Philosopher's Cornered, issue 10

    Once, when our family was traveling with a real-estate agent in her car, our young son asked, “Hey, what kind of car is this?” I looked over at the steering wheel, saw the Chrysler logo, and waited for her answer. She said, “Look around you. Do you see any clues?” Surprised, I asked, “Are you a teacher?” “I was,” she said.
     This attitude of encouraging students to discover for themselves best characterizes good teaching. The teacher should lead a student to answer his own question based on what he already knows. This isn't a new idea; Socrates taught by asking questions of the student. Polya is probably the best modern proponent of leading students to solve problems themselves.
     The first requirement for teaching through discovery is that the teacher love discovery. It's the attitude of wanting to work things out yourself—to make them your own ideas. It's the fun of playing around and the thrill of a treasure hunt. If the teacher just parrots what he has been told, he hasn't really made the material his own—he doesn't understand it in his gut. And discovery is the process of problem solving—the skill his students will need as practitioners in their field.
      So a good teacher would make a good practitioner. But a teacher must go beyond problem solving—he must be aware of what's going on in the process. So (with apologies to G. B. Shaw) we can say that those who can, do, and those that know how they can, teach. Being aware of the discovery process is not easy. It's a fuzzy process that moves rapidly, with tenuous connections and alternative attempts. We're probably not conscious of all the tools we're using, and, if we are, do we know how we know them?
     The next step is to lead the students to solve problems on their own—usually through questions. The challenge is to choose the questions to neither give too much away nor leave the students stymied. And they must be natural questions—ones that the students could come up with themselves. (I was once quizzing my young son on his multiplication tables. "What's 3 times 4?" I asked. When he hesitated, his sister tried to help by prompting him with a question: "How old is Lisa?"   Well, he knew Lisa was 12, but it certainly wasn't a natural question.) The questions are usually based on how the teacher himself had come to a solution, but the teacher must remain flexible to new approaches by the students.
     Finally, the teacher needs to figure out why the teaching isn't going very well—why a student needed so much help. Was the teacher assuming the student had knowledge or experience that he didn't actually have? Was the teacher unaware of some of the steps or knowledge he himself used? Was the student awed by the teacher's ability and unwilling to display his ignorance? When the student himself is not aware of what he's lacking, the teacher must probe with questions, be a good listener, and use his imagination to guess the problem.
     In summary, a good teacher must

Practice discovering solutions to problems by playing around and developing an intuitive grasp of concepts.
Be aware of the steps taken in the discovery—what background knowledge, analogies, and similar experiences were used.
Devise a way to lead the student to a similar discovery through questions and exercises, leaving as much as possible to the student.
Assess what's wrong with the teaching process—why it was so slow and why it ended up with the teacher explaining rather than the student discovering. Perhaps the teacher needs to get better at the first two steps.

Teaching through discovery is not easy. It takes twice as long as simply showing the solutions. It takes getting yourself out of the way—not showing off, but letting the student have the fun of discovery. It's hard work, but then, anything really effective usually is. And the rewards are great!
And that's my philosophy.