University of Texas Bulletin No. 2706: February 8, 1927 THE TEXAS MATHEMATICS TEACHERS' BULLETIN Volume XI, No. 2 PUBLISHED BY THB UNIVERSITY OF TBXAS AUSTIN Publications of the University of Texas Publications Committees: GENERAL: FREDERIC DUNCALF E. K. McGINNIS J. W. CALHOUN H. J. MULLER J. L. HENDERSON G. W. STUMBERG HAL C WEAVER OFFICIAL: W. J. BATTLE R. A. LAW H. Y. BENEDICT F. B. MARSH C. D. SIMMONS The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue, the last two the position in the yearly series. (For example, No. 2201 is the first bulletin of the year 1922.) These comprise the official publications of the University, publications on humanistic and scientific sub­jects, buUetins prepared by the Division of Extension, by the Bureau of Economic Geology, and other bulletins of general educational interest. With the exception of special numbers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to University Publica­tions, University of Texas, Austin. University of Texas Bulletin No. 2706: February 8, 1927 THE TEXAS MATHEMATICS TEACHERS' BULLETIN Volume XI, No. 2 PUBLISHED BY THE UNIVERSITY FOUR TIMES A MONTH, AND ENTERED AS SECOND-CLASS MATTER AT THE POSTOFFICE AT AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24, 1912 The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of a free govern­ment. Sam Houston Cultivated mind is the guardian genius of democracy. • • • It is the only dictator that freemen acknowl­edge and the only security that free­men desire. Mirabeau B. Lamar University of Texas Bulletin No. 2706: February 8, 1927 THE TEXAS MATHEMATICS TEACHERS' BULLETIN Volume XI, No. 2 Edited by C. D. RICE Associate Professor of Applied Mathematics and MARY E. DECHERD Adjunct Professor of Pure Mathematics MATHEMATICS STAFF OF THE UNIVERSITY OF TEXAS P. M. Batchelder Helma L. Holmes H. Y. Benedict Goldie P. Horton J. W. Calhoun R. L. Moore C. M. Cleveland M. B. Porter A. E. Cooper C. D. Rice Mary E. Decherd W. J. Trjitzinsky E. L. Dodd H. S. Vandiver H. J. Ettlinger This Bulletin is open to the teachers of mathematics in Texas for the expression of their views. The editors assume no r-2sponsibility for statements of facts or opinions in articles not written by them. CONTENTS Foreword ___________________________________________________. ..Editors ---------------------------------5 Report of National Committee_______________ Editors ---------------------------------8 Requirement in Mathematics for High- School Graduation_____________________________ ____ Miss Elizabeth Dice__ _______ ll A High-School Course of Study________________P. H. Underwood______________ l5 Correlated Mathematics for the High School -------------------------------------------------------R. E. Baker________________________30 "Algebraic Cut-Outs"_________ _ ,_________ ___________ Miss Rebecca Goldstein____35 Denver Public Schools·----------------------------· _Editors ________________ ___ ___________ 33 Textbooks in High-School Mathematics___J. E. Burnam_ _______ __________ _44 Methods of Defining the Trigonometric Functions ________________________________________________ Miss Mary E. Decherd___-49 FOREWORD The high school as it has been developed in our edu­cational system is a unit that has been set off as an inde­pendent element and is not related as it should be to the work of the grades below or to the college that follows. Many new subjects are introduced in the high school that should be more closely related to the grades below. The break between high school and the grades in methods and subject matter is in many cases as striking as that be­tween high school and college. Some of the difficulties found in mathematics by students in the high school may be noticed: (1) The use of letters to represent numbers. (2) The symbols of operation. (3) The negative number. (4) Logical proofs about geometrical magnitudes and properties about which the student has not had sufficient preparation. The lack of correlation between high school and grades is felt especially by the pupil in making the transition from arithmetic of definite numbers to the general statements of algebra. In a great majority of the high schools of our State this transition is made without any preparation whatever in the use of letters to represent numbers in the last year of arithmetic. This transition should be made slowly and intelligently. The student should become accus­tomed to the use of letters and be able to think mathe­matically in terms of such symbols. This work in arith­metic should not be dignified by the word algebra. A better name would be literal arithmetic. At this place may be given simple equations with sets of selected problems in which negative numbers are not involved. It is easily seen that we may use letters to great advantage in the subject of percentage, interest, proportion, mensuration, and many of the simpler business problems. A preliminary training in the use of letters in the last year of the grades will prepare the student for the be­ginning of formal algebra in the first year of the high school. The negative number which is so difficult to the beginner may then be given and formal algebra introduced. By thus leading the student into one difficulty at a time he is able to progress more intelligently than when the double abstraction of letters and negative numbers are given at the same time. This pedagogical mistake is made so far as we know in no other country except in our American schools. Another pedagogical error made in our schools is to define for the first time certain geometrical magnitudes in the second or third year of the high school and then begin in the same lesson to give logical proofs of properties and relations of those quantities. In educational systems of other countries two to three years preliminary training is given before logical or formal proofs are attempted. This is generally done by definitions, drawing, . constructions, measurements and considerable training in the computation of areas and volumes. There are many constructions, measurements, and drawings that should be made to familiarize the student with the quantities about which he is to make later logical deductions. The various attempts to introduce intuitional or objec­tive geometry in the grades below and lead in this way through constructions and measurements to logical deduc­tions has, in general, not become a success in our school systems. On account of this omission we have been forced to delay in our high schools the study of deductive geom­etry until the student has acquired a sufficient mental ma­turity to make any progress at all with the limited amount of material given in high-school texts. By placing geom­etry so late in the curriculum, algebra has been forced into the first two years of the high school and often com­pleted when the student is least mature. When the student is not mature enough to understand the logical processes necessary in algebra, it becomes fo him a piece of mechanism that will turn out results, but to him there is no real training. Work under such conditions is a loss in time and energy for both pupil and teacher. For this reason there is a growing feeling that the last year in algebra should come later in the curriculum than now shown in the majority of our high schools. Another error we make pedagogically is that we teach each branoh of mathematics too often w:thout any relation it may have to other subjects in mathematics. While one is studied the others are forgotten. The practice of teach­ing in the so-called "closed compartments" is in great con­trast with the correlated methods of other countries in which the relations of the various branches are shown at each step and the student is ready to use, intelligently, at any time what he has already learned in another branch. In the courses of study noticed in this bulletin it will be shown how the schools are meeting and overcoming the various difficulties in curriculum. We shall be glad to receive for our next bulletin any new or ideal course of study or any suggestion that will help in the revision of high-school curricula. REPORT OF THE NATIONAL COMMITTEE "The National Committee on Mathematical Requirements was organized in the late summer of 1916, under the auspices of the Mathematical Association of America, for the purpose of giving national expression to the movement for reform in the teaching of mathematics, which had gained considerable headway in various parts of the country, but which lacked the power that coordination and united effort alone could give. The results of the com­mittee's work and deliberations are presented in a report published by the Bureau of Education, Washington, D. C." Those wishing the report may obtain it by writing to the Bureau, asking for Bulletin 1921, No. 32, and sending 10 cents for each copy. This report is full of information and good suggestions for those wishing to take up the study of reorganization of the high-school curriculum. To meet the requirements that may be met in arranging and organizing the material in a high-school course we quote the following from the report: The Junior High School.-In view of the fact that under this form of school organization, pupils may be expected to remain in school until the end of the Junior High School period instead of leaving in large numbers at the end of the eighth school year, the mathematics of the three years of the Junior High School should be planned as a unit, and should include the material recommended. There re­mains the question as to the order in which the various topics should be presented and the amount of time to be devoted to each. The committee has already stated its reasons for not attempting to answer this question. The following plans for the distribution of time are, however, suggested in the hope that they may be helpful, but no one of them is recommended as superior to the others, and only the large divisions of material are mentioned. PLAN A First Y ear.-Applications of arithmetic, particularly in such lines as relate to the home, to thrift, and to the vari­ous school subjects; intuitive geometry. Second Y ear.-Algebra; applied arithmetic, particularly in such lines that relate to the commercial, industrial and social needs. Third Y ear.-Algebra; trigonometry; demonstrative geometry. By this plan demonstrative geometry is introduced in the third year, and arithmetic is practically completed in the second year. PLAN B First Year.-Applied arithmetic (as in plan A) ; intuitive geometry. Second Y ear.-Algebra; intuitive geometry; trigonom­etry. Third Year.-Applied arithmetic; algebra; trigonometry; demonstrative geometry. By this plan trigonometry is taken up in two years, and the arithmetic is transferred from the second to the third year. PLAN C First Year.-Applied arithmetic (as in plan A) ; intui. tive geometry; algebra. Second Y ear.-Algebra; intuitive geometry. Third Year.-Trigonometry; demonstrative geometry; applied arithmetic. By this plan algebra is confined chiefly to the first two years. PLAN D First Year.-Applied arithmetic (as in plan A); intui­tive geometry. Second Y ear.-Intuitive geometry; algebra. Third Year.-Algebra; trigonometry; applied arithmetic. By this plan demonstrative geometry is omitted entirely. PLAN E First Year.-lntuitive geometry, simple formulas, ele­mentary principles of statistics, arithmetic (as in plan A). Second Year.-Intuitive geometry, algebra, arithmetic. Third Year.-Geometry, numerical trigonometry, arith­metic. Senior High School.-"In the majority of high schools at the present time the topics suggested can probably be given most advantageously as separate units of a three­year program. However, the National Committee is of the opinion that methods of organization are being expe­rimentally perfected whereby teachers will be enabled to present much of this material more effectively in combined courses unified by one or more of suC1h central ideas, functionality and graphic representation. "As to the arrangement of material the committee gives below four plans which may be suggestive and helpful to teachers in arranging their courses. No one of them is, however, recommended as superior to the others." PLAN A Tenth Year.-Plane demonstrative geometry, algebra. Eleventh Year.-Statistics, trigonometry, solid geom­etry. Twelfth Year.-The calculus, other elective. PLAN B Tenth Year.-Plane demonstrative geometry solid geom­etry. Eleventh Year.-Algebra, trigonometry, statistics. Twelfth Year.-The calculus, other elective. PLAN C Tenth Year.-Plane demonstrative geometry, trigonom­etry. Eleventh Year.-Solid geometry, algebra, statistics. ':l.'welfth Year.-The calculus, other elective. PLAN D Tenth Year.-Algebra statistics, trigonometry. Eleventh Year.-Plane and solid geometry. Twelfth Year.-The calculus, other elective. REQUIREMENTS IN MATHEMATICS FOR HIGH-SCHOOL GRADUATION MISS ELIZABETH DICE The high schools of Texas virtually require two years of algebra and one year of plane geometry for graduation. For the slow or immature child, the result is two or three years of mechanical shifting of one set of symbols into another set of symbols and one or two years of rebellious memory work. For the bright or more mature child, the result is three years of unrelated mathematics. In algebra, if the students are slow or immature, the teachers untrained, or the classes large, competitive drills, interesting equations, and thoughtful questions are dis­couraged. Thought problems, which should introduce algebra because they show the need for equations and me­chanical manipulations, are not only a small part of the adopted textbook but also are poorly arranged, awkwardly worded, and out of date. In plane geometry, the assignment of formal proofs of the congruency of certain angles and triangles before the students are taught to draw triangles or read angles or point out corresponding sides encourages unintelligent memory work, develops a dislike for logical reasoning, and discourages, electing fourth-year mathematics. The boys and girls who, despite the lack of a knowledge of intuitive geometry, understand the formal proofs of the text­book enjoy the rare opportunities o: discussing original proofs and the different cases of these and other problems. The majority of the teachers of mathematics in the secondary schools of Texas have had no training in the teaching of mathematics, have not gone beyond freshman mathematics in college, and have chosen the subject be­cause the papers are easy to correct, or because mathematics combines nicely with administrative work, or because of a vacancy in this department. These teachers do not know what to stress as a foundation for higher mathematics, they can neither answer the questions nor properly direct the energy and interest of prospective mathematicians, and have to follow the textbook no matter how poor it may be. Even those teachers who know mathematics and who, if given the opportunity to introduce new divisions of the subject or if given the same students two successive terms, know hoW' to teach, cannot help students to remember prin­ciples which are rarely applied. Do you remember a poem which you did know but have not repeated for two years? Can you name the presidents in order-or out of order? Can the average college graduate, three months after re­ceiving his degree, extract the cube root, arrange a magic square or solve without review any problem in calculus? Cube root, the magic square, and integration are used as frequently by college students as are factoring, radicals and quadratics by high-school students, and advanced appli­cations are no less difficult for the college students than the elementary principles are for the students of the high school. Texas high schools had the same requirements for gradu­ation in 1900 yet they sent to college students who knew more algebra than the students of 1926. Why? (1) The classes are larger. "The maim, the lame, and the blind," mathematically speaking, are in the high schools now. They take a great deal of the time which was given to the bright children. (2) The students are less mature. The mental age exceeds the chronological age-on the surface. Algebra which wiil stick for two years is not a surface subject. (3) The children of 1926 have less time. While the chil­dren of 1900 interested themselves in solving problems in pure mathematics, the children of 1926 are studying applied mathematics: the radio, the mechanism of an automobile, wiring the garage, etc. In, fact, so many of the results of mathematics do they know and understand and appreciate that they should not only be forgiven for not knowing the underlying principles but also should be given teachers, books, laboratories, and classmates to make mathematics one of the most enjoyable courses in the secondary schools. Some of the mathematical requirements for high-school graduation should be : teachers trained to teach mathe­matics, textbooks which deal with problems of this century, required and elective courses, and different classes or at least different assignments varying with the ability of the children. The ability of the child may be determined by a combination of the I. Q., grades from grammar school, and grades for the first few weeks in high school. Two years of general mathematics in which mental arith­metic, the language of algebra, intuitive geometry, ancl fundamental notions of trigonometry are combined in a way which the less gifted or vocationally inclined children can understand and appreciate should be made a require­ment for graduation. Half of this course may be given the third or fourth year thereby supplementing the mental age or the mathematical inaptitude with the chronological age. The more gifted children, mathematically speaking, espe­cially prospective engineers, can take the essentials of the course in general mathematics in one year. The two groups can, with different assignments, use the same textbook for the first year. In the small high school the two groups, even though in the same class, can be given different assign­ments. A course of related mathematics with more advanced algebra and formal geometry may complete the two-year requirement for the second group. The children wlhose work is satisfactory the second year in the second group may elect third, or fourth, or third and fourth year mathematics. The elective courses, since the children have been more or less balanced by the first two years, may be less closely re­lated courses. Certainly the latter half of the fourth yP.ar should be a preparation for college mathematics. The first group, that is the group not mathematically inclined, should, if their work for the first year is above the average, have the privilege of enrolling in the second group. Otherwise they should be required to take a seccnd year in general mathematics. The second group should be shifted to the first group any time occasion may arise. Briefly, the men and women who plan the mathematical requirements for high-school graduation should shift the children intelligently rather than allow the boys and girls to shift one set of symbols into another set of symbols unin­telligently. HIGH SCHOOL COURSE OF STUDY P. H. UNDERWOOD Ball High School, Galveston, Texas First Year. First Term. Ap31lied Arithmetic. Intuitive Geometry ARITHMETIC I. Review fundamental operation of fractions and dec­imals. The meaning the terms: sum, remainder, dif­ference, product, quotient, factor, multiple, least common multiple. The ability to express by words and by symbols any one of the terms, multiplier, multiplicand and product in terms of the other two. The ability to express by words and by symbols any one of the terms, division, dividend, quotient, and remainder in terms of the other three. In a simple division problem how is the quotient affected if the dividend is multiplied by a number? How is it affected if the divisor is multiplied by a number? How, if both divi­dend and divisor are increased in the same proportion? Reduction of common fractions to decimals and of dec­imals to common fractions. An appreciation of the fact that a common fraction in its lowest terms can be exactly expressed as a decimal only when its denominator contains no other prime factors than twos and fives. Short cuts in multiplication and division by 25, 125, etc. Tests of divisi­bility by 3, 4, 6, 8, 9. II. Percentage. Finding a per cent of a quantity. Finding a quantity if a per cent of it is given. Finding what per cent one quantity is of another. Finding the per cents equivalent to common fractions. III. Drill in the formulas l=prt A=p(l+rt). IV. Ratio and proportion. Application to indirect meas­urement. Easy examples in drawing to scale. V. Square root. Pythagoras' Theorem. c2=a2+b2 • VI. Areas: Rectangle, square, parallelogram, triangle, trapezoid polygon with study of the formulas for perimeters and areas. Circumference and area of the circle. P=2a+2b, A=lw, A=1f2bh, A=S2 , A=1f2(a+b)h C=2r.r, A=7TT2, A=\IS(S-(J,)(s-b)(s-c). VII. Volumes and surfaces of the prism, pyramid, cylin­der cone and sphere with the corresponding formulas: V=lwh, V=Bh, V=l/sBh, V=7r1"2h, V=1/.