# Moore method

The Moore method is a deductive manner of instruction used in advanced mathematics courses. It is named after Robert Lee Moore, a famous topologist who first used a stronger version of the method at the University of Pennsylvania when he began teaching there in 1911.

The way the course is conducted varies from instructor to instructor, but the content of the course is usually presented in whole or in part by the students themselves. Instead of using a textbook, the students are given a list of definitions and theorems which they are to prove and present in class, leading them through the subject material. The Moore method typically limits the amount of material that a class is able to cover, but its advocates claim that it induces a depth of understanding that listening to lectures cannot give.

## The original method

F. Burton Jones, a student of Moore and a practitioner of his method, described it as follows:

Moore would begin his graduate course in topology by carefully selecting the members of the class. If a student had already studied topology elsewhere or had read too much, he would exclude him (in some cases, he would run a separate class for such students). The idea was to have a class as homogeneously ignorant (topologically) as possible. He would usually caution the group not to read topology but simply to use their own ability. Plainly he wanted the competition to be as fair as possible, for competition was one of the driving forces. […]

Having selected the class he would tell them briefly his view of the axiomatic method: there were certain undefined terms (e.g., 'point' and 'region') which had meaning restricted (or controlled) by the axioms (e.g., a region is a point set). He would then state the axioms that the class were to start with […]

After stating the axioms and giving motivating examples to illustrate their meaning he would then state some definitions and theorems. He simply read them from his book as the students copied them down. He would then instruct the class to find proofs of their own and also to construct examples to show that the hypotheses of the theorems could not be weakened, omitted, or partially omitted.

When the class returned for the next meeting he would call on some student to prove Theorem 1. After he became familiar with the abilities of the class members, he would call on them in reverse order and in this way give the more unsuccessful students first chance when they did get a proof. He was flexible with this procedure but it was clear that this was the way he preferred it.

When a student stated that he could prove Theorem x, he was asked to go to the blackboard and present his proof. Then the other students, especially those who had not been able to discover a proof, would make sure that the proof presented was correct and convincing. Moore sternly prevented heckling. This was seldom necessary because the whole atmosphere was one of a serious community effort to understand the argument.

When a flaw appeared in a 'proof' everyone would patiently wait for the student at the board to 'patch it up.' If he could not, he would sit down. Moore would then ask the next student to try or if he thought the difficulty encountered was sufficiently interesting, he would save that theorem until next time and go on to the next unproved theorem (starting again at the bottom of the class).

—(Jones 1977)

The students were forbidden to read any book or article about the subject. They were even forbidden to talk about it outside of class. Hersh and John-Steiner claim that, "this method is reminiscent of a well-known, old method of teaching swimming called 'sink or swim' ".

## Current usage

After Moore became an associate-professor at University of Texas at Austin in 1920, the Moore method began to gain popularity. Today, the University of Texas at Austin remains a strong advocate of the method and uses it in various courses within their mathematics department, including:

• The University of Chicago offers the following Moore method classes: honors calculus, analysis, algebra, geometry, and number theory along with one or two Moore method electives each year.
• Professor Arnold Lebow uses the Moore method in his Advanced Calculus, Probability, and Discrete Structures courses at Yeshiva University in New York.
• Professor Bryan Snyder at Sault Ste. Marie, Michigan's Lake Superior State University, has introduced the Moore Method to the university in a course named "Fundamental Concepts of Mathematics."
• Professor Ronald D. Taylor at Berry College in Rome, Georgia successfully uses the Moore method in his Real Analysis course.
• The Physics Department of Berry College successfully uses the Moore method in numerous upper level courses.
• Professor Don Chalice at Western Washington University regularly uses a modified Moore method in all the upper level courses he teaches. He has done so for many years; as such, his influence has spread the Moore method to many other courses at WWU. See references below.
• Professor Lawrence Fearnley of Brigham Young University has, over the course of several decades, thoroughly implemented the Moore method in several of the analysis, topology and Calculus courses.
• Professor Mike Brilleslyper of the United States Air Force Academy uses the Moore Method to teach Real Analysis.
• Professor Ed Parker of James Madison University uses a modified Moore Method in Calculus and Analysis courses.
• Professor Elena Marchisotto of California State University, Northridge uses a modified Moore method in her "Foundations of Higher Mathematics" course.
• Many topology professors in the Mathematics Department of Auburn University use varying modifications of the Moore method.
• Professor David W. Cohen of Smith College implemented a modified Moore method for courses in Infinite-dimensional Linear Algebra and Real Analysis.
• Professor Vladimir N. Akis of California State University, Los Angeles uses the Moore Method to teach graduate Topology courses.
• Professor Thomas Wieting of Reed College uses a Moore method in his Real Analysis and Differential Equations courses.
• Professor Glenn Hurlbert of Arizona State University, uses the Moore Method to teach Introduction to Proofs, Combinatorics, and Linear optimization courses, and has written a Springer textbook to facilitate its use in Linear Optimization.
• Professor Gordon Johnson of University of Houston utilizes the Moore method to instruct Calculus and Analysis courses.
• Professor Genevieve Walsh of Tufts University uses a modified Moore method in her Point-set topology course.
• Different instructors have used the Moore method at Canada/USA Mathcamp to teach various topics on algebra, topology, number theory, logic, and set theory.
• Mike Cullerton used a modified Moore Method to teach the quadrangles unit of a high school Geometry class at Ute Creek Secondary Academy in Longmont, CO. Students were enthusiastic and discovered all the material normally covered in the text on their own. (And then some.)
• Professor Dylan Retsek uses this method at Cal Poly San Luis Obispo to teach Calculus, Introduction to Proofs, and Real Analysis.
• Professor Padraig McLoughlin uses this method at Kutztown University of Pennsylvania to teach Calculus, Set Theory, Foundations of Mathematics, Real Analysis, Topology, and Probability & Statistics.

## Quotations

• "That student is taught the best who is told the least." Moore, quote in Parker (2005: vii).
• "I hear, I forget. I see, I remember. I do, I understand." (Chinese proverb that was a favorite of Moore's. Quoted in Halmos, P.R. (1985) I want to be a mathematician: an automathography. Springer-Verlag: 258)

## References

• Cohen, David W., 1982, "A modified Moore method for teaching undergraduate mathematics", American Mathematical Monthly 89(7): 473-474,487-490.
• Jones, F. Burton, 1977, "The Moore method," American Mathematical Monthly 84: 273-77.
• Parker, John, 2005. R. L. Moore: Mathematician and Teacher. Mathematical Association of America. ISBN 0-88385-550-X.
• Wall, H. S. Creative Mathematics. University of Texas Press. ISBN 0-292-71039-9.
• Chalice, Donald R., "How to teach a class by the Modified Moore Method." American Mathematical Monthly 102, no. 4 (1995), 317-321.
• Hersh, Reuben and John-Steiner, Vera, "Loving + Hating Mathematics". ISBN. 978-0-691-142470