When the Son of Man comes in his glory, and all the angels with him, he will sit on his glorious throne. All the nations will be gathered before him, and he will separate the people one from another as a shepherd separates the sheep from the goats. He will put the sheep on his right and the goats on his left. Then the King will say to those on his right, “Come, you who are blessed by my Father; take your inheritance, the kingdom prepared for you since the creation of the world. For I was hungry and you gave me something to eat, I was thirsty and you gave me something to drink, I was a stranger and you invited me in, I needed clothes and you clothed me, I was sick and you looked after me, I was in prison and you came to visit me.”
Then the righteous will answer him, “Lord, when did we see you hungry and feed you, or thirsty and give you something to drink? When did we see you a stranger and invite you in, or needing clothes and clothe you? When did we see you sick or in prison and go to visit you?”
The King will reply, “Truly I tell you, whatever you did for one of the least of these brothers and sisters of mine, you did for me.”
–The Bible, New International Version (2011), The Book of Matthew, Chapter 25, Verses 31-40
It has been said that God looks after little children and fools. That is undoubtedly true in the case of my coming to know and love Bill Mahavier and the Moore Method. My graduate career began abysmally. I do not remember, in either Moore-style courses or ones taught by the traditional lecture method, proving a single theorem during my first two quarters at Emory University (these took up the fall of 1966 and the winter of 1967). All of my mathematical work to this point had consisted of looking at the work of professional theorem-provers, that is, mathematicians and textbook-authors. I had never proved a theorem myself. I had about as much chance of doing so as I would have had of riding a bicycle for the first time when all I had done before that was to watch other kids ride theirs.
But a miracle occurred during the spring of 1967. The third quarter of my Topology I course, like the first two, was taught by Bill. The class consisted of eight students. Three of them, two men of my age and a fifteen-year-old high-school student, clearly had “cracked the code” of mathematics. For the first two or three weeks of the quarter, they took turns in going to the board and presenting correct and, for the most part, clear and even elegant proofs to the theorems. Meanwhile I sat among the other five students, dutifully copying down what was being put on the board. At the beginning of class during the third or fourth week of the term, Dr. Mahavier announced that he intended to split the class. He would, he said, continue to meet five of us at the regular class-time, but he would see the other three one hour later. The three later-meeters were, of course, the three hot-shots. I immediately concluded that Dr. Mahavier had pulled the classic “sheep and goats” maneuver, and it was clear to me who were the goats. I felt humiliated, and I was furious.
Nevertheless, I decided to play the game. So, apparently, did the rest of the Goats. We began to meet without the Sheep, and I actually enjoyed the Sheep’s absence. I wondered, however, would we Goats now rise to the challenge? Would we really, given more time and no pressure from the Sheep, prove theorems?
We would and we did. The five of us began to take fairly regular turns in going to the board, presenting our arguments to one another, and often being correct. Our confidence grew with our successes, to the point where we could absorb the disappointment of having a flaw in our reasoning pointed out on Wednesday and then return on Friday with a correction and the rest of the proof. We were rolling! But this exultation was quickly followed by questions. What would happen at the beginning of the fall term and Topology II? Would Dr. Mahavier reunite the Sheep and the Goats, and if so would the old pattern of the Sheep dominating the Goats return?
Dr. Mahavier did indeed reunite the class. The old pattern, however, did not return. While we Goats had been honing our skills at proving theorems and presenting our proofs, Dr. Mahavier had kept the Sheep from getting ahead of us in the sequence of theorems for the course by assigning them “diversionary” problems of an enrichment nature. As a result, as the fall of 1967 began, the Goats found themselves at the same point as the Sheep in the sequence of theorems. Even better, it became readily apparent that the Goats had become Sheep. The usual pattern now was for all of the members of the reunited class to share uniformly in the presentation of arguments and in the satisfaction of having arguments certified as correct by the class. The miracle was complete.
Bill Mahavier’s Sheep-and-Goats Maneuver is the most dramatic demonstration that I have ever experienced of the effectiveness of the Moore Method, and of the dedication and generosity of its practitioners. Seemingly without a second thought, Bill had assigned himself an extra course during Spring 1967, at no extra pay, to help five struggling would-be mathematicians, none of whom might “pan out.” (When, some years later, I “reminded” Bill of what he had done and how it had transformed my life, he surprised me by first looking at me blankly and then stating that he had no memory of the incident. As I reflected on this exchange, my surprise turned to understanding. Taking on an extra course, gratis, was no more extraordinary for Bill than servicing a motorcycle. It was simply something that he did to solve a problem, in this case helping some students who needed a jump-start.) I seriously doubt that, without Bill, I would ever, at Emory or anyplace else, have learned to practice, teach, and enjoy mathematics as I do, much less earn a Ph. D. in it. Bill’s love for students and teaching lives on in all of us whose lives he has touched.