This dissertation his been 69-15 949 microfilmed exactly as received 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 1964-196 6 .............. 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,