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PRUITT, Ralph Lewis, 1914- AN ANALYSIS OF TYPES OF EXERCISES IN PLANE GEOMETRY TEXTS IN THE UNITED STATES FROM 1878 TO 1966.
The Ohio State University, P h .D .,1969
Education, historyJ
University Microfilms, Inc., Ann Arbor, Michigan
(2) Copyright by RALPH LEWIS PRUITT
1969 AN ANALYSIS OF TYPES OF EXERCISES IN PLANE GEOMETRY TEXTS IN THE UNITED STATES FROM 1878 TO 1966
DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By Ralph Lewis Pruitt, A.B., M.S. * * * * -x- -x-
The Ohio State University 1969
Approved by
,tion ACKNOWLEDGMENTS The writer wishes to express his sincere gratitude to the many persons who offered encouragement and assistance during the period of this research. I must extend a special note of gratitude to Dr. Nathan Lazar whose unique suggestions and timely advice have made the completion of this research possible. Finally, I wish to express to the members of my family my most sincere thanks for the love, understanding, and assistance I received from each of them during the entire period of study and work which led to the completion of this research. VITA
September 24, 1914 . . . Born - Athens, Georgia ' 1936 ...... A.B., Talladega College, Talladega, Alabama 1936-1941...... Mathematics Instructor in Public Senior High Schools in Georgia
1947 ...... M.S. Atlanta University, Atlanta, Georgia 1947-1962...... Assistant Professor - Associate Professor - Chairman, Department of Mathematics, Albany State College, Albany, Georgia 1963-1 9 6 6 ...... Teaching Assistant, Department of Mathematics, The Ohio State University, Columbus, Ohio 1964-1966 ...... Assistant Professor, Department of Mathematics, Capitol University, Columbus, Ohio 1966 Assistant Professor, Department of Mathematics, Cleveland State University, Cleveland, Ohio
PUBLICATIONS "Some Observations in Elementary Science." Georgia Teachers Herald, April 1955. "The Promotion and Teaching o.f Mathematics in Secondary Schools and Colleges." Faculty Studies Bulletin, Albany State College, May 1957. iii VITA
FIELDS OF STUDY Major Field: Education Studies in Mathematics Education. Professors Harold P. Fawcett and Nathan Lazar Studies in Higher Education. Professor Everett R. Kircher Studies in Secondary Education. Professor Jack R. Frymier
iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii VITA iii
LIST OF TABLES viii
LIST OF GRAPHS ix LIST OF ILLUSTRATIONS x Chapter
I. INTRODUCTION 1 Types of Exercises in Plane Geometry Textbooks History of Exercises in Plane Geometry Textbooks Purpose of the Study ■ Importance of the Study Description of the Procedure Limitations of the Study Related Historical Investigations Chapter Summary Organization of the Remainder of the Study
II. SIGNIFICANT RECOMMENDATIONS 3 REPORTS AND PROGRAMS 17 Recommendations and Reports from 1900 to 1922 The Perry Movement Recommendations Contemporary with the Perry Movement The Reports of the International Commis sion on the Teaching of Mathematics The Report of the National Committee of Fifteen Summary Recommendations and Reports from 1923 to 1939 The Report of the National Committee on Mathematical Requirements v TABLE OF CONTENTS
Chapter Page The Advent of Emphasis on Patterns of Reasoning in Everyday Life Studies that May Have Influenced Emphasis on Everyday-Life Exercises in Plane Geometry Texts Summary Recommendations and Reports During the 1940's The Report of the Joint Commission of the Mathematical Association of America and the National Council of Teachers of Mathematics The Report of the Committee on the Function of Mathematics in General Education The Report of the Committee on Essential Mathematics for Minimum Army Needs The Report of the Commission on Post- War Plans Summary Recommendationsj Reports^ and Experimental Programs of the 19501s The Trend Toward Fewer Everyday-Life Exercises About Physical Situations in Plane Geometry Textbooks The Committee on School and College Study of Admission with Advanced Standing The Commission on Mathematics of the College Entrance Examination Board The Secondary School Curriculum Committee Other Forces Which Probably Affected Everyday-Life Exercises About Physical Situations and Patterns of Reasoning in Plane Geometry Textbooks
Experimental Programs The University of Illinois Committee on School Mathematics The School Mathematics Study Group Reports Made in the 196o*s The Report of the International Congress of Mathematicians The Report of the Cambridge Conference on School Mathematics vi TABLE OF CONTENTS Chapter Page Chapter Summary
III. REPORT OF ANALYSES 1...... 47 The Wentworth* Wentworth-Smith* Welchons-Krickenberger* Welchons- Krickenberger-Pearson Textbooks Chapter Summary IV. REPORT OF ANALYSES I I ...... 74 The Stone-Millis-Mallory-Meserve- Skeen Textbooks Chapter Summary V. REPORT OF ANALYSES III...... 86 The Wells-Hart-Schult-Swain Textbooks Chapter Summary
VI. REPORT OF ANALYSES I V ...... 100 Selected Textbooks* 1903 to 1965 Chapter Summary VII. REPORT OF ANALYSES V ...... 130 Selected Textbooks* 1963 to 1966 Chapter Summary VIII. GENERAL SUMMARY AND CONCLUSIONS...... 142 Summary Conclusions Suggestions for Further Study Implications
BIBLIOGRAPHY...... l48
vii LIST OF TABLES Table Page 1. A Comparison of the Total Number of Exercises and the Number of Applied Exercises in 28 Representative Plane Geometry Texts...... 6 2. Combined Data from the Analyses of the Wentworth-Smith-Welchons-Krickenberger- Pearson Series ...... 69 3. Combined Data from the Analyses of the Stone-Millis-Mallory-Meserve-Skeen S e r i e s ...... 83 4. Combined Data from the Analyses of the Wells-Hart-Schult-Swain Series ...... 97
5. Combined Data from the Analyses of Selected Texts - 1903 to 1 9 6 5 ...... 125 6. Combined Data from the Analyses of Selected Texts - 1963 to 1 9 6 6 ...... 139
viii LIST OF GRAPHS
Graph Page I. A Comparison of the Everyday-Life, Physical Situation, and Patterns of Reasoning Exercises from Data of Table 2 ...... 71 II. A Comparison of the Everyday-Life, Physical Situation, and Patterns of Reasoning Exercises from Data of Table 3 ...... 84 III. A Comparison of the Everyday-Life, Physical Situation, and Patterns of Reasoning Exercises from Data of Table 4 ...... 98 IV. A Comparison of the Everyday-Life, Physical Situation, and Patterns of Reasoning Exercises from Data of Table 5 ...... 127 V. A Comparison of the Everyday-Life, Physical Situation, and Patterns of Reasoning Exercises from Data of Table 6 ...... l4o
ix LIST OF ILLUSTRATIONS
Figure Page 1. Two Congruent Triangles ...... 2 2. Two Parallel Lines Cut by a Transversal . . 2 3. A Triangle Along a Ruler...... 52 4. Design of a Circular Window ...... 53 5. Parallel Lines Formed by Moving a Carpenter's Square Along an Edge of a B o a r d ...... 75 6 . A Design of a Maltese C r o s s ...... 90 7. A Trefoil Inscribed in a Circ le...... 101 8 . Similar Triangles - Measurement of an Inaccessible Distance ...... 102
9. A Log from Which a Rectangular Beam is C u t ...... 102 10. Similar Triangles - Measurement of an Inaccessible Distance...... 108
x CHAPTER I
INTRODUCTION Types of Exercises in Plane Geometry Textbooks
Most modern geometry textbooks contain three kinds of exercises. These exercises are: (1) the formal geometric type which are sometimes referred to as originals; (2 ) applications to physical situations in everyday life; and (3 ) applications to valid and invalid patterns of reasoning in everyday life. Some writers refer to the latter two types of exercises as practical applications. They originate from everyday-life situa tions and thus could adequately be called everyday-life exercises. Formal geometric exercises are those which present a statement or statements of given geometric information and a geometric statement or statements to be proved. Applications to physical situations in everyday life are those which contain facts about real life to which geometric definitions, concepts, and theorems are to be applied. Applications to valid and invalid reasoning in everyday life are those which deal with patterns of reasoning about real life situations, which do not have any mathematical or physical content. Exercises are usually based on the theorems and the axioms of the texts. The formal geometric exercises often make use of geometric figures and they are solvable by means of definitions, axioms and theorems. Some of them are actually minor theorems of formal geometry. Examples of this type of exercise are the following:
Given: BD bisects /B, AB = BC Prove : A A B D = ACB d Ti
2> Figure 1.— Two congruent triangles In the adjacent figure, we know that
/I - /2 = 1800 and /I = /3 /3 - 72 = T8 0 ° State the axiom applied.2
Figure 2.— Two parallel lines cut by a transversal
1William G. Shute, William W. Shirk, and George F. Porter, Plane Geometry (New York: American Book Company, 1957), p. "3'5" ---- 2 Ibid., p. 22. 3 Applications to physical situations in everyday life are taken from such areas as industry, art, nature, engineering, and surveying. Examples of this type of exercise are: If a ladder, whose foot rests on a horizontal plane and top against a vertical wall, slips down find the locus of its middle point.3 An orchardist, setting out trees, first places a tree at each end of a row. How then may he locate the other trees of that row so that they will be in a straight line, without, stretching a rope between the end trees ?4 Applications to valid and invalid patterns of reasoning in everyday life usually present first some statements of fact about a nonmathematical situation and those statements are sometimes called premises. From these statements conclusions may be derived. Examples of this type of exercise are: The following exercise represents a simple reasoning situation. Is the conclusion good? Poor? Doubtful? Explain your rating. The Boy Scouts are to be admitted free to the football game Saturday. Jim isn't a scout so he will not get in free.5
^G. A. Wentworth, Elements of Geometry (Boston: Ginn and Company, 1 8 7 8 ), p. 127. ^Webster Wells and Walter W. Hart, Modern Plane Geometry (Boston: D. C. Heath and Company, 192bJ, p. 24. 5Leroy H. Schnell and Mildred Crawford, Clear Thinking - An Approach Through Plane Geometry (ifew York: iiarper and Brothers, 193^); P- l£. 4 If we accept what is given (the "if" statements) as true in the following exercise, which conclusion ("then" statements) must also therefore be true? IP (a) On long trips Grace never drives after it is dark enough to require car lights, and if (b) She would like to reach home as soon as possible to start her vacation, and if (c) She still has 120 miles to go when the sun sets; THEN (a) She will drive the remaining distance that night. (b) She will take a room for the night and complete the trip the following morning.° Exercises belonging to this category will often be referred to in this study as "exercises on patterns of reasoning."
History of Exercises in Plane Geometry Textbooks Early geometry textbooks contained no exercises at all. Euclid's Elements which were used as the major geometry texts for almost 2000 years had no exercises. Exercises began to appear in geometry textbooks after the middle of the nineteenth century and their number gradually increased in textbooks published after that time. The first exercises included in geometry texts were the formal geometric type. By 1900 at least 1000 exercises of this type were included in geometry textbooks. By 1930 about
6 Ibid., p. 1 9 . 2000 exercises were included in most geometry texts, some of which were of the second type mentioned above. By i960 many geometry texts included 3000 or more exercises, some of which were of the second and third types mentioned above. Table 1 appears in a study completed by Shibli? in 1932. Shibli made a comparison between the total number of exercises and the number of applied exercises in 28 representative plane geometry texts and found that Playfair's Euclid, published in 1843, had no exercises. He found Olney's text of 1872 to contain 287 exercises, 27 of which were applied exercises. He also found that certain books published at the beginning of the twentieth century contained 1000 or more exercises but no applied exercises and that each of certain books published in 1 9 2 9 , 1930, and 1931 contained over 2000 exercises with the number of applied exercises being 2 3 9 * 112, and 107 respectively. Hassler and Smith, in discussing a typical geometry text published in 1862, state that "On the final pages of the book there was assembled a list of 'Exercises in Euclid', 417 in all, which were other theorems of pure
?J. Shibli, Recent Developments in the Teaching of Geometry (State College, Pennsylvania: J. Shibli Publish ing Company, 1932), p. 1 6 5 . TABLE 1
A COMPARISON OF THE TOTAL NUMBER OF EXERCISES AND THE NUMBER OF APPLIED EXERCISES IN 28 REPRESENTATIVE PLANE GEOMETRY TEXTBOOKS
Total Year Author Number Applied Exercises Exercises
1843 Playfair's Euclid 00 00 1870 Chauvenet 2 8 l 00 1872 Olney 237 27 1875 Davis 90 00 1877 Wentworth 47 00 1886 Wells 4oi 00 1888 Wentworth 479 00 1899 Wentworth 603 00 1899 Milne 737 00 1901 Schultz - Sevenoak 1032 00 1904 Dure11 1205 00 1908 Wells 303 00 1910 Stone - Millis 800 206 1910 Wentworth - Smith 1027 70 1913 Schultz - Sevenoak 1552 122 1916 Durell - Arnold l64l 125 1916 Stone - Millis 1179 224 1916 Wells - Hart 1348 48 1923 Smith, D. E. 9^5 124 1925 Schultz - Sevenoak - Schuyler 1357 125 1925 Seymour 1644 94 1926 Wells - Hart 1674 94 1926 Clark - Otis 1676 90 1927 Strader - Rhoads 1959 118 1929 Hawkes - Louby - Touton i4oi l4o 1929 Stone - Mallory 2156 239 1930 Durell - Arnold 2015 112 1931 Morgan - Foberg - Breckenridge 2103 107 7 geometry.."® In discussing a book published in 1912 which was used in many schools they say that the book contained "... 400 exercises that relate to abstract or formal geometry, and 470 other exercises that include simple questions, solutions of simple equations, and about a score of applications to trades and industries."9
Applications to physical situations in everyday life began to appear in geometry texts at the beginning of the twentieth century and their number gradually increased until the middle of the century. Applications of valid and invalid patterns of reasoning in everyday life began to appear in geometry texts about 1938 and they continued to increase in number in texts published from that time to the present.
Purpose of the Study This study is focused on exercises in plane geometry textbooks that are: (1) applications to physical situa tions in everyday life; and (2) applications of valid and invalid patterns of reasoning in everyday life. Special attention is given to the applications of valid and
®Jasper 0. Hassler and Rolland R. Smith, The Teaching of Secondary Mathematics (New York: The Macmillan Company, 1937), p. 314. 9Ibid., p. 315. invalid patterns of reasoning in everyday life. Exercises of the above two types will hereafter be referred to as "everyday-life exercises." The purpose of this study is to determine the trend of everyday-life exercises as a part of the content of 44 plane geometry textbooks dating from 1878 to 1 9 6 6 . Certain questions naturally arise in the mind of one whose interest is in this direction. Among them are the following: 1. What are some of these exercises? 2. How may they be classified with respect to everyday-life activities? 3. How many of them are included in various plane geometry textbooks? 4. What percentage are they of the total number of such exercises in a given plane geometry textbook?
5. Do they possess equal emphasis with exercises dealing with logical proof and abstractions? 6 . What percentage of the exercises from everyday life are exercises on patterns of reasoning?
Importance of the Study
In 1938 a committee of the Mathematics Section of the Society for the Promotion of Engineering Education collected a list of mathematical applications to engineering. In 1942 the National Council of Teachers of Mathematics published its Seventeenth Yearbook, entitled A Source Book of Mathematical Applications, which included an extensive list of everyday-life exercises arranged according to topics usually treated in various mathematical subjects. There seems to have been no attempt on thd part of anyone to trace everyday-life exercises in plane geometry over an appreciable period of years. Shibli's work indicated only a beginning of such exercises in plane geometry textbooks. Research designed to do this for plane geometry that includes texts now being used in many of our secondary schools can be helpful to professional educators as they plan plane geometry courses. Classroom teachers can make use of the exercises included in this research to maintain pupil interest and to motivate learning in plane geometry courses. Students of mathematical history can gain some grasp of the developments in plane geometry from the
research, and textbook writers may find it a helpful source for material designed to develop mastery of mathematical processes and principles in plane geometry.
Description of the Procedure Analyses of 44 plane geometry textbooks dating from 1878 to 1966 have been made to determine the trend in the percentage of the total number of everyday-life exercises that are applications to physical situations and also the 10 percentage of the everyday-life exercises that are applications of patterns of reasoning. The analyses have been made in separate chapters. Chapters III, IV, and V present examinations of early and subsequent editions of the same text. When editions by the original author or authors ceased to be published, editions were selected for examination that were based on the earlier editions or considered as their successors. For example, in Chapter III, the G. A. Wentworth series of plane geome'try texts, dating from 1878 to 1 8 9 9 , are examined. Succeeding texts to the Wentworth series, dated 1910, 1911, and 1913 by George Wentworth and David Eugene Smith are examined. The Welchons-Krickenberger texts and one by Welchons, Krickenberger, and Pearson, dating from 1933 to 1 9 6 1 , are examined as successors to the Wentworth and the Wentworth-Smith series. Considera tion is given only to the sections on plane geometry where textbooks include sections on both plane and solid geometry. Chapters VI and VII consist of examinations of two sets of textbooks, most of which are by different authors. These examinations are made to determine whether the trend of everyday-life exercises in these books tends to agree with the trend observed in the analyses of Chapters III, IV and V. Books that were available, not necessarily all texts published during the period of the study, were used. Classifications of the exercises and where they may be found in the textbooks are given with the examination of each individual textbook. The classification of an exercise is determined from its content. For instance, an exercise dealing with the reflection of light or with revolutions and rates in motion situations, is classified as a physics exercise. One dealing with air planes and flying is classified under aeronautics. Exercises which seemed to fit no particular classification are considered miscellaneous. A summary of the data on the exercises and their classifications conclude the individual examinations. Summarizing tables are included at the end of the examinations of each series of texts showing authors' names, titles of texts, dates of publi cation, the total number of exercises, the number and percentage of the total number of exercises that are everyday-life exercises, the number and the percentage of the everyday-life exercises from physical situations, and the number and the percentage of the everyday-life exercises that are classified as exercises on patterns of reasoning. Graphs are also used to compare the summary data for each analysis. Tabulations of the analyses may be studied in the office of Dr. Nathan Lazar, Professor of Education, The Ohio State University, Columbus, Ohio. Only summaries and comments on these tabulations will appear in the 12
study. Each of the percentage computations has been made to the nearest tenth percent.
Limitations of the Study This study has at least four obvious limitations. First * the classifications of the everyday-life exercises are not mutually exclusive. For example* the following exercise might be classified as an exercise on measure ment of an inaccessible distance or it might be classified as a surveying exercise : The shadow of a chimney is 36 yards long. At the same time the shadow of a stake 2 yards high is 1.5 yards long. How high is the chimney?10
Second* no consistent attempt was made to sub class ify the exercises on patterns of reasoning into their various categories. Third* only textbooks published in the United States are examined. Fourth* the study is limited to the years 1878 to 1 9 6 6 .
Related Historical Investigations
Numerous historical investigations have traced certain developments in the teaching of plane geometry and a few of them have used the method primarily employed in this study. Historical investigations that presented
10Webster Wells and Walter W. Hart* Modern Plane Geometry (Boston: D. C. Heath and Company* 192b)* p. 166. 13 some findings relating to everyday-life exercises in plane geometry are reviewed here. Shibli's11 work has been mentioned; it included an analysis of the aims that were receiving emphasis in the teaching of plane geometry. He found that "appreciation of the beauty of geometric form in nature, art, and industry" and "the development of'mental habits and attitudes needed in life situations" were important aims. Lundberg12 synthesized a review of the literature on geometry as a high school subject in the United States and the findings from an analysis of 65 textbooks copy righted between 1821 and 1950 to trace significant influences affecting geometry as a secondary school subject. He found that there was a trend toward establishing a concrete basis in plane geometry through more introductory material, concrete illustrations, more numerical and applied exercises, and an increase in the number of illustrations and applications to life situations.
Hlavaty13 traced the changes in the philosophy and
llShibli, pp. 213-217. 12Harold G. Lundberg, "Significant Influences Affecting Geometry as a Secondary School Subject." (unpublished Ph.D. dissertation, George Peabody College, 1951).
■^Julius H. Hlavaty, "Changing Philosophy and Content in Tenth Year Mathematics." (unpublished Ph.D. disserta tion, Department of Education, Columbia University, 1950). 1 4 content of tenth year mathematics, investigated their / f causes, analyzed theirfoperation, and evaluated their results using authoritative recommendations and an ■ ' \ f analysis of 18 representative textbooks copyrighted from 1937 to 19^8. Twelve of the textbooks contained exercises on life situations. Two of the aims emphasized in the average plane geometry textbook were; (1) to develop the power to think in life situations as well as in .geometry; and (2) to show the role of geometric facts, principles, and methods in life and nature. A typical text, he concluded, would include a broad selection of applied problems ranging from the carpentry shop type to nuclear physics, accompanied by clear explanations and first-rate illustrations, and a section devoted to clear thinking in everyday situations. Pitzerl4 analyzed five textbooks, copyrighted from
1940 to 19513 and the Seventeenth Yearbook of The National Council of Teachers of Mathematics (1942) to determine-the definitions, postulates, and theorems essential to the proof of the applications, and to list the concepts from mathematics and other fields which must be understood to solve the application exercises. Among
l^George Lott Pitzer, "An Analysis of the Proofs and Solutions of Applications in Geometry" (unpublished Master's Thesis, Dept, of Education, Ohio State Uni versity, 1 9 5 3 ). his findings were the following: 1. The applications available in textbooks and source books are many and varied, but an abundance of applications does not guarantee a variety since many are alike except for wording. 2. Many nonmathematical understandings are needed in working with applications. 3. Not all textbooks include the information needed to prove the applications they contain, but many include information not used in applications. 4. Plane geometry textbooks vary greatly in the number of applications they contain. Some of the findings of the above investigations will be substantiated in the pages which follow, but in general the findings of these investigations do not accurately reflect the trend of everyday-life exercises in plane geometry textbooks.
Chapter Summary In this chapter, background information pertinent to the study has been discussed - the purpose, the importance, the limitations, a description of the procedure of the study, as well as related historical investigations, have been presented.
Organization of the Remainder of the Study Chapter II of the study presents certain committee and individual recommendations and reports that have some relevance to this study. Chapters III^ IV* and V present textbook analyses and deal with examinations of 25 plane geometry textbooks by one author or a team of authors. Fifteen are from the series by Wentworth^ Wentworth-Smithy and others published from 1878 to 1 9 6 1 ; five are from the series by Stone- Millis and others published from 1916 to 1959; and five are from the series by Wells-Hart and others published from 1915 to 1964. Chapters VI and VII present more analyses and deal first with 14 plane geometry textbooks by various authors from 1901 to 1965j and second^ with analyses of 5 text books by various authors from 1963 to 1 9 6 6 . The latter five were separately analyzed to gain special insight into what most recent texts reveal regarding everyday-life exercises. Chapter VIII presents the summary and conclusions of the study^ and a few suggestions are made regarding similar studies which might be undertaken. CHAPTER II
SIGNIFICANT RECOMMENDATIONS, REPORTS, AND PROGRAMS Recommendations and reports by committees and individuals and of several experimental programs have had much influence on the content of subsequently published plane geometry textbooks.
Recommendations and Reports from 1900 to 1922 The Perry Movement In 1901, John Perry, in an address before the British Association of Mathematics Teachers, attacked the use of Euclid as a text and made a strong plea for a complete divorce from its rigidity. He advocated: . . . steady emphasis on the practical uses of mathematics as found in mechanical drawing, physics, chemistry, and engineering, the teaching of the: general principles of mathematics through experi ments and concrete examples, the acceptance of many propositions in geometry without proof, or the deferment to a later time when logical proof could be established, and more extended use of graphical methods.1 Perry's recommendations began what later became
Central Association of Science and Mathematics Teachers, A Half Century of Science and Mathematics Teaching (Oak Park, Illinois : Central Association of Science and Mathematics Teachers, Inc., 1950), pp. 4o-4l, quoting from the Perry Speech.
17 18
known as the "Perry Movement" on everyday-life exercises
in the teaching of geometry.
Recommendations Contemporary with the Perry Movement Contemporary with Perry's recommendations were those of Tannery^ in Prance and by Klein3 in Germany. Tannery recommended that some objectionable proofs be replaced by common sense discussions and exercises and Klein recommended closer correlation between mathematical sub jects by use of the function concept and functional thinking in exercises. Influenced by the Perry Movement in England, Dr. E. H. Moore^ of the United States in an address before the American Mathematical Society in 1902 advocated emphasis on practical applications and the use of the laboratory method of instruction. Like Perry, he suggested the practical uses of many postulates in plane geometry and that the more philosophical treatment of propositions be given later. These recommendations represent the beginning of a
^Charles H. Butler and P. Lynwood Wren, The Teaching of Mathematics (3d ed.j New York: McGraw-Hill Book CoT, Inc., 19b0 ), p. l4. 3 Ibid., p. 14. ^Jasper 0. Hassler and Rolland R. Smith, The Teaching of Secondary Mathematics (New York: The THcmTTIan-T3o7T-T^371T™P^-™I^Bl 19 reform movement in the teaching of mathematics which aimed to prepare students for everyday life as well as for college mathematics. Some plane geometry textbooks published after this time could be assumed to contain more exercises of the everyday-life type.
' The Reports of the International Commission on the ’Teaching of Mathematics' The reform movement drew widespread attention. David Eugene Smith of the United States, suggested the appointment of an international commission on the teaching of mathematics. This commission was appointed by representatives from various countries in 1908 at Rome, Italy, and its reports were published in the United States between 1911 and 1 9 1 8 . An analysis of the reports by the American members of the International Commission reveals marked tendencies toward change in the teaching of plane
geometry had taken plac e . 5 Noteworthy among these
tendencies were efforts to omit proofs too obvious or too difficult and the endeavors to include more applications to problems from the sciences and practical life.
The Report of the. National Committee of Fifteen on Geometry Syllabus In the United States in 1909* the Secondary Depart ment of the National Education Association and the American Federation of Teachers in the Mathematical and
5]3utler and Wren, p. 11. 20
Natural Sciences authorized a joint committee of the association and the federation to study and report on the problem of a syllabus for geometry. The committee formed was called the National Committee of Fifteen on Geometry Syllabus. On the basis of its study the committee felt that there had indeed been a tendency to overemphasize difficult abstract applications of various theorems in geometry exercises. It recommended in its provisional report that geometry courses include more concrete exer cises, along with a reasonable number of abstract exercises, with careful attention to their distribution and gradation. It suggested that material for" concrete exercises could "be found in other subject fields, such as architecture, design, indirect measurement, and any other source available to the individual teacher. The final report of the committee, published in 1912, listed the following specific claims for the study of geometry: 1. Geometry is an exercise in logic, and in types of logic not generally met in other courses of the secondary school, and yet types which occur in usually simple settings and which are easily carried oyer into the actual affairs of life.'
6 Ibid., p. 1 3 . Citing The Committee of Fifteen: '^Provisional Report on Geometry Syllabus," School Science and Mathematics, Vol. XI, 1911* pp. 509-53TI 7Final Report of the National Committee of Fifteen on Geometry Syllabus, The Mathematics Teacher, Vol. V., No. 2 (December, 1912), p. 42. 21
2. Geometry cultivates space intuition and appreciation of and control over, forms existing in the material world, which can be secured from no other topic in the high school curriculum.8 The recommendations of this committee represent the first organized effort of mathematics educators in the United States to move toward practical values in the teaching of plane geometry.
Summary At the beginning of the twentieth century a reform movement that directed more attention to practical values in the teaching and studying of plane geometry started in Europe and later spread to this country. Perry, Tannery, and Klein contributed to the European movement. Moore and Smith made notable contributions in the United States. The Committee of Fifteen on Geometry Syllabus organized the effort in this country. The recommendations of these committees and individuals influenced textbook writers to include more exercises from everyday life and the practical sciences in their plane geometry textbooks.
Recommendations and Reports from 1923 to 1939 The Report of the National Committee on Mathematical Requirements The reform movement was well under way in this
8 Ibid., p. 43. 22 country by 1 9 1 6 , though not recognized in many sections. The Mathematical Association of America saw the need for a nationally coordinated and united effort and, in 1 9 1 6 , organized the National Committee on Mathematical Require ments for this purpose. The final report of this committee, entitled The Reorganization of Mathematics in Secondary Education, was popularly known as the 1923 Report. The widespread involvement and participation that the committee had encouraged gave its final report the distinction of being the first major national effort to improve mathematics instruction in this country.
The Report stated that the aims of mathematics instruction were practical, disciplinary, and cultural. These aims greatly influenced the number and categories of exercises that were included in subsequently published plane geometry textbooks, as particularly indicated by the following: 1. To develop familiarity with geometrical forms common in nature, industry, and life; 2. To develop understanding of the elementary properties and rela tions of these forms, including the mensuration of these forms; 3. The acquisition, in precise form, of terms in which the quantitative thinking of the world is done such as ratio and measurement and dependence of one quantity upon another; 23 4. The development of ability to think clearly in terms of such concepts; 5. The acquisition of mental habits and attitudes which will make the above training effective in the life of the individual; 6 . The development of understanding of the idea of relationship or dependence; 7. Appreciation of beauty in the geometrical forms of nature, art, and industry; 8 . Ideas of perfection as to logical structure, precision of statement and of thought, logical reasoning, discrimina tion between true and false, etc.; 9. Appreciation for the power of mathematics.9 The 1923 Report recommended that knowledge be acquired in terms of ideas and concepts used in quanti tative thinking needed in the everyday world. It also directed emphasis in the teaching of geometry toward appreciation of the beauty of geometrical forms in every day life. A significant outgrowth of the 1923 Report was the recognition of the fact that the concepts and the prac tice of deductive thinking, usually confronted in plane
9The National Committee on Mathematical Require ments, The Reorganization of the Mathematics in Secondary Education (boston: Houghton Mifflin and Company, 1923), pp. 9-io. geometry should and could be generalized and applied to problems of a nonmathematical nature. This development represented a trend away from the existing educational psychology that there was automatic transfer of training from one field to another and that the study of mathematics disciplined the mind toward logical reasoning in other situations. Thorndike and Woodworth10 published experimental results in 1901, long before the 1923 Report, showing that almost no transfer of training from one field to another occurred unless identical elements in both fields were involved. Thorndike's work appeared to encourage the teaching of only what might be immediately useful. Judd11 presented evidence in 19Q8, also before the 1923 Report, showing that transfer could occur under proper circumstances. In Chapter IX of the 1923 Report investigations made by Harold Rugg and Vevia Blair are reported which support the opinion that transfer is not automatic but that the teacher, the curriculum, and the method of teaching are factors in transfer. It appeared then that transfer was neither automatic nor inevitable,
10E. L. Thorndike and R. S. Woodworth, "The Influence of Improvement in One Mental Function upon the Efficiency of Other Functions," Psychological Review, Vol. VIII, (1901), pp. 247-261, 3«4-39$, 553-564. 11Charles H. Judd, "The Relation of Special Training to General Intelligence," Educational Review, Vol. XXXV, (1908), pp. 28-42. 25 but that certain ways of bringing it about were necessary. However* despite the findings of Thorndike, Woodworth, and Judd, various other positions were held among educa tors at the time of the 1923 .Report regarding transfer of training.
The Advent of Emphasis on Patterns of Reasoning in Everyday Life The 1923 Report gave new impetus to the controversy between those making claims for automatic transfer and those seeing the need to make use of adequate teaching procedures and curriculum content to facilitate transfer. During this controversy, the long held view with regard to disciplinary values in the study of plane geometry in the United States began to give ground to emphasis on practical aims. In fact a new dimension in the emphasis on practical aims began to receive attention. The major aim in the teaching of plane geometry for a long time had been to develop logical reasoning, but the 1923 Report indicated the beginning of a trend toward patterns of reasoning in everyday-life situations through better curricula and teaching methods. Consequently it could be expected that plane geometry textbooks published in the ensuing years would not only include more everyday-life exercises but exercises on patterns of reasoning in every day life. Meanwhile, as the controversy on transfer of training 26 continued, various mathematics educators made significant statements regarding the need for more exercises from everyday-life situations including those which might develop good reasoning in everyday life. Stone commented in 1930: Today every teacher of geometry would no doubt list the following aims: training in logical thinking, practical uses of the subject; its preparation for other mathematics and a certain cultural v a l u e . 12
Upton also said in 1930 that: Our great aim in the tenth year is to teach the nature of deductive proof and to furnish pupils with a model of all their life thinking.13
In 1933* Christofferson recommended "more emphasis on the system of formulated reasoning and its application to nongeometric as well as geometric situations."!^ Beatley, from responses to a questionnaire sent to 101 American and European mathematics teachers, found Almost unanimous agreement that demonstrative geometry can be taught so that it will develop the power to reason logically more readily than other
I2John C. Stone, "A One Year Course in Plane and Solid Geometry," The Mathematics Teacher, Vol. XXXVI,' (April, 1930), ^ C . B. Upton, "The Use of Indirect Proof in Geometry and Life," Ibid., p. 132.
l^H. C. Christofferson, Geometry Professionalized for Teachers (Menasha, Wisconsin': George Banta Publishing Company, 1933 )> p. 2. 27 subjects and that the degree of transfer of this logical training to situations outside geometry _ j. is a fair measure of the efficacy of the. instruction. -3 Fawcett said in 1 9 3 5 : . . . in view of the fact that any worthwhile transfer results only where there is conscious teaching for it, does it not seem desirable to extend the study of proof beyond the narrow confines of geometry and examine some of the countless propositions that are met in other areas of life?l6
Shanholtl7 noted a trend toward the'development of patterns of reasoning in the teaching of geometry and advocated emphasis on the processes of thought and reasoning used in the study of geometry which could be applied to one's everyday experience. In 1938,
Breslichl8 cited "ability to reason in life situations" as one objective in the teaching of geometry on which there seemed to be general agreement. Christoffersonl9
•^Raiph Beatley, "Third Report of the Committee on Geometry," The Mathematics Teacher, Vol. XXXVIII,(1935), p. 1 0 . "I Harold P. Fawcett, "Teaching for Transfer," The Mathematics Teacher, Vol XXVIII, (1935), p. 466. !7Henry H. Stanholt, "A New Deal in Geometry," The Mathematics Teacher, Vol. XXIX, (February, 1936), p p . 67-74.
r, Breslich, "The Nature and Place of Objec tives in Teaching Geometry," The Mathematics Teacher, Vol. XXXI, (November, 1938), p. '309. ^H. C. Christofferson, "Geometry as a Way of Thinking," The Mathematics Teacher, Vol. XXXI, (April, 1938), pp. I V T - m ------28
entered the controversy again in 1938 in an attempt to show how, when, and where the pattern of thinking in geometry could be and was used in nongeometric situations.
Studies That May Have Influenced Emphasis on Everyday-Life in Plane Geometry TexTs~ To close this discussion on the controversy on transfer of training four studies have been examined that probably had some bearing on the everyday-life exercises
included in plane geometry textbooks during this time. Lazar^0 accepted the point of view in a study completed in 1938 that transfer is a form of generalization and that the real problem of transfer is to organize training so that it will be carried over into other fields. He attempted to show that it is possible to utilize geometry as a medium for making the pupils conscious of the existence of logical patterns of valid and invalid reasoning in mathematics as well as in the thinking of everyday life. In agreement with the recommendation of the 1923 Report that "the practice of deductive thinking usually confronted in plane geometry should and could be generalized and applied to problems of nonmathematical nature," he suggested generalizations of the definitions
Nathan Lazar, The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of geometry, (New York: Nathan Lazar, 1 9 3 8 ) The Mathematics Teacher, Vol. XXXI, (1938), pp. 99-H3, 15'8-T74’,' 216-2'4'0. The Mathematics Teacher, Vol. XXXX, (1947), pp. 225-23$. 29 for the concepts of the converse, the inverse, and the contrapositive of a theorem. His suggestion came as a result of an examination of the definitions of the converse, inverse, and contrapositive in 93 textbooks. Relatively few of these texts contained definitions of all of these concepts and defects existed in them among the texts which did. As a result he presented proposals for more generalized definitions of these concepts, demonstrated logical advantages in their use, and recommended that students be encouraged to develop state ments of them from fields outside mathematics. Fawcett2^ used nongeometric exercises from insurance, sports, tax laws, school life, home life, political affairs, court decisions, reasoning in science, speeches, newspaper editorials, and advertisements to develop in students an understanding of the nature of proof and to show that students were able to transfer many character istics of good reasoning in geometry to these nonmathe- matical situations. Evaluation of the procedure showed that the effects of this kind of teaching were felt in other classes and school activities, in student-parent discussions at home, and that the students involved scored far above the median on statewide geometry tests.
21Harold P. Fawcett, The Nature of Proof. National Council of Teachers of Mathematics, Thirteenth Yearbook (New York: Bureau of Publications, Teachers College, Columbia University, 1938). 30
Ulmer22 completed a study in 1939 in which he used problem situations, many of which were from everyday life and other fields, as an effective medium for providing varied experiences conducive to effective thinking in these situations. Gadske2^ compared experimental and control groups in demonstrative geometry for the purpose of determining their relative effectiveness in developing critical thinking abilities in high school pupils. He found that major emphasis upon critical thinking, rather than acquisition of knowledge and manipulation of geometric content, had given the students of the experimental groups not only a satisfactory understanding of the subject matter of geometry but also a more effective method of thinking through problems encountered in nonmathematical situations.
Summary The reform movement became more widespread in the United States as a result of the 1923 Report of the National Committee on Mathematical Requirements. The
9^Gilbert 9 Ulmer, "Teachingit Geometry to Cultivate Reflective Thinking: An Experimental Study with 1239 High School Pupils." (unpublished Doctoral Dissertation. Lawrence, Kansas: University of Kansas, 1939)* 2 3Richard E. Gadske, "Demonstrative Geometry as a Means for Improving Critical Thinking." (unpublished Ph.D. dissertation, Department of Education, Northwestern University, 1940). major aim in the teaching of plane geometry was still the development of logical or deductive reasoning. Investi gations included in the 1923 Report regarding automatic transfer of training not only helped to continue the transfer of training controversy but also led to appreci able success for those who sought in accord with recommendations of the Report, to find ways by which transfer could take place. More emphasis on practical aims in the teaching of plane geometry meant more exercises in plane geometry texts from everyday-life situations. However a concern that everyday-life exercises should help to development logical reasoning, the long-standing major aim in the teaching of plane geometry, caused much attention to be given to exercises on patterns of reasoning in everyday life. Four studies were examined which supported the attempts to apply geometric ideas, concepts, and methods to nonmathematical situations of everyday life. It was assumed that plane geometry text
books following the 1923 Report would include more exercises on patterns of reasoning in everyday life as well as more everyday-life exercises. 32 Recommendations and Reports During the 1940's
The Report of the Joint Commission of the Mathematical Association of America and the National Council of Teachers of ^Mathematics In 1935 the Mathematical Association of America and the National Council of Teachers of Mathematics appointed the Joint Commission of seven university teachers and seven secondary school teachers. The Joint Commission made its report in 1940. It recommended the organization of the mathematics curriculum in terms of seven major subject fields : numbers and computation, geometric form and space perception, graphic representation, elementary analysis, logical (or "straight") thinking, relational thinking, and symbolic representation. Each of these was subdivided into the five categories: basic concepts, principles, and terms, fundamental processes, fundamental relations, skills and techniques, and applications.
Included in the report are the following recommenda tions : 1. There should be a gradual and continuous development of the ability to recognize and to use geometric facts, concepts, and principles in everyday-life situations.24
2. There should be "a clear under standing of the meaning of the
24ihe Place of Mathematics in Secondary Education, Fifteenth Yearbook (Washington, D.C. : National douncil' of teachers of Mathematics, 1940), p. 6 6 . 33 basic terms and the ability to recognize their actual occurrences and their bearings on life situa tions. Examples of such terms are: assumption or postulate, proposi tion, converse, conclusion.25 (Note the resemblance of this recommendation to Lazar’s study and recommendations).
3. There should be "a gradual increasing development of ability to manifest coherent, logical thinking in everyday- life situations.26
4. There should be "a clear grasp of what is meant by a deductive science; and the ability to apply the method of postulational thinking in life situations not specifically mathematical.27
Recommendations of this nature supported the efforts of those who sought ways of bringing about transfer of training from understandings obtained in the study of plane geometry to everyday-life situations. These recommendations no doubt had much influence on the number and types of everyday-life exercises included in the plane geometry textbooks that were being published.
The Report of the Committee on the Function of Mathematics in General Education The Committee on The Function of Mathematics in General Education was appointed by the Commission on Secondary Curriculum of the Progressive Education Associa tion in 1932. Its report was published in final form in
2 5Ibid., p. 6 7 . 2 6 ibid., p. 6 8 . 2 7 ibid., p. 9 5 . 34
19^0 under the title, Mathematics in General Education. The report held that mathematics instruction must contri bute to four aspects of living: personal living, immediate personal social relationships, social civic relationships, and economic relationships. Excerpts from the report pertinent to this study are the following: 1. The recognition of familiar elements in new contexts, which contributes to the satisfaction of the successful student, also influences his apprecia tion of geometric form as seen in the world around him. Mathematical instruc tion may enhance appreciation of geometric form as it occurs in art, nature, industry, and architecture. In the application of geometric construc tion to artistic design the student has an opportunity to exhibit his esthetic taste and to create new combinations more or less expressive of his personality.2 o
2. Fundamentally the end sought (in training for logical reasoning) is for the student to acquire both a thorough understanding of certain aspects of logical proof and such related attitudes and abilities as will encourage him to apply his under standing in a variety of life situations.29 Here again is continuing evidence of the concern for emphasizing the use of activities and experiences of everyday life in the study of plane geometry as a way of
28committee on The Function of Mathematics in General Education, Mathematics in General Education (New York: D. Appleton Century Company, 1940), p. 49.
29Ibid., p. 188. 35 transferring training to patterns of reasoning.
The Report of the Committee on Essential Mathematics for Minimum Army 'Needs Deficiencies in mathematics discovered among inductees into the war training program of World War II caused the United States Office of Education and the National Council of Teachers of Mathematics to cooperate in an effort to meet this emergency. One of two committees appointed for this purpose was the Committee on Essential Mathematics for Minimum Army Needs. In 1943, this Committee reported that instruction in geometry should afford students competence in using geometric form where and when needed in practical situations. ■30
The Report of the Commission on Post-War Plans In 1944., the National Council of Teachers of Mathe matics appointed the Commission on Post-War Plans to plan effective programs in secondary school mathematics for the post-war period. In the Commission's first report of 1944, a plan of organization of mathematics courses was presented. From its second report of 1945; the Commis sion issued a guidance pamphlet naming twenty-nine items
30]flational Council of Teachers of Mathematics, Report of the Committee on Essential Mathematics for Minimum Army Needs, The Mathematics Teacher, Vol. XXXVI, (April, 1943), pp. 243-2821 necessary to the development of functional competence in mathematics. These items^l pointed to the need of mathe matical competence for the ordinary affairs of life as a part of a general education appropriate for the majority of the secondary school, population.
Summary The recommendations and reports of this period continued to support the efforts of those seeking adequate means of obtaining transfer of logical and deductive reasoning to everyday-life situations. The controversy on transfer of training appeared to have ended in the general acceptance that transfer is not automatic and that attempts to develop ways to obtain the transfer of patterns of logical reasoning to everyday-life were in order. The reports and recommendations made during the 1 9 3 0 's and the 1940's listed many of the categories into which everyday-life exercises in plane geometry textbooks might be expected to fall.
Recommendations, Reports and Experimental Programs of the 1950 's' The Trend Toward Fewer Everyday-Life Exercises About Physical Situations in Plane Geometry ^Textbooks Recommendations, reports, and comments by mathematics
S^See Commission on Post-War Plans, Guidance Report, The Mathematics Teacher, Vol. XXXX, (1947)* PP. 315-339* for a listing of these items. 37 educators in the 1 9 5 0 's seemed to begin a counter trend that would result in fewer everyday-life exercises about physical situations, but not necessarily fewer on patterns of reasoning, in plane geometry textbooks. It is anticipated that the examinations of textbooks published during these years will reveal the trend regarding exercises on patterns of reasoning. In discussing the status of the mathematics of 1950 in the United States, Kinsella stated: Mathematical proof was confined to geometry. Very few students acquired the notion that the same deductive structure (as that of geometry) also applied to arithmetic, algebra, and all areas of mathematics.32
The Committee on School and College Study of Admission With Advanced Standing In 1953j the Committee on School and College Study of Admission with Advanced Standing said in its report to the Mathematical Association of America: . . . certain bright students are wasting time in the school as set up at present, and they could profitably spend some time anticipating the work they would normally do in their first year of college.33 Thus, this Committee recommended that emphasis on
32john J. Kinsella, Secondary School Mathematics (New York: The Center for Applied Research in Education, Inc., 1965 )j p. 13.
33h . W. Brinkman, "Mathematics in the Secondary School for the Exceptional Student," The American Mathematical Monthly, Vol. LXI, No. (May, 1954), p. 319* citing the committee report. 38 deduction as in the plane geometry of the tenth grade be retained and extended to other types of mathematics, such as algebra, analytic geometry, and trigonometry. A concern of the Committee was that superior students were being neglected when they were denied the opportunity to acquire understanding of the deductive structure of plane geometry as applied to other areas of mathematics. Concern for the superior student came from the feeling of this Committee and others that we needed more and better mathematicians.
The Commission on Mathematics of the College Entrance Examination Board Probably the most authoritative recommendation of a program for mathematically superior students was made by the Commission on Mathematics of the College Entrance
Examination Board. In 1959j the Commission presented a nine-point program on mathematics which was designed for the "college capable" and which included the following two points : 1. An understanding of deductive reason ing in other fields as well as geometry. 2. An appreciation of mathematical structure.34
•^College Entrance Examination Board, "Report of the Commission on Mathematics (Program for the College Capable)." (New York: College Entrance Examination Board, 1959), p. iii. 39 The Commission made the following statement which seems to express its view on everyday-life exercises : Mathematics, and consequently deductive methods, can be applied to life only in those life-situations that are capable of accurate transformation into mathematical models. These situations, though of tremendous importance, are far from frequent in the everyday lives of high school students.35
The Secondary School Curriculum Committee The report of the Secondary School Curriculum •Committee^ of the National Council of Teachers of
Mathematics which also was made in 1959 concurred with the Commission on Mathematics with respect to its emphasis on the deductive structure of geometry. Emphasis on deductive structure, the Committee held, meant emphasis on the study of the basic principles or properties common to all systems of mathematics with the understanding that some of these systems might not be concerned with our familiar numbers.
Other Forces Which Probably Affected Everyday-Life Exercises About Physical Situations and Patterns of Reasoning in Plane Geometry Textbooks Other forces were also at work during the first half of the 1 9 5 0 's that affected changes in high school mathe matics and probably influenced fewer everyday-life
35lbid., p. 2 3 . 36Report of the Secondary School Curriculum Committee, The Mathematics Teacher, Vol. LII, (1959 )* pp. 4o4-4o8. 4o exercises about physical situations and patterns of reasoning in plane geometry textbooks. Most important among these forces were: 1. The rapid growth of mathematics during the past one hundred and fifty years; 2. The revolutionary development of science and technology during the twentieth century; 3. The historical tendency for college and university mathematics to move downward to lower grades; 4. An awareness of the great technological and mathematical progress of Russia; 5. A great increase in the collaboration among mathematics teachers at the college and high school levels; 6 . The huge financial support given by the federal government and large foundations to the improvement of mathematics education; 7. The emergence of vigorous and imaginative leadership in mathematics education in various universities and professional organizations.37
What happened in mathematics instruction as a result of such forces as described above resulted in the changes in content and approach that we now refer to as the "Revolution in Mathematics".
Experimental Programs The revolution was fully under way by 1959. Many experimental programs were being introduced that presented
37Kinsella, op. cit.j p. 15. 4i ideas and approaches to mathematical content that had been suggested earlier in the decade. Two38 Qf these are mentioned briefly here to indicate the effect they probably had on the number and kind of everyday-life exercises about physical situations and patterns of reasoning in plane geometry textbooks.
The University of Illinois Committee on School Mathematics The University of Illinois Committee on School Mathematics (UICSM) was organized in 1951 but did not produce its first experimental textbooks until 1958* The material of these texts followed the new philosophy and made no formal recognition of the subdivisions of algebra, geometry, and trigonometry. The geometry unit included very few everyday-life exercises. Emphasis was given to the study of structure, the importance of sophisticated and precise language, the development of concepts, and student discovery through mathematical exercises. The student discovery exercises and the precise expression of ideas in words and symbols received the greatest amount of attention.
The School Mathematics Study Group The School Mathematics Study Group (SMSG) was
an analysis of eight of these programs see An Analysis of New Mathematics Programs published by the National Council of Teachers of Mathematics, Washington, D.C., 1963. appointed by the American Mathematical Society in 1958 to work on a program for the improvement of the teaching of mathematics in secondary schools. The first course materials were developed in 1959 and they were based on the recommendations of the Commission on Mathematics of the College Entrance Examination Board. Plane and solid • geometry and algebra are integrated and few everyday- life exercises are included in the geometry course, but strong emphasis is given to mathematical structure, to the development of concepts, and an intuitive approach to precise definitions arid terminology. These programs and similar ones developed during this period seemed to concern themselves with structure, new approaches to concepts, precise definitions and terminology, and discovery exercises. These changes in emphasis no doubt affected the number and type of every- day-life exercises on physical situations and patterns of reasoning in plane geometry textbooks.
Summary Committee reports and comments by individuals discussed in this section indicate a trend toward fewer everyday-life exercises about physical situations in plane geometry textbooks of the 1950's. Brief comments were made about the UICSM and SMSG programs, two of the many experimental programs developed in the latter years of this decade. These programs included few everyday-life exercises about physical situations in their materials on geometry, while those on patterns of reasoning increased.
Reports Made in the 196q 's Two major reports have been made in the 196o's and neither has made any recommendation favoring more everyday-life exercises on physical situations in geometry textbooks. In fact, their recommendations move farther away from this emphasis than those made in the 1 9 5 0 's.
The Report of the International Congress of Mathematicians The report of the International Congress of Mathe maticians at Stockholm, Sweden, in 1962 was concerned about the attempts of twenty-one nations to modernize mathematics instruction in secondary schools. These countries reported as their concerns in the teaching of geometry such topics as the following: the study of geometric transformations, an introduction of metric ideas, an introduction to analytic geometry, the study of geometry as an axiomatic system, the inclusion of non-
Euclidean geometry, the study of relations, with special emphasis on equivalence relations and order relations, and a systematic study of cardinal n u m b e r s . 38
38j0hn G. Kemeny, "Report to the International Con gress of Mathematicians," The Mathematics Teacher, Vol. LVI (February, 1963)5 pp. 6 6 -78"! 44
The Report of the Cambridge Conference on School Mathematics The Cambridge Conference Report^9 was made in 1963* It recommends more radical revisions than any experimental program yet emerging from the "Revolution in Mathematics". Although the report recognizes that all mathematical ideas are applications of some sort and that many relate to the physical sciences or to other aspects of real life, its ambitious subject matter proposals seem to indicate lack of concern about everyday-life exercises on physical situations.
It is likely that adherence to the proposals of these two programs by textbook writers would mean even fewer everyday-life exercises on physical situations than might be found in a text published in the early 1950's.
Chapter Summary
Recommendations and reports from 1900 to 1963 have been presented in the light of their possible influence on the number and type of everyday-life exercises in plane geometry textbooks. Experimental programs of the 1950's were mentioned and two were discussed which probably have affected everyday-life exercises in geometry textbooks. At the close of the nineteenth century the
39cambridge Conference on School Mathematics, Goals for School Mathematics. Published for Educational Services, Inc., (Boston: Houghton Mifflin Company, 1 9 6 3 ). disciplinary value of plane geometry was dominant and textbooks of this time could be assumed to have few everyday-life exercises. At the beginning of the twentieth century a reform movement began in Europe and spread to the United States which emphasized the practical values of plane geometry. The movement took on wide scope and emphasis in this country with the publication of the 1923 Report. Investigations appearing in this report continued the controversy over whether emphasis on disciplinary values meant automatic transfer of logical reasoning patterns to other fields and life situations or whether logical reasoning patterns in such situations might better be developed by different teaching procedures and curriculum content. Numerous comments made during the controversy were presented and four studies were reviewed which offered suggestions and recommendations toward obtaining adequate transfer of logical reasoning to everyday-life situations. The controversy extended through the 1 9 3 0 's, but emphasis on everyday-life exercises increased, and everyday-life exercises on patterns of reasoning began to receive atten tion in 1 9 3 8 . During the 194o's it appeared to be generally accepted that transfer was not automatic. The recommen dations and reports of this period supported the efforts of those who sought ways of bringing about the transfer of 46
logical reasoning and deductive thinking to the ordinary affairs of life. Reports and recommendations of- the 1950's directed the attention of plane geometry instruction to the superior or mathematically capable student, the relevance of the deductive structure of geometry in other mathe matical fields, and new concepts and procedures pursuant to preparing students for effective confrontation with the problems of an advanced technological society. This "Revolution in School Mathematics 11 was supported by many experimental programs. Two of them were briefly discussed. Neither emphasized everyday-life exercises of physical situations. Both gave attention to exercises on patterns of reasoning. Reports and recommendations of the 1 9 6 0 's extend beyond the scope of this study. They appear to be heading toward another reform movement or a second revolu tion that would place advanced topics and concepts in geometry at much lower levels and disperse them throughout the mathematics curriculum from grades K-12. CHAPTER III
REPORT OF ANALYSES I One series of plane geometry textbooks will be analyzed in this chapter. Careful consideration has been given to selecting textbooks that have been written by the same authors or succeeding authors.
The Wentworth, Wentworth-Smith, Welchons-Krickenberger, and Welchons- Krlckenberger-Pearson Textbooks
G. A. Wentworth, Elements of Geometry, 1 8 7 8 . A publisher's note appears at the front of this book which includes the following: This book is thought to be the first printing of Wentworth's, Plane Geometry, although the title reads, Elements of Geometry . . . The book is printed on heavier paper than that usually used in many books which tends to support the opinion that it is a first edition. There are 117 exercises in this text of which just two are of the type that are here considered everyday-life exercises. These two exercises are shown and classified below.
Examples of Everyday-Life Exercises
Classification Exercise Miscellaneous - If a ladder, whose foot rests on a hori zontal plane and top against a vertical
47 I
48
wall, slips down; find the locus of its middle point, p. 1 2 7 j no. 1 1 . Surveying - How many acres are contained in a triangle whose sides are respectively 6 0 , 7 0 , and 80 chains? p. 2 5 0 , no. 3 .
Thus, to the nearest tenth percent, 1.7 percent of the exercises in this text are everyday-life exercises on physical situations and none are on patterns of reasoning.
G. A. Wentworth, A Textbook of Geometry, Revised Edition, 1 8 8 8 . Eleven of 437 exercises in this text are everyday- life exercises on physical situations. This number
amounts to 2.5 percent of the total number of exercises.
Examples of Everyday-Life Exercises
Classification Exercise Building - A house is 40 feet long, 30 feet wide, 25 feet high to the roof, and 35 feet high to the ridge-pole. Find the number of square feet in the entire exterior surface, p. 6 9 , no. 1 3 0 .
Carpentry - Find the side of the largest square that can be cut out of a circular piece of wood whose radius is 1 foot 8 inches, p. 2 3 9 , no. 419. Mapmaking - On a certain map the linear scale is 1 inch to 5 miles. How many acres are represented on this map by a square the perimeter of which is 1 inch? p. 1 8 2 , no. 2 9 1 .
Tiling - How many tiles 9 inches long and 4 inches wide will be required to pave a path 8 feet wide surrounding a rec tangular court 120 feet long and 36 feet wide?- *p. 1 9 2 , no. 2 9 7 . 4 9 Time-Telling - How many degrees (are) in the angle formed by the hands of a clock at 2 o’clock? 3 o'clock? 4 o'clock? 6 o'clock? p. 2 5 , no. 3 * Number of Everyday-Life Exercises by Classifications Building 1 Carpentry 1 Mapmaking 1 Miscellaneous 1 Surveying 5 Tiling 1 Time-Telling 1 TOlAL IT Surveying exercises amount to almost half of the everyday-life exercises in this text. None of the everyday-life exercises are on patterns of reasoning.
G. A. Wentworth, A Text of Geometry, Revised Edition,1 8 9 1 . Twelve (2.5 percent) of 479 exercises on plane geometry found in this text are everyday-life exercises on physical situations. Examples of Everyday-Life Exercises Classification Exercise Building - How long must a ladder be to reach a window 24 feet high., if the lower end of the ladder is 10 feet from the side of the house? p. 1 7 7 , no. 2 5 . Surveying - A straight railway passes two miles from a town. A place is four miles from the town and one mile from the railway. To find by construction how many places answer this description, p. 1 1 6 , no. 108
Number of Everyday-Life Exercises by Classifications
Building 1 Carpentry 2 50
Mapmaking 1 Miscellaneous 1 Tiling 1 Time-Telling 1 Surveying 5 TOTAL T2 The surveying exercises exceed other types of everyday-life exercises in this text as in the previous text. None of the everyday-life exercises are on patterns of reasoning.
G. A. Wentworth, Geometry, Revised Edition, 1898. The exercises on plane geometry in this text are exactly the same as those in the previous text. There are 12 of the 479; which again is 2.5 percent, on physical situations. None are on patterns of reasoning.
G. A. Wentworth, Plane and Solid Geometry, 1899• The number of plane geometry exercises in this text increases, but the percentage for everyday-life exercises, which again is 2.5 percent, remains the same as that for the two previous books. Two new classifications, engi neering and physics, appear in this text, and the contents of some of the exercises change. Examples of Everyday-Life Exercises Class ification Exercises Engineering - The span (chord) of a bridge in the form of a circular arc is 120 feet, and the highest point of the arch is 15 feet above the piers. Find the radius of the arc. pp. 241-250, no. 527. 51 Physics - The diameter of a bicycle wheel is 28 inches. How many revolutions does the wheel make in going 10 miles ? pp. 241-250, no. 523* Surveying - The length of each side of a park in the shape of a regular decagon is 100 yards. Find the area of the park, pp. 241-250, no. 510. Number of Everyday-Life Exercises by Classifications Building 2 Engineering i Mapmaking 1 Miscellaneous 1 Tiling 1 . Time-Telling 1 Physics 2 Surveying 6 TOTAL 33“ Again no exercises are on patterns of reasoning.
George Wentworth and David Eugene Smith, Wentworth's Plane Geometry, Revised, 1910. This book represents the beginning efforts of George Wentworth and David Eugene Smith to carry forward the works of G. A. Wentworth. There are 1027 exercises in the text, and 65 (6.0 percent) are everyday-life exercises. It can be observed from the examination of this text that the various recommendations by Perry and Moore suggesting that more exercises be included in textbooks were
receiving attention. Examples of Everyday-Life Exercises Classification Exercises Designing - Find the area of a fan that opens out into a sector of 120 degrees, the radius being 9 3/8 inches, p. 2 2 5 , no. 9 . Drafting - Show that if we place a draftsman's triangle against a ruler and draw AC, and move the triangle along as shown in the figure and draw A' C', then AC || A' C' .
Figure 3*— A triangle along a ruler
Sports - A baseball diamond is 90 feet square on a side. Draw the plan, using a scale of l/l6 inches to a foot. Locate the pitcher 60 feet from home plate, p. 1 3 , no. 2 0 .
Engineering The circumference of two cylindrical steel shafts are respectively 3 inches and 1-5- inches. The area of a cross section of the first is how many times a cross section of the second? p. 24l, no. 1 0 . Number of Everyday-Life Exercises by Classifications Architecture 1 Building 3 Carpentry 2 Classroom Sit . 2 Designing 7 Drafting 4 Engineering 12 Gardening 1 Mapmaking 4 Miscellaneous 5 Physics 4 Sports 2 Surveying 12 Tiling 1 Time-Telling 4 Travel 1 I'OlAL bb Surveying exercises increase, but exercises of other types also gain, such as engineering and designing. 53 None are on patterns of reasoning.
George Wentworth and David Eugene Smith, Wentworth1s Plane and Solid Geometry, 1911. Sixty-seven (6.4 percent) of the 104l plane geo metry exercises are the type sought. Aside from an increase in the number of exercises in this text there began to appear exercises representing a broader perspec tive of everyday-life situations. Examples of Everyday-Life Exercises Classification Exercise Drafting - A circular window in a church has a design similar to the accompanying figure. Draw it, making the figure twice its size. pp. 145-148, no. 2 8 .
Figure 4.— Design of a circular window
Foundry Work - A water wheel is broken and all but a fragment is lost. A workman wishes to restore the wheel. Make a drawing showing how he can construct a wheel the size of the original. pp.l45-l48, no. 3 0 . Building - From a circular log 16 inches in diameter a builder wishes to cut a column with its cross section as large a regular octagon as possible. Find the length of each side. p. 247,no. 2 7 , Number of Everyday-Life Exercises by Classification Architecture 1 Building 2 Carpentry 2 54
Classroom Sit. 2 Designing 4 Drafting 11 Engineering 8 Foundry Work 4 Gardening 1 Mapmaking 5 Miscellaneous 3 Physics 4 Sports 2 Surveying 13 Tiling 1 Time-Telling 4 ' TOTEE------57“ Again, there are no exercises on patterns of reasoning.
George Wentworth and David Eugene Smith, Plane and Solid Geometry, 1913. This text shows the continuing increase in the total number of plane geometry exercises, but two less everyday-life exercises than the previous text. There are 1059 exercises of which 65 are the everyday-life type. The percentage of everyday-life exercises is 6.1 percent. Examples of Everyday-Life Exercises Classification Exercise Carpentry - The area of the cross section of a beam 1 inch thick is 12 square inches. What is the area of the cross section of a beam of the same proportions and 1 1/8 inch thick? p. 235 > no. 9 . Physics - Find the diameter of a carriage wheel that makes 264 revolutions in going a half mile. p. 2 5 5 * no. 2 Mapmaking - In drawing a map of a triangular field 75 rd., 60 rd., and 50 rd.,respec tively, the longest side is 1 inch long. How long are the other two sides? p. 1 7 1 j no. 5 . 55 Miscellaneous - A five cent piece is placed on the table. How many five cent pieces can be place around it, each tangent to it and tangent to two of the others? ■ Prove it. p. 247, no. 1 Physics - Find the length of the belt connec ting two wheels of the same size, if the radius of each wheel is 18 inches, the distance between the centers is 6 feet, and 4 inches is allowed for sagging, p. 2 6 0 , no. 6 l. Number of Everyday-Life Exercises by Classification
Building 3 Carpentry 2 Designing 8 Drafting 4 Engineering 8 Gardening 1 Mapmaking 5 Miscellaneous 10 Physics 5 Sports 2 Surveying 13 Tiling 2 Time-Telling 2 TOTAL " ' ■ 6b Surveying exercises continue to be the largest number of everyday-life exercises. None of the everyday life exercises are on patterns of reasoning.
A. M. Welchons and W. R. Krickenberger, Plane Geometry, 1933. This text reflects the recommendations made by committees and mathematics educators that plane and solid geometry be used and that courses make some provisions for individual differences among students. There are no separate sections on plane and solid geometry and the exercises are graded for Groups A, B, and C. Group A exercises are for a minimum course, Group B exercises are for a medium course, and Group C exercises are for a maximum course. Review questions and chapter tests are included but very few everyday-life exercises are included in either. Pictures showing uses of geometric ideas in life situations are included which could serve to motivate learning. Only the exercises, however, are examined in this study. There are 1915 exercises in the text, and 134 (7.0 percent) are everyday-life exercises. Exactly one such exercise is found in the chapter tests (Test 7* p. 9 8 , no. 3: "A wheel makes 30 revolutions per minute. Through how many degrees does it turn in one second?") Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - Are handkerchiefs ever bisected when they are ironed? Explain, pp. 16-17, no. 2.
Pattern of - Tell which is the hypothesis and which Reasoning is the conclusion. "If Elmer Smith lives in Chicago, he lives in Illinois, pp. 29-30, no. 10. Sports - The shortstop fields a batted ball at a point one third the distance from second base to third base. How far must he throw the ball to make an out at first base? pp. 244-245* no. 1, part B. Physics - If the drive wheels of a locomotive are 60 inches in diameter, find the number of revolutions a minute they 57 make when the engine is going 60 miles an hour. pp. 317-319* no. 9, part B. Number of Everyday-Life Exercises by Classification Astronomy 1 Botany 1 Building 12 Carpentry 3 Designing 17 Drafting 4 Engineering 8 Mapmaking 1 Miscellaneous 19 Navigation 1 Physics 16 Patterns of Reasoning 7 Sports 6 Surveying 36 Tiling 1 Time-Telling 1 tOfAL lST
First Appearance of Everyday-Life Exercises on Patterns of Reasoning in This get of Textbooks' Seven exercises on patterns of reasoning appear in this first text by Welchons and Krickenberger, apparently due to opinions resulting from the transfer of training controversy. Additional classifications such as astronomy, botany, and navigation are included. More exercises of the miscellaneous type are included than in the previous texts, but surveying exercises continue to be the most numerous of everyday-life exercises.
George Wentworth and David Euguene Smith, Plane Geometry, 1 9 3 8 .
The copyright for this book was jointly held by Maude Simpson Wentworth, daughter of George Wentworth, and David Eugene Smith who, no doubt, collaborated in an effort to further extend the Wentworth series. Prior to this publication a series was begun by Welchons and Krickenberger which was destined to succeed the Wentworth series. This test includes 108l exercises, 75 (6.9 percent) of them everyday-life exercises. New features of this text are sections entitled "Appendix" and "Applications of Geometry." In the section entitled "Applications of Geometry," 13 everyday-life exercises are included. Examples of Everyday Life Exercises Clas s ificat ion Exercise Sports - A baseball diamond is a sqdare 90 feet on a side. Draw the plan using a scale of 1/16 inch to a foot. Locate the pitcher 60 feet from home plate, p. 1 3 > no. 2 0 . Drafting - In drawing a map of a triangular field with sides 70 rd., 60 rd., and 50 rd., respectively, the longest side is drawn 1 inch long. How long are the other two sides drawn? p. 1 7 1 * no. 5 . Surveying - From two adjacent sides of a rectangular field 60 rd. long and 40 rd. wide a road is cut 4 rd. wide. How many acres are cut off for the road? p. 1 9 6 , no. 1 1 . Pattern of Kate decides to buy a dress if either Reasoning - her father or her mother gives her the money. Kate buys the dress. What conclusions do you draw as to the money? p. 2 8 6 , no. 2 . 59 Number of Everyday-Life Exercises by Classifications Building 3 Carpentry 7 Designing 6 Drafting 10 Engineering 9 Foundry Work 2 Gardening 1 Miscellaneous 9 Physics 4 Patterns of Reasoning 4 Sports 2 Surveying l4 Tiling 1 Time-Telling 3 U m L ------75 Four exercises on patterns of reasoning appear in this last edition of the Wentworth texts for the first time.
A. M. Welchons and W. R. Krickenberger, Plane Geometry, Revised Edition, 1940. In this revised edition, the authors include topics on space geometry, everyday reasoning, and analytic geometry as additional features in recognition of the latest accepted developments in the teaching of plane geometry. This is the first text of the set that contains the concept of the contrapositive of a theorem, and the number of exercises increases by more than one thousand. There are 2844 exercises and 242 (8.5 percent) are everyday-life exercises, fifty-five (22.3 percent) of them are patterns of reasoning exercises. 6o
Examples of Everyday-Life Exercises Classification Exercise Miscellaneous < Are handkerchiefs ever bisected when they are ironed? Explain, p. 18, no. 2 . Designing - Draw a five-pointed star (pentagram). How many angles are formed inside the figure? Which of these angles seem to be equal? p. 1 1 , no. 7 . Pattern of Explain how each of the following Reasoning - proverbs is the result of inductive reasoning: a. All is not gold that glitters. b. A bird in the hand is worth two in the bush. c. A rainbow at night is the sailor's delight, p. 2 6 , no. 4. Pattern of State the contrapositive of the Reasoning - following statement: "If a woman is beautiful, she is young." p. Ill, no. 2 . Measurement - Find the area in square miles of the state of Utah. (Obtain your measure ment from a map of Utah), p. 212, no . 2 . Number of Everyday-Life Exercises by Classifications Astronomy 1 Botany 1 Building 3 Carpentry 27 Designing 10 Drafting 3 Engineering 11 Farming 6 Field Artillery2 Geography 1 Measurement 22 Miscellaneous 15 Navigation 3 Physics 34 Patterns of Reasoning 55 6l
School Life 4 Surveying 27 Tiling 2 Time-Telling 5
For the first time in this set of texts the number of exercises on patterns of reasoning exceeds the number of everyday-life exercises in any other classification. Fifty-five (22.3 percent) of the everyday-life exercises are on patterns of reasoning and 187 (77*7 percent) are on physical situations.
A. M. Welchons and W. R. Krickenberger, New Plane Geometry, 1952. This text is a revision of texts published in 1933j 1938, 1940, 1943, and 1949. The grading of exercises for minimum, medium, and maximum courses continues. Review questions, chapter tests, and pictures are included as in the previous text. A chapter entitled, "Additional Topics," appears at the end of the text patterned after the section on "Applications of Geometry" in the previous
Wentworth-Smith text. The number of exercises included in the text increases to 2901 and 352 (12.1 percent) are everyday- life exercises. The chapter entitled "Additional Topics" includes 94 exercises, and 65 of these are everyday-life
exercises. Examples of Everyday-Life Exercises Clas s ifIcatIon Exercise Miscellaneous - When men set fence posts, why do they first set the end posts? pp. 11-1 2 , no. 5 • Pattern of In a western city some people say that Reasoning - if it rains on Easter Sunday, it will rain on each of the next seven Sundays. How do you think this conclusion was made? Do you believe that it is valid? p. 3 3 * no. 2 . Pattern of If two automobile parts are made from Reasoning - the same mold, are they congruent? p. 6 6 , no. 5 *
Carpentry - The rafters of a roof make an angle of 35° with the level line. What angle do the rafters form at the ridge? Why" are the rafters of equal length? pp. 134-135> no. 10. Aeronautics - A pilot finds that the bearing of an airport from his plane is 137°• What is the bearing of his plane from the airport? pp. 500-5 0 1 , no. 7 .
Number of Everyday' •Life Exercises by Classifications
Aeronautics 31 Artillery Firing 11 Astronomy 1 Botany 1 Building 12 Carpentry 14 Designing 13 Drafting 11 Engineering 17 Mapmaking 22 Measurement 1 Miscellaneous 31 Navigation 1 Physics 33 Patterns of Reasoning 87 Sports 7 Surveying 53 Tiling 1 Time-Telling 5 TOTAL Patterns of reasoning exercises again exceed the exercises in other classifications. Eighty-seven (24.8 percent) are on patterns of reasoning and 265 (75*2 per cent) are on physical situations.
A. M. Welchons and W. R. Krickenberger, New Plane Geometry, 1956. This text follows the pattern of the previous text. A total of 2911 exercises are included, and 359 (12.3 per cent) are everyday-life exercises. The chapter entitled
"Additional Topics" includes 124 exercises, and 71 of these are the everyday-life type. Examples of Everyday-Life Exercises
.Classification Exercise Pattern of A farmer in Ohio experimented on the Reasoning - use of lime as a fertilizer for corn. (Inductive) In each hill of one row he placed a small amount of lime. In the next row he used no lime. The fertilized row gave the better yield. The next year he repeated the experiment and obtained the same result. Should he conclude that lime benefits his corn crop? p. 33* no. 1. Pattern of See if you can draw a valid conclusion Reasoning - from the following: (Deductive) "(1) John's dog barks whenever a stranger enters the yard. (2) John's dog is barking, p. 3 6 , no.l Carpentry - Make a plumb level and bring it to class, pp. 91-9 2 , no. 7 . 64
Physics - A man sees his eye reflected by two plane mirrors at an angle of 6 0 0 . Construct the path of the ray of light which starts from his eye and after being reflected by the two mirrors returns to his eye. pp. 165-1 6 6 , no. 9<
Architecture - Find the height of a circular arch having a radius of 5 feet if the height of the arch is one fourth its span. p. 3 8 2 , no. 10.
Aeronautics - An airplane has an air speed of 100 mph. If the wind is blowing east at 20 mph, how far east can the plane go and return in 7| hours? p. 498, no.3. Number of Everyday-Life Exercises by Classifications Aeronautics 31 Architecture 5 Artillery Firing 12 Astronomy 3 Botany 1 Building 4 Carpentry 30 Designing 10 Drafting 5 Engineering 9 Industry 4 Logistics 1 Maps and Map Reading 29 Miscellaneous 25 Measurement 9 Navigation 2 Physics 36 Patterns of Reasoning 92 Sports 8 Surveying 40 Time-Telling 3 TOTAL 5 W Patterns of reasoning exercises exceed by far the number of exercises in any other classifications. Ninety- two (25.6 percent) are on patterns of reasoning and 267 65 (74.4 percent) are on physical situations
A. M. Welchons and W. R. Krickenberger, Plane Geometry, Revised Edition, 1 9 6 1 . This text does not include as many exercises as the previous text. There are 2823 exercises, and 331 (11.7 percent) are everyday-life exercises. This represents a decrease of 0.6 percent in everyday-life exercises. The last chapter is entitled "Applications of Plane Geometry," and 66 of its 91 exercises are the everyday-life type. Examples of Everyday-Life Exercises Classification Exercise
Pattern of Tell which is the hypothesis and which Reasoning - is the conclusion: "Radio reception is best if the weather is clear." pp. 44-455 no. 3 . Carpentry - Why is a brace placed on a gate. Can you give another way to brace a garage door? Would a rope take the place of the iron brace in the stepladder? pp. 79-8 0 , no. 5 .
Surveying - What is the area of a tract of land in the shape of a right triangle if its hypotenuse is 52 feet and one of the other sides is 40 feet? pp. 207- 208, part B, no. 4. Artillery If the range of a field artillery gun Firing - is between 1000 yards and 5000 yards, and the gun can be turned through an angle of 135 degrees, draw the locus points where the projectiles may fall, pp. 295-2 9 6 , Part C, no. 3. Number of Everyday-Life Exercises by Classification Aeronautics 33 Architecture 6 Artillery Firing 4 66
Astronomy 3 Botany 1 Building 6 Carpentry 30 Designing 9 Drafting 3 Engineering 11 Industry 3 Mapmaking 27 Measurement 11 Miscellaneous 33 Navigation 2 Physics 36 Patterns of . Reasoning 6l Sports Surveying Time-Telling TOE Although the total number of everyday-life exercises decreases, the exercises on patterns of reasoning remain the largest category. Sixty-one (18.4 percent) are on patterns of reasoning and 2 7 0 (81.6 percent) are on phys ical s ituat ions. This is the only text in this set of texts which shows a rise in the number and percentage of exercises on physical situations and a decline in the number and per centage of exercises on patterns of reasoning.
A. M. Welchons, W. R. Krickenberger, and Helen R. Pearson, Plane Geometry, 1 9 6 1 . Striking color features and more appropriate pictures appear in this text to make it more attractive to the student. More exercises appear than in any previous text of this series, but the percentage of everyday-life exercises continues to decrease. There are 67 2964 exercises, and 332 (11.2 percent) are everyday-life exercises. The chapter entitled "Additional Topics" includes 55 exercises, 33 the everyday-life type. Examples of Everyday-Life Exercises Classification Exercise Pattern of Name the hypothesis and the conclu Reasoning - sion. "If you can run the 100 yard dash in 10 seconds, you are a good sprinter." p. 51> no. 1.
Pattern of Tell what assumption is implied. Reasoning - "Teachers should be retired when they become 65 years of age." pp. 71-72, no. 1.
Building - How many tiles each 1 inch wide and 2 inches long will be required for a fireplace hearth 2 feet wide and 5 feet 4 inches long, allowing 10 per cent of the surface for cement? p. 242, no. 7
Navigation - If the bearing of a ship from a lighthouse is 52 degrees 18 minutes, what is the bearing of the lighthouse from the ship? p. 546, no. 5» Time-Telling - True or false: "At 3 o'clock the hands of the clock form an obtuse angle." p. 2 9 , no. 4. Number of Everyday-Life Exercises by Classifications Architecture 7 Aeronautics 11 Astronomy 3 Artillery Firing 3 Botany 1 Building 8 Carpentry 26 Designing 11 Drafting 1 Engineering 16 Industry 4 Map Reading 5 Mapmaking 5 Measurement 18 Miscellaneous 32 Navigation 4 Physics 34 Patterns of Reasoning 100 Sports 7 Surveying 36 Time-Telling 5 TOTAL 3 3 2 “ Exercises on patterns of reasoning again far out number the other classifications. One hundred (30.1 per cent) are on patterns of reasoning and 232 (69.9 percent) are on physical situations. TABLE 2
COMBINED DATA PROM THE ANALYSES OF THE WENTWORTH-SMITH-WELCHONS- KRICKENBERGER-PEARSON SERIES
0 0 ra ra <£ <£ P p £ •rH •H •rH •rH <£ <£ O co
G. A. Wentworth Plane and Solid 1899 603 15 2.5 15 100 - - Geometry Wentworth & Smith ro Wentworth's Plane H O 1910 65 6.3 65 100 - - Geometry
Wentworth & Smith Wentworth * s Plane 1911 104l 67 6.4 67 100 ■ - - & Solid Geometry
Wentworth & Smith Plane & Solid 1913 1059 65 6.1 65 100 - - Geometry O' SJ o' sj O' SJ s; O' S} £P CD CD CDS'sJ CD CD CD p cd p CD P H 4 H 4 H 4 H 4 H 3 4 H c+ O cw o era o m o CP) o c+ CP) o 31 3* CD 13* CD & CD 3" CD J3* S P 3* O o 4 O 4 O 4 O 4 O O 4 O 3 3 3 3 3 4 3 M CD CD CD CD c+ cd 1 1 1 1 1 3* l 8 w W 4 H 4 f? H 8? H- H- H- H- H- o o O n O to O p? PT f* pr tv tv P CD CD P CD H<3 P 3 3 3 3 3 c+ 3 1 1 1 l 1 3* 1 hd hd tel 3! W hd hd hd H H CD CD CD H H l-J P SB s: S! 3 2*< Continued 2— TABLE CD CD 3 < H H HH H H H VO VO VO VO VO VO VO Publication CTV ov VJ1 VJl •pr 00 OO H H ov ro O 00 oo Date ro ro ro ro ro H H Number of VO 00 vo VO 00 O VO ov ro H o -Pr CO H Exercises -P- u> HH ■Pr H VJl Number of 00 OO 00 00 ro H U) (JO VJ1 on •pr -S] OO Everyday-Life ro H VO ro ro VJl •Pr Exercises H H HH H H ro ro co OV -a •• •• • t • Percent of ro -SI (JO H VJl vo O Everyday-Life ro ro ro ro H I- 1 oo CT\ Ch 00 -si ro Number of ro o -si VJl -S! H -a Physical Sits. ov 00 -si -SI -si VO vo VO H •Pr VJl VO ■P- -Pr Percent of •••• • • • VO 0v -p- ro -Si 00 Physical Sits. H O ov VO CO VJl Number of OH ro -VI VJl -pr -si Patterns of Reasoning U> • H ro ro ro O 00 VJ1 ■p- ro VJl vjl Percent of • • •• • • • Patterns of H -pr Ov oo 00 00 ro Reasoning . GRAPH I A COMPARISON OP THE EVERYDAY-LIFE, PHYSICAL SITUATION, AND PATTERNS OP REASONING EXERCISES PROM DATA OP TABLE 2 0 * tot mt J9oo Icfrb fizo Trio y&frfz s Patterns of Eve ryday- Li fe Physical Situation Reasoning Exercises Exercises. . . . Exercises— V 72 Chapter Summary Table 2 and the graph of its data indicate that the general trend in the percent of everyday-life exercises in this series of plane geometry textbooks has been upward. The first Wentworth text included just two such exercises. The Wentworth-Smith texts increased the everyday-life exercises to above 6 percent and showed, a .3 percent decrease in them in the 1913 text. The Welchons-Krickenberger texts show a definite increase in the percentage of everyday-life exercises from 7.0 in 1933 to 12.3 percent in 1956. This 1933 text shows the first instance of patterns of reasoning exercises in this set of texts. The 1940 Welchons-Krickenberger text, which is a revision of two previous texts, reflects the influence of recommendations cited elsewhere in this study that were made by individuals and committees regard ing the inclusion of everyday-life exercises in mathe matics courses. This influence also appears in the 1952 and the 1956 texts. The last two texts examined show a slight decline in the percentage of the total number of everyday-life exercises. They also show a decline in the number of exercises included in the last chapters where a concentration of everyday-life exercises had previously occurred. Everyday-life exercises in surveying are predominant in the earlier editions of this series of texts. In the Wentworth-Smith texts patterns of reasoning exercises appear for the first time in the 1938 edition. The Welchons-Krickenberger texts which followed the Wentworth- Smith texts have exercises on patterns of reasoning in each edition beginning with the first edition of 1 9 3 3 . Exercises on patterns of reasoning exercises outnumber the other types for the first time in the 1940 text, and this continues to be the case in subsequent editions. Although many more everyday-life exercises appear in the later texts than in the earlier ones of this series, there seems to be some decline in the percentage of such exercises on physical situations and an increase in those on patterns of reasoning in the most recent texts. CHAPTER IV REPORT OF ANALYSES II Analyses of a series of five texts by the same or succeeding authors are reported in this chapter. The Stone-Millis, Stone-Mallory, Mallory, Mallory-Meserve-Skeen Textbooks John C. Stone and James F. Millis, Plane Geometry, 1 9 1 6 . A statement in the preface of this text says that the text represents: . . . an attempt to motivate the teaching of geometry in the secondary school, and to make it function in the life of the student, but utilizing as exercises a large number of applied problems of geometry such as are actually encountered in real life. There are 1174 exercises in the text and 226 of these are everyday-life exercises. No exercises on patterns of reasoning in everyday life appear in this text. Examples of Everyday-Life Exercises Classification Exercise Drafting - What principle is used in ruling parallel lines on a drawing board by means of a T - square? p. 35> no. 13. Carpentry - What geometry principle does a carpenter use when he marks off parallel lines on a board by moving 74 75 one blade of his square along an edge of the board and marking along the other? p. 3 6 , no. 3 * E Figure 5*— Parallel lines formed by- moving a carpenter's square along an edge of a board Designing - The Greek design above is called a meander. Make a copy of it. How are the parallel lines drawn, p. 3 6 , no. 4. Miscellaneous - A robber hid some treasure by a road side and equidistant from two trees. Explain how to find it. p. 8 9 , no. 11. Sewing - A piece of cloth 36 in. wide is to be cut on a bias, at an angle of 45° with the edge. How long will the bias cut be? p. 2 0 1 , no. 8 . To the nearest tenth, 10.7 percent of the exercises are everyday-life exercises. Number of Everyday-Life Exercises by Classifications Aeronautics 1 Architecture 13 Astronomy 6 Carpentry 18 Designing 32 Drafting 6 Engineering 12 Farming 1 Forestry 2 Foundry Work 2 Gardening 2 Measurement 19 Miscellaneous 12 Navigation 9 Photography 1 Plumbing 3 Sewing 2 Sports 5 Surveying 63 Time-Telling 1 Tool and Die Work 2 t 6TAl 1225 Surveying exercises are dominant but a reasonable number of the everyday-life exercises are applications to designing, measurement, and carpentry. None of the everyday-life exercises are on patterns of reasoning. John C. Stone and Virgil S. Mallory, Modern Plane Geometry, 1 9 2 9 , 1940. Each unit of this text contains groups of everyday- life exercises. There are 2731 exercises and 272 (9.6 per cent) of these are the everyday-life type. Only 6 (2.3 percent) of the everyday-life exercises are on patterns of reasoning while 2 5 6 (97.7 percent) are on physical situations. ^ Examples of Everyday-Life Exercises Classification Exercise •* Surveying - A bridge is to be constructed over a river so as to be the same distance from two towns A and B which are 10 miles apart and on opposite sides of the river. If the distance from A to the river is 2 miles, and from B to the river is 5 miles, draw a sketch and show how to locate the bridge, p. 8 7 , no. 2 0 . Miscellaneous A six-inch ruler moved so that its ends always are on two adjacent edges of your paper. Find the locus of its mid point. p. 144, no. 1 1 . Physics A fly-wheel makes 40 revolutions per minute. Through how many degrees of 77 arc does a point on the rim travel in 1 second? p. 2 1 7 * no. 5 » Time-Telling - Lines are drawn on the face of a clock from 12 to 4 to 7 to 2. How many degrees are in each angle formed? p. 225* no. 3. Engineering - A canal is 28 ft. deep, 120 ft. wide at the top, and 90 ft. wide at the bottom. What is the area of the cross-section of it? p. 333* no. 4. Number of Everyday-Life Exercises by Classifications Aeronautics 1 Architecture 2 Auto Mechanics 1 Carpentry 18 Designing 26 Drafting 12 Engineering 17 Mapmaking 7 Metallurgy 1 Measurement 21 Miscellaneous 28 Navigation 16 Physics 4 Plumbing 4 Patterns of Reasoning 6 Sewing 2 Sports 5 Surveying 86 Time-Telling 5 TOTAL ” 562' Surveying exercises also outnumber other classifi cations in this text. John C. Stone and Virgil S. Mallory, New Plane Geometry, 1941. The authors say they fulfill the recommendations of the 1923 Report in this text. The text included 3487 exercises and 247 (7.1 percent) are everyday-life exercises. 78 Examples of Everyday-Life Exercises Classification Exercise Pattern of If it rains, the ground is wet. The Reasoning - ground IS wet. Can you conclude that it has rained? Give reasons, p. 3 8 , no. 5 • Pattern of Each of three students worked a Reasoning - problem at the blackboard. On taking their seats each discovered an error in a problem and each raised his hand to correct the error. Assuming that no one discovered an error in his own problem, what is the greatest number of problems correct? p. 8 3 , no. 11. Carpentry - Why is a roof sufficiently braced by a board nailed across each pair of rafters? p. 9 2 , no. 3 • Surveying - Three towns, A, B, and C, are so situated that C lies on the perpendi cular bisector of the segment joining A and B. If C is 8 miles from A, how far is it from B? p. 112, no. 3* Designing - Divide a circle into equal arcs. By joining every third point in order, form an eight-pointed star. p. 1 8 2 , no. 4. Number of Everyday' •Life Exercises by Classifications Aeronautics 1 Architecture 12 Carpentry 12 Designing 31 Drafting 4 Engineering 21 Mapmaking 8 Measurement 11 Metallurgy 1 Miscellaneous l4 Navigation 5 Physics 15 Patterns of Reasoning 17 Sewing 1 79 Sports 7 Surveying 78 Time-Telling 7 Tool and Die Work 2 ""TOTSE------247"“ Seventeen (6.7 percent) of the everyday-life exercises are on patterns of reasoning and 230 (93.3 per cent) are on physical situations. An increase in the number of exercises on patterns of reasoning appears in this text, but surveying exercises remain dominant and the total number of everyday-life exercises is less than in the text examined previously. Virgil S. Mallory, New Plane Geometry Revised Edition,1943. The author claims to have met the recommendations of the Joint Commission Report in this text. There are 2866 exercises in the text and 240 (8.3 percent) of them are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Pattern of An old problem: A man traveled from Reasoning - A to B by first going half the distance from A to B; then going half the remaining distance. If he continued this, each time going half the remain ing distance toward B, would he ever reach his destination? p. 1 6 , no. 8 . Pattern of James Wilkins is 8 years old. Every Reasoning - year that he can remember it has rained on July 4. Why can he not logically draw the conclusion: It always rains on July 4? p. 375 no. I 8o Surveying - The angle of elevation of a church steeple from a point 100 ft. away is 36°. How high is the steeple? p. 299* no. 5 . Navigation - A boat has sailed 20 min. in a north easterly direction. How far east has it gone? How far north? p. 300, no.17* Drafting - Draw a rectangle representing a rec tangular field that is 1200 ft. long and 489 ft. wide to a scale of 240 ft. to an inch. What are the dimensions of the drawing? p. 2 0 2 , no. 7 . Physics - Two forces, one of 80 lb. and the other 96 lb., are exerted on an object at A at an angle of 30°. Find the value of the resultant force and the angle it makes with the given force, p. 3 2 8 , no. 31. Number of Everyday-Life Exercises by Classifications Aeronautics 12 Architecture 11 Art 1 Carpentry 13 Designing 23 Drafting 5 Engineering 13 Mapmaking 5 Measurement 14 Miscellaneous 14 Navigation 5 Physics 12 Patterns of Reasoning 18 Sewing 1 Sports 7 Surveying 77 Time-Telling 7 Tool and Die Work 2 TOTAL 24o The number of exercises on patterns of reasoning is about the same in this text as in that previously examined. Eighteen (7*5 percent) are on patterns of 81 reasoning and 222 (92.5 percent) are on physical situa tions. Surveying exercises continue to outnumber other classifications. The number of everyday-life exercises increases, but the percentage that this number represents of the total number of exercises shows a decrease. Virgil S. Mallory, Bruce E. Meserve, Kenneth C. Skeen, A First Course in Geometry, 1959• It is the claim of the authors that this text is modern in philosophy, content, and pedagogy. The number of exercises increases by more than 1200. There are 4089 exercises and 317 (7.8 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Designing - See how many patterns you can find by looking at the way bricks (or concrete blocks) are used in making walls. Make a drawing of each pattern, p. 4l, no. 9 . Pattern of Give a converse and tell whether or not Reasoning - the converse is true for the following: If a man lives in San Francisco, he lives in California, p. 95* no. 7« Surveying - Pirates buried a treasure 60 ft. from a certain tree and 80 ft. from a certain straight path. Show how to locate the treasure. When are there two possible locations? Only one? More than two? p. 220, no. 7. Mapmaking - It is 62 miles from Denver to Colorado Springs. What should be the distance on a map drawn to a scale 20 miles to an inch? p. 250. Navigation - After a ship sails for 3 hours at an average speed of l8 miles per hour on a course 30° west of souths how far west of its starting point is it? p. 295i no. 1 8 . Number of Everyday-Life Exercises by Classifications Aeronautics 1 Architecture 1 Art 2 Building 2 Carpentry 8 Designing 16 Drafting 5 Engineering 5 Journalism 1 Mapmaking 18 Mapreading 18 Marksmanship 1 Measurement 6 . . . Metallurgy 1 Miscellaneous -27 Navigation 4 Physics 25 Physiology 1 Patterns of Reasoning 89 Salesmanship 1 Sewing 1 Sports 5 Surveying 76 Time-Telling 2 TOTAL 31? Exercises on patterns of reasoning outnumber other everyday-life exercises in this text. Eighty-nine (28.1 percent) of the everyday-life exercises are patterns of reasoning exercises and 228 (71.5 percent) are on physical situations. Despite the tremendous increase in the number of exercises^ the percentage of these that are everyday-life exercises decreases from 8.3 percent in the previous text examined to 7.8 percent in this text. TABLE 3 COMBINED DATA FROM THE ANALYSES OF THE STONE-MILLIS-MALLORY-MESERVE-SKEEN SERIES Authors Texts Patterns Patterns of Seasoning Publication dumber of Percent Percent of Patterns of Seasoning Date Number of Exercises Physical Sits Number Number of Everyday-Life Percent of Everyday-Life Number of Percent of Physical Sits Exercises Stone & Millis Plane Geometry 1916 1174 226 10.7 226 100 - - Stone & Mallory- Modern Plane Geometry 1940 2731 262 9.6 256 97.7 6 2.3 Stone & Mallory New Plane Geometry 1941 3487 247 7.1 230 93.3 17 6.7 Mallory New Plane Geometry 1943 2866 240 8.3 222 92.5 18 7.5 Revised Mallory, Meserve, A First Course in 1959 4089 317 7.8 228 71.9 89 28.1 and Skeen Geometry u>00 A/u m 6 3 D A COMPARISON OF THE EVERYDAY-LIFE THE OF COMPARISON A vrdyLf' hscl iuto Reasoning Situation Physical Everyday-Life' Exercises Exercises Exercises Exercises OF REASONING EXERCISES FROM DATA OF TABLE 3 TABLE OF DATA FROM EXERCISES REASONING OF GRAPH II GRAPH > PHYSICAL SITUATION PHYSICAL .... Exercises atrs of Patterns 9 AND PATTERNS PATTERNS AND I UD --- -- oo Chapter Summary The authors of the textbooks examined in this series seem to have made some attempts to include the current recommendations in their texts. Table 3 and the graph of its data show that the total number of exercises increased, but, although this increase meant more every day-life exercises, the percent of such exercises tended to decrease. No exercises on patterns of reasoning appeared in the first text examined in this series. A few exercises on patterns of reasoning appeared in the 19^0 text, and there were more in the later texts. The 1959 edition contained far more exercises on patterns of reasoning than any previous editions. CHAPTER V REPORT OF ANALYSES III Analyses of five more texts by the same or succeeding authors are reported in this chapter. The Wells-Hart, Hart, Hart-Schult-Swain Textbooks Webster Wells and Walter W. Hart, Plane Geometry, 1915. This text begins with the usual arrangement of topics in plane geometry text of its time by placing definitions, rules, and axioms at the beginning of the course. It includes 1248 exercises and 100 (8.0 percent) of these are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - When walking along a straight line, are you constantly moving in the same direction or not? Answer the same question if you are walking along a curved line. p. 5 j no. 7 . Time-Telling - At what hour do the hands of a clock form a straight angle? p. 1 5 , no. 5 9 * Sports - Draw a line to represent the path of a baseball when the pitcher throws an "out-curve". p. 2 7 * no. 9 6 . Carpentry - The rafters of a saddle roof make an angle of 40° with a level line. What 86 87 angle do the rafters form at the ridge? p. 5 8 , no. 115. Designing - Construct in full size the design for a four inch square tile. Make the decorative arcs 3/8 in. wide. p. 95 > no. 8. Surveying - Find the angle of elevation of the sun whan a monument whose height is 360 ft. casts a shadow 400 ft. high? p. 1 7 8 , no. 117. Number of Everyday-Life Exercises by Classifications Architecture 3 Carpentry 12 Designing 21 Drafting 3 Engineering 6 Farming 2 Landscaping 3 Measurement 6 Miscellaneous 8 Navigation 1 Physics 4 Sewing 2 Sports 3 Surveying 23 Time-Telling 3 TOTAL 100 More exercises appear in surveying and designing in this text than in other classifications of the every- day-life exercises. None are on patterns of reasoning. Webster Wells and Walter W. Hart, Modern Plane Geometry, 1926. The authors state that they have attempted to include in this text the recommendations of the Committee on College Entrance Requirements and the National Committee Report of 1923. The text includes 1687 exercises and 112 (6.6 percent) of them are everyday-life exercises. 88 Examples of Everyday-Life Exercises Classification Exercise Engineering - If four towns, A, B, C, and D are situated so that no three can be connec ted by one straight road, how many roads must be constructed if each town is to be connected to the others by a straight road? Illustrate by a drawing, p. 5* no. 6. Horticulture - An orchardist, setting out trees, first places a tree at each end of a row. How then may he locate the other trees of that row so that they will be in a straight line,-without stretching a rope between the end trees? p. 24, no. 1 0 9 . Measurement - How does the ratio of one quart to one gallon compare with the ratio of one peck to one bushel? p. 114, no. 6l. Sports - How far is the pitcher of a baseball nine from each base when he is in the center of the box? p. 176, no. 111. Designing - A designer wishes to make a pattern for the octagonal top of a taboret whose longest diagonal is to be 18 in. Make a scale drawing of the octagon, letting 1 in. represent 3 in. p. 2 3 8 , no. 22. Surveying - The sides of a triangular field are 10 .rd., 8 rd., and 9 rd., respectively. What is the area of the field? p. 300, n o . 22. Humber of Everyday' •Life Exercises by Classifications Aeronautics 2 Architecture 4 Astronomy 2 Building 6 Carpentry 13 Designing 22 Drafting 8 Engineering 2 Horticulture 3 Industry 5 Measurement 6 Miscellaneous 3 Physics 1 Sewing 1 Sports 5 Surveying 26 Time-Telling 3 tfOlAL l'l'2 Again in this text as in others, there are more exercises in surveying and designing than in other classi fications. There are no everyday-life exercises on patterns of reasoning. Walter W. Hart, Plane Geometry, 1950. This text is a revised edition of the earlier texts in this sequence of books. The author states that it is designed to help teachers guide pupils to mastery of the subject of geometry and of related subjects to the extent of the pupils' capability. Some fundamental concepts of solid geometry are included on pages 352 to 361. Chapter 8 gives an introduction to analytic geometry and Chapter 9 is on patterns of reasoning. There are 2085 exercises included and 218 (10.4 percent) of these are everyday-life exercises. Examples of Everyday-Life Exercises Class ification Exercise Miscellaneous - Try to obtain for your geometry note book pictures in which geometrical shapes appear, p. 1, no. 2. Pattern of What conclusion do you reach from the Reasoning - following statements ? Is the conclusion a true one? 1. All cows seen on an automobile trip had black and white markings; 2. No cream rises to the top of bottles of milk bought at a certain store, p. 2 7 , no. 2 Pattern of What is the converse of the following Reasoning - statement? Is the original statement true? Is the converse true? "If Mr. Smith owns a Buick, he owns an auto mobile." p. 8 5 , no. 2. Designing - Construct a Maltese cross having the dimensions indicated, p. 1 6 5 , no. 20. Figure 6.— A design of a Maltese Cross Gardening - By a drawing represent a rectangular flower bed. Draw the locus of plants that are equidistant from the two long sides, p. 173, no. 7. Surveying - How far is it from one corner to the opposite corner of a rectangular lot that is 6 rods wide and 8 rods long? p. 213, no. 10. Pattern of Some girls do not like to study Reasoning - geometry. What faulty generalization is based on this fact? p. 2 9 9 , no. 2. Pattern of Complete the final statement: Reasoning - "Wisconsin is north of Illinois. Illinois is north of Kentucky. There fore ..." p. 301, no. 1. Pattern of Write the converse of the following Reasoning - statement. If the statement and the converse of it are true, what other statements are true? If both are not true, what other statement is true? "If a boy is a senior, he is a student in school." p. 309, no. 11. Number of Everyday-Life Exercises by Classifications Aeronautics 11 Architecture 3 Astronomy 3 Building 5 Carpentry l6 Designing 13 Drafting 8 Engineering 5 Gardening 4 Measurement 9 Metallurgy 2 Miscellaneous 29 Navigation 1 Physics 4 Patterns of Reasoning 71 Sewing 3 Surveying 26 Tiling 3 Time-Telling 2 W A t ------215- Exercises on patterns of reasoning outnumber by far the everyday-life exercises of other classifications in this text. Seventy-one (32.6 percent) are on patterns of reasoning and 147 (67.4 percent) are on physical situations. Walter W. Hart, Veryl Schult, and Henry Swain, Plane Geometry and Supplements, 1959. The authors of this text say that it offers a fundamental course in plane geometry and that it includes supplementary units on some of the ideas recommended by committees and by the Commission on Mathematics of the College Entrance Examination Board. Units relating ideas of plane geometry to solid geometry and analytic geometry appear throughout the text. Chapter 8 is on trigonometry 92 and Chapter 13 is on patterns of reasoning. The authors recommend that gifted students study the supplementary units. There are 2147 exercises included in the text and 243 (11.3 percent) of these are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - During the year, try to find for your notebook pictures showing geometrical forms in natural objects, p. 3 » no.l. Pattern of Three straight roads connect villages Reasoning A, B, and C in pairs. Is AC longer or shorter than AB -f BC? p. 3 8 , no. 8. Pattern of Is the converse of the following state Reasoning ment true?: "If a pupil is late arriving at school, he is tardy." p. 100, no. 9. Surveying - From the top of a cliff 240 ft. high, the angle of depression of a horseman on the plain below is 25°. How far is the horseman from a point on the level of the plain directly below the observer? p. 249, no. 5* Aeronautics - From an airplane flying at an elevation of 8500 ft., the angle of depression of a landing field is 18° 30'. Find the ground distance to the field, p. 252, no. 9 . Horticulture - How many tulip bulbs are needed for a circular flower bed that is 10 ft. in diameter if 24 sq. in. on the average are allowed for each bulb? p. 3 0 6 , no. 7 . Measurement - A cylincrical gas holder is 60 ft. in diameter and 50 ft. high. How many cubic feet of gas will the tank hold? p. 3 5 6 , no. 5. 93 Pattern of Write a conclusion and discuss its Reasoning - truth or falsity: "If a student is to study physics, he should have at least three years of mathematics. Joe is going to study physics. There fore ..." p. 3 8 0 , no. 12. Pattern of A. Write the converse, inverse, and Reasoning - the contrapositive for the following statement; B. Tell which of the four statements are true or probably true; "If you live in southern Florida, you do not need a fur coat." p. 3 8 6 , no.l4. Number of Everyday-Life Exercises by Classifications Aeronautics 2 Architecture 3 Artillery Firing 1 Astronomy 2 Building 10 Carpentry 15 Designing 4 Drafting 2 Farming 2 Gardening 2 Horticulture 1 Industry 7 Measurement 33 Miscellaneous 20 Navigation 2 Physics 11 Patterns of Reasoning 90 Sewing 3 Sports 10 Surveying 22 Time-Telling 1 lOTAL 243 Exercises on patterns of reasoning outnumber by far other everyday-life exercises in this text and measurement exercises exceed the exercises in surveying. Ninety (37*0 percent) are on patterns of reasoning and 153 (6 3 .O percent) are on physical situations. Walter W. Hart, Veryl Schult, and Henry Swain, New Plane Geometry and Supplements, 19b4. In this text the supplementary units again appear, and again are recommended for the gifted students. Chapters on trigonometry and reasoning appear as in the previous text. The authors also propose in this text a departure from traditional subject matter toward emphasis on the current efforts to modernize the organization and development of elementary geometry for-high schools. Consequently, innovations such as the following are included: 1. Point, straight line, plane, and space appear as undefined elements. (p. 13). 2. Order of points is introduced with out a formal definition. (pp. 13, 14, 1 6 ). 3. Geometrical figures are defined as sets of points. (pp. 13, 15 > 1 9 ). 4. Postulates replace some concepts formerly defined. (pp. 17, 1 8 , 23). 5. Discovery exercises are included. 6. Proofs of some propositions are omitted as suggested by Moore. There are 2l6l exercises in the text and 246 (11.4 percent) of these are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - What geometrical shape does a baseball have? Name another object that has the same shape, p. 12, no. 11. 95 Pattern of "If an automobile has defective Reasoning brakes, it is unsafe." (a) What is the converse statement? (b) Is the converse statement true? p. 100, no. 11. Pattern of State all the possible converses: Reasoning - "If an automobile engine is in good condition, if it has a supply of gasoline, if its ignition system is in working order, then, it will run when the starter is pressed." p. 117, no. 6. (Note: This exercise allows for three converses as pointed out in Lazar's study of 1 9 3 8 . ) Measurement - How long must a tent rope be to reach from the top of a 12 ft. pole to a point that is l6 ft. from the foot of the pole? p. 2 2 9 , no. 4. Surveying - A road from town A north 20 miles to town B turns 30° west of north and goes 50 miles to C. How long would a new road be to go directly from A to C? p. 257j no. 7 . Pattern of In the following statement give what Reasoning - you think is the hidden assumption and tell whether you think the assumption is usually true, usually false or neither: "Since Mr. Simpson's hair is gray, he must be quite old." p. 375, no. 6. Pattern of What conclusion follows by indirect Reasoning - proof from the information given: "There are six problems on a page from which a study assignment has been made. Ruth knows that four problems have been assigned for homework. She is sure that the number 5 was not assigned and number 1 was assigned on a previous lesson." p. 3 8 0 , no. 6. Number of Everyday-Life Exercises by Classifications Aeronautics 1 Architecture 3 Artillery Firing 2 Astronomy 1 Building 7 Carpentry 15 Designing 3 Drafting 1 Engineering 5 Farming 3 Gardening 3 Horticulture 1 Industry 8 Measurement 32 Miscellaneous 36 Navigation 2 Physics 11 Patterns of Reasoning 82 Sewing 4 Sports 9 Surveying 17 TOTAL 246 Exercises on patterns of reasoning again far out number the everyday-life exercises of other classifications. Eighty-two (33*3 percent) are on patterns of reasoning and 164 (66.7 percent) are on physical situations. td sl s ! go go p r £P CD CD % 4 4 4 H t—1 c+ CO c+ CO c+ c+ H H p1 a * 3 v CO CO o P p 4 H - CO H * CO 8? 8? CO o P o P o o P* P1 &3 P P FF H H 4 4 H c+' c+ c+ c + V* !> 8? a CO 4 ) ►d T) i> CD P H H 8= H co a P p p< P P P CD U ►O T) H CD g 4 g o ►d H CD P S H P 3 H3 CD P CD TJ £ CD B CD P O o H O X CD c+ S 3 P 3 e+ P CO CD CD CD CO c+» c+ c + . g c + co O 4 4 4 3 « CD $ c+ 8? O k ! 4 3 Ca CD bd c+ co « HH HH H VO 00 OO OO OO Publication ov On on ro H Date p- 00 o oo on ro ro ro HH Number of tT» H H o OO ro t-* Ov p- 00 00 P" Exercises CO H -4 on -SI 00 I ro ro ro H r—1 Number of wg p- p- H H o Eve ryday-Life 1-3 oo U> 00 ro o I Exercises CO o HH H oo Percent of •H ••••HO 00 p- 00 P" OO o Everyday-Life 6I CO HH HHH Number of CT\ on -P- H O g p- 00 -o ro O Physical Sits. Oo CT\ OO H H Percent of CO o\ 00 o O bd • • • o O Physical Sits. W -S] o P- H M Number of CO 00 oo i 1 ro o H Patterns of Reasoning u> 00 00 Percent of U) -01 ro •• • i 1 Patterns of 00 o oo Reasoning H H U Af 6 & fts too A COMPARISON OF THE EVERYDAY-LIFE3 PHYSICAL SITUATION, AND PATTERNS PATTERNS AND SITUATION, PHYSICAL EVERYDAY-LIFE3 THE OF COMPARISON A Everyday-Life Everyday-Life Exercises OF REASONING EXERCISES FROM DATA OF TABLE 4 TABLE OF DATA FROM EXERCISES REASONING OF ______' hscl iu- Reasoning Situa- Physical tion tion Exercises. . Exercises . GRAPH III GRAPH atrs of Patterns ----- oo Chapter Summary The number of exercises in these texts were found to increase. The percent of the total number of exercises that were everyday-life exercises tended to in crease , which indicates some variance from findings in the two previous series examined. Everyday-life exercises in surveying were dominant in the earlier editions of the textbooks. Exercises on patterns of reasoning in everyday-life appeared in this series and increased in number and percent to the point that they became the dominant type by 1950. This phenomenon in each of the three series examined was probably due to the general acceptance, resulting from the transfer of training controversy, that reasoning in everyday life was a major objective in the teaching of plane geometry. A reasonable number of the everyday- life exercises were found to be in measurement and designing, although numerous other classifications were noted. It appeared that the authors sought to implement in their texts many of the current professional recommenda tions and suggestions. CHAPTER VI REPORT OF ANALYSES IV Analyses of representative plane geometry textbooks copyrighted from 1903 to 1965 wi.ll be reported in this chapter. Some of these texts are successors to earlier editions. Comparisons between earlier and later editions will be made in these cases. Selected Textbooks, 1903 to 1965 Alan Sanders, Elements of Plane and Solid Geometry, 1903• There are 741 exercises in the plane geometry section of this text and five of them are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - If a line be drawn on this page parallel to the upper edge, show that it is parallel to the lower edge, p. 3 6 , no. 1 0 9 . Surveying - The shadow cast by a church steeple on level ground is 27 yd.j> while that cast by a 5 ft. vertical rod is 3 ft. long. How high is the steeple? p. 173> no. 7. Designing - Three radii are drawn in a circle of radius 2a, so as to divide the circum ference into three equal parts; and, with the middle of these radii as centers, arcs are drawn each with the 100 101 radius a, so as to form a closed figure (trefoil). Show that the length of the perimeter of the trefoil is equal to that of the circle, and find its area. p. 246, no. 42. Figure 7.— A Trefoil inscribed in'a circle. Number of Everyday-Life Exercises by Classifications Designing 1 Miscellaneous 3 Surveying 1 TOTAL ----- 5“ To the nearest tenth, just 0.7 percent of the exercises in this section of the text are everyday-life exercises on physical situations. None are on patterns of reasoning. Arthur Schultz and Frank L. Sevenoak, Plane Geometry Revised by Schultz, 1915. There are 1558 exercises in this text and 137 (8.8 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - What kind of a surface is represented by each wall of a room? p. 4, no. 4. Surveying - Wishing to determine the distance across a pond, A B, we place a stake at a convenient station C, and by sighting from A, we locate A in the prolonga tion A C, and make A C ■ C A 1. Similarly we produce B C to B 1 so that B C ■ C B'. Which line must we measure 102 to obtain A B? Prove your statement, p. 2 6 , no. 132. B1^ ------\ ?A' N. / A B Figure 8.— Similar triangles - the measurement of an inac cessible distance. Measurement - A rectangular field is 24 yd. long and 15 yd. wide. Find the area. p. 208, n o . 1 0 8 6 . Carpentry - From a given round log a rectangular beam is to be cut so as to make its cross section A B C D as large as possible. Construct A B C D. p. 2 8 5 , no. 8 9 . Figure 9 .— A log from which a rectan gular beam is cut. Number of Everyday -Life Exercises by Classifications Architecture 6 Carpentry 3 Drafting 8 Designing 10 Engineering 12 Mapmaking 7 Measurement 16 Miscellaneous 3 Navigation 6 Photography 1 Physics 8 Sports 2 Stonecutting 1 Surveying 51 Time-Telling 3 trotfAL 137 103 The number of surveying exercises exceed the everyday-life exercises in other classifications. None of the everyday-life exercises are on patterns of reasoning. Royal A. Avery, Plane Geometry, 1925. There are 1034 exercises in this text and 57 (5*5 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - What kind of angle is formed at the corners of your book cover? What kind of angle is formed by two adjoining spokes of an automobile? p. xxxiii, no. 5. Photography - A photograph 2 1/4" by 3 1/4" is enlarged so that it is 4 3/4" wide. How long is it? A figure 1" long on the original is how long on the enlarge ment? p. 2 2 9 , no. 5. Architecture - Architects in drawing their plans use a scale of 1/4 inch to a foot. If a room represented on these plans is 4 1/4" by 3 3/4", what are the dimen sions of the room? p. 2 2 9 , no. 6. Sports - A tennis court is jQ ft. long and 36 ft. wide. What is the area of the playing field? p. 245, no. 2. Physics - If the diameter of a locomotive wheel is 76 inches, find the number of revolu tions per minute made by the wheel when the locomotive is going 54 miles per hour. (1 mile = 5280 ft.), p. 286, no. 3. • 104 Number of Everyday-Life Exercises by Classifications Architecture 2 Building 3 Carpentry 1 Designing 3 Engineering 4 Miscellaneous 3 Photography 2 Physics 3 Plumbing 1 Sports 7 Surveying 26 Time-Telling 1 Tinsmithing 1 toTTl — 57 Surveying exercises are the largest in number of everyday-life exercises in this text. None are on patterns of reasoning. Arthur Schultz, Frank Sevenoak, and Elmer Schuyler, Plane Geometry, Revised by Schuyler, 1933• The earlier edition by two of these authors had 1558 exercises of which 137 (8.8 percent) were everyday- life exercises. There are 1471 exercises in this text and 149 (10.1 percent) of them are everyday-life exercises. There were no everyday-life exercises on patterns of reasoning in the earlier text, and there is just one (0.7 percent) in this text. Examples of Everyday-Life Exercises Classification Exercise Physics - Over an angle of how many degrees does a spoke of a wheel sweep when the wheel makes 1/4 of a revolution? 1/6 of a revolution? 2 revolutions? p. 7> no.4. 105 Pattern of Indicate the hypothesis and the con- Reasoning - elusion of the following statement: "If iron is heated, it expands." p. 15, la. Time-Telling - What is the angle between the hands of a clock at 12 min. past 11? At 6:30? At 8:45? p. 97j no. 1. Surveying - How far is a man from his starting point, if he first travels 5 miles north, then 12 miles east and then 4 miles south? p. 2 3 2 , no. 7* Tinsmithing - Find the number of square inches of tin that would be wasted in cutting the largest possible circular disk from a piece in the form of an equilateral triangle 12 in. on a side. p. 259 no. 9* Number of Everyday-Life Exercises by Classifications Architecture 7 Carpentry 5 Designing 7 Drafting 9 Engineering 12 Mapmaking 7 Miscellaneous 1 Navigation 7 Physics 11 Patterns of Reasoning 1 Surveying 78 Time-Telling 4 TOTAL 149 One everyday-life exercise (0.7 percent) in this text is on patterns of reasoning, while those on survey ing far outnumber other everyday-life exercises. One hundred forty-eight (99*3 percent) are on physical situations. io6 Ernest R. Breslich, Plane Geometry-Purposeful Mathematics, 1938. There are 1864 exercises in this text and 197 (10.6 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - Why do people like to cut across the corner of a lot rather than walk around it? p. 1 8 , no. 2. Pattern of Complete the cycle by stating the con Reasoning - clusion that follows: "Cars with defective brakes are unsafe for driving. John’s tires are defec tive." p. 4o, no. 1. Sports - The goal lines on a football field are perpendicular to the sides of the field. Prove that they are parallel, p. 121, no. 6. Pattern of Find the hidden assumptions: Reasoning - "Some children eat until they become ill. Ted is absent from school today. I believe that he ate too much and is ill." p. 1 2 9 , no. 3. Surveying - A road is to be laid out cutting two given parallel roads so that one of the interior angles is 5 times as large as the other interior angle on the same side. Find the number of degrees in each angle, p. 133» no. 1 Navigation - A steamer is traveling east at the rate of 16 miles an hour while the wind is driving it northward at the rate of 4 miles an hour. From a drawing determine the distance and direction of the steamer from the starting point after one hour of travel, p. 154, no . 10. Number of Everyday-Life Exercises by Classifications Architecture 4 Building 5 Carpentry 9 Designing 3 Drafting 3 Engineering 14 Gardening 1 Measurement 4 Miscellaneous l4 Navigation 2 Physics 16 Patterns of Reasoning 38 Sports 7 Surveying 77 tfotfAL 19T" The number of exercises on patterns of reasoning is greater in this text than in texts examined previously in this chapter, but surveying exercises again outnumber all others. Thirty-eight (19*3 percent) are on patterns of reasoning and 159 (80.7 percent) are on physical situations. Leroy H. Schnell and Mildred Crawford, Clear Thinking - An Approach Through Plane Geometry, 19387 This was probably the first textbook designed to give more emphasis to patterns of reasoning in everyday life than earlier texts. Four of its twenty units are assigned to this emphasis. Unit One is entitled "Everyday Reasoning;" Unit Five is entitled "Assumptions Unit Nine is entitled "Hints for Improving Your Reasoning in Everyday Situations;" and Unit Seventeen is entitled "Reasoning in Nonmathematical Situations." There are 738 exercises in the text and 233 (31.6 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Pattern of We live in a country which is rich in Reasoning - material goods, but thousands of people are literally starving or are so poor that life holds little happiness for them. Possibly you are one of this group or you know someone who is. Would intelligent, unprejudiced reason ing, on the basis of facts, help solve this problem? p. 7 > no. 1. Locus - Imagine you are telling someone where to place the stand for a speaker's microphone in an empty rectangular room. It is not to be placed against a wall. What exact directions would you give so that there could be no mistake about the location, p. 5 6 , no. 1. Surveying - A surveyor wishes to find the distance XY, across a march. He lays off XB and AY, making AR = RY and XR = RB. How can he measure the distance across XY by measuring another line? Prove your answer, p. 1 2 6 , no. l6. A Figure 10.— Similar triangles - Measure ment of an inaccessible dis tance . Pattern of Katherine is younger than Mary. Five Reasoning - years from now, which one will be older p. 3l4, no. 4. Measurement - A rectangular vacant field is 30 yards by 40 yards. What distance is saved by walking diagonally across the field? p. 348, no. 4. 109 Number of Everyday-Life Exercises by Classifications Carpentry 2 Cooking 1 Designing 6 Locus 15 Measurement 21 Metal Industry 1 Miscellaneous 13 Navigation 1 Patterns of Reasoning 158 Sewing 1 Surveying 10 Time-Telling 4 tfOTAL 53T One hundred-fifty eight (67.8 percent) of the everyday-life exercises are on patterns of reasoning and 75 (32.2 percent) are on physical situations. This represents the largest percentage found on patterns of reasoning in any of the texts examined. George Birkhoff and Ralph Beatley, Basic Geometry, 1941. This text is a bit different in its approach from the usual text of its time. It organizes emphasis on the meaning of "proof" and "demonstration" and other fundamental geometric ideas around five fundamental postulates and seven basic theorems. There are 700 exercises in the text and 117 (16.7 percent) are everyday- life exercises. Examples of Everyday-Life Exercises Classification Exercise Pattern of Point out what is given and what is Reasoning - to be proved in the following: "If your dog is barking, there is a 110 stranger on the premises." p. 24, 5g. Pattern of Point out the error in the reasoning Reasoning - in the following: "A traveler reported that a coin had recently been unearthed at Pompeii bearing the date 70 B.C." p. 31j no. 10. Measurement - A tomato can is 5 inches high and 4 inches in diameter. How far is it from the center of the can to a point on the rim? p. 1 0 3 * no. 20. Architecture - An architect submits a design for a memorial window drawn to the scale of 1 to 20. Compare the area of the window as drawn with the area of the actual window, p. 2 0 7 * no. 1. Pattern of Mr. Barkley, a dealer in coal, fire Reasoning - wood, and fuel oil, delivered a cord of firewood by mistake at the home of Mr. Burnham.' Mr. Burnham knew that he had ordered no firewood but burned it nevertheless. Must he pay for it? p. 2 7 3 , no. 10. Number of Everyday-Life Exercises by Classifications Architecture 2 Carpentry 5 Designing 5 Drafting 12 Engineering 4 Farming 1 Mapmaking .3 Measurement l4 Physics 3 Pattern of Reasoning 66 Time-Telling 2 TOTAL lTT“ More than 'half of the exercises are on patterns of reasoning, and none on surveying. Sixty-six (56.4 per cent) are on patterns of reasoning and 51 (43.6 percent) are on physical situations. Ill Royal A. Avery, Plane Geometry, 1947. One of the textbooks examined previously in this selection is by Avery. There are 2133 exercises in this text while the earlier one had 1034. There are 128 (6.0 percent) everyday-life exercises in this text, but only 57 (5.5 percent) in the earlier text. There is no doubt that the increase of the number of exercises in the later text is noteworthy. Though the percent in crease of everyday-life exercises may not be considered a significant departure, this case does lend some support to the opinion that the number of exercises in textbooks increased during this time and everyday-life exercises tended to decrease. The comparisons of surveying and patterns of reasoning exercises in everyday-life will be mentioned later. Examples of Everyday-Life Exercises Classification Exercise Carpentry - Why is a sagging gate or screen made rigid by a diagonal brace? p. 1 5 6 , no. 2 . Pattern of All voters are citizens. Mr. Brown Reasoning - is a citizen. Mr. Brown (is, is not, is not necessarily) a voter, p. 164, no. 2 3 6 . Architecture - It is required to construct a circular arch with a span of 60 feet, the height of its center is 15 feet above the level of the plane from which the arch is contructed. Find the radius, p. 314, no. 9 112 Sports - A swimming pool is 60 ft. long, 20 feet wide, 3 feet deep on one end and 8 feet deep at the other end. Find the number of square feet in the bottom of the pool. p. 363 j no. 7* Pattern of We constantly hear statements similar Reasoning - to the one which follows. Is the assumption sufficient for the conclu sion?: "The subject must be poorly taught, because so many pupils fail." p. 401, no. 7. Physics - The diameter of a bicycle wheel is 28 inches; the wheel makes 2 3/7 revo lutions for each revolution of the pedal. How many revolutions of the pedal must be made when riding a distance of 3 miles? p. 433> no. 1 8 . Number of Everyday-Life Exercises by Classifications Architecture 5 Astronomy 3 Carpentry 11 Designing 1 Drafting 3 Gardening 1 Mapmaking 1 Measurement 3 Miscellaneous 2 Nivigation 3 Physics 13 Photography 3 Plumbing 1 Patterns of Reasoning 14 Sewing 2 Sports 7 Surveying 51 Tiling 1 Time-Telling 2 TOTAL ' 128 In the earlier text there were 26 exercises on i ■ surveying, but 51 in the later one. The earlier-had no exercises on patterns of reasoning and the later text 113 has 14 (10.9 percent) and ll4 (89.1 percent) on physical situations. Rachel Kenniston and Jean Tully, Plane Geometry, 1953. The authors proposed as their first objective in this text "to point out the geometric aspects of our environment. . . . There are 2158 exercises included and 273 (12.7 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Time-Telling - Through what angle does the minute hand rotate in five minutes2 p. 2 8 , no. 21. Pattern of Jane is the same age as Mary, Ruth is Reasoning - also the same age as Mary. What do you conclude about the relative ages of Jane and Ruth? p. 59 5 no. 1. Pattern of List the facts you accept, and the Reasoning - assumption you believe most probable : "Tom attends school in a town thirty miles from home. His father decides it is best to buy him a car." p. 72, no. 2. Navigation - If a ship which is sailing due west changes its course to northeast, through what angle does it turn? p. 75* no. 5. Taxes - Discuss the exercise from the point of view of proportion. If possible state the two equal ratios involved: "Taxes of $40 were assessed on a house valued at $5000; the tax was 8 mills on a dollar." p. 284, no. 5 . -'-Rachel P. Kenniston and Jean Tulley, Plane Geometry (Boston: Ginn and Company, 1953 )j P. v. 114 Number of Everyday-Life Exercises by Classifications Aeronautics 19 Art 1 Building 7 Carpentry 9 Designing 9 Drafting 9 Electronics 1 Engineering 7 Foundry Work 2 Gardening 3 Insurance 1 Journalism 1 Miscellaneous 21 Music 1 Navigation 4 Photography 1 Physics 7 Patterns of Reasoning 113 Sports 2 Surveying 39 Time-Telling 6 "TOTAL------55T" Far more everyday-life exercises in this text are on patterns of reasoning than any other classification. One hundred-thirteen (4l.O percent) are on patterns of reasoning and 160 (59.0 percent) are on physical situations. Leroy H. Schnell and Mildred Crawford, Plane Geometry - A Clear Thinking Approach, 1953• An analysis of a textbook written earlier (1938) by these authors has already been presented. The 1953 edition, directed much emphasis toward patterns of reasoning in everyday-life and nonmathematical situations. The 1938 edition included 738 exercises, but the 1953 edition included 2113. Two hundred thirty-three (31.6 per cent) of the exercises in the earlier text, and 291 115 (13.8 percent) in the later text are everyday-life exercises. One hundred fifty-eight (67.8 percent) of the everyday-life exercises in the earlier, and 156 (53.6 percent) in the later text are on patterns of reasoning. Seventy-five (32.2 percent) in the earlier, and 135 (46.4 percent) in the later one are on physical situations. Examples of Everyday-Life Exercises Classification Exercise Pattern of Hate the conclusion in the following Reasoning - argument as good if it follows as a necessary result of the first two state ments ^ or as poor if it does not necessarily follow: "All boys must pass three subjects to be eligible for the baseball team. Frank is playing on the baseball team. Therefore Frank passed three subjects." p. 7, no. 3. Time-Telling - When the hands of a clock point to 3 o’clock what angle is formed? p. 5 6 , no. 3 . Pattern of Analyze the Preamble of the Constitu Reasoning - tion. List the assumptions back of it. What implications do these assumptions have for us today? p. 1 0 2 , no. 8 . Locus - Show that locus is involved whan a ship's position at sea is given as latitude 4oo north and longitude 30° west. p. 2 8 0 , no. 6 . Measurement - A rectangular vacant field is 30 yards by 40 yards. What distance is saved by walking diagonally across the field? p. 323, no. 4. Humber of Everyday-Life Exercises by Classifications Aviation 5 Chemistry 1 Designing 5 116 Investment 7 Locus 20 Measurement 40 Miscellaneous 17 Navigation 1 Physics 5 Real Estate 3 Patterns of Reasoning 156 Salesmanship 3 Surveying 25 Time-Telling 3 ~toee----- 251" The increase in the number of exercises in the later text by these authors is in keeping with what was generally occurring in plane geometry textbooks. There are more everyday-life exercises in the later text* but the percentage they represent of the total number of exercises decreases. In like manner, the percentage of the everyday-life exercises decreases although it is much larger than that for other geometry books published during the period under examination here. William G. Stone, Avery*s Plane Geometry, Revised, 1959* The text from which this revised edition was written (Avery, Plane Geometry, 1947) contained 2133 exercises of which 128 (6.0 percent) were everyday-life exercises, with l4 (10.9 percent) patterns of reasoning, and 114 (8 9 .1 percent) on physical situations. This revised edition contains 2335 exercises of which 132 (5.6 percent) are everyday-life exercises with 26 (19.7 percent) on patterns of reasoning, and 106 (80.3 percent) 117 on physical situations.- The comparison reveals an increase in the number of exercises, a slight increase in the number of everyday-life exercises, a slight decrease in the percent the everyday-life exercises represent of the total number of exercises, and an increase in both number and percent of the exercises on patterns of reasoning. Examples of Everyday-Life Exercises Classification Exercise Pattern of State the conclusion we are forced to Reasoning - accept, if we accept the given state ments : "All engineers must study mathematics. John is an engineer. ’ Therefore, ..." p. 1 8 6 , no. 3. Sewing - Ethel has a piece of linen 10 1/2 inches square, which she wishes to use in making a handkerchief. In hemstitching the edge she pulls the thread so that the hem shall be 1/4 inch wide. How should she plan her work? p. 2 5 3 > n o . 12. Surveying - A tower casts a shadow 110 feet long when a vertical 10-foot pole cast a shadow 8 feet long. How high is the tower? p. 308, no. 3 . Sports - In a baseball game a boy caught a ball on the foul line 120 feet beyond first base. What is the distance from the point where the ball was caught to third base? p. 320, no. l4. Architecture - A circular arch over a door has a span of 3 feet; the highest point of the arch is 12 inches above the line connecting the ends of the arch. What is the radius of the circular arch? p. 323* no. 9 . 118 Number of Everyday-Life Exercises by Classifications Architecture 3 Astronomy 3 Building 4 Carpentry 6 Designing 4 Drafting 2 Engineering 2 Measurement 5 Miscellaneous 4 Navigation 3 Photography 2 Physics 19 Patterns of Reasoning 26 Sewing 1 Sports 5 Surveying 40 Time-Telling 3 TOTAL 1'32' The number of surveying exercises is greatest in this text but a good number are on patterns of reasoning. Wilson Goodwin, Glen Vannatta, and Harold P. Fawcett, Geometry— A Unified Course, l§6l. Since one of the authors of this text had done a study emphasizing the development of logical reasoning using nongeometric situations * it was anticipated that this text would have a high percentage of everyday-life exercises and that the greatest number of these would be on patterns of reasoning. There are 1569 exercises in the text and 138 (8.8 percent) are everyday-life exercises. Chapter One is on inductive reasoning. Chapter Four includes deductive reasoning, and Chapter Twenty is on coordinate geometry. 119 Examples of Everyday-Life Exercises Classification Exercise Pattern of Determine whether or not the two Reasoning - statements lead necessarily to a third statement: (a) "All sophomores are bright] (b) I am a sophomore." p. 5 5 * no. 1 . Pattern of Criticize the following analogy: Reasoning - "If Mary is allowed to go to the party, I should be permitted to go." p. 32^ no. 1 . Mus ic - The following are the vibrations per second (scientific pitch) for eight notes of the major music scale of beginning at middle C: C DEF GA BC 256 288 320 341 1/3 384 426 2/3 48o 512 (a) What is the ratio of middle C to the C which is an octave higher in pitch? (b) What is the ratio of middle C to E of the major scale? (c) What is the ratio of F to G? p. 3 H j no. 1 1 . Number of Everyday-Life Exercises by Classifications Aeronautics 2 Agriculture 1 Building 3 Butchering 1 Farming 2 Gardening 2 Measurement 7 Metallurgy 4 Miscellaneous 15 Mus ic 1 Navigation 2 Patterns of Reasoning 43 120 School Life 9 Sports 3 Surveying 4l Time-Telling 1 Tinsmithing 1 TOTAL “T B S As anticipated valid and invalid reasoning patterns were emphasized, though almost an equal number of exercises were on surveying. Forty-three (31.2 percent) are on patterns of reasoning and 95 (68.8 percent) are on physical situations. Kenneth Henderson, Robert Pingry, George Robinson, Modern Geometry - Its Structure and Function, 1 9 6 2 . The title of this text seems to imply that much emphasis will be given to the deductive structure of plane geometry and that probably few of its exercises might be — classified as everyday-life exercises. This was actually found to be the case. There are 5830 exercises in the text and just 72 (1.2 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classificati Exercise School Life Use a book and your desk top to illus trate a dihedral angle, p. 2 7 * no. 3 . Geography - State the number of elements in the following set: "The set of states in the United States." p. 172, no. 1. Pattern of State the contrapositive of the Reasoning - following statement. In an informal way show that the statement and its contra positive are equivalent conditionals : "If Mr. Jones is a member of the PTA, then he is a parent or a teacher." p. 19^* no. 6 . 121 Measurement - Describe how you could calculate the height of a tree by measuring the sha dow of a yardstick, p. 234, no. 1 2 . Aeronautics - An airplane climbs 1000 feet in traveling one mile over the ground. Find its angle of climb to the nearest degree, p. 5 1 5 * no. 1 0 . Number of Everyday-Life Exercises by Classifications Aeronautics 1 Building 4 Carpentry 2 Drafting 1 Engineering 4 Farming 3 Geography 8 Measurement 2 Metallurgy 1 Miscellaneous 8 Physics 5 Plant Life 1 Patterns of Reasoning 5 School Life 8 Sewing 5 Surveying 13 Time-Telling 1 tT'OtTAL 72 Although this text has a small percentage of everyday-life exercises many of the illustrations and pictures refer to everyday-life situations. Five (6.9 percent) are on patterns of reasoning and 67 (93.1 per cent) are on physical situations. Wilson Goodwin, Glen Vannatta, and Harold Fawcett, Geometry— A Unified Course, 1 9 6 5 . This text is a revision of the 1961 text by these authors. They state in the preface that this edition reflects the most recent ideas in the teaching of geometry. Chapters on inductive and deductive reasoning and coordinate geometry are included, as in the first text. This text includes topics on the language and concepts of sets which were not in the first edition. More emphasis is put on deductive structure and student discovery of geometric relationships than in the first text. There are 2347 exercises in this text compared to 1569 in the first. One hundred eighty-seven (7-9 percent) of the exercises in it are everyday-life exercises compared to 138 (8.8 percent) in the first. Seventy-one (38.0 percent) of the everyday-life exercises in this text are on patterns of reasoning and 43 (31.2 percent) are on patterns of reasoning in the first. One hundred sixteen (62.0 per cent) in this text and 95 (68.8 percent) in the first are on physical situations. This comparison shows increases in the number of everyday-life exercises , the number on patterns of reasoning, and the percent on patterns of reasoning, but there is a decline in the percent of every day-life exercises on physical situations. Examples of Everyday-Life Exercises Classification Exercise Pattern of ^The student council of Leesville High Reasoning - School adopted a regulation that each student organization in school must be sponsored by a teacher. A group of students in this school wished to organize a Reading Club and invited the school librarian to be their sponsor. In view of the regulation 123 adopted by the council is it possible for her to serve in this capacity? p. 14, no. 1 . School Life - A teacher at a university may also be a student at that university. Does this individual represent an inter section or a union of the set of teachers and the set of students ? p. 2 9 , no. 7 Pattern of Write the following statement in the Reasoning - "if - then" form. Identify the antecedent by the letter £_ and the consequent by the letter c[_. Write the converse of the statement and determine its truth if you can: "A resident of Missouri is a resident of the United States." p. 87 ^ no. 1 7 . Physiology What type of symmetry is represented by the skeleton of the human body? p. 2 2 1 j no. 3 . Measurement - The lateral area of a flower pot in the form of a frustrum of a regular hexa gonal pyramid is 108 square inches. The lengths of the sides of the bases are 2 and 4 inches. Find the slant height, p. 5 2 3 j no. 3 . Number of Everyday ■Life Exercises by Classifications Aeronautics 1 Astronomy 30 Building 8 Butchering 1 Carpentry 3 Engineering 4 Farming 2 Gardening 2 Measurement 6 Metallurgy 9 Miscellaneous 9 Music l Navigation 1 Physiology 1 Pottery 1 Patterns of Reasoning 71 School Life 4 Sports 4 Surveying 26 Tiling 1 Tinsmithing 2 “TOTEE IBT" Exercises on patterns of reasoning outnumber other everyday-life exercises. The number of exercises in astronomy and surveying, in order, are next largest. TABLE 5 COMBINED DATA FROM ANALYSES OF SELECTED TEXTS - 1903 TO 1965 0 0 CO CQ p -P a •H •H •H •H ft 0 ih Ch ft co Sanders Elements of Plane 1903 741 5 0.7 5 100 — — and Solid Geometry Schultz and Plane Geometry re 1915 1558 137 8.8 137 100 Sevenoak vised by Schultz R. A. Avery Plane Geometry 1925 1034 57 5.5 57 100 Schultz 3 Seven- Plane Geometry Re 1933 1471 149 10.1 148 99.3 0.7 oak 3 and vised by Schuyler Schuyler E. R. Breslich Plane Geometry 1938 1864 197 10.6 59 80.7 38 19.3 Schnell and Clear Thinking— An 1938 738 233 31.6 75 32.2 158 67.8 Crawford Approach Through Geometry Birkhoff and Basic Geometry 1941 700 117 16.7 51 43.6 66 56.4 Beasley TABLE 5 - Continued Authors Texts Publication Number Number of Date Exercises Number of Exercises Percent of Everyday-Life Everyday-Life Number Number of Physical Sits Percent of Number Number of Physical Physical Sits Patterns Patterns of Reasoning Percent of Patterns of Reasoning R. A. Avery Plane Geometry 19^7 2133 128 6.0 114 8 9 .I 14 10.9 Kenniston and Plane Geometry 1953 2158 273 12.7 160 59.0 113 41.0 Tully Schnell and Plane Geometry— A Crawford Clear Thinking Approach 1953 2113 291 13-8 135 46.4 156 53.6 Stone Avery's Plane Geometry 1959 2335 132 5.6 106 80.3 26 19.7 1 CT\ i— I — Goodwin, Van- 1 1569 138 8.8 95 68.8 43 31.2 nata, and Fawcett Geometry— A Unified Course CO in 00 Henderson, . Modern Geometry— Its 0 1962 72 1.2 67 43.1 5 6.9 Pingry, and Structure and Function Robinson Goodwin, Van- Geometry— A Unified 1965 2347 187 7.9 116 62.0 71 38.0 nata, and Course ll>A T.Trt 4“ 4- Be/? PATTERNS AND SITUATION, PHYSICAL EVERYDAY-LIFE, THE OF COMPARISON A (CO vrdyLf Pyia iuto Reasoning Situation Physical Everyday-Life OF REASONING EXERCISES FROM DATA OF TABLE 5 TABLE OF DATA FROM EXERCISES REASONING OF Exercises Exercises ____ xrie. . Exercises- . . . Exercises. 'y&ARS • atrs of Patterns GRAPH IV GRAPH 127 128 Chapter Summary Fourteen plane geometry texts have been analyzed in this chapter. Although some of the textbooks are by different authors, Table 5 and the graph of its data show that their contents represent interesting trends with respect to: 1 . the total number of exercises included; 2 . the number of everyday-life exercises; 3 . the percent of the total number of exercises that are the everyday-life type; and 4. the number and percent of exercises on pat terns of reasoning with respect to the total number of everyday-life exercises. The total number of exercises and the total number of everyday-life exercises tended to increase from the earlier to the later texts. The percent of everyday-life exercises with respect to the total number of exercises increased from 1903 to 1953 but thereafter tended to de cline. The first appreciable number of exercises on valid and invalid patterns of reasoning in everyday life appeared in the texts of the late 1 9 3 0 's and tended to increase thereafter. An exception to parts of the foregoing state ments was Modern Geometry, Its Structure and Rmction, (1962), by Henderson, Pingry, and Robinson. It contained the largest number of exercises of all the books examined, yet it had fewer everyday-life exercises than the other texts except Saunders' text of 1903 and Avery's text of 1925. It exceeded only Saunders' text in the percent of everyday-life exercises with respect to the total number of these exercises. It exceeded only the first four texts examined in this series as to patterns of reasoning exercises. It exceeded only three texts of this series in the number of exercises on physical situations. This is probably due to its strong emphasis on deductive struc ture and other ideas recommended for the teaching of geometry in recent years. CHAPTER VII REPORT OF ANALYSES V The findings in Modern Geometry - Its Structure and Function (1 9 6 2 ) by Henderson, Pingry, and Robinson, stimulated the selection of five more recently published texts for examination copyrighted from 1963 to 1 9 6 5 . These examinations were made on the premise that they might reflect the most recent recommendations on the teaching of geometry and elucidate present trends in geometry textbooks. Selected Textbooks, 1963 to 1 9 6 6 . Frank Morgan and Jane Zartman, Geometry-Plane-Solid- Coordinate, 1 9 6 3 . There are 4227 exercises in this text and 343 (8.1 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - Whan children break a candy bar in two pieces, they sometimes argue over the 'bigger half'. Mathematically, is there such a thing as a 'bigger half'? Explain your answer, p. 9 , no. 9 . Time-Telling - Through how many degrees does the minute hand of a clock move between noon and 12:45 P.M.? p. 3 8 , no. 3. 130 131 Pattern of Can the conclusion be justified? Give Reasoning - a reason for the answer? "Mrs. Cook, will not patronize Ken's Market because a bag of their potatoes was underweight." p. 5 5 * no. 5 . Pattern of State the hypothesis and the conclu Reasoning - sion: "If it snows, I shall wear over shoes." p. 5 9 * no. 2 . Carpentry - A board has parallel edges and a carpenter makes a cut perpendicular to one edge. Will the cut be per pendicular to the opposite edge? Why? p. l4l, no. 3. Designing - Find an example of a design made only of lines. Bring it to class and be prepared to explain what was done to make it more pleasing to the eye than simple, evenly spaced lines, p. 1 9 7 * no. 4. Measurement - Find the number of acres in a triangu lar field whose sides are 1 7 , 2 5 , and 28 rods, respectively. One acre = 160 sq. rd.. p. 433* no. 5 . Number of Everyday •Life Exercises by Classifications Aeronautics 5 Age 2 Astronomy 3 Building 5 Carpentry 11 Designing 39 Drafting 5 Engineering 9 Farming 2 Gardening 3 Machinist and Industry 2 Mapreading 1 Measurement 71 Metallurgy 3 Miscellaneous 50 Navigation 5 Physics 20 Patterns of reasoning 6 l 132 Sports 8 Surveying 33 Time-Telling 5 iotal ■" 343 Exercises on measurement and patterns of reasoning exceed those in other classifications in this text. Sixty-one (17.7 percent) are on patterns of reasoning and 282 (82.3 percent) are on physical situations. Harry Lewis, Geometry - A Contemporary Course, 1964. This text contains 1967 exercises and 71 (3.6 per cent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Miscellaneous - Define the underlined word in two ways: (a) to make the sentence true and (b) to make the sentence false. A dog is a man's best friend, p . 6 , no . 2 . Pattern of In the following state whether the Reasoning - reasoning is correct or incorrect. Justify your answer in either event. "If Mr. Strong is elected senator, then our taxes will be reduced. Mr. Strong is elected senator. our taxes will be reduced." p. 8 8 , no. 1 . Architecture - A circular archway is to be built such that the distance between the endpoints of the arch is to be 14 times as long as the height of the arch. If the radius of the circle of this arch is 20 feet how high is the arch? p. 505 j> no. 1 1 . Measurement - Two cubic feet of liquid plastic is poured into a rectangular mold whose base is 20 feet by 10 feet and allowed to cool. How thick will the sheet of plastic be? p. 624, no. 1 0 . 133 Number of Everyday-Life Exercises by Classifications Aeronautics 2 Architecture 1 Carpentry 1 Measurement 13 Medicine 1 Miscellaneous 30 Physics 1 Patterns of Reasoning 18 Sports 2 Surveying 1 Tiling 1 TO T A L 7T“ Exercises of a miscellaneous nature outnumber those in other classifications, with patterns of reasoning second and measurement third. Eighteen (25.4 percent) are on patterns of reasoning and 53 (74.6 percent) are on physical situations. Ray Jurgenson, Alfred Donnelly, and Mary Dolicani, Modern Geometry - Structure and Method, 1 9 6 5 . This text contains 4788 exercises and. 210 (4.4 per cent) are everyday-life exercises. Examples of Everyday-Life Exercises Classifications Exercise Calendar - Use the roster method to specify the following set. State whether the set is finite or infinite. "The months whose names begin with the letter A. " p. 45j no. 1 . Pattern of Tell whether the process used in the Reasoning - following is inductive, intuitive, or neither: "A girl looks into several robins' nests and says, 'All robins' eggs are blue." p. 5 5 , no. 2 . 134 Pattern of Is this thinking inductive or Reasoning - deductive?: "It will rain on Christmas again this year since it has done so for the last five." p. 8 8 , no. 4. Surveying - A person at a window 15 ft. above the sidewalk looks at a street light 12 ft. above the sidewalk. Is the angle at the person’s eye an angle of ele vation or an angle of depression? p. 303r no. 3 . Aeronautics - Who gains altitude more quickly, a pilot traveling 400 m.p.h. and rising at an angle of 30°, or a pilot traveling 300 m.p.h. and rising at an angle of 4o°? How much more quickly (in m.p.h.) does he gain altitude? p. 3 0 5 * no. 1 8 . Number of Everyday-Life Exercises by Classifications Aeronautics 2 Astronomy 2 Building 1 Calendar 6 Carpentry 9 Drafting 2 Engineering 1 Measurement 18 Miscellaneous 19 Photography 1 Physics 9 Patterns of Reasoning 107 Science 1 Smeltering 1 Sports 2 Surveying 24 Teaching 1 Time-Telling 4 TOTAL 21o More than half (50.9 percent) of the everyday-life exercises are on patterns of reasoning with surveying and measurement following. A total of 103 (49.1 percent) are on physical situations. 135 I^ron RosskoffHarry Sitomer., and George Lenchner, Modern Mathematics - Geometryj 1 9 6 6 . This text contains 1593 exercises and 75 (4.7 per cent) of them everyday-life exercises. Examples of Everyday-Life Exercises Clas s ification Exercise Pattern of Assume the given statements are true. Reasoning - Write a final statement that follows from the given statements and name the deductive law used: "(a) If it rains, I shall study. (b ) It is raining." p. 2 3 > no. 3 . Sports - A baseball diamond is a square whose sides are each 90 ft. long. What is the distance from the catcher to second base? Approximate the distance to the nearest tenth of a foot, p. 303> no. 12. Photography - A photograph is 2 inches by 2 3/4 inches. It is to be enlarged so that its longer side will be 22 inches. How long will the shorter side be? p. 305* no. 3. Tinsmithing - How much tin is wasted in cutting out the largest possible circle from a square piece of tin 4 inches on each side? p. 448., no. 11. Physics - A force of 200 pounds is applied to a rock to loosen it. If the force is applied at an angle of 6 0 ° to the horizontalj what is the effective vertical force? The effective horizontal force? p. 524 j no. 6. Number of Everyday-Life Exercises by Classifications Aeronautics 2 Astronomy 2 Carpentry 2 Designing 1 Industry 2 Measurement l4 136 Navigation 1 Physics l4 Patterns of Reasoning 29 Sports 3 Surveying 1 Time-Telling 1 Tinsmithing 3 M ------75 Exercises on patterns of reasoning are the largest in numberj with physics and measurement poor seconds. Twenty-nine (38.7 percent) are on patterns of reasoning and 46 (61.3 percent) are on physical siutations. Richard Anderson, Jack Garon and Joseph Grimillion, School Mathematics, Geometry, 1 9 6 6 . The following statement on the preface sheet of this text indicates the recommendations emphasized in the text: Portions of this book have been either taken directly from or based upon the text Geometry, issued by the School Mathematics Study Group. . . .2 There are 3163 exercises in the text and 143 (4.2 percent) are everyday-life exercises. Examples of Everyday-Life Exercises Classification Exercise Pattern of If possible write a new sentence that Reasoning - follows logically from the two sentences. Otherwise write "None." "If a student is in Miss Smith's class, Miss Smith is his teacher. John is in Miss Smith's class." p. 11, no. 1. 2Richard Anderson, Jack Garon, and Joseph Gremillion, School Mathematics Geometry (Boston: Houghton Mifflin Company, ±9bb, preface sheet). 137 Measurement - (a) Using 8 1/2 inches as a unit of distance, find the length and width of a sheet of paper measu ring 8 1/2 x li inches. (b) Repeat part abusing 11 inches as a unit. p. 47, no. 8 . Sports - Is the surface of a football a convex set? Is the Interior of a football a convex set? p. 8 7 , no. 2 . Pattern of We say that two people can converse Reasoning if they speak the same language. Is the ability to converse an equi valence relation? p.l45, no. 9 . Measurement - A bug crawls 5 inches north, then 2 inches east, then 1 inch north, and finally 6 inches east. How far is the bug from its starting point? p. 4o4, no. 5. Drafting The lengths of the sides of a triangu lar lot are 120 feet, 130 feet, and 150 feet. If the lot is drawn to scale so that 1/4 inch represents 5 feet, what will be the scale drawing? p. 4l8, no. 8 . Number of Everyday' -Life Exercises by Classifications Astronomy 5 Carpentry 2 Drafting 2 Engineering 1 Machine Work 1 Measurement 33 Miscellaneous 16 Navigation 3 Physics 10 Patterns of Reasoning 52 Shopping 1 Sports 4 Surveying 13 TOTAL' WT Exercises on patterns of reasoning exceed others, and measurement exercises are second. Fifty-two (36.4 percent) 138 are on patterns of reasoning and 91 (63*6 percent) are on physical situations. A review of Table 6 and the graph of its data indicate no consistent trend in the number of exercises in these five texts5 though they have more than many earlier texts. A general trend might be considered to be downward. The table shows that there was a tendency for the percent of everyday-life exercises to decline, the percent of exercises on patterns of reasoning to increase5 and those on physical situations to decline. C-I S p 3§* UUP I? isi o £ p P i p m 0 0 4 s: P 4 c+ P i CD 4 p h p era H* 4 (W P 4 o o p P cn c+ P O Q 03 Hj H- P p 3 P 4 CD O P H, p H m P W P P v P H O P Pi H,c OW P P M o 3 1 Pi 1 Pi g i CO H H H H VO VO VO VO VO CO ov CSV OV CSV OV Publication ov CSV VJI p- 00 Date o CSV oo H -P H -p H vji -] VO ro Number of CO CSV VO 00 csv ro OO 00 00 -0 -o Exercises £ 0 ro 00 t-3 -P H —VI p- Number of UJ vji O M 00 Everyday-Life @ Exercises H X p - P^ -P 00 OO Percent of i-€ • • •• Co ro -3 p- ov I-1 Everyday-Life 1 H ro Number of H vo p- O VJI 00 VO H CSV 00 OO ro Physical Sits. OV OO CSV csv •P- -V! 00 i-3 U) H-> VO Percent of • .•P- •ro Phys ical Sits. o csv LO H1 OV 00 VO Number of ov -CSV H H ov Patterns of VJI ro O 00 H ro VO -0 Reasoning VJI ro H Percent of u> U) O VJI -0 ov 00 • •• Patterns of • VO p- -VI - 6£l UuM 321^ /CO vrdyLf Pyia iuto Reasoning Situation Physical Everyday-Life A COMPARISON OF THE EVERYDAY-LIFE, PHYSICAL SITUATION, AND PATTERNS PATTERNS AND SITUATION, PHYSICAL EVERYDAY-LIFE, THE OF COMPARISON A xrie Exercises Exercises ------OF REASONING EXERCISES FROM DATA OF TABLE TABLE OF DATA FROM EXERCISES REASONING OF o Y GRAPH V GRAPH ...... Exercises atrs of Patterns 6 ------. 4 ^ Chapter Summary Five textbooks have been examined in this chapter. They are copyrighted from 1963 to 1966 and are by different authors. In most recent texts the greatest percentage of the everyday-life exercises in any one classification is the patterns of reasoning classification. These examinations seem to support the opinion that the number of exercises increased in the later texts and support the findings of other research discussed. They also seem to support the opinion that the percent of everyday-life exercises in textbooks tended from the late 1950’s to decrease as time elapsed and the percent on patterns of reasoning tended to increase. The latter findings appear to substantiate the assumptions made earlier in this study regarding the influence of professional recommendations on everyday-life exercises in plane geometry textbooks. CHAPTER VIII GENERAL SUMMARY AND CONCLUSIONS Summary An attempt has been made in this research to determine the trend of everyday-life exercises in plane geometry from an analysis of 44 geometry textbooks published from 1878 to 1 9 6 6 . Everyday-life exercises are defined as exercises relating to the physical world of man and to nongeometric situations involving valid and invalid patterns of reasoning. Historical and background information converting emphasis in the teaching of plane geometry from its early emphasis on discipline to practical values was discussed. This development influenced significant increases of everyday-life exercises in plane geometry textbooks. The recommendations, reports, studies, and experimental programs that effected the trend were specifically discussed. Recent recommendations, within the last two decades, seem to have redirected attention away from emphasis on everyday-life exercises, except those on valid and invalid patterns of reasoning. The shift involved more attention to mathematically capable students, more advanced topics at lower levels, the use 142 143 of sets as a unifying concept, the study of deductive structure, the use of precise terminology and definitions, the integration of ideas from plane, solid and analytical geometry and trigonometry, and the use of more exercises to encourage students to discover geometric ideas for themselves. This emphasis has been called "modern" mathematics. The analyses of the textbooks were given in five chapters. In Chapter III, 15 textbooks from one series were examined. In Chapter VI, 14 textbooks were examined. In each of Chapters IV, V, and VII five textbooks were examined. Each series examined in Chapters III, IV, and V consisted of a chronologically ordered set of books by the same author(s), or succeeding authors, published within'the period on which the study was conducted. The fifteen books examined in Chapter III were from the Wentworth-Smith-Welchons-Krickenberger-Pearson series; the five examined in Chapter IV were from the Stone- Millis-Mallory-Meserve-Skeen series; and the five in Chapter V were from the Wells-Hart-Schult-Swain series. Among the l4 texts examined in Chapter VI were some written by the same or succeeding authors. Comparisons were made between earlier and later texts with regard to everyday- life exercises they contained where this condition existed. The last five in Chapter VII were recent books by various authors which were examined to determine 144 whether their content in everyday-life exercises reflected the more recent professional recommendations concerning the teaching of geometry. Conclusions On the basis of the textbooks analyzed the following conclusions are drawn: 1. Authors generally attempted to implement major professional recommendations which preceded their textbooks. 2. The number of exercises in plane geometry textbooks increased. This agrees with findings in other investigations men tioned in this study. 3. The number and classifications of everyday- life exercises also increased. The most pronounced increases appeared in textbooks published during and after the first decade of the twentieth century. 4. The percent of the total number of exercises which were everyday-life exercises increased in textbooks published from 1878 to the latter part of the 1 9 5 0 's and tended to decrease thereafter. Most recent textbooks have almost as few such exercises as textbooks published before 1 9 0 0 . 5. Most of the earlier textbooks, those published from 1888 through the latter part of the 1 9 3 0 's included more everyday- life exercises on surveying than on any other type. Exercises on patterns of reasoning in everyday life generally exceeded others after the later 1 9 3 0 's. 6 . Everyday-life exercises on patterns of reasoning first appeared in textbooks of the early 1 9 3 0 's and increased thereafter to the point that most recent textbooks include a larger percentage of such 145 exercises than any other type. The ideas of "modern" mathematics have received more attention in these textbooks. Specifically, the above findings indicate that the trend of everyday-life exercises in the textbooks examined was upward from 1878 to approximately 1959 and downward from that time to the present. They also indicate that everyday-life exercises on surveying were dominant up to about 1938 and that since that time exercises on valid and invalid patterns of reasoning in everyday life have been dominant. This dominance has prevailed despite the limited number of everyday-life exercises included in the most recently published geometry textbooks. Suggestions for Further Research During the course of this research numerous illus trations and pictures were observed which emphasized the use of geometry in everyday-life situations. It might be of interest to investigate the trend of this develop ment in the teaching of geometry. One might also look at experiments over a period of years in the use of color and transparent overlays in geometry textbooks and their effect on learning. Tracing the development of a single - *•. — — idea often produces a panoramic perspective of the total picture. Implications The results of this study seem to have implications l46 for mathematics teachers, mathematics textbook writers, and others who wish to afford imaginative and effective leadership in mathematics education. It appears inevitable, and appropriately so, that the demands of a changing society will determine changes in the content and methods of teaching mathematics. Rapid advances in science and technology, as well as the explosion of knowledge and population we are experiencing in our society demand attention as we try to prepare ourselves and young people to face the realities of productive and meaningful existence. The role of mathematics teachers might be better played by keeping abreast of changes and developments, and by reorienting their teaching procedures accordingly to make use of both old and new professional content and methods proposals deemed suitable for their teaching situations. It is good that experimental psychologists are continuously seeking a better understanding of how we learn, for their findings should serve as allies- in the efforts of teachers and professional mathematicians. Mathematics textbooks writers will probably con tinue to be influenced by the recommendations of leaders in mathematics education. It is reasonable to expect that in the near future mathematics textbooks will follow some, if not all, of the proposals of the Cambridge Conference on School Mathematics. This could mean that 147 plane geometry as a separate subject would become extinct in mathematics teaching. A few integrated textbook series now available seem headed in this direction.* Some would probably not accept this as a desirable approach. It does appear to be an effective way to emphasize the important inner-connections and relationships which exist among the several branches of mathematics. Leadership in mathematics education as used here includes leaders in mathematics, in education, and in experimental psychology. It is the opinion of the writer that leaders in mathematics education have the most challenging, enviable, responsibility in the development of mathematics teaching and learning, for they must recognize fully the demands to be met and propose in detail how older mathematical ideas should be assessed and which new ones, properly presented, will best help to make intelligent and useful citizens as time moves on. Finally the statements above are not intended to imply that teachers, textbooks writers, and leaders in mathematics education should work independently of each other. Nothing would be more detrimental to the cause of better mathematical intelligence and its practical deployments than independent, perhaps quarrelsome, efforts by these three groups. *One such series is the following: H. Vernon Price, Philip Peaks, and Philip S. Jones, Mathematics - An Integrated Series. New York: Harcourt, Brace and World, Inc., 19b5 . BIBLIOGRAPHY Books Textbooks Anderson, Richard, Garson, Jack W., and Gremillion, Joseph G. School Mathematics Geometry. Boston: Houghton Mifflin Company, 1 9 6 6 . Avery, Royal A. Plane Geometry. Boston: Allyn and Bacon Publishing Company, 1925. Plane Geometry. Boston: Allyn and Bacon Publishing Company, 1947. Birkhoff, George, and Beatley, Ralph. Basic Geometry. Chicago: Scott, Foresman and Co., 1941. Breslich, Ernst R. Plane Geometry. Chicago, 111.: Laidlaw Brothers, 1938. Goodwin, Wilson, Vannatta, G. D., and Fawcett, Harold P. Geometry - A Unified Course. Columbus, Ohio: Charles E. Merrill Books, Inc., 1 9 6 1 . Geometry - A Unified Course. Columbus, Ohio: Charles E. Merrill Books, inc., 1 9 6 5 . Hart, Walter W. Plane Ge ome t ry. Boston: D. C. Heath and Co., 1950'. Hart, Walter.W., Schult, Veryl, and Swain, Henry. Plane Geometry and Supplements. Boston: D. C. Heath and Co., 1959. New Plane Geometry and Supplements. Boston: C. C. Heath and Co., 1964. Henderson, K. B., Pingry, R. E., and Robinson, G. A. Modern Geometry - Its Structure and Function. New York: McGraw-Hill Book do., 19b£. Jurgeson, Ray C., Donnelly, Alfred J., and Dolciani, Mary P. Modern Geometry - Structure and Method. Boston: Houghton Mifflin Co., 1965• 148 149 Kenniston, Rachel P., and Tully, Jean. Plane Geometry. Boston: Ginn and Co., 1953. Lewis, Harry. Geometry - A Contemporary Course. Princeton, J1. : D . Van No s trand Company, Inc., 1964. Mallory, Virgil S. New Plane Geometry, Revised. Chicago: Benjamin H. Sanborn and do., 1943. Mallory, Virgil S., Meserve, Bruce E., and Skeen, Kenneth Co. A First Course in Geometry. Syracuse, N.Y. : The L. W. Singer Co., 1$59. Morgan, Frank M., and Zartman, Jane. Geometry - Plane - Solid - Coordinate. Boston: Houghton Mifflin Company, l9t>3. Price, H. V., Peaks, P., and Jones, P. Mathematics - An Integrated Series. New York: Hareourt, Braee, and World, inc., 1 9 6 5 • Rosskopf, Mfyron F., Sitomer, Harry, and Lenchner, George. Modern Mathematics - Geometry. Morristown, N.J.: Silver Burdett Co., 19bb. Sanders, Alan. Elements of Plane and Solid Geometry. New York: American Book Co., 1963. Schnell, Leroy., and Crawford, Mildred•G. Clear Thinking- An Approach Through Plane Geometry. New York: Harper and Brothers, 1936. Plane Geometry - A Clear Thinking Approach. Hew York: McGraw-Hill book Co., inc., 1953• Schultz, Arthur, and Sevenoak, Frank L. Plane Geometry. New York: The Macmillan Co., 1915. Schultz, A., Sevenoak, F., and Schuyler, E. Plane Geometry, Revised by Schuyler. New YorFT The JVlacmillan Co., 1933* Stone, John C., and Millis, James F. Plane Geometry. Chicago: Benjamin H. Sanborn and Co., I916V Stone, John C., and Mallory, Virgil S. Modern Plane Geometry. Chicago: Benjamin H. Sanborn and 'Co., l94o. 150 . New Plane Geometry. Chicago: Benjamin H. Sanborn and. Co., 19^1 . Stone, William C. Ave ry1s Plane Ge ome t ry. Boston: Allyn and Bacon, Inc., 1959• Welchons, A. M., and Krickenberger, W. R. Plane Geometry. Boston: Ginn and Co., 1933. ______. Plane Geometry, Revised Edition. Boston: (Jinn and Co., l$4o. New Plane Geometry. Boston: Ginn and Co., T952. ]_9 5 6 _New------Plane Geometry. Boston: Ginn and Co., ______. Plane Geometry, Revised Edition. Boston: Ginn and Co... 1961. Welchons, A. M., Krickenberger, W. R., and Pearson, Helen R. Plane Geometry. Boston: Ginn and .Co., 1 9 6 1 . Wells, Webster, and Hart, Walter W. Plane Geometry. Boston: D. C. Heath and Co., 1915. Modern Plane Geometry. Boston: D. C. Heath and Co., I9 2 6 . Wentworth, G. A. Elements of Geometry. Boston: Ginn and Heath Publishing Co., 1878. A------Textbook of Geometry. Boston: Ginn and Co., A Textbook of Geometry. Boston: Ginn and Co., 1891. Geometry, Revised Edition. Boston: Ginn and Co., 1898. r B 9 9 :Plane------and Solid Geometry. Boston: Ginn and Co., Wentworth, George, and Smith, David E. Wentworth1s Plane Geometry, Revised. Boston: Ginn and Co., 1910. ______. Wentworth's Plane and Solid Geometry. Boston: (Jinn and Co., 19 H . 151 ______. Wentworth's Plane and Solid Geometry. Boston: Ginn and Co., T913. ______. Plane Geometry. Boston: Ginn and Co., 1938. Methods Books Breslich, Ernest R. The Technique of Teaching Mathematics in Secondary Schools. Chicago: University of Chicago Press, 1930. Brown, Claude H. The Teaching of Secondary Mathematics. New York: Harper and brothers, 1953- Butler, Charles H., and Wren, F. L. The Teaching of Secondary Mathematics. New YorlTi Me draw Hill Book 'do'.',' T'9'60. ______. The Teaching of Secondary Mathematics. New York: McGraw Hill Book Co., T9 6 5 . Christofferson, H. C. Geometry Professionalized For Teachers. Menasha, Wisconsin: deorge Banta Publishing Co., 1933. Davis, David R. The Teaching of Mathematics. Reading, Mass.: Addison-Wesley Publishing Co., 1951. Fehr, Howard F. Secondary Mathematics - A Functional Approach for Teachers. Boston: D. C. Heath and TSor; 1 9 3 1 7 ------Hassler, Jasper 0., and Holland R. Smith. The Teaching of Secondary Mathematics. New York: The Macmillan do., 1937.' ------Kinney, L. B., and Purdy, C. R. Teaching Mathematics in the Secondary School. New York: Rinehart and Co., ■Kc.V"T952l ------Meserve, Bruce and Sobel, Max. Mathematics for Secondary School Teachers. Englewood dliffs, N. J . : Prentice-Hall, Inc., 1 9 6 2 . Minnick, J. H. Teaching Mathematics in the Secondary Schools. Englewood dliffs, itf. J . : Prentice-Hall, inc., 1 9 3 9 . Schorling, Raleigh. The Teaching of Mathematics. Ann Arbor, Michigan! Hie Ann Arbor Press, 1 9 3 6 . 152 Willoughby, Stephen S. Contemporary Teaching of Secon- dary School Mathematics. New York: John Wiley and Sons, me., 1^6 Y. Young, J. W. A. The Teaching of Mathematics. New York: Longmans, Green, and C o . , Inc., 1924. History Books Cajori, Plorian. A History of Mathematics. New York: The Macmillan Co., 1931. Central Association of Science and Mathematics Teachers, Inc. A Half Century of Science and Mathematics Teaching. Oak Park, 111.: Central Association of Science and Mathematics Teachers, Inc., 1950. Eves, Howard. An Introduction to the History of Mathe matics . ifew York: Rinehart and Co., Inc., 1953- A Survey of Geometry, Vol. I. Englewood Cliffs, If. J7! Allyn and Bacon Co., 19^3. Sanford, Vera. A Short History of Mathematics. Boston: Houghton Mifflin Co., 1930. Shibli, J. Recent Developments in the Teaching of Geometry. State College, Penna.: J. Shibli Publishing Co., 1932. Smith, D. E. History of Mathematics. Vols. I, II. Boston: (Jinn and Co., Vol'. T7 1923j Vol. II, 1925. Yearbooks National Council of Teachers of Mathematics. "A General Survey of Progress in the Last Twenty-Five Years," First Yearbook. Edited by Raleigh Schorling. New ^ork: National Council of Teachers of Mathematics, 1926. . "The Teaching of Geometry," Fifth Yearbook. Edited by Ralph Beatley. New York: feureau of Publications, Teachers College, Columbia University, 1930. "The Nature of Proof," Thirteenth Yearbook. Edited by Harold P. Fawatt. New York: Bureau of Publications, Teachers College, Columbia University, 1938. 153 "The Place of Mathematics in Secondary Education," 'Fifteenth Yearbook. Edited by W. D. Reeve. New York: Bureau of Publications, Teachers College, Columbia University, 19^0. "A Source Book of Mathematical Applications," “Seventeenth Yearbook. Edited by W. D. Reeve. New York: Bureau of Publications, Teachers College, Columbia University, 19^-2. "Emerging Practices in Mathematics Education," Twenty-Second Yearbook. Edited by John R. Clark. Washington, b.C.: National Council of Teachers of Mathematics, 195^-. "Insights into Modern Mathematics," Twenty- Third Yearbook. Washington, D.C.: National Council of Teachers of Mathematics, 1957. "The Growth of Mathematical Ideas," Twenty- Tburth Yearbook. Washington, D.C.: National Council of Teachers of Mathematics, 1959. Other Books Davis, Robert B. The Changing Curriculum: Mathematics. Washington, D.C.: Association for Supervision and Curriculum Development, National Education Associa tion, 1 9 6 7 . Kinsella, John J. Secondary School Mathematics. New York: The Center for Applied Research in Education, Inc., 1965. Payne, Joseph N., and Goodman, Frederick L. Mathematics Education. Vols. I and II. Ann Arbor, Michigan: University of Michigan Campus Publishers, 1 9 6 5 . Schaaf, William L. A Bibliography of Mathematical Education. Forrest Hills, W.Y. : Stevinus Press, l94l. Articles and Periodicals Beatley, Ralph. "Third Report of the Committee on Geometry," The Mathemetics Teacher, Vol. XXVIII (November, 1935). 154 Breslich* E. R. "The Nature and Place of Objectives in Teaching Geometry*" The Mathematics Teacher* Vol. XXXI (November* 193^ )• Brinkman* H. W. "Mathematics in the Secondary School for the Exceptional Student*" The American Mathematical Monthly* Vo. LXI* No. 5, (May* 1954). Christofferson* H. C. "Geometry as a Way of Thinking*" The Mathematics Teacher* Vol. XXXI (April* 1 9 3 8 ). Commission on Post War Plans. "The Second Report*" The Mathematics Teacher* Vol. XXXVIII (May* 1945). Fawatt* Harold P. "Teaching for Transfer*" The Mathe matics Teacher* Vol. XXVIII (December* 1935). Hlavaty* Julius H. "The Nature and Content of Geometry in the High Schools *" The Mathematics Teacher* Vol. LII (February* 1959). Judd* Charles H. "The Relation of Special Training to General Intelligence*" Educational Review* Vol. XXXVI (June* 1908)1 Kemeny* John G. "Report to the International Congress of Mathematicians*" The Mathematics Teacher* Vol. LVI (February* 1 9 6 3 ). Lazar* Nathan. "The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry*" The Mathematics Teacher* Vol. XXXI (1938). . "The Logic of the Indirect Proof in Geometry: Analysis* Criticisms and Recommendations*" The Mathematics Teacher* Vol. XL (1947). Meserve* Bruce. "New Trends in Algebra and Geometry*" The Mathematics Teacher* Vol. LV (October* 1 9 6 2 ). National Association of Secondary School Principals. "New Developments in Secondary Mathematics*" The Bulletin of the National Association of Secondary School Principals (May, 1959). National Council of Teachers of Mathematics. "Report of the Committee on Essential Mathematics for Minimum Army Needs." The Mathematics Teacher, Vol. XXXVI (April* 1943). 155 Secondary School Curriculum Committee of the National Council of Teachers of Mathematics. "The Secondary Mathematics Curriculum," The Mathematics Teacher, Vol. LII (May, 1959). Stanholt, Henry H. "A New Deal in Geometry," The Mathe matics Teacher, Vol. XXIX (February, 193671 The Committee on Geometry Syllabus. "Final Report of the National Committee of Fifteen on Geometry Syllabus," The Mathematics Teacher, Vol. II (December, 1912). Reports Cambridge Conference on School Mathematics. Goals for School Mathematics. Boston: Houghton Mifflin and '(537; T^bT. ------College Entrance Examination Board. Report of the Commission on Mathematics. New York: College Entrance Examination Board, 1959. Lazar, Nathan. The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of tieometry. A report of research. New York: Nathan Lazar, 1938. National Council of Teachers of Mathematics. An Analyses of New Mathematics Programs. Washington, D.C. National Council of Teachers of Mathematics, 1 9 6 3 . Progressive Education Association. Mathematics in General Education. A Committee report. New York: Appleton-Century-Crafts, Inc., 1940. The International Commission on the Teaching of Mathematics. Mathematics in the Public and Private Secondary Schools of the United States'! Washington, D. C. : U.S. Office of Education, Bulletin 1 6 , 1911. The National Committee on Mathematical Requirements. The Reorganization of Mathematics in Secondary Education. iBoston: Houghton Mifflin and Co., 1923. Unpublished Material Gadske, Richard E. "Deomonstrative Geometry as a Means of Improving Critical Thinking." Unpublished Ph.D. dissertation, School of Education, Northwestern University, 1940. 156 Hlavaty, Julius H. "Changing Philosophy and Content in Tenth Year Mathematics." Unpublished Ph.D. dissertation, Teachers College, Columbia Univer sity, 1950. Lundberg, Harold. "Significant Influences Affecting Geometry as a Secondary School Subject." Unpublished Ph.D. dissertation, Department of Education, George Peabody College, 1951. Pitzer, George L. "An Analysis of the Proofs and Solu tions of Applications in Geometry." Unpublished Master's thesis, College of Education, The Ohio State University, 1953. Ulmer, Gilbert. "Teaching Geometry to Cultivate Reflec tive Thinking: An Experimental Study with 1239 High School Pupils." Unpublished Ph.D. dissertation, School of Education, University of Kansas, 1939. Pamphlet The Ohio State University. "The Role of Geometry in the Secondary School," College of Education Bulletin. Columbus, Ohio: The Ohio State University, 1948.