THE TREND OF THE CONTENT OF FIRST-YEAR ALGEBRA TEXTS BY ELINOR BERTHA FLAGG B. S. University of Illinois, 1921 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN EDUCATION IN THE GRADUATE SCHOOL OF THE UIVERSITY OF ILLINOIS 1922 URBANA, ILLINOIS £ 53 UNIVERSITY OF ILLINOIS THE GRADUATE SCHOOL JUriG cL , _JQ2<- 1 HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION by El ino r Bert h a Flag g ENTITLED The. .Trend of the Content of First.- Year Algebra- Texts — BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE DEGREE OF Master Qf__SclerLQa_J.n Educ atl.on Recommendation concurred iiP Committee on Final Examination* •Required for doctor’s degree but not for master’s A i hz <0 jl Digitized by the Internet Archive in 2015 https://archive.org/details/trendofcontentofOOflag TABLE OF CONTENTS Chapter Page I Introduction 1 II The Problem and the Method of this Investigation 15 III The Content of Twelve First Year Algebra Texts E4 IV Summary • The Trend of the Content 56 Chapter I Introduction Discussions of the secondary school curriculum of today never fail to provoke lengthy debates concerning mathematics. These debates center around the following questions. What topics and processes should be taught? to whom? how much? and why? Such questions have become increasingly important because of the interests and capacities of the pupils now entering high school. Formerly students attending secondary schools were a highly se- lected group. Their purpose in attending these schools was to prepare for college and courses were planned to meet the demands which the college made for entrance. The function of the modern high school, however, cannot be stated as simply as this. Older educators coped with a simple industrial situation while the modern educator faces an extremely complex situation. The watch- word of the modern high school, as Johnston quotes, is M equal opportunity for all the children of all people” • * This means that today all pupils of the high school age have a right to demand an education; and in this day of de- mocracy ** America demands that we educate the whole group. * Johnston, G. H. — Modern High School, Chapter I, P. 10 Jessup, W. A. -- "The Greatest Need of the Schools -- Better Teaching.” Journal of N. E. A., Vol. X, No. 4, Pp. 71 - 73, April 1921 2 With this ideal our high school enrollment necessarily Includes pupils from many walks of life . We no longer have only hoys and girls of exceptional ability. Thorndike points out * that our pupils of today differ from those of twenty-five years ago in their experiences and interests and in their capacities and abilities. They are born into a more complex social and industrial organization. In- dustries and industrial centers have enlarged; more children are found in large cities where they are experiencing various modern inventions. The automobile, the wireless, and innumer- able changes in factory machinery are new in the last quarter of a century. As a result of these changing conditions the enrollment of the high school has changed both quantitatively and qualita- tively. Thorndike estimates * that the number of high school pupils in 1918 was six times that in 1890, while the population had not doubled in that time. He also shows that for every one hundred children who reached the age of fourteen there were about three and one-half times as many entering high school in 1918 as in 1 890 ; therefore we may say that where one pupil in ten entered high school in 1890, in 1918 one pupil in three entered. The interests of these pupils were more varied in 1918 than in I89O; some came because the school lav/ compelled them to stay in school up to the age of fourteen or sixteen: ** * Thorndike, E. L. — "Changes in the Quality of the Pupils Entering High School," School Review, Vol . XXX, No. 5, Pp 355-359, May 1922 ** U. S. Bulletin, Bureau of Education. "Part-Time Education of Various Types, No. 5, 1921 I . . • . . formerly these pupils would have gone to work instead of entering high school. Others come to prepare for college, or simply to "go thru" high school. With both industrial and college interests in the high school, the value of the traditional college preparation courses for all pupils came to be questioned. It was felt that the new interests present in the high school should be recognized, that something should be changed. Hester, * in an article on "Eco- nomics in the Course in Mathematics from the standpoint of the High School," says, "We all know there are topics in the high school which experience does not justify spending time on, but what they are we are not quite agreed." As a result a discontent arose concerning the traditional mathematics course. He goes on to say that the outstanding causes of dissatisfaction with our mathematics courses are two: (1) the general attack upon the doctrine of formal discipline and (2) the rapid increase of in- dustrial and practical education in the elementary and secondary schools. These attacks indicated that mathematics, of long standing as the best example of a logically organized subject, should be psychologized, reconstructed, and reorganized to place greater emphasis upon the life-mathematics needed by the average pupil. The interests and capacities of the pupil should be given attention. Educators are offering various suggestions for improve- ment. There are those who would not have courses in mathematics * Hester, Frank 0. -- "Economics in the Course in Mathematics from the Standpoint of the High School," School Science and Mathematics, Vol. XIII PP. 751 - 757 V. * . ' 4 changed. This conservative group would keep the traditional content and would find the hope of improvement in a more adequate preparation of teachers. One radical group would teach only those topics and processes that the pupil will actually use. Another radical group would eliminate mathematics from the required courses. Still another attempt at change is noted in the move- ment for "unified mathematics" . Two outstanding publications along this line are Breslich's three volumes, First-Year, Second- Year and Third- Year Mathematics, and Rugg and Clark's "Funda- mentals of High School Mathematics" . Breslich's Course . These three hooks are based on the viewpoint that there are fundamental operations in geometry and algebra -- distinct bodies of material in the two traditional subjects which ought in some way to be merged into a coherent course. "When the various branches of mathematics are treated as separate subjects," Breslich says ,*" there is a tendency for each to take on the rigid form of the final science. This tends inevitably to a certain formalism in mode of presentation. Such formalism is not the best method for the beginner. Correlation helps to avoid excessive formalism." Concerning the presenta- tion of material Breslich adds, "For correlated mathematics it is relatively easy to adopt a method of approach that is induc- tive. In Geometry the peculiar properties of the appropriate figures as studied and the results are then combined into a * Breslich, E. R. — First Year Mathematics (1916) Author's preface - ~ : . 5 theorem. This brings about an easier and a much better under- standing than a beginner can obtain from a logical proof. Axioms usually assumed to be self-evident are in the following pages illustrated in order to make their mea.ning apparent and vital. Not until toward the end of the geometry of the first year does the demonstration take the form of a logical proof. Even then the method of proof is informal, the aim being to convince the student by the truth of the theorem, to enable him to U3e a theorem to establish other facts, and to prepare him gradually for the formal geometry of the second course. Algebra is intro- duced as a natural means of expressing facts about numbers and gradually becomes a symbolic language especially well adapted to stating the conditions of a problem in a natural and helpful way. The growing difficulty and complexity of problems then lead to the necessity of learning how to manipulate algebraic symbols. The symbolism of algebra thus becomes a highly clarifying instru- ment of problem analysis and problem solving. The laws of alge- bra are carefully illustrated thus avoiding the danger of symbol- juggling without insight into the real meaning. "There are certain processes which belong together logically but which should be separated in treatment because they make difficulties for the beginner. Hence, wherever the processes are not needed as instruments of instruction, they are taught separately; for example, the meaning of positive and negative numbers, the laws of signs, and the operations with positive and negative numbers are not studied until the pupil has become thoroughly familiar with unsigned literal numbers and . . r - , . - - , 6 with the operations of laws of such literal numbers. " The material to be given in Rugg and Clark’s text was determined by a thorough investigation on "Scientific Method in the Reconstruction of Ninth Grade Mathematics" . Rugg and Clark’s Investigation of Ninth Year Mathe - matics . This study shows that one- third of our instructional attention in the first -year algebra is devoted to formal types of material. One-sixth of the formal examples of text books are based upon the four fundamental operations, and another sixth of the text is devoted to special products and factoring. One- fourth of the entire problem material is devoted to the equation "which nearly all mathematics teachers embrace as the central operation of algebra" . Their analysis of the use of current ninth-grade mathematics in other high school subjects and in oc- cupational activities shows that it is impossible to defend the large amount of attention to formal material based on the funda- mental operations and absolutely impossible to defend this em- phasis upon special products and factoring, on the basis of use. It is their opinion that more than one half of the subject matter that is taught will never be used by the vast majority, neither in the high school, in occupational or other life activities beyond the school, or even by that fraction of percent of the population that engages in the various scientific professions. Desiring to have mathematics taught for greater usefulness they recommend the following mathematical creed: "to develop in the pupil the ability to use intelligently the most powerful devices . . . . . . 7 of quantitative thinking: the equation, the formula, the graph and the properties of the more important space forms." Fundamental to the improvement of the present situation Rugg and Clark enumerate these two important steps: (1) The thorough overhauling of the course of study in mathematics, the elimination of non-essential material, the addi- tion of the types of real mathematics not now a part of the ninth grade course, and the construction of a continuous mathematical course, worked out around two basic principles, one mathematical and the other psychological. (2) The improvement of methods of teaching mathematics to ninth grade students. Ideally this demands better training of mathematics teachers. For this improvement we need a new type of textbook, a wordy textbook, a real applied psychology for the teacher, standardized tests with which to check up at intervals throughout the year the results obtained from instruction, and practice devices perfecting the former skills. Rugg and Clark’s text, bearing the title "Fundamentals of High School Mathematics", includes first, selected material which is now in the traditional first year course, and second, much material which is either not taught at all to high school students or is taught to only a limited portion in some advanced year. The tv/o principles controlling their selection of subject matter we re social worth and thinking value. The new course of study based on the criterion of social worth must include, Rugg and Clark say, "training in (a) the use of letters to represent • < . . 8 numbers; (b) the use of the simple equation; (c) the construction and evaluation of formulas; (d) the finding of unknown distances by means of (l) scale drawings, (2) the principle of similarity in triangles, (3) the use of the properties of the right triangle (ratios of the sides as in the Hypotenuse Rule and in the cosine and the tangent of an angle) ; (e) the preparation and use of statistical tables and graphs to represent and compare quantities (this includes the group of elementary statistical measures.)” The writers of the text, however, are of the group that regard "thinking" outcomes as coordinate in importance with the more common social utility. The entire course has been organized around the central core of "problem-solving". "Even the purely formal materials themselves," they say, "have been so organized, wherever possible, as to provide an opportunity for real thinking and not mere habit formation." Also, an attempt has been made to build the course in such a way as to "contribute constantly to ability in the expression and determination of relationship." In this book the content of the course was selected to i satisfy "rigorous criteria of either social worth or definitely established thinking value, or both". * The presentation of ma- terial of this type led to "(*) vast economy of time by excluding non-essential operations and forms; (2) introduction of new ma- ter ial not commonly attempted or successfully taught in the first year course, for example, (a) the use of statistical measures, tables, and graphs to represent and compare quantities, (b) the * Rugg and Clark, Fundamentals of High School Mathematics (19*9) Preface - - . . .• “ 9 organization of a whole course about the central theme of relation- ship and the systematic organisation of three methods represent- ing and determining relationship -- the graphic method, the tabu- lar method, and the equational or formula method, (c) systematic teaching methods of indirect measurement, for example, the find- ing the unknown distance by scale drawings, similar triangles, and the use of properties of the right triangle." The Investigation of the National Committee . Another attempt to reorganize high school mathematics is indicated by the National Committee on Mathematical Requirements in a report on "The Reorganization of the First Courses in Secondary School Mathematics." This committee, working under the auspices of the Mathematical Association of America, published this report which was preliminary in February 1920. A full report bearing the title "The Reorganization of Mathematics in Secondary Education" was published the following year. The material contained in the preliminary report is of primary interest to this thesis. Since it contains all the material basic to the later report and also the material on which the problem of this investigation is founded, a summary of only the preliminary report will be given here . The National Committee suggest the most desirable train- ing to be included in ninth and tenth year mathematics, taking secondary school mathematics as a whole, for they say "at the present stage in the discussion, agreement as to what should constitute the work of the ninth and tenth years separately may 1 I 10 . be difficult to secure". The material considered desirable was determined by two fundamental principles. "(1) The primary purposes of the teaching of mathematics should be to develop those powers of understanding and analyzing relations of quantity and of space which are necessary to a better appreciation of the progress of civilization and a better under- standing of life and of the universe about us, and to develop those habits of thinking which will make these powers effective in the life of the individual. "(2) The courses in each year should be so planned as to give the pupil the most valuable mathematical information and training which he is capable of receiving in that year with little reference to the courses which he may or may not take in succeed- ing years . "The general topics under which all details should fall are the formula, graphic representation, the equation, measure- ment and computation, congruence and similarity, demonstration. To impart these ideas, ’practical 1 problems, real to the pupil, I must be connected with his experience and interest. To Unify the i course, the idea of functional relations is sufficient. "Continued emphasis throughout the course must be placed on the development of power in applying ideas, processes, and principles to concrete problems rather than to acquisition Of mere facility or skill in manipulation. On the side of alge- bra, the ability to analyze a problem, to formulate it mathematical- ly, and to interpret the result must be dominant aims. Drill in algebraic manipulation should be limited to those processes and * . t . . , 11 to the degree of complexity required for a thorough understanding of principles and for probable applications either in common life or in subsequent mathematics. It must be conceived throughout as a means to an end, not as an end in itself. Within these limits skill in algebraic manipulation is important and drill in this subject should be extended far enough to enable students to carry out the fundamentally essential processes accurately and expeditiously.'* Minimum essentials recommended by the National Committee . This list of topics and processes is considered by the committee to be the essential part of high school algebra. These topics and processes are those contributing to the development of the “basic principles" quoted above. * (1) The formula, its meaning and use • a. As a concise language. - b. As a shorthand rule for computation. c . As a general solution. d. As an expression of the dependence of one variable on another variable. (2) The graph and graphic representations in general - their construction and interpretation: a. As a method of representing facts (sta- tistical, etc.) b. As a method of representing dependence. c. As a method of solving problems. (3) Positive and negative numbers -- Their meaning and use : a. As expressing both magnitude and one of two opposite directions or senses. b. Their graphic representation. c. The fundamental operations applied to them * Much of the same type of information is given concerning geometry as has here been summed up regarding algebra, but since this in- vestigation is concerned primarily with algebraic material, it was not considered essential that the geometric part be reported. 12 (4) The equation — its use in solving problems: a. The linear and the quadratic equation in one unknown, their solution and applications . b. Equations in two variables. 1 . As expressing a functional relation, with numerous concrete illustrations 2. As making possible the determination of the unknowns. c. Ratio, proportion, variation. (5) Algebraic technique: . . a. The fundamental operations. b. Factoring. The only cases that need con- sideration are monomial factors, the difference of two squares, the square of a binomial, and trinomials of the second degree that can easily be factored by trial. c. Fractions. d. Exponents and radicals: 1. Laws for positive integral exponents. 2. Radicals should be confined to the simplification of expressions the form a 2 b and a and to the numeri- c b cal evaluations of simple ex- pressions involving the radical sign. 3» Extracting the square roots of numbers . Optional topics . When schools can cover satisfactorily the work suggested, the Committee favors introducing the follow- ing topics and processes earlier than is now customary: (1) Meaning and use of fractional and negative exponents . (2) Use of logarithms and other simple tables. (3) Use of the slide rule. (4) Simple work in arithmetic and geometric progressions (on account of their importance in financial and in scientific thinking.) (5) Simple problems involving combinations, and prob- ability (on account of their frequent occurrence in daily life and their thought provoking qualities) (6) Elementary ideas concerning statistics. . 13 The Committee recommends that all topics and processes which do not directly contribute to the development of the powers mentioned in the fundamental principle should be eliminated from the course. Accordingly the list of topics and processes quoted be- low is recommended "to be omitted", from the work of the first two years i (1) Highest common factor and lowest common multiple, except in the simplest cases involved in the addition of simple fractions. (2) The theorems on proportion relating to alteration, inversion, composition and division. (3) Literal equations, except such as appear in common formulas, such as may be necessary in the deriva- tion of formulas, the discussion of geometric facts, or to show how needless computation may be avoided. (4) Radicals except as indicated elsewhere. (5) Extraction of square roots of polynomials. (6) Cube root. (7) Theory of exponents. (8) Simultaneous quadratic equations in more than two unknowns . (9) Pairs of simultaneous quadratic equations. (10) The theory of quadratic equations (remainder and factor theorems, etc.) (11) Binomial theorem. (12) Arithmetic and geometric progressions. (13) Theory of imaginary and complex numbers. ( 14) Radical equations, except such as arise in dealing with elementary formulas. (15) All equations of degree higher than the second. 14 History and biography, the committee says, should he used incidentally for emphasizing the idea that mathematics has been, and is a growing science. Summary . These investigations and texts show a vast change from the time when high school algebra was taught only to fulfill college entrance requirements. They sho?/ an attempt to meet present day conditions; along with preparing pupils for college the high school today trains many boys and girls who enter industry after one year in the high school. These pupils need especially the topics and processes that will help them in their work. Recognizing this need for connecting subject matter with life, that is a vital school problem today, these investigations recommend modifications of the traditional formal mathematics, and the text books of Breslich, and Rugg and Clark give evidence of changes from the formal teaching of algebra and geometry. * ' . X > . . ■ ■ . . Chapter IX The Problem and the Method of this Investigation Professor L. V. Koos * in an investigation concerning secondary school mathematics points out that textbooks dominate the content and organization of courses in mathematics. Results from responses to questionnaires concerning the degree to which high school teachers followed the textbooks showed the following deviations from the plan of the text: Deviations from the plan of the text. Number of texts. Omissions Additions Shifts of order None No answer 21 6 21 42 19 Of the ninety responses received, sixty-three reported no change in the textbook material or merely changed the order of intro- duction of topics, twenty-one omitted part of the text, and only six high school teachers added to the content presented in the text . Professor Koos* investigation also shows that elementary algebra is almost always a first-year high school subject. ’’Plane geometry,” he says, ” is markedly a second-year subject, but is reported in some schools in the third year, or in the latter half Koos, L. V. — ’’The Administration of Secondary-School Units (Supplementary Educational Monographs, Vol. I, No. 3, Whole No. 3) Chicago: The University of Chicago Press, July 1917. 16 of the second year and the first half of the third. Advanced alge- bra appears most commonly in the third and fourth years, but in a few schools in the second. Solid geometry appears in the third and fourth years and trigonometry in the fourth year.” Hence, it is concluded (1) that high school teachers cling closely to the text for the material taught in algebra courses, and (2) that algebra is essentially a first-year high school subject. The report of the National Committee on Mathematical Requirements is the most recent study dealing with present day standards in secondary school mathematics. Because it is modern, the topics and processes recommended therein have, for the purpose of this investigation, been considered as the criterion for our modern high school; hence for our present purpose, a text which meets the recommendations of the National Committee is meeting the requirements of the modern high school. It is the problem of this investigation to determine whether the trend of the content of algebra texts is toward meet- ing the requirements of the modern high school as they are stated in the recommendations of the National Committee. The trend of the content may be known by a historical study of the material contained in a number of text3, and a comparison of the content of texts through a number of years with the content recommended by the Committee, will indicate whether the trend is toward the recommendations of the Committee or not. Statement of Procedure . For the purpose of obtaining information relative to the trend of the content of algebra texts. 17 twelve one-volume first-year algebra texts representing a period of thirty-nine years were selected for study. These were not chosen in any scientific way; they were selected because of wide usage and with the advice of certain persons whose acquaintance with the field of secondary mathematics has extended over a number The texts and the date of publication are as follows: 1 . Wentworth Elements of Algebra 1881 2 . Milne - High School Algebra 1892 2 • Wells - Essentials of Algebra 1827 4 . Wentworth First Steps in Algebra 1898 5 . Wells - Algebra for Secondary Schools 1906 6 . Young and Jackson - First Course in Elementary Algebra 1908 7 . Hawkes, Luby and Touton - First Course in Elementary Algebra 1910 8. Wells and Hart -- First Year Algebra 1912 9 . Milne -- First Year Algebra 1915 10. Slaught and Lennes - Elementary Algebra 1915 1 1 . Hawke s , Luby and Touton - First Course in Algebra 1917 12 . Durell and Arnold - First Book in Algebra 1920 The wide use of Wentworth, Milne, Wells, and Hawkes, Luby and Touton is evident from the fact that they have been re- vised and with the exception of Wentworth's text they are still in use. Young and Jackson also is still in use. Slaught and Lennes is a comparatively new text, but it has been revised once ' 18 and hag found wide usage. Durell and Arnold was selected as the newest first-year algebra. Method of Examining; Texts . In Chapter I the topics "to be included" and those "to be omitted" were quoted from the report of the National Committee, and it is not necessary that they be re-quoted here. It is sufficient to state here the studies based on these two lists of topics. A study was made of (1) Topics and space included which the Committee recommended "to be included", (2) Topics and space included which the Committee recommended "to be omitted", (3) The percent of drill exercises, or examples, and also the percent of verbal exercises or problems, (4) The space given to history and biography. For each study it was necessary to set up criteria for each topic in order that a uniform comparison might be drawn. Some topics may be definitely placed under one of the headings in either the list "to be included" or the list "to be omitted" by merely glancing at the name of the topic. For example, no criteria is essential for the topic "square root of polynomials"; this heading in itself places the topic. Other topics of this kina, requiring no statement of criteria, are parenthesis, fundamental operations, positive and negative numbers, exponents and radicals, highest common factor and lowest common multiple, binomial theorem, cube root, and progressions. But merely naming a topic or process does not always place it as a topic "to be included" or one "to be omitted", since it is frequently found that one subdivision of a topic is 19 desirable and another is undesirable. For example, under "equa- tions" some topics are "to be included" and others are "to be omitted" . It is necessary to divide this topic into those sug- gested by the Committee: linear and quadratic equations in one unknown, and linear and quadratic equations in two unknowns as approved by the Committee, are "topics to be included"; while literal equations (except such as appear in common formulas), simultaneous equations in more than two unknowns, pairs of simul- taneous quadratic and radical equations are "topics to be omitted". As illustrations of both types of exercises, the following examples are given: Topics "to be included": linear in one unknown, quadratic in one unknown, linear in two unknowns, quadratic in two unknowns, and literal equations that are formulas, radical equations that are formulas, Topics "to be omitted": literal equations, as simultaneous equations in more than two unknowns, as ICx - 5= 3x +• 30 x 2 ~ 5x = 24 (3x -t y - 1 1 (5x - y - 13 (xf y ^ -3 ( xy — 54 - a- -h b 2 = - 1 V — lwh 5 - 1/2 gt 2 2x f a - 7a - x :!2x - yfz - 5 i3x +■ 2y - 3z — 7 Ax - 3 y - 5z — -3 20 pairs of simultaneous quadratics of either of these two types: (1) two equations of the second degree, as ( 2 ) one equation of the second degree , a 3 radical equations, as 3 - T =“\/5c Other topics "to he included” and "to he omitted" which are not entirely clear in themselves and the criteria for these topics and processes are as f ollows : Formula, "Formula" is considered "to he included" where the topic is specifically given or where it is implied and various formulas are given "as a concise language, as a shorthand rule for computation, as a general solution, or as an expression of the dependence of one variable on other variable s" . For example, material of this type should he included: (1) Find the formula for the area of a rectangle; (2) Express this rule in words : A= tt r Graph . "Graph" is considered to include all material on the graph and graphing which is given anywhere in the text. Illustrations of various types of material given under this topic are as follows: (1) Determine the length of the rivers in Fig- ure R (given in the text^ ; (2) Draw graph of the equation L ^ 1 ; (3) G-et the temperature readings in your own school district tomorrow and draw the graph! (4) Draw the graph of y . x Select at least four negative and four positive values 2 5 ~ 1 * «. ' . ' . t ... v* i . . . . 21 of x, and from them determine the corresponding values of y. What sort of graph do you obtain? Ratio, proportion and variation * Under ’’Ratio, pro- portion and variation”, the National Committee recommends that the pupil gain a "clear, working knowledge of the idea of ratio and its uses, and of what is meant by saying that a variable is proportional to another variable”; hence material of this nature is included under "topics to be included”; the meaning of technical terms, such as "inversely proportional to" and "mean proportional “ , is "to be included". Theorems on alternation, inversion, composition and division are topics "to be omitted” in accordance with the sug- gestions of the National Committee. Factoring . The Committee recommends that only these four types of factoring be presented in algebra; (1^ the monomial type, as a (b4- c) — ab 4- ac (2) the difference of two squares, as a 2 - b“— (a+-b)(a - b) (3) the square of a binomial, as ( a t- b j 2 = a- +- 2ab +- b 2 (4) trinomials of the second degree that can be easily factored by trial, as x 2 4-2x - 15 - (x 4-5) (x - 3). These types are included in the list of topics "to be included" . To other types such as those given below the Com- mittee gives no consideration. Consequently they are omitted from the present investigation. 22 (1) the common compound type, as ax + ay + bx f by = la + b) (x-f y) (2) the sum or difference of two cubic, as a? +- b^ — ( a t- b) ( a- - ab 4- b 2 ) Sections of texts on "Equations solved by factoring" , are included also under the topic "Factoring", since they are given primarily for practice in factoring. " Fractions " includes the four fundamental operations with fractions and also that space given especially to fractional equations. The Report says, "the fundamental operations with fractions should be considered only in connection with simple cases." Those fractions are called simple which have whole numbers or letters in the numerator and denominator, as 2x“ - 2y ~ ; 5x^ 5y5 and fractions of the following type are called complex: 3ab x ba /: 'b “x2~~ " Theory of Exponents 11 is considered to be included in a text if the topic of exponents includes a theoretical explana- tion of negative and zero exponents. "Theory of quadratic equa- tions" is interpreted to mean a theoretical explanation of roots. Drill and Verbal Exercises . For the study of the per- cent of drill and verbal exercises, a criterion was set for each type. The two types are referred to as examples and problems. The word "example" is used to designate the drill type of exercise, which calls definitely for certain algebraic operations. The word "problem" is used to designate those exercises in which the pupil is required to determine what operations are to be performed. 23 This type requires some "reasoning" on the part of the pupil. Problems are always stated in verbal form, while examples of the drill type usually are not. "Problems“ are frequently referred to as practical or verbal exercises. For example, the following exercises are drill exercises or examples: (1) Add, 8ab, -9cd, ~6ab, and +- 4cd (2) Solve, x - 3 =12; 7y = 3y - 12 (3^ G-et the temperature readings in your own school district tomorrow, and draw the graph. Those following are problems: (1) A's age is three fifths of B*s age; but in 16 years A's age will be five sevenths of B’s-age. Find their ages at present. (2) The base of a rectangle is 9 feet more and the altitude is 8 feet less than the side of a square. The area, of the rectangle exceeds the area of the square by 15 square feet. Find the dimensions of the rectangle. (3) What number increased by 11 equals 19? With these criteria each text was examined, first, to discover which topics and processes were included of those recom- mended by the Committee "to be included" and “to be omitted" and the amount of space given to them; second, to discover the per- cent of examples and problems; and third, for the space given to history and biography. The examination of texts included all pages except those given to the preface, the index, the table of contents, and pages on which answers are printed. Those pages that did not classify under "topics to be included" or “topics to be omitted” were omitted from any consideration. A few topics of this type are "Inequalities," “Limits," and "Identities and Equations and Condition." . - Chapter III The Content of Twelve First Year Algebra Texts Following the procedure outlined in Chapter II, the content of the texts was studied. The purpose of the present chapter is to present the material in each text. Tables have been arranged showing (1) the topics and space included which the Com- mittee recommended "to be included", (2) topics and space given that are recommended by the Committee "to be omitted", (3) the percent of drill and verba.1 exercises, and (4) the space given to history and biography. Topics "to be included " . Table I shows how many of the topics and processes which the Committee recommends "to be in- cluded, “ each text has included in its content. A topic marked with a plus sign, thus (+) , is included in the text named at the top of the column, while a minus sign, thus (*•), indicates that a topic has been omitted from the text designated at the top of the column. We see from Table I that the formula, as a concise language, was not included until 1898; the formula, as an ex- pression of the dependence of variables, was omitted until 1908. No graphing of any kind was included up to 1897; at that date, graphs and graphing were included in the "appendix" to Wells' text; in 1898 Wentworth* s text did not offer graphs, but after 1906 the topic was given a place in the content of each text studied. "Positive and negative numbers" as expressing magnitude and direc- tion was given in Wentworth ‘ s text published in 1881, but was ! . ■ 3 i~ Table I Detailed Table of Topics Included in Texts ffent- Went- ■ £oung Hawke s Wells Slaught Hawke s Dure 11 worth Milne Wells worth Wells and Luby and Milne and Luby and Topic Jack- and Hart Lennes and Arnold son Touton Touton 1881 1392 1897 1898 1906 1908 1910 1912 1915 1915 1917 1920 Formula 4 4 4 4 a. As a concise language b. Shorthand rule of - - - 4r 4 4 4 ■+ *+ 4 4 4 + computation 4 - 4- 4- 4 - 4 c . As a general solution d. As expression of the ■r • 4- 4- 4- 4 4 4 4 4 -4” dependence of 4 variables - - - - - + — +* 4 4 Graph a. As a method of repre- 4 + 4 4 senting facts - - ■h - b. As a method of repre- 4 presenting dependence - - 4 - 4- 4- *+- 4 4 4 4 c . As a method of solv- ing problems - 4- "V" 4 4 4 Positive and negative numbers a. Expressing magnitude 4- 4 4 4 4 and direction -V - 4- 4- 4 b. Graphic representation - - - - 4 4 4 4 4 4 4 c. Fundamental operations 4 -4- -+- 't i~ 4 “4* 4 4 4 4 The Equation a. Linear quadratic in 4- -4 4 one unknown -t + 4 4 4 4 4 4 b. Equations in two 4- variables : 4 4- 4 4 4 4 4 4 -4 1 As expressing a func tional relation 2 Determination of un knowns 4 4- 4- "4 4 4 4 4 4 4 4 c. Ratio, proportion and -4- 4- 4 4- 4 variation 4 4 4 4 4 4- 4 Algebraic technique . 4- 4- a. Fundamental operations b. Factoring 4 4- 4 4 4 4 4 4 -f 1 Monomial 2 Difference of two 4 4- -f 4" 4- 4“ i 4 4 4 4 4 4 4- squares \ 4 r 4 4 4* r 4 3 Square of binomial T 4 f *T" 4 4- by trial 4- 4 4 4- 4 4 ■f -f- ~ir 4 — f' c. Fractions d. Exponents and 4: 4- 4 4 4 4 4 4 4 4 4 -v "4~ 4 4 radicals -t- 4- r 4 4 4 4 Number of topics included 15 12 18 16 20 20 20 2 1 20 2 1 21 21 Percent of topics included 71 57 86 76 95 95 95 10U 95 100 100 100 26 omitted from Milne's text in 1892; in 1897 the topic was again included and is given in all texts published later than that date. Graphic representation of positive and negative numbers was omitted until 1906 and since then it has been offered. The re- maining topics, as the checks in Table I indicate, are all in- cluded in all t?/elve texts that were examined. At the bottom of each column a summary has been made of the number of topics each text contains, which, in accordance with the Committee's recommendations, should be given. Reading the column headed "Wentworth 1381“ as an example, we see that in Wentworth* s text published in 1881 fifteen topics of the possible twenty-one are included, and in the next column twelve of the twenty-one are given. The lowest line, "Percent of topics in- cluded," indicates that seventy-one percent of these topics were included in 1381, fifty-seven percent in 1892, eighty-six percent in 1397, and so on. This table (Table I). does not mean that only those topics and subtopics that are approved by the Committee are in- cluded; it simply indicates that at least these topics are given. For example, under "Exponents" and "Radicals” the Committee re- commends that "the laws for positive integral exponents should be included," and that "the consideration of radicals should be con- fined to the simplification of expressions involving the radical sign." In almost every case, topics more advanced and more com- plex than these are included, but the check in the table remains the same as if only the required topics and processes were given in the text. In Table III and Table (a table of conclusions) 27 attention is given to the amount of space given to topics and pro- cesses that are unnecessary. Table II showing the space given to topics which the Committee recommends u to be included" is more explicit. This table shows In terms of the total space in the text the percent of space that each text, indicated at the top of the column, has given to the topic indicated in each line. Thus, Wentworth (pub- lished in 1881) gives .07 percent of the total space in the text to the formula. This text gives no material on the graph, .9 per- cent to positive and negative numbers, 2.8 percent to linear equa- tions in one unknown, and so on down the column. Reading the lines from left to right, we find .07 percent of the space in the text was given to the formula in 1881, none in 1892, and .2 per- cent in 1897; thereafter the space given to the formula varies irregularly between 1 percent and 4.9 percent of the total space in the text. The "graph", which was not included until 1897, was given 2.3 percent of space at that date, but was omitted entirely in 1898 . After 1906 the space given to this topic varied irregular- ly from 2 percent in 1906 to 10 percent in 1920. Pointing out the changes in texts where revised editions have been published, we can make no general conclusion with respect to "graphs" : neither edition of Wentworth contains any material on the graph; the first edition of Milne ( 1892) gives none of its space to graphs and the later edition (1915) gives 7.8 percent; Wells in 1897 gives 2.3 percent space to the graph, in 1906, 2 percent, and in 1912 in the text revised by Wells and Hart 5.9 percent of the space was devoted to graph. t , . Tab] Detailed Table Show] Unnecess 1 .e III .ng Sp jary T ace Given to opics i .. Went- Went- Young Hawke s Wells SI aught Hawke s Durell worth Milne Wells worth Wells and Luby and and Luby and Topics Jack- and Hart Milne Lennes and Arnold son Touton Touton 1881 1892 18*7 1898 1908 1908 1910 1912 191o .1915 1917 1920 Highest common factor and lowest * ' - * • common multiple 4.3 2.y 4.0 3.o 1.2 2.0 1.0 1.9 1.6 1 .4 1.4 .4 Proportion . o .9 .4 .4 ' .3 0 .9 .7 .4 .3 1 .2 0 Literal equations 3. i .7 1.8 1.8 1.9 .2 1.6 1 .5 1 .4 .7 2.9 1.5 Radicals 3.3 5.4 2.3 4.0 4.5 1 . 6 4.y 0 2.4 2.4 5. 1 4.3 Square root of polynomials • 6 .8 .8 1 .0 1 .0 .8 1.0 .7 .8 1 . 1 .9 .6 Cube root 1.5 .8 2.0 1.1 1.8 1 o .2 .4 .3 0 .5 .07 Theory of exponents 1.2 2. 1 1.9 .8 2.2 .3 .3 .2 .2 0 0 0 Simultaneous equations in more than two unknowns .8 1.9 1 .8 1. 1 2.9 2 . 0 1 .0 1.0 .9 1 .4 1.3 .6 Pairs of simultaneous quadratic equations 1.5 1.7 2.3 1.7 1.7 0 .2 0 1 .2 1 .2 .8 0 Theory of quadratic equations 2.2 u 1.4 2.8 0 0 0 0 0 .08 0 Binomial theorem 2.8 5.4 1.8 4.9 2.6 .9 0 0 .9 1.7 0 0 Progressions 4.0 4.8 5. 1 3.9 4.4 U 0 0 0 0 0 0 Theory of imaginary and complex numbers .3 0 0 1 .4 0 0 0 0 0 0 0 0 Radical equations 1 .6 1.0 .5 .9 .6 ' *07 1.7 0 2.4 .5 .5 0 Equations of degree higher than second .4 .7 .0 1 . 1 .7 0 .07 0 .7 .5 .08 .3 Total pages 325 367 357 407 458 323 329 321 334 352 325 335 Total percent 28.6 ,28.7 25. 1 29. 1 28.6 10. 17 13.7 6.4 13.2 1 1 .2 14.7 7.8 29 The space given to "positive and negative numbers" in- creases from .9 percent in 1 38 1 to 2.4 percent in 1920 , with two higher percents of space given to it in 1915s 9.6 percent in one text and 6.3 percent in another. Reading the two extremes in dates we see that the space given to the "linear equation in one unknown" increases from 2.8 percent in 1381 to 11 percent in 1920. This increase is a gradual one showing at only one place a percent of space lower than in 1881. This exception is in 1906 when 2.6 percent were given. "Quadratic equations in one unknown" is included in every text, and is given an irregular increase in space. With the ex- ception of the text published in 1381 "linear equations in two variables" is given more space in the six later published texts than in the older ones. One text ( 1908) fails to include "quad- ratic equations in two unknowns." Noting the two extremes in dates, we observe that the oldest text (1381) gives just as much space to this topic as the newest one; the greatest percent of space is 2,3 in 1915* Treatment of the topic "exponents" is ex- cluded from one text ( 1892 ). The line marked "total pages" is read in the following manner: Wentworth ( 1 83 1 ) consists of 325 pages of mathematical material exclusive of pages devoted to a preface, table of contents, index, and answers. The line begin- ing "total percent" points out that 47 percent of the total space in the text has been given to topics and processes that ought to be included. Topics "to be omitted ." For the purpose of showing the content of each text that is "to be omitted," as the National 1 - . ■ . • ■. : ..i ■ • ii. t 30 Committee recommends, Table III has been arranged. Using the first column as an example, information presented there is read in this manner: Wentworth's text published in 1881 gives 4.3 percent of its total space to "highest common factor" and "lowest common multiple"; .6 percent of the space is given to proportion, 3.1 percent to literal equations and so on down the column to the last item indicating that .4 percent of the texts space is given over to equations of degree higher than second. In the following line, we have given the total number of pages in the text, 325 in this case; here, as in Table II "total number of pages" excludes the preface, table of contents, index and answers. The line beginning "total percent of space that should be omitted" indicates that 28.6 percent of the space is devoted to topics that are "to be omitted" in accordance with the suggestions recommended in the Report . Observing the first line in Table III a gradual decrease is noted in the space given to "highest common factor and lowest common multiple" from 4.3 percent in 1881 to .4 percent in 1920. The next two topics, "proportion" and "literal equations" also show a decrease. The space given to "radicals" increases from 3.5 percent in 1881 to 4.3 percent in 1920. However, lower per- cents are shown at dates previous to 1920, for example, Wells and Hart's text (1912) contains no "radicals" to be omitted, Young and Jackson in 1908 gives 1.6 percent, and the two texts in 1915 each give 2.4 percent of their space to "radicals." "Theory of exponents" loses space gradually and after 1915 we find the topic omitted entirely. Three texts, those published in 1908, 1912, and : * « • • , . ; . High* Propc Lite} Radic Squa: Cube Theo} SimiL Pairi Theo: Binoi Prog: Theo; Radi' Equa Tota Tot a 1 82 192G omit "pairs of simultaneous equations." Four of the twelve texts examined included "theory of quadratic equations," one of these texts was published as recently as 1917* The two texts published in 1915 give .9 percent and 1.7 percent of their space to the "binomial theorem", while four texts, published in 1910, 1912, 1917 and 1920 omit this topic. "Progressions" is excluded after 1996. Wentworth's two texts are the only texts including the "theory of imaginary and complex numbers", making 1898 the latest date when this topic was included. Only two texts omit "radical equations" as the National Committee recommends? these two were published in 1912 and I 92 O. "Equations of degree higher than second" is included in every text but two ( 1908 and 1912 ;. The figures in this line show no general decrease in the space given to this topic . Drill and Verbal Exercises . From the summary of the Report of the National Committee (Chapter 1), it has been ob- served that algebra should include some exercises of each of two types (1) drill and ( 2 ) verbal exercises. Table IV shows the re- sults of such a study of the numbers of drill and verbal problems in each text. Using the line beginning Wentworth (1881) as il- lustrative, the number of drill and verbal exercises is pointed out as follows: Wentworth's text published, in 1831 contains 2158 examples, and 342 problems, or verbal exercises; this placed in percents reads 86 percent drill and 14 percent verbal exercises are given in this text. The next line shows 4038 drill and 549 verbal exercises in Milne's text published in 1882; or changed to percents, this is 88 percent drill and 12 percent verbal exercise 33 Table IV Drill and Verbal Exercises in Texts 1 f = — “ Text Date Number of exercises Percent of exercises Drill Practical Drill Practical Wentworth 1881 2158 342 86 14 Milne 1892 4058 549 88 12 Wells 1897 3027 328 90 10 Wentworth 1898 3015 475 86 14 Wells 1906 3296 335 69 1 1 Young and Jackson 1908 4228 464 90 10 Hawke s, Luby and Touton 1910 3077 61 1 64 16 Wells and Hart 1912 2932 537 85 15 Milne 1915 3815 661 85 15 SI aught and Lennes 1915 4118 662 86 14 Hawke s, Luby and Touton 1917 3276 661 83 17 Durell and Arnold 1920 3333 t- 00 c\ 80 20 , ■ Pages of hi bio^ Page pic tui Total pages and bi< Percent of * Portrait 6 3/4 pages 35 The fa,ct that no history or "biography was given in any of the texts studied up to 1910 is shown in Table V. Using the column headed Hawkes, Luby, and Touton (1910) as an illustration, we read the table thus: The edition of Hawkes, Luby and Touton, published in 1910, contains twelve and one half pages of history and biography, and six pictures of mathematicians, making eighteen and one half pages in all of historical material. This material occupies 5.6 percent of the total space of the text. In this tabulation a page picture was considered first as a topic in itself and later combined in the total number of pages and the percent of space given to history and biography. These five tables present in detail the content of the twelve texts studied. The next step in this investigation is to sum up the tables for drawing conclusions as to the trend of the content of algebra, texts. Chapter XV Summary: The Trend of the Content The purpose of Chapter IV is to sum up the information tabulated in Chapter III into a form such that the trend of the content of the twelve texts may he observed. Table VI is a sum- mary of the five preceding detailed tables; each column gives the information secured concerning the total content of the text named at the top of the column; for example, the first column points out that Wentworth ( 1881 ) contains (1; 71 percent of the 21 topics recommended bjr the Committee ‘‘to be included”, and (2) all of the 15 topics that ought "to be omitted"; (3J 46.9 per- cent of the total space in the text is given to topics and pro- cesses that should "be included", (4; 28.6 percent of the total space is given to topics and processes that ought to be omitted, according to the National Committee, (5) 86 percent of all the exercises in this text fall under the drill type of exercise, and the other 14 percent are exercises of the verbal type; (6) no history or biography is given in this text. Other columns are read in a similar manner. These same results are presented also in graphical form. Figure 1 shows the percent of topics and the percent of space given to topics that are "to be included" from 1881 to 1920; Figure 2 shows the percent of topics and processes that are "to be omitted" and the percent of space given to these topics during the same years; and Figure 3 indicates graphically the . . >• ■j . • ■ ( 1 ) Perce thal (2) Perce thal (3) Perce thal (4) Perce thai (5; Perce (6) Perce ( 7 ) Perce torj 3 ? Table VI Summary of Detailed Tables Topics of Tables Went- worth 1881 Milne 1892 Wells 1897 Went- worth 1898 Wells 1906 foung and Jack- son 1908 Hawke s Luby and Touton 1910 Wells and Hart 1912 Milne 1915 SI aught and Lennes 1915 Hawke s Luby and Touton 1917 Durell and \rnold 1920 (U Percent of topics included that should "be included" -71 57 86 76 95 95 95 100 95 100 100 100 ( 2 ) Percent of topics included that should "be omitted" 100 b7 87 100 93 60 73 47 73 67 73 47 (3) Percent of space to topics that are "to be included" 46.9 47.3 41. 1 50.2 32.0 52.8 62.5 68.4 60.4 50.2 64.5 68.7 (4) Percent of space to topics that are "to be omitted" 28.6 28.7 25.1, 29.2 28.6 10.2 14.4 6.4 10.3 10.1 15.8 7.7 (5) Percent of drill exercises 06 88 90 86 89 90 34 85 85 86 83 80 (6) Percent of verbal exercises 14 12 10 14 1 1 10 16 15 15 14 17 20 (7) Percent of space to his- tory and biography 0 0 0 0 0 0 5.6 1 .2 .2 4.4 4.4 0 percent of drill and verbal exercises included in the texts examined. From a study of Table VI and these graphs we may as- sume conclusions concerning the trend of the content of algebra texts . Topics and processes "to be included ." Table VI shows the range in percents of space given to topics and pro- cesses "to be included" from .369 to .750. The lowest percent in the six latest published texts is higher than the highest percent found in the six older texts. From 1681 to 1920 we find a change in the space given to topics "to be included" from 46.9 percent to 72.1 percent with the two extremes, 36.9 percent and 75.0 percent falling at 1905 and 1917 respectively. The upper portion in Figure 1 represents the percent of topics in- cluded of the twenty-one topics which the National Committee recommends "to be included." This line also shows a rise from 1881 to 1920 almost parallel to the rise in the percent of space that should be included; in 1861, 71 percent of these topics and processes were included, while in 1920, 100 percent were given. Summing up the facts concerning the topics and processes that are "to be included", a decided increase is noted in (1) the number of topics and processes included, and in (2) space given to these topics and processes. Topics and processes "to be omitted ." Table VI in- dicates that 28,6 percent of the total space in the text pub- lished in 1881 was given to material which the Committee con- siders unnecessary, in 1698 we find 29.2 percent, and in 1908, 10 percent of the page were used for non-essential matter; in q+b *nj r -ff + 42 1912, 6.4 percent, and in 1920, 7.7 percent. Thus we see the two extremes, represented in 189b and 1912, are 29*2 percent and 6.4 percent. The lower line in Figure 2 represents these results: the greatest percent of unnecessary material in 1898, and the least in 1912. Even though these variations cause the graph to he irregular from 1881 to 1920 there is a decrease of 20.9 per- cent indicated in the amount of material presented that ought "to be omitted." A comparison of the two graphs represented in Figure 2 shows that the percent of topics "to be omitted" has decreased along with the decrease in the percent of space given to topics that should be omitted. The upper line in Figure 2 shows a decrease of 53 percent in the number of topics included; in 1881, 100 percent of the topics were included that the National Committee recommends to be omittedJ while in 1920 only 47 percent were given in the text. In conclusion, it is noted that the number of topics "to be omitted" has decreased 53 percent and that the space given to these topics has decreased 20.9 percent. Drill and Verbal Exercises . The lines in Table VI, indicating the percent of drill and verbal exercises in each text, show a rather marked agreement among authors as to the ratio of the two types of exercises. The two extremes are in- dicated in the texts published in 1397 and 1908 (both giving the same percent of drill and verbal exercises) and in the text published in 1920. In 1897 and 1908 we find 9p percent of all the exercises are examples and 10 percent are problems. In 1920, 80 percent of the total number of exercises are examples and . « 43 20 percent are problems. Taking the two extremes in dates, we find that in 1881 86 percent are drill examples, and 14 percent are problems; and in 1920, as quoted above, we have 80 percent drill and 20 percent verbal exercises. Reading only these two results, we might say that there is a little tendency to have our texts contain less of drill and mor*e of application material; it might be said that our texts are becoming more “wordy" , as the National Committee recommends. Dividing the texts examined into two classes, the older six and the newer six, as before, we might also say that there is a slight tendency for the newer texts to give fewer drill examples and more verbal problems than the older texts. Figure 3 indicates the percent of drill and practical exercises in the year indicated on the lower line. This graph points out that the number of drill exercises is always at least 60 percent greater than the number of verbal exercises. The upper line shows a decrease of 6 percent in the number of drill exercises included in the texts; the lower line indicates the percent of practical exercises, and shows an in- crease of 6 percent in the number of practical exercises given. Summarizing the content of the twelve texts we would say that the trend of algebra texts is slightly toward the suggestion me.de by the Committee: "Continued emphasis throughout the course must be placed on the development of power in applying ideas, processes, and principles to concrete problems, rather than the acquisition of new facility or skill in manipulation." History and biography . These topics have been in- cluded in texts only recently* it is observed in Table VI that 44 up to 1910 no history or biography was given in any text. The text published in 19 10 , however, gave 5.6 percent of its space to these topics; in I 9 I 2 , 1,2 percent was given to history and biography; in I 9 I 5 one text shows .2 percent and another 4,4 percent; and in 1920 no history or biography was given. Dis- tinguishing between the six older and the six newer texts, we may say that the newer texts tend to include more history and biography than the older ones. However, there is no gradual in- crease evident in the amount of space given to these topics. Concluding statement . Regarding the preliminary report of the National Committee on Mathematical Requirements as repre- sentative of the material that should be taught in the secondary schools of today, and basing our conclusions on this examination of twelve first-year algebra texts, the following statements are made concerning the trend of the content of first-year algebra texts since 1381 : There is a tendency to?/ard omitting and giving less space to topics and processes which the National Committee recommends "to be omitted." There is a tendency toward including and giving more space to those topics and processes which the National Committee recommends be included. The percent of drill exercises tends to decrease slightly, and the percent of verbal exercises increases slightly indicating a relative increase in wordy exercises as the Committee recommends. The tendency to include history and biography is slight, but gives evidence of some thinking in historical terms. Bibliography Bikle, C. E. -- The Aims in Teaching Algebra and How to Attain Them. Mathematics Teacher, V: I, 1908, P. 77* Breslich, E. R. -- First-Year Mathematics (I 9 I 6 ), Author's Preface. ~ . Carmichael, P. . D. -- Mathematics and Life, - The Vitalizing of Secondary Mathematics. School Science and Mathematics, V: 15, P. 105. Cobb, H. E. -- Current Educational Movements and General Mathe- matics. V: lo, 1916, P. 4lp. Crathorne, A. R. -- Algebra from the Utilitarian Standpoint. School Science and Mathematics, Vil6, P. 4lb. Hester, Frank 0. — Economics in the Course in Mathematics from the Standpoint of the High School. School Science and Mathematics, V: 13, P. 751* Jessup, W. A. -- The Greatest Heed of the Schools Better Teaching. Journal of N. E. A., Vi 10, No. 4, Pp. 71 - 73, April 1921. Johnston, C. H. -- Modern High School, Chapter I, P. 10. Koos, L. V. -- The Administration of Secondary-School Units ( Supplementary Educational Monographs, V: I, No. 3, YiThole No. 3) Chicago: The University of Chicago Press, July 1917.-- Minutes of the Mathematics Section of the California High School Teachers Association, School Science and Mathematics, V ; 13, P. ^29. Minutes of the Thirteenth Annual Meeting of the Central Association of Science and Mathematics Teachers. School Science and Mathematics, V: 14, 1914, P. 172. Monroe, W. S. -- An Experiment in the Organization and Teach- ing of First-Year Algebra. School Science and Mathematics, V: 12, 1912, P. 225. Monroe, W. S. -- Measuring the Results of Teaching, Chapter von Nardoff, C. R. — Mathematics for Service. The Mathe” matics Teacher, V: 2, 1909 - 10, P. 16. National Committee on Mathematical Requirements, Preliminary Report of, -- The Reorganization of the First Courses in Secondary School Mathematics. United States Secondary School Circular No. 5, 1920 (February). National Committee on Mathematical Requirements -- The Re- organization of Mathematics in Secondary Edu- cation. United States Bulletin, 1921, No. 32. Pierce, Harriet R. -- The Value of Mathematics as a Secondary School Subject. School Science and Mathematics, V: 16, P. 780. Rugg and Clark -- Fundamentals of High School Mathematics (1919) Preface. -- - - Rugg and Clark -- Scientific Method in the Reconstruction of Ninth Grade Mathematics. Supplementary Edu- cational Monographs, V: 2, No. 1, Whole No. 7, April 1916. Thorndike, E. L. -- Changes in the Quality of Pupils Entering High School. The School Review, May 1922, P. 355. United States Bulletin, Bureau of Education -- Part-Time Edu- cation of Various Types, No. 5, 1921.