### Dissertation Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

The University of Texas at AustinMay 1998

*Copyright by Joseph William Eyles, 1998*

## Dedication

To those individuals who prefer learning and teaching mathematics Texas-style.

## Acknowledgements

In the fall of 1972, I enrolled in a calculus class at Auburn University taught by Dr. Coke S. Reed that changed the world, at least for me. I started the course intending to be an engineer, and as much as I disliked school, by the time I finished, I wanted to be a mathematics teacher. However, it took earning a master's degree under the direction of Dr. Jack B. Brown to give me the confidence that I could teach in the highly interactive style, which had precipitated my turning point. In addition, I gained the confidence I needed to solve the ill-posed problems of the industrial sector to which I've belonged for most of the past sixteen years. Thanks to the assistance of Dr. John W. Neuberger and Dr. John Ed Allen at The University of North Texas, I re-discovered my love of mathematics teaching and became more aware of the raging debates on mathematics teaching methods.

The desire to understand the issues in these debates ushered me to the Mathematics Education Department at The University of Texas at Austin and produced this installment in hopefully a much greater knowledge of the teaching methods emanating from the mathematics departments here during the tenures of Hyman J. Ettlinger, R.L. Moore, and H.S. Wall. I credit my supervising professor Dr. L. Ray Carry, and doctoral committee members Dr. Ralph W. Cain, Dr. O. L. Davis, Jr., Dr. Susan B. Empson, Dr. Peter W.M. John, Dr. D. Reginald Traylor, and Dr. Diane L. Schallert, not only for their guidance in the pursuit of this dissertation, but their influence in creating the environment in the College of Education which encourages study of even the most unorthodox of questions. It is a pervasive atmosphere that elicited unmitigated enthusiasm and support from the staffs of The Office of Executive Vice-President and Provost, The Office of the Registrar, The Center for American History, and the School of Natural Sciences.

Dr. John W. Neuberger, who is still practicing his own version of the Texas Method after over forty years, not only suggested R.L. Moore's calculus course as starting point in my research, but has been continually available with suggestions and clarifications when I needed them. Dr. Ben Fitzpatrick, Jr., now Professor Emeritus at Auburn University, who has written and spoken around the globe about R.L. Moore's teaching, provided my introduction to many of the interviewees and many sources of information used in this project. Dr. Albert C. Lewis, mathematical historian and one of the creators of the R.L. Moore Collection, provided much needed insight into the location of reference material as well as frequent encouragement.

The results of this research include glimpses into the lives of individuals who were willing to give up their precious time and energy to contribute, but to also give up some of their privacy. Whether or not they have been directly responsible for one or more lines of text in this document, they have added perspective or technical assistance to the author's pen. These individuals are:

Ms. Carol Bartlemehs, Mr. Arval W. Bohn, Dr. Jack B. Brown, Ms. Jane Isenhower Brown, Dr. Ramon Burstyn, Mr. Jerry Chandler, Mr. Sam Creswell, Dr. Richard T. Eakin, Dr. Austin Gleeson, Dr. Jo Heath, Dr. John Hinrichsen, Dr. J. P. Holmes, Dr. Sonya Ingwersen, Ms. Mary Long, Mr. Harry Lucas, Jr., Ms. Jean Donaldson Mahavier, Dr. William S. Mahavier, Dr. W. Ted Mahavier, Mr. Ruben Martinez, Dr. Lee C. May, Dr. Walter E. Millett, Ms. Barbara Neuberger, Mr. Valmer Norrod, Dr. Joel O'Connor, Dr. G. Edgar Parker, Ms. Cecilia Price, Dr. Coke S. Reed, Dr. Jack W. Rogers, Ms. Janet Shaw Rogers, Mr. Jimmie N. Russell, Mrs. Peggy Sackett, Dr. Michel Smith, Ms. Gail Sneeden, Dr. Michael Starbird, Dr. Douglas R. Stocks, Mr. Willie G. Swenson, Jr., Dr. William R. Transue, Dr. Sam Young, and Dr. Phillip L. Zenor.

It is also important to note that my availability to pursue this research in the first place was due to the encouragement and financial support provided by my managers: David Leitch at Motorola and Greg Wetli and Paul Satre at IBM and to the patience and moral support of my family.

## R.L. Moore's Calculus Course

Publication No._____________

Joseph William Eyles, Ph.D.

The University of Texas at Austin, 1998

Supervisor: L. Ray Carry

R.L. Moore (1882-1974) distinguished himself not only as a research mathematician, but also as a professor of mathematics at The University of Texas from 1920 until he retired at age 86 in 1969. In 1964, the Mathematical Association of America produced the film "Challenge in the Classroom: The Methods of R.L. Moore," which credits Moore as one of the most prolific producers of research mathematicians in history. In 1973, The University of Texas named its new Physics-Mathematics-Astronomy building Robert Lee Moore Hall in honor of, according to Moore student R.L. Wilder, "not just Moore's eminence as a research mathematician but his achievements as a great teacher." The "Moore Method" is described by noted mathematician and author Paul Halmos as the "right way to teach anything and everything."

The past decade can be characterized as a time of initiation of broad reform in mathematics education from primary grades to the college level through what has come to be known as The Calculus Reform Movement. Proponents of mathematics education reform emphasize that the change in how mathematics is taught is at least as important as the change in which mathematics is taught. Some of the proposed teaching practices resemble, if not model, the teaching practices used by R.L. Moore a half century ago.

While the "Moore Method" has been well publicized, his pedagogical techniques are described in the context of graduate level and upper division mathematics courses. Although the fact that Dr. Moore taught calculus on a regular basis is often mentioned in the literature, it is not clear how he employed his techniques in this typically non-theorem-proving course. The purpose of this research was to describe Dr. Moore's calculus course and to investigate how well it met objectives of the current reform movement.

Two descriptions of Moore's calculus course are presented, one edited from notes he made, but never published, and a second drafted from interviews with former students. A comparison of final grades in a physics course indicate that Moore's approach to calculus instruction was not detrimental to later study in applied areas.

## Table of Contents

- List of Tables
- List of Figures
- CHAPTER 1 - INTRODUCTION
- R.L. Moore
- The Moore Method
- Moore's Commitment to Teaching Calculus
- Calculus Reform Movement
- Rationale for the Study
- Statement of the Problem
- CHAPTER 2 - REVIEW OF THE LITERATURE
- Introduction
- R.L. Moore
- Education
- Early Career
- The Moore Method
- Pure Math versus Applied Math
- Pure Academics
- Axiomatics
- The Calculus Reform Movement
- Recent History
- Implementing Reform
- Assessing Reform Efforts
- R.L. Moore's Calculus Course
- Conclusions
- CHAPTER 3 - COURSE DESCRIPTION
- Historical Research Methods
- Data Collection
- Sample
- Data Analysis
- Calculus in Moore's Own Words
- Forward
- My calculus class
- Editor's Impressions
- Student Evaluations
- Introduction
- Starting at the Same Place
- Defining Terms
- Stating Problems
- As Much Time As They Need
- Encouraging Competition and Creativity
- Class Dismissed
- Summary
- Comparison of Student Achievement in Physics
- Calculus Prerequisite
- Physics Requiring Calculus
- Data Description and Analysis
- CHAPTER 4 - CONCLUSIONS
- Suggestions for further study
- Appendix A - Letters
- Appendix B - Interview Materials
- TEXAS METHODS OF TEACHING MATHEMATICS
- GOALS AND GUIDELINES: ORAL HISTORY ASSOCIATION
- Agreement
- Possible Interview Questions
- Appendix C - Interview Transcripts
- Appendix D - Class Notes
- Appendix E - Moore's undervisningsmetod
**Moore's Teaching Method**- Introduction
- A Description of Moore's Method
- Analysis of Moore's Method
- Conclusions
- A few voices on Moore's Method
- References
- Bibliography
- Vita

### List of Tables

Table 3.1: Number of students in Moore's calculus class by academic year

### List of Figures

Figure 3.1: Drawing 1 from Dr. Moore's notes.

Figure 3.2: Drawing 2 from Dr. Moore's notes

Figure 3.3: The graph of f(x) = (x-1)(x-2)(x-3)

Figure 3.4: G_{1}, G_{2}, and G

Figure 3.5: Dr. Moore's drawing of graph of y from group1

Figure 3.6: Dr. Moore's drawing of the graph of y from group2

Figure 3.7: Dr. Moore's Drawing of the inscribed rectangle.

Figure 3.8: f(x) = x^{2}

Figure 3.9: Graph of y = x3 - 6x2 + 11x - 6

Figure 3.10: Tin can problem

Figure 3.11: Field in track problem

Figure 3.12: The wine glass problem

Figure 3.13: Taking a ladder around a corner

Figure 3.14: Area and anti-derivatives

Figure 3.15: F(x) is the area of D_{x}

Figure 3.16: f(x) = sinx

Figure 3.17: f(x) = e^{x}

Figure 3.18: f(x) = tanx

Figure 3.19: Density of a table top

### CHAPTER 1 - INTRODUCTION

Known for his non-traditional teaching techniques and axiomatic approach to mathematics, R.L. Moore taught calculus during many of his 49 years as Professor of Mathematics at The University of Texas. In the current climate of calculus reform that is engaging mathematics faculty from colleges and universities across the U.S. it may be beneficial to consider the calculus course taught by a well known reformer. Because of the dearth of published information available, the purpose of this study was to generate a description of Dr. Moore's calculus course and to investigate the success of his calculus students in light of the goals of current reform efforts.

**R.L. MOORE**

R.L. Moore (1882-1974) distinguished himself not only as a research mathematician, but also as a professor of mathematics at The University of Texas from 1920 until he retired at age 86 in 1969. In 1964, the Mathematical Association of America produced the film "Challenge in the Classroom: The Methods of R.L. Moore," which credits Moore as one of the most prolific producers of research mathematicians in history. In 1973, The University of Texas named its new Physics-Mathematics-Astronomy building Robert Lee Moore Hall in honor of, according to Moore student R.L. Wilder, "not just Moore's eminence as a research mathematician but his achievements as a great teacher" (Wilder, 1974). After nearly thirty years since R.L. Moore was forced into retirement, his way of teaching known as the Moore-Method or the Texas-Method is still emulated in Universities throughout the United States.

**THE MOORE METHOD**

The Moore-Method has been called the axiomatic method, the Socratic method, and a modification of the German seminar (Whyburn, 1970.) It is recommended by noted mathematician and author Paul Halmos (1987) as "the right way to teach anything and everything." There are those who would beg to differ with Halmos. Cain, Carry, and Lamb (1985) proposed the use of a "Pure Mathematics" program organized around an axiomatic curriculum for that top ten percent of the population that is able to pursue mathematics for its own sake. Morris Kline (1977) claims that "generalization for the sake of generalization can be a waste of time" (p. 43) and attributes responsibility of poor mathematics instruction in college to the axiomatic approach in "Pure Mathematics" research. Kline (1977) cites Richard Courant, John von Neumann, and other well known research mathematicians in their calls for "close interconnection between mathematics, mechanics, physics, and other sciences" (p. 155) and "the rejuvenating return to the source" (p. 157) in his impassioned plea for a problem-centered, application-based approach to teaching mathematics.

**MOORE'S COMMITMENT TO TEACHING CALCULUS**

R.L. Moore began teaching calculus when he ordered W. E. Byerley's __Differential Calculus__ from The University of Texas and taught himself around the age of fifteen. One year later he entered The University of Texas as a freshman. When Dr. Moore returned to Texas in 1920 as Professor of Pure Mathematics, Calculus was to become a mainstay in his teaching load. Even after he was placed on modified service he continued teaching calculus along with four other courses every semester until he was forced to retire in 1969. During his 68 years as an educator, he influenced thousands of students, some that went on to teach calculus. This course that Morris Kline envisioned as the introduction to applied mathematics was important to R.L. Moore and his influence may have been important to the way calculus has been taught over the past century.

**CALCULUS REFORM MOVEMENT**

The past decade can be characterized as a time of initiation of broad reform in mathematics education. The National Council of Teachers of Mathematics (NCTM) has led the reform movement at the kindergarten through twelfth grade level, beginning with the publication of __Curriculum and Evaluation Standards for School Mathematics__ in 1989 and continuing with many projects to promote and monitor the implementation of the new vision for mathematics education. During this time reform was also taking shape on college campuses under the title of The Calculus Reform Movement. Proponents of mathematics education reform emphasize the change in how mathematics is taught is at least as important as the change in which mathematics is taught. Some of the proposed teaching practices resemble, if not model, the teaching practices used by R.L. Moore more than a half century ago.

**RATIONALE FOR THE STUDY**

Although advocated decades before, The Calculus Reform Movement dates its official beginning with the 1986 Tulane Calculus Workshop and the publication of its proceedings, __Toward a Lean and Lively Calculus__. Following much discourse and debate, a 1992 National Science Foundation workshop formulated a list of objectives for good Calculus instruction. Number one on the list is "transfer techniques of calculus to other disciplines and novel situations." This same objective was expressed for pre-collegiate mathematics instruction as part of the "mathematical connections" goals of the NCTM __Standards__ in 1989.

In __Why the Professor Can't Teach__ (1977), Morris Kline lays total blame for the downfall in mathematics education on the separation of mathematics from science through axiomatic training of mathematical researchers. While other mathematics educators such as Cain, Carry, and Lamb (1985) are not as extreme in their views, they see applications receiving lower priority in a "Pure Mathematics" or axiomatic program than they would in an "Applied Mathematics" or problem-centered approach. Since R.L. Moore was a calculus teacher who is often cited in the literature as a proponent of the axiomatic method, we may be able to get some measure of the effectiveness of this teaching approach toward the transfer goal of the Calculus Reform Movement by studying the achievement of Moore's calculus students in a course that requires calculus.

**STATEMENT OF THE PROBLEM**

While the "Moore Method" has been well publicized in the literature on mathematics and mathematics teaching, his pedagogical techniques are described in the context of graduate level and upper division mathematics courses. Although the fact that Dr. Moore taught Calculus on a regular basis is often mentioned in the literature, it is not clear how he employed his techniques in this typically non-theorem-proving course. Since no more than brief mention of his calculus course has been published, the primary goal of this research was to acquire a comprehensive description of R.L. Moore's Calculus course.

A secondary objective of this study was to measure the ability of students who took calculus from R.L. Moore to transfer techniques of calculus to other disciplines. Comparisons were made on the performance of Moore's calculus students with non-Moore calculus students in one of the first undergraduate physics courses requiring knowledge of calculus for admission. The final grades of students taking Physics 335, Intermediate Mechanics, at The University of Texas between 1949 and 1956 were used to determine the level of success in the students' ability to transfer the techniques of calculus to this particular physics course.

[End of Chapter One]

*This dissertation may be viewed in its entirety at The Center for American History or at the PCL (call number: DISS 1998 EY47) on the UT-Austin campus . Current students, staff, and faculty may access this dissertation using the UMI digital library accessible via UTNetCAT*