In the days of teaching
I had a dream about this last night, so thought I’d blog the memory (of the actual events, not the dream).
About 35 years ago, I was teaching at a small private liberal-arts college and was asked to take over a Freshman Math Tutorial at the beginning of their second semester.
A word about class format: the college used the “Great Books Program”, and Freshman math was devoted for about the first two-thirds to the study of Euclid’s Elements (in translation) and then the Almagest of Ptolemy (also in translation). The tutorial consisted of the tutor (in this case, me) and 10-12 Freshmen (co-ed). The tutor’s job was to ask the right questions so that the students, in discussing the answers and thinking about the implications, would develop an understanding of the focus of study.
In studying Euclid’s Elements, there would occasionally be a general discussion, but mostly a student would go to the board and “demonstrate” one of the theorems or constructions: stating the proposition, setting out a diagram and stating what specifically is to be proved or constructed, doing any necessary construction, and then going through the proof. The student demonstrating could be asked questions by the other students (or the tutor): to clarify a step, to explain why something actually followed from what had been said, etc. Questions such as those were common.
Demonstrating the theorems led to a deeper understanding for all, and provided the student doing the demonstration with useful practice in presenting abstract material, answering questions, thinking on his/her feet, etc.
Unfortunately, when I arrived the students were struggling, and were having trouble understanding what was what. I went over the parts of a proposition, and then explained that I would not be calling upon individual students to demonstrate, nor would I ask for volunteers. Instead, I would use random selection, putting the choice in the lap of the gods. I took out a pack of cards, discarded the Joker and face cards, and thus had four cards—one of each suit—for each of the 10 students in the tutorial. We agreed on which student would be represented by the 10, the 9, and so on down to the ace.
I would draw a card at random, and that student would go to the board to demonstrate: easy, impartial, etc. Those were the rules.
The next day I drew a card, and the so-designated student said, “I’m sorry, I’m not prepared.” I assured her that her lack of preparation didn’t matter. She just had to go to the board and demonstrate the theorem, and if she had difficulties, the class would help.
It was rather painful. She couldn’t even state the theorem without help. But, step by step, the students around the table helped and eventually we got through it. It took so long that the next student had relatively little time, but was prepared and got through the demonstration.
The next class meeting (we met MWF), I drew the cards again, and—lo!—the same young woman was first up. She was a little reluctant, but I went over the rule again—your card comes up, you go to the board—and she trudged to the board, where the reason for her reluctance became clear: she had somehow figured that since she had demonstrated in the previous session, she would not be called on in this session. But random selection is random, and…
Once more we had an exceedingly painful and slow demonstration, the students and I providing help at every single step. It was grating, but we got it done.
At that point, the entire class realized that, when their number came up, prepared or not, they would be going to the board. And I realized that when a student was randomly selected, I should remove one of his/her cards from the deck, to make it somewhat more likely that someone else would be selected—but at least one card was always present for each student. After everyone had demonstrated, the deck was restored, shuffled, and we began again.
The result, of course, was that every student started to prepare every proposition for every class: anything was better than having to stand in front of the clas and struggle through—not knowing one’s lines, as it were. And, of course, by preparing every theorem for every class, learning the theorems became easier and easier: patterns became obvious, each student quickly learned to see which parts of a new theorem were the tricky bits and which were more or less boilerplate. And they got better and better.
We finally got to Book XIII, which is difficult. After some preliminary propositions, the main task of that Book is the construction of the five regular (aka “Platonic”) solids: those 3-dimensional figures whose faces are congruent regular polygons. This was not easy stuff—and in fact the usual process followed in the other tutorials was to quietly notify one student that he or she would be demonstrating a particular theorem and give them a few days to prepare. I, of course, would have none of that: we used the cards.
Because I was also director admissions, I occasionally would have to go on a trip. I was away on a trip during Book XIII, and another tutor, whom I’ll call Miss Leonard, took the class for me. She entered the room with the class, they sat down around the table, and she asked which theorem was next. It was XIII:13—“To construct a pyramid, to comrephend it in a given sphere, and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid.” (The pyramid in this case is the tetrahedron: a 4-sided shape with each face being an equilateral triangle.)
“Okay,” said Miss Leonard. “Who’s demonstrating this?”
Consternation! “The cards!” the students said. “Didn’t Mr. Ham give you the cards?”
Miss Leonard was puzzled—I had forgotten to tell her about this little tactic. “No,” she said. “No cards.”
Quick discussion among the students, and then they said, “Say a number from 1 through 10.”
Miss Leonard: “Eight.” A few sighs of relief and one woman, with an enormous groan, got up and went to the board—and did the construction and demonstrated the theorem flawlessly.
Miss Leonard was some impressed, because from the questions and comments of the other students it was clear that any of the students could have demonstrated the proposition.
Needless to say, perhaps, the random-selection card method did not catch on with my colleagues, who tended to be traditionally minded. But it really, really worked.