Choosing Which Student Goes Next
In a lecture course there’s no need to choose among the students: all students simply take notes, though from time to time one may ask a question for clarification. Some courses, however, require students to actively participate in class. Such classes use some form of the Socratic method, in which students do the talking or presentation, with the teacher acting as a coach or guide, asking occasional questions and keeping the discussion productive.
Choosing who will next participate
In discussions classes students talk more than the teacher. The teacher often calls upon students — to go to the board to demonstrate a theorem in geometry, or to read their translation of a text, or to give their thoughts about a passage in an assigned reading (with those thoughts then discussed by the students). Or students may perform experiments individually or in teams and then gather to discuss their results and conclusions and questions.
If the class is purely an open discussion, as of a book that has been assigned (and hopefully read), students typically speak at will, with the teacher moderating to ensure that students are not interrupted and stay on track, and also to bring into the discussion students who have been silent.
Some material, however, naturally calls for sequential student performance — demonstrating a sequence of theorems, for example. The teacher then commonly chooses a student to present the next theorem. Typical methods, arranged by pedagogical utility, are:
- The teacher asks for a volunteer and chooses someone from among those who volunteer by raising their hands (sometimes, especially in elementary schools, also waving their arms and saying, “Me! Me!”). This method is common, but I would rate it as the worst approach pedagogically.
- The teacher picks a student to answer, going in strict rotation (in alphabetic order on last name, for example, or according to a seating chart), so that each student in turn is asked to participate. The weakness of this method is the predictability of participation, so that many students will pay no attention to what is going on in the class and instead focus on preparing what they will do when it becomes their turn.
- The teacher picks a student to answer but skips around. The weakness here is that teacher bias (conscious or not) will follow a pattern so that that selection becomes to a degree predictable. For example, the teacher may grow to favor some students who have given good responses (and thus those students get more practice, becoming better and better, leaving behind their fellows, who would profit from more attention). Or the teacher may unconsciously choose students in a kind of rotation — moving across the room, for example — so students have an idea when they should pay attention and when they can zone out. Or the teacher’s selections might be influenced by unconscious racial or gender bias, which some students will doubtless notice.
- The teacher uses random selection, so that neither the students nor the teacher knows who will next be called upon. When this method is used, all students must be prepared to respond, since any could be up next.
When I was asked to teach a small second semester freshman class, I decided to use random selection. At that college, teachers taught a class for the entire year, but in this case the teacher had to withdraw and I was tapped to take over the math class. I was told privately that he had left the class demoralized with the students struggling.
The class I taught
In this math class the course began with the study of Euclid’s Elements (plane and solid geometry and some number theory), followed by a study of the first five books of Ptolemy’s Almagest (geometric astronomy). Classroom time was spent in student presentations and discussion rather than teacher lectures. The teacher’s role was that of a coach and guide, ensuring the discussion was focused, constructive, and orderly.
The teacher’s main responsibility was to ask good questions (rather than provide answers) and to see that each student could be heard (that is, that an overbearing student would not dominate the discussion and that a shy student’s voice would be heard).
When I became their teacher, the students were well into the Elements. Students would study the theorems on their own time and in class would singly go to the board and present a theorem to the class—that is, state what was to be proved, draw the appropriate diagram, do any necessary geometric construction, and prove the theorem through logical argument. (The study of geometry thus served as a vehicle for studying a kind of knowledge and methods of proof. Students learned through practice the skill of proving propositions and patterns of proof, and mastered mathematical concepts through discussion.)
When one student was demonstrating a theorem at the board, those seated around the large table could ask for clarification, offer suggestions if the student got stuck, and discuss ideas raised by the theorem.
Random selection method
One session of the math class allowed enough time to demonstrate and discuss four or five theorems in the early books of the Elements, though by book 13 (the construction of the five Platonic solids) the theorems were long and complex and one theorem — demonstrated and discussed — might require a full class session.
I explained that I would not choose students to demonstrate theorems. Instead, the choice would be left to the gods. (The freshman year was heavily devoted to the study of Greek classics — not only in math but also in seminar discussions of books by classical authors (the Iliad, the Odyssey, works by Plato, Aristotle, et al.) and in language (students studied classical Greek and translated texts, beginning with Plato’s Meno).)
My class had 12 students, so I brought a deck of playing cards. I discarded the Kings, so the deck had Ace through Queen (12 cards) in each suit. Each student had a card value assigned: Ace, 2, 3, and so on.
I told the students that fate would determine who would take the next theorem. I would shuffle the deck and deal out cards to identify five different students (ignoring duplicate cards). For example, if the cards dealt were 2, 7, Ace, 8, 2, 8, and 4, the second 2 and 8 were ignored, and the students assigned to 2, 7, Ace, 8, and 4 would demonstrate the next five theorems, in that order.
Each class began with a new deal — no carry forwards from the previous class — so if we didn’t get to the fifth person, they were off the hook until their card came up again.
Each time a student demonstrated a theorem, I removed from the current active deck one of that student’s cards, thus reducing the likelihood that the student would be called upon again. However, I did not remove the last card for the student so that it was still possible the student might be called upon, just less likely. Once all students had participated, the deck was fully restored, four cards per student, and the process continued.
The rule was that, when their card came up, they must go to the board to demonstrate: that was their inexorable fate. Whether they were prepared or not was irrelevant. If their card came up, they went to the board because the gods had decided.
If they encountered difficulties, the other students would help them through parts they didn’t know or didn’t understand, but the student at the board was responsible for leading us through the theorem, with or without assistance. I repeated that the choice was not mine or theirs: it was in the hands of the gods, and it had to be accepted. Declining was not an option.
The very first card that day was for a student who asked to pass, because (she said) she was not prepared. I reiterated the rule: if your card comes up, you go to the board, prepared or not. So she (reluctantly) went to the board and asked the other students for help to state the theorem. It seemed that she may not have even looked at the theorem before class. Her classmates had to help her along at every step of the way, through the entire theorem. It was painful, and it took most of the class session, but we got through it. The next student whose card had come up was (thankfully) better prepared and did a reasonable job.
In the next class meeting, purely by chance the first card up was for the same student who had been so unprepared the previous session. Unfortunately the student had somehow assumed that since she had just demonstrated, she would be safe for a class or two and again was totally unprepared. She (and we) had to suffer through another painful demonstration with constant help required to get through it, though this time went a little smoother because we all knew better how to handle the situation. Still, it was doubtless embarrassing for the student.
After that awkward beginning, things improved quickly. The students knew that in each class their card might turn up and they would have to demonstrate a theorem — any one of the theorems for that session—and so they had to prepare all the next several theorems (or else suffer embarrassment at the board).
And (of course) the more they consistently studied and prepared, the better they understood the material and the easier it became to prepare. They started to recognize patterns, but instead of patterns of teacher selection, they recognized the patterns of the theorems — patterns of proof, the patterns of theorem sequence. The Elements started making sense to them, and when we got to Ptolemy, they were ready for it and felt they were on firm ground: they knew what they were doing and could understand the sequence of theorems and could appreciate each other’s demonstrations. (The Almagest is fairly tough going.)
One anecdote to show the class’s progress and the strength of this approach: When the class was in book 13 of the Elements, the construction of the five Platonic solids, I had to be away, so I asked another teacher to take over the class. Book 13’s proofs are so long and complex that teachers normally took aside a student and privately told him or her to be sure to be prepared for the theorem for the next session. So my substitute asked, “Who is doing this theorem?” (It happened to be the construction of the icosohedron.)
“The cards!” the class exclaimed. “Didn’t Mr. Ham give you the cards?”
The substitute was puzzled — I had grown so accustomed to the procedure I had forgotten to mention it — so a student said, “Name a number from 1 to 12.”
“Eight,” she said.
One student groaned loudly, went to the board, and (as the substitute told me) hit it out of the park: a perfectly polished presentation. And the class’s comments, questions, and discussion made it clear that any of students could have presented that theorem well. They were all prepared because they had no idea whom the gods would choose that day.
Why it works
Random selection works well for two main reasons:
- The students understand that being selected has nothing to do with the teacher. Because the teacher is removed from the equation, being chosen is neither favoritism nor punishment, nor is it the result of any bias.
- Because any student might be selected, every student studied and prepared, and as a result their studies became progressively easier as they absorbed more of the structure of the subject and gained more knowledge with which to work.
I did discover that I had to make all the selections at the beginning of the class session. When I drew the card at the beginning of each presentation, some students during the presentation would covertly the next theorem to be ready for it, since they might be called upon. Knowing who was tapped for the current class session meant that the others could relax and pay full attention to the presentation. Of course, some who were chosen were well prepared and didn’t require any last-minute cramming, and as the weeks went on, being well-prepared became the rule.