THIS IS AN ORIGINAL MANNOIRIPT IT MAY NOT BE COPIED WITHOUT THE AUTHOR’S PERMISSION
CHANGES IN MATERIALS AND METHODS IN ELEMENTARY ALGEBRA FROM 1829 to 1929, INCLUSIVE
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Dean of (the Graduatfe School.
CHANGES IN MATERIALS AND METHODS IN ELEMENTARY ALGEBRA FROM 1829 to 1929, INCLUSIVE
THIS IS AN ORIGINAL MANUSCRIPT IT MAY NOT BE COPIED WITHOUT. THE AUTHOR’S PERMISSION
THESIS
Presented to the Faculty of the Graduate School of The University of Texas in Partial Fulfill-
ment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Ira Irl Nelson, 8.A., M.A.
Austin, Texas
June, 1932
355741
PREFACE
This study represents an attempt to trace changes in materials and methods in elementary algebra throughout a period of one hundred years, beginning in 1829. The use of textbooks for primary source material is based upon the assumption that the textbooks of any period constitute the best index of the practices of that period. The changes noted in materials and methods will be those resulting from refinement of subject matter, also those brought about by the influence of the psychology of learning and teaching.
I wish to acknowledge mjr indebtedness to Professor J. L. Henderson for his constant encouragement and able assistance in the carrying on of this study; to my wife, Mrs. Grace Lightfoot Nelson, for her sympathy and help at all times; and to my brother and sister, Mr. and Mrs. J. E. Nelson, of San Antonio, Texas, without whose material aid the work could not have been carried on.
Austin, Texas June, 1932
TABLE OF CONTENTS
Page
Chapter I Aims, Scope, Materials and Methods of the Study .... 1
Aim and Scope of the Study 1
Materials for the Study 2
Methods of the Study ....... 6
General Plan of the Study 7
Previous Studies of This Nature ... 8
Chapter II Contrast Between English and French Texts 11
Nature of Early American Algebra ... 11
Treatment of the Equation . .... 19
Treatment of Negative Numbers .... 20
”Wordy n Textbooks ........ 22
The Theoretical versus the Practical . . 23
Examples and Problems for Practice . . 24
Chapter Summary 25
Chapter 111 Order and Space in Seventy- Seven Texts 27
Purpose and Methods of the Chapter . . 27
Order of Topics in Seventy-Seven Algebra Texts 31
Page
Tabulation of the Data ..... 33
Extent of the Changes in Order and Space ........ 34
Junior High School Textbooks and the Topics of Algebra .... 38
Four General Types of Changes . . 43
Topics That Have Disappeared ... 44
Topics Unchanged as to Order and Space 50
Topics That Have Changed Position in the Course 56
The Appearance of New Topics ... 62
Warren Colburn’s Introduction to Algebra (1829) 70
Order of Topics in Two Modern Textbooks 74
Some Special Cases 80
Chapter Summary 84
Chapter IV Negative Numbers . . . . 86
I Introduction 86
Purpose of the Chapter 86
History of Negative Numbers in America . . 86
The Placement of Negative Numbers . 90
Space in Texts D e voted to Negative Numbers 100
Page
11. Meaning and Significance of Negative Numbers 102
Examples from Early Texts . . . . 102
The Realness of the Negative Number . .... 105
The size of Negative Numbers . . . 106
Formal Definitions of Positive and Negative Numbers .... 11l
The Subtraction Operation and the Negative Number .... 115
The Algebraic Scale 119
Generic Names for Positive and Negative Numbers 126
Double Use of the -4-and - signs . 128
Approach to the Study of Negative Numbers . . . . . . 133
Illustrations of the Use of Positive and Negative Numbers 135
From the Deductive to the Induc- tive ....... 138
Examples of Special Devices Employed Dy Authors . 140
Chapter Summary ....... 151
Chapter V The Equation ..... 154
Purpose of the Chapter .... 154
Ward’s Treatment of the Equation . 155
Page
Importance of Equations in Algebra . . 164
The Reduction of the Equation . . . 170
Transposition ......... 175
Definitions of the Equation .... 178
Organization of Subject Matter on Equations . . . 181
Solution of Word Problems 184
Expressing Conditions of a Problem as Equations 185
Solution and Discussion of the Equation . 194
Solution of Simultaneous Linear Equations 198
Solving Quadratic Equations .... 201
Simultaneous Quadratic Equations . . . 211
Chapter Summary ........ 215
Chapter VI The Formula 218
Purpose of the Chapter ...... 218
Four Authorities on the Formula . . . 218
How the Formula Came into Algebra Text s 221
Definitions of the Formula . . . . 224
What the Topic ’‘Formulas” Includes . . 225
Graphs 227
Page
Some Special Devices for Teaching Formulas 229
Chapter Summary 233
Chapter VII Exponents 235
Purpose of the Chapter 235
Exponents in Two Early Textbooks . . 235
Most Common Method of Treatment of Exponents 238
Special Chapter on Exponents in Textbooks 240
Power and Exponent 242
Types of Exercises on Exponents . . 246
Chapter Summary 249
Chapter VIII General Summary 251
TABLES
Table */ Page
I Distribution of Publication Dates of Textbooks 4
II Order Assigned to Topics in English and French Algebra Textbooks .... 17
111 Space in Pages Assigned to Topics in English and French Algebra Textbooks 18
IV Order of Topics in Seventy-Seven Algebra Textbooks ........ 31
V Pages of Space Devoted to Topics in Seventy-Seven Algebra Textbooks . . 32
VI Contrast of Order Assigned to and Space Given to Topics in Early and Modern Textbook 36
VII Topics That Have Disappeared ..... 45
VIII Topics That Have Remained Practically Unchanged . . . ... . .51
IX Changes in Simple Equations, the Four Operations, and. Radicals .... 57
X The Appearance of New Topics . .... 64
XI Placement of Positive and Negative Numbers 91
XII Changes in Placement of Positive and. Negative Numbers ....... 92
XIII Illustrations of Positive and. Negative Numbers with Frequencies . .136
Table
Page
XIV Space in Certain Texts Devoted to Translating English Into Algebra 192
XV Methods of Solving Simultaneous Linear Equations Used in Twenty-Nine Textbooks «... 199
XVI Methods of Solving Quadratic Equations Used in Twenty-Nine Textbooks 202
XVII Exercises and Problems Involving Simultaneous Quadratics in Twenty-Five Textbooks 211
XVIII Frequency of Cases of Simultaneous Quadratics in Thirty Textbooks . . 214
DIAGRAMS
Diagram
Page
I Inequalities - Frequency, Order, Space . . 46
II Imaginaries - Frequency, Order, Space . . 47
111 H.C.F. and L.C.M. - Frequency, Order, Space . . . 48
IV Fractions - Frequency, Order, Space . . 52
V Quadratics - Frequency, Order, Space . . 53
VI Simple Equations - Frequency, Order, Space 58
VII Four Fundamentals - Frequency, Order, Space 59
VIII Radicals - Frequency, Order, Space . . 60
IX Graphs - Frequency, Order, Space . . 65
X Formulas - Frequency, Order, Space . . 66
XI Negative Numbers — Frequency, Order, Space 67
PLATES
Page
Plate
I From Rugg and Clark's Fundamentals of High School Mathematics, TTlu st rating Te aching’ofTegat ive Numbers 141
II From Rugg and Clark 1 e Fundamentals of High School Mathematics, 11 lustrat ing Teaching of life gat iv e Numbers . 142
11l From Newcomb’s College Algebra, Device for Te acting Negative Numbers ...» 144
IV From Collins’ Practical Flementary Algebra. Multiplication of Signed Numb ers 145
V From Fite’s College Algebra, Graphical definition of Iterations with 146
VI From Hedrick’s Algebra for Secondary Schools. Pictorialßuie o? Addi-t ion and Subtraction of Signed Numbers 147
VII From Gugle ’s Modern Junior Mathema- tics* Oppo sif ivene s s in Fairs of lumbers of Ideas 148
VIII From Drushel and Withers’ Junior High School Mathemat ical*Tssen~ £ials, fli Year. Test for the Negative Number Idea . 150
IX From Myers 8 First Year Mathematics. Turning-Tendency in Operations with Signed Numbers 152
Plate
XIII
X From Ward’s Young Mathematician 1 s Guide. The Enigma . . . . . 157
XI From Ward’s Young Mathematician’s Guide. The Enigma 158
XII From Ward’s Young Math emat ic i an ’s Guide, Simple Equations . . 160
XIII From Ward 1 s Young; Mathematician’s Guide. Simple Squat ions . . 161
XIV From Ward’s Young Mathematician’s Guide. Quadratics . . . 163
XV From Ward 1 © Young Mathematician 1 © Guide. Quadratics .... 165
XVI From Rugg and Clark’s Fundamentals of High School Msthema'tics.~~ We Balance in Teaching Equations 172
XVII From Rugg and Clark’s Fundamentals of High School Mat hem atic s Rules for Solving Word ¥roblems. 189
XVIII From Schorling and Clark’s Modern Algebra, First Course. Direc-~ t ions ioi’ "SoTv d Problems . 190
XIX From Swenson’s High School Mathematics. Completing the Square Geometrically. ...... 210
XX From Simpson’s Treatise on Algebra. * a——ata— ■n—-nm—rrw* aw— m» h— li— it—tM Early Appearance of Modern Exponent Terminology 239
CHAPTER I AIMS, SCOPE, MATERIALS AND METHODS OF THE STUDY
Aims and Scope of the Study.
The purpose of this study is to trace the changes in subject matter and methods in elementary algebra from 1829 to 1929, inclusive. The term, ’’elementary algebra”, is here taken to mean a first course in the subject, such as would be covered in about a year’s work in the average modern high school. The study shows that the elementary algebra of 1829 was considerably different from that of 1929; but by constant use of the phrase ’’first course in algebra”, it is possible to have a continuous element throughout the century and one in which striking changes may be traced.
As a background, certain aspects of elementary algebra as it existed prior to 1829 are considered. This does not widen materially the scope of the study, but it helps greatly in understanding the significance of the course as it existed in 1839. This pre-
liminary survey also does much to establish a comprehensive basis for the study in general.
Materials for the Study.
There are two types of records that may be used for studying changes in elementary algebra. One consists of what various people have written about the course as it has been taught from time to time in the schools; also, such courses of study as have appeared in announcements and catalogues are included here. The other type of source material includes the algebra textbooks that have appeared and have been used in the classroom. For various reasons, the second type of material, which is in fact primary source material, is much to be preferred over the first. In the first place, it is easier to get at. In the second place, it is more definite. In the third place, it comes much nearer than any other kind of material to representing the course as it actually is, or was. Rugg and Clark, in their study published in 1918, give this last fact much prominence. They made a canvass of the practices of high-school teachers of mathematics with respect to the degree to which they deviate from the textbooks. Their findings confirmed those which Koos reported in 1917 and are summed up in the
following quotations:
.... textbooks almost always completely determine the specific subjectmatter that is taught to students in the course.l
We have shown, therefore, the tremendous influence of the textbook on the course in mathematics — the powerful leverage of the small group of ’adopted authors’ on the educational careers of children in our public schools. It seems quite clear that a person who is desirous of really effecting permanent improvements in the subject-matter learned by children must embody his ideas in the form of a textbook. *
Textbooks furnish the basic material for this study. About one hundred of them are involved at one time or another. In order to show how these textbooks cover the period from 1829 to 1929, there is presented the following distribution of dates of publication of the texts used in gathering the data of Chapter 111.
In addition to these texts, three typical French texts are treated in Chapter 11. Complete lists of textbooks used appear in the bibliography* If the findings of Rugg and Clark quoted above are to be taken as valid, it is not unreasonable to claim that the
textbooks examined furnish the best possible picture of the changing subject matter and methods in elementary algebra for the last one hundred years.
Included in the list of textbooks examined are a few College Algebras. It is interesting to note that the order of the more common topics does not differ materially in these books from that in the more elementary texts. The chief difference lies in the inclusion of additional topics and in the intensiveness of the treatment. Also, a number of English textbooks are involved, especially in the 1875 - 1890 period. It is to be observed that these do not differ materially from the American texts of the same period.
In addition to the textbooks, other types of materials enter into the study from time to time. These consist for the most part of books on the psychology and teaching of algebra and of reports of committees. These materials play a considerable part in the interpretation of the da.ta gathered from the textbooks. Finally, the results accruing from twenty years of teaching elementary algebra, of observing the teaching of others, and
of training prospective teachers of elementary mathematics inevitably enter considerably into the whole study.
1 Rugg, H. 0., and Clark, J. R.: Scientific Method in the Reconstruction of Ninth- -rgJe Mainematics, p. 20.
“Ibid., p. 23.
Date Interval Number of Textbooks Used Before 1829 5 1829 - 1839 2 1840 - 1849 2 1850 - 1859 3 I860 - 1869 5 1870 - 1879 6 1880 - 1889 10 1890 - 1899 6 1900 - 1909 13 1910 - 1919 14 1920 - 1929 12
TABLE I Distribution of Publication Dates of Textbooks
Methods of the Study.
In general, each part of the study embodies two phases, namely, the gathering of the data, and the analysis and interpretation of the same. For the gathering of the data, the textbooks have been examined separately for each topic included. In spite of many handlings of the books made necessary, this makes for economy in the long run, since the attention of the examiner can be focused upon one section of each book and all others can be ignored for the time being. Furthermore, a condition of saturation upon any particular topic may be more quickly reached in this way.
Laborious as the gathering of data may be, it does not present such serious difficulties as does its interpretation* In the present study, all available reliable information is brought to bear upon this task of interpretation. In such a situation, one does not hesitate to invoke the aid of the psychologist, both general and special; and the results of the experiments in the teaching of algebra are especially helpful. The extent to which these aids are used will appear in the different chapters.
General Plan of the Study
The study may be divided into two parts. The first includes Chapters II and 111. Chapter II deals with the two influences that have played a great part in determining the contents and methods in elementary algebra in the United States. In this chapter, appear the results of an analysis of certain early French and English algebra textbooks, done with a view to discovering what each group has contributed to our modern elementary algebra. In Chapter 111, changes in the order of topics and in the amounts of space devoted to the various topics are studied. Chapters II and 111, then, are of a general nature and may well be considered as making up Part I of the study.
Part II includes a critical and detailed study of the changing subject matter and methods in certain topics of elementary algebra# Topics treated are positive and negative numbers, the equation, the formula, and the theory of exponents#
It is in this part of the study that the ele- ment of method enters most prominently. The various phases of methods appearing in the textbooks are noted and evaluated in the light of established facts and principles.
Previous Studies of This Nature.
The analyzing of the contents of textbooks is rather a common practice. However, considerable search has revealed only one study of any consequence similar to the present one. Rugg and Clark in their Scientific Method in the Reconstruction of Ninth-Grade Mathematics (1918) have a chapter (Chapter II) entitled ”An Inventory of Ninth- Grade Mathematics”. The data for this chapter came from nine algebra textbooks then in use in the high schools of the University of Illinois High-School Conference and in the high schools of the North Central Association. Rugg and Clark limited their use of this material to a comparison of the amounts of formal material and of verbal problems. They concluded that too much emphasis was placed on formal exercises and that
. . . . roughly, one-fourth of the entire problem material is devoted to the equation which nearly all mathematics teachers will openly embrace as the central operation of
2 algebra.
Rugg and Clark’s third chapter bears the title, ”How Algebra Became the Ninth-Grade Course”. This involved an analysis of the contents of nineteen algebra textbooks, the publication dates of which ranged from 1751 to 1881. The method of analysis was very much like that used in Chapter HI of the present study. In fact, Rugg and Clark’s form of tabulation is used here with some modification.
Rugg and Clark 1 s study involved twenty-seven textbooks and dealt primarily with the relative emphasis placed on formal and thinking material. Furthermore, this study was only a part of a larger study and was made to serve the purposes of the latter. The present study is more comprehensive in its purpose and scope. Not only are many more textbooks involved, but changes in the teaching of the main topics of elementary algebra are considered in detail.
In an unpublished thesis written at the University of Texas in August, 1926, Miss Mary Ethel Adams treated the subject of graphs in elementary algebra. A large part of her data was gathered by examining algebra textbooks. Because of the thoroughness of Miss Adams 1 study, the topic graphs is included only incidentally in the present study. Observations made in the course of the study tended to confirm her findings.
In an unpublished thesis, written at the University of Texas, Mr. J. F. Hulse reported results of examining twenty-four algebra textbooks with publication dates ranging from 1898 to 1927. His findings with regard to materials on graphs harmonized in general with those of Miss Adams.
2 Ibid., p. 33.
CHAPTER II THE HERITAGE OF AMERICAN TEXTBOOKS IN ALGEBRA
Nature of Early American Algebra.
The root stock of the American people came from England, and it was but natural that English influence should predominate in the shaping of early American institutions; for example, the Latin Grammar School was transplanted almost unchanged from England to America, and our early Academy had its prototype in England*
In colonial and early national times in America, algebra was taught only in the colleges and corresponded roughly to what is now included in a high school course. This early algebra followed the English model almost exclusively, English textbooks furnishing the basis for the usual college courses.
Between 1776 and 1815, a large number of books in elementary and a few on higher mathematics were published in America. Many of them were reprints of English works, while others were compilations by American writers, modelled after English patterns. French and
German authors were almost unknown.
But in the early part of the nineteenth century, another influence made itself felt. This influence came from France and was strengthened by the fact that English mathematicians had recognized the superiority of the French mathematics over their own.
The improvements in mathematical textbooks and reforms in mathematical instruction were due to French influences. French authors displaced the English in many of our best institutions. It is somewhat of a misfortune, however, that we failed to gather in the full fruits of the French intellect. We followed in the path of French writers whose works had ceased to be the embodiment of the later results of French science; many of the works which we adopted were beginning to be I behind the times* when introduced into America. We used works of Bezout, Lacroix, and Bourdon. But Bezout flourished before the French Revolution, and Lacroix wrote most, if not all, of his books before the beginning of this century. In 1821 Cauchy published in Paris his Cours d*Analyse. If thoughtful attention and study had been given by our American text-book writers to this volume, then many a lax, loose, and unscientific
method of treating mathematical subjects might have been corrected at the outset. The wretched treatment of infinite series, as found in all our text-books, excepting the most recent, might have been rejected from the very beginning. 5
Such French textbooks as were used in America appeared usually in translation. The best examples of these were Farrar 1 s translation of Lacroix, appearing in 1818 for the use of Harvard students, and Davies’ translation of Bourdon (1844), which was much abridged from an earlier translation by Lieutenant Edward C. Ross. 6 Both of these books were examined
in the course of this study, and the translations were compared with the originals.
In what respects did French textbooks excel the English? Cajori repeatedly states that this superi- ority was most pronounced, but in no instance does he specify as to the particulars in which English books were inferior to those of the French. Davies, in the introduction to his translation of Bourdon, gives his readers some inkling as to the differences in the following words:
It has been the intention to write in this work, the scientific discussions of the French, with the practical methods of the English; that theory and practice, science and art, may mutually aid and illustrate each other. 7
While the contrast made in this quotation is still couched in general terms and will need particularizing, yet it serves the inquirer as a pointer to guide him in searching for particulars.
With a view to answering more definitely the question which opened the paragraph immediately preceding, some early English and French textbooks in algebra were examined, analyzed, and contrasted. A list of these books follows:
1. Ward, John: Young Mathematician’s Guide (1706)
2. Hammond, Nathaniel: Elements of Algebra (1772)
3. Simpson, Thomas: A Treatise of Algebra (1800)
4. Bonnycastle, John: A Treatise on Algebra (1820), Second Edition.
5. Peacock, George: A Treatise on Algebra (1830)
6. Bezout, Etienne: Gours de Matheinatiques, Troisieme Bartle TlSlS}
7. Bourdon, M.: Elements D’Algebre, Tenth Edition (1S48) “
8. Lacroix, S. F.: Elements D l Algebre, 51st Edition (1654) “
Obviously, the first five texts listed are English, the last three French. The wide variation in dates was not thought to be of any special significance. Earlier editions of the textbooks of Bourdon and Lacroix were not available, but there is every reason to believe that the later editions listed above do not differ materially from earlier ones. In fact, comparison of the 51st edition of Lacroix with an English translation (Farrar’s, 1818) of an early edition showed that the two are almost identical.
One way to bring out rough contrasts in algebra textbooks is to examine their tables of contents so as to see differences and similarities in the topics treated and in the order in which topics are introduced. In order to do this most advantageously for the eight texts listed above, Table II is presented, showing for the most common algebraic topics the order of introduction and the number of pages devoted to each topic.
4 Cajori, Florian: History of Mathematics in the United States, p. 45. '
s lbid., p. 99.
6 Davies, Charles: Elements of Algebra, p. 3.
7 rbid., p. 4.
Topics i puo lAJ mpso n Pea G oc k ) t -K C * o Sr 'b U 3 3 O Q u p 1. Notation 1 1 1 1 1 1 1 1 2. Four Operations 2 2 2 2 2 2 1 2 3 3. Involution 3 3 3 4 4 8 8 8 4. Evolution 4 4 4 5 6 5 6 9 9 5. Simple Equations (one unknown) 7 6 7 8 13 6 2 4 2 6. Fractions 5 5 3 3 3 3 4 7. Simple Equations (two unknowns) 9 8 15 7 4 5 5 8. Radicals 6 5 6 7 9 10 9* Exponents 5 10 10. Negative Numbers 7 3 6 6 11. Progressions 8 9 7 8 9 11 15 12. Quadratics 10 8 9 14 10 5 7 7 13. Binomial Theorem 10 18 11 8 14. Logarithms 10 12 12 12 16 15. Proportion 9 9 9 14 16. Imaginaries 6
TABLE II Order Assigned to Topics in English and French Algebra Textbooks
Topics ( “V t a £ £ Simpler? * ■+_ 4 'A o a qq — J/ Q 0 kj cr 4 0 c * c o p O X J 0 -1 1. Notation 2 4 7 8 9 41 8 14 2. Four Operations 7 39 28 11 102 28 5 33 33 3. Involution 6 7 6 4 3 6 6 23 4, Evolution 3 9 3 30 28 11 15 7 13 5. Simple Equations (one unknown) 8 10 6 27 30 16 20 16 17 6. Fractions 8 6 14 8 11 12 15 7. Simple Equations (two unknowns) 14 6 24 17 17 11 27 8. Radicals 4 23 6 3 12 26 9. Exponents 15 5 10. Negative Numbers 2 2 17 9 11. Progressions 6 6 13 10 19 18 16 12. Quadratics 8 16 19 20 61 14 80 49 13. Binomial Theorem 3 6 17 13 19 14. Logarithms 30 33 18 42 19 15. Proportion 4 24 4 3 16. Imaginaries 4
TABLE 111 Space in Pages Devoted to Topics in English and French Algebra Textbooks
Treatment of the Equation.
A glance at Tables II and 111 will reveal some differences in English and French practices as to the order of topics and in amounts of space devoted to each topic. Perhaps the most noticeable of the differences is to be found in connection with the position assigned to the equation. In the English texts this topic is assigned places ranging from seven to thirteen and is always made to follow such topics as involution, evolution, radicals, and fractions. This practice evidently is based upon the theory that the student should master the various mechanical manipulative elements of algebra before he tries his hand at the solution of equations, which is really the fundamental task of the subject. On the other hand, all three of the French writers introduce the equations early; in no case does it come later than fourth in the order of topics, and in two cases it is in second place. In Chapter 111, it is shown that what might be called the English arrangement of topics persisted until about 1910. The early use of the equation, however, is in exact accord with approved modern practices and can be completely justified on psychological grounds. Consequently, it may be said that the French texts were superior to the English texts with regard to this particular aspect of the arrangement of topics.
The foregoing remarks refer primarily to simple equations involving one unknown. As can be seen from Tables II and 111, they apply quite as readily to simple simultaneous equations as to quadratic equations. Furthermore, more aspects of equations were considered by the French than by the English. For example, the following phases of the equation are treated by Lacroix (the titles are anglicized):
The equation in general. Simple equation with one unknown. Simple equation with two unknowns. Systems of simple equations. General formulae for the solution of first degree equations. Quadratic equations. Equations in the quadratic form. Reduction of equations. Commensurable roots of an equation. Roots of a numerical equation. Exponential equations. 8
s Lacroix, S. F.: /l&nents D ‘Alg^bre.
Treatment of Negative Numbers
Tables II and
11l show that the French algebraists were more progressive than the English in their understanding and use of negative numbers. It will be shown in Chapter IV that the extension of the number system to include negative numbers is one of the most striking changes in elementary algebra. It will emerge that the topic positive and negative numbers first began to take a prominent place in American textbooks published from 1880 to 1890. It is significant, then, that only one of the English texts involved in Tables II and 111 gives the topic a special place in the course and that in this text it is assigned to seventh place. The other English authors buried the topic piecemeal, considering parts of it in connection with other topics. In sharp contrast to these practices, or lack of practices, one finds all three of the French writers giving a special place to the consideration of positive and negative numbers. Furthermore, the topic is introduced in the third, sixth, and sixth order respectively in the three books, and the topic is discussed more at length than in the English books. That this French treatment of the negative is more in harmony with present-day practices is revealed by the investigations reported in Chapter IV. Here again, then, one is forced to conclude that French thinking and practice were superior to those of the English.
"Wordy” Textbooks.
Harold 0. Rugg has pointed out that students beginning the study of algebra do not think naturally in terms of the algebraic symbolism which characterises the thinking of the mathematician. He says:
Children, however, do their thinking in terms of detailed word-symbols; only laboriously do they take on the more abbreviated methods of symbolic thinking which are typical of mathematical manipulation. The solution is clear: we need ’wordy* textbooks — textbooks in which the transition from thinking in detailed word-symbols to that in abbreviated letter or algebraic symbols and in their manipulation is made so gradually as to keep, step by step, just ahead of the pupil’s mental advance. y
French, authors did just this. Consciously or unconsciously, they eased their readers over this trying transition process. Though they were expert in the use of algebraic symbology, their textbooks are wordy. They are careful, meticulous, even verbose, in their detailed development of the theory of algebra. Thus they instinctively adapt their discussions of the subject to the needs of the reader.
9 Rugg, H. 0., and Clark, J. R.: Fundamentals of High School Mathematics, p. iv.
The Theoretical versus the Practical.
Davies referred to the ”scientific discussions” of the French and the ’’practical methods” of the
That this characterization is accurate is amply shown by an examination of the textbooks. The French are theoretical, mathematically rigorous, while English textbooks read almost like handbooks of detailed directions for doing things. For example, Hammond in his Elements of Algebra (1772) tells his reader how to solve equations ’’when the unknown Quantity is to several Powers in one equation, and but to the first Power in the third Equation,” also ’’the Manner of resolving them, when the unknown Quantities are to several Powers, in both Equations’’. 11 These directions for doing the various tricks of algebra did not always cohere into a logical, sequential development of the subject.
In general, then, the French writers were concerned primarily with logic, the why of algebra, while the English authors put more stress on the how of the subject.
quotation, p. 14.
N.: Elements of Algebra, pp. 243-257.
Examples and Problems for Practice.
So intent were the French upon the logic of algebra that they forgot, or ignored, the principle that students learn most effectively to do (and to understand) by doing. They included only such problems and examples as were necessary for their own purposes in developing the principles of algebra. The examples were for the most part included in the continuous discourse of the discussion and were rarely displayed as they are in all modern textbooks. Only one set of examples for students to solve was found in the three French texts. This set appeared on page 52 of Bezout*s Cours de Mathematiques, Troisieme Partie, and consisted of seven exercises. All other exercises were worked out by the various authors.
In the books of Ward, Hammond, Simpson, and Peacock, all exercises and problems were solved by the authors. However, whole sections of these books were devoted to the solution of extra problems, not at all necessary for the logical sequence. Here again is an instance of the English stress upon the practical doing. In Young’s Treatise on Algebra (1838) there are problems for the students to solve. For example, pages 62 to 68 present a group of word problems leading to simple equations involving one unknown. Again, in Bonnycastle’s Treatise on Algebra Volume I (1820), which is modern in some respects, there are plenty of problems left for the student to solve. On pages 98 to 100 are given twenty-five “examples for practic H , involving for the most part equations of various types. Thus is strengthened the conviction that French algebra texts were logical and English texts psychological, comparatively speaking. The degrees to which these characteristics entered into later American textbooks will be considered in succeeding chapters.
Chapter Summary.
Early American textbooks in elementary algebra are the results of many influences, the most important of which came from English and French sources. Until about 1815,
American textbook writers followed the lead of the English authors, but after that date French influence became very strong. Translations of the books of Bourdon, Bezout, and Lacroix were used as textbooks in some of the leading American schools.
An examination of typical early English and French algebra textbooks shows that the French were inclined to be theoretical and logical and that the English were practical and psychological, comparatively speaking* French mathematical thought, as revealed in the textbooks, was in advance of that of the English. Many of the features of the early French textbooks have a distinctly modern trend. Particularly is this true of their treatment of the equation, in their insight into negative numbers, and in the "wordiness 1 ’ of their books. However, English texts excelled in the practice of providing examples and problems for the student.
CHAPTER III CHANGES IN THE ORDER OF AND SPACE DEVOTED TO THE TOPICS OF ELEMENTARY ALGEBRA
Purpose and Methods of the Chapter.
This chapis a report of a survey made with the purpose of determining the changes that have taken place since 1830 in the order of topics in elementary algebra and in the amount of textbook space devoted to them*
The data come from seventy-seven textbooks on elementary algebra. The publication dates of these texts are shown in Table I, page 4. A glance at this table will show that the sampling of textbooks is typical, in so far as the date distribution is concerned. In addition to this, the books are chosen for the most part because the contents of each were representative of the elementary algebra taught in the period in question.
The position and amount of space allotted to each topic in any particular textbook is determined in most cases by an examination of the tables of contents. For the early textbooks, this method of analysis would not serve, for the simple reason that no tables of contents were furnished by the authors. In such cases, the data are secured simply by turning through the books.
In fact, judging the contents of a book by its table of contents is open to criticism. If a topic is not mentioned in the table of contents, it does not always follow that the topic is entirely omitted from the book; it may be treated incidentally, fragmentarily, or in connection with other topics. However, the topics considered in this survey are of such significance that they should be classed as major topics; and if an author considers them as such, he will give them major classification and list them in his table of contents. It is with this realization and understanding that the table of contents is relied on so implicitly in this survey. The table of contents is freely supplemented, especially in cases of doubt, by reference to the body of the text.
The topics considered in the survey are listed here:
1. Notation and Definitions. 2. Four Operations. 3. Involution. 4. Evolution. 5. Simple Equations (x). 6. Fractions. 7. Simple Equations (x,y). 8. Radicals. 9. Exponents. 10. Negative Numbers. 11. Progressions. 12. Quadratics. 13. Binomial Theorem. 14. Logarithms. 15. Proportions. 16. 17. Formulae. 18. Factoring. 19. H.C.F. and L.C.M. 20. Inequalities. 21. Graphs. 22. Trigonometric Ratios. 23. Function Ideas.
Other topics are frequently included in elementary algebra textbooks, but those listed above are by far the most common and the most important. Some of the titles used have to be interpreted in a very broad way. For example, the first topic, “Notation and Definitions”, is made to cover a multitude of items. In many cases, it amounts to an introduction to algebra. The topic proportion is made to include ratio and variation. Finally each topic is so interpreted as to include problems in which are applied the principles developed in the preliminary
discussion of the topic* This last extension of meaning applies particularly to the various types of equations, namely, simple equations involving one unknown, simple equations involving two unkncwns, and quadratic equations involving one or more unknowns.
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TABLE IV Order of Topics in Seventy-Sevm Algebra Textbooks
Notation & Defini- tion Four Operations 1706 1772 1800 1820 1829 a H P ft O o 15 1830 o o o cd CD ft 9 102 1838 b0 a 2 o 41 28 1842 co P 44 A CD ft 6 23 1842 >» A cd P CD 8 >»CD Cdft ftW 38 1855 0) ft J* O CD 6 26 1857 (D ♦H 8 o o 11 30 1859 1862 1864 1866 cd CD ft P CD CD P O 16 22 1866 >» £ cd <D P a a CD g >» CD Cd ft ft N 39 1869 1872 1872 01 p ♦H 44 P (D ft 6 26 1875 1876 1879 P P CD O P m p P cd •H 8 ft CD O ft ft W 18 40 1879 ft p o ch P cd CD 21 46 1881 1882 1883 1885 CD 44 P cd o 14 45 1885 1885 1887 1887 • p ft P P o P o o +3 p p P o CD cd 15 41 1889 8 cd ft cd p o 6 26 1890 1890 1892 ft Sp ft bO ft «H P P ft M 8 34 1894 1898 1899 1900 >* CD P £ O Q 7 37 1900 CD td p £ Ui P •H O ft m 23 55 B02 1902 L905 1908 1908 P a CD P 5 S ft p ft Ui 15 61 1908 1908 1909 1909 Nicholson o CO 1909 P CD § C cd o N P P P CD CD ft P ft o 14 1910 1911 1911 1911“ 1912 1912 1912 1913 1915 ft o •h 01 CD P ft 41 1915 1016 1.917 (n p p • cd Ui * .• ft Ui • • p ft 19 1919 ♦ - Ui CD • P ft o • p p UI ft 15 34 1919 p ft p ft S ft P * o p ft CD ft cd ft ft < 3 9 1920 ft p CD ft P O P P £ 7 49 1920 1920 1920 44 P co O 1 bD bD 5 21 1923 o6 -CD P a P CD P Opp P P O POO CD p. bdpp p cd oi WOft 13 65 1923 1924 1924 1925 1926 1926 1929 P p cd 2 7 *3 £ O S § 4 39 g 01 ft 8 CD 7 28 cd o >* £ O m 8 11 P cd a P CD CD •H 8 > CD Cd r—1 QH 26 26 p o co a •H ft O ft 11 24 Ui CD cd * Q ft 9 21 CD ft cd P CD > 23 37 CD O P ft 11 29 44 O CD ft 18 25 nd § "cd w 23 27 r*» CD P O 11 27 p p CD O > P Ct ft CD cd CD 4 34 p CD P ft O Ui 16 28 ft p 9 cd 38 o •H 8 CD cQft ft cd ft o CD 14 24 s o o £ CD 24 34 CD * bo Ui (D ft ft CD O O 8 32 3d p p O rd £ O P o P rP CD o £5 Ui 27 31 33 CD -P ft P CD O P (D P P P CD 01 ft 18 25 a o P o P ft o o CD P P £ CD CD ft 14 36 >* ft 9 p p P CD O P £ (D P 8 P CD CD ft ft 15 46 p g cd p 8 ft (D 8 ft Ui 29 39 £ . 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P ’H * Q fee ft 64 p o CD P CD £ Ui 17 40 CD A O CD CD •'Cd ft - -p bO ♦H P 8 CD CD ft 54 Stone, New Math* Thorndike do P P 44 O P ft cd O ft Ui o 25 Involution 6 7 6 4 4 3 5 6 24 4 13 9 16 12 6 5 10 5 16 1 10 8 5 5 17 4 7 11 3 5 6 4 4 4 3 3 5 6 8 4 6 3 6 4 4 2 - 2 1 3 Evolution 3 9 3 30 11 28 11 13 13 20 6 33 16 26 21 17 18 19 11 24 11 17 10 23 13 40 11 15 4 12 11 28 13 9 12 14 9 15 16 14 3 19 17 17 25 13 14 9 12 7 8 18 10 8 7 14 5 14 11 Simple Equations (x) 8 10 6 27 26 30 16 18 25 40 19 30 27 16 26 30 36 11 24 15 23 36 50 39 36 20 30 27 s 26 24 25 22 13 39 13 33 44 30 14 18 19 15 18 11 23 36 9 46 P o p o 21 35 31 12 53s 44 s 35 34 37 28 7 19 21 15 73 20s 69s 56 s 35 s 6 27 75 s 22s 60 Fractions 8 6 14 16 8 11 19 27 8 15 18 19 23 31 25 25 23 14 20 29 29 42 28 15 14 35 39 26 17 25 15 15 24 4 14 25 27 14 30 30 23 21 40 22 55 36 22 s i s i 21 8 29 s 128 25 11 9 20 33 7 19 18 30 30 28 26 p 22 Simple Equations (x,yj 14 6 6 24 17 25 15 30 13 10 35 22 28 15 20 16 29 16 17 20 17 19 16 15 25 21 26 12 13 26 - 15 34 13 10 31 30 • 7 27 27 18 30 21 21 40 8 30 CD P CQ P 19 31 28 25 16 23 9 18 23 13 12 27 13 12 33 22 22 4 20 w 21 Radicals 4 23 6 18 10 3 I 17 29 27 28 26 12 9 15 17 29 22 25 29 18 22 47 21 21 11 18 6 18 18 16 22 12 12 25 35 10 22 O O 22 12 11 45 7 10 2 38 19 17 9 6 25 6 17 13 CQ CO 4 Exponents 5 15 1 J 5 4 7 3 2 8 13 12 10 21 5 26 8 13 8 25 4 12 39 2 2 15 9 20 20 CD i s 10 9 4 8 5 7 2 3 6 p 4 Negative Numbers 2 6 1 7 3 13 16 15 6 11 7 9 Q Q 3 5; 13 9 11 5 18 24 18 13 10 11 4 30 24 30 5 9 7 8 28 13 CD 35 Progressions 6 6 13 11 10 18 12 9 14 20 15 21 18 15 12 20 20 11 23 22 24 12 13 20 23 18 21 14 18 24 13 19 10 16 18 13 18 6 20 10 16 25 21 25 CD CD 9 9 13 8 17 16 D i Quadratics 8 16 19 46 20 61 40 25 45 33 40 49 38 61 27 30 60 33 32 28 35 27 26 14 30 42 22 44 22 27 42 39 38 21 10 41 32 17 34 55 8 60 53 46 125 57 45 S e S e 26 31 50 30 40 10 2e 10 11 32 6 33 27 15 20 27 15 24 23 Binomial Theorem 3 6 13 17 10 7 6 10 17 5 4 5 20 5 4 19 7 12 10 20 8 13 3 7 16 18 4 4 6 7 6 4 CD Logarithms • 30 17 33 18 27 18 18 17 14 11 9 14 9 41 14 23 17 20 26 14 8 37 17 22 23 14 23 19 19 15 6 19 3 15 20 24 23 8 15 CD Proportion 4 24 4 10 16 13 7 7 10 20 8 20 21 17 11 23 21 21 9 16 12 17 17 17 10 19 21 21 13 14 26 15 25 14 13 4 9 20 11 15 21s 9 14 15 22 7 5 19 20 15 6 16 12 33 Imaginaries 4 9 1 3 4 5 6 13 10 7 6 6 9 9 8 6 3 13 9 6 12 10 Formul as 8 2 28 2 7 9 10 10 26 6 25 16 6s 10 40 s s 14 Ils 8 Factoring 20 5 7 16 6 3 5 5 6 9 16 6 14 14 14 24 9 21 14 13 23 20 12 23 26 26 26 43 35 s 35 17 18 298 lie 44 7 23 22 7 19 4 26 25 14 44 s 24 H.C.F. & L.C.M. 9 3 12 12 10 3 13 14 13 12 4 7 6 6 9 10 9 15 11 13 11 15 2 5 15 15 8 30 11 2 8 6 23 4 3 6 7 Inequalities 4 2 6 2 2 3 4 3 5 6 3 4 2 5 6 4 6 7 3 4 3 6 15 Graphs 14 15 20s 12 17 6 24 16 39s 12s 15 22 15 23 5 15 32 10 39e 13s 28 s 47s 29 s 17 39s 58 24 15 i Trigonometric Ratios 16 4 24 18 25 Function Idea 35 s 7 26
TABLE . V Pages of Space Devoted to Topics in Seventy-Sevm Algebra Textbooks
Tabulation of the Data.
The data derived from the seventy-seven textbooks are displayed in Tables IV and V. Table IV shows the order of topics in each of the seventy-seven textbooks. The topics are numbered consecutively in the order in which they are treated in the book. If topics other than those under consideration appear in any particular text, the number or numbers corresponding to the extraneous topic or topics are omitted from the tabulation. If a topic is treated in a spiral or cyclic manner, it is assigned a position corresponding to its first appearance, and it is labelled «s H (spiral).
Table V shows the number of pages devoted to each topic by each author. In oases of spiral treatment the number in Table V refers to the total number of pages devoted to the topic.
Certain textbooks do not fit at all into this form of analysis. They are Myers, First Year Mathematics (1909), Nicholson, School Algebra (1909), Thorndike, Junior High School Mathematics, Book Three, (1926)* Spaces for these textbooks are provided in the tables, but the data are considered in a special discussion.
Extent of the Changes.
To run the eye from left to right along these two chronologically arranged tables is enough to convince one that sweeping changes have taken place since 1830 in the order of topics in elementary algebra and in the amount of space allotted to them. The remainder of this chapter is devoted to a somewhat detailed analysis of these changes; and as a sort of introduction to this analysis it is proposed to present a contrast between the arrangement of subject matter in an early textbook and that in a modern one. This is in the nature of a "before and after taking” picture; what happened in between the before and after will be revealed to some extent fey later discussion.
Young’s Elementary Treatise on Algebra is chosen as typical of the early algebra textbooks. This book was written by an Irish college professor, was published in 1838, and, according to Cajori, was used extensively as a textbook, even in the American colleges. In spite of its college origin and flavor, in its scope it is not unlike those texts now being written for and used in high
school algebra classes. The subject matter of Young 1 s textbook is contrasted with that of Smith, Foberg, and Reeve’s General High School Mathematics, Book JI, which was published in 1925. This text is typical of those in which is attempted a unification of the arithmetic, algebra, plane geometry, and trigonometry of the high school mathematics course. Book One, which is considered here, is composed almost entirely of algebraic subject matter.
The position and space given each topic is shown below in parallel columns for the two textbooks*
At this point, it is sufficient to call attention to a few of the striking differences in the two tables of contents given above. They are:
1. Graphs, directed numbers, the formula, and numerical trigonometry are new topics. The first three occupy the first three positions in the 1925 textbook.
2. More attention is given to the simple equation in the modern text. Fractional equations are given a special place.
3. Involution and evolution are treated very differently in the two texts.
4. Some of the more traditional topics (series, progressions, binomial theorem, indeterminate equations) appear in the older text but not in the modern one.
Topic Order of Topic Space Given to Tonic Young (1838) Smith, F.,and Reeves (1925) Young (1838) S'., F., and R. (1925) 1* Four Operations 1 5 18 54 2. Fractions 2 6 11 26 3. Involution 3 11 3 4 4. Evolution 4 12 8 10 5. Linear Equations (one unknown) 5 4 16 51 6. Linear Equations (two unknowns) 6 10 17 20 7. Ratio and Propor- tion 7 8 5 16 8. Progression 8 — 10 None 9. Quadratic Equa- tions 9 12 46 24 10. Surds 10 7 None 11. Imaginaries 11 8 None 12. Binomial Theorem 12 —• 17 None 13. Logarithms 13 18 None 14. Series 14 34 None 15. Indeterminate Equations 15 17 None
TABLE VI Contrast of Order Assigned to and Space Given to Topics in an Early and a Modern Textbook
Topic Order of Topic Smith, Space Given to Topic Young (1838) F.,and Reeves (1925) Young (1838) S. ,F.,and R. (1925) 16. Graphs 1 None 24 17. Directed Numbers 2 None 28 18. The Formula B11 III * 3 None 14 19. Numerical Trigonometry 9 None 18
Junior High School Textbooks and the Topics of Algebra
Some special consideration should be made of the parts of several sets of junior high school textbooks included among the books examined. Strictly speaking, these books are not textbooks on algebra, just as they are not textbooks on arithmetic nor on geometry. They furnish about the best existing examples of that arrangement of middle grade mathematics that has been characterized by such terms as generalized, unified, correlated. This organization of subject matter plays havoc with the traditional logical sequence of topics, so much so that some of these topics are often lost in the new order of things.
The reasons for studying these textbooks here are two in number: first, they include what is roughly equivalent to the material of first year algebra; second, they have undoubtedly had considerable influence upon both the selection and arrangement of subject matter in algebra and upon the methods of teaching this subject matter. Consequently, no study of the changes in elementary algebra since 1829 would be complete without some mention of the junior high school textbooks.
In the early stages of any movement, the theories and practices are fluidic, individualistic, and almost devoid of what might be called standardization. This is certainly true of the movement toward unifying the mathematics of the junior high school. In sharp contrast to the standard first year algebra course of twenty-five years ago, one finds the contents of the various sets of junior high school mathematic textbooks showing a great amount of variety. It is the purpose here to show the contents of a representative set of junior high school mathematics textbooks in order to see what happens to the topics cf algebra therein. There is no standard set of such books. Hence it matters little which set is chosen for display; and the set actually used is selected because it is at hand and is probably as good as any other. There follows, then, the table of contents of Drushel and Withers’ Junior High School Mathematical Essentials (1924),
for the Seventh, Eighth, and Ninth school years.
Seventh School Year:
Topics Pages I. Using graphs in problem solving ... 12 11. The Table solution in problem solving . 7 111. The Equation in problem solving ... 6 IV. Reviewing methods of problem solving. . 5 V. Practice tests for accuracy and speed • 13 VI. Saving time in computing 12 VII. Common business forms, practices, and problems 16 VIII. Percentage . 9 IX. Using Percentage . 9 X. Protecting one’s life and property . . 17 XI. Collecting and distributing public money ....... 9 XII. Constructing and measuring lines and angles ........ 18 XIII. Constructing and measuring surfaces • . 9 XIV. Computing the contents and surfaces of solids 9 XV. Reviewing the year’s work . . . . . 21. *
Eighth School Year:
Pages
I. A new use for graphs in problem . solving 8 II• Short tests in the fundamentals for speed and accuracy . .... 15 111. Speed and accuracy in computation through short methods . .... 11 IV. Squares and square roots . .... 9
Pages V. Using Squares and square roots ... 9 VI. Similar triangles 8 VII. Learning and using formulas in problem solving 30 VIII. Metric measurement . 10 IX. Money and banking ....... 11 X. Trade and transportation ..... 20 XI. Travel 11 XII. Economy and thrift 15 XIII. How money earns money 20 XIV. Savings accounts . 7 XV. Investments ....... 21 XVI. Reviewing the year’s work .... 22.
Ninth School Year:
Topics Pages
I. Learning a new kind of number . . • 6 11. Applying the fundamental processes to positive and negative numbers • 22 111. Applying the fundamental processes to positive and negative numbers (continued) . ... 48 IV. Saving time by knowing short methods 28 V. Algebraic fractions 28 VI. Learning more about formulas ... 9 VII. Learning more about graphs .... 13 VIII. Solving simultaneous linear equations by algebraic methods ... 22 IX, Powers and roots 32 X. Learning more about the right triangle ...... 28
XI. Quadratic equations .... 15 XII. Variation and proportion . . 15 XIII. Reviewing the essentials of junior high school _. mathematics .... 30.
It is evident that much of the material appearing in these three volumes is not algebraic. Yet the tables of contents as they appear above show what the junior high school textbooks do to the topics of algebra. Some of them, as fractions and quadratic equations, remain almost intact in the ninth school year, some of them have disappeared, and some have been moved back to the seventh year. Excellent examples of the last are found in such topics as formulas, equations, and graphs. Problem solving by these three means greets the pupil at the very beginning, and he never loses sight of them. Formulas play a very important role in the development of the subject of general mathematics.
Formula interpretation and manipulation of the character developed . . • should give
pupils power to grow naturally into the more general mathematics of the ninth year and should prepare pupils to read with understanding the easy formulas of physics which they may encounter in their general science. 15
1 ? Drushel, J. A., and Withers, J. W.: Junior High School Mathematical Essentials, Seventh School Year, pp. viii - xi.
13 Drushel, J. A., and Withers, J. W.: Junior High School Mathematical Essentials, Eighth School pp. lx- xiii 7
•^Erushel, J. A., and Withers, J. W.: Junior High School Mathematical Essentials, Ninth School Year, pp. viii - xii.
15 Drushel, J. A., and Withers, J. W.: op. cit. Eighth School Year, p. vii.
Four General Types of Changes.
A considera- tion of the facts brought out in the preceding sections and an examination of Tables IV and V show that in general four major changes have taken place in the order of and emphasis on the topics of elementary algebra:
I. Certain topics have practically disappeared from the course.
11. Certain topics have just about held their own.
111. Certain topics have changed position in the course.
IV. Certain topics have appeared in the course since 1829.
Topics That Have Disappeared.
The best examples of topics that have ceased to be integral parts of the elementary algebra course are inequalities, imaginaries, highest common factor and lowest common multiple, involution, and the binomial theorem. The last two topics should be considered together, since raising a binomial to any power is one phase of the subject of powers in general.
Table VHand Diagrams I, 11, and 111 show graphically the changes that have affected inequalities, imaginaries, and highest common factor and lowest common multiple, respectively.
*’’Order” and ’’space” refer to averages in the various texts.
The changes in the treatment of the three topics, inequalities, imaginaries, and highest common factor and lowest common multiple, are very similar one to
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another. It appears from Diagrams I, 11, and 111 that textbooks published before 1850 contained little or no material bearing on these topics. Beginning about 1850, the percentages of textbooks containing these topics steadily increased, slightly more space was devoted to them, with little variation in the order of the topics. This increase in the prevalence of these topics continued until about 1910, at which time a decline set in, and the topics disappeared from tables of contents about 1920. The Committee on Mathematical Requirements recommended in 1923 that the highest common factor and lowest common multiple and imaginaries be omitted from the algebra course.
No mention is made of inequalities; it is reasonable to conclude that the Committee assumed that this topic should be omitted. The present study, then, shows that in practice the elimination of the three topics in question preceded the recommendations of the Committee.
Committee on Mathematical Requirements, The Reorganization of Mathematics in Secondary Education, p. 27.
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Dates Inequalities Imaginaries H.C.F. andL.Cu. $ Order Space Order Space $ Order Space Before 1829 0 0 0 20 6 5 0 0 0 1830 - 1839 0 0 0 0 0 0 0 0 0 1840 - 1849 0 0 0 50 9 8 0 0 0 1850 - 1859 66 10 >5* 3* 33 9 1 66 8 5 1860 - 1869 20 7 6 20 12 3 100 3.8 10 1870 - 1879 50 9.3 2 1 3 16.6 8 4 83 5.8 10 1880 - 1889 33 10.6 4 44 11.2 8.5 88 5.1 9.7 1890 - 1899 66 9.2E 5 50 13.6 6.3 100 5.4 10.6 1900 - 1909 63 11.7 5.1 63 14.5 8.1 63 5.5 12.6 1910 - 1919 22 8 11.3 22 15 9.3 28 6 5 1920 - 1929 0 0 0 0 0 0 0 0 0
TABLE VII Topics That Have Disappeared
Topics Unchanged as to Order and Space.
Certain algebraic topics have remained practically unchanged for one hundred years as to order and space allotted to them. The best examples of this are to be found in the topic labelled “Definitions and Notation*’ in Tables IV and V, fractions, and quadratic equations. Table VIII and Diagrams IV and V show graphically the same supporting evidence for this statement.
No tabular data are presented for "definitions and notations". It is shown in Tables IV and Vto be so uniformly in first place in all books published
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before 1920, that no graphical devices are necessary. The space devoted to this topic varies widely, but this is to be expected, since this preliminary chapter serves as a dumping ground for miscellaneous items.
The striking changes taking place about 1930 in the practice of handling definitions and systems of notation has a psychological explanation. The authors of the older texts evidently held the theory that a beginning student in algebra could not accomplish anything until he had learned the language of algebra. Consequently, each author made his first chapter to consist of a definition or description of every symbol, term, or idea that was to be used anywhere in the course. Inevitably, this process became for the student merely a matter of memory; the definition was learned quite apart from the use to which the thing defined was to be put.
Modern authors have changed this. They proceed in accordance with the psychological principle that a thing is best understood if it is used. Consequently, modern textbooks in algebra do not confront their readers at the very beginning with a formidable array of definitions; definitions are given and symbols introduced when they are needed.
As to the topic, ’’Fractions”, Table VIII and Diagram IV show that practically all textbooks treat it as a separate topic, the exception in the interval 1910 to 1929 being charged for the most part to the junior high school textbooks. The position assigned to fractions has changed gradually from about three to eight. Comparatively speaking, this is not a striking change; and it can be accounted for partly by the fact that certain new topics, such as graphs and formulas, have been placed before fractions and that equations now precede this topic also. The amount of space given to the topic fractions has not changed materially. All of this is not to be taken to imply that the treatment of fractions in current textbooks is not different from that found in texts published one hundred years ago.
As a topic, ’’Quadratics’ 1 , has varied even less than has fractions as to frequency, order, and space. Table VIII and Diagram V indicate that, except for the junior high school texts, practically all books have quadratics as a topic, that the order has
changed almost not at all, and that the space does not vary any more than would be expected in such a wide variety of textbooks. Further discussion of quadratics will be found in a later chapter devoted to a discussion of the equation. (See Chapter V.)
T)/ciCirGt m rtf- fractions - Space.
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Dates Fractions Quadratics Order Space 1 Order Space Before 1829 80 3*4 11 80 9 21 vl830 - 1839 100 3 9.5 100 12 40 1840 - 1849 100 3 23 100 9.5 32.5 1850 - 1859 100 3.3 14 100 10 36 1860 - 1869 100 4.8 24 100 11.6 41 1870 - 1879 100 5.1 26 100 11.1 36 1880 - 1889 100 7.5 24 100 11.5 30 1890 - 1899 100 6.3 19 100 12.3 27 1900 - 1909 91 5.9 29.3 100 11.1 47 1910 - 1919 71 7.6 18 64 10.3 27 1930 - 1939 62 8.3 26 82 11.1 23
TABLE VIII Topics That Have Remained Practically Unchanged
Topics That Have Changed Position in the Course.
An outstanding change can be seen in the shifting of topics from one position to another in the arrangement of the course. Illustrations of the working of this tendency can be found by examining such topics as involution, evolution, simple equations, the four operations, and radicals. Because of their greater importance, the last three will be considered more fully, and changes affecting them will be shown by means of tabular data. These data are given in Table IX and Diagrams, VI, VII, and VIII.
The term ”simple equations” is here limited to first degree equations involving one unknown, but changes in this topic are almost exactly paralleled
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by those in first degree equations involving two or more unknowns. The fact that textbooks published from 1910 to 1919 and from 1920 to 1929 show only 92% and 95%, respectively, containing the topic simple equations is to be accounted for by the presence of several sets of Junior high school texts in these groups of texts. The sets of books written for the junior high school, while containing for the most part the equivalent of a year’s work in algebra, are organized along “generalized, correlated, unified 1 * lines and do not always preserve the identity of the traditional topics.
As can be seen from Table IX and Diagram VI, the topic simple equations has steadily risen to an earlier place in the order of topics. In fact, it is not unusual to find this topic in second place. It occupies that place in one-half of the texts published since 1914 and examined in the course of this study. Also, the amount of space given over to simple equations has steadily increased from 15.4 pages (average) to 44.3 pages (average). Further discussion of simple equations will be found in Chapter V.
The four fundamental operations once firmly ensconced in second place are now relegated to about fifth place. This has been caused by the general shifting of the
topics, more especially by the introduction of graphs, formulas, and negative numbers, and by the earlier introduction of the equation. The tendency to give increased amounts of space to the four fundamental operations may be ascribed to the practice of studying the four fundamental operations with positive numbers only and again later with both positive and negative numbers. In addition to the data of Table IX, Diagram VII should be considered in connection with the four operations.
Radicals is one of the old reliable topics. Table IX and Diagram VIII make it clear that this topic has waxed and waned, insofar as its frequency of appearance as a separate topic is concerned, also with regard to the amount of space devoted to it. On the other hand, the position of the topic in the course has steadily declined, that is, it has been placed farther and farther from the beginning of the course*
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Dates Simple Equations Four Operatione Radicals Order Space $ Order Space * Order Space Before 1829 100 5.8 15.4 100 2.1 20 60 5.7 10 1830 - 1839 100 9 23 100 2 65 0 0 0 1840 - 1849 100 4 21.5 100 1.5 30.5 100 8 14 1850 - 1859 100 5.6 29.6 100 2 27 100 7.9 14 1860 - 1869 100 6.1 27 100 1.8 29 100 10 21 1870 - 1879 100 7 26.5 100 2 31 100 10.5 26 1880 - 1889 100 5.3 26.4 100 2.1 33 77 10 .2 22 1890 - 1899 100 3.5 28.6 100 2.8 34 50 12.1 14 1900 - 1909 100 4.5 21 100 2.8 49.7 72 10.8 19 1910 - 1919 100 2.8 27.2 100 3.8 34 66. 10 19 1920 - 1929 91 3.6 44.3 72 4.9 46 70 10.1 13.8
TABLE IX Changes in Simple Equations, the Four Operations, and Radicals
The Appearance of New Topics.
Graphs, formulas, and negative numbers have been chosen as typical examples of new topics which have appeared in the course since 1829. Before considering these topics a word will be said about some minor examples of this type of topic. For example, the trigonometric ratios began to appear about 1915 and were found in several of the modern textbooks. Amounts of space ranging from four to twenty-four pages were found devoted to this topic. The subject of elementary statistics was found occupying a distinct place in four texts, all published since 1922. The amount of space given over to this subject varied from five to twenty-five pages. It appeared in two cases in connection with graphs. As a third example of these minor new topics may be mentioned the function idea I *. Its first appearance as a distinct part of the course is almost impossible to detect; but at any rate it is given some prominence in the Rugg-Clark Fundamentals of High School Mathematics (1920), in Drushel and Withers Essentials of Junior High School Mathematics (1924), and in the Schorling-Clark Algebra First Course, (1929).
Table X and Diagrams IX, X, and XI display the data for graphs, formulas, and negative numbers.
Note: One symbol used in Table X needs explanation. In the order and space columns for graphs and the space column for formulas, all in the date interval, 1920-1929, appears the symbol ”i H . This is used merely to indicate that these
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items are M indefinite”. The indefiniteness is due to the ways in which the topics are treated in some texts. The treatment becomes so spiral in some cases that it is impossible to determine either the order or the amount of space. Indeed, it will be seen later that some modern authors view graphs not as separate topics; they are looked upon as means for treating other algebraic materials. The graphs for these data will be discontinuous at the points corresponding to the dates mentioned.
The story of graphs (See Table X and Diagram IX) as they appear in the textbooks examined is quickly told. They began to appear in some textbooks during the decade 1890 - 1899. To be specific,they are found for the first time in Wentworth’s First Steps in Algebra (1894) and in his New School Algebra (1898). Graphs became in the 1910 - 1919 period, and have remained so. So far as can be determined, the position of graphs rose nearer and nearer to the top, while the space allotted to it increased somewhat. A special chapter will be devoted to graphs in a later part of this study.
Formulas have had a career somewhat checkered, as is shown in Table X and Diagram X. They, like radicals, have waxed and waned. Their appearance before 1829 is to be credited to Warren Colburn. But it should be said that the use of the word H formula” prior to 1910 was different from that in the times after that date. The early formulas were hardly more than problems generalized. They could even be called solutions to general literal problems. The interpretation of the data relative to formulas should be done with this distinction in mind. Formulas have been introduced earlier and earlier in the course, and considerable space is devoted to them. A special discussion of formulas will appear later in this study.
Negative numbers were considered somewhat definitely by Hammond in his book published in 1772 and by Colburn in his Introduction to Algebra (1829). They did not begin to become a clearly distinguished topic until about 1900 and did not reach the class until the period, 1920 - 1929, They have consistently occupied about the third place in the course, and space devoted to them has gradually increased. Chapter IV of this study treats of negative numbers.
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Dates Graphs Formulae Negative Numbers Order Space Order Space $ Order Space Before 1829 — — 20 7 8 33.3 6 4 1850 - 1839 — — *** — — — — — — 1840 - 1849 —• — —• — — — — 1850 - 1859 — — — — — — — 1860 - 1869 — — — 20 3 2 — — — 1870 - 1879 — — — 33.3 7 15 —> — 1880 - 1889 — — — 11 3 7 11 3 8 1890 - 1899 33.3 16 14 16.6 3 9 50 2.6 11 1900 - 1909 18 6.5 16 —■ — —• 45.4 2.4 9.6 1910 - 1919 100 7.1 17.1 38.4 2.6- 15.4 84.6 4.7 11.6 1920 - 1929 M0 . 3.91 i 72.7 2.5 1 100 3.5 16.7
TABLE X The Appearance of New Topics
Warren Colburn's Introduction to Algebra (1829).
That Warren Colburn was a man who saw far ahead of his times is evident to anyone who has examined his Intellectual Arithmetic and his Introduction to Algebra < The following quotation refers primarily to his textbook on arithmetic:
Thus in Colburn’s books, published a century ago, we find methods proposed that are in entire accord with the psychology of the present day. The use of objects, the ’inductive* approach, the employment of practical problems, procedure from the simple to the complex, insistence upon understanding before memory, the precedence of mental work over written work, and the demand for personal responsibility of pupils in the original solution of problems — all of which Colburn seems to have advocated — are now accepted as correct principles of teaching.
That the Introduction to Algebra has about the same characteristics as those found in the arithmetic texts is shown in part by the following:
This innovation, made first by Warren Colburn, of introducing the pupil to algebra by a large number of ’intellectual exercises 1
- a series of problems requiring reasoning from the pupil, and only that simple, ’intuitiv 1 use of the equation which pupils beginning algebra can actually make — was, from the standpoint of a real psychology of learning, one of the most significant contributions to the teaching of algebra. It is even doubtful if present practices can compare favorably with the results of this remarkable insight.lB
As can readily be seen from the tables and diagrams of this chapter, Colburn’s Introduction to Algebra was strangely modern in many ways, at least insofar as the selection and arrangement of materials are concerned. In order to make this point even more emphatic, there is presented herewith a free paraphrase of Colburn’s table of contents, or at least of such parts of it as might appear in a present-day first year course.
Topics
Pages
1. Questions producing simple equa- tions involving one unknown . . . 26
2. Practice in putting questions into equations ♦ . 8
Pages
3. Questions producing equations with two unknown quantities . . . 4
4. The four fundamental operations • • 16
5. Fractions . 16
6. More equations and problems .... 10
7. Negative quantities, explanation of them 9
8. Negative exponents 2
9. General formulas 8
10. Questions producing equations of the second degree 2
11. Square root 12
12. Questions producing pure equations of the second degree 1
13. Extraction of the third root ... 10
14. Questions producing affected equa- tions of the second degree ... 12
15. Powers and roots in general .... 15
16. Fractional exponents and irrational quantities ........ 5
17. Binomial Theorem . 11
18. Progressions 11
19. Logarithms 21. 19
Even from such a superficial analysis coming from the table of contents, it is easy to see that this textbook has a number of features which would fit nicely into a modern textbook. Some of them are:
1. Early introduction of the equation and frequent return to it.
2. The careful explanation of negative quantities.
3. Spiral treatment of exponents.
4. The use of general formulas.
5. Introduction of square and cube root when they are needed.
6. Study of powers and roots in general.
Thus, Colburn forsook the guiding star of logic in his presentation of the elements of algebra and tried to arrange these elements in the order in which they could best be learned by the student. That to a large extent he succeeded in this undertaking, according to modern standards, only makes to grow the wonder that it required nearly one hundred years for succeeding textbook writers to learn to follow in his footsteps.
17 Henderson, J. L.: Materials and Methods in the Middle Grades, pp. 288 -
18 Rugg, H. A., and Clark, J, L., op. cit,, p. 43.
T 9 x Colburn, Warren: Introduction to Algebra, pp. 6—B.
Order of Topics in Two Modern Textbooks.
Having considered at some length the arrangement of topics in Colburn’s book, published one hundred years ago, it will be of interest to examine with equal care the contents of two modern textbooks in first year algebra. The first of these, Thorndike’s Junior High School Mathematics, Book Three (1926), is chosen because the author, although not a professional mathematician, has made a special study of the psychology of algebra and has set down in another connection what he considered the best order of topics in elementary algebra. In the textbook named above, the topics appear in the following order:
Topics Pages
!♦ Review: Expressing rules as formulas and solving problems by equations . 29
11. Algebraic expressions and formulae: addition and subtraction . . 22
111. Algebraic expressions and formulae: multiplication and division; products and factors ... 46
IV. Fractions: fractional equations and formulas 34
V. Powers and roots: Reviews .... 30
VI. Expressing relations by tables, graphs, and formulas 25
Pages
VII. Equations expressing important mathematical relations: variation; inverse variation; equations of straight lines; sets of equations 19
VIII. Solving quadratic equations ... 7
IX. Ratio and proportion 10
X. Tangent, sines, and cosines . . . 10
XI. The general principles of algebra: review 51
XII. Logarithms and other labor saving devices 26. 20
devices
Certain well-known topics which do not appear in this list are not omitted from the book, as it would seem. For example, the topic positive and negative numbers appears at pages 18 - 25 and is given adequate treatment. Also, the book is liberally supplied with practice material, reviews, summaries, and tests.
The arrangement here is modern, especially in that the formula is given much prominence. Even more prominent is the place given to the formula by Thorn- dike in the suggested order of topics appearing in his Psychology of Algebra. This last arrangement of topics is presented here because it seems to be that ’’far-off event” toward which all elementary algebra is moving.
Order of Topics from Thorndike 1 s Psychology of Algebra
DIVISION A: THE FORMULA
(With such concepts and computation and manipulation as are needed.)
1. Easy formulas.
a. To understand. b. To evaluate, c. To make.
2. Formulas with parentheses and complex fractions. (As above.)
3. Formulas containing radicals. (As above.)
4. Formulas with abstract quantities, (As above.)
5. Formulas with negative numbers. (As above.)
6. Formulas: to ’’change the subject”.
DIVISION B: THE PROBLEM
Any quantitative issue can be solved by framing a suitable special formula or set of formulas and solving.
7. Genuine problems naturally treated a having one unknown.
8. Genuine problems naturally treated as having two or more unknowns.
a. Solved by substitution. b. Solved by addition or subtraction (treated as a short-cut.)
DIVISION C: RELATED VARIABLES
9. Statistical graphs.
a. To understand and interpret graphs on concrete real issues, e. g., cost of living, growth of population, practice curves, in 4 + quadrant only. b. To make such graphs.
10. Mathematical graphs.
a. The coordinate system, with extension to - + , - -, and + - quadrants. b. Important curves, e. g., y Kx, y - K per cent of x, y - y = y = x 3, y =
11. Direct and inverse proportion.
a. Equation y = kx and graph, b. Equation y - k/x and graph.
12. The general linear graph.
a. Equation y- kx b.
13. Simultaneous linear equations.
a. To find constants for the general linear equation y kx -b, given two sets of values of x and y.
14. Square root.
a. Of numbers. b. Of trinomials.
15. Trigonometric ratios.
a. Tangent, sine, cosine, of angles in Ist quadrant.
16. Logarithms.
a. For ease of computation. b. Slight theory of logs, based upon graph of y 10 .
17. Quadratic equations in 1 unknown.
a. By graph. b. By completing the square. c. By formula.
18. The general notion of variation: summary and systematization.
DIVISION D: ABSTRACT FORMULAS
19. The series.
a. Arithmetic. b. Geometric.
20. Fractional and negative exponents.
21. The binomial theorem.
21 22. Abstract formulas.
The other modern text selected for the examination of the order of topics is the Schorling-Clark Modern Algebra, First Course (1929). It is chosen because of its general excellence and because it is the heir of the Rugg-Clark, Fundamentals of High School Mathematics (1918), the material of which was selected and arranged by careful experiments. Schorl ing and Clark present their topics in the following order:
Topics
Pages
I. How to express the relation between numbers that change together 26
11. Using the formula; the axioms; the use of the equation in problem solving * 23
111. How to apply important relations of the right triangle .... 26
IV. The use of positive and negative numbers 36
V. Learning to apply the four fundamental operations to algebraic numbers 26
VI. Products and factors 25
VII. More practice in solving equations 29
VIII. Some important principles of arithmetic extended to algebraic fractions 23
Pages
IX. Literal and fractional equations • 26
X. Simultaneous linear equations . . 22
XI. How numbers change together; variation and proportion • . . 37
XII. Roots and powers 26
XIII. How to solve equations of the second degree 24
XIV. What everyone should know about statistics 26
2? XV. The measurement of angles ... 5.
The 1929 status of the various important topics as they are depicted in Diagrams I - XI is fairly well shown in the books of Thorndike and Schorling and Clark. One notable exception may occur to the reader. Graphs do not seem to be displayed as much as their importance would warrant. This can be explained by the fact, already pointed out, that graphs are no longer looked upon as a separate topic but as a tool for the treatment of other topics.
Mention has already been
20 Thorndike, E. L.: Junior High School Mathematics, Book Three, p. ix.
21 Thorndike, E. L.: The Psychology, of Algebra, pp. 309 - 311.
2? 'Schorling, R., and Clark, J. R., Modern Algebra, First Course, pp. vii - viii.
Some Special Cases.
made of the fact that a few texts do not lend selves readily to the method of analysis employed in this study. The reason for this is that their subject matter is organized according to plans entirely different from that which shows itself in a more or less orderly sequence of such distinct topics as are listed in Tables IV and V. This makes it impossible to include the material of these texts in the two tables just mentioned. B r ief descriptions of the plans of these books will now be given.
Hall and Knight’s Elementary Algebra (1892) furnishes a good example of the spiral treatment of algebraic Material. For example, there are “simple equations” and “harder equations”, some 120 pages apart. Again, “factors” and “harder factors” are separated by about 100 pages. Several other topics are treated in like manner. This gives the topic a scattered appearance, but it at least is an attempt to introduce various phases of a topic at times when the pupil is ready for them.
Nicholson’s School Algebra (1909) follows a plan that is both topical and spiral. The book is divided into two parts, “primary algebra” and “high school
algebra”, and to a large extent the second part is a re—view of the first. It is not so much a matter of introducing the pupil to new phases of algebra as it is one of showing him extended views of phases already studied. This is the essence of the spiral plan.
Myers l First Year Mathematics (Third edition, 1909) is “an interweaving of the more concrete and the easier portions of the first courses in both algebra and geometry”, to quote the words of the author appearing in the preface of the book. As this book is the result of one of the very first attempts to unify the mathematics of the high school, its contents are presented somewhat in detail.
Topics
Pages
I. General uses of the equation . . 14
11. Uses of the equation with perimeters and areas . . . 16
111. The equation applied to angles . 32
IV. Positive and negative numbers . 27
V. Beam problems in one and two unknowns ... 18
VI. Problems in proportion and similarity . 30
VII. Problems on parallel lines . . . 16
VIII. The fundamental operations applied to integral algebraic expressions 28
IX. Practice in algebraic language General arithmetic ..... 29
X. The simple equation in one unknown 24
XI. Linear equations containing two or more unknowns. Graphic solution of equations and problems 28
XII. Fractions ....... 13
XIII. Factoring, Quadratics, Radicals. . 48
XIV. Polygons, Congruent Triangles, Radicals ....... 43.
This pioneer textbook did much to alter the arrangement of topics in algebra. Quite apart from the presence of some geometry, other striking features are easily identified. What the author calls the ’’early and persistent use of the equation” is perhaps the most noticeable of these. Then there is the
“early, varied, and systematic used of pictured and graphic modes of rendering algebraic truths vivid and appealing to beginners.” Finally, some of the more formal topics of algebra are omitted, and others are arranged in unusual ways (for those days); as, witness the linking of factoring, quadratics, and radicals and the spiral treatment of the last-named topic.
23 Myers, G. W.: First Year Mathematics, p. vii.
Chapter Summary.
Many changes have taken place since 1829 in the order of topics in elementary algebra textbooks and in the amount of page space devoted to the various topics. The junior high school textbooks have had a good deal to do with these changes, especially those occurring since 1917. The correlation of algebra with other branches of mathematics and the liberal views of writers of junior high school mathematics textbooks are responsible for most of the changes from this source. In general, an examination of textbooks written from 1829 to 1929 reveals that such topics as inequalities, imaginaries, and highest common factor and lowest common multiple have practically
disappeared from the first year algebra course. Notation and definitions, fractions, and quadratic equations have remained almost unchanged as to position and amount of space. Certain topics have experienced radical changes in order and space; for example, simple equations new come much earlier in the course than formerly, the four operations come later, and the topic, radicals plays a smaller part in the course than it once did. Perhaps the most striding and significant difference in the algebra course in 1929 and in 1829 is due to the appearance of new topics. Negative numbers began to appear as a distinct topic about 1880, graphs about 1890, and formulas in the modern form about 1910. There had been earlier so-called formulas in the shape of generalized literal problems. Warren Colburn’s Introduction to Algebra (1829) was very advanced in many respects, since it took later authors many years to catch up with his leads. A considerable number of modern algebra textbooks follow the findings of modern psychology in the selection and arrangement of topics in the first year algebra course.
CHAPTER IV NEGATIVE NUMBERS
I. Introduction
The Purpose of This Chapter.
It is the purpose of this chapter to determine the steps by which the teaching of negative numbers has developed as a phase of elementary algebra. The evidence is collected from seventy-seven algebra textbooks, the publication dates of which are fairly evenly distributed over the century from 1839 to 1929. A few texts published before 1829 are also considered. Both subject matter and methods of teaching are considered, and particular attention is given to indications of the growth of ideas of negative numbers as part of the complete algebraic scale.
History of Negative Numbers in Algebra,
Ideas about the nature of positive and negative numbers developed with remarkable slowness in the early days of algebra*
The Hindus were the first to recognize the existence of absolutely negative numbers .... The difference between f and - numbers was brought out by attaching to the one the idea of “assets”, to the other that of “debts”, or by letting them indicate opposite directions. 24
Bhaskara (born 1114 A. D.) recognized both positive and negative roots of quadratic equations but stated that the negative root was not to be taken, ’’for it is inadequate; people do not approve ?5 of negative roots”.
Four hundred years after the time of Bhaskara, Cardan of Milan referred in the Ars Magna (1545) to negative numbers and to the square root of the negative as numeri ficti or falsi. He tolerated negative roots of equations only when they admitted of interpretation as ’’debitum”. Descartes, in his
trie of 1637, gave a real interpretation for the negative. He represented a geometric curve by an equation and made the convention that perpendiculars on opposite sides of a line of reference and intercepts on opposite sides of a point of reference should have the opposite algebraic signs. This convention gave the negative a new position in mathematics; it was made indispensable, placed on an equal footing 27 with the positive number.
Sir Isaac Newton (1642 - 1727) was the first to let a letter stand for any number, negative as well as positive. In such a formula as a(b 4- c) _ab + ac, the predecessors of Newton would restrict the letters to represent any positive number, while Newton regarded the letters as representing any numbers whatever either positive or negative. This was of very great importance, since it greatly reduced the number of formulas required. During the seventeenth and eighteenth
centuries, negative numbers came only gradually into use. They were practically forced upon the mathematicians of those times without any conscious volition on their part, and, in fact, in spite of much active opposition. J. W. Young quotes Professor Klein to the effect that the algebraic symbols were more reasonable than the men who employed them.
24 Cajori, Florian,: A History of Elementary Mathematics, p. 101.
25 Ibid., PP. 101 - 102.
26 Fine, Henry B.: The Number System of Algebra, p. 115. ”
37 1b1d.. p. 119.
28 Slaught and Lennes, First Principles of Algebra, pp. 50 - 51.
pg Young, J. W.: Fundamental Concepts of Algebra and Geometry, pp. 11l - 112.
The Placement of Negative Numbers.
As a separate topic negative numbers, or rather positive and negative numbers, had no place at all in textbooks published prior to 1887. The subject was treated incidentally in connection with subtraction or with the negative results sometimes secured from the solution of certain equations. Frequently, the negative numbers were simply defined in the introduction, and no attempt was made to discuss them at length.
In Simon Newcomb’s Algebra (1887) the first chapter, consisting of eight pages, was entitled ’’algebraic numbers and operations”, and of this chapter the first three pages were devoted to a discussion of algebraic numbers. Wentworth in his Shorter Course in Algebra (1887), also devoted three pages in chapter one, entitled ’’quantity and number”, to algebraic numbers. Wells* College Algebra (1890) included the treatment of positive and negative numbers among the preliminary definitions. Hall and Knight, in their Elementary Algebra (1892), started the practice of treating positive and negative numbers in a separate chapter, but it was not until 1908 that this practice became well-nigh universal. Between 1892 and 1908 the majority of authors included their discussions of positive and negative numbers among their introductory remarks.
One is interested here in considering the place in the course assigned to the discussion of positive and negative numbers. And in the light of the findings noted just above, it is evident that it will be necessary to consider only those textbooks written not earlier than 1908. Of such texts, thirty-five have been examined.
In these texts four plans with regard to the placing of the topic positive and negative numbers are followed. These plans with their frequencies are given below in Table XI.
It can be seen from Table XI that authors of textbooks written in recent years are of the opinion that the extension of the number system to include negative numbers should be made early in the course; that it should be preceded at most by a study of equations and graphs. It is interesting to note that texts written during the early years of the period from 1908 to the present followed the practice of giving positive and negative numbers priority over both equations and graphs. Gradually, equations came to precede positive and negative numbers, and at the present time the accepted place for the latter topic seems to be that provided for it by plan number four in Table XI.
These facts are shown in a more striking manner in the following table. The plans referred to are those shown in Table XI.
The data of this table show a fairly striking correlation between time and the prevalence of the four plans* The last two plans are shown to be strongly in favor since 1914.
The National Committee on Mathematical Require-30 ments, w working under the auspices of the American Mathematical Association, suggested in their 1923
report the following arrangement of the topics involved in the foregoing discussion:
1* The formula
2. Graphs
3. Positive and negative numbers
4. The equation
This particular recommendation of the Committee does not seem to have been received with much favor, since not a single textbook was found in which the suggested order was followed*
What is the best placement for the subject of positive and negative numbers? In addition to the practices at textbook writers analyzed above, evidence from three other sources will be presented.
Nunn, an English authority on the teaching of algebra, would delay the extension of the number system until the following topics have been presented:
!♦ Making, interpreting, and evaluating of formulas. 2. Factorization (common monomial and difference of two squares) and the converse expansions. 3. Easy fractions. 4. Changing the subject of a formula. 5. Functionality as found in proportion. 6. Use of the trigonometric ratios. 7. Graphs. 8. Square root. 9. Radicals. 10. Approximate formulas for the squares and cubes of the sum and difference of two numbers. 11. Mean, median, and quartile in statistical data.
Nunn bases this arrangement upon the following considerations:
There are two good reasons for this procedure. The first is that the expression of generalized statements by formulae and the use and manipulation of numbers accompanied by signs are two distinct processes which have no necessary connection with one another. Each
has its own difficulties for the beginner, difficulties which are best overcome if faced The second reason is that the two processes are not only different in kind; they depend upon the presence of characters in the child’s mind which begin to ripen at different ages. It is easy enough to make a boy or a girl apply mechanically the rule that like signs produce plus, unlike signs minus; but if the rule is to be used with intelligence, the teacher must be able to appeal to logical powers and interests which have rarely emerged at the age when lessons in algebra begin. Observation will, in fact, show that the pupil who has been early taught the properties of positive and negative numbers rarely uses them spontaneously in his thinking. His mind works freely only among signless numbers with their familiar properties. This fact is itself a strong indication that numbers accompanied by signs have been taught prematurely.
These arguments do not apply- to American conditions as strongly as may appear on first reading. This is due to the fact that in England, students are introduced to the subject of algebra much earlier than is the practice in American secondary schools. Furthermore, this question of the emergence of logical powers and interests brings up psychological
problems and controversies that have never been settled conclusively.
Rugg and also present positive and
negative numbers much later in the course than is usually the case. They would arrange the topics as follows:
1. How to use letters to represent numbers in solving problems. 2. How to use the equation. 3. How to construct and use algebraic expressions. 4. How to find unknown distances by means of scale drawings. 5. How to find unknown distances by using similar triangles. 6. How to find unknown distances by using right triangles. 7. Tables and graphs. 8. How to represent and determine the relation between numbers that change together. 9. Use of positive and negative numbers.
This arrangement is the result of experimental evidence and because of this fact it is worthy of special consideration. John R. Olark, co-author
with Mr. Rugg of the Fundamentals of High School Mathematics (1921) is also co-author of the Schorling-Clark Modern Algebra First Course (1929). In this latter text the placement of positive and negative numbers is quite different from that in the Rugg-Clark book and approaches more nearly to the usual practices. Schorling and Clark’s arrangement of the early topics is shown here:
1. Formulae and equations. 2. Relations of the right triangle. 3. Positive and negative numbers.
Thorndike in hie Psychology of Algebra suggests that negative numbers should be introduced early. He would include them in what he calls his “Division A: The Formula”. The sub-divisions of this Division show the place of the negative number:
1. Easy formulas
a. To understand b. To evaluate c. To make
2. Formulas with parentheses and complex fractions
(As above)
3. Formulas containing radicals (As above)
4. Formulas with abstract numbers (As above)
5. Formulas with negative numbers (As above)
6. Formulas: to ’’change the subject” 33 (As above)
Thorndike gives four reasons for preferring his arrangement of the early topics over those of Nunn and Rugg and Clark.
First, there are many generalizations and statements made into formulas which are only half truths until the comprehension of the negative number permits them to run the entire gamut of values ....
Second, the failure to introduce the negative number reasonably early often makes necessary the writing of two formulas where one should suffice ... To make two formulas grow where one should suffice does not benefit the pupil.
In the third place, an interference factor is to be reckoned with when a child taught to make and use numerous algebraic expressions on the assumption of positive numbers only is required to readjust to the concept and use of negative numbers in the same or similar expressions. In case he has so learned his algebra that it yields
him satisfaction as a tool of operation, his resentment at the new upsetting doctrine will be pronounced - and possibly the keener the more diligently he has studied the earlier algebra ....
Again, there is on the part of most pupils a readiness to acquire new points of view at the beginning of the course in algebra of which it is worth while to take advantage. By most American children a, new subject, a new teacher, a new classroom, a new textbook, a new group of fellow students are met in one happy readjustment. All surroundings thus conspire to amenability to revolutionary doctrine concerning what has seemed one of the fixed items of the universe of abstract truths, namely, that numbers start at 1 (for a few children, at zero) and progress by unit increase to a very large sum. To take advantage of this flood tide in the affairs of pupils seems sound psychology. 34
It seems, then, that the evidence bearing on the placement of positive and negative numbers does not lead to any conclusive settlement of the problem. Both Nunn and Thorndike appeal to the principles of psychology to support their contentions, but their conclusions are very different. Also, the only body of experimental data available leads to conclusions that are at variance with accepted practices. In the face of such conflicting evidence, the inquirer can only reserve judgment and hope that further study and experiment will lead to some definite conclusion as to the place of negative numbers in an elementary algebra course. Surely this is a fertile field for careful experimental investigation.
30 National Committee on Mathematical Requirements: The Reorganization of Mathematics in Secondary Schools, pp. -24.
31 Nunn, T. Percy: The Teaching of Algebra, p. 53.
32 Rugg, H. 0., and Clark, J. R.: Fundamentals of High School Mathematics, p. xv.
'Z’Z Thorndike, E. L.: The Psychology of Algebra, p. 309.
'Thorndike, E. L.: op. cit., pp. 316-317.
Plan Frequency 1. Included in the Introduction 5 2. Preceded by an introduction 12 3. Preceded by an introduction and Equations 12 4. Preceded by an introduction, Equations and Graphs 4
TABLE XI Placement of Positive and Negative Numbers
19/o /9!° - l$iz 1912- /9!+ 19/♦' (9/6 19/6- /913 /9/S- 1920 /92O- f9Zt f92Z- 1924- 192 9- f92Q~ /92& f93<> P/an 1 z / z / • P/<jn Z / 2 / I / / Plein J z I z Z z / z / Plan 4- / / z f
TABLE XII Changes in Placement of Positive and Negative Numbers 1908 - 1930
Space in Texts Devoted to Discussion of Negative Numbers.
Some understanding of the changing attitude toward negative numbers during the century from 1829 - 1929 may be gained by noting the varying amounts of space in textbooks devoted to the treatment of positive and negative numbers. In the texts published in the second and third quarters of the nineteenth century, there were mere mentions of negative numbers in connection with certain preliminary definitions. The operation of subtraction and the examination of certain roots of equations also involved references to negative numbers. Due to the absence of indexes in these older textbooks, it was by no means easy to find any mention of negative numbers. The subject was never given a separate heading in the table of contents. In contrast to this practice, one finds in books published since 1890, and particularly in those published since 1910, whole chapters devoted to positive and negative numbers. The shift of emphasis is away from the four fundamental operations to the meaning and significance of positive and negative numbers and to the use of these numbers in the four operations. For example, in Simpson’s Treatise of Algebra there are only a sort of definition of the negative number and incidental references to it in the rules for the four operations. This book was written in 1800. In Milne’s Elements of Algebra (1894) only one page is given to the subject of positive and negative numbers; a definition, two illustrations, and six exercises appear on this page. In the Modern Algebra First Course by Schorling and Clark (1989), ”the use of positive and negative numbers” is the subject of chapter IV, which covers thirty-six pages. The treatment includes many examples of the use of these numbers in concrete situations, numerous exercises furnishing practice in their use, and careful and detailed development of understanding and skill in the addition, subtraction, multiplication, and division with algebraic numbers. These examples of extremes give a picture of the change in relative importance of this topic. The progress from one extreme to the other was gradual and steady, measured in terms of space in the textbooks.
II. The Meaning and Significance of Negative Numbers
Examples from Early Texts.
In the textbooks published before 1825 there is scant evidence of any conception of negative numbers as a real and integral part of the algebraic scale of numbers. References to negative numbers are few, and for the most part the minus sign is associated only with the subtraction operation. In John Ward’s Young Mathematicia 1 s Guide (1706), published in an eleventh edition in 1762, the following statements appear:
. . . the Signs by which Quantities are chiefly managed are the same, and have the same Signification, with those in the first Part (Arithmetic)♦ (p. 144)
. . . the sign 4 is the Affirmative Sign, and therefore all leading Quanties (spelling not corrected) are understood to have it, as well as those that are to be added . . . But the sign - being the Negative Sign, or sign of Defect, there is a Necessity of prefixing it before that Quantity to which it belongs,
wherever that Quantity stands. (p. 145)
• . . +2a taken from -j-3a is the same with -2a added to +3a ♦ . . to subtract a Negative Quantity from an Affirmative, will be the same as to add an Affirmative Quantity to an Affirmative. (p. 149)
In these statements the negative number appears only in connection with the idea of subtraction.
Thomas Simpson in his Treatise of Algebra (1800) has even less to say about the negative number. He includes no special discussion of the negative, merely defining negative quantities as those to which the minus sign is prefixed 35 and mentioning negative
numbers in his rules for the four fundamental operations* This use of the term quantity is typical of the instances in which the early writers used number and quantity loosely and interchangeably. Strictly speaking, a quantity or a magnitude is anything capable of being measured; a number is a symbol expressing the number of times a certain unit is contained in the quantity. It follows then that a quantity
does not lend itself readily to the prefixing of negative or minus signs. This error is not entirely limited to the works of the early writers.
In one of the earliest algebra textbooks written in America, Warren Introduction to Algebra (1829), the following observations about negative numbers are found:
They arise from the necessity of expressing subtraction by a sign because it cannot actually be performed on dissimilar quantities . . . they are only positive quantities subtracted, and in their nature they differ in nothing from positive quantities. (p. 113)
A negative number is the result of subtracting a larger number from a smaller one. (p. 114)
A negative quantity cannot be a quantity less than nothing, but it implies a contradiction. It answers to a figure of speech frequently used. (p. 114)
Here at least is an attempt to grapple with the problem of defining and characterizing the new kind of numbers, although there might well be a question as to the amount of light shed on the subject. It is to be noted that the terms quantity and number are used in a synonymous sense, that the operation of subtraction is inextricably bound up with the negative number, and that there is a sense of the
unreal connected with the latter.
35 Simpson, Thomas: Treatise of Algebra, p. 2.
The Realness of the Negative Number.
Apparently, this conviction that negative numbers were not real, expressed by Cardan in his use of the phrase numeri floti, had been completely transcended by 1850, if one may judge by the discussions of negative numbers found in the textbooks. Some of the early authors, however, seemed to think it necessary to call attention to the real quality of these numbers, while more modern writers take it for granted. For example:
A negative quantity is just as much a reality as a positive quantity, the only difference being that in the case of a positive quantity the effect is to increase the quantity with which it is connected, and in that of a negative quantity the effect is to diminish the quantity with which it is connected. 36
But though there can be no such thing as a negative number, we can conceive the real existence of a negative quantity.
Negative numbers can always be interpreted.3B
The illustration of positive and negative numbers from concrete situations and the geometric representation of the algebraic scale of numbers had much to do with the strengthening of this feeling of reality. Each of these practices will be discussed more at length later.
S. P.: New Elementary Algebra (1879), p. 28.
J. H.: Elementary Algebra (1885), p. 12*
38 Hall and Knight, Elementary Algebra (1892) p.
The Size of Negative Numbers.
In the study of negative numbers the difficulties which the older thinkers found or raised probably remain, like an undispersed fog,to obscure the path of the beginner today. One of the most fruitful sources of these troubles lies in the failure to realize that the difference between positive and negative is not synonymous with the difference between greater and less.
This confusion of two distinct ideas vitiates • . . the example with which beginners have made first acquaintance with negative numbers in every generation
since the end of the sixteenth century. If a person (we aie told) who possesses /5 is to estimate nis wealth as then a person who has no money in his purse but owes /5 must write his wealth down as B -5”, for he has less than nothing. How the dlffi culty which the beginner has to face here lies not in the abstruseness of the new idea but in its absurdity; for he is asked as the price of admission to the new subject, to give up the conviction of common sense that there cannot be anything less than nothing. So unfortunate a result might well have suggested a careful scrutiny of positive and negative numbers with the object of determining whether it is inevitable, or whether the true nature of the numbers has been misapprehended. 39
It is the usual thing for authors of textbooks to include this idea of size in their thinking about negative numbers in their relation to positive numbers. Furthermore, that there has been a change in the attitude toward this matter is shown by the following quotations:
A negative quantity cannot be a quantity less than nothing. 40
The phrase, less than nothing, cannot carry an intelligible TSea.4l
Negative numbers are not less than nothing. 42
No quantity can be less than nothing. 43
Quantities on one side of 0 are positive, those on the other side are negative. Positive quantities are greater than 0, negative quantities are less than 0. 44
Numbers less than zero are called negative numbers. 45
Negative numbers are numbers smaller than zero. 4b
Every negative number is less than zero. 47
Negative numbers are numbers less than zero. 4b
Numbers less than zero are called negative numbers. 49
As revealed by these quotations, opinions with regard to the size of negative numbers changed squarely about sometime during the last quarter of the nineteenth century. Why this reversal of position? Why are the authors writing before 1875 unanimous in saying that a negative number cannot be less than nothing, while those writing after 1900 almost define the negative as a number less than zero?
The explanation is not far to seek. One is confronted here with the ever-present confusion of the difference between positive and negative numbers with the difference between greater and less. The earlier writers were thinking more of size in terms of concrete numbers, representing a collection of objects, for example. With this in mind, a number less than nothing simply has no meaning. As Ray puts it, ’’the phrase, less than nothing, cannot carry an intelligible idea.”
In order to explain the attitude of the later writers toward this question, it will be necessary to anticipate somewhat. In a later section of this chapter there will be discussed the development of the practice of representing geometrically the complete algebraic scale, consisting of the positive numbers, the negative numbers, and zero. These numbers are shown as corresponding to points on an unending straight line, positive numbers to the right of zero, negative numbers to the left. On this scale, the relative sizes two numbers are deter mined by their relative positions on the scale. One number is greater than another if it stands to the right of the other on the scale. This convention is supported by the fact that the positive numbers may be made to represent increasingly large groups of objects as the numbers are taken further and further to the right on the scale.
The quotation from Downey, given above, shows how closely the two ideas of position and size are related. For the sake of clearness, the quotation is repeated here:
Quantities on one side of zero are positive, those on the other side are negative. Positive quantities are greater than zero, negative quantities are less than zero. $0
It will be shown later that no representation of the geometric-algebraic scale was found in books published prior to 1882. It seems reasonable to conclude, then, that this practice gave a pictorial and visual basis for the idea of order, which gradually replaced that of size. Concepts and terms die hard, and one finds the modern writers using the word size where the word in its old sense would have no meaning. In this way, the statement ’’every negative number is less than zero” is interpreted to mean ’’every negative number lies to the left of zero on the scale.”
39 Nunn, T. Percy, op. cit. pp. 159 - 160.
Colburn, Warren: Introduction to Algebra (1829), p. 114.
41 Ray, Joseph: Algebra, First Part (1848), p. 44.
Joseph: New Elementary Algebra (1866), p. 27.
4S Robinson, H. N.: New Elementary Algebra (1875) p. 35.
44 Downey, J. F.: Higher Algebra (1900), p. 20.
45 Fisher. G. E., and Schwatt, J. I.: Secondary Algebra (1904), p. 19.
Arthur: Elements of Algebra (1912), p. 4.
47 Fite, W. B.: College Algebra (1913), p. 2.
E. 1., and Carpenter, P. A.: A First Course in Algebra (1923), p. 15.
40 Hamilton, Samuel, Bliss, R. P., and Kupfer, Lillian, Essentials of Junior High School Mathematics (193?),p71W. “
50 Downey, J. F.: op. cit. p. 20.
Formal Definitions of Positive and Negative Numbers
Not all authors of algebra textbooks
attempt a formal definition of negative numbers. Of the seventy-seven textbooks examined, eighteen contained what might be called definitions. These definitions fall readily into four classes.
1. Those which may be characterized as 11 super ficial”.
Examples:
”A negative number is one whose sign is - . ” ”Negative numbers are numbers smaller than zero.
2. Those definitions which stress the effect of combining the negative number with other numbers.
Examples:
“Any quantity which, when put with another quantity of the same kind, neutralizes it wholly or in part, is, or may "be called, a negative number. $3
• . . . a quantity that increases the quantity considered is called a positive quantity; and a quantity that decreases the quantity consid-
ered is called a negative quantity. 54
3. Those definitions which employ the formal algebraic terminology.
Examples:
”-b can be defined as a symbol, and such that the sum of +b and -b is 0.” 55
. therefore, (-b) tb = 0 gives a definition of -b.” $$
4. Those definitions making use of the geometric representation of the algebraic scale.
Examples:
Numbers that represent points to the left of the reference point are called negative numbers .... 87
These definitions are arranged roughly in classes corresponding to the chronological order in which they appeared in the texts. That there has been progress is evident from the nature of the types of definitions, and that the progress has carried forward to a modern viewpoint can be shown by comparing the examples in the last two groups above with the following definition of negative numbers by J. W. Young:
If a and b are two positive numbers, and a<b, the expression a - b has no meaning in the system of numbers thus far considered. The expression b - a in this case is, however, a definitely determined positive number, c. The expression a - b is then placed equal to a new symbol -c, which is called a negative number. $8
51 Nicholson, J. W.: School Algebra, p. 19.
5? “’Schultze, Arthur: Elements of Algebra, p. 4.
53 Smith, J. H.: op. oit. p. 13.
54 Wentworth, G. A.: First Steps in Algebra p. 33.
5 5 B O yd, James H.: College Algebra, p. 43.
ssRietz, H. L., and Crathorne, A. R.: College A3g ebra. p. 3.
57 Fite, Benjamin: College Algebra, p. 2.
58 Young, J. W.: Fundamental Concepts of Algebra and Geometry, p. 109.
The Subtraction Operation and the Negative Number
Reference has been made to the close association existing between the subtraction operation and the negative number. It will be pointed out later that one of the most common approaches to the idea of the negative number consists of calling to the student*s attention the impossibility of subtracting a larger arithmetical number from a smaller one and the need for some new kind of number which will make this operation possible. This ever-present subtraction operation, then, furnished the clearest and the most frequently used contact with the negative. Hence, it is not strange that the understanding of the negative was somewhat limited by this relationship.
It appears that for a time there was in use a symbol the very existence of which represented an intermediate stage in the extension of the scope of subtraction from the limited to the unlimited, that is, from arithmetical to algebraic subtraction* This symbol can be exhibited best in the expression:
a b
Hackley, in a text published in 1885, defines the symbol in the following terms:
The sign denotes the difference between two numbers when it is not known which is greater. 59
Robinson (1875), Schuyler (1883), and John B. Clark (1885) used the symbol in the same sense. It was not found in any textbook published since 1885. This would indicate that by this time the expression a- b had acquired its present meaning. With the extension of the number system to include negative numbers, this expression a - b does not depend upon the relative values, or sizes, of a and b for its full and complete meaning.
It is interesting to note that this rather indefinite and now defunct symbol for subtraction has been used in recent years by sane authors to indicate similarity.
Another sign which was used in the early text-
books, but which was not found in any text published since 1875, was referred to as the ’’essential sign”. Its meaning may be made clear by a quotation from a text published in 1864. References are to the signs in the expression a - (-b) = a + b.
The sign immediately preceding b is called the Sign of Quality; the sign preceding the parenthesis is called the Sign of Operation; the sign resulting from the operation is called the Sssential Sign. 60
The term ’’essential sign” is used in the same sense by Peck in his Manual of Algebra (1875).
It is clear that some of the early writers failed to distinguish between the two elements, namely, the subtraction operation and the negative number standing alone. For example, in a text published in 1838, there appears:
. . . those with the sign - are
called negative or subtractive
Davies (1864) also uses the phrase ’’negative or subtractive”. James Bryce goes still further:
When the sign - is prefixed to a quantity standing alone, ... it indicates that there are circumstances which require that quantity to be subtracted, if there were anv quantity going before it. 62
Sanford wrote in 1879:
The principal point of difference between a positive and a negative quantity is that a positive quantity is one which is always to be added. A negative quantity, on the contrary, is one always to be subtracted.
After 1885 writers seemed to realize more and more clearly that it was not necessary to make the negative number lean for support upon the subtraction operation, and statements like those just quoted are not to be found in their books.
59 Hackley, Charles, W.: A Treatise on Algebra, p. 2,
GO Davies, Charles: University Algebra, p. 26.
61 Young, J. R.: Algebra, p. 14.
62 Bryce, James, Treatise on Algebra, p. 2.
63 Sanford, S. P., New Elementary Algebra, p. 26.
The Algebraic Scale.
Even the earliest writers of algebra textbooks in this period extend ing from 1829 to 1929 were inclined to refer in one way or another to the ’’algebraic scale”, although that particular term was not used consistently until the 1880*s. The most usual way of referring to negative numbers in the early days was to point out that, as algebra differed from arithmetic by the representation of quantities by letters , so did it differ further from arithmetic in the use of a new kind of number, the negative number. It was usually pointed out further that this gave to algebra a wider field of operation than was peculiar to arithmetic. But there was found no textbook published prior to 1875 which gave a clear statement, or picture, of the algebraic scale, such as became quite common later.
It is probably true that no one writing on the subject of negative numbers at the present time would fail to solicit the aid of the geometric representation of the algebraic scale of numbers, by which is meant the scale including the positive and negative numbers. This is one device that has done much to make the mastery of the negative number idea so much easier for the beginner in later times than it was for those who so slowly discovered negative numbers in earlier times. It was observed in the course of this study that practically every book published ance 1882 made use of this device in some form.
A picture of the scale as it is commonly used now is given at this point:
etc.,-4 -3 -2 -1 0 -*■! t 3 +4 . etc.
The use of this scale brings into prominence direction as a property or quality of these numbers* More will be said about this property in later sections of this study.
A very near approach to the geometric representation of the algebraic scale was found in a
textbook written in 1875. The author must have had a mental picture of the scale, although he presents no such graph as is shewn above. For example, he says:
Positive and negative numbers are counted in opposite directions; hence, the difference, or space between them, is their apparent sum.
Words in this statement which almost delineate the geometric-algebraic scale are:
1. ’’Counted” - The connotation here is one of enumerating successive unit-lengths laid off on a line.
2. ’’Direction” - That is, to the right and left of a reference point, in this case zero.
3. ’’Space” - The idea represented by this word is not algebraic, but geometric. Standing alone, its presence in the statement is sufficient justification for the conclusion
that the writer had before his mind’s eye the geometric-algebraic scale•
The next step in the development of this geometric-algebraic scale in the textbooks is typified in a textbook published in 1882. In this book the algebraic numbers are simply written successively in a way to indicate that they extend indefinitely in opposite directions from zero.
—6, —5, —4, —3, —2, —l, 0, +l, +2, 4-3, 4-4, 4*5, +Q
One notes especially the absence of the horizontal straight line upon which the unit-lengths 65 are laid off, supposedly.
It is the inclusion of this line that marks the third and final essential step in the evolution of the pictorial representation of the algebraic scale. The first sample of the complete scale was found in a textbook published in 1883. Its appear- ance is approximated in the following diagram:
-5 ,-4,-3, -2 , -1 , 0 , 41, *2 , +3, +4 , *5
Of eleven other texts that were published from 1885 to 1894 only three graphed the algebraic scale. However, practically every textbook published since 1894 represents the algebraic numbers in seme geometric fashion.
In a few cases the algebraic scale is shown in a vertical position at first, the approach being through the well-known scale on a thermometer* One example of this practice is found in Hedrick’s Algebra for Secondary Schools (1908). Another example is given by Wheeler in his First Course in Algebra (page 20). It is reproduced here to illustrate the more ornate form in which the scale is sometimes depicted.
Long before the algebraic numbers were shown geometrically, authors of textbooks spoke frequently of the oppositeness of positive and negative numbers. There is evidence to show, however, that they were not thinking so much of opposite directions as they were of other properties. Just what these opposite properties were is by no means clear. Some statements from early texts are given to illustrate these points.
Negative quantises embrace those that, are, in their nature, the opposite of positive quantities.
. . . the office of the negative sign is to denote a state, condition, or effect, directly opposite to that denoted by the positive sign. 68
The terms positive and negative are merely relative in many cases. They indicate opposition between two classes of quant itles.
Positive and negative numbers may be regarded as used to denote opposite qualities, effects, or conditions of quantities. 70
Modern authors are somewhat more definite in their statements with regard to this opposition.
Positive and negative numbers are opposite s 71 of each other in time, direction, or character.
In order to clarify and emphasize this idea in the pupil’s mind, modern authors frequently give a number of exercises on this phase of the subject of the negative number. Excellent illustrations of such exercises are to be found in Hart 1 s Junior High School Mathematics, Book Two (1922) and in Keal and Phelp’s Eighth Grade Mathematics (1917). In the former text,fifteen exercises are included, and in the latter, twenty.
s^Robinson, H. N.: Hew Elementary Algebra, P. 34e
6 sSeaver, E. P., and Walton, G. A.: Franklin Elementary Algebra, p. 45.
RR Schuyler, Aaron: Complete Algebra, p. 21.
an Ray, J.: Algebra First Part, p. 46.
George R.: Elements of Algebra, p. 17.
CO Loomis, Elias: Treatise on Algebra, p. 12.
70 Greenleaf, Benjamin: New Elementary Algebra, p. 25.
71 Edgerton, E. 1., and Carpenter, P. A.: A First Course in Algebra, p. 16.
Generic Names for Positive and Negative Numbers.
Evidence gathered from the textbooks examined would indicate that about 1908 authors began to apply certain general, inclusive names to positive and negative numbers, that is, names other than ’’algebraic” numbers. Strictly speaking, the term algebraic numbers includes more than merely positive and negative numbers; for example, imaginary numbers are included in this term. Consequently, if one refers- 1 to positive and negative number as slgebraic numbers, he is speaking correctly in a sense, yet the two categories are not exactly equivalent.
Three generic names are applied to positive and negative numbers. They are, arranged in order of frequency of appearance in the texts:
Signed numbers Relative numbers Directed numbers
The term signed numbers is far more common than is either of the other two. Relative numbers, used in Young and Jackson’s Elementary Algebra (1908) and in Palmer’s Practical Mathematics (1913), would probably mean less to a beginner than would either of the other terms. ’’Signed numbers”, as a name, seems to stress a superficial characteristic, a mere symbol standing for a quality in the number so signed.
All things considered, the term ’’directed numbers” refers to the most fundamental characteristic of these numbers. For direction implies order, and order is fast coming to displace the idea of size, in speaking of numbers.
The upshot of their investigations is the discovery that over a large part of the field of mathematics the fundamental idea is not magnitude but order . . . .
The reader will see why the name ’’directed numbers” is given in this book to numbers
accompanied by plus or minus signs. Such a number alwaysreTers, explicitly or implicitly, to a term of an ordered series, and the sign always indicates the direction in which the term is to be sought starting from a certain point of reference or zero ....
. ... in every case in which positive and negative numbers are applicable, direction in one of its senses must be present and constitutes the essence of the conditions which make their use appropriate.
72 Nunn, T. Percy: op. cit., pp. 160 - 161.
Double Use of the + and - Signs.
Practi- cally every author who discussed positive and negative numbers at any length felt called upon to explain the double use of the 4- and - signs. The explanation usually took the form of saying that, whereas, in one case the signs indicated that operations were to be performed, in the other case they were signs of quality. As to the reasons for using + and - to indicate quality, two different attitudes were much in evidence. In some cases, after the two series of numbers on each side of zero in the algebraic scale were identified, the students were told that as a conye numbers to
the right of zero would be accompanied by the 1 sign, those to the left by the - sign, but that frequently it was immaterial which was selected for the positive series. In other cases
the double use of the signs was justified by the close relation existing between the subtraction operation and the negative number; in fact, statements were found to the effect that a negative number in a sense expressed a subtraction operation; for example:
A negative number, such as —3, indicates a reversed subtraction, there being nothing, when it stands alone, from which to subtract it. It is in nature always a subtrahend.
A negative number represents an unperformed subtraction.
How is the student to know in a, specific case whether a sign is one of operation or of quality? Textbook writers use one of two ways to distinguish between the two. The first is by means of a set of conventions which would cover all cases; for example, in the expression +3 - (+4) the two positive signs are signs of quality, and the - sign is a sign of operation. Whenever there might be uncertainty the term with its sign is enclosed in parentheses. If a term stands alone, its sign, expressed or not, is one of quality. This is by far the most common way of distinguishing between the two meanings of the 4 and - signs.
However, a number of authors make use of another device. They print the signs of operation in ordinary type, but the signs of quality are printed in
small type above and to the left of the term modified in each case. Where this symbology is employed one finds expressions like the following:
"3 + ”4 - + 6 = ?
The following books contained these small signs:
1. Milne, Academic Algebra (1901).
2. Fisher and Schwatt, Secondary Algebra (1904).
3. Collins, Practical Elementary Algebra (1908).
4. Milne, Algebra (1908).
5. Slaught and Lennes, First Principles of. Algebra (1915).
In every case in which the small signs of quality were used they were dropped sooner or later. Furthermore, the act of dropping these small signs was not always performed gracefully, as the following quotation will show:
. ... in addition, the signs + and - denoting quality have the same meanings as the signs + and - denoting arithmetical addition and subtraction. For example:
+ 1 means o+l and '1 means 0-1; + 5 means o+s and *5 means 0-5; etc.
Hence, in finding the sum of any given numbers, only one set of signs, + and - , is necessary,
and they may be regarded either as signs of quality or as signs of operation. . . 7o
+ 00 - + 3O = -30. Thus can replace + 7O, and 00 - + 3O can replace "30. and the small signs, + 7-*7, and
All things considered, this practice of using for a time the small signs of quality seems to be of doubtful value. The two chief objections to it are:
1* Xt is not easy to get rid of these signs, once their use is begun* They prove to be somewhat an “old man of the sea”.
2. This practice seems to be a violation of the psychological principle that in general an element should not be introduced into a learning process if it must be discarded later*
?3 Hedrick, E. R.: Algebra for Secondary Schools, p. 14.
J. C. and Millis, J. F.: Essentials of Algebra, p. 32.
Hawkes, H. E., Luby, W. A., and Touton, F. C.: Complete School Algebra, p. 16.
76 Milne, W. J.: Standard Algebra, p. 26.
Collins, J. V.: Practical Elementary Algebra, p. 19.
The Approach to the Study of Negative Numbers.
It is a truism that in teaching much depends upon the way in which a new subject is introduced to the student. The teaching of negative numbers is no exception to this general rule, and it is evident that writers of textbooks have given this matter much thought.
In general, there are two common ways of approach to the idea of the negative, and in each the writer, or teacher, is seeking to show the learner the need for a new kind of number. The two approaches may be characterized in this way:
1. There is the approach through calling the student’s attention to examples from his own experience of the use of numbers of opposite qualities. Some of the most common examples are furnished by the thermometer scale and the latitude convention. More will be said about these illustrations in a later section.
2. The other commonly used approach is by means of pointing out to the pupil that a large number cannot be subtracted from a smaller one. He is reminded,for example, that 2-5
may be written 2-2-3, which yields -3, and that a new kind of number has arisen; or, he may be shown that the operation indicated by 2 - 5 can be performed by counting five units to the left of zero into the uncharted realms of the new type of numbers♦
Of course, there is nothing to prevent the use of both of these approaches. But in any case one of them will serve as the student*s first point of contact with the new subject, and that is the sense in which the word ’’approach” is used here.
The question as to which of the approaches is better can be settled only by careful experiment. Theoretically, the first has much to recommend it in preference to the second. Assuming that the illustrations used are within the experience of the pupils, it seems better to make use of this apperceptive basis than to employ the subtraction operation, in which the pupil may have little or no interest. As a matter of observed fact, the majority of modern textbooks use the first approach.
Now and then, one finds authors using other ways of approaching the subject. For example, Phillip KeHand in his Elements of Algebra (1876) makes use of a word problem which yields a negative solution. Stone and Millis in Essentials of Algebra (1905) use 3x4- 7=2x +4, which leads to x= 4- 7. This is a complicated form of the second approach described above. Both of these schemes would probably have less contact with the pupil’s interests and experiences than either of the common approaches.
Illustrations of the Use of Positive and Negative Numbers
As stated in the previous section, concrete illustrations of the use of numbers of of opposite qualities are often used as an approach to the study of these numbers. Even when these illustrations are not used in this way, .they are included as a means of clarifying and fixing the idea of positive and negative numbers.
As the seventy-seven textbooks were examined a tally was kept of the uses made of these illustrations. The results of the tally are shown in Table XIII.
Ligda questions the value of some of the most common of these illustrations, pointing out that some children in southern latitudes have never experienced negative temperatures and that losses of hundreds of dollars are entirely beyond the ken of the average beginner in algebra. He suggests the use of time before and after the pupil 1 s birth, the two cases of trains running at different rates, and the example of the stairway extending above and below a certain reference level. 7 ®
The relative values of these illustrations would depend entirely upon the background or experiences of the pupil or pupils, and it would be for the teacher to explore and evaluate this background and select the illustrations accordingly.
The ”scores-in-games” illustration is of recent vintage, no example of its use being found in books published before 1912. The following extract from the Schorling-Clark Modern Algebra First Course (1929) will show how the transition is made, for example, from or ’’three in the hole”, to -3.
Who Won? Who Won? Ruth Jack Ruth Jack First score . . . . 2 3 +2+3 Second score ... (3) 1 -3+l Third score .... 3 © +-4 -2 Fourth score ... 1 2 +1 +2 Fifth score .. . © ® -2 -1 Sixth score .... ® -1 f 3 Total score . . .
The -»• and - sign way is more commonly used in writing positive and negative numbers in everyday affairs.
78 Ligda, Paul, The Teaching of Algebra, pp. 95-105.
7Q Schorling, R., and Clark, J. R.: Modern Algebra First Course, p. 78.
Illustration Frequency Thermometer scale 30 Direction in general 21 Gain - loss 19 Credit - debit 17 Latitude 10 Time (A.D. and B.C.) 8 North-south 7 Scores in games 6 Heat-cold 2 Weight-levity 1
TABLE XIII Illustrations of Positive and Negative Numbers with Frequencies
From the Deductive to the Inductive.
One of the most striking changes in algebra teaching from 1829 to 1929 is to be found in the substitution of the inductive for the deductive method of treatment. It has already been made clear that in early textbooks it was the custom merely to define negative numbers and to use them in operations, with little emphasis placed on an understanding of the significance of the negative. In contrast to that treatment modern authors were seen to approach the idea of the negative in ways that were always careful and sometimes elaborate. The result of these preparatory measures was that the pupil was ready to frame his own definitions and even most of the principles governing the use of negative numbers.
In no connection is this contrast more marked than in the formulation of the rules for the four fundamental operations with signed, or directed, numbers. Tie rules for signs in multiplication will serve as an illustration of the radical change in methods.
In Rackley’s Treatise on Algebra (1855) the rule of signs in multiplication is given abruptly, without any sort of preparation.
RULE OF SIGNS IN MULTIPLICATION
The product of quantities with like signs is affected with the sign 4 ; the product of quantities with unlike signs is affected with the sign - ; or t multiplied by + and - multiplied by - give + ; 4 multiplied by - and - multiplied by + give - ; or like signs produce t and unlike signs - . The continued product of an even number of negative factors is positive; of an uneven number, negative.
The rule is followed by six examples, all worked out by the author, to illustrate the working of the rule. No exercises are given for the student.
A good example of the inductive treatment of this same topic is to be found in the Rugg-Clark Fundamentals of High School Mathematics. The steps in the derivation of the rules in that text may be outlined thus:
1. The four ways in which signed numbers must be multiplied ( +x+ ,+ x-, -x+, and -x -)
2. Illustrative examples from the saving and wasting of money. (Summary of the results from these examples)
3. Further examples from thermometer readings. (Summary as before)
4. Rules given in ’’completion” form, the student to complete them from his study of the foregoing illustrations.
The treatment of this topic is so typical in general of the modern methods and is such a complete example of the inductive method that it is reproduced in photostatic form and may be seen in Plates I and 11.
The two examples just presented represent the two extremes of method. However, a gradual change from the deductive to the inductive method of treating all phases of the negative is most noticeable in the textbooks published from 1829 to 1929.
80 Hackley, Charles W.: Treatise on Algebra, p. 15.
Examples of Special Devices Employed by Authors.
The monotony of the older algebra textbooks was almost wholly unrelieved by charts or graphs of any kind. On the other hand, modern writers make the fullest possible use of graphical aids. A few of these will be shown by way of illustration. They appear in Plates 111 to IX.
The first graphical device shown (Pla,te III) is taken from Newcomb’s College Algebra (1887;. It is interesting because of the unique appearance of the directional aspect of positive and negative numbers and because of the clear representation of the algebraic scale.
Plate IV, taken from Collins’ Practical Elementary Algebra (1908), shows a typical graphical explanation of the law of signs in multiplication of signed numbers. It represents a worthy attempt to make pupilft’ ideas less vague and more meaningful by means of impressions coming through the sense of sight.
Plate V portrays a method used by Fite, in his College Algebra (1913), to define the operations graphically. One cannot escape the impression that the definition of multiplication might fail to make for clarity in the mind of the learner.
In Plate VI is shown Professor Hedrick’s pictorial rule for addition and subtraction tf signed numbers. This graph appears in his Algebra for Secondary Schools (1908). In spite of the unusual use of the words ’’backwards” and ’’forwards”, such a device should be of great assistance to the pupil.
Plate VII shows a striking example of the care.
ful approach to the subject of positive and negative numbers through development of the idea of ’’oppositeness” in pairs of numbers or ideas. The completion chart at the bottom of the page is typical of the exercises provided by authors of modern algebra textbooks for the strengthening of concepts. The author of this particular text (Gugle, Modern Junior Mathematics, Book Three (1921)) follows the chart by the two questions:
If all the foregoing ideas are considered positive ideas, what one name might you give to all their opposites?
If the sign plus belongs with the positive ideas, what sign would naturally show their opposites, or negative ideas?
Then follows a reference to and a picture of the thermometer scale, which leads directly to definitions of positive and negative numbers.
Plate VIII comes from Drushel and Withers’ Junior High School Mathematical Essentials, Ninth School Year (1926)* It contains another form of the test for the -y and - number idea. This test had been preceded by a discussion of the thermometer scale and a description of the number scales to the right and left of zero. It is followed in
the text by a set of twelve exercises, the first two of which are shown on the plate, designed to give the pupil practice in the use of positive and negative numbers.
Plate IX shows a page from Myers* First Year Mathematics (1909) and illustrates a unique device for employing the operation of multiplying positive and negative numbers. There is involved here the so-called ”turning-tendency”, which is used often and effectively by the author. The direction idea furnishes the basis for classifying the various quantities involved. Weights exert a positive or negative force according as they pull down or up; lever arms are positive to the right and negative to the left; and counter-clockwise turning-tendencies are positive, clockwise turning-tendencies are negative. In the text, there appear many drill exercises for experimenting with the forces and levers.
PLATE I Illustrating the teaching of negative numbers From Rugg and Clark’s Fundamentals of High School Mathematics
_ PLATE II .. Illustrating the Teaching of Negative Numbers. From Rugg and Clark 1 s Fundamentals of High School Mathematics
PLATE 111 D e vice for Teaching Negative Numbers. From Newcomb’s College Algebra,.
PLATE IV Multiplication of Signed Numbers F r om Collin’s Practical Elementary Algebra
PLATE V Graphical Definition of Operations with Signed Numbers From Fite’s College Algebra
PLATE VI Pictorial Rule for Addition and Subtraction of Signed Numbers From Hedrick’s Algebra for Secondary Schools
PLATE VII Oppositiveness in Pairs of Numbers of Ideas F r om Gugle * s Modern Junior Mathematics
PLATE VIII Test for the Positive and Negative Idea From Drushel and Withers’ Junior High School Mathematical Essentials, Ninth School Year
Chapter Summary
Clear ideas about and defin ite practices with negative numbers have developed very slowly. Since 1829, writers of textbooks have assigned a more important place in the elementary
algebra course to negative numbers. Until recently, authors of textbooks have been prone to doubt the reality of negative numbers, to have vague ideas as to the size of negative numbers, and to confuse the negative number with the subtraction operation. The algebraic scale,represented geometrically, came into common use about 1880. In teaching negative numbers, authors of modern algebra textbooks make an approach through the common experiences of the pupils. The methods are characterized by the use of concrete illustrations of or analogies to negative numbers, and the entire procedure is usually inductive in sharp contrast to the deductive methods found in earlier textbooks. Special graphic devices are common in modern books, and these devices emphasize the direction idea in relation to positive and negative, numbers.
PLATE IX Turning-Tendency in Operations with Signed Numbers F r om Myers’ First Year Mathematics
CHAPTER V TREATMENT OF THE EQUATION IN THIRTY ALGEBRA TEXTBOOKS
Purpose of the Chapter.
The aim of this chapter is to report the results of an examination of those parts of thirty algebra textbooks which deal with the equation. The publication dates of the textbooks examined range from 1762 to 1929 and are fairly well distributed over the interval between these two dates. A list of these books appears in the appendix.
A background for the study is laid by examining somewhat in detail the treatment of equations in a textbook written in 1706 and published in an eleventh edition in 1762. Practices in defining and reducing equations are to be examined. The selection of subject matter and the methods recommended by authors will be considered critically in connection with simple equations involving one or more unknowns, with quadratic equations, and with such topics as inequalities
and indeterminate equations. Formulas, a topic closely related to equations, will be reserved for treatment in a later chapter.
Ward's Treatment of Equations.
John Ward wrote his Young Mathematician 1 s Guide in 1706 and corrected it in 1722. The eleventh edition, used in this study, was published in 1762. This book was used extensively as a textbook in England and even in America. 81 As a sort of background for the study
of the treatment of the topic Equations in textbooks published between 1829 and 1929, it is proposed to examine briefly the sections of Ward’s book dealing with the aforesaid topic. The material is so different from that found in modern textbooks and the terminology is so unique that photostatic copies of certain pages are presented in order that the reader may get a better idea of Ward’s treatment than could come from discussion alone.
In the first place, Ward has only two chapters devoted to expositions of equations. His fifth chapter covers eight pages and is concerned with “the nature of equations and how to prepare them for a solution”. In this chapter, after some preliminary discussion, he shows his reader how equations may be reduced by addition, subtraction, multiplication, division, involution, and evolution. Strictly speaking, these operations are designed merely to prepare the equation for solution.
Chapter VIII bears the title, "Of substitution, and the solution of quadratick equations.” It, like Chapter V, covers eight pages. Chapter IX deals with the solution of ’’problems exemplified by variety of numerical questions”. This chapter is very long and includes some thirty-five problems, all of which are worked out by the author. It is not within the scope of this study to discuss in detail the nature of these problems; but the last problem is of such a unique character that it is reproduced here in Plates X and XI. When one contrasts this ’’enigma" with the problems appearing in modern algebra text-
books, he can not fail to discern something of the progress that has been made in this respect. The enigma is properly named; the series of seven puzzles finally leading to the solution “Soli Deo Gloria” might well challenge the powers of a Sherlock Holmes.
Plates XII and XIII show something of Ward 1 s ways of handling simple equations* Particular attention is called to the last paragraph but one on Plate XII (page 177 in Ward’s book). Here the “ingenious Mr. Scooten” is mentioned as having done something to help in “casting off, or getting away Quantities out of an Equation”, even when the unknown quantity is “so mixed and entangled with those that are known”. Plate XIII shows Ward’s methods of reducing equations by addition, subtraction, and multiplication. Succeeding pages carry instructions as to reduction of equations by division, involution, and evolution. Ward’s terminology is interesting. He uses the small vowels to represent quantities sought and small consonants for the known quantities. Furthermore, the exponent convention is not used in expressing the second power of a symbol; the second power of “a” is simply written “aa”. In other parts of the book, the author uses exponents for powers higher than the second.
Ward recognizes only three cases of ’’quadraticks”. They are:
aa + 2ba = de aa - 2ba = de 2ba - aa » de
Plate XIV shows the methods for solving these equations. The first method is by a curious sort of substitution for the purpose of ’’casting off the lowest terms,” the second by "compleating the square!’. It is to be noted here that only one root is obtained for each quadratic equation. This results from the failure to give the square root the double sign. The method of obtaining the second root is shown by the quotation below and by Plate XV.
But every adfected Equation hath as many Roots .... either real or imaginary*, as are the Dimensions ... of its highest Power; and therefore, the Quantity a, in this Equation, hath another value . .; which may thus be found.
The given Equation is aa + 32a = 4644, and its Root a =54. Let these two Equations be made equal or equated to 0, viz. to Nothing.
The last part of this example and other examples are shown in Plate XV. An index to the author’s understanding of negative numbers is to be found in the statement (See Plate XV) that it seems impossible that ’’Affirmative Quantity should be equal to a Negative Quantity”, which statement was provoked by seeing a = -86, as an expression of the second root of the equation.
Ward concludes his treatment of the types of equations that are included in modern elementary algebra by showing on pages 236 and 237, a method of solving quadratics by the ’’universal Method of continued series”. This process is so cumbersome and involved that it would never be included in any modern elementary course.
Cajori, F.: History of Mathematics in the United States, pages 2fc, &2.
82 Ward, John: Young; Mathematician* s Guide (1760), p. 199.
PLATE X The Enigma From Ward’s Young Ma themat ici an 1 s Guide
PLATE XI The Enigma From Ward’s Young Mathematician’s, Guide
PLATE XII Simple Equations From Ward’s Young; Mathematician’s Guide
PLATEXIII Simple Equations From Youn^lZathemat ician 1 s Guide
PLATE XIV QUADRATICS From Ward’s Young Mathemat io ian’s Guide
Importance of Equations in Algebra.
Authors of the thirty algebra textbooks examined for their treatment of equations are practically unanimous in holding that the equation is one of the most important, if not the most important, element of algebra. Before examining their statements on this point, however, two dissenting opinions will be noted. The two mathematicians holding these opinions are Englishmen, a fact which may point to a rather fundamental difference in the English and American conceptions of the subject matter of elementary algebra.
Whitehead, the eminent English mathematician and philosopher, wrote apparently for the layman a small book on mathematics in which he dealt with some of the foundation ideas of the subject. In particular, he compares the formula and the equation as to their real importance in mathematics and decides in favor of the former. He says:
The idea of the undetermined ’’variable’ 1 as occurring in the use of ” or ’’any” is the really important one in mathematics; that of the ’’unknown" in an equation, which is to be solved as quickly as possible, is only of subordinate use, though, of course, it is very important. One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the textbooks with the solution of equations.
Nunn, an English authority on the teaching of mathematics, writes in very much the same vein:
In history equations began as conundrums, and the school tradition has not lifted them to a much higher level of intellectual dignity. The pupil may become skilful in compelling ”x” to reveal the value hidden in a symbolic statement of baffling complexity; he may become acute in threading the intricate mazes called "problems” which the ingenuity of the textbook writer has set in his path. Yet in the end he may still be only an expert solver of conundrums. He may have gained but an imperfect idea either of the practical or of the scientific importance of processes which he has learnt to handle for mere artificial purposes.
Nunn avails hlms elf of the power of the conundrum to stimulate intellectual activity, however, and in a certain section of his exercise book on algebra, there appear conundrums like the one following
I am thinking of a number. I multiply it by 3 and add 14. The result is 23. What is the number? 88
This type of exercise is made to lead immediately to a universal rule in the shape of a formula.
In sharp contrast to these views are those of the authors of textbooks examined. Those who express themselves on the importance of the equation leave id doubt as to their opinion on this matter. Those who make no definite statements seem to take the importance of the equation for granted. A few pertinent statements will be quoted.
The most useful part of algebra is that which relates to the solution of problems. This is performed by means of equations. ~ c
The equation is the distinguishing feature of algebra.
The most important thing in mathematics: the EQUATION. 88
The greatest help it (algebra) offers us in solving problems is the use of the equation. 8 $
Perhaps the best proof that modern authors consider the equation of paramount importance is to be found in the fact that they use it throughout their books. The data of Chapter 11, especially those shown in Tables IV and V, show that this topic is now introduced as the first, second, or third topic of the course and that much more space is now devoted to it than was formerly the case.
83 Whitehead, A. N.: An Introduction to Mathematics, pages 17 - 18.
84 Nunn, T. Percy: The Teaching of Algebra, p. 77.
85 Nunn, T. Percy: Exercises in Algebra, p. 70.
86 Ray, Joseph: New Elementary Algebra (1866), p. 90.
S' 7 Sanford, S. P.: New Elementary Algebra (1879), p. 136,
88 Rugg, H. 0., and Clark, J. R.: Fundamentals of High School Mathematics (1920), p. 13.
B sEdgerton, E. 1., and Carpenter, P. A.: First Course in Algebra (1923) p. 54.
PLATE XV Quadratics F r om Ward’s Young; Mathematician’s Guide
The Reduction of the Equation.
The operations performed upon a simple equation in order to reduce it are all based upon the axioms. All of the textbooks introduce the axioms in one form or another. The four axioms, one for each of the four fundamental operations, represent the minimum list of axioms, although axioms for involution and evolution are sometimes added. Newcomb gives the essence of the axioms in the following words:
Similar operations upon equal quantities give equal results. $$
Olney (1881) and Wells (1885) epitomize the axioms in very much the same way.
Gerrish and Wells (1902') bring into their state ment of the axioms the idea of ratio.
The ratio of two equal numbers is not changed when the members of the equality are subjected to the same operation.
Given 16 = 16 16+2 = 16+2 16-2 =l6-2 16 x 2 = 16 x 2 16+2 = 16+2 (16 t - (16) 2 jl6 = vi 6
Given a - a a + b - a +b a-b = a-b a•b- a • b a-r b - a 4 b Z i- a = a /a s -
The ratio of the numbers, 1:1, is not changed. ®1
Rugg and Clark (1920) lead their students to ’’discover” the axioms in an inductive way* Plate XVI shows how these authors introduce the equation, how they use the balance device, and ho they lead up to a statement of the axioms. The material shown in Plate XVI is completed on the next page of the text by the following:
Exercise 11
By thinking of the equation as a you should be able to complete the following statements. Fill in the blanks with the oropar words..
1. Any number may be subtracted from one side of an equation if ? is ? from the other side.
2. Any number may be added to one side of an equation if ? is ? to the other.
3. One side of the equation may be multiplied by any number if the other side is ? by the ? .
4. One side of an equation may be divided by any number if the other side is ? by the ? .92
This treatment is in sharp contrast to that found in earlier textbooks wherein the axioms are stated categorically and left for the student to apply.
Schorling and Clark (1929) refer to the axioms as '’keys’ 1 to the solution of the equation. They employ an inductive approach very similar to that just described for the Rugg-Clark text. 93
It seems, then, that it is the common practice to make some use of the axioms in the solution of simple equations, however much the wording and applications of the axioms may vary. Before closing this discussion, however, it will be of interest to examine an opinion with regard to the use of axioms that is diametrically opposed to that which leads to the practice just described. Nunn says:
Two points in the exposition of ch.x should receive special attention. The first is that the rules for removing a number from one side of the formula to the other are not based upon any axioms of equality. This departure from traditional procedure needs but little justification. A boy is told that when 7 is added to a certain number the sum is 12, and at once states that the unnamed number was 5. It will not be pretended that he reaches this result by reflecting that ”if equals be taken from equals the remainders are equal”, nor that if he could not reach it unaided an appeal to that axiom would help him to conviction. Children can, in fact, solve such concrete riddles years before they can appreciate the abstract axiom. Moreover, the pupil will see with perfect clearness that the mode of solution of this problem - ’’take 7 from 12 and you have the other number is perfectly general; that is, that its validity as a process does not depend upon the specific numbers involved in it. $4
Here again is a good subject for experimentation to determine the best possible use of the axioms in the solution of simple equations.
90 Newcomb, S.: Algebra for Colleges and Schools, (1887), p. 85. ~
91 Gerrish, 0., and Wells, Webster: The Beginners' Algebra (1902) p. 49.
92 Rugg, H. 0., and Clark, J. R.: op. cit.,p. 24.
. n , T o Schorl mg, R., and Clark, J. R.: Modern Algebra First Course (1929), pp. 30 - 34."
94 Nunn, T. Percy: The Teaching of Algebra, p. 78.
PLATE XVI The Balance in Teaching Equations FTom Rugg and. s Fund.amentals of Sign School
Transposition.
Transposition is an operation that merits brief special mention. It is, of course, one of those transformations frequently necessary in the reduction of an equation and is based upon the axioms of addition and subtraction. Much difference of opinion exists among teachers as to the best practices as to transposition with beginning students. Some teachers would prefer merely to give the rule with some proof of its reasonableness and then allow students to transpose terms more or less mechanically in accordance with the rule. Other teachers insist that the student should be required to add or subtract the proper terms from both sides of the equation, in order to get the misplaced terms in the desired position. The complete process would be about as follows:
Solve 3x 4- 10 = 13 -10 -10 3x - 3 .•. x = 1
Add -10 (or subtract 10) to both sides of the equation.
If equals are added to equals, the sums are equal*
In some cases this complete process is required of students throughout the first year, or at least for a semester.
The books in almost every case merely show the need for transposing in the process of reducing the equation. There usually follows a statement of the rule with a sort of proof. The following is typical
To solve 3x — 8 x f 12
We can . . . transpose any term from one side to the other by simply changing its sign. This we proceed to shew.
Subtract x from both sides of the equation, and we get
3x - x - 8 = 12 (Axiom 2)
Adding 8 to both sides, we have
3x - x 12 f 8
It is evident that similar steps may be employed in all cases. Hence we may enunciate the follow ing rule:
Rule. Any term may be transposed from one side of the equation to the other by changing its sign. $5
Following this proof, exercises are usually provided, to solve which the student is expected to transpose mechanically in accordance with the rule. In fact, the advantage of transposing in such a manner is often stressed, as witness the following:
The transposition of a term is moving the term from one member of an equation to the other member. We shall see that when a term is transposed, the sign of the term must be changed.
Ex. 1. Find the value of x in x -
PROCESS WITHOUT TRANSPOSITION
We have given x - 5 - 7 5 5
Adding sto each x-7 + 5 of the equals or x = 12 Ans.
PROCESS WITH TRANSPOSITION
We have given x - 5 - 7
Transposing 5 to the right hand member x - 7 + 5 of the equation and changing its sign x - 12 Ans.
Hence transposition is a short way of adding equal numbers to two members of an equation. The labor saved by means of transposition is more evident when several terms are to be
96 transposed at the same time.
It is evident that an enormous saving of time and effort can be accomplished by means of mechanical transposition. Also, requiring students to apply the axiom completely at all times amounts to requiring them to make use of an element which must be discarded later. Consequently, it would seem best to allow students to use the rule and to take what measures are necessary to see that they do not forget the raison d’etre for the rule.
OR "Hall, H. 8., and Knight, S. R.: Elementary Algebra, (1892), p. 49.
96 Durrell, J., and Arnold, E. E.: A First Book in Algebra (1920), pp. 118 - 119.
Definition of the Equation.
It is always good form to define the terms used in any discussion. Writers of algebra textbooks as a rule give some sort of definition of the equation when they first introduce that topic. The form of the definitions varies. Those definitions appearing in the older books are interesting.
An equation is, when two equal quantities,
differently expressed, are compared together, by means of the sign ~ placed between them. [Punctuation not corrected.!
Bonnycastle refers to the practice of putting two expressions equal to each other and adds, ”which formula in that case, is called an equation.” 98
The majoritsr of authors follow the form used by Colburn (except for the punctuation):
The expression of equality between two quantities, is" called an equation. 99
A few definitions hardly deserve to be classed as such. An example:
The expression 3x + x - 32 is what is called an equation.
One text contains a definition that makes a distinction between equation and identities, something that the definitions just quoted fail to do.
An equality which exists only for particular values of certain letters representing unknown quantities is called an equation.
Ordinarily, however, this distinction is made after the definition is given, and the definition is thus limited to the so-called‘tequation of condition. 1 ’ All authors seek to clarify their definitions by the use of examples.
"'Simoson, Thomas, A Treatise of Algebra (1800), p. 57.
SS Bonnycastle, John: Algebra (1820), p. 70.
Warren: op. cit.,p. 13*
Edward: First Principles of Algebra (1881), p. 88.
101 Beman. W. W., and Smith, D. E.: Academic Algebra (1902;, p. 10.
Organization of Subject Matter on Equations.
Examination of algebra textbooks reveals that there developed in the course of the century 1829 - 1929 what may well be called an orthodox way of arranging the various phases of the subject matter bearing on equations. This orthodoxy became pronounced as early as 1880. The arrangement referred to was an outgrowth of the logical attitude toward the subject of algebra. This attitude has been referred to before; it leads to the practice of giving first thought to the logical sequence of topics rather than to the sequence by which children can best master the material.
The most common order was about as follows:
1. Simple equations involving one unknown.
2. Problems involving the same.
3. Simple equations involving two unknowns.
4. Problems involving the same.
5. Simple equations involving three or more unknowns.
6. Problems involving the same.
7. Indeterminate equations.
8. Inequalities.
9. Radical equations.
10. Quadratics involving one unknown.
6. Pure. b. Affected.
11. Problems involving the same.
12. Quadratics involving more than one unknown.
13. Problems involving the.same.
14. Equations in the quadratic form.
15. Theory of quadratic equations.
Fractional equations were frequently the subject of an additional chapter. In some texts items 11 and 13 were combined, and indeterminate equations and inequalities were not always in the same positions. But for the most part, the topics were arranged as shown above. For one who studied elementary algebra during the period 1880 - 1910 this statement willneed no proof.
The few exceptions to this accepted arrangement are interesting. Colburn (1829) devotes the first sixty pages of his book to problems involving one and two unknowns. Sanford (1879) makes his problems precede the more formal exercises, at least in the early part of the book.
There can be little doubt that this orthodox arrangement of subject matter developed with little attention being given to the ways in which children learn, except insofar as the authors tried to proceed from the simple to the complex. Rugg and Clark pointed out in 1920 that mathematical sequence and learning difficulty should control the organization of all material in algebra. 102 They substituted an
experimental procedure for the practice of making textbooks at the desk. But before 1920, considerable thought had been given to this problem of presenting algebra in an order most conducive to ease of learning on the part of the pupil.
Some interesting treatments of the equation came from this new attitude. An example is presented from Myers’ First Year Mathematics (1909).
1. General uses of the equation.
2. Uses of the equation with perimeters and areas.
3. The equation applied to angles.
4. Beam problems involving one unknown.
5. Simple equations involving one unknown.
6. Simple equations involving two unknowns.
7. Quadratic equations involving one unknown, (Taught along with factoring)
It should be pointed out that Myers 1 book is a text on general mathematics and that some difference in the treatment of equations might be expected. Nevertheless, the arrangement shown here is fairly typical of the new practices. The approach to the topic is new, and equations are employed more widely as a mathematical tool. The old logical arrangement is followed in part only. The formal, manipulative exercises are not always made to precede the word problems.
i a n O Rugg, H. 0., and Clark, J. R.: O£. cit.,p. viii.
The Solution of Word Problems.
All authors of algebra textbooks give a prominent place to the solution of problems. A statement by Ray, quoted on page 168, makes equations a means to the end of solving problems. It is not the purpose here to enter into an analysis of the problem materials of elementary
103 algebra. That task has been well done by others.
In the present study, it is the aim to consider rather the means by which the student is led to set up an equation, or equations, from the verbal statement of the conditions of the problem. The interpretation of results coming from the solution of the problem will be included also.
103 Especially by Rugg and Clark and by David Eugene Smith. See also Powell, Jesse Jerome, A Study of. Problem Material in High School Algebra, Contributions Fo’’WucaH’o” H0T405, Teachers College, Columbia University.
Expressing Conditions of a Problem as Equations.
In the solving of word problems, the first step is, of course, to translate the English into algebraic symbology, in other words, to express the conditions of the problem as an equation, or equations. This is also the most difficult step, a fact which is apparently recognized by the authors of all textbooks examined in this study. Professor Judd says on this point:
Another common difficulty in both arithmetic and algebra arises from the fact that in many cases the examples to be solved are stated in words and must be translated into numbers, or algebraic symbols before the strictly mathematical operations can begin. When a pupil makes a mistake in solving verbal problems, the trouble can frequently be traced to his lack of understanding of the words and phrases rather than to his lack of knowledge of the mathematical procedure involved.
The psychological theory on which algebra texts base the use of verbal problems is that the pupil must learn to apply his mathematics by * discovering in various complex situations the opportunity for mathematical manipulation. In other words, the textbook aims" to confront the pupil with realistic conditions for the use of mathematics in order to give training in application, which is a form of generalization. 104
In order to aid the student in this difficult task, textbook writers make use of three main devices:
1. The working of numerous examples by the author.
2. The formulation of sone thing in the nature of a rule for the translation of the words
into equations.
3. Furnishing practice for the student in this translation before the complete solution is undertaken.
Frequently, as in Wentworth’s books, the problems are classified into groups, and a typical solution precedes each group of problems.
The rule or directions for the formation of the equation from the conditions of the problem varies somewhat in form from text to text. Colburn (1829) states that no rule can be given for ‘’putting questions into equation”, but he gives the following ’’precept”, which leaves a feeling of inadequacy:
Take the unknown quantity, and perform the same operations on it, that it would be necessary to perform on the answer to see if it was right. When this is done, the question is in equation. 105
Most of the older books have a rule or precept of this general nature. Gradually, this practice of trying to reduce this process to a formal rule was followed less and less often. Directions of a very different order were given in later books. Beman and Smith (1902) put theirs in the form of questions:
1. What should x represent? 2. For what number described in the problem may two expressions be found? 3. What is the algebraic form of each of these? 4. How do you state the equality of these expressions in algebraic language?
The directions given by Rugg and Clark (1920) are labelled "How word problems are solved by equations" and display the wordiness of the Fundamentals, of. High School Mathematics. These directions are shown in full in Plate XVII. It is difficult to see hew much more could be done for the student than is done here. Plate XVIII shows the directions given by Schorling and Clark in Modern Algebra First Course. There is added here in the illustrative problem the device of tabulating the facts set forth in the statement of the problem. Many teachers of first year algebra use this device in all work with problems. Its effectiveness is doubtless due to the fact that it organizes the data, that it makes the data visual, and that it reduces problem solving in part to mere routine.
The third device for aiding the student in setting up his equations is that of furnishing exercises for practice in translating English into algebraic symbology. It is a truism that students who are wea,k in reading ability are seriously handicapped at this point. When a student is told to ’’get in mind very clearly what is known” and to ’’recognize what is to be found out”, he is in bad straits if he cannot read intelligently a clear straight-forward English sentence. At all times, teachers do well to supplement the help given by authors of textbooks by diagnosing and remedying where necessary the reading ability of the student.
Even assuming average reading ability on the part of the student, many textbook writers seem to realize that he will need much practice in extract- ing equations from written statements. Some textbooks are outstanding in this respect. These texts are listed below in Table XIV in chronological order, together with the amount of space in pages devoted to this type of exercises.
The Hall and Knight text, the pioneer in providing this type of exercise, offers a great variety of ques-
tions requiring the use of symbology. Some examples are given here.
What must be added to x to make £? A man has a crowns florins, how many shillings has he?
What is the age of a man who y years ago was m times as old as a child then aged x years? I° 7
Swenson (1924) provides exercises both for translating from English into algebra and for the reverse process. Examples of the latter are:
What do each of the following equations assert: lOr - 130, lOr = 70?
A can build k feet of fence in one day and B 5 feet less than A. What do the following represent: k -5, 10k, 7 (k -5), 30k? 108
Other texts provide exercises here and there for this translation process. Schorling and Clark offer a test in translation which requires of the student to express in a formula or an equation the relations stated verbally in the problem. The test appears in two forms. A typical exercise from Form A of this test is shown here.
Two trains started 400 miles apart and traveled toward each other. The first traveled x miles an hour. The second traveled 15 miles an hour faster. Write the algebraic expression which tells how far apart they were at the end of 4 Hours.
104 Judd, C. H.: Psychology of Secondary Education, pp# 93 - 94.
Warren: op. cit M p. 37.
106 Beman, W. W., and Smith, D. E.: O£. cit,., p. 14.
107 Hall, H. S., and Knight, S. R.: op. cit., pages 57 - 63.
Swenson, John A.: op. cit., p. 38.
109 Schorling, R., and Clark, J. R. * op. cit., pages 233 - 237.
PLATE XVII Rules for Solving Word. Problems From Rugg and Clark Fundamentals of High School Hatnematics
PLATE- XVIII Directions for Solving Word Problems From Schorling and Clark*s Modern Algebra, First Course
Text Date of Publi- cation Number of pages Hall and Knight, Elementary Algebra 1892 6 Wentworth, New School Algebra 1898 3 Gerrish and Wells, Beginners 8 Algebra 1902 5 Schultze, Elements of Algebra 1912 5 SIaught and Lennes, First Prin- cip les of Algebra 1912 2 Stenson, High School Mathematics 1924 3
TABLE XIV Space in Certain Texts Devoted to Translating English into Algebra
Solution and Discussion of the Squation.
Once the conditions of a word problem are expressed in an equation, the equation must be solved in the usual ways. A difference is to be noted here between procedures in older books and those in many books published recently. Formerly, it was the custom to dwell at length upon the formal solution of equations and to reserve the application of equations to the solving of problems for treatment later. This is the logical sequence — to acquire dexterity in handling a tool before the tool is to be used in a life situation. In contrast to this practice many modern textbooks start with the problem and discover ways of solving equations as the equations arise. This practice arises from a desire to make the most of the challenge of real problems. The object is to motivate the learning of algebra, and the formal, manipulative aspects of the subject are kept in their proper place, namely as means to the end of solving problems.
The older authors were prone to make much of the discussion of problems. This discussion usually took the form of casting the problem into the literal form and then making various suppositions as to the relative values of the given numbers, and explaining the results. Courier and rate problems in general furnish the favorite material for this sort of work. The discussions are almost always of a most theoretical nature. As an example, the following is taken from Wentworth:
Two couriers are travelling along the same road, in the same direction. A travels m miles an hour, and B travels n miles an hour. At 12 o’clock, B is d miles in advance of A. When will the “ couriers be together?
This yields an equation in x and a solution x - d m - n Various cases are then considered, such as m <n, m= n, m > n, and d -0, and the results of each supposition are noted. As an exercise in abstract reasoning this has undoubted value.
However, very little of this is found in modern texts; apparently, the determination of the authors to be practical automatically excludes such topics from the course.
Another opportunity for discussion of equations arises in connection with certain roots for quadratic equations, particularly negative roots. This has been mentioned earlier in connection with the discussion of negative numbers (See page 87 ). The necessity for interpreting negative roots is urgent in the solution of problems. In the earlier books, students were required to grapple with this question; the authors told them that a negative root is
sometimes the answer to another analogous problem, formed by attributing to the unknown quantity a quality directly opposite to that which has been attributed to it. m
There is a tendency to avoid such questions in the more modern texts. This is true also of what has been called the theory of quadratic equations. This phrase covers certain considerations as to the nature of the roots of a quadratic and the relations between the coefficients and roots. This phase of the study of the quadratic equation was once considered as an integral part of the beginner’s course in algebra, but it no longer has a place there. It does not appear in the textbooks written by Rugg and Clark
(1920), Durrell and Arnold (1920), Edgerton and Carpenter (First Course, 1923), and Schorling and Clark (First Course, 1929). Incidentally, of the books examined, Hedrick’s Algebra for Secondary Schools (1908) was the first to make general use of the term discriminant in reference to the expression b x - 4 ac, derived from solving the general quadratic + bx + c - 0.
G. A.: New School Algebra (1898), p. 203.
■^Greenleaf, B.: Elementary Algebra, p. 276.
The Solution of Simultaneous Linear Squations.
Simultaneous linear equations have long been a part of elementary algebra. Each of the thirty texts examined contains this topic in some form. From time immemorial, there have been three methods of solving such equations, and in modern times, there has been added a fourth. Table XV shows the extent to which these methods are used in the textbooks. The data of Table XV come from only twenty-nine textbooks; the data for Bonnycastle‘s Algebra (1829) is incomplete and cannot be used.
♦Wentworth 1 s New School Algebra (1898).
The following obvious facts may be noted in Table XV:
1. Addition or subtraction is used most frequently.
2. The older authors seemed to think it advisable to give the student as many ways as possible for solving linear equations in two unknowns. Many of the later authors dropped the use of the comparison method.
3. No one method was used in all texts. Special mention goes to the graphical methods of solving these equations. The first book in which the method appears is Wentworth’s Nev/ School Algebra (1898). However, this material is contained in an appendix and has the effect of making a separate topic of graphs. Hedrick (1908) was the first author of those whose books were examined who made of graphical solution a part of his regular section on simultaneous linear equations. Only one of ten texts published from 1906 to 1929 failed to use this method.
Simpson (1800), true to the deductive standards, regales his readers with three lengthy rules, one for each of the traditional methods. He follows with eleven “examples I *, most of them completely worked out by the author. His reader is a reader only. Oolburn (1829) provides nothing in the way of drill material for students. His simultaneous linear equations occur only as they are used to solve problems.
Beman and Smith (1902) classify comparison as a special form of substitution. Some authors, for example, Schultze (1912), recommend certain special devices, such as considering and as the unknown x y
quantities instead of x and y.
In sharp contrast to the strictly deductive procedure of Simpson (noted above) and others, one finds Rugg and Clark (1920) developing the principles of the addition or subtraction method by the use of illustrative examples, each of which is discussed and analyzed at length. The rules, or principles, are derived “from these examples’*.
lIP Rugg, H. 0., and Clark, J. R.: op. cit., p. 247.
Methods of Solving Number and Percent of Texts Using Through 1880 1881-1905 1906-1929 Totals No. No. $ No. No. Addition or Subtraction 10 90.9 8 100 10 100 28 96.5 Substitution 9 81.8 6 75 9 90 24 82.8 Comparison 9 81.8 6 75 2 20 17 58.6 Graphically 0 0 1* 12.5 9 90 10 34.5
TABLE XV Methods of Solving Simultaneous Linear Equations Used in Twenty-Nine Textbooks
Solving Quadratic Equations.
In Chapter 111, it was shown that quadratic equations as a topic has not changed materially in order and amount of space during the last one hundred years. (See Table IV.) Nevertheless, there have been changes on the inside, so to speak. It is the purpose of this section to point out some of these changes. Here as before, no attempt is made to analyze thoroughly the problem material appearing in the textbook.
Table XVI shows the extent to which the various methods of solving quadratics are used in the twenty-nine textbooks examined. Here, as in Table XV, the data from Bonnycastle 1 s book could not be used.
♦Wentworth: New School Algebra, (1898).
Table XVI records the following facts:
1. Completing the square by the first method (making the coefficient of x - unity, etc.) is by far the most commonly used method.
2. The Hindoo method of completing the square has just about disappeared from the textbooks.
3. The formula and factoring are used more frequently now than in the older books. The latter method was not used in any text published prior to 1881.
4. John Ward used a cumbersome method of solution by continued series. No other author made use of this method.
5. As in the case of simultaneous linear equations, to Wentworth goes the honor of first solving quadratics by graphs. This feature appears in the last chapter of his New School Algebra (1898).
The figures in Table XVI for the period 1906- 1929 show that nearly all modern textbooks teach at least three ways of solving quadratics. Completing the square by the first method is always included, the formula and factoring methods appear in the majority of books, and the graphical method is used in half of them. This multiplicity of methods of solution is not in harmony with Thorndike’s ideas on the question which he expresses in the following words:
When one method of accomplishing a task is all that is needed, only one is taught. Thus, in solving a quadratic equation, the method of completing the square is, for pupils in Grade 9, adequate for all purposes of mathematical insight and practical service.
Simpson (1800) has the poorest treatment of quadratics of all the books examined. He used the term ’’quadratic”, but he in no ways shows his reader a systematic solution of such an equation. In a section devoted to ’’the resolution of equations of several dimensions”, he plunges at once into a consideration of cubic and biquadratic equations, a favorite occupation of the mathematicians of his day.
Colburn’s treatment of quadratic equations is, as would be expected, unique. He introduces the subject by means of ’’questions” (problems) which lead to pure equations of the second degree. These equations having arisen, he immediately proceeds to grapple with the task of solving them. This in turn leads to the felt need for extracting the square root of numbers and algebraic expressions. This latter process is taught then and there. Finally, the student is inducted into the mysteries of completing the square. The medium of instruction is the problem throughout.
Peck (1875) is different in that he uses the formula as the sole means of solving the quadratics.
Some of the authors have striking ways of explaining and clarifying the process of completing the square. Ray is conservative:
We will now explain the principle b}r which the first term of this equation (x + 2px » q) may always be made a perfect square.
The square of a binomial is equal to the square of the first term, plus twice the product of the first term by the second, plus" the square of the second.
If now, we consider x + 2px as the first two terms of the square of a binomial, x x is the square of the first term (x), and 2px, the double product of the first term by the second; therefore,
If we divide 2px by 2x, the quotient, p, (half of the coefficient of x) will be the second term of the binomial, and its square, p, added to the first member, will render it a perfect square. To preserve the equality, we must add the same quantity to both sides. This gives
1"I 4- x + 2px t P = q + p .
Newcomb (1887) states part of the rule on a unique way. Instead of saying ’’add the square of one-half the coefficient of x” > he puts it ’’add one-fourth the square of the coefficient of x to both members of the equation.”
Hall and Knight (1890) present a superior
explanation of this process of completing the square.
The equation x - 36 is an instance of the simplest form of quadratic equations. The equation (x - 3)*" = 25 may be solved in a similar way; for taking the square root of both sides, we have two simple equations,
x - 3 =±s
Taking the upper sign, x - 3 - -+5, whence x -8; taking the lower sign, x - 3 = -5, whence x = -2. .• . the solution is x -8, or -2.
Now the equations (x-3) = 25 may be written - 6x +(3) =25, or x x - 6x - 16.
Hence, by retracing our steps, we learn that the equation - 6x = 16 can be solved by first adding (3) or 9 to each side, and then extracting the square root; and the reason why we add 9to each side is that this quantity added to the left side make it a perfect square. 11$
Gerrish and Wells (1902) provide extensive
exercises on completing trinomial squares, as a preparation for the solution of quadratics. For example:
1. The following examples call attention to that fact in the structure of a trinomial square:
2yx -4 y\ x\ 2yx 4
2. Add a term to each of the following numbers, and make of each a trinomial square:
x - IOOx
3. Complete the square and preserve the equation:
x 4 x - 15 2 2
4. Complete the square, preserve the equality, and simplify:
xl4x - 11 5
5. Take the square root of both members and find the values of x.
-6x -i- (3) r -4
6. Solve the given equations:
Bx _ 21 .117 5 ~ 5
Swenson (1924) makes an approach to the task of completing the square through a geometric figure. The first exercise is:
1. Figure 44 contains one square and two rectangles, with dimensions as indicated.
(a) Find the area of the square and the area of each rectangle.
(b) What is the total area of the figure?
(c) What is the side and the area of the missing corner square, needed to make the whole figure a square?
(d) After this corner square has been added, what is the side and the area of the big square?
(e) Show by means of multiplication that the correct relation exists between the side and the area.
The extent to which this idea is followed up and elaborated is shown in Plate XIX, which represents pages 220 and 221 in Swenson* s oook. No other text of those examined used this idea of areas in developing the process of completing the square.
113 Thorndike, E. L.: Junior High School Mathematics, Book Three, p. v.
114 Ray, Joseph: op. cit., p. 195.
118 Newcomb, S.: op. cit., p. 176.
He 8., and Knight, S. R.: op. cite, p. 179.
117 Gerrish, M., and Wells, W.: Beginner 1 s Algebra (1902), pp. 117 - 120.
J. A.: High School Mathematics, p. 219
44
PLATE XIX Completing the Square Geometrically From Swenson’s High School Mathematics
Method of Solving Number and Percent of Textbooks Using Through 1880 1881-1905 1906-1929 Total No* * No. No. 4 No. Completing Square,First Method 8 72 8 100 10 100 25 89.6 Completing Square, Hindoo Method 5 46 2 25 2 20 9 31 Formula 5 46 4 50 7 70 16 55.2 Factoring 0 0 5 62.5 7 70 12 41.4 Graphically 0 0 1* 12.5 5 50 6 20.7 Continued Series 1 9 1 3.4
TABLE XVI Methods of Solving Quadratic Equations Used in Twenty-Nine Textbooks
Simultaneous Quadratic Equations.
Here is a topic which does not now receive the attention once accorded to it. The following table will show the number of exercises and problems involving simultaneous quadratics in twenty-five of the thirty texts examined.
Table XVII (continued)
TABLE XVII (continued)
* Problems mixed with those involving one unknown. Complete title in appendix.
From the table it is evident that during the period from 1890 to 1912 a great deal of space was given to simultaneous quadratics. Wentworth’s First Steps and Gerrish a,nd Wells’ book are much too brief and elementary to include this topic. However, that it is omitted by Rugg and Clark and by Schorling and Clark in their first year books is significant. Another fact revealed in Table XVII is the swing away from exercises to problems. This is in line with the tendency in modern elementary algebra to emphasize the thought material rather than that which is merely formal in its nature.
Table XVIII shows the frequency of appearance in the thirty textbooks of certain so-called cases of simultaneous quadratic equations.
Strictly speaking, the first case is not one of simultaneous quadratics in spite of the fact that it appears most often in the texts. The older authors were inclined to use all three of these cases and to give considerable attention to special devices. These special devices, sometimes called artifices, called for the exercise of much ingenuity in the solution of exercises, the practical value of which one is inclined to doubt. Some modern authors consider only the first
case listed in Table XVIII. In fact, in one text it is stated that
Systems of equations both of which are quadratic do not often occur.
Perkins (1854), Hackley (1855), and Ray (1866) do not classify simultaneous quadratics but recommend the use of special devices only. Hedrick (1908) makes extensive and effective use of graphic aids. Wells and Hart (1923) devote only three pages to the topic and label it ’’supplementary”. Swenson (1924) also relegates it to a position in the supplement.
119 n T n A Edgerton, E. 1., and Carpenter, P. A.: First Course in Algebra (1923), p. 306.
Textbook* Date Number of Ex- ercises Number of Problems Young 1838 20 17 Perkins 1854 7 24* Hackley 1855 33 39* Greenleaf 1866 39 32* Textbook* Date Number of Ex- ercises Number of Problems Ray 1866 19 14 Venable 1869 21 25 Peck 1875 25 5 Sanford 1879 23 14 Olney 1881 17 14 Wells (Academic) 1885 81 18 Newcomb 1887 15 12 Hall and Knight 1890 71 35 Wentworth (First Steps) 1894 None None Wentworth (New School) 1898 70 17 Beman and Smith 1902 115 25 Gerrish and Wells 1903 None None Hedrick 1908 76 23 Schultze 1912 98 17 Slaught and Lennes 1912 20 35* Rugg and Clark 1920 None None Durrell and Arnold 1920 20 27 and Carpenter 1923 43 30 Textbook* Date Number of Ex- ercises Number of Problems Wells and Hart 1923 10 24 Swenson 1924 16 None Schorling and Clark 1929 None None
TABLE XVII Exercises and Problems Involving Simultaneous Quadratics in Twenty-Five Textbooks
Case Frequency One equation simple one quadratic 18 Both homogeneous 9 Both symmetrical 7
TABLE XVIII Frequency of Cases of Simultaneous Quadratic Equations in Thirty Algebra Textbooks
Chapter Summary
John Ward ’ s Young Mathematicia 1 s Guide (1760) is typical of the texts published prior to 1829, insofar as the treatment of the equation is concerned. In this book, about sixteen pages are devoted to this topic, the terminology is archaic, the methods of solving equations are cai- siderably different from those in vogue now, and all of the word problems are worked out by the author. All authorities are agreed that the equation is of great importance in algebra, but some of them place it second in importance to formulas. Modern authors are inclined to pay much attention to the inductive discovery of the axioms governing the reduction of the equation. Transposition is given much prominence as a laborsaving device. Since about 1915 writers of algebra textbooks have broken away from the usual arrangement of the subject matter dealing with equations and now try to organize it more in accordance with the laws of learning. Modern authors give the student much more aid in solving word problems that did the authors of es.rlier texts.
The so-called discussion of an equation, an exercise very common in early books, is usually omitted in modern books. Simultaneous linear equations are solved most commonly by the ’’addition or subtraction” method; the graphical method has been used by some authors of books written since 1900. Quadratics have been solved most frequently by the first method of completing the square. Factoring and graphs are recommended for this purpose by authors of books written since 1900. Simultaneous quadratics receive much less attention now than formerly. This topic is usually omitted entirely or limited to the case in which one equation is simple and the other is quadratic.
CHAPTER VI THE FORMULA
Purpose of the Chapter.
It is the purpose of this chapter to report the analysis of the treatment of the formula in twenty algebra textbooks, all published since 1899. The growth of the formula as a part of elementary algebra will be traced, and a characterization of the present status of the topic will be attempted. In particular, the formula will be considered in its relation to the equation.
Four Authorities on the Formula.
Before entering into the discussion of the formula as it appears in the textbooks, it is appropriate to consider briefly the conclusions of four men who have made studies of this subject.
Nunn’s attitude toward the formula has been noted in connection with the discussion of the equation (See p. 167). He says further:
. . . it is hardly extravagant to say that facility in working, interpretation, and
application of formulae is one of the most important objects at which early mathematical studies can aim. 120
In his Exercises in Algebra (1914), Nunn includes forty-five pages of exercises on the formula.
A. R. Congdon made recently a study at Columbia University to determine the mathematical training needed in high school for success in certain college subjects, especially physics, that do not presuppose a knowledge of college mathematics. One conclusion drawn from this study is
. . . that so far as subject matter is concerned, increased emphasis should be given to ... . manipulation of for- mulas using capital letters, Greek letters, primes, and subscripts. 121
Thorndike arranges the various algebraic abilities into five groups or sections in the order of their usefulness. In Section I appear:
Ability to read simple formulas, including knowledge of positive integral exponents, of fa or , {Ta or , of single parenthesis, and of the mission of the x sign and of 1 as coefficient. Ability to evaluate and solve such formulas, including the interpretation of negative numbers as answers, when the unknown forms one member of the equation.
Section II includes:
The ability to read formulas involving a parenthesis within a parenthesis and complex fractions .... The ability to transform formulas (after evaluation) where the desired number has a coefficient or is found in the denominator
Professor A. R. Crathome writes:
Great emphasis would be placed on the formula and all sorts of formulas could be brought in ... . The pupil should think of the formula as an algebraic declarative sentence that can be translated into English. Then evaluation leads up to the tabular presentation of the formula. Mechanical ability in the manipulation of symbols should be encouraged, through inversion of the formula. We have here also the beginnings of the equation, when our declarative sentence in changed to the interrogative. 124
120 Nunn, T. Percy: Op. cit., p. 63.
121 Congdon. A. H.: Training in High-School Mathematics Essentials for Success m Certain College Sub jec t s, p.
122 Thorndike, E. L.: Psychology of Algebra, p. 90.
I^s lbid.. p. 91.
12 %rathorne, A. R.: ”Required Mathematics, ” School and Society, vol. VI, No. IS2 (July 7, 1917),
How the Formula Game into Algebra Texts.
The development of the formula belongs to a very late stage in the development of mathematics. It requires a much higher form of thinking to see that the area of any.triangle can be expressed by A. ab than to find the area of a particular lot whose base is two hundred feet and whose altitude is fifty feet. Hence, it was very late in the race’s development that letters were used in expressing rules. 125
In algebra textbooks published prior to 1900, the formula was not considered as a topic, that is, there was no attempt made to cultivate the formula for its own sake. To be sure, a few formulas appeared, but they were considered only in such connections as:
1. The solution of interest problems (e.g., i - prt).
2. The solution of a quadratic equation by a formula.
3. The treatment of the topic ’’variation”.
4. The treatment of arithmetical and. geome trical progressions.
In textbooks published since 1900, the formula has come to occupy more and more space. The first two books examined in this study, Taylor’s Elements of Algebra (1900) and Tanner’s Elementary Algebra (1904), contained no material on the formula save that of the types described just above. J. V. Collins’ Practical Elementary Algebra (1908) was the earliest text examined in which the formula came in for special treatment. Collins includes in his book a chapter entitled ’’Applications of Algebra”, which involves, among other things, twen- ty-three exercises in framing and using formulas, ten exercises in finding nun® rical values from formulas, and five exercises in translating formulas to rules. Of the books examined only three of those published since 1908 fail to assign a special place to the formula. These three are Hedrick’s Algebra for Secondary Schools (1908), Schultze’s Elements of Algebra (1912), and Slaught and Hennes’ First Principles of Algebra (1912). In these books formulas appear solely as material for practice in connection with other topics. As a rule, algebra textbooks published since 1900 have chapters, or sections, dealing exclusively with the formula, and in addition formulas are used liberally in other sections of the books. Notable examples of these modern practices are to be found in the Rugg-01ark Fundamentals of High School Mathematics (1920) and in Wells and Hart’s Modern First Year Algebra (1923). The authors of tjhe latter book give first mention to the increased emphasis on the formula as one of the new features of the text. They say:
In Chapter I, the formula is made the means of transition from arithmetic to algebra.
It is emphasized throughout the text as its importance demands. 126
R., and Reeve, W. D.: General Mathematics, p. 296.
12 $Wells, W., and Hart, W. W.: Modern First Year Algebra, p. iii.
Definitions of the Formula.
Collin’s Practical Elementary Algebra (1908) is the earliest book of those examined to carry a definition of the formula. This definition is:
A formula is a rule or law expressed in symbols.
Practically all texts published since 1908 contain some sort of definition of the formula, most of them being almost identical with that given by Collins. A more elaborate form of definition is illustrated by the following:
A formula is a rule of computation expressed by means of arithmetic and literal numbers, connected by mathematical signs which s£ll what must be done with the numbers.
J. V.: Practical Elementary Algebra T p. 66.
128 Wells, W., and Hart, W. W.: o£. art., p. 3.
What the Topic "Formulas" Includes.
Nunn
writes:
Cultivation of formulas involves, four distinct elements: (a) practice in analyzing arithmetical processes and rules of procedure; (b) practice in symbolizing results of analysis; (c) practice in interoreting given pieces of symbolism; (d) practice in ‘substitution’. The first two constitute the art of formulation; the second two the art of using formulae.
This statement appears in a book published in 1914, and no more complete characterization of the scope of this topic of formulas has been found. In fact, Nunn, in his theory and in his practice, was somewhat in advance of his American contemporaries. It was at least 1920 before American textbooks began to treat formulas as fully as Nunn had done in his Exercises in Algebra (1914). This is particu larly true of what Nunn calls the “art of formulatio 11 , by which he means the making of formulas, the expressing of relations, or ’’rules of computation”, in symbolic form.
Nunn and later American authors make much of the process of changing the subject of a formula. This phrase refers to such operations as are involved in solving the formula t - l/l~ for 1 o.r for g.
The extent to which modern algebra textbook writers have covered the field marked out by Nunn can be indicated by setting down some of the phases of the formula treated in two volumes of a series of books published in 1929.
From Schorling and Clark, Modern Mathematics Eighth School Year: Pages
1. Expressing relations between numbers of formulas 1 2. Making formulas 3
♦
Pages 3. Graphing formulas 2 4. Interest formulas 2
From Schorling and Clark, Modern Algebra First Course:
Topic Pages 1. Practice with formulas 10 2. Graphing formulas ...... 5 3. Solving formulas ...... 2 4. Solving formulas by axioms ... 4 5. Centigrade and Fahrenheit thermometers 3 6. Fractions in formulas ..... 3 7. Equations and formulas 1 8. Construction of formulas .... 3 9. Solving formulas ...... 1 10. Variation and proportion .... 36 (Formulas used throughout this topic) 11. Evaluating formulas ...... 1
In addition to these phases of formulas, the cumulative reviews contain much material involving miscellaneous formulas.
129 Nunn?, T. Percy: The Teaching of Algebra, p. 63.
Graphs.
of the stress placed by some authors upon the practice of representing graphically the relationships of a formula. Apparently, there are two reasons for giving this such prominence. First, the graph clarifies and makes more significant the real meaning of the formula, and second, the graph frequently enables one to read at sight the value of one element of the formula when the value of the other is given or known. Edgerton and Carpenter’s First Course in Algebra (1923) and Schorling and Clark’s Modern Algebra First Course (1929) contain excellent treatments of gm. phing formulas. In the former text the “thermometer” formula is graphed and discussed in a very effective manner (p. 217). Schorling and Clark develop, explain, and illustrate the steps to be taken in graphing a formula. They are:
1. Making a table of pairs of numbers that will satisfy the formula. 2. Locating points on the graph paper. 3. Connecting the points. 4. Checking the graph.
Perhaps special mention should be made
Schorling, R., and Clark, J. R.: op. cit_., pp. 19 - 21.
Some Special Devices for Teaching Formulas.
Authors of modern algebra textbooks give considerable attention to the effective teaching of formulas. For example, it is a common practice to intro duce the subject by attempting to show the pupil the very great superiority of the formula over a cumbersome rule. An illustration of this is shown in the following:
’’Shorthand” Method or The Formula.
Ordinary Word Method
1. A rule for making coffee is 1 tablespoon for each two persons and 1 for the pot.
t_n + 1
2. The cost in cents of sending a telegram with more than 10 words in it is found by multiplying the number of words by 2 and adding 15.
c - 2n + 15
3. The number of units in the perimeter of a square equals four times the number of units in the diameter.
P =4s
4. The number of units in the circumference of a circle is 3 1/7 times the number of units in the diameter.
C' 3 D
5. The number of cents postage required to send a parcel into the third zone is ob- tained by multiplying the number of pounds in its weight by 4, grid adding 3 to this result.
C x 4w 3
6. The number of square units in the area of a rectangle is equal to the number of linear units in its length, multiplied by the number of linear units in its width.
A c Iw
Similar devices are to be found in Drushel and Withers 1 Junior High School Mathematical Essentials Eighth School Year, p. 62 and in Smith, Foberg, and Reeves’ General High School Mathematics, Book One, p. 55.
Formal tests on formulas, some of them timed and standardized, appear in most of the books published since 1920, especially in those written by Rugg and Clark, Smith, Foberg, and Reeve, and Schorling and Clark. A unique form of such a test, or possibly a drill exercise, is found in Durrell and Arnold and is presented here:
Copy the following tabulation and fill in the vacant place with the needed rule or formula:
Subject Brief Rule Formula Area of Floor No. sq. yd. in floor (no. ft. in length) x (no. ft. in width) 4- 9 Percentage b _ P vr Area of circle Area of circle equals TT times the radius squared Circumference Birr Division Dividend diminished by remainder product of divisor and quotient Multiplication M - P m Number of bricks, B = 6 in wall 11 Area of circle r _ 132
In order to cause the student to think about his formula, to see all of its possible relationships and implications, authors sometimes resort to thought questions involving hypothetical cases. Such questions are well exemplified by the following exercise in which a rule is to be expressed as a formula:
2. (a) The perimeter ... of a rectangle equals twice the sum of the base and altitude. (b) How many variables may the formula in (a) contain? (c) Draw a graph which will show the relation between the perimeter and the base of any rectangle whose altitude is 2 feet and whose base is not longer than 15 feet. (d) Draw a graph which will show the relation between the perimeter and the altitude of any rectangle whose base is 4 feet and whose altitude is not longer than 16 feet. (e) How is the perimeter affected if the base and the altitude of a rectangle are both doubled? Multiplied by 10? 133
Finally, one gets back to Nunn’s delineation of the whole field of cultivating the formula in
the following list of steps recommended to the student in ’’gaining control of the formula”:
Cultivating and gaining control of a formula means
1. Analyzing an arithmetical situation so as to see the rule of procedure. 2. Translating the rule into a formula. 3. Solving the formula for any letter in terms of all the others. 4. Evaluating the formula.
131 Schorling, R., and Clark, J. R.: op. cit., p. 5.
F., and Arnold, E. E.: A First Book in Algebra, p. 319.
J. A.: High School Mathematics p. 264.
134 Schorling, R., and Reeve, W. D.: General Mathematics, p. 279.
Chapter Summary.
Authorities in the teaching of algebra are agreed that working, interpretation, and application of formulas should occupy a prominent place in the first year course. In algebra textbooks published before 1900 the formula was considered only as an adjunct to other topics. By 1908 the topic formulas was allotted the place it deserves in textbooks. The formula is defined in various ways by writers of algebra texts, the most common definition stating, in substance, that the formula is a general law or rule expressed in symbols. The study of formulas is ordinarily made to include the construction or making of formulas and the use of formulas, the latter consisting of inter pretation and substitution. Writers of modern algebra texts make much use of the graph in connection with the study of formulas. Other helpful devices are used, the most common being the contrasting of statements in ordinary language with the same statements in formula symbology, drill exercises in the form of incomplete charts, and practice in changing ordinary rules into formulas.
CHAPTER VII EXPONENTS
Purpose of the Chapter.
This chapter contains a report regarding the treatment of exponents in thirty-two textbooks published for the most part between 1829 and 1929. Three books were published prior to 1829 and are included merely to furnish a background for the study.
The method of collecting data was similar to that described in previous chapters of this thesis. Each textbook was examined somewhat minutely for material bearing on the subject of exponents. Care was exercised to locate and study not only special chapters or sections devoted to exponents but also isolated references to the topic in the form of definitions or illustrations.
Exponents in Two Early Textbooks.
Credit is given to Herigone, a French mathematician, for having grasped the idea of an exponent and for introducing a rather good notation. As early as 1484, another
French mathematician, Chuquet, had had some idea of an exponent and had written expressions involving a form of negative exponent and also the zero exponent. His ideas, however, did not spread far. Other attempts to introduce general exponents were made between that time and the time of Newton. To Newton must be given credit for having finally fixed the present form of writing the various kinds of exponents. 135
Simpson’s Treatise on Algebra (1800) and Ward’s Young Mathemat ician 1 s Guide (1762, eleventh edition) are selected as carrying examples of the treatment of exponents before 1829. Ward makes no mention of exponents in his preliminary definitions and descriptions of notation. His exercises for multiplication and division involve no higher power than the third, and the second and third powers of ”a”, for example, are written ”aa” and ”aaa”, respectively. In connec tion with the treatment of involution a sort of definition of ’’power” is given, although the term exponent is not used.
EXAMPLES
I<s K& 3 l$ y
a aa aaa aaaa aaaaa
-a aa -aaa -aaaa -aaaaa
the Root, or single Power Square, or second Power. Cube, or third Power. Biquedrat, or fourth Power Sursolid, or fifth Power.
1 2 3 4 5
Note, the Figures placed in the Margin, after the Sign (Q) of Involution, shew to what Height the Root in involved; and are called indices of the Power; and are usually placed over the involved Quantities, in order to contract the Work, especially when the Powers are any thing high.
j a ~ aaaaa a _ — _aaaaaa a 4 a 3 b 3
a = a a — aa a 3 — aaa a* x aaaa
Thus
And
136
The words ”any thing high” are interpreted by Mr. Ward as applying to the fourth and higher powers. Consequently, these powers of ”a” are written a/, a , a*, etc. Ward mentions fractional exponents only <2 incidentally in saying that root of a may I be obtained from a = a .
Simpson 1 s attitude toward exponents in interesting chiefly because of the terminology used. Throughout practically the entire book, the exponent terminology is avoided, and powers of letters are written by repeating the letter. Near the end of the book (pages 304 and 305) the author, half-apologetically, informs his readers that there is another way of ’’expressing the high powers of any Quantity”. After describing the new-fangled symbol, he resolves a certain ’’question” using the new terminology. This quaint discussion is shown in full in Plate XX.
135 Wells, W., and Hart, W. W.: Modern First Year Algebra (.1923), p. 301.
•I Ward, John: Young Mathematician 1 s Guide (1862), p. 155.
Most common method of treatment.
Examination of textbooks shows that there developed a certain logical order of treatment of the topic exponents. The order of events was about as follows:
1. Definition of power and exponent in a preliminary chapter.
2. The law of exponents for multiplication.
3. The law of exponents for division.
4. The law of exponents for involution.
5. The law of exponents for evolution.
Naturally, these items were parts of virtually every text. They did not follow each other immediately but were introduced at appropriate points in the development of the course. In later texts the practice of defining all terms in a preliminary chapter is not followed, and the definitions of “power” and “exponents” are introduced as they are needed.
In addition to these ccmmon elements in the treatment of exponents, there are in certain texts additional features. The most important are:
1. A special section or chapter on exponents.
2. Somewhat detailed treatment of fractional, negative and zero exponents.
Each of these is discussed further in the present chapter.
PLATE XX Early Appearance of Modern Exponent Terminology From Simpson’s Treatise on Algebra
Special Chapter on Exponents.
A few of the earlier books, in addition to the usual fundamentals of exponents, devoted a special chapter to “indices”. Colburn (1829) and Venable (1869) are examples of such books. But this practice became well-nigh universal during the period from 1890 to 1910. Hall and Knight, in their Elementary Algebra (1892), include a very theoretical discussion of the theory of indices. This special chapter appears in all but one of eight other textbooks examined in this study and published between 1890 and 1910. Few of the newer books contain such a chapter.
This special treatment varies from book to book in its degree of completeness. Usually from eight to twelve pages are included in this chapter, unless the subject of radicals is included in the chapter. The order of treatment is frequently as follows:
1. RecapituMion of the laws of exponents.
2. Generalization of the laws of exponents.
3. Proof of the laws of exponents.
4. Special and detailed treatment of fractional, negative, and zero exponents.
5. Multiplication and division with all kinds of exponents.
6. Many and varied exercises.
Examples of exercises presented by various authors are presented later in this chapter. As
to the ’’proof” of the generalized laws of exponents, it may be said that first-year students of algebra are almost certain to find them incomprehensible. Such proofs are found in the books of Hall and Knight (1892), Wentworth (1898), Taylor (1900), and Brooks (1909).
Almost all of the textbooks published since 1910 contain no proofs of the generalized laws of exponents. Wells and Hart refer to fractional, negative, and zero exponents by saying:
. ... it can be proved that negative, fraction, and zero powers of a number can be used like positive integral powers .... 137
137 Wells, W., and Hart, W. W.: Modern First Year Algebra (1923), p. 299.
Power and Exponent.
There is a point of definition on which writers of algebra textbooks differ to some extent. The easiest way to define a positive, integral exponent is to say that it
indicates the number of times the ’’base” has been multiplied by itself. Most authors do not qualify this definition, thereby making it apply to all exponents. When the student comes to the fractional, negative, and zero exponents, he must revise «his ideas of the meaning of exponents. He may be told by the author that
1 SR Power has lost its original meaning.
To multiply an expression by itself ”-3” or ’’O” times has no meaning.
A better practice is to put the student on his guard from the beginning. Wentworth says:
If the exponent is a whole number, it shows the number of times the given number is taken as a factor. 139
Even this statement fails to provide for negative exponents, which may be whole numbers, that is, integers.
Olney makes a sharp distinction throughout between exponents that are positive integers and other types. After explaining that m 2" means m x m, he says:
The pupils must not get the idea that all exponents mean just what has now been explained.
This is the case only when the exponent is a whole number without any sign, or with the 4 sign. 140
Beman and Smith’s shift from this "primitive idea of exponents" is ingenious:
THE MEANING OF THS NEGATIVE INTEGRAL EXPONENT
199. The primitive idea of power ( § 8) was
a product of equal factors. The primitive idea of exponent was the number which showed how many equal factors were taken. According to this primitive idea the
3 3d power of a meant aaa, written a , 2d power of £ meant aa, written
but there was no first power of a,, because that is not the product of any number of a’s, nor was there any zero power, fractional power, or negative power.
3 */ But since a means aaa, or a 4 a, and " aa, " a 3 +a, I 2. . ’ . it is reasonable to define a as a, or a * a, and, “ ” ” ” ” a° as 1, ” a 4a, H If I! If H H H a * ” i ” 1 -fa, a fi»nh»» a H 1 H i a? - H S- “t a, and, in general, to define a as 1 , n being a positive integer.
200. For this reason we define
a / to mean a,
a° " ” 1,
a'" " " 1 a*"
i 43 n being a positive integer and a not 0 .
Venable, G. S.: Elementary Algebra (1869), p. 187.
IS9 Wentworth, G. A.: New School Algebra (1898), p. 6.
4 %lnev, Edward: First Principles of Algebra (1884), p. 15.
W. W., and Smith, D. E.: Academic Algebra (1903 J p. 209.
Types of Exercises on Exponents.
Very decided changes are discernible in the kinds of problems or exercises offered by the various authors. In general, this evolution has been from the simple to the complex and back to the simple. About the middle of the century, 1839 - 1929, the exercises began to come in more and more complex form, some of them involving unusual and even absurd operations with all sorts of exponents. Apparently, the idea of the probability of use in practical operations was lost sight of entirely.
Some of the most startling of these exercises are presented here:
* -z_ 2'”' x 2"" 7
(Hall and Knight (1892) p. 234.)
MH [mt
(Beman and Smith (1S02) p. 221.)
MW
(Milne (1908), p. 241.)
(Brooks (1909), p. 23?.)
Such complicated exercises serve ’’manipulative” purposes only, but they are fairly concrete in comspa.rison with the following:
o / '6 „ 2*- . , 3 D U = Q D XT y Z 56 z
(Hackley (1846), p. 55.)
It is difficult to see how a circle, a new moon, and a third symbol which defies description, and with all three involved as they are here, could have any, connection with the quantitative affairs of daily life. Such an exercise could spring only from the theory that any kind of practice material is acceptable, regardless of whether it has any application to practical affairs.
Thorndike’s Junior High School Mathematics, Book Three (1926), which is an algebra, text for the ninth grade, furnishes an example of the saner selection of drill and illustrative material. In fact, simplicity and practicability are the outstanding characteristics of the treatment of exponents in this book. For example, the student is told:
Exponents like 5 and 3 will rarely 2 de met. Treat £hem just as any exponents are treated.
In this book, the sections dealing with exponents are filled with an abundance of simple exercises, most of them numerical. Examples of Thorndike’s problems involving exponents follow:
IS'T =-20
3y z z" x syz* -15 y s z
(2 x y 1 z / = ?
The Schorling-Clark Modern Algebra First Course (1929) gives another illustration of present day practices, insofar as exponent exercises are concerned. The numerical phase of the subject is stressed, and all of the exercises are simple and practical.
It is to be noted especially that in these modern books not much stress is placed on fractional, negative, and zero exponents. For example, in Durrell and Arnold’s First Book in Algebra (1919), the treatment of exponents in the main body of the text is limited to definitions of power and exponent and statements of the laws of exponents for multiplication and division. Even the laws for involution and evolution are omitted. In a supplementary chapter entitled ’’Additional Topics”, literal and fractional exponents are treated briefly and simply. There is no mention of zero and negative exponents.
142 Thorndike, E. L.: Junior High School Mathe matics. Book Three, p. 133.
Chapter Summary.
The present notation in the use of exponents is found in typical English and American algebra textbooks published during the century 1829-1929. In the English textbooks immedi- ately preceding this period, this notation was not common. Most textbooks published during the period from 1829 to 1929 followed a rather rigid order in presenting the various phases of the subject of exponents. In textbooks published from 1890 to 1910, there is noticeable a decided tendency to give a special chapter to exponents and to lay stress on zero, negative, and fractional exponents. The majority of textbook writers define an exponent as something that indicates the power to which a base has been raised. A few authors, however, point out from the beginning that an exponent does not always indicate a power. Chronology seems to have no bearing on these differences in definitions.
In textbooks published from about 1875 to about 1910, the problems and exercises involving exponents tended toward the extreme in complexity. In modern textbooks, the exercises are much simpler and have a much more practical nature.
CHAPTER VIII GENERAL SUMMARY
An examination of algebra textbooks published from 1829 to 1929, inclusive, leads to the following conclusions:
1. Early French algebra textbooks were inclined to be theoretical and logical, while contemporaneous English algebra textbooks were of a more practical nature.
2. These characteristics seem to have entered into American algebra textbooks from French and English sources.
3. Inequalities, imaginaries, and highest common factor and lowest common multiple have practically disappeared from the first year algebra course.
4. Notation and definitions, fractions, and quadratic equations remained almost unchanged as to position and amount of space, during the period, 1829-1929, inclusive.
5. Simple equations come earlier in the course than was formerly the case, the four operations come later, and radicals play a smaller role than formerly.
6. Negative numbers began to appear as a new topic about 1880, graphs about 1890, and formulas in the modern form about 1910.
7. A considerable number of modern elementary algebra textbooks follow the findings of modern psychology in the selection and arrangement of topics
8. Ideas and practices involving negative numbers developed very slowly. Until about 1850, most writers of algebra texts had vague ideas concerning the reality and size of the negative number, and they confused it with the subtraction operation.
9. The algebraic scale of numbers, represented geometrically, appeared in textbooks about 1880.
10. Much ingenuity is shown by textbook writers in the. use of devices for teaching the negative number concept.
11. All authors are agreed as to the great import ance of the equation in elementary algebra.
12. Usually the pupil is led to ’’discover' 5 the axioms governing the reduction of an equation.
13. Since 1315 textbook writers have organized the material on equations more in accordance with the laws of learning.
14. Simultaneous linear equations are solved most
commonly by the ’’addition or subtraction” method; also by graphs since about 1900.
15. Quadratics have been solved most frequently by the first method of completing the square, also by factoring and graphs since about 1900.
16. The attention given to simultaneous quadratics has been greatly reduced.
17. Formulas are assigned to a special chapter by the authors of practically all algebra textbooks published since 1908.
18. The study of formulas is usually made to include the construction of formulas and their use. The latter consists of interpretation and substitution.
19. Many devices for teaching the use of formulas are employed by textbook writers. Most of these feature in one way or another the value of the formula as a time saver.
20. Algebra textbooks published during the period 1829-1929 show the use of the present exponent notation. Cruder notations were found in English textbooks published just prior to this period.
21. In textbooks published from 1890 to 1910, there is a tendency to treat exponents as a separate topic used to put great stress on elaborate exercises involving zero, negative, and fractional exponents.
22. The most common definitial of an exponent is that it is something that shows the power to which a base is raised. A minority of authors point out from the beginning that an exponent does not always indicate a power.
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I
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Schuyler, Aaron: A Complete Algebra, Van Antwerp, Bragg and Company, New York, //83.
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