Moore method
Encyclopedia
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 advocates point to a depth of understanding that listening to lectures cannot give.
, a student of Moore and a practitioner of his method, described it as follows:
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:
In addition:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
courses. It is named after Robert Lee Moore
Robert Lee Moore
Robert Lee Moore was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics.-Life:...
, a famous topologist who first used a stronger version of the method at the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
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 advocates point to a depth of understanding that listening to lectures cannot give.
The original method
F. Burton JonesF. Burton Jones
F. Burton Jones was an American mathematician, active mainly in topology....
, 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)
Current usage of the Moore Method
After Moore became an associate-professor at University of Texas at AustinUniversity of Texas at Austin
The University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in 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:
- Number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- Real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
- Point-set topology
- Algebraic topologyAlgebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
- Automata theoryAutomata theoryIn theoretical computer science, automata theory is the study of abstract machines and the computational problems that can be solved using these machines. These abstract machines are called automata...
- Mathematical LogicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
In addition:
- The University of ChicagoUniversity of ChicagoThe University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
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 course at Yeshiva UniversityYeshiva UniversityYeshiva University is a private university in New York City, with six campuses in New York and one in Israel. Founded in 1886, it is a research university ranked as 45th in the US among national universities by U.S. News & World Report in 2012...
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 CollegeBerry CollegeBerry College is an American accredited, private, four-year liberal arts college located in Mount Berry, unincorporated Floyd County, Georgia, north of Rome. It was founded in 1902 by Martha Berry.-Location:Berry College is located on U.S...
in Rome, Georgia successfully uses the Moore method in his Real Analysis course. - The Physics Department of Berry CollegeBerry CollegeBerry College is an American accredited, private, four-year liberal arts college located in Mount Berry, unincorporated Floyd County, Georgia, north of Rome. It was founded in 1902 by Martha Berry.-Location:Berry College is located on U.S...
successfully uses the Moore method in numerous upper level courses. - Professor Don Chalice at Western Washington UniversityWestern Washington UniversityWestern Washington University is one of six state-funded, four-year universities of higher education in the U.S. state of Washington. It is located in Bellingham and offers bachelor's and master's degrees.-History:...
regularly uses a modified Moore method in all the upper level courses he teaches. He has done so for many years. 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 UniversityJames Madison UniversityJames Madison University is a public coeducational research university located in Harrisonburg, Virginia, U.S. Founded in 1908 as the State Normal and Industrial School for Women at Harrisonburg, the university has undergone four name changes before settling with James Madison University...
uses a modified Moore Method in Calculus and Analysis courses. - Professor Elena Marchisotto of California State University, NorthridgeCalifornia State University, NorthridgeCalifornia State University, Northridge is a public university in Northridge, a neighborhood in the San Fernando Valley area of Los Angeles, California, United States....
uses a modified Moore method in her "Foundations of Higher Mathematics" course. - Many topology professors in the Mathematics Department of Auburn UniversityAuburn UniversityAuburn University is a public university located in Auburn, Alabama, United States. With more than 25,000 students and 1,200 faculty members, it is one of the largest universities in the state. Auburn was chartered on February 7, 1856, as the East Alabama Male College, a private liberal arts...
use varying modifications of the Moore method. - Professor David W. Cohen of Smith CollegeSmith CollegeSmith College is a private, independent women's liberal arts college located in Northampton, Massachusetts. It is the largest member of the Seven Sisters...
implemented a modified Moore method for courses in Infinite-dimensional Linear AlgebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and Real AnalysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
. - Professor Vladimir N. Akis of California State University, Los AngelesCalifornia State University, Los AngelesCalifornia State University, Los Angeles is a public comprehensive university, part of the California State University system...
uses the Moore Method to teach graduate Topology courses.
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)
External links
- The Legacy of Robert Lee Moore Project.
- Links to biographical material and the Moore method.