Moore first used his Method while teaching point set topology to graduate students at the University of Pennsylvania, as an introduction to the material they would need for their research work. He subsequently used it for undergraduate teaching, including with students whose major was not mathematics. It was Moore’s time as a graduate student at the University of Chicago, however, where he first came into contact with the development of matheamtical pedagogies.
The University of Chicago had been founded in 1894, and among its first faculty members was the psychologist and education reformer John Dewey, who, in 1896, founded the University of Chicago Laboratory Schools, which aimed to teach subjects using Dewey’s empirical peda-gogy. Dewey believed that experience was the only source of knowledge, and his pedagogy focussed on letting students experiment with whatever topic they were working on, though the limits of this were not clearly defined. The Head of the Department of Mathematics at Chicago, E. H. Moore (no relation), who was also the president of the AMS at the time, developed his own ideas based on Dewey’s theories. In his retirement address to the AMS in December 1902, E. H. Moore proposed the ‘Laboratory Method’ of teaching mathematics and physics to undergraduates, stressing the following:
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The teacher should lead up to an important theorem gradually in such a way that the precise meaning of the statement in question, and further, the practical — i.e. computational or graphical or experimental — truth of the theorem is fully appreciated; and furthermore, the importance of the theorem is understood, and indeed the desire for the formal proof of the proposition is awakened, before the formal proof itself is developed. Indeed, in most cases, much of the proof should be secured by the research work of the students themselves. — Moore, 1967, p. 419Dewey and E. H. Moore’s laboratory system of instruction did not meet expectations, and in 1904 Dewey left Chicago. By that point, however, the laboratory system had come to the attention of R. L. Moore, who subsequently took an interest in it (Parker, 2005, p. 55). While Moore quickly dismissed the idea of the laboratory system, it represented the first contact that
Moore had with those working on the forefront of pedagogical theory.
E. H. Moore further influenced R. L. Moore by being the principle member in a triumvirate of mathematicians at Chicago, the other two being Oskar Bolza and Heinrich Maschke. Between them, these three fostered a competitive environment among the students in the department, one which Moore’s Method sought to emulate. So, while Moore did not teach at Chicago, it was undoubtedly an important time for the young postgraduate student and one that influenced his own teaching.
Moore never explicitly codified his Method, and the ‘Moore Method’ as it exists today is the result of work by his students that have subsequently used it in their own teaching. We rely on the accounts of his graduate students to describe it, in particular those of Jones (1977, pp. 273–278) and Coppin et al. (2009). The basic rules of the course were as follows (adapted from Parker, 2005, p. 100):
1. There were no textbooks for the course and none was to be consulted.
2. Students would prove theorems from given axioms and present their proofs to the class.
Students were not to discuss their proofs between themselves until such a presentation has taken place.
3. Once a student had presented a proof the rest of the class would offer comment and criticism.
4. Competitiveness was key, and students were encouraged to come up with alternative proofs to the same theorems.
5. Much emphasis was placed on logic.
To foster fair competition between students, those who took his courses were required to have no background in the topic being studied. Moore would expel students whom he discovered to have read material outside that which they were given in class. On receiving a letter from his future graduate student Mary-Elizabeth Hamstrom, asking for advice on reading to do the summer before she joined the University of Texas in 1948, Moore’s reply included the sentence “I wish you had never taken a course in Real Variable Theory and that you had read even less about point set theory than I imagine you have.” (Parker, 2005, p. 245)
Once a class was selected, it would begin with the Method being outlined to students:
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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 was to start with[…] An example or two of situations where the axioms could be said to apply (e.g., the plane or Hilbert space) would be given. — Jones, 1977, p. 274After students had a written down the axioms and motivating examples, Moore would read definitions and theorems from his books. He then let students go away and find proofs of theorems, and to construct examples demonstrating the important aspects of a theorem’s statement.
At subsequent meetings, students would be called upon to prove theorems in an order chosen by Moore. These proofs were completed at the board, with weaker students being given the opportunity to prove a theorem before a stronger student. Moore was strict in preventing
students from interrupting the presenter; only once they had completed a proof or got stuck was the class invited to comment. If a student failed to complete a proof Moore would invite another student to offer a proof, or allow the original student to return in the following class to present a solution:
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Quite frequently when a flaw would appear in a proof everyone would spend some time (possibly in class) trying to get an example to show that it couldn’t be“patched up,” i.e., a counterexample to the argument (even though the theorem might be correct). — Jones, 1977, p. 275
Classes proceeded in this way for the rest of the course. Students submitted written solutions to all problems, but took no final examination. The course mark awarded by Moore took into account both their written solutions and presentations. While Moore did not impose solutions on students, and made sure each put in a lot of effort to solve the problems with which they were presented, he was not a silent partner in their teaching:
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Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. — Mahavier, 1999, p. 339)The motivation for the Method was to develop in students an understanding of the way in which mathematical research proceeds, where the development of one’s own ideas is central.
At times in his graduate topology courses students would be presented with Moore’s own
theorems to prove, and where a student found a more elegant proof than Moore’s, would be credited in his subsequent writing.