I am often asked by others why I teach and why I love it so much. As a response to this question, I have written a short essay about my views on teaching and thinking.
Vignettes on Teaching
Dinner. My brother and I both have Ph.D.’s in mathematics largely because of dinner time conversations when we were growing up. Not until adulthood did I learn that some people have dinner without math puzzles and questions that begin, “In that Scientific American article about porpoises learning to speak, do you think...?” Some people, I am informed, talk about their day over dinner. My family, however, considered major and minor problems of all sorts. Philosophical questions, mathematical puzzles, scientific concerns, historical, aesthetic, and ethical issues were all part of the daily fare. Each discussion was lively. Novel points of view were acknowledged and appreciated. Thinking was what was done. My parents always expanded intellectually, always embraced new ideas, and always sought to learn and invent. Their curiosity and constant learning spread to everyone around them. Their habits of asking questions, thinking about issues, and continuing to learn throughout their whole lives simply made life more interesting for everyone. People who think for themselves and who delight in seeking insights and answers appear to me to live more interesting lives. My parents gave me clear models of the added richness from thought, and their examples certainly inspired me to want to pass on those habits to others.
Medieval art. One thing I learned in school was that important lessons come from unlikely sources. I was a mathematics major in college, but one of my seminal educational experiences occurred in a Medieval Art History class. This course involved looking at cathedrals and old paintings and saying things about them. Generally, no particular thought came to my mind when looking at a medieval painting, but I had noticed that others were able to draw deep insights from the malproportioned figures. One day the elderly professor, Miss Frisch, showed us a painting and turned to me. “Mr. Starbird,” she said, “What do you see in this picture?” I did my best to imitate the types of answers I had heard to such questions. Hoping to hit the right tone, I began something like, “The painting seems to me to represent the aspirations of a soulless humanity for higher meaning.” Professor Frisch was a distinguished scholar. Years of experience and refinement restrained her from saying, “Cut out the bull, Starbird.” Nevertheless, she conveyed that sentiment rather effectively by repeating the original question, “What do you see in this picture? Just tell us what you see.” To me this question was a breakthrough. She was asking me to give the direct evidence of my own eyes. If I saw small heads and long hands, stiff joints or similar triangles, those were the features that she wanted to hear about. Any hope I had for developing an appreciation of art must be built on noting features I personally saw in pictures. Learning and pointing out what others have seen is a different skill, but I recall that day as a step in learning to see with my own eyes.
Miss Frisch, taught us to look afresh at ideas that others had examined for centuries, and taught us to expect sometimes to gain new insights that others had missed for generations. She taught us that what we see ourselves is important and that we should develop our own capacity for independent insight. She also taught me that people can learn life-changing lessons from any field—medieval art or mathematics.
What I teach. I teach the joy of thought.
The inner lives of other people are mysteries to me. Societal customs or deeper human drives dictate that we keep our inner experience private. What is the character of the stream of consciousness of another person? I don’t know. But from outward signs, I judge that many people miss chances to enjoy ideas and develop them for themselves. Trying to see the world with my own eyes has been an experience rich with reward and fascination for me. Parents and teachers taught me the habit of taking independent views and I try to encourage students to think for themselves.
Watching. We live in a world where watching is a large part of the norm of daily existence. Watching TV, watching sporting events, watching the evening news. Times spent watching consume a large part of our days. Schools and colleges tend also to emphasize watching. Classes are frequently times for students to watch someone else do something. Good students are those who watch well. These habits of passivity are hard to break. Our educational system systematically rewards attentive passivity. A high proportion of the experience that our students receive through their education encourages them to look for authorities and to view learning as the acquisition of facts. Yet our society needs leaders who think independently, people who can cope with novelty and find ways to solve problems that have not been clearly posed or answered.
Keeping up. Knowledge is being developed or discovered at a faster rate than ever before. In the technical fields especially, the rate of increase in knowledge is exploding. Each year hundreds of thousands of articles are published in research journals in mathematics, biology, chemistry, and computer science. How do we as educators face the daunting task of keeping up?
The answer is, we cannot keep up. Keeping up is a feeble dream and has been for decades. Now it is all the more important for us to empower our students to be able to deal with situations and knowledge that they do not know and that no one knows. The accelerating production of knowledge means we must abandon the hope of keeping up with new knowledge and must instead take seriously the job of teaching students the skills of navigating in a sea of change. Change is the world for which we are attempting to prepare students. Some knowledge and skills are certainly fundamental and will always be useful, but many specific skills that were previously the bedrock foundation of an educated person must now be looked upon with skepticism. We must try to teach life lessons that students will keep and use in settings we do not foresee.
Mathematics presents a great example of a field in which the idea of what is fundamental and important must now be a subject of debate. Calculators and computers are far better than people at doing much of the skilled mental labor that was previously viewed as absolutely basic to an understanding of mathematics. Multiplication and division are simple examples, but many far more sophisticated skills as well are now within the range of inexpensive devices. How much of a student’s effort in learning mathematics should be devoted to learning skills that can be done better by a machine?
Knowledge comes and goes, but hatred lasts forever. What will my students retain from my class after 25 years?
Attitudes far outlast facts. Knowledge comes and goes, but hatred lasts forever. Perhaps the joys of thought and discovery last forever too.
Bring school home. I try to get students to bring their learning and thinking into their daily lives and make it part of who they are. Students should discuss their mathematics at the dinner table. Perhaps instead of testing our own students, we should test their roommates. If our students have explained the ideas to their roommates and families, then they have brought the course into their daily lives. For me a primary goal of teaching is for students to view education as part of their daily habit of seeing their real world.
Discovery learning. In graduate school I took a course taught in a way that seemed peculiar at the time. The teacher simply handed out questions at the beginning of the semester. The answers to the questions embodied the examples, definitions, and theorems comprising a body of mathematics. Each day the professor called on specific students and asked them to go to the blackboard and explain the answers to that day’s questions. In one of the early days of the semester, several students were at the board presenting answers to questions that had been advertised as being rather difficult. When the students presented their answers, the teacher praised them for their good work in settling the question, and I said to myself, “I can do that.” Somehow it was necessary to be in a situation where independent work was requested for me to realize what it was in mathematics and that I could do it. I was not alone. Others can do it too, but frequently they don’t. They are not asked and it has been discouraged in many settings for a long time. I learned later that this teaching method in mathematics was pioneered between 1920 and 1970 by a UT professor R.L. Moore.
Producers, not consumers. This discovery teaching method promotes the long-term goal of education, namely, getting students to be lifelong learners. When students discover important ideas on their own, they come to view themselves as producers of knowledge, not merely consumers of other people's knowledge. If students have seen themselves generate central, significant ideas on their own, then for the rest of their lives they will respect their own ability to produce knowledge. Not every class needs to emphasize personal discovery, but that experience can be an important piece of education that can affect how students learn, work, and live.
Learn from yourself. In each person's hierarchy of respect for wisdom and insight, no one stands higher than him or herself. People listen to themselves. This statement is not denigrating nor cynical. An idea you yourself conceive must fit into the rest of your world view. You cannot generate a notion that does not fit into your own mental frame since the generation of thoughts is an act of your individual mental condition.
On the other hand, it is easy for a teacher to say things that do not fit into the students' view. The advantage of getting students to discover ideas on their own is that they not only learn the isolated concept, but they must have adjusted the surrounding territory to accommodate that idea.
The conclusion of these observations is that one method of effective teaching is to get students to explain the course content to themselves.
You may feel that that is impossible. If you are teaching new things to students, how can the students explain those ideas to themselves? They don't know the ideas. However, that is exactly the trick. Students are in the position of creating a world view just as humanity as a whole was when the course's central themes were first discovered. The goal then is to create artificially a setting in which the desired insight is most obvious. That is, make the students aware of examples, related insights, and techniques for exploration that will work. Frequently, new ideas are inherent in a set of existing notions. Force the pre-conditions and the conclusions will follow—often accompanied by the unexpected additions of the students' individual insights and idiosyncrasies.
It’s what the students do that counts. Good teaching can never be judged by watching the teacher. It's what the students do that counts. If a teacher creates a world where students make discoveries and crystallize ideas for themselves, then the teacher has been effective. I see the teacher's role as assembling for the students an unusually rich collection of building blocks—examples, techniques, and information. Then the teacher must stand back while the students struggle to assemble the pieces into knowledge and wisdom. Sometimes students just miss insights; sometimes they seem to grasp them only to let them slip away; sometimes they endure the frustrations of partial insight. These moments are important moments for the teacher to stay mum and give the students the joy and satisfaction of personal discovery as understanding takes root in its own wandering way.
Little changes magnify over a lifetime. Contributions to students during a semester are small in their fractional content of our students' lives, but a tiny habit of independent analysis repeated over a lifetime can make the difference of a lifetime. If people like to think and like to explore new ideas, the amount of education they give themselves over their lifetime will be vastly greater than any amount of material covered in a semester’s class. I preach the joy of thought and hope my students can enjoy thought's pleasures for their lifetimes.