Moore Method

“Since the roots of the problems described above run so deep, it is imperative that potential solutions (such as the Moore method) be implemented early in students’ careers—and not just for students planning to become mathematicians” (p. 686).

This film features interviews with Moore and shows him in the classroom. It is the only attempt he made to publicize his teaching method.

This is a sample reel, prepared by George Paul Csicsery, for a proposed new video on Dr. Moore and the Moore Method. It includes excerpts from interviews with mathematician Paul Halmos and theologian James W. McClendon.

"Today's world needs creative minds and in India the need for such minds is desperate. We should experiment with any method which has a promise of enabling the students to face unfamiliar situations with confidence. The world is so dynamic today that mastery of facts has become secondary to mastery of techniques of acquiring knowledge." Kapur was a professor at Indian Institute of Technology, Kanpur, and Vice-Chancellor of Meerut University.

"The 1999 interview with Paul Halmos (1916-2006) that forms the backbone of I Want To Be A Mathematician was initiated to gather some comments from Halmos about R.L. Moore for the Educational Advancement Foundation’s R.L Moore Legacy Project."

"I will not, I told them, lecture to you .... They stared at me, bewildered and upset--perhaps even hostile. ... They suspected that I was trying to get away with something, that I was trying to get out of the work I was paid to do. I told them about R.L. Moore, and they liked that, that was interesting. Then I gave them the basic definitions they needed to understand the statements of the first two or three theorems, and said 'class dismissed'. It worked."

More than twenty-five influential teachers, top researchers, inventors, and leaders of industry attest to the life changing rewards that began for them in a classroom taught by IBL and the Moore Method.

"[I]mplementation of the Common Core State Standards for geometry cannot be successful until our teachers themselves gain a modern mastery of the subject that is consistent with these standards."

A seminar course for junior-senior mathematics majors is described. The topic is continued fractions, taught by a modified Moore Method, where the focus is on students creating their own mathematics.

"What might be viewed as highly suspect (if not downright immoral and debilitating) in the light of present-day prizing of cooperative learning, was a life-enhancing source for an unusually large number of students. In fact, R.L. Moore single-handedly turned out this century's leading set-theoretic topologists." (p. 42) The author was a student of E.E. Moise (Moore PhD, 1947) at the Harvard Graduate School of Education.

"... many mathematicians, including those who criticize constructivism, revere R. L. Moore as an outstanding teacher of mathematics.... The goal of this article is to show that Moore's method aligned with a constructivist approach."

Course Resources

(See also the Journal of Inquiry-Based Learning in Mathematics which publishes university-level course notes that are freely downloadable, professionally refereed, and classroom-tested.)

"These chapters are written in a very different style, which is motivated in part by the ideal of the Moore method of teaching topology combined with ideas of VIGRE programs in the US which advocate earlier introduction of seminar and research activities in the advanced undergraduate and graduate curricula" (p. vi).

From the blurb: "Written in a modified R.L. Moore fashion, it offers a unique approach in which students construct their own understandings. However, while students are called upon to write their own proofs, they are also encouraged to work in groups." Reviewed by Robert A. Fontenot. American Mathematical Monthly 108, 2001, 179-182 (on JSTOR)

"Those experienced in the Moore method, I believe, will appreciate and be able to use this book just on the basis of the first two chapters on set theory, but they write their own sets of notes for topology and won't need this one. To those not experienced in the Moore method, I recommend this book as a means of introduction to the method and say, 'try it, you'll like it.'" (R.R. FitzGerald, from review in Amer. Math. Monthly 79(1972), 920-921.)

"The Moore Method is an idea with many fruitful aspects. Let us not throw out the whole idea because it has some difficult points, rather let us search for a wider application of the good aspects. Moise has written an excellent book; it should make it easier for the problem course approach to find a larger place in the undergraduate curriculum." (From the review by Carl C. Cowen in the Amer. Math. Monthly, 91(1984): 528-530.)

A text "designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). The result of this approach will be that students:"An extremely ambitious and attractive new textbook." –Review by John J. Watkins, The Mathematical Intelligencer 31, 2009, 82-83.

A demonstration of the Moore Method by way of video clips, notes, and commentary on a class taught by William S. "Bill" Mahavier at Emory University in 2008.

"In an effort to show prospective mathematics majors that mathematics is a vital and beautiful subject, Levine organizes his book around active participation by students in a four-stage scheme for doing mathematics: experimentation, conjecture, proof, and generalization." -- Review by Robert A. Fontenot. American Mathematical Monthly 108, 2001, 179-182 (on JSTOR)

Dedicated to R.L. Moore, this text follows a "developmental" course which "entails letting the student develop a body of mathematical material under the guidance of the professor."

“From a pedagogical standpoint this book was inspired by our involvement in the Legacy of R.L. Moore Project …. When we set out to write this book, we wanted to capture the spirit of a Moore method course, but we also wanted to make sure that the resulting text was accessible to a non-mathematical audience.” (p. x) Online review by Raymond N. Greenwell.

"[T]his is a Moore-style text whose proper use depends on the confluence of a patient instructor open to the Moore technique ... with motivated and intelligent students. ... Highly recommended." (Judith Roitman, from review in The Journal of Symbolic Logic, 52(1987), 1048-1049. JSTOR link.) See also The American Mathematical Monthly, 95, 1988, p. 844 (on JSTOR ) for Roitman's response to the "vituperative review" by C. Smorynski.

From review by Marion Cohen: "I also thoroughly approve of her format, which is consistent with the Moore Method...."

"[O]ne can easily convert this text to a collection of problems in classes where the Moore Method is used."

Makes use of a new set of axioms that draw on a modern understanding of set theory and logic, the real number continuum and measure theory. Written for an undergraduate axiomatic geometry course, the book is particularly well suited for future secondary school teachers. It covers all the topics listed in the Common Core State Standards for high school synthetic geometry.

"The Moore Method ... implicitly supported me to put into practice an extremely non-traditional approach [in a multivariable calculus course] where traditions are highly ubiquitous. However, I changed the method to such a great extent that one can hardly recognize Moore anymore." The author is at Shahid Behesti University, Tehran.

“The main purpose of this approach is to encourage readers, in the well known educational method of R.L. Moore, to try hard to prove results for themselves” (cover). Review in American Mathematical Monthly on JSTOR

“Highly influenced by the legendary ideas of R. L. Moore, the author has taught several generations of mathematics students with these materials, proving again the usefulness and stimulation of the Moore method” (cover). The author was a student at Princeton 1949-52 (Ph.D. under Emil Artin) and was introduced to the Moore method there by Ralph Fox.

Personal Experiences as Students and Teachers

"The evidence is clear that the discipline and rigor learned through [the Moore method] have lasting effects. I can still remember the thrill of discovery and the triumph of presentation of a basic theorem as a freshman 47 years ago. ... I will fashion a course [from the text] in which I will include as many of the applications as possible, and on Fridays I will convert the class into a Moore method setting, with students proving theorems from a separate list I have generated." Vick, a professor at the University of Texas at Austin, is a third generation doctoral descendant of Moore.

“We adopted a computer version, so to speak, of the famous R.L. Moore method of teaching. ... It is easy to implement such a method in a computer framework. A typical example would be a presentation of fifteen elementary statements about the geometrical relation of betweenness among three points. Students are asked to select no more than five of the statements as axioms and to prove the rest as theorems” (p. 208).

“I have taught, or rather the students have taught themselves, a full syllabus of the course … using a small group discovery method. … In a sort of Moore method approach, the students are given examples to work out with guidance, a form of programmed prodding towards a solution.” (p. 85)

"Before [my husband John and I] got interested in mathematics education as a research subject, we were mathematicians teaching at least some upper-division and graduate mathematics courses using the Moore Method."

The author, professor of mathematics at the University of Birmingham in England, describes a first-year course, "Development of Mathematical Reasoning," which has proven popular and effective for mathematics students entering the university.

"[T]he students learned the underlying mathematics of program derivation and learned to apply it, by presenting proofs and derivations on a daily basis. Professorial intervention in the classroom was minimal. Our experience has been that students learn otherwise difficult material better, and are better able to put it into practice, with this teaching technique than they would have been able to do in the typical classroom."

“One of us (Barnett) took courses taught using the method in the departments both of mathematics and of computer sciences while in Austin in the 1980s. We describe the method as it was experienced at that time” (p. 85).

"Avers' instructional program parallels the work of the nationally known R.L. Moore of the University of Texas. Both are able to get their students emotionally involved with their subject by taking away their textbooks." (From the editorial introduction, p. 227.)

History

"Perhaps because of the distinctiveness of the teaching methods that Mooreâ€™s students had experienced in his classes, many of his students were active in educational issues. Specifically, many of them became leaders in the MAA, participating actively in the educational issues of their day."

A one-semester undergraduate course started by Bing at Wisconsin and continued by R.E. Fullerton, S.C. Kleene, and R.F. Williams. The notes were used by C.B. Allendoerfer at the University of Washington and by W.L. Duren, Jr., at Tulane University.

Mathematics - Sample Work of Some Who have Experienced the Moore Method

Quotes R.L. Moore in the epigram to Chapter 2: "That student is taught the best who is told the least."

Dedicated to Moore's student E.C. Klipple "who taught me real variables by the R.L. Moore method at Texas A&M in 1944."

Biography of Moore and His Students

Little had been known about Moore's third PhD student Anna Mullikin (1922) who published only one research paper, her dissertation, and became a high school teacher. This article sheds new light on her life and shows how influential her mathematics and her teaching were. For a demonstration of Mullikin's Nautilus, see Demo Collection.

Zettl describes his research but includes a brief biographical account of his family's harrowing escape from a Yugoslavian concentration camp and eventual immigration to the US. Thanks to the Moore Method used in classes that he took, he found his weak mathematics background to be no great disadvantage since he was on the same initial footing as everyone else.

Written during Dr. Moore’s lifetime but not authorized by him, this account by a member of the Moore school of mathematics is a major resource for information about his life and career. The author interviewed early students and colleagues. Chapters by W. Bane and M. Jones list publications by Moore and his mathematical descendants.(See Paul Halmos's review in Historia Mathematica 1(1974), 188-192.) His audio recordings of interviews with Anna Mullikin, Blanche Bennett Grove, F. Burton Jones, George Hallett, and others, can be accessed at the Archives of American Mathematics website

This is the most comprehensive biography to date and, in contrast to D. R. Traylor’s 1972 account, it was able to make extensive use of Dr. Moore's papers and the oral history resource in the Archives of American Mathematics

A principal research biostatistician with DuPont corporation talks about how he went from topology to statistics and "how Dr. Moore’s influence continues in this new career." In the course of a challenging and politically sensitive research position Dr. Green shows how important qualities such as persistence, self-reliance, and clear thinking, as well as a sense of humor, have proven to be valuable lessons he took from Dr. Moore’s classes. (See also his interview with B. Fitzpatrick above.)

Mentions of Moore and the Moore Method

Ager, Tryg, A., “From interactive instruction to interactive testing,” in Artificial Intelligence and the Future of Testing ed. Roy O. Freedle, (Lawrence Erlbaum Associates, 1990).

From the section “Example of finding-axioms: mathematical conjecture”: “VALID has one other exercise type that I would like to discuss. This is by far the most complex type of problem in the course. Based on an idea of R. L. Moore and modified for interactive use in VALID, there are seven ‘finding-axioms’ exercises, of which the following is the simplest” (p. 40).

Dreyfus, T., and Eisenberg, T., "On different facets of mathematical thinking," in The Nature of Mathematical Thinking, eds. R. J. Sternberg, T. Ben-Zeev (Lawrence Erlbaum Associates, 1996), pp. 253-284.

Includes a basically sympathetic account of the Moore Method but finds cooperative learning, viewed as "humanizing Moore," to be more congenial, especially for less motivated students.

Eisenberg, Theodore. “Some of My Pet-Peeves with Mathematics Education,” Mathematics & Mathematics Education: Searching for Common Ground, Michael N. Fried, Tommy Dreyfus (eds.), Springer (2014), 35-44.

"My heroes in those days were mathematicians who had a sincere interest in teaching ... [including] R.L. Moore who had this new (to me) way of teaching by pitting student against student in competitive situations. (I did not like the competitive part of Moore's teaching, but one certainly could not argue with his success. And so in mathematics education classes we talked about how to humanize Moore's method.)"

Garrity, T.A. All the Mathematics You Missed: But Need to Know for Graduate School, (Cambridge University Press, 2002), pp. 77-78.

Discusses significance of point-set topology and its changing role in the curriculum over the last 70 years with a special mention of its use by Dr. Moore at the University of Texas.

Hersh, Reuben, and John-Steiner, Vera, Loving + Hating Mathematics, (Princeton University Press, 2011).

"The stories of Clarence Stephens and Robert Lee Moore embody two different, opposed strains in American education .... full integration of previously excluded groups ... requires more than mere legal equality, it demands transformative teaching methods." (p. 299)

Knuth, Eric J., " Secondary School Mathematics Teachers' Conceptions of Proof," Journal for Research in Mathematics Education 33, 2002, 379-405.

" As undergraduates, do prospective teachers have opportunities to experience and discuss these roles of proof? The Moore Method of teaching, for example, which is used by some mathematicians, provides undergraduate students with just such an experience." (p. 400)

Laursen, Sandra L., and Rasmussen, Chris, "I on the Prize: Inquiry Approaches in Undergraduate Mathematics," International Journal of Research in Undergraduate Mathematics Education 5, 2019, 129-146.

Mentions the historical role of the Moore Method and the Educational Advancement Foundation.

Lucas, J.R. The Conceptual Roots of Mathematics: An Essay on the Philosophy of Mathematics. (Routledge, 2000).

The author, a Fellow of Merton College, Oxford, argues for a form of logicism in which much of mathematics is grounded in transitive relations instead of natural numbers or set theory. After looking at Alfred North Whitehead’s failed attempt to found geometry and topology on a mereological basis, i.e. a theory of whole and part, he considers the transitive relation of ‘being embedded in’ as utilized by R.L. Moore.

Milnor, John, “Growing up in the old Fine Hall,” in Prospects in Mathematics

ed. Hugo Rossi (American Mathematical Society, 1999), p. 3

“The person who was closest to me in the early years [at Princeton University] was Ralph Fox. … I particularly enjoyed the course in point set topology which he taught by a form of the R. L. Moore method: He told us the theorems and we had to produce the proofs. I can’t think of a better way of learning how to make proofs and how to learn the basic facts of topology – it was a marvelous education.”

Morrel, J.H. “Why lecture? Using alternatives to teach college mathematics,” in Teaching in the 21st Century: Adapting Writing Pedagogies to the College Curriculum, (Routledge (UK), 1999), pp. 29-48. Online purchase http://www.questia.com/

“In order to adapt this [the Moore method where students ‘understood the topics extremely well and had a lot of practice in writing and explaining mathematics’] to an undergraduate setting, in which the time constraints and required syllabi mitigate against the use of such a method, I have used a modification of this approach … ” (p. 38).

Palombi, Fabrizio, and Rota, Gian-Carlo, Indiscrete Thoughts, (Birkhäuser, 1997).

"The core of graduate education in mathematics was Dunford's course in linear operators. Everyone who was interested in mathematics at Yale eventually went through the experience, even such brilliant undergraduates as Andy Gleason, McGeorge Bundy, and Murray Gell-Mann. The course was taught in the style of R.L. Moore ..." (p. 29).

Raiffa, H. “Game theory at the University of Michigan, 1948-1952,” in Toward a History of Game Theory ed. E. Roy Weintraub, (Duke University Press, 1993).

“I took a course called ‘Foundations of Mathematics’ with Professor Copeland, who taught in the R. L. Moore style: students are challenged to act like mathematicians, to convince themselves and others of the veracity of some plausible conjectures, to concoct starkly simple illuminating counterexamples, to generalize, to speculate, to abstract. No books were used. All the results were proved by the students. … I became hooked. Even though I didn’t know Leonard J. (Jimmy) Savage at the time, he also became enthralled in the same type of teaching program by being forced to act like a mathematician. I decided to become a pure mathematician and pursue a Ph.D. degree” (p. 166).

Ross, Arnold E., "Creativity: Nature or Nurture? A View in Retrospect," in N. Fisher, et al., eds., Mathematicians and Education Reform, 1989-1990, (American Mathematical Society, 1991), pp. 39-84.

Ross's early education in the USSR he likens to the Moore Method. Its spirit of “explanation and justification” continued to characterize his own approach to teaching mathematics. However, he mistakenly attributes its origins in the USA to E.H. Moore. (More on this misunderstanding can be found in the 1999 videotaped interview with Ross available at the Archives of American Mathematics.

Samuelson, Paul A., Inside the Economist's Mind: Conversations with Eminent Economists, (Blackwell, 2006).

Robert Aumann on taking real variables from the logician Emil Post at City College in the 1940s: "It's called the Moore method—no lectures, only exercises. It was a very good course." (p. 329)

Schmitz, S., and East, K. “Using proof pedagogy to scaffold pre-service teachers' application of child development,”Teacher Education & Practice, 31(2018), 63–80.

“After learning about the Moore method, it occurred to us that this kind of thinking was exactly what we wanted pre-service teachers to do with development. We wanted our students to apply knowledge of development rather than simply learn about development.” (65)

Selden, A., and Selden, J. “Tertiary mathematics education research and its future,” in Teaching and Learning of Mathematics at University Level: An ICMI Study, ed. Derek Holton, (Springer, 2001), pp. 255-274.

A section on the Moore Method suggests that courses making use of it “could provide interesting opportunities for research in mathematics education.”

Shier, D.R., and Wallenius, K.T. Applied Mathematical Modeling: A Multidisciplinary Approach, (CRC Press, 1999).

“The modeling approach in applied mathematics has much in common with the discovery methods used in pure mathematics, such as the famous R. L. Moore approach” (p. 14).

Szenberg, M., and Ramrattan, L.B.,eds., Collaborative Research in Economics: The Wisdom of Working Together, (Springer, 2017).

“Collaboration is formed from the desire to follow or imitate the leader. ... The mathematic discipline provides such an example in regard to the Moore method.” (Introduction by the editors, pp. 15-16.)