What's Missing?

By Dr. Eugene Maier

A number of years ago, dissatisfied with the way I was teaching mathematics—filling the blackboard with proofs and procedures while students dutifully recorded everything I wrote—I was looking for other ways of conducting classes. Mathematics educators of the time were urging that mathematics be "taught from a discovery approach, an approach that encourages the learner to manipulate devices, to play mathematical games, to gather data, and to form his own conclusions."

Such practices seemed promising to me—at least, they took the students out of the passive role they assumed in a lecture setting. I visited a classroom where, I was told, I could see these methods in action. I came away with mixed feelings. The students were willing participants. They carried out the activities as instructed but, in the end, nothing much seemed to happen. The classroom milieu was pleasant, it had a sense of industry about it, but there was no electricity in the air—no lights were coming on. I wondered what was missing.

A few days later, reading Rollo May's The Courage to Create, I discovered what it was. There was no encounter between the students and the subject matter at hand. Everything had been laid out so nicely that the students were proceeding step by step, as if they were following a recipe. Everyone was comfortable—and nothing creative was happening.

According to May, creativity—the process of bringing something new into being—always entails an encounter; an encounter between a highly involved individual and some aspect of his or her world. And, May continues, this encounter brings with it an anxiety, "a temporary rootlessness, disorientation."

"That's it," I thought. The mathematics classroom should be a place where learners, whatever their age, encounter their mathematical world in a way that expands and enlarges it—that brings something new into being. My job, as the teacher, was to set the stage for this encounter, and to provide a safe and supportive environment if anxiety ensued.

The issue became: How does one bring about this encounter? It didn't occur when I demonstrated a procedure and asked the students to practice it twenty times. Or when I presented a flawless demonstration of some theorem. Or when I led them through a series of small steps designed to get them to arrive at a foregone conclusion. The students weren't encountering their mathematical worlds. They were being presented with some textbook version of the mathematical world and expected to quietly absorb it.

On the other hand, I found the encounters I wanted can occur. They happen most frequently when I am able to frame a mathematical situation in some context that connects with my students' worlds and, at the same time, provides pathways to new territory. Posing a problem that catches their interest and moves them into this new arena, and then leaving them to their own devices, precipitates the encounter. The boundaries of their knowledge are challenged; ways to extend them are conceived and explored.

The process does generate anxiety—or, disequilibrium, as we call it around here. Both the anxieties of students who are unsure of their thinking or afraid of being wrong, and one's own concerns that students become involved and learning takes place. But the anxiety vanishes in the light of the first "Aha!"