I do not believe it is an exaggeration to say that when I was a graduate student at Auburn University in Ben Fitzpatrick's topology class, or later as a more advanced student at the University of Georgia in B.J. Ball's topology class, the discovery method of teaching opened up the world of mathematics to me. On a more basic level, this method taught me how to think. As a beginning graduate student, no one had ever before asked me to figure out for myself the meaning of a definition, without supplying me with examples. No one had ever before asked me to prove a statement, without showing me a similar proof. And certainly no one had ever before asked me to investigate what might be true and what might be false about a new (for me) mathematical concept. My mathematical maturity, that before had increased incrementally as I learned and memorized more material, now grew in leaps and bounds to the point where it was no longer possible for me not to become a research mathematician. I loved those classes; they were exciting, demanding and absorbing. The difference in the depths of my understanding of concepts, between those learned in a lecture-memorize class versus those learned by my integrating the idea into my own work in a discovery class, was profound. I thought about a mathematical idea, dreamed about it, and thought about it some more, until the idea belonged to me. I firmly decided at that time, and I still firmly believe, that there is no more powerful way to teach or learn mathematics than the discovery method.
I have used the discovery method in my own teaching of advanced classes for decades now and I offer the following observations; some are mine and some are copied from my teachers, who in turn copied some of them from theirs: R.L. Moore, H.S. Wall, or H.J. Ettlinger at the University of Texas.
(I) In order for a discovery class to work, it is important to establish competition between the students and to prevent competition between the students and the instructor. There are a number of techniques I use to accomplish this.
- At the beginning of the course, besides the usual admonition not to use outside references or to seek help from any person, I make it clear that the class is responsible for finding any errors in the proofs that are presented. Occasionally, an invalid proof is put up and no student sees the error. I either tell the class that someone should have a question, or I use the time-honored technique of telling the student at the board that I "don't understand" a certain claim, and would she please go over that part once more.
- When the student presenting her proof at the board looks to me for approval, I deflect the question to the class simply by looking at the other students to read their reactions. Likewise, when a student asks me a question about the proof being presented at the board, I look to the person at the board for an answer.
- I never go to the board after a student has presented a valid proof of a theorem and show the class a more elegant proof. I stifle myself. This is the hard part about teaching by the discovery method. After the class is well established and the students are brimming with self-assurance, I allow myself to make general comments, to a student who has just presented a valid proof, such as "when you prove something for n = 1, then for n = 2, and then you say 'etcetera', change the proof to an induction proof", or "when you have a proof by contradiction within a proof by contradiction, see if a direct proof lurks within". But early in the term I always appear pleased with any valid proof, even if it has fourteen unnecessary cases.
- Each day I ask the students for theorems in reverse order; that is, the student with the least number of successful presentations is asked first, then the student with the second least number is asked second, etc. (I know, I know, I should use induction here.) This relatively non-threatening, but public, ordering of the students helps to foster a healthy competition.
- It often happens that part of a student's proof will serve as a useful lemma. After the presentation I point out the lemma and I name it after the student (like Smith's Second Lemma), and the others are encouraged to use the lemma whenever they need it. It isn't long before the students are inventing their own lemmas, partly because they see the efficiency in not re-proving facts and partly because they like to have lemmas named after them.
- I offer the students a wide range of theorems: easy, moderately difficult, and difficult. Without the easy theorems, the percentage of class participation will be lower; not just for the obvious reason that weak students can only prove easy theorems, but because talented but insecure students like to start out with proofs they know for certain are correct.
(II) I have found that there are some important considerations that predict which classes are likely to succeed with the discovery method.
- The class size makes a difference. If there are fewer than four students, they simply do not have enough time outside of class to prepare original material to take up, say, one-third of the class time. With a class of four to perhaps eight, I expect to present a lot of enrichment material, such as set theory, to take up the slack, but the class can succeed. With a class of more than roughly twenty-four, a sit back and watch mentality can take over, and obviously it takes more courage to present work to a packed roomful of faces.
- The age of the students is relevant. There are many exceptions of course, but my experience is that average American freshmen are not independent enough, nor do they have the level of abstraction needed, to readily prove theorems; but juniors and seniors do well. Another consideration: I have found that average juniors and seniors are not quite ready for prove-or-disprove statements. They still need the security of knowing that the statement is true. But later, on the graduate level, students thrive when they are asked to prove or disprove statements.
- Sometimes a talent disparity threatens the success of the class. That is, there is one student who is so much better than the others that she threatens to take over the class. My solution is to offer the talented student a directed reading class on the same subject. She will benefit from the extra attention and the class will not miss her.
- Prerequisite courses with overloaded syllabi do not work well with the discovery method. If time is short, in a lecture-memorize class, the instructor can work a few examples (on some topic) to demonstrate technique and beg off proving the theorems that justify the technique. But in a discovery method class, topics cannot be lightly skimmed in this way.
In summary, the discovery method does not always work. The class size, the ages of the students, or the amount of material that must be covered, may make it inadvisable. But when conditions are right, the discovery method can ignite students' enthusiasm for mathematics as can no other method, and it can give them the confidence and the power to independently discover, question, analyze, and conquer, new ideas. What could be better than that?