April 1998. Revised 15 April 1999.

Its use is not just that History may give everyone his due and that others may look forward to similar praise, but also that the art of discovery be promoted and its method known through illustrious examples. (Leibniz 1 )

Texas's most prominent mathematician and teacher of mathematics, R. L. Moore, who taught at the University of Texas at Austin from 1920 to 1969, expressed his educational philosophy by the statement: "That student is taught the best who is told the least." 2 Moore's paradoxical encapsulation of his belief ran the risk of giving a hostage to fortune, but to those who have seen the 1965 Mathematical Association of America film devoted to his teaching method, it is an apt summary. No one acquainted with Moore's method would think, for example, that he wished to distance himself from students or diminish his interaction with them; on the contrary, he was in effect requiring students to join him in doing, and indeed creating, mathematics and this, in turn, required the highest degree of devotion on his part.

Attempts to analyze just how and why Moore's method worked so effectively have not been completely successful--even those students of his who have documented their experience have yet to do so in a way that all the others find entirely satisfying. Moore was not one for explaining or justifying what he did, he simply did it; his one-sentence credo is not just a summary of his method, it is virtually his sole verbal expression of it. Though he taught teachers of high school mathematics during summer classes at the university, I don't believe he saw himself as an educational reformer. 3 The term "Moore school" has been used but this refers mainly to the particular branch of mathematics (point-set topology) that he and his students developed. The term "Moore Method" has evolved to refer to the teaching method independently of the subject being taught. The one thing that everyone, Moore student or not, agrees on is his success--unparalleled in the history of the U.S.--as measured by the prominence of so many of his students who became themselves mathematicians and by the positive effect he has had on those who did not become professional mathematicians. 4

However, it has been my experience that even the Moore Method is better known among mathematicians than among teachers in mathematics education. For whatever reason, the educational literature in general seems to be short on examples of what constitutes good teaching: this observation comes admittedly from my narrow knowledge of the area, but the words of David Hawkins at a 1966 conference on discovery learning seem still appropriate to keep in mind:

The good teaching I have observed, teaching by teachers who are accustomed to major success, owes little to modern theories of learning and cognition and much to apprenticeship, on-the-job inquiry, discussion, trial--ceaseless trial--within a common-sense psychological framework ....

... it seems to me the better part of wisdom to find the good school situations--not the better third but the better one per cent--and engage in close observation and intellectual resonance; then try to recreate such situations and make them more abundant and reproducible, no holds barred. Nothing about this process is easy, but it can be done because it is done--but rarely. (p. 4) 5

Undoubtedly Moore's example, albeit as a university professor but also as a teacher of high-school teachers during summer sessions, is an important one - certainly one of the one percent that Hawkins suggests we should study. But we know next to nothing about what influenced Moore: how much, for example, is due to outside influences of his own teachers and colleagues, compared with crafting a method from his own teaching experience. Also, to what extent was the method he evolved a product of a particular view of the nature of mathematics? I think this kind of information is useful to have in addition to the direct (or as direct as possible) observation of his classroom because it would contribute towards showing that Moore's method is not an isolated, irreproducible phenomenon.

Just after the turn of the century seems a good starting point to look for answers to some of these questions. In 1903 John Dewey, the American pragmatist philosopher and educator, published a paper on the teaching of mathematics (geometry in particular) that was in response to a paper by George Bruce Halsted. They could well have known each other before: in the 1880s Dewey and Halsted were at Johns Hopkins University and attended the lectures in logic given by Charles S. Peirce, the principal founder of the American pragmatic school of philosophy.
6
The articles were both published in the *
Educational Review*, the main journal of the profession. The basic issues they raised are familiar to us still and I suspect their discussion can serve as well as anything as a background to the discussions ever since in mathematics education in this country. This exchange is appropriate as a background in particular to R. L. Moore's method of teaching.

I do not think that Moore could easily have avoided this particular exchange at the beginning of his long career. In 1902, when Halsted's paper appeared (in the December issue), Moore was teaching at a high school in Marshall, Texas. He had been a student of Halsted's at the University of Texas and kept in touch with him for many years after he left Texas. We know that Moore was well aware of Halsted's paper since Halsted referred to him in it (though not by name) and sent him a copy. 7 The paper by Dewey, who was at the University of Chicago at this time, was published in the spring of 1903, about the time that Moore received an offer of a fellowship at Chicago. I do not know for certain if Moore saw Dewey's paper but it would seem to be at least a good possibility. 8

One of Halsted's long-term projects was to push for increased rigor in the teaching of geometry. He was a principal propagator of the new non-Euclidean geometries into the English-speaking world through his translations of Lobachevskii and Bolyai and through his many journal articles intended for teachers of mathematics and a broader lay public. In this article in the *
Educational Review*
of December 1902 he is preparing the way for his next college textbook, a "rational geometry" with which he intends to make much of the spirit if not the content of David Hilbert's *
Foundations of Geometry*
of 1901 accessible to beginners. "Geometry," he says, "always relied upon for training in the logic of science, for teaching what demonstration really is, must be made worthy [of] the world's faith. There must be a text-book of rational geometry really rigorous."

We are no longer content to bear with superficially clear statements which seldom if ever lead into actual error--nor does it suffice to start with inaccurate statements and, as we advance, to modify them so as to bring them into accord with our wider vision and our more stringent requirements. We must from the beginning bring up ourselves and our pupils on, not only the truth, but the whole truth. (457)

Halsted quotes approvingly John Perry of the Perry Movement, perhaps the best known mathematics education reform movement in England which had gotten underway a couple of years earlier: "Now, in my experience there is hardly any man who may not become a discoverer, an advancer of knowledge, and the earlier the age at which you give him chances of exercising his individuality the better." (In fact it is in this connection that Halsted cites the "young man under twenty [who] proved that of the great Hilbert's 'betweenness' assumptions, one of the five is redundant" (459).) However, Halsted disapproves of what he sees as Perry's "personal antipathy to 'a perfect logical system'" 'deduced logically from simple fundamental truths.' He quotes Perry: "As soon as we give up the idea of absolute correctness we see that a perfectly new departure may be made in the study of mathematics." This is anathema to Halsted who cites Hilbert on how the demand for greater rigor leads to greater simplicity and ease of understanding.

Halsted gives a number of examples of what he regards as misstatements in current texts as well as in the standard treatments of Euclid (relying for this on Bertrand Russell's recent article in the *
Mathematical Gazette*
on the teaching of Euclid
9
). He includes his favorite example for beginners, the definition of the straight line. We know from accounts of his students that Halsted liked to start off with his freshmen by asking them to give their definitions or understandings. He would then make what for many of them was a memorable issue out of the "definition", sure to be presented, that a straight line was the shortest path between two points--an unhappy and illogical definition as Halsted would have it. This is one of the things that must go. The teacher, Halsted says, "has no right to dump into a helpless scholar what he knows to be trash" (458).
10

Though these quotes do, I believe, convey Halsted's main point, it is important to note that he begins and ends his paper with an admission that a preliminary course in geometry does not call for full rigor and that a "sensuous rather than a rational" approach is appropriate. But such an approach, he maintains, is already available in present texts and the real task is how to introduce on this basis the new rigor. This is a matter for "the genius, the tact of the author of the new text-book of rational geometry" (i.e. Halsted).

Dewey in his paper, entitled "The Psychological and the Logical in Teaching Geometry" gently points out the lack of specific guidance that Halsted provides: Halsted admits the need for a preliminary, intuitive basis in learning geometry but is not very helpful at indicating just where and how a transition should take place. He seems to condemn such approaches as currently exist. Dewey agrees with Halsted that, whatever the preliminary approach is, it should not inculcate "mental habits and preconceptions which have later on to be bodily displaced or rooted up in order to secure a proper comprehension of the subject" (217). But how is this to be done? Dewey writes that he is "disturbed" by reading Halsted's stricture about the necessity from the beginning of bringing up "ourselves and our pupils on not only the truth, but the whole truth." Dewey responds:

To try to put into practice such a method is to develop impossible text-books and train impossible teachers. *
Towards*
the whole truth with all our heart; on it, no, because it is a meaningless requirement. The need and demand for teaching arise from the fact that the whole truth is not there to build upon. For a mind to build upon the very thing which it is the goal of its endeavor to obtain is surely the Irish bull of pedagogy.

The question of the correct definition of the straight line is a matter of mathematics alone. ... But there is need for making clear the fact that the content of a given book or lesson for a given grade of pupils is not a matter of mathematics alone. It is a psychological matter as well, ... (219)

Dewey then proceeds to turn Halsted's favorite example of the straight line not so much against Halsted as into a positive argument for cooperation between mathematician and teacher. I think this must be one of the clearest and most masterly examples of how philosophy (in this case pragmatism), psychology, mathematics, and logic can contribute to addressing the pedagogical issue of how learning takes place. This, at least, is the pure Dewey, as opposed to the views that have also come to be associated with him through some of his followers. The basic tenets are those upon which constructivism and discovery methods are based. Here let me just quote the beginning of the first of a two-pronged argument:

What the student actually learns is not what is in the mind of the author or teacher who propounds the definition: What the student learns is what the proposition means to him. This is certainly a truism. But it becomes a vital statement when we recall that what a given statement means to a pupil depends absolutely upon the interaction set up between the topic presented and the habits which the pupil brings with him to it. (219-220)

I believe this is a good statement of the basic assumption shared by proponents of a wide range of reform movements. Dewey goes on to enunciate in effect what I take to be a key tenet of the radical constructivist today:

... I must express my doubt whether the definition [of the straight line, say] of even the competent mathematician is absolutely logical in any other sense than answering to the status of mental growth, mastery of methods and problems reached by him, in the same sort of way that the relatively rough and ready notion of the beginner answers to the growth and needs of the latter. (224-225)

Dewey is here attempting to diminish the distinction between the psychological and logical. In the process he is asserting that learning and creating mathematics ought to be regarded as essentially the same process and that this comes from the observance of mathematics as it is actually practiced. On this view, there is no such thing as conveying mathematics through teaching, there is only a shared learning process, and this brings us to Moore's credo of being "told the least".

What was Halsted's response? As Dewey seems to allow, it is hard to see that Halsted, in spite of his usual dramatic rhetoric, would in practice necessarily disagree with anything Dewey has said. But the logician Dewey appears to have caught him in a logical inconsistency so that it is not clear exactly what Halsted is preaching, let alone what he practices.

Halsted was a certifiably successful and inspiring teacher that students, whatever their capabilities, remembered long afterward. 11 But the implication of Dewey's reply is that perhaps Halsted's method was really most appropriate for those who already had a disposition for imbibing "method" and "logical abstraction" at an early age. At any rate, we can be sure that Halsted had a response (even if unpublished) and continued with his usual confidence; he went on to publish his rational geometry and other textbooks in a similar vein.

The main thing is that here is a debate echoing ones that have occurred before and after in this country and elsewhere, in particular echoing the present debate over constructivist and discovery methods. I am hypothesizing that R. L. Moore was an observer of this debate. In 1903 he is between Texas and Chicago. In a footnote to his paper Dewey adds that since he wrote the article the Chicago professor E. H. Moore, America's most prominent mathematician of the time, had published (in March 1903) an address "On the Foundations of Mathematics" 12 which Dewey implies is supportive of the theme of his paper and thanks him for "various suggestions". It was E. H. Moore who earlier praised R. L. Moore's betweenness proof and arranged for him to come to join the distinguished group at Chicago.

I have not fully developed the possibilities here, but I think it is already possible to see in R. L. Moore's method a balancing on the one hand of Halsted's inspirational and rigorous version of student-centered learning with, on the other, the Chicago school's attentiveness to the intuitiveness and uncertainties of mathematical practice. The Texas Moore was heir to a tradition, an uncommon and challenging tradition of pragmatic enquiry, of trial and error, in the search for a better method of learning.

The Legacy of R.L. Moore Project

1.
*
Mathematische Schriften,* C. I. Gerhardt, ed., vol. 5, p. 392. Translated by André Weil in his *
Oeuvres,* vol. 3, "History of mathematics" (1978).

3. [There is a letter from Moore, in the 1950s?, in which he describes himself as not a pedagogue. This is quoted by Mrs. Whyburn in one of her articles.]

4. This latter is a less well-documented measure of success: those students, especially undergraduates, who did not go on to become professional mathematicians, but whose knowledge and appreciation of mathematics meant that it would never be a "foreign" subject to them. Unfortunately these students are not so quantifiable as Moore's graduate students and hence all we really have to go by are the numerous anecdotal accounts. Nearly everyone I know in or related to the "Moore School" at Texas has stories of undergraduate students who "happened" into one of Moore's classes and found temselves amazed, first that this subject in Moore's hands could be the same subject they had been taught in high school, and second that they enjoyed it.

5.
Hawkins, David. "Learning the Unteachable". Shulman, Lee S. and Keislar, Evan R., editors.*
Learning by Discovery: A Critical Appraisal,*
Chicago: Rand McNally; 1966; pp. 3-12. Proceedings of a conference, Stanford University and Social Science Research Council. This work appears to have been the high point in the organizational history of the discovery method. It seems never to have had the coherence as a group or movement as the School Mathematics Study Group (SMSG), for example, had. There was virtually no agreement at this conference about the meaning of "discovery" and whether there was a well-defined method whose cause they were aiding.

6. Both Halsted and Dewey obtained their doctoral degrees from Johns Hopkins University, Halsted in mathematics 1879 and Dewey in philosophy in 1884.

8. I do not know of any more direct connection between Moore and Dewey. [There is the reference in, I believe, Veblen and Young's book to Dewey in connection with the concept of categoricity.]

9.
Russell, Bertrand. "The Teaching of Euclid". *
Mathematical Gazette.*
1902; 2(33). Reprinted in *
Mathematical Gazette*
55(1971), special centenary issue.

10.
Halsted does not give an alternative definition for the straight line. In his textbook, *
Rational Geometry: A Text-Book for the Science of Space,*
(New York: John Wiley, 1904, 2d ed. 1907), he follows Hilbert in starting with points, lines (or "straights" as he terms them), and planes as undefined initial elements.