment Number of pupils solving maximum assignment Total. Figure II FIRST SECTION. ALGEBRA CHAPTER TWO DIVISIONS OF ELEMENTARY ALGEBRA It is advisable in any course of study that the teacher have a definite outline of the successive stages in the development of the subject, and that these be further subdivided into their smaller units. These general topics may be called Units of Instruction and the subtopics, Units of Recitation. The course of study here outlined is that suggested by the New York State Education Department in its 1910 syllabus. A. Units or Instruction. I. Introduction. II. Positive and negative numbers. III. Addition. IV. Subtraction. V. Multiplication. VI. Division. VII. Factoring. VIII. Common factors and multiples. IX. Fractions. X. Simple equations. XI. Graphic representation. XII. Involution. XIII. Evolution. XIV. Radicals. XV. Quadratic equations. B. Units of Recitation. Divisions of Elementary Algebra 21 The subdivisions do not imply that only one recitation is to be given to each topic, but rather that all the recitations for the length of time needed will center around this special topic. I. Introduction. The material given under this head varies in different texts, but it will at least contain : Units of Recitation : 1. Symbols of algebra. 2. Literal numbers. 3. Historical notes. 4. Definitions and notation. n. Positive and Negative Numbers. Units of Recitation : 1. Explanation and illustration of signed numbers. 2. The addition, subtraction, multiplication, and division of signed numbers. HI. Addition. Units of Recitation: 1. Addition of monomials. 2. Addition of polynomials. IV. Subtraction. Units of Recitation: 1. Subtraction of monomials. 2. Subtraction of polynomials. 3. The parenthesis. V. Multiplication. Units of Recitation: 1. Multiplication of monomials by monomials. 2. Multiplication of polynomials by monomials. 3. Multiplication of polynomials by polynomials. 4. Special cases. 22 Supervised Study in Mathematics and Science VI. Division. Units of Recitation: i. Division of monomials by monomials. 2. Division of polynomials by monomials. 3. Division of polynomials by polynomials. VII. Factoring. Units of Recitation: 1. To factor a monomial. 2. To factor a polynomial. 3. To factor a polynomial whose terms may be grouped to show a common polynomial factor. 4. To factor a trinomial which is a perfect square. 5. To factor the difference of two squares. 6. To factor a trinomial in the form of x 2 +px+q. 7. To factor a trinomial in the form of ax 2 +bx+c. 8. To factor the sum or difference of cubes. 9. To factor the sum or difference of the same odd powers of two numbers. 10. To factor the difference of the same even powers of two numbers. 11. To factor by the factor theorem. 12. To factor by special devices. VIII. Common Factors and Multiples. Units of Recitation: 1. Highest common factor. 2. Least common multiple. IX. Fractions. Units of Recitation : 1. Reduction to higher or lower terms. 2. Reduction to an integral or mixed expression. 3. Reduction to similar fractions. Divisions of Elementary Algebra 23 4. Addition of fractions. 5. Subtraction of fractions. 6. Multiplication of fractions. 7. Division of fractions. 8. Complex fractions. X. Simple Equations. Units of Recitation : 1. Equations with one unknown. 2. Equations involving fractions. 3. Literal equations. 4. Problems. 5. Simultaneous equations. a. Elimination by addition or subtraction. b. Elimination by comparison. c. Elimination by substitution. 6. Literal simultaneous equations. 7. Problems. 8. Equations with three or more unknowns. 9. Problems. XI. Graphic Representation. Units of Recitation: 1. Graphs of statistics. 2. Graphic representation of linear equations. XII. Involution. Units of Recitation: 1. Involution of monomials. 2. Involution of polynomials. 3. Involution by the binomial theorem. Xm. Evolution. Units of Recitation: 1. Evolution of monomials. 2. To extract the square root of polynomials. 24 Supervised Study in Mathematics and Science XIV. Radicals. Units of Recitation: i. To reduce radicals to their simplest form. 2. To reduce a mixed surd to an entire surd. 3. To reduce radicals to the same order. 4. Addition and subtraction of radicals. 5. Multiplication of radicals. 6. Division of radicals. 7. Involution and evolution of radicals. 8. Rationalization. 9. Square root of a binomial quadratic surd. 10. Radical equations. XV. Quadratics. Units of Recitation: 1. Pure quadratic equations. 2. Affected quadratic equations. 3. Solution by a. Factoring. b. Completing the square. c. The formula. 4. Literal quadratic equations. 5. Radical equations in quadratics. 6. Problems. 7. Simultaneous equations involving quadratics. a. One simple equation, one involving the second degree. b. Two homogeneous equations of the second degree. c. Symmetric equations of third or fourth degree, readily solvable by dividing the variable member of one by the variable member of the other. 8. Problems. Divisions of Elementary Algebra 25 Time Table for the Term. — Below is a suggested time table for the year's work : ALGEBRA : 40 weeks First Term 1 st week Introduction 2d week Addition 3d, 4th, 5th weeks Subtraction and parenthesis 6th and 7th weeks Multiplication 8th and 9th weeks Division and review 10th to 15th week Factoring 16th week Common factors and common multiples 17th to 20th week Fractions Second Term 21st to 24th week Simple equations 25th week Graphs 26th and 27th weeks Involution 28th and 29th weeks Evolution 30th to 34th week Radicals 35th and 36th weeks Quadratics 37th to 40th week Review Factors Modifying the Foregoing Arrangements. — Local conditions will necessarily determine the emphasis to be placed on the respective units of subject matter. If a course in general mathematics is followed, it is obvious that considerable modifications will be required. Some classes are more mathe- matically-minded than others and this fact should affect emphasis. Whatever units are found to be rarely of value even in advanced studies of mathematics should be practically ignored. There is not time in public school work for the elaboration of the useless. Many textbooks in mathematics are now so arranged that certain designated sections may be omitted without affecting the continuity. 26 Supervised Study in Mathematics and Science LESSON I THE INSPIRATIONAL PREVIEW Purpose. — The purpose of such a lesson is to arouse in the child the will to learn, to awaken interest in the study of algebra, and to outline in a general way some of the things of interest which will be studied during the course. Need. — Coming from the grades with no very clear idea of what high school means and assigned to new subjects, the very names of which are often strange, it is no wonder that the young pupil is not only lacking in any particular interest in his new work, but may even have a natural antipathy toward it from the start, unless interest be aroused through this inspirational preview. It is important in meeting any class for the first time that the teacher get en rapport with the pupil as quickly as possible. It is well in place to give a simple talk on the history, practical value, and general content of the subject, — a sort of advertis- ing or " selling" talk. The careful traveler will always plan his trip ahead in order that he may be prepared to note all the important and interesting things that may He in store for him ; otherwise, many things would escape his attention. So the preview is a sort of bird's-eye view of the course, a cranking up of the pupil's interest, preparatory to a good start and a run. Method. — Simple language should be used. The class is composed of immature boys and girls to whom big words and phrases mean little; the talk should be more to the child than about the subject. The essential thing is to make the children feel at home, to arouse enthusiasm for the subject, and to make them look forward to their work in algebra with pleasure. Divisions of Elementary Algebra 27 When a strange word or important date is given, it will have more effect upon the class if it is written upon the blackboard at the time that it is mentioned. Prearranged work upon the board, simply referred to in passing, does not rivet their attention so well. When the preview is completed, it might be well to ask a few questions concerning what has been said and give, if neces- sary, any further details needed to make the subject clear. Ask the pupils to jot down the data you have placed upon the board and tell them to hand in the next day a simple state- ment of what has been said. This is not so important from the standpoint of the material as it is from the standpoint of inculcating at once the necessity of paying attention, of being specific regarding facts ; and of the implication that mathe- matics is closely correlated with English in the clearness of its exposition. Historical. — A little of the history of the subject will arouse the pupils' interest in the age and romance of algebra. It is best not to go much into detail because of the complexity of its historical development. It will be better also to intro- duce each new topic, when studied, by its individual history. Many of the recent texts in algebra have historical notes scattered through the book as a help to humanizing the sub- ject. Pictures of famous mathematicians add to the interest of these historical references. A framed portrait of one or two mathematicians of note, or a statuette, placed in the room, will give an added atmosphere to the study of the subject. Origin of the Word "Algebra." Its name is derived from the title of a book which the Arabs introduced into Europe in the ninth century. The full title of the book was " Al-jebr w' al muqabalah," of which the first two syllables have been cor- 28 Supervised Study in Mathematics and Science rupted into the present term, algebra. In the original tongue it referred to the process of transposing terms in the manipu- lation. Hence the solving of problems was early considered the main business of this subject. Contributors to the Subject. Modern algebra has not come down to us in its present composition, but like the automobile and every other invention, it is the result of years of growth and of contributions of many minds. The early Egyptians and the ancient Greeks of the "golden age" had some con- ception of the equation and have left their imprint upon its development. Heron of Alexandria about ioo b.c. made the greatest advance in its development up to that time, but Diophantus, a fourth-century Greek, was the first to write an entire book upon the subject. He emphasized indeterminates, which are even now called Diophantine after him. But as stated above, the book whose title has given the subject its modern name was the first general treatise of importance. The modern founder of algebra was a Frenchman, named Vieta, who, in 1591, gave the science the technical symbolism which is in use to-day. Among other modern contributors were Wessel of Norway, Gauss of Denmark, and Sir Isaac Newton of England. Interesting Incidents. The history of the subject abounds in many interesting episodes. It is said 1 that Sir William Hamilton, an Englishman, who had been working for years on a certain problem, was one day taking a stroll with his wife, when the solution suddenly flashed into his mind, and he at once engraved upon a stone in Brougham Bridge, which he was crossing at the time, one of the fundamental formulas of 1 "Science-History of the Universe," VIII; Mathematics, Current Litera- ture Publishing Company, 191 2. Divisions of Elementary Algebra 29 modern algebra, called quaternions. This bridge has ever since been called Quaternion Bridge. An Italian named Tartaglia, who claimed a certain alge- braic discovery, was challenged by another famous mathe- matician by the name of Fiori. The contest was to see which one could solve the greatest number of a collection of thirty problems within thirty days. Tartaglia, by using his new discovery, which by the way is now a matter of common knowledge, i.e. the cubic equation, solved the entire thirty within two hours' time. He celebrated his triumph by com- posing some verses, but, according to the custom of the times, he kept his discovery a secret for many years. Practical Value of Mathematics in General and Algebra in Particular. — From the very earliest times, the study of mathematics has been considered primarily practical. Mathe- matics has been, in some form or other, the bulwark of the education of the Chinese, the Arabian, the Assyrian, the Jew, the Greek, the Roman, and every modern race. One might as well try to conceive of life without atmosphere as to try to separate the influence of mathematics from human life and endeavor. Dr. Eugene Smith of Columbia University says that " if all mathematical knowledge were eliminated, civilization would be demoralized, factories would stop for want of ma- chinery, and life would revolt to chaos." * This is the era of machinery. A machine, mathematically wrong, is a failure. Long before the machine is put on the market, however, it has been the subject of painstaking effort on the part of the inventor, the draftsman, the pattern- 1 "Mathematics w> Training for Citizenship"; Teachers College Record, May, 1917. 30 Supervised Study in Mathematics and Science maker, the mechanic, and the promoter. Each one has applied his knowledge to his labor, and the finished product is the re- sult of the accumulated researches and experiences of these men in turn. Imagine the invention of the steam engine, the sewing machine, the typewriter, the adding machine, the lathe, the printing press, the automobile, the airplane, with- out an expert mathematical training on the part of the inventors. The recent war with its wonderful though terrible inventions, which sprang into being from all sides, illustrates most forcibly the value of mathematical ability, because every machine from the gas mask to the submarine involved the exercise of mathe- matical genius. From the moment of its first conception in the mind of the inventor to the actual firing of the first shot on the battlefield, the giant field gun is a product of mathe- matics. The battle of Messines Ridge was won by engineers who skillfully tunneled the hills. Dr. Nichols, of the Uni- versity of Virginia, contributed to our wonderful shipbuilding exploit by successfully solving an equation to the ninth degree. 1 Moritz shows our commercial dependence on mathematics. Thales, the ancient mathematician, was by profession a mer- chant, and yet he studied mathematics for the intrinsic value it held for him. Such modern business problems as equation of payments, theory of interest, valuation of debenture bonds, amortization of interest-bearing notes, life insurance mor- tality tables, distribution of dividends, casualty insurance, and a thousand others, have their solution through the ap- plication of pure mathematics. Statistical work, which is used in hundreds of transactions and enterprises of every kind, is all mathematics. Business executives say they prefer 1 C. E. White in School Science and Mathematics, January, 1919. Divisions of Elementary Algebra 31 mathematically trained men because they are more methodical, exact, and resourceful, and, therefore, efficient. 1 Algebra Has an Important Position. From a strictly utilitarian standpoint, algebra has a firm claim for an im- portant position in our program of studies. It offers the only means of solving many of the problems connected with engineering, architecture, navigation, surveying, meteorology, geology, physiology, and even psychology, if we accept the Weber-Fechner law, 2 astronomy, physics, chemistry, and many other branches. No mechanic or artisan can read intelligently his trade journal, a technical book, an article in the encyclopedia, without a knowledge of the universal language of algebra. There are 27,000 volumes in the Naval Observatory at Washington which absolutely require a mathematical training for their perusal. 3 Statistics of all kinds are given in equations or mathematical formulas. The graphical treatment is employed by the economist, the business expert, the physician, the dietitian, and men of hundreds of other professions. The handling of pig iron in the modern foundry is a result of long continued analytical experiments based on algebraic formulas. Algebra Is Necessary to Secure a Higher Education. It is indispensable to the future student of astronomy, physics, chemistry, and higher mathematics. The modern engineering world and all technical schools are forever closed to him with- out a mastery of this subject. Just the other day the writer had an illustration of this. A young man who had been 'R. E. Moritz, "Our Relation of Mathematics to Commerce"; School Science and Mathematics, April, iqiq. 2 C. E. White in School Science and Mathematics, January, 1919. ' 3 Schultze," The Teaching of Mathematics"; TheMacmillan Company, 1912. 32 Supervised Study in Mathematics and Science recently graduated from high school found that it was possible for him to go to college but he had been allowed to go through this school without algebra and geometry, and he found to his dismay that he was unable to matriculate because of this deficiency in his education. Many a young man seems for- ever doomed to a nominal wage and a position near the foot of the ladder because he failed to fit himself for higher positions by the careful study of mathematics. A noted inventor once said that he had no use for algebra 01 higher mathematics, but in the next breath he admitted that he hired experts to work out for him all problems involving advanced mathematics. Mathematical physics and mathe- matical chemistry are important branches of science to-day, and every phase of electrical engineering is intimately bound up with algebra and its manipulations. The writer once knew a man who lost a fine position as assistant contractor because he did not know algebra, al- though he was an expert workman. His would-be employer felt he could not afford to have a man in this responsible position who was ignorant of this subject, as occasions some- times arose which demanded a knowledge of it. A man must know algebra to advance in the automobile business. A modern blue print with its mechanical and mathematical symbols looks like a Chinese puzzle to the lay- man, but to the trained mind of the skilled mechanic it is as plain as the printed page. And so we might multiply the examples of the practical value of algebra indefinitely, but enough has been said to arouse in the mind of the pupil the strong suggestion that here is a subject which has a bearing, both directly and indirectly, upon his future as well as upon the progress of the world. Divisions of Elementary Algebra 33 Bird's-eye View of the Course. — A few minutes might profitably be spent in outlining the semester's work. Tell the class that there will be a certain amount of formal work, similar to that done in arithmetic, such as learning how to add, subtract, multiply, and divide algebraic terms and quan- tities, finding factors, reducing fractions, and manipulating equations, in order that we may arrive at the solution of problems. Explain that this is necessary in order to become familiar with the tools of algebra, just as the carpenter must learn how to use the hammer, the saw, the square, and the compass, before he can build a house. Explain that from time to time, as a topic is finished, we shall have a sort of field day, when we may exhibit our work, give special reports of certain phases of the subject, and invite our friends to see the things we have accomplished. Tell the pupils also that we shall occasionally have contests, in which we shall choose sides to see which side can win. This may be done with fine results, as will be illustrated later, upon the completion of the work in factoring. Again, explain that during the year the class will learn how to solve various problems which will be drawn from business, agriculture, the vocations, etc. Tell them that they them- selves will also be expected to make up and solve problems, and that data drawn from current events, like an election or a ball game, will be used. Conditions for a Successful Preview. — Too much must not be attempted in this first lesson. The word " inspirational " very graphically illustrates the underlying motive for this lesson. The preceding material may well be added to or in part eliminated, as seems best in the judgment of the teacher himself. It is given simply to suggest some of the things that 34 Supervised Study in Mathematics and Science may make this first meeting of the class an inspiration and a forward look. Illustrations concerning the practical value will have more force if they have come under the actual observation of the teacher. Indeed, the teacher of algebra may well keep a note- book in which such material and experiences may be collected and added to from time to time. Such a teacher will soon accumulate a valuable set of illustrations, and one that will have variety and modern application. The historical notes will also vary with the teacher ; he must ever be on the alert for interesting incidents to use in this connection. Such material, aside from the cases which come under his own observation, will be found in magazine articles, newspaper articles, and the experiences of others and will be drawn from the pupils themselves. LESSON II UNIT OF INSTRUCTION I. — INTRODUCTION Lesson Type. — A Lesson in Correlation Program or Time Schedule The Review 5 minutes The Assignment 30 minutes The Study of the Assignment 25 minutes Purpose. — The purpose of this lesson is threefold: (1) to gain the confidence and free expression of the pupil by getting him to do something he knows how to do ; (2) to review and reemphasize some of the fundamental operations ; and (3) to link together arithmetic and algebra by showing their inter- relationship. Divisions of Elementary Algebra 35 The Review. — Since this is the first review work of the year and there has been no assignment, a few questions like the following might be asked, suggested by the " Inspirational Preview " : How did we get the word algebra ? Name some countries which have contributed to the devel- opment of this subject. For what is Vieta important ? Relate an interesting incident connected with the history of this subject. Name some professions which presuppose a knowledge of algebra. The Assignment. 1. Information given by the teacher regarding the (a) func- tion and (b) applications of algebra. 2. Review of the fundamental processes in arithmetic. 3. Recognition of the interdependence of algebra and arithmetic. The Function of Algebra. Sir Isaac Newton called algebra " Universal Arithmetic." Comte defined arithmetic as the sci- ence of values and algebra as the science of functions. Alge- bra deals historically and primarily with number. Primeval man, desiring to count his possessions, used various forms of tallies. The ancient Roman used the pebble, setting aside one for each article counted. The ten fingers or digits were early used in counting, thus giving us the term used in numeration, and later evolving into the ten digit or decimal (i.e. decern, ten) system. Gradually numbers became of interest because they allowed combinations. We therefore use symbols to stand for or represent things, later substituting the thing itself. Thus we are enabled to derive a general statement which may be applied to all similar cases. 36 Supervised Study in Mathematics and Science We have done something of this in our arithmetic when we used the statement BXR equals P. Any problem in percentage may be reduced to this formula or some variant of it. When the substitutions are made, the problem may be solved. As in the case of percentage we have used the first letter of the word indicated, so in algebra we also represent the unknown by letters. But in that case, not knowing what the words may be, we do not try to use the initial letter. It has become customary to employ the least used or last letters of the alphabet, x, y, and 2. We find, therefore, that algebra becomes a general science while arithmetic remains a particular science, and, though they may be said to resemble each other in some respects, in reality algebra becomes a new and more general method of manipulating quantities. Applications of Algebra. The fundamental processes of addition, subtraction, multiplication, and division of whole numbers and fractions form the foundation stone of the formal work in algebra as well as in arithmetic, but we shall find that their functions and applications in this new science reach a breadth which is impossible in arithmetic. In solv- ing problems in arithmetic we are always limited to things concrete, but in algebra, through the use of the unknown, we are at liberty to sweep the whole field in our solution. It is for this reason that algebra assumes a universal value of its own. Review of the Fundamental Processes in Arithmetic. — A half hour's time may well be spent in reviewing the funda- mental operations in arithmetic and in bringing out the errors which are very common to most first year pupils, such as the product when multiplying a number by unity or zero, the manipulation of simple fractions, mixed numbers, reduction Divisions of Elementary Algebra 37 of a fraction to unity in cases like f =1, cancellation, etc. It will amaze one to find out the number of pupils who, sup- posedly ready for algebra, cannot do correctly some of the simplest fundamental operations in arithmetic. It will sur- prise not only the teacher but the pupil as well, since he is apt to feel that, having passed the final preacademic examina- tion in this subject, he is somehow endowed with a supreme and unfailing knowledge of arithmetic for all time. The teacher may apply some of the operations of arithmetic to algebra, but it is best not to do too much of this at first. For instance, after the fraction i has been reduced to 1, we might take 4r an d show that it reduces to ix, or after a review of the multiplication of fractions, as iXi=i, we might show thatiXf=f, keeping in mind that we are not at present interested so much in the teaching of algebra as we are in showing that our present knowledge of arithmetic will be of constant help to us ; and in emphasizing that we are not going to work in a strange land but will have with us old friends. Interdependence of Algebra and Arithmetic. The nomen- clature is similar to that of arithmetic and even the mechani- cal method of setting down the problem, as in division, will be the same. We have already used the simple equation in the lower grades when in teaching addition we give such problems as " two and what make five? " In arithmetic we call this the Austrian method, but in algebra it becomes finding the un- known. We have learned that we cannot add together 5 apples and 3 oranges except to say that we have 5 apples plus 3 oranges. So in algebra we cannot add 5a and 3& except by indicating Sa plus 36. 38 Supervised Study in Mathematics and Science There is an intimate relation between exercises in removing the parentheses in arithmetic and algebra, such as (18 — 2 -^4X2), in which the order of performing the operations and rules for removing the signs of aggregation are identical. In the use of the question mark to indicate what is desired, we have in arithmetic simply anticipated the use of the un- known x in algebra, in such an example as this : 8 8 8 The graph is used in many exercises in arithmetic to show data which are used in the problem. These statistical graphs are a very important phase of elementary algebra. (See Unit of Instruction XI, Chapter Two.) We have already referred to the use of the formula in arithmetic. In addition to BxR equals P, we have a number of others relating to problems in interest, as P equals I-r-($iXrXt). Again, in measurement we have A equals bXh; circumference of a circle, or 0, equals irXD. Many texts in arithmetic also to-day have a section covering simple linear equations of one unknown which are solved algebraically. The Study of the Assignment. — I or Minimum Assignment. Ten or more examples to illustrate the common errors made in arithmetic as suggested in the paragraph on fundamental processes. 1. 48X0= 7. i+i+H 2. oXio£= 8. 4|~2i = 3- 4|XStV= »■ 12-4X3+6 = 4. A+f+2i= 10. 48-3* = 6. f-f= 11. 6. fXiXf= Divisions of Elementary Algebra 39 II or Average Assignment. Eight or ten examples to illustrate the similarity of arith- metic and algebra as explained in the paragraph on inter- dependence. 12. 32 and what make 50? 13. 17 and (x) make 25, what is a;? 14. 23 and 32 give? 15. 67 times 7 give (x) ? 16. 7 books plus 3 chairs plus 2 books plus 6 chairs equal (?) books plus ( ?) chairs. 17. s6+ 3 c+2&+ic = (?)6 + (?)c. 18. 3»+7x = how many x ? 19. If 6 = 5 and area = 120, find the altitude in formula, area = bXh. III or Maximum Assignment. 20. Mention some other similar cases of algebraic applications in arithmetic. 21. Give some formulas besides those stated, which you have had in arithmetic. 22. Bring to class some illustration of the use of the graphic means of showing statistics. 23. Report on the origin of symbols of operation. (Slaught and Lennes' "Elementary Algebra," p. 7, and Hawkes, Luby, and Touton's "First Course in Algebra," pp. 4, 5.) The Silent Study Period. — As soon as the time for the study period arrives, the pupils should commence work upon the lesson assigned for the next day. This assignment in three sections has been fully explained in Chapter One. The complete assignment should always be placed upon the board before the class assembles, so that there may be no delay in commencing to work. 40 Supervised Study in Mathematics and Science Since the exercises above suggested are simple, the pupils will probably not have much trouble, but they should under- stand that, in case anyone finds difficulty with the work, he has the privilege and is expected to raise his hand for aid. The teacher may then step quietly to his desk and find out what the difficulty is. It is important that the teacher and the pupil realize that the teacher is not expected to do the pupil's work for him. The teacher's part is to direct attention to his difficulty. The obstacle should be skillfully cleared up through the redirected effort of the pupil along the right path. If the point raised seems likely to be a stumblingblock to others, the teacher may step to the board and, calling the attention of the class to the difficulty, make necessary expla- nations which will at once be a help to all. It often happens that some little point was overlooked in the explanation before the class, which may now be made clear. Encourage the pupils to do as much of the assigned lesson as possible during the study period. The more the pupil does in the classroom, the better will he understand his work. Summary on the Review. — Each day should provide some definite review of the preceding day's work, some definite advance in the mastery of the subject, and some definite work assigned. The pupil then becomes conscious of something positive having been accomplished. Nothing is more detri- mental to the morale of the pupil than for him to feel that a day's work or a recitation period has been wasted or, at least, has passed without some definite advance. When our pupils realize that nothing will be allowed to interfere with the day's work, we shall find that they will be more anxious to eliminate their absences. Friday or the day before or after Divisions of Elementary Algebra 41 a vacation or some circumstance is allowed often to break up the routine of steady, purposeful work. With the proper attitude and evaluation of each day's importance, however, the teacher can make every meeting of the class, no matter under what disadvantages, a distinct step forward. The morale of the allied army was at high pitch until the Rhine was reached, and then it became a matter of anxiety to the com- manders, because the soldiers felt that their object had been attained and that, with no further advance being made, they were merely marking time. The review should be short, snappy, and purposive. It really takes the place of the old recitation, as such, and since it is much shorter in time, there must be intensive work done. It should be a re-view of the previous day's work, a clearing house for all the difficulties encountered in the study of the assignment, and an opportunity for the pupils to view from a new and broader angle the work studied the preceding day. Summary on the Assignment. — This is the portion of the period devoted to the explanation of the new lesson. The teacher should do most of the board work, the pupils following the operations at their seats. As far as possible the work should be developed through the pupils, since they will then become active participants and their interest peculiarly acute. Except on special occasions, all board work should be elimi- nated, so far as the pupils are concerned. Much time, chalk, and patience are lost when all or part of the class are working at the board. The inequality of time needed by various pu- pils, and the ease with which the brighter ones, out of a job temporarily, turn to things not connected with the subject, cause dismay to the teacher and an undesirable diversion for the rest of the class. 42 Supervised Study in Mathematics and Science But when the teacher himself develops the work on the board, and the class follows the operations on paper, every pupil is of necessity alert and attentive, because he may be called upon at any time for some point. The entire class is therefore kept up to a high pitch of intensive work. The blackboard as a means of visualizing a demonstration before the whole class has a distinct value, but as a common working ground it is open to criticism. Summary on the Study of the Assignment. — As has already been stated, the various assignments should be placed upon the board before the meeting of the class. Let them be definite, concise. In the case of references, the exact title and page of the book referred to should be given. The use of the assignment sheets has been fully explained in Chapter One and need not be repeated here. Besides giving the pupils something definite to do at once, this first assignment of work will enable the teacher almost immediately to single out the poorer pupils and those who are likely to be the workers of maximum ability. Plans must be made at once to take care of both classes, — something to which the supervised study period is especially well adapted. A re- arrangement of the seating of the class will be made after a few days, as bad results sometimes come from the premature an- nouncement that permanent seats have been assigned. A workable plan is to put the slow workers on one side of the room, the rapid workers on the other side, and the rest of the class between them. If the class can be approximately so divided early in the term, no especial attention will be called to this classification, and the reasons therefor may remain a secret with the teacher. This arrangement will make it much easier to reach the two extremes of the class and give to them Divisions of Elementary Algebra 43 special aid. It will also be a distinct help in administering the details of board work, individual instruction, and the use of supplementary material. Summary on the Silent Study. — The importance of getting to work at once without loss of time should be explained in the beginning of the term. The pupil should be made to feel that every minute is valuable, and that waste of time will not be countenanced. Various devices for keeping track of the completion of the different sections of the assignment will be given in succeeding lessons. The object of these sectional assignments is solely to aid the teacher in developing to the utmost each individual pupil in the class. This section of the supervised study period should be the most important part of the hour, because it is here that the pupil comes into personal contact with the teacher and receives first hand the kind of aid he needs. At the same time a certain amount of sympathetic relationship is developed on the part of both teacher and pupil. LESSON III UNIT OF INSTRUCTION I. — INTRODUCTION Lesson Type. — A How to Study Lesson Program or Time Schedule The Review 5 minutes The Assignment 30 minutes The Study of the Assignment 25 minutes Purpose. — The crux of the supervised or directed study period consists in the definite directions for the proper study of the subject in question, and careful explanations of just how to study rather than what to study. It is, therefore, 44 Supervised Study in Mathematics and Science valuable during the course and especially at the beginning, that very definite rules for study should be outlined and in- sisted upon. The Review. — Subject Matter. A summary of the pre- vious day's assignment. Method. Call for questions on the exercises assigned. Ask for a few leading facts brought out by this assignment, such as : i. What was Newton's definition of algebra? Comte's definition of arithmetic ? ( Show their pictures to the class.) 2. How did we get the word decimal? 3. Mention some formulas used in arithmetic. 4. What makes algebra a broader subject than arithmetic? Give a few examples like those in assignments I and II. Call on someone who completed the maximum assignment of yesterday's lesson to give the answers to the questions in this assignment. The Assignment. — 1. Methods of study. 2. The system of supervised study explained by the teacher. 3. The technic of the textbook in algebra. 4. Definite instructions in how to study algebra. Methods of Study. — (a) External conditions. In the first place a correct physical environment is necessary. One must be in a comfortable seat, with good light coming over the left shoulder, breathing fresh air, and in a room of proper tem- perature. The pupil must have the necessary tools, as paper, well-sharpened pencil, ruler, textbook, etc. Next, the pupil must put himself in the proper attitude toward the subject and concentrate his mind upon his work. And then, with a determination and expectation to succeed, he is ready to com- mence his work. All these prerequisites will in time become Divisions of Elementary Algebra 45 automatic if the teacher frequently calls the pupil's attention to them. It is easier to talk about concentration of mind than it is to achieve it, but the pupil should be carefully shown that the better control one has over his ability to keep his mind from wandering, the better student he will be and the sooner will he be able to master the task in hand. It takes will power and practice, but the teacher cannot emphasize too strongly the inestimable value of this acquirement, if once formed even to a limited degree. As McMurry suggests, 1 one of the best methods of acquiring concentration is through the employment of time tests, which require undivided attention. When we must do a certain thing within a definite time, we concentrate upon it. With sufficient practice, this may become habituated. (b) Technical factors. 1. The first technical factor in proper studying is the sensing of the problem. If the pupil simply goes at his work with a view of covering a certain amount of prescribed ground, without a realization of the problem involved, his study will degenerate into a mechanical grind. Every assignment, as stated before, should have some definite object or problem, around which the lesson will re- volve. In the first lesson, as outlined in these pages, it was "Is algebra practical?" In the second lesson it was "Is algebra entirely new?" and to-day it is "What is the best way in which to study algebra? " 2. The second factor, after the recognition of the motive of the lesson, is bringing to bear upon it all our present knowl- edge of the problem and then supplementing the problem from our text and possibly other books and sources. 3. We next seek to master the supplementary material by 1 F. M. McMurry, "How to Study" ; Houghton Mifflin Co., 1909. 46 Supervised Study in Mathematics and Science constantly referring to our present knowledge and associating our new facts with data already known. This may mean the employment of the memory, which, if rightly used, will be- come a means and not an end. Too much of our studying resolves itself into memorizing alone, and it then becomes a detriment. But as Miss Earhart well says, " Memory must not be substituted for thought but be based on thought." * There are three kinds of memorizing: purely arbitrary memorizing, memorizing based on reasoning, and remember- ing the sequence rather than the things themselves. 4. The fourth and last factor is the application of our data and material in the solving of our problems or in making our new power a part of ourselves. The Supervised Study Organization. A few words about the supervised study period may now be given. Each day's work will be divided into three parts : the review, the assign- ment, and study of the assignment. During the review, the previous day's work will be re-viewed in summarized form and any difficulties cleared up. This will usually take about 15 minutes. The new work will be explained during the second time division, i.e. that of the assignment. This will take about 20 minutes but may vary in amount. The study of the new lesson will take place the last 25 minutes, and dur- ing this period the pupil will be expected to do as much of the new work as possible. Explain that he may feel free to raise his hand if he finds need of help, that he must not expect to have the teacher do his work, but only redirect him to find his own trouble, or lead him to see wherein his line of procedure is erroneous. The study period is not an opportunity for 'Lida B. Earhart, "Teaching Children to Study"; Houghton Mifflin Co., 1909. Divisions of Elementary Algebra 47 getting someone else to do the pupil's task but an opportunity for getting proper directions so he may be able to do it himself. Explain that the assignment will be placed daily upon the board, that it will be in three parts, and that each pupil will be expected to do the first two parts and as many pupils as possible to do the third. The assignments should be so arranged that much of the work may be done during the period itself. The exact amount, of course, will depend on the pupil. It has been found in work of this kind that pupils are proud to excel their classmates. There are only the rarest instances of pupils deliberately doing only the minimum assignment. The Open Book. The pupils are asked to open the text- book in algebra while the teacher explains the structure. Comparison is made between title-page and the cover. A few words regarding the position or personality of the author, the name of the publisher, and the date of the book's publication may arouse some interest in the author as a second teacher of the class ; for, of course, the author of a textbook is to be regarded as a teacher. Have a pupil read the preface and then ask him a few questions which will bring out the reason for such an intro- duction to the book. Turn to the table of contents and note the various topics and subtopics of the subject. Compare this with the table of contents of some other text in algebra. Note the number of pages one or two units of instruction cover in your book and in some other text. If any topics are to be omitted from study, mention the fact. If your text has answers, give a few words as to their use and abuse. Teachers differ in their opinions regarding the 48 Supervised Study in Mathematics and Science value of the printed answers, and, if it is impossible for the teacher to train the class to make them a side issue and not the most worn portion of the book, it is clear that their publi- cation is a serious mistake. Now open the book at the first page, calling attention at the same time to the fact that the paging of the book proper commences at this point. Announce that the work for the next day will begin here. This review of the make-up of the book proper may seem irrelevant to the study of algebra and a waste of time, but aside from the value of the general knowledge thus gleaned, it serves as an introduction to the text with which the pupils are to have intimate acquaintance during the year. It is well that they know something of the nature of the tool with which they are going to work. It often happens that things learned incidentally in connection with a subject will be of greater educational value than the subject matter itself. After all, our children are coming to school primarily to be educated and secondarily to learn algebra, Latin, or any other partic- ular subject. It is through these subjects that we hope to attain the ends of education. 1 Instructions in How to Study. A few mimeographed directions may now be distributed to the class, and, after necessary explanations, the pupils may be told to insert them in their books for future reference. Explain that you may add other directions from time to time as the class progresses, and suggest that each pupil should feel free to make any suggestions for the enlargement of the list. 1 " The Textbook— How to Use and Judge It" by Hall-Quest, The Macmillan Company, 1018, gives a full discussion of what might well be covered in teach- ing pupils how to learn to use academic tools. Divisions of Elementary Algebra 49 The list which follows is by no means perfect or complete ; it is simply suggestive : Suggestions for Effective Studying Be sure you understand the assignment. Study the meaning of the type of problem you are to solve, as suggested in its name, i.e. highest common factor, addition of radicals, etc. Recall the teacher's explanation of the new work. Study again the type form or example of the new problems. Understand thoroughly what is wanted before you begin to use your pencil. Avoid guesswork. Take time to think. Do not rush into an exercise trusting to luck you will strike it right. Be sure you are right ; then go ahead. Be sure you set the exercise down correctly on your paper. Work carefully ; it is easier to avoid mistakes than it is to find them. When you find a new application, study it until you master it. Expect each new problem to be different from the one preceding ; else, we would never advance. In case you cannot proceed, raise your hand. Do not expect the teacher to find your mistake but to direct you to find it yourself. Be neat in your work. A good workman is known by his neat performance. Slovenly habits of work lead to slovenly habits of thought. The Study of the Assignment. — The assignment for to- morrow will be in one section only. It will consist of some questions on the points of to-day's lesson, in regard to the attitude of study, factors of study, the technic of the text- book and the list of directions on how to study. These questions will help to focus the study on the essential features and to prevent wrong conclusions. A few sample questions are given : What is the proper temperature for a living room? Why should the light come over the left shoulder ? 5o Supervised Study in Mathematics and Science Suggest a good method of practice to attain concentration of mind. What was the problem of to-day's lesson in biology? Which of the three kinds of memorizing do you use in relating the incidents of a ball game? Name some sources of supplementary material aside from the textbook. Is the preface necessary in every book ? Which do you think the author compiled first — the table of contents or the index? Give your reasons. Suggest any other directions than those given to you on the printed list. Which one of those given do you think would save you the most work, if carefully carried out ? BEING PAPER AND PENCIL TO-MORROW LESSON IV unit of instruction i. — introduction Lesson Type. — An Inductive and How to Study Lesson Program or Time Schedule The Review 15 minutes The Assignment 20 minutes The Study of the Assignment 25 minutes The Review. — Subject Matter. The questions on the previous lesson. Method. Write the various questions assigned yesterday upon slips of paper and have these in a loose pile, face down, upon the teacher's desk. Call on some pupil to come to the front of the room, to pick up one of the slips at random and, after reading it aloud, to proceed to answer it. If the class accepts this answer as correct and complete, ask someone else to repeat the process, and so on until all the questions have Divisions of Elementary Algebra 51 been answered. If any question is not answered acceptably, replace the slip in the pile. This method of review will further help to break down the barrier between pupil and teacher, to accustom the pupils to talking before the class, to teach clearness and accuracy of expression, and to test the judgment of the value of the answer, thus giving all something to do. Furthermore it helps to review thoroughly the essential points of the preceding day's lesson. Note. — The lessons in algebra will not be based upon any special textbook, but the directions will be found applicable to any textbook upon the market. Copies of all the modern texts are upon the teacher's desk, and constant refer- ence is made to these either for supplementary examples or other material. Inasmuch as the present writer is interested chiefly in presenting a variety of schemes for teaching and training pupils in economical and effective methods of study, it is hoped that the point of view herein developed will be compre- hensive enough to include the situations that may arise in the use of any text in algebra. The Assignment. — 1. Instructions in how to study the printed page. 2. Treatment of illustrative material. 3. Treatment of class exercises (a) oral, (b) written. Instructions in How to Study. The pupils will open their texts and have their attention directed to the " definitions." It will be noted that this is the first unit of instruction as listed in the table of contents. (In the divisions of algebra outlined in Chapter Two, it is given as a unit of recitation under introduction; authors differ in the arrangement of the material.) The first paragraph is read carefully and the central or important point or problem discussed. The pupils will readily select the essential point of this paragraph. If there are any words which are new or not clearly understood, they should 52 Supervised Study in Mathematics and Science be immediately defined by the teacher. Before we can com- prehend the sentence, we must know the meaning of its com- ponent parts. Explain the use of italics and heavier type. These take the place of the emphases in oral speech. These mechanical means call the attention of the reader to the im- portance of the word or phrase and should be specially noted by the pupil. If the sentence or paragraph is not clear at the first reading, reread it until the thought is mastered. Insist on the impor- tance of making reading thought-producing, and not simply a mechanical pronunciation of words. The language of mathematics is absolute and therefore cannot be read rapidly or slurringly ; every word means something. Now have someone reproduce the paragraph in his own words. Call upon a number of pupils to do the same thing, thus bringing out in various degrees of perfection the mean- ing of the assignment, and setting up a little rivalry for the best work. Emphasize the facts that we know what we can reproduce in our own words and that, when reproduced word for word like the text, we are thinking more of the mechanical reproduction than we are of the thought to be reproduced. The above outlined study of the paragraph might well be applied to the printed page of any book which is a subject of study, although some authorities strongly advise that the first reading of the page or section be made as a whole in order to get the general sense of the material. In algebra, however, since the textual matter is localized in its meaning, the pre- reading might be dispensed with. The reader will note also that the first three steps, mentioned in the preceding lesson as the order in which a subject should be studied, have been Divisions of Elementary Algebra 53 followed, i.e. the point of view or problem, the data or mate- rial, and its mastery. Its application, as is often the case, will be made later. It often happens that we accumulate material through these steps for some length of time before we finally bring it together in the fourth or concluding step. Treatment of Illustrative Material. When we come to illustrative material, the example should be reworked on paper. Otherwise the pupil will mechanically read the oper- ation, think he understands it, and in a short time find that it has slipped away from his consciousness. This reworking of the example on paper will also help to fix it firmly in his mind and to establish each step thoroughly as it is written down, provided always that the pupil does the work with the motive of understanding the operations as they are evolved. Since this is an illustrative or model lesson, the teacher will also do the work which he will expect the pupils to do for themselves in their future study. For instance, suppose this formula is given : a=bXh. The meaning of the symbols is studied and then the value of the letters in a specific case is given and put upon the board. Thus, a equals 20 ; h equals 5. The question is, what is the value of 6? The substitutions are now made in this formula and the solution performed. This operation should be repeated a number of times with varying values for the letters. Two things are being done in this operation : the pupil is learning how to interpret the printed word by his clarified perception, and he is also learning the fundamental character- 54 Supervised Study in Mathematics and Science istics of algebra, the broad application of the algebraic func- tion. It might be well at this point to request some pupil to turn to the introduction of Milne's Standard Algebra 1 and read what that author has to say concerning it : The basis of algebra is found in arithmetic. Both arithmetic and algebra treat of number, and the student will find in algebra many things that were familiar to him in arithmetic. In fact, there is no clear line of demarcation between arithmetic and algebra. The fundamental principles of each are identical, but in algebra their application is broader than it is in arithmetic. The very attempt to make these principles universal leads to new kinds of numbers, and while the signs, symbols and definitions that are given in arithmetic appear in algebra, with their arithmet- ical meanings, yet in some instances they take on additional mean- ings. . . . In short, algebra affords a more general discussion of number and its laws than is found in arithmetic. Since with this introduction the pupil has an idea of the manner in which he should study, the teacher should further encourage him to proceed alone in his study. Questions need to be asked from time to time, however, to make sure that the pupil is following the directions and getting the right ideas. To illustrate, after the class has studied some paragraph or section, ample time having been allowed for the use of the dic- tionary, etc., ask some questions about it, and then call upon someone to state the problem involved ; someone else to restate it in his own words ; and others to supplement it from their own knowledge if possible. Thus we more and more throw the pupil upon his own resources but always with the proper methods of procedure before him, and careful supervision on the part of the teacher to see that he gets the correct interpreta- tion. He will eventually acquire the habit of study as outlined 1 American Book Co., 1914. Divisions of Elementary Algebra 55 above, which may be of more lasting value to him than the algebra itself. Treatment of Class Exercises. The treatment of exercises to be worked in class will differ somewhat from that of illustrative material. Suppose we wish to take up such exercises as the following : Read and explain : 1. a+b. 3. aXb. 2. a-b. 4. a-i-b. We have now accumulated our material and are ready for its application, or the fourth step. Here are definite examples of what we have been studying about up to this point. All study is for an end. As the final end of algebra is the solution of problems, so an intermediate step in the attainment of this end is the ability to perform the mechanical processes which will later be involved in their solution. Class exercises will therefore be of two kinds, (a) oral and (b) written. (a) Oral exercises. Some pupil is told to rise and read the first example. He is then asked to analyze or tell the meaning of it, which should be something like this : Two general mem- bers of different values are added together by indication, a added to b. The teacher should insist on complete answers, told in technical terms and in simple English. Clear thinking and clear expression will thus be unconsciously habituated by the pupil. (b) Written exercises. Exercises like the following, how- ever, may preferably be treated in a wholly different manner. Some such procedure as outlined here may be used or some modification of it : 56 Supervised Study in Mathematics and Science If = 3, b = 2, and c=£, find the value of each of the following : 1. f i- 3b-. c a+b 3. V^T 6. oX-- Most of the board work should be done by the teacher himself. The pupils should remain at their seats and either work on paper or tell the teacher what to write upon the board. This elimination of board work by the pupils will result in a more efficient use of the time of the period, as all members of the class will be either at work or on the lookout for possible questions. Every mark put upon the board should first be supplied by some member of the class and accepted by all as correct. Thus each member becomes personally inter- ested in the operations and alert to give directions or to detect errors. The class is thus kept up to a high tension and inten- sive work may be accomplished. The board work becomes a check and not a key, and the pupils feel that they have had a real part in its development. To illustrate, the work on the first example given above will proceed like this : the teacher will ask someone to read the example and to explain how to make the substitutions. He will then write it upon the board, directing the pupils to do like- wise on their papers. Another pupil will then be called upon to tell what is to be done next. As the pupil states the various steps, the teacher will place them upon the board, the class meanwhile doing the same on their papers. The pupil reciting will say something like this : The expression, 6a, means that the literal number a is taken 6 times, or that a multiplied by Divisions of Elementary Algebra 57 6 constitutes the term in the numerator, and that the product is to be divided by b. Unless we give these literal numbers some values, the actual division can only be indicated as in the example. But if we assign some arbitrary values to the literals, we may substitute these values in the expression, perform the necessary operations, and reduce to its simplest term. In this case, since a is given the value 3 and b the value 2, we find that 6a is equivalent to 18, and this divided by 2, or the value of b, gives us the result, or 9. The work on the blackboard will appear as follows : 6a 6X3 18 A — - = - = — = 9. Ans. b 2 2 In this way, it would be well for the teacher to work on the board, with the assistance of the pupils, these six examples, in order that the pupils may learn how to handle written exer- cises. When written work is next required, it will be safe to assume that they will know how to go about their work, after one or two typical demonstrations have been given by the teacher. Work of this kind is oral or cooperative studying and, should characterize every general assignment. The Study of the Assignment. — Assuming a list of 30 graded exercises in the textbook in use, assign as I or Minimum Assignment. Exercises 1-20. II or Average Assignment. Exercises 21-30. III or Maximum Assignment. Exercises 15-20 on page 46 of Ford and Ammerman's First Course in Algebra, 1 or exercises on page 7 of Slaught and Lennes' Elementary Algebra. 2 Value of Outside Work. Many of the introductory lessons will be along the line of the foregoing, the majority of the 1 The Macmillan Company. 2 Allyn and Bacon. 58 Supervised Study in Mathematics and Science exercises being worked in the class under the supervision of the teacher. Each day a short assignment should be made, based on the ground covered and preferably taken from outside texts, especially the maximum assignment. This ought to be in the nature of a review of the work done in class and should be short enough to allow the majority of the class to complete all three assignments. These examples may be handed in the next day, but the best way for the teacher to make sure that the principles are thoroughly understood is to work out on the board through the minimum workers, or those who only completed the minimum assignment during the study period, a few typical examples during the review. The pupil must realize that the teacher is interested not so much in what the pupil has done as in what he can do now. If pupils could be made to know that work done outside of class is important only in so far as it makes them capable of doing something the next day in class, the incentive for getting other people to do their work would be greatly diminished. The out- side work must be insisted upon, unless the periods are long enough to have all the work done in class, but the credit should always be allowed largely upon the ability to do similar work again in class. It is the same rule that applies through the walks of life. The stenographer, the carpenter, the printer, the dentist, the worker of every sort is not paid for the record he has made in speed or the house he has built or the books he has printed or the bridge work he has done, but for his ability to do similar work again. To be sure, the experience has made him proficient, but we pay for results and not for the practice that has made the results possible. The one is indispensable, but the other is the criterion by which all of us are judged. Divisions of Elementary Algebra 59 The next two or three lessons in the textbook may be worked out in a manner similar to the above. The amount of time spent on the work will of course depend on the book used and on the teacher. He may condense it into a shorter period or take even longer. The material given is merely suggestive and no teacher is expected to follow it verbatim. In fact, such a procedure would probably pre- determine the failure of supervised study, because more than anything else its successful operation depends on the origi- nality and individuality of the teacher himself. No system has been or ever will be evolved which automatically may be operated by someone and without change or adaptation be a success for everyone else. All that may be hoped for any method is that it be suggestive ; its final application and adap- tation, in the last analysis, lies with the teacher himself. In the words of Miss Simpson, author of a companion book in this series, "it is imperative that teachers adapt rather than adopt the methods suggested in these lessons." 1 LESSON V unit of instruction dx — addition Lesson Type. — An Inductive Lesson Program or Time Schedule The Review 15 minutes The Assignment 20 minutes The Study of the Assignment 25 minutes The Review. — Upon the completion of each unit of in- struction, it is advisable to re-view the unit in toto. This may take the form of an oral or written review. Various methods 1 " Supervised Study in History " ; The Macmillan Company. 60 Supervised Study in Mathematics and Science should be used. In reviewing Positive and Negative Numbers, the following is suggested : Method. Write upon the board a large number of exercises covering the various phases of this topic, and call on different members of the class for the answers. These should be writ- ten upon the board in the proper place. Send a pupil to the board and as the answers are given have him write them down if he considers them correct. If he calls one correct when it is wrong, he must take his seat and another be sent to the board in his place. Thus the pupils are tested on their ability to solve the problems and, also, on their ability to judge correct results. It might be well to select someone to call on the dif- ferent pupils to recite, the teacher noting, however, that all or most of them are given a chance. When the pupil at the board makes a mistake, then the leader should take his place, and so on. Thus the review becomes socialized. It will provide interest for work that too often is needlessly weari- some. The Assignment. — i. The arithmetic preview. 2. Recognition of the problem. 3. Explanation of the type form. The Arithmetic Preview. As intimated in the second les- son (page 36), the pupils' present knowledge of arithmetic should always be drawn upon when possible to illustrate the new work. A few examples in adding numbers are followed by the implied addition of literal terms. The induction should be made by the class and the Commutative Law of Addition deduced. After a few attempts, a workable law will be developed by some such questioning as this : how much is three and five ? five and three ? a and b ? b and a ? Ask whether it makes any difference in what order numbers or Divisions of Elementary Algebra 61 letters are added. If the answer is " no," ask someone to state this principle in a sentence. Answer : Numbers can be added in any order. Tell the class that this is a law of order or the Commutative Law. Then broaden this principle when two or more numbers are grouped, as (5 plus 6) plus 4. What is the sum? (5 plus 4) plus 6? The sum is the same. Then we broaden the above law to include groups. Ask someone to state the revised principle. Answer : Numbers may be added in any order or group. Tell them that this is the Associative Law of Addition. Recognition of the Problem. Ask the pupils what is meant by a term, a monomial. The above illustrations are all monomials. Therefore the first problem under Addition will be addition of monomials, which becomes our first problem. Explanation of the Type Form. Place these examples upon the board : Add: 1. 3 2. 3 boys 3. 30 _5_ 5 boys .50 There will be no difficulty with the first two. Ask in 3, what a stands for. Someone will say " boys." But might it not stand for girls or houses or almost anything? The class will readily see that a may stand for anything and therefore the answer to the example will be 8a. Repeat with other simple monomials, all positive. Then put these examples on the board : Add: 1. 5 2. s dollars 3. 5a 2 — 2 dollars — 20 With their previous knowledge of positive and negative numbers, the class will see that in each case the coefficient is 3. 62 Supervised Study in Mathematics and Science ASSIGNMENT AND STUDY SHEET Subject Elementary Algebra Period 2d Date September 7, 1921 Unit of Instruction Addition (III) Unit of Recitation Addition of monomials (I) Unit of Study Examples 1-25, text Lesson Type Inductive Review: Positive and Negative numbers. Exer- cises from Wheeler's Examples in Algebra, pages 4-7. Memoranda Examples on board. One writes answers which others give. Change when mistake is made. Work out laws. Assignment : 1. Arithmetic preview. *. Recognition of new problems. 3. Explain type form. What are monomials? Add : 3 3 boys 3a S 5 boys 50 . Minimum Exercises 1-20 In text. 2. Average Exercises 21-25 In text. 3. Maximum Exercises 43-50, Wells and Hart's New High School Algebra, page 38. Involve fractions and deci- mals. Study: See that the signs are correctly copied. Number of pupils solving minimum assignment Number of pupils solving average assignment Number of pupils solving maximum assignment Total Figure III 7 22 1 30 Divisions of Elementary Algebra 63 Now call on some pupil to stand and solve the first exam- ple, which may be : Add : 2a ja Similar exercises may be given orally. They may well be supplemented with others from the board until the principle is well understood. Then a typical example like the following may be developed on the board and the class set to work on the study of the new assignment : Add: 2%, \x, —x. The Study of the Assignment. — Assuming the textbook in use gives a list of 25 similar exercises, make the following assignments : I or Minimum Assignment. Exercises 1-20. II or Average Assignment. Exercises 21-25. III or Maximum Assignment. Exercises 43-50, page 38, in Wells and Hart's New High School Algebra. 1 These exercises are similar but a little more difficult, involving fractions and decimals. The Silent Study Period. — The Assignment Sheet. The division of the assignment into three parts, as suggested in Hall-Quest's book on Supervised Study, has been fully ex- plained in Chapter One of the present volume, which should be reread. The assignment numerals only should be used when designating the sections in placing the assignment upon the board and this should always be done before the class as- sembles. A sample sheet, made out to conform to this lesson, is given in full on page 62. It will aid the teacher materially 1 D. C. Heath and Co. 64 Supervised Study in Mathematics and Science if he will make these sheets out conscientiously during the term. There will then be no confusion or waste of time if additional exercises are needed during the class period. Many precious moments are saved by a little foresight and planning. A lack of prearranged plans may also break down the morale of the class. Pupils are quick to respond to fine or poor executive ability when either is exhibited by the teacher. Napoleon was one of the world's greatest generals because he had the absolute confidence of every soldier under him. Teachers are generals of a little school army and the morale of the one is analogous to that of the other. The Completion of the Assignment. The minimum and average assignment should cover the amount of work that the teacher would ordinarily give under the old method of only one assignment for all. That is, the ordinary lesson for the next day would be about fourteen exercises in the text. But these have been broken up into two sections, the first of which should be worked by all within the 25 minute study period; if not, then the pupils who fail to complete this part need special attention. All the class is expected to have completed the average assignment before the next day and some pupils will do so before the end of the period. Those having trouble will have their work taken up during the review. The maximum assignment is designed primarily for the brighter and quicker pupils, those who are capable of doing more than the average amount of work. They should be given an opportunity of trying more difficult applications of the day's work. Not many will complete this part and it should not be demanded from all ; but when done, some system of giving extra credit should be used. A good method Divisions of Elementary Algebra 65 is to add half a credit to the monthly grade for every day that the maximum assignment was done correctly. In case a pupil did this correctly every day for a month, it would only mean ten extra credits, which might raise the grade from 80 to 90. Very few would attain this maximum advance grade, however. But the teacher must be careful not to give too much credit to this advance work and so discourage the slower worker ; it should be the aim always of the teacher to en- courage each one to do his best all the time. The Teacher's Duty during the Study Period. As soon as the study period begins, all start to work on the next day's assign- ment. The rate of speed will soon become uneven. Some will experience no difficulty and will advance rapidly; others will be in trouble at the outset. For the latter, the up- raised hand will quickly bring the teacher with aid. Thus help comes when it is needed and at the time that the correct direction or word of helpful explanation will do the most good. The teacher must, of course, ever be on the alert to see that his help is corrective or suggestive and not simply a crutch. It should be directive and not simply finding the mistake for the pupil. The teacher, in quietly moving among the pupils, will note many wrong methods and incorrect habits of work which he can tactfully correct. Many small but important things, such as legible handwriting, neatness, care- ful arrangement, accuracy in copying the example, may be brought to the attention of the child at the time that he is working. It is a case of striking when the iron is hot. Occasionally a glaring error and its possible results may be called to the attention of the class ; for instance, the danger of mistaking a poorly formed 6 for a o if the loop is not care- fully attached to the bend of the figure below the top. Draw 66 Supervised Study in Mathematics and Science the attention of the class to the fact that this little error invalidates the whole later process. This leads excellently to an explanation of the value of carefully checking the work as one proceeds. If each step is carefully gone over and checked for errors of omission or commission, before the next step is taken, valuable time may be saved. It is easier to avoid mis- takes than it is to find them. (See rule, page 49.) Verification. — A minute or so before the close of the period, check up the work done in class by the pupils. Various meth- ods may be employed, two or three of which will here be explained. Others will readily suggest themselves to the teacher. , First Method. Have some printed slips like this : Name. Class Period- Examples completed Assignment completed Time spent outside of class on to-day's lesson Teacher's check Figuke IV Such a form can be filled out by the pupil in a minute's time ; the teacher can collect them and the next day file them with the papers handed in. It can then be noted how much of the lesson was done in class, and how that amount cor- responds with the examples done outside of the class period. This method will give the teacher the names of the pupils, Divisions of Elementary Algebra 67 who failed to complete the minimum assignment in class, and these should have special attention the next day. This check- ing will also serve to tell the teacher whether his assignments are too long, too short, or about right. If the class does not approximately conform to the percentages mentioned in Chapter One, page 16, something is wrong in the assignment and an analysis of the situation should be made. Incidentally, this plan will take care of the roll call. Second Method. Have the pupils hand in all exercises com- pleted at the end of the period. Then the data may be computed by the teacher. The next day the examples worked outside of class may be handed in and filed with the others, which will thus constitute the completion of the assignment. Third Method. Just before the close of the period, call on all who have completed the. minimum assignment to stand or raise their hands ; the teacher can either take down the names or have the pupils hand in their names on slips of paper. Then in like manner the names of those who have done the average and maximum assignments may be obtained. Fourth Method. Have the pupils check on their papers the point they had reached when the period terminated; then when these papers are handed in next day, the teacher may compile his own lists. LESSON VI unit of instruction ill— addition Lesson Type. — An Inductive Lesson Program or Time Schedule The Review 15 minutes The Assignment 20 minutes The Study of the Assignment 25 minutes 68 Supervised Study in Mathematics and Science The division of the time of the class period, as stated at the beginning of each lesson, is that followed in the Canton High School, Canton, N. Y., where supervised study has been in operation since 191 5. The day is divided into five periods of one hour each. Longer periods, which would allow the pupil to do all his studying in school, would be ideal, but in many schools such a program could not be administered. If possible, the time schedule should be amplified, but the above proportion of time seems very adaptable, where different arrangements cannot be made. The Review. — Subject Matter. Addition of monomials. Method. The object of to-day's review is (a) to assist those pupils who had trouble with the assignment and (b) to give additional drill in this work to those who had no special difficulty but who were unable to complete the work. Those who have completed the threefold assignment and who have mastered the addition of monomials should] be allowed to proceed at once with the advance assignment in polynomials. This will serve as an inducement for intensive work and will encourage the brighter pupils to work harder and to solve all the exercises during the period if possible. This number will be small if the assignments have been care- fully planned. Those who have done part of the maximum assignment should be instructed to complete it. Now that we have the more advanced pupils working on the next lesson, or on the more difficult examples of the maxi- mum assignment, we can turn our attention to those who did not complete the minimum assignment or who had more or less difficulty. These pupils may be treated in different ways. If several failed on the same problem, they may be sent to the board to work on it under the supervision of the teacher. The Divisions of Elementary Algebra 69 teacher can then watch their work and soon note the trouble. As soon as a pupil finishes one, he should commence on the next with which he experienced difficulty, and so on. This method is not advised, however, for reasons already stated against board work. A better method would be for the teacher himself to work out the problem, with the pupil directing him what to do, the others meanwhile following the process at their seats. Pupils should never be sent to the board, however, to work, for the benefit of others, examples which they solved them- selves. Board work has been inordinately stressed in mathe- matics. When a pupil can do a thing, he should not be asked to do it again ; it is his ability to do something which he could not do before which will make him advance. On the other hand, such exercises, written out on the board and afterwards read for the benefit of those who could not do them, will be of practically no value to them. Pupils can learn best by doing the work for themselves. As soon as the problems giving trouble have thus been solved, the remainder of the review period should be devoted to working additional ones of like nature, until this difficulty has been mastered. If there are still those who do not seem to be able to under- stand the problem, they should be given individual attention during the remainder of the period. The teacher must feel that his special task is to help the less capable ; children have varying degrees of ability and it is the peculiar province of the supervised study scheme that the backward ones are thus given special attention and brought up to the standard of the class as quickly as possible. The old process of the elimination of the dull pupil must give way to the new idea of reaching him 7o Supervised Study in Mathematics and Science through a study of his particular difficulties and applying the proper stimulus which will enable him to " find himself." The Assignment. — i. Explanation of the method of add- ing polynomials. 2. Recognition of the new problem and its attendant rule. Explanation of How to Add Polynomials. As in arithmetic we can only add or subtract terms of the same kind, so in algebra like must come under like, before we can add or sub- tract. In the example, Add: a+^y and 2a— $y, we write it as follows : a+4y 30- y and add each term separately. Thus a plus 2a equals 30, and 4y plus minus 53/ equals minus y. If the order were different in the example, we would be obliged to rearrange the terms so that the a's would come under the a's, and the y's under the y's. Write another similar example on the board and ask some- one to direct the work. Put on another and send someone to the board to work it. If all claim to understand the opera- tions, pass on to the development of the problem involved in this lesson. Recognition of the New Problem and Its Rule. Ask what kind of expressions these are that the pupils have been manipu- lating. After you get the right term, polynomials, ask what is being done with them. Then ask someone to state the problem of to-day's lesson. The answer should be " Addition of Polynomials." It must not be forgotten that every day's lesson should have an object or problem; to-day it is Addition of Poly- Divisions of Elementary Algebra 71 nomials. Various schemes may be used to emphasize it. One which has been used with success is writing it upon the board with yellow crayon. Thus it stands emblazoned in the mind of the child ; and, noticing it a number of times during the hour, he cannot forget that there is a definite object in view, and that the work in hand leads up to its understanding and solution. Again, if the special problem under consider- ation has not been thoroughly mastered, it remains upon the board and in this way the aim of the lesson is even more deeply imprinted upon the pupils' understanding. Pupils like to advance and if they find that another day must be spent upon some topic, — say addition of monomials — because they did not master it, renewed efforts will be made to move on to some- thing new. Develop the rule for adding polynomials by some such analytical method as this : ask why we put the various terms involving a in one column, and what kind of terms these are. Develop the definition of similar terms. When the example has been set down, what do we do? Draw a line and add each column, connecting them with their signs. The rule has thus been developed. Have some pupils state it in full. Repeat with different ones until you have something like this : Rule. — Arrange the similar terms in the same column, add each, and connect the resulting terms by their proper signs. Work two or three examples on the board, asking pupils to apply this rule by specific reference to the terms in the ex- ample thus solved. Such mechanical drill is necessary in all study of mathematics but, after all, there is only one object of drill, i.e. to grasp the principle involved, — and if by any means this may be done quickly, it ought in all justice to be employed. Mathematics should not become a master but a servant. 72 Supervised Study in Mathematics and Science The Study of the Assignment. — I or Minimum Assignment. Exercises 2-23, in textbook. II or Average Assignment. Exercises 24-27, in textbook. III or Maximum Assignment. Exercises 14-20, on page 41, Durell's School Algebra. 1 Verification. — Especial attention should be given to the pupils who were yesterday on the minimum list. For ex- pediency, the back of the assignment sheet used yesterday might be employed to record the names of these pupils. It might be a good plan, as soon after the organization of the class as possible, to reseat the pupils, placing those habitually in the minimum classification at the front of the room where they may be easily watched. Care should be taken, however, not to name the groups in such a way as to embarrass any pupil. The teacher will use tact under all circumstances. Certainly a gain in facility of class management should not be achieved at the loss resulting from humiliating or embar- rassing any member of the class. If the teacher is unable to locate a pupil's particular diffi- culty on account of illegible figures or general confusion of data, such a pupil may be sent to a side blackboard and there given a private lesson in some of the fundamentals of study. If he seems to have no conception of the problem, let him analyze it for the teacher, telling him what each term is and what it signifies. Make him comprehend the make-up of each term of the expression ; ask him why he puts it in a certain place, why he draws the fine under it, how he treats signs in adding, etc. Such individual instruction takes time but is well worth while if thereby some boy or girl is saved from failure. One or two such private lessons like this each day, while the class 1 Charles E. Merrill Co. Divisions of Elementary Algebra 73 is at work upon the assignment, will serve to keep the teacher pretty busy, and yet the intensified effort will well repay the expenditures of time and trouble. The satisfaction of joy over seeing the weak pupil become strong is a great reward in itself. Such a case is like the physician's. It needs special study, painstaking oversight. But surely the restoration to health and the happy development of a "case" is a deep professional satisfaction. LESSON VII unit of instruction x. — the equation and problems Lesson Type. — An Expository and How to Study Lesson Program or Time Schedule The Review 15 minutes The Assignment 20 minutes The Study of the Assignment 25 minutes The Review. — Subject Matter. Simple equations. Method. Give a speed test in reviewing simple equations. Have a large assortment of exercises on the board, or on mimeographed sheets ; see to it that the class is provided with paper and pencils ; set all at work on the minute. See how many exercises can be worked correctly in some definite length of time, say five minutes. When the time is up, have all stop immediately. Read the answers and ascertain how many attained a mark of 100. Collect all the papers and put the names of those having all correct upon the board. In looking over the papers which fall under 100, the teacher can note just where the trouble lies with the individual pupils and can remedy it the first chance he has. 74 Supervised Study in Mathematics and Science Such a review, based on the time element, serves to strengthen the pupil's ability to concentrate, puts snap into the work, and lends an element of interest, as all young people like anything that savors of a contest. There are a number of excellent standardized algebra tests now on the market, which the teacher may use to advantage in this work. These tests were primarily constructed that there might be given an opportunity to teachers to compare the work done in their classes with that in other school systems. Since it is a fact that teachers will differ more or less markedly in their ordinary grading of examination papers and in their judgment of pupils' ability, the employment of tests, which have been used by a large number of teachers and the results of which have been standardized, gives an excellent means of evaluating the work in any class. But aside from this, these tests have other values for the teacher. They point out beyond doubt where weaknesses exist and allow a scientific basis for constructive work. Again, they arouse a vital inter- est in the results among the pupils tested because they like to know how their progress compares with that of other schools. The avidity with which pupils will strive to raise the standard of the school and of themselves offers the best inducement for intensified work in the classroom. (a) The Rugg and Clark tests. These consist of booklets, containing a series of sixteen tests on the various types of algebraic operations from one on collecting terms to one on quadratic equations. These may be given at one time at the completion of the work in algebra, but the author has found them of greater value in checking up his pupils on the com- pletion of each type process and noting wherein the pupils are weak, thus affording an opportunity for his diagnosing each Divisions of Elementary Algebra 75 individual case and permitting him to give more drill in those processes that seem to need it. (6) The Hotz scales. These are in the form of sheets covering all the processes in algebra, including problems. The sheet on addition and subtraction gives examples in adding and subtracting terms, expressions, fractions, and radicals. The one on equations and formulas gives examples in simple equations, simultaneous equations, fractional, radical, and quadratic equations, and equations involving the manipula- tion of formulas. There are other sheets treating multiplica- tion and division, problems, and graphs. These tests are primarily useful in testing a class at the close of the work in algebra, as a means of comparison with standard scores. Used in connection with the Rugg and Clark tests, they form a valuable system of accurately and scientifically testing the progress of the class. The Assignment. — 1. Explanation of algebraic repre- sentation. 2. The algebraic equations applied to a concrete problem. 3. Definite rules for studying and solving problems. 4. Analyses of several simple problems. The Representation of Concrete Things Algebraically. Some such questions as the following lead up very logically to the study of the problem by means of algebraic representation : 1. Express the sum of five and three ; of a and b. 2. Express the difference of five and three ; of a and b. 3. What number increased by three is equal to eight? 4. What number diminished by three is equal to two? 5. How do the last two questions differ from the first two ? The pupils will see that in the last two questions something is lacking which is to be found. Tell them to indicate this 76 Supervised Study in Mathematics and Science unknown by some letter, as x. Then the third question stated in terms of the known values and the unknown values, will read : z+3 = 8, and, after solving by transposition of terms, *=5; the fourth question will read : x-s = 2, or, after solving, x = s- 6. If a pencil costs five cents, what will three cost? 7. If a pencil costs n cents, what will three cost? 8. Express the fact that a tablet costs five cents more than a pencil in both 6 and 7. 9. Express the fact that two pencils and a tablet cost fifteen cents. Ans. 2^+5 = 15. 10. Solve and find the cost of one pencil. Ans. n==$. These questions and others of like nature may be read to and be answered by the class; the result will be that the pupils will gradually sense the fact that by using literal num- bers, we are able to represent many things in a manner that we could not do otherwise. When definite values are assigned to letters, so that they will for the moment stand for some- thing concrete, they take on an entirely different meaning. Very strange "4X plus $x" may sound to a boy, but when we let x stands for dollars it assumes a very sensible and familiar form. Bring out the fact that the mechanical work preceding the study of problems has aimed at enabling the class to manipulate the resulting algebraic representation of some- thing concrete. Divisions of Elementary Algebra 77 Application of the Equation to a Concrete Problem. Let us take this simple equation, x plus 5 equals 12. Have someone analyze it. It means that 5 added to some number unknown will give us 12. By the law of transposition of terms in equa- tions we solve, and x equals 7. Now suppose we have this problem: What number in- creased by 5 will be equal to 12 ? What are we trying to find? A certain number. Then since this is unknown, we will for the moment let x stand for it or equal it. How do we repre- sent increased value? By adding. Then how may we indicate the expression "number increased by 5"? Since x stands for the number, it will be x+5. But according to the remainder of the statement, it is equal to 12 ; then x+5 = i2. And solving, we find that x equals 7, or what we wanted to find. In like analytical manner take up several similar problems, such as : What number diminished by or increased by or exceeded by, etc., equals something? Lead the pupil in each case to see, through a prior arithmetical representation if necessary, the algebraic representation of the same. Definite Directions for Solving Problems. At this point either give the pupils the following definite steps, previously mimeographed, or have them written upon the board and copied by the pupils. a. Read the problem very carefully ; study it until you know its every meaning. Close the book to see whether you can state it to yourself, silently. Then reopen the book to see whether you were right. b. Decide what is the thing wanted and represent it by x. c. If more than one unknown is involved, represent them by some other letters. d. Express in algebraic language each of the conditions men- tioned in the problem. 78 Supervised Study in Mathematics and Science e. Make an equation of the two statements that express the same conditions. /. Solve for the unknown. g. There should be as many equations as there are unknown quantities. Illustration of the Directions. Given this problem : What number diminished by 8 is equal to 12? By a : We mentally analyze this to be : What number is there which will be equal to 12 or become 12 after 8 has been taken away? By b : Number is the thing wanted ; therefore let x equal the number. By c : Only one unknown is implied in this problem. By d: x minus 8, and 12 are two expressions concerning the unknown. By e : They are equal, therefore, a;— 8=12. By /: x= 20, or what was desired. Analyses of Several Problems. The teacher, through dif- ferent pupils, will then work out in similar analytical form, the analyses of several related problems. Insist on the above mentioned steps being followed each time; pupils must be taught how to study problems and not merely how to solve them. The pith of the whole thing lies in the ability of the pupil to read the problem intelligently and to understand it so thor- oughly that he can tell it in his own words. Pupils are apt to commence work before they fully comprehend what is given and what is wanted. The Study of the Assignment. — I or Minimum Assignment. Exercises 1-16, in text. (All of these should have been analyzed in class but not worked.) 1 1 or Average Assignment. Exercises 17-21, in text. (These have not been analyzed.) Divisions of Elementary Algebra 79 III or Maximum Assignment. Exercises 26-30, page 176, Vosburgh and Gentleman's Junior High School Mathematics, Second Course. After this preliminary lesson on how to study problems, all of which should be very similar and not too difficult, it is advisable to take up problems in the following manner : Have each day a typical problem on the board ; as soon as the class assembles, let all read it over carefully and study it for a few minutes. Then call on someone to state what is wanted, someone else to state the expressions, someone to make the equation, and someone to solve it. Call on a number of different members to explain various phases of it, making the problem an object of class study and analysis. If the problem is simple in principle, it is often found profitable to have someone make up a similar problem. This method of having the pupil make his own problem and then solve it will be found an excellent means of getting him interested in this kind of work. Before the end of the year the problems which pupils will make up by themselves will astonish the most experienced teacher. Data may be supplied from current events, such as elections, ball games, business statistics, etc. The author cannot recommend too highly this method of spending each day a few minutes on a problem and then pro- ceeding with the regular work. It serves to keep the principles of solving problems ever before the class rather than for short, intermittent periods. Pupils do not tire of them but will really look forward to this phase of the day's work. Solving problems becomes a habit and what is quite generally con- sidered the hardest feature of algebra loses this aspect, be- cause the children have become so used to problem solving that it has become " second nature." Occasionally, an advance 80 Supervised Study in Mathematics and Science lesson may be given on problems only ; but the above plan has been found, after careful trial, to cover their treatment adequately and well. LESSON VIII UNIT OF INSTRUCTION VH. — FACTORING Lesson Type. — A Socialized Lesson Program or Time Schedule The Review 30 minutes The Assignment 30 minutes The Review. — Subject Matter. Factoring ; all cases. Method. Have the boy and girl, who received the highest mark on the last grade card in algebra, choose sides. When this has been done, the lines should be placed as in the old fashioned " spelling-down bee." Commencing with the leaders and taking them alternately from the two sides, the teacher sends the pupils in turn to the board to work an exercise in factoring. Excellent material will be found in the numerous textbooks on the teacher's desk. If the exercise is worked correctly, the pupil returns to his place in the line and one from the opposite side goes to the board. If a pupil misses an exercise, he must take his seat and during the re- mainder of the contest work out all the exercises on paper and hand them in later to the teacher. In this way a large number and variety of exercises may be worked, and the pupils be tested for their skill in recognizing correct answers, for it is evident that the teacher should not take too active a part in reviews of this kind. The pupils know that they are expected to pass judgment on what is worked at the board. If serious confusion results, the teacher is resorted to as judge of the court of appeals. Divisions of Elementary Algebra 81 When one side wins, i.e. has factored down its opponent, the assignment is taken up. The Assignment. — The following miscellaneous exercises in factoring, taken from various texts, will have been written upon the board. The first eleven will illustrate the different type forms of factoring, and a question or remark is set opposite each to direct the pupil in analyzing it. (Look out for a common factor.) (Difference of what? Rule?) (What type form have you here ?) (Be careful in your grouping.) (The parentheses indicate what?) (Is this a perfect square ?) (Note the powers ; odd or even ?) (How may this be made a perfect square ?) (As a last resort, use factor theorem.) (How may such an example be best treated ?) 11. a 3 "— a 36 . (Keep in mind what 3W and 36 are.) Then give the pupils an equal number of examples illus- trating all the various phases of factoring but in a different order, of course, and with no remarks. Tell the pupils to indicate, in addition to the answer, the type form which each illustrates, as: a 3 +b 3 = (a+b)(a?— ab+b 2 ). Ans. Sum of cubes. x n+1 +x=x(x n +i). Ans. Common term. Supply the pupils with copies of Wheeler's Algebra and refer them to page 70. Tell them to select all the exercises that illustrate some type form which is mentioned, such as Factor : 1. 3^-3*. 2. a 3 — 1. 3. x 2 + 17a; +72. 4. ac—ax— 4bc+4.bx, 5. (x+a) 2 — (x— a) 2 . 6. 4a?+4ab-\-b 2 . 7. m s +n & . 8. a*+b*+aW. 9. x 3 — 'jx+6. 10. r 6 —s e . 82 Supervised Study in Mathematics and Science the difference of squares, and note them on their papers without working them. For example, Nos. 3, 6, 10, etc. Then make a list of all the examples which illustrate some other case in factoring, and so on, covering all the principal cases. This may be carried out to any degree the teacher wishes, the idea being to acquaint the pupils thoroughly with the different type forms, to practice judgment in associating the example with its type form, and therefore in selecting the method which must be applied for its solution. LESSON IX UNIT OF INSTRUCTION IX. — FRACTIONS Lesson Type. — A Deductive and How to Study Lesson Program or Time Schedule The Review 15 minutes The Assignment 20 minutes The Study of the Assignment 25 minutes The Review. — Subject Matter. Multiplication of fractions. Method. Write a number of exercises on the board, illus- trating the previous lesson on fractions. Send two pupils to the board, telling the others to work the examples on their papers. Then let the two at the board work the same example simultaneously. The one who solves it first may take his seat. As soon as the second pupil has solved it, the teacher sends- a third pupil to the board to work the next exercise with him. In this way the second pupil, who had trouble, gets additional drill. If this method is continued the poorer ones will remain the longest and therefore get the most practice. Meanwhile all the others are busy. The pupils are expected Divisions of Elementary Algebra 83 to solve the problems as rapidly as possible, the teacher during the meantime helping those at the seats who are experiencing difficulty. To vary this method, it is sometimes well to allow the one remaining at the board to choose his next opponent. The Assignment. — 1. Definitions of complex fractions. 2. Directions for their solution. Complex Fractions Defined. A fraction containing one or more fractions in the numerator or denominator, is called a complex fraction. For example : x _y_ a T In other words, it means that the quotient obtained by dividing x by y is divided by the quotient obtained by dividing a by b. It may be set down like this : x _ a y ' b and it then becomes similar to fractions in to-day's lesson. But sometimes the numerator of the complex fraction may itself be a mixed number or a series of fractions or another complex fraction, in which case it becomes necessary to follow out certain definite directions. These are : a. Simplify the numerator. b. Simplify the denominator. c. Divide the first result or quotient by the second. Illustration : 84 Supervised Study in Mathematics and Science Bya: x +?- = 3x+y . 3 3 3 By6: a+ b - = ^±. y 3 3 Bye: 3^±Z ^3^±& or 3£HL + _3_ = 3£±y. Am 3 3 3 3 a + b 3 a + The Study of the Assignment. — I or Minimum Assign- ment. Exercises 2-1 1, in text. 7/ or -4uerage Assignment. Exerdses 12-17, m text. III or Maximum Assignment. Make up and solve five complex fractions. The Silent Study. — The three steps as outlined above should be written upon the board with colored crayon so that the pupils may have them plainly in view. If they will care- fully follow out each step as applied to each exercise, the class will have no difficulty. If they do have trouble, it will be from carelessness. Most of the difficulty in fractions comes from the pupil's own illegible figures. The exercises on account of the awkward shape of their graphic representation are easily confused unless care be taken to set them down in good form and adhere to logical order in their solution. The teacher, by passing around among the pupils, will be able to note any such errors and he should avail himself of the op- portunity to correct them. In case someone has difficulty and calls upon the teacher for directions, unless the difficulty is easily found, it will be better for him to start anew, with the teacher overseeing that appli- cation is made of the three successive steps. If any help is given, it should be only to direct properly the application of these rules to the exercises under consideration. Divisions of Elementary Algebra 85 LESSON X A Red Letter Day Program in the Nature of a Field Day I. Parade. — Each pupil in the class is to be assigned some rule covered in algebra during the first term, which he is to recite upon being called to the front of the room. For instance, the teacher, or judge of the parade as he might be designated, will announce : To add two algebraic numbers. The pupil who has been assigned this rule will come forward and answer : " If they have like signs, add the absolute values and prefix the common sign ; if they have unlike signs, find the difference of the absolute values and prefix the sign of the numerically greater." (Milne.) When all have been called on, the judge might award a prize, of no intrinsic value, to the pupil who made the best appearance and recited the rule in the most distinct voice. II. Races. — (Select three qualified pupils to act as judges.) 1. Multiplication Race. Method. Have two exercises in multiplication, exactly alike, upon the board. Send two pupils to work on them ; the one getting his done first and correctly wins. 2. Division Race. Method. Similar to above, but using different pupils. 3. Championship Race in a Multiplication and Division Contest. Method. Give each of the winners in the first two races the same exercise, which will be a combination of multipli- cation and division. The one solving it correctly is considered the champion, and if deemed advisable may be awarded some prize, such as a colored ribbon. 86 Supervised Study in Mathematics and Science 4. Grand Relay Race in Removing Parentheses. Method. Put two exercises involving the removal of a number of signs of aggregation, upon the board. Pick out a relay team of at least as many pupils for a side as there will be complete operations. Start one from each side at the same instant and as soon as one operation is complete, let the next in order take his place. The side that first gets the exercise done correctly wins. It is suggested that two pupils be al- lowed to choose sides for this, members going to the board in the order chosen. HI. Game of Factoring (modeled after baseball). Method. Select two teams of nine pupils each, preferably of pupils who have not taken part yet in the program, except the parade. These again may be chosen as noted above for the relay race. Also select an umpire. Each team will be composed of a pitcher, catcher, etc., as in a ball game. These positions will probably be best selected or assigned by the teacher, who will also explain the duties of the players and the rules of the game. The pitcher will read the exercises in factoring which will have been handed to him by the teacher. The catcher will try to tell the type form of the exercise before the batter can do so. The batter will tell the type form of the exercise as soon as he can. For instance, if the example is: a?+2ab+b 2 , he will say : " a perfect square." Each baseman and fielder will have been assigned some type form and his duty will be to solve the exercise by its application as soon as the batter refers it to him. For instance, the short- stop may have been assigned " a perfect square " as his position, so that as soon as the batter gave the exercise this classification, Divisions of Elementary Algebra 87 the shortstop will solve it. If he correctly solves it, the batter is out ; if he cannot solve it, the batter makes a home run. If, however, the catcher gives the correct form before the batter does, and the fielder can solve it correctly the batter is also out ; if the fielder fails in this case, it is called a strike and the batter has another chance. Three such strikes will put him out. Again, if the batter gives the wrong type form, it counts a strike. Failure to understand the exercise at the first read- ing constitutes a foul ; the first two count as strikes, as in baseball. The game may be varied as to number of innings, accord- ing to the length of time that is available, but probably three will suffice for this program. As already mentioned, there should be an umpire to call strikes, fouls, etc. The teacher himself may act as referee in case of dispute. One or two score keepers may also be selected. IV. Picnic. — Method. Each of the following typical exercises in fractions may be considered to represent different articles of food, and the ability to solve them correctly will give the pupil a helping of each kind. Inability to solve the last one, for instance, which represents ice cream, would de- prive him of this dish. All pupils are given paper and told to solve the exercises which are placed upon the board. In parentheses is indicated what each exercise represents. The picnic may be held after school, if the period is not long enough, which probably will be the case. It would also be difficult to determine the amount of food needed before then. 88 Supervised Study in Mathematics and Science Exercises I , , 2S -+ 1 + 2. 1+5 4-c 2 Simplify : Simplify : Simplify : Simplify : Simplify : 3c*- 9 e-54 1 — fi-» -1— T f Reduce to mixed number : — 3 ■ s c — 4 c + 2 16 — c 2

74~7S; technic of textbook in, 47-48 ; textbook in, 51 ; time table for, 25 Al-jebr w 'al muqubalah, 27 Allen, L. M., 10 Amortization of interest-bearing notes, 3° Angles, properties of, iio-in, 123; questions on, 123; theorem for vertical, 112, 117 Animals, snapshots of, 196 Answers, use of, 48; complete, 55 Ants, 182 Aphids, 182 Apparatus, construction of, 152 ; physi- cal, 219-223; special, 151; tinker- ing with, 150; use of gymnastic, 204; wireless, 226 Aquarium, 171 Arches, 108 Archimedes, 216-217, 221 Architecture, 31, 137 Arithmetic, common errors in, 36; Comte's definition of, 35 ; examples in> 37) 38-39; formulas used in, 36; interdependence of algebra and, 37-38; nomenclature of, 37; pupil's present knowledge of, 60-61, 77 ; review of fundamental processes in, 36-37 Articles, magazine and newspaper, 34, 160, 186, 226, 227 Assignment, 6 (see also each lesson outlined); aim of, 12; average, 15, 64, 67, 132, 151; completion of, 12-13, 64. explanation of the new, 133; importance of, 12; maximum, 15, 64, 67, 68, 117-118, 122, 132, 151, 199; minimum, 14-15, 64, 67, 68, 129, 132, 151; nature of, 12; study of the maximum, 121- 122; study of, 12, 133, 162-165, 167-168, I73-I7S, 177-178, 183-184, 191-192, 202, 215-218, 222-223; m 234 Index summary of, 41 ; summary on the study of, 42 ; the threefold, 14-15, 151 ; time allotted to, 12 Assignment sheet, how to make, 14, 63-64, 72, 151 ; how to use, 15-16, 128, 130; illustration of, 19, 62; object of, 13-14; the threefold, 14 Astronomy, 31 Attention, individual, 69 Authorities, varied opinions of, 155 Automobile, 30, 219 Axioms, 105 Bacteria, 156 Banking, 8, 182-183, J 86, 191 Bibliographies, 186 Biennial, 171 Biology, animal, 156; conducting a field trip in, 179-181; correlation of English and, 205 ; divisions of, I 5S _I S6; equipment of classroom in, 177-178, 196, 201; human, 156; lessons in, 213; plant, 155-156; problems of, 159, 166-167; survey of the course in, 159-160; use of notebooks in, 162 ; valuable lessons of, 157-160 Bird houses, 196-197 Birds, 156, 160; stereopticon lecture on, 196; stories about, 187; study of, 178 Blackboard, use of, 16, 27, 39, 40, 41, 42, 47, 56-57, 60, 63, 69, 71, 72, 77, 79. 80, 82, 84, 87, 92, 94-95, in, 113, 117, 119, 122, 123-125, 129- 130, 134-135, 152, 158, 164, 169, 171, 184, 187, 190-191, 193, 196, 200-202, 202-203, 219-220; use of the spherical, 145 Blood, 156 Bonds, valuation of debenture, 30 Bones, 156, 197 Book, the open, 47-48 Books, 160, 186, 193, 226; supple- mentary, 186 Bordeaux mixture, 192 Botany, 155, 195 Bridges, 108 Bulletins, 186 Buoyancy, 216-217 Burbank, Luther, 159 Business, statistics of, 79 Busy work, 15 Butterfly, 182 Cage, 157, 177,183 Calories, 207 Camera, hunting with, 197 Cancellation, 37 Canton, N. Y., 10, 13, 68 Capitol, at Washington, 108 Carbon, 163, 164 Cards, index, 143-144; problems written on, 191 ; reviewing Book I through use of, 134-135; use of the divided, 114-116, 119-122 Carelessness, 14 Carpenter, 58 Catcher, 86, 87 Caterpillar, tent, 193 Ceilings, steel, 108 Charts, 196; making, 151, 171, 173, 174, 175, 182, 184, 187, 195 Checking, value of, 66, 245 Chemistry, 31, 203 Children, educating all the, 151 Circle, 105-106, 109, 145; circum- ference of, 38 Circulation, 156 Class, testing progress of, 75 Cochineal bug, 192 Codling moth, 186, 192, 193 Coefficient, 61 Coleoptera, order of, 190 Coloration, protective, 158 Comte, 35 Concentration, 44-45, 74, 108 Cone, 109 Contents, table of, 7, 47 Contests, 33 Index 235 Contractor, 32 Corn, kernel of, 171, 173 Cotton-boll weevil, 192, 193 Course, 7 ; bird's-eye view of, 26, 33, 109 Course of study, evaluation of, 20, IO S> *S5> 2I 3J minimum essentials, 208 Court, the class as a, 89-91 Crayfish, 182, 195 Crayon, use of colored, 84, 190, 196, 201, 219; waste of, 41; yellow, 71 Credit, awarding extra, 64-65, 101 Crustaceans, 156, 160, 181-182 ; ques- tions on, 182 Current events, 33 Curriculum, 6; college preparatory, 7; domestic science, 7 Cylinder, 109 Dandelions, 181 Darwin, Charles, 182 Definitions, 21, 51, 105 Density, 216-218; formula for, 217 Descartes, 218 Devices, 6, 9, 43, 71 Diamonds, 164 Dictionary, use of, 170 Dietitian, 31 Digestion, organs of, 156 Digits, 35 Diophantus, 28 Diptera, order of, 190 Discipline, 180; formal, 108 Dividends, distribution of, 30 Doughnuts, 88, 207 Drawings, 109, 136, 177, 188, 195, 2or, 222 Drill, 82; function of, 42; impor- tance of, 71 Earhart, Lida, 8, 46 Earth, size of, 107 Education, 29, 48 Egyptians, 28, 107, no Elections, 33, 79 Embryo, 172-173, 176 Encyclopedia, 31 Endosperm, meaning of, 172 Engineering, 30, 31, 107 English, technical, 205; use of pure, 205 Enthusiasm, arousing, 26 Entomologist, class, 193 Environment, correct, 44; definition of, 170; importance of, 158; of the pupil, 178, 186; varieties of, 171 Equation, 73-75; applications of, 77; cubic, 29; quadratic, 20, 24, 55, 75, 92-93; simple, 20, 23, 38; study of, 75-77 Equipment, 6, 151, 196, 201 Euclid, 107, 109, no Evolution, 20, 23 Examination, a sample, 98-100, 206- 207; an analysis of the suggested, 207-209 ; value of the suggested, 100-101 ; criticism of the ordi- nary, 208; final, 15; formal, 4; grading, 74; object of, 97; pre- academic, 37 ; regents, 3, 208-209 ; standardized tests as, 97-98; written, 9, 97-98, 166 Excursions, field, 162, 194; how to conduct, 1 79-181; importance of, 179 Exercises, oral, 55; treatment of, 55, 225; written, 55-57 Exhibition, or "red letter day" lesson, method of conducting, 136; ob- ject of, 135 ; place for, 135 ; prepar- ation for, 135-136; program of, 136-137 Existence, struggle for, 169-170, 171, 180 Experiences, 34, 58 Experiments, how to conduct, 174- 175, 175-178; home, 178, 186 Explanations, 40 236 Index Fabre, Henry, 138, 182 Factoring, 8, 20, 22; exercises in, 81-82; game of, 86-87; lesson on, 80-82 Factors, highest common, 20, 22; modifying, 25; technical, 45 Failures, causes of, in mathematics, 3-4 Federal Bureau of Entomology, 159 Figures, rectilinear, 105, 109, 110-135 Fiori, 29 Fishes, 136, 160 Flowers, 155. *77 Fly, house, 193 ; tachina, 192 ; ichneu- mon, 183, 191 Flytrap, 196, 197 Foods, 156 Forests, 155 Formulas, algebraic, 31, 108; arith- metic, 36, 38, $3 Fractions, 20, 22-23, 36-37, 63, 75 ; complex, 83-84; definition of com- plex, 84; lesson on, 82-84; multi- plication of, 82-83 Frog, 156, 195 Functions, 158 Games, ball, 33, 79, 86-87 Geology, 31 Geometricians, lives of, 136 Geometry, plane, 213; applications of, 132; bird's-eye view of course in, 109; deduction of a proof in, 1 1 2-1 14; discipline of, 108; divi- sions of, 105-106; history of, 107; meaning of the word, 107, 109; originals in, 127-128; practical value of, 107-iog ; review questions in, 109-110; steps taken in prov- ing a proposition in, 114; study of originals in, 124-128; suggestions for studying, 114-116, 118 Geometry, solid, 141, 145-146 Germination, 174; experiments in seed, 176-177 Goethals, George W., 159 Grade, testing for a final, 98 Grades, arithmetic in, 37 ; supervised study in, 146 Graphite, 163 Graphs, 20, 23, 38, 75 Grasshopper, 157-158, 182; dissection of the, 187, 195 Gravity, specific, 217-218, 219 Guidebooks, textbooks as, 161, 183 Habitat, 158 Hall-Quest, Alfred L., 6, 14, 63 Hamilton, Sir William, 28 Handwriting, 65, 135 Health, 157-158, 203; restoration of, 73 Herbarium, 177 Heron of Alexandria, 28 Home work, n, 57-58; value of, 58, 186-187 Hotz' scales, 75 House fly, 193 Ice cream, 87-88 Ichneumon fly, 183, 191 Index, card, 143; units of, 7, 20, 47, 51, 105, 144, 155-156, 213 Insects, 156, 160, 182-183; a game about, 193-194; beneficial, 183, 189; biting, 1 go; characteristics of, 182, 189; classification of, 183, 189; economic loss from, 192; harmful, 183, 189, 193-194; in- teresting incidents concerning, 159, 182; list of supplementary topics on, 192; pictures of, 194; sucking, 190 Insurance, casualty, 30 Interest, arousing the pupils', 160; problems in, 38 ; theory of, 30 , Inventions, 29-30 f Involution, 20, 23' Iodine, 171 Index 237 Iron, 163 Italics, use of, 5a' Koch, Dr., 159 Laboratory, supervision of work in, 150, 219-223 Lantern, stereopticon, 197, 226 Leaves, 155, 167 Legibility, importance of, 65, 135 Lesson, aim of the, 71 ; definitions of various types of, 8-9; private, 72; purpose of a socialized, SS Lesson types: correlation, 34-43, 189-192, 204-206; deductive, 117- 122, 129-133, 202-204; deductive and how to study, 82-84, 110-116; examination, 95-101, 206-209; ex ~ pository and how to study, 73-80, 92-95, 214-218; how to study, 43-50, 123-129, 161-165, 171-17S, 181-187, 197-199, 223-225; in- ductive, 59-67, 67-73, 165-168, 169-171, 175-178, 225-227; induc- tive and how to study, 50-59; laboratory, 187-189, 200-202, 219- 223; preview, inspirational, 26- 34, 106-110, 156-160; red letter day (see Program), 85-88, 135- *37> IQ 5 -I 97> socialized, 80-82, 179-181, 192-194; socialized re- view, 88-91, 134-13S Librarians, 144 Library, contents of biologic, 186 Life, biology, the study of, 157 Lincoln, Abraham, 108 Lines, parallel, 105 Loci, 105 Logic, 108 McMurry, Frank M., 45 Magazines, 34, 160, 186, 226, 227 Mammals, 156, 160 Man, existence of, 157; study of, 160, 167 Manuals, laboratory, 1 51-15 2 Material, accumulation of, 55, 179- 181, 186; available, 155; exami- nation of biologic, 161 ; importance of a variety of, 167; source of, 45, 5°. 79. 149, 160, 178-179; supple- mentary, 14, 45, 80, 132, 151, 165; treatment of, 53-55, 150, 167 Mathematicians, pictures of, 27 Mathematics, characteristics of, 3; contributors to, 28; English of, 27; failures in, 3 ; language of, 52; mastery of, 71 ; practical value of, 29-32; severity of, 4, 224 Matter, functions of living, 158 Memoranda, 14, 62, 130, 144 Memorization, kinds of, 46; power of, 5; reliance on, 116 Memory, employment of, 121, 224; function of, 46 ; overdeveloped, 4 Mensuration, 108 Meteorology, 31 Method, adaptation of, 59 ; Austrian, 37 Microscope, use of, 188-189, 200-201 Milkweed, 181 Mimeograph, use of, 48, 73, 77, 127, 151-152 Mistakes, common, 65; correction of, 60, 65, 128-129, I 3 2 , J 69; ex- amples of common, 38 ; glaring, 65 ; how to avoid, 65-66; how to find, 84, 94, 214 Monitors, pupils as, 92 Monocotyledon, derivation of word, 172 Monomials, addition of, 21, 61, 68-70; division of, 22; factoring, 22, multiplication of, 21 ; subtraction of, 21 Morale, 41, 64 Morey, 158 Moritz, R. E., 30-31 Mosaics, 108 Mosquito, 183, 186, 193 238 Index Mountains, 169 Moving-picture machine, 226 Multiples, common, 20, 22 Muscles, 156, 198; biologic effect of exercise on, 204-206; questions on, 199 Museum, making a, 178 Myers, G. W., 9 Narcotics, 156 Nature, dissimilarities in, 167; tran- quillity of, 169 Naval Observatory, at Washington, 31 Navigation, 31 Neatness, importance of, 135 Newspapers, 34, 160, 186 Newton, Sir Isaac, 28, 35 New York State Education Depart- ment, lantern slides from, 197 ; statistics of, 3, 20; syllabus of, 20, 1SS-1S6 Notebooks, criticism of the work in, 133; display of the best, 133, 195- 196; how to study the returned, 132; loose-leaf, 15, 162; manip- ulation of, 128-129, 1 3° _I 3 2 ) J 62, 184; pupils', 125-126; record- ing experiments in, 164-165, 168, 174-177, 179, 219-222; the teach- er's, 34; value of, 129 Notes, historical, 34, 105 Numbers, literal, 21 ; positive and negative, 20, 21, 60, 61; signed, 21 Operations, fundamental, 34, 36 Originals, rules for studying, 128; study of, 125-128 Outline, for the study of insects, 194 Panama Canal, 159 Paraffin, 218 Paragraph, study of the, 51-52, 54, 188, 198; writing a, 166 Paramecium, 195 Parentheses, 21 ; removing of, 38, 86 Parthenogenesis, 192 Patterns, tile, 108 Payments, equation of, 30 Percentage, 8, 36 Period, length of, 10-n, 68, 150; management of, 17-18 Philosophy, school of, 108 Phosphorus, 163, 165 Photography, 222 Photos, 226 Physician, 31, 73, 186, 226 Physics, 31, 213, 222, 223; lantern slides on, 226; teacher of, 226 Physiography, 213 Physiology, 31, 155-156, ^95 Picnic, an educational, 87-88 Plant, electric power, 226 Plants, cellular structure of, 155; classification of, 171 ; problems of, 170; study of, 157, 177 Plumule, 172 Pointer, use of the, 124 Polygons, areas of, 105-106; regular, 105-106 ; similar, 105-106 Polynomials, 68; addition of, 21, 70- 71; division of, 22; factoring, 22; multiplication of, 21 ; rule for adding, 71 ; subtraction of, 21 Postulates, 105 Potato bug, 193 Press, hydrostatic, 226 Preview, inspirational, 26, 34, 106- 110, 141, 156-160, 182; conditions for a successful, 33-34; in arith- metic, 60-61 ; method of, 26, 157- 160; need of, 26; purpose of, 26, 106-107, 156-157; questions on, 35 Principal, 136 Prizes, 85 Problems, analysis of several, 78 ; ap- plying the equation to, 77 ; business, 30-31 ; conception of, 72 ; daily study of, 79, 166-167; difierent Index m kinds of, 96-97, 223; directions for studying, 77-78, 127, 223-224; questions leading up to the study of, 75 ; recognition of, 61-62, 70- 71; sensing, 45, 168; solving, 36; standardized tests on problems, 75; statement of, 112 Program (see Lessons, types of) : pur- pose of, 19s; "red letter day," 85-88, 136-137. 195-197 Program of studies, 2 ; place of ad- vanced mathematics in, 141 Proportion, 105-106 Protractor, use of, 137 Psychology, 31 Pupils, characteristics of, 13, 72-73 ; classification of, 16, 18, 98; collec- tion of specimens by, 160; criti- cism of work by, 203 ; elimination of, 13, 69, 70; embarrassment of, 72; free expression of, 34; grading of, 18; grouping, 72; guidance of, 13; ingenuity of, 152; judgment of, 82 ; judgment on part of, 135, 191 ; maximum, 137, 189; name of, 14; responsibility of, 177, 184; seating of, 17, 42, 72; self-reliance of, 123, 18s Pyramids, 107, 109 Pythagoras, 107, no Quadrilaterals, 105 Quartz, 218 Quaternion Bridge, 29 Questions, how to ask, 187, 191 ; im- portance of asking skillful, 131 Quiz, importance of oral, 6, 166; method of the, 200, 202-203 Race, championship, 85 ; division, 85 ; multiplication, 85; relay, 86 Radicals, 20, 24, 75, 88 Reading, necessity of intelligent, 78; supplementary, 151, 186 Reasoning, faculty of, 5 ; undeveloped powers of, 4 Recitation, the complete, 205; the unsupervised, 5 ; types of, 8 ; units of, 8, 21-24, 5 1 . 105-106, 165, 213 Record of work, 14, 16 Recreation, 15 Regents academic examinations, 3 Resourcefulness, of pupils, 69, 152, 208 Results, checking, 18, 66^67 Review, methods of, 11-12, 35, 44, SO-31. 59-6o, 68-70, 73-75, 80-81, 82, 83, 88-91, 92, iio-in, 117-118, 123-125, 129-132, I34-I35. 165- 166, 169, 171, 175, 181-182, 187, 189-191, 192, 193-194, 197-198, 200, 202-203, 204-205, 214-215, 219, 223 ; nature of the, 41, 46, 161, 175; purpose of, n; socialized, 60, 88-91, 134-135; summary of the, 40-41 Roll call, 17, 67 Romanes, George, 159, 182 Romans, 29, 35 Roosevelt Dam, 108 Roots, 155 Rugg and Clark's tests, 74-75 Ruler, 44 Sanitation, 156 Schedule, daily lesson, amplification of, 68; divisions of, 10-11 ; impor- tance of, 17; sample sheets, 19, 62; time (see Lessons, types of) School, the average, 122; the cor- respondence, 144 Schoolroom, 9, 144 Schultze, Arthur, 4 Science, algebra, a general, 36; fas- cination of, 214; function of the study of, 205 ; importance of super- vised study in, 150, 173; popularity of, 149; practical aspect of, 149; the study of, 149 Scorekeeper, 87 Seeds, 155, 171-175, 176-178; dis- persal of, 181 240 Index Semester, 141 Sheets, mimeograph, 48, 73, 77, 127, 151-132 Shipbuilding, 30 Signs, 72 Simpson, Mabel E., 13, 59 Smith, Dr. Eugene, 29 Snapdragon, 181 Solar system, 108 Spheres, 109 Square, method of completing the, 93 ; rule for completing the, 93; the perfect, 86-87 Squash bug, 193 Standardized tests, 6, 97-98; Hotz', 75; the Rugg and Clark's, 74, 97; value of, 74 Statements, loose, rg9 Station, pumping, 226 Statistics, 39 Stems, 155 Stenographer, 58 Stereopticon, 197, 226 Stereoscopic views, 145 Stimulus, supplying the proper, 70 Strayer, George D., 8 Study period, function of, 12-13; management of, 17, 39, 43, 63-67; organization of, 46-47 ; the logical culmination of, 146; the teacher's duty during the, 38, 65-66, 118; work done outside of the, 57-58 Study, how to, instruction in (see also how to study lessons), 43-49, 51-55, 78, 114-116, 127, 128, 131, 161-165, 172-173. 184-198, 214-215; pur- pose of lessons on, 43-44, 161 Study, cooperative, 57 ; correct habits of, 54; methods of, 44-46, 114-T16, 119-122, 163-165, 188-189; sum- mary on the silent, 43; the period of silent, 39-40, 63-66, 84, 94-95, 116, 110-122, 133, 184-186, 188- 189; units of, 8; value of outside, 57-58, 186-187 Studying, cooperative, 57 ; rules for, 49 Submarine, 218 Sulphur, 163-165 Superintendent, 136 Supervised study, function of, 5-6, 43-44, 69, 119, 151; Hall-Quest on, xiii-xvi; installation of, 10-n; management of, 17; meaning of, 149; organization of, 46-47, 68; relationship of, 6; technic of, 6; value of, 5, 42, 150, 173 Surveying, 31, 107, no, 137 Symbols, 21, 35, 54; meaning of, 53; origin of, 39 Symmetry, 106 Sympathy, 6, T2, 13, 43, 96, 146 Table, time, 25, 156 Tachina fly, 192 Tartaglia, 29 Teacher, 205 ; activity of, 80 ; check- ing up work by, 66-67; duty of, 40, 65-66, 121, 124, 131, 144, 184, 189, 245; helps for, 63-64; judg- ment of, 33, 134; leadership of, 152; opportunity of, 168; origi- nality and individuality of, 59; preparatory work of, 171-173, 179- 180; tactfulness of, 72; the pupil as teacher, 198 ; use of standardized tests by, 74-75 Technic, mastery of, 95, 146 Terms, significance of, 72; transposi- tion of, 28, 76-77, 93, 94 ; use of, 6 Tests, 6, 142, 166; Hotz', 75; prob- lems as real, 96-97; real, 95-97; Rugg and Clark's, 74, 97; speed, 73-74; standardized, 6, 97-98; time, 45; written, 97-98 Textbooks, 109; arithmetic, 38; study of, 47, 149, 183-184; sup- plementary, 51, 57-59, 91, 122, 186-199; use of, 44, 161, 203; varying characteristics of, 167; verification of, 177, 223 Index 241 Thales, 30, 137 Theorem, 112, 117; explanation of the new, 118; review of the, 134- i3S Thoroughness, importance of, 135, 144, 165, 174 Time {see also each lesson outlined), allotment of, 17, 150; amount of, 59, 150; efficient use of, 6, 174-175 Toad, value of, 186, 192 Tools, 44, 152 Triangles, 105, 109, 123; definition of, 118 Trigonometry, plane, 141, 145-146 Type forms, explanations of, 61 ; dif- ferent, 82 Types, classification of exercise, 9 Unknown, use of, 36, 76-78 Variation, meaning of, 168 Verification, importance of, 149 ; methods of, 66-67, 72-73, 93-94, 120, 176, 202 Vieta, 28, 35 Volume, 2T7-218 Weber-Fechner law, 31 Weeds, 170, 186 White, C. E., 30 Wiley, Dr. Harvey, 159 Wings, classification of insects ac- cording to, 1 go Work bench, 152 X-ray, 226 Zoology, 155, 183, 195