Bob Case receives Tepper Haimo Award
for Excellence in Teaching

Professor Bob Case of our Department of Mathematics was selected to receive the prestigious Deborah and Franklin Tepper Haimo Award for Excellence in University or College Teaching. It was presented by the Mathematical Association of America (MAA) at the annual meeting held in San Antonio in January, 1999. Professor Case was invited to deliver an address at the meeting, the text of which appears below.

During the last decade, issues of mathematics teaching, and in fact of teaching in general, have come to be dominated by one question: How can we as teachers foster a thorough transition to student-active and student-centered learning?

We've moved over the last generation from an almost complete reliance on exposition to a place where the student's own activity and participation is seen as central. Rather than the instructor being simply the dispenser of knowledge, the roles of coach, questioner and even provocateur have emerged. Previously the instructor was viewed as the source and font of knowledge and the student a kind of vessel to be filled.

But if the student is now seen as the active center of learning, this makes the teacher into an architect of learning experiences and a facilitator. The whole process is based on the perception that students want to be lifted to a place where they can above all be creative and productive.

In coming to this view of things, it is surprising how we've worked our way through the ideas and strategies of some of the principal Greek thinkers. During the last generation, the approaches associated with Euclid, Socrates, Aristotle and Plato have come onto the scene one after the other. Not that we've dropped all previous methods when we've come to the next, in a faddish way, but rather there's hopefully been a conscious effort to integrate the best of each. In other words, the process has been cumulative.

Euclid. Those of us in school a generation or more ago looked forward to the polished presentations of outstanding lecturers. The Euclidean rigor and the logical structure of the exposition were certainly powerful and beautiful in many respects. But this is not the way most people learn mathematics, and those who try to bring the style of the lectures they received as graduate students to their own teaching are severely jolted. This doesn't mean we must abandon rigor, but where we go as teachers is dependent on the purposes of the course and the situation of the students. As one experienced math teacher put it: "Tell them the truth. Tell them nothing but the truth.But, for God's sake, don't tell them the WHOLE truth!"

Socrates. In the 1970's some of us began to utilize the method known as Socratic Dialogue, which roughly means confronting the students with a chain of questions which elicits in them the discovery of an idea or a method. The flow of mathematics is coming from the students rather than the instructor. Of course, the instructor has broken down the mathematics in advance and has thought of the main paths the question-answer process may take. While every effective teacher comes upon some of the techniques of Socratic Dialogue through experience, my colleague Carla Oblas and I, who had been using Dialogue extensively, wanted to see what happened when it is used EXCLUSIVELY in a college classroom. We got a small internal grant from the University so that I would teach two sections of integral calculus, one entirely by discovery through dialogue, the other one entirely by traditional lecture or exposition. Professor Oblas would come to the class to keep me honest: no declarative sentences at all from the instructor in the Socratic Dialogue section; no questions at all in the lecture sections.

The effect was dramatic. The Dialogue section fared substantially better on the standard final exam for the course, but there was much more to it than that. The Dialogue students became very active and curious as time went on. Often they would not wait for the next question, but sensed the direction in which things were moving. They had long discussions about the central ideas, then, once these were over, would move very decisively in the applications of the ideas. By contrast, the lecture students settled down to the traditional passive note-taking and took very little initiative. In their evaluations of me as a teacher, I was rated quite highly. On the other hand, I wasn't rated so highly by the Dialogue students ("after all, we learned this ourselves"). The Dialogue students wanted to keep studying math "this way". They rated the course highly -- the instructor had become relatively invisible. In the spirit of Socrates, we had moved to a radical shift in the direction of the mathematics, and to a position of the student's sense of authority over the subject. With Aristotle, in the 80's, we took another step.

Aristotle. Greek thought's moving to a direct contact with objects and with experimentation is often attributed to Aristotle. Mathematics courses paralleled this dimension by including student projects based on actual applications. Students began to analyze the growth of crystals, to express in differential equations their measurements of the heating and cooling of various materials, and so forth. I remember one student who examined the diminishing of copper concentrations in Boston harbor as anti-pollution programs began to take hold. Not only did he put the information in a mathematical form not foreseen by the authorities, but he showed that the concentrations were not moving toward zero as anticipated (this phenomenon would then be a candidate for new projects). Students worked together to produce polished expositions, a much more lifelike exercise in view of the future requirements of their employment. Moreover, the use of computers and graphing calculators became integral to courses, providing the tools to deal with real data, and the power to explore and make conjectures in a way that would have been unthinkable twenty years ago.

Plato. As we approach the millennium, what could become the keystone of mathematics learning in the future is moving into place. It is the subtle process involved in the instructor assisting the student to be an AUTONOMOUS learner. And this requires an even greater renunciation of the teacher's traditional visible authoritative role than that involved in Socratic Dialogue. The key strategy of this instruction is student group problem-solving. Answers are not given and the students must justify their positions. Of course the curriculum is very carefully prepared by the instructor. Results are striking, both in student attitudes and student learning. I make the parallel with Plato's political thinking, in that every citizen of his Republic was to be given the opportunity to develop his/her potential to the utmost, to have the fullest access to education so that eventually the wise leaders of society could emerge from the entire population. This society would be no longer based on heredity, but on equal access. Education must lead to true autonomy and independent thought and authority over the subject matter --leading to the discerning citizen constantly developing and growing intellectually. In mathematics, the form that has great promise to promote this growth involves problem-solving with justification and exposition coming from the student.

Raphael's painting "The School of Athens" included all the Greek pioneers of thought -- even from different periods -- in one comprehensive scene. Similarly, contemporary mathematics instruction can be thought of as weaving the educational characteristics of these thinkers into a single seamless process.

By the way, the small size of the math classes at Northeastern constitutes a gem of an educational policy that makes possible student activity at the level described above. It is crucial that the University has not yielded to the practice -- widespread in other institutions -- of having large lecture sections in mathematics.

We want to move forward with a sense of balance. The traditional description of teaching as "the sharing of the fruits of contemplation with others" ("contemplata aliis tradere") lays out three fundamental facets, not one or two:

1. Original research (contemplata);
2. Strategies for teaching(tradere);
3. Sensitivity to the needs and aspirations of students (aliis).

In his inimitable way, Yogi Berra described baseball by saying that "Ninety percent of the game is fifty percent mental". In teaching, one hundred percent of what we do it one hundred percent mental. But it involves three distinct intelligences to correspond to the three facets of the definition of teaching:

• For research, pure mathematical intelligence (contemplata);
• For communication, a technical or performance intelligence to utilize and develop teaching strategies (tradere);
• For relating to students, a social intelligence (aliis).

Neglect of any one of these three is to court decline for our profession and ultimately for our discipline.

Finally, a word about the crucial area of university outreach. Concern for students (the "aliis" in the description of teaching) reaches not only to those currently in universities and colleges, but to potential students, especially minorities and new immigrants in public schools in the inner cities of the nation, who are being denied access to strong mathematics courses. Their plight has been compellingly delineated in such books as Jonathan Kozol's "Savage Inequalities". University departments have an obligation to reach out to cooperate with public school teachers and systems to bring about fundamental equity and to help develop strong programs,and the MAA's Board of Governors has recently underlined this mandate. New immigrants and low-income and minority students sitting today in public school classrooms in the cities are the future mathematicians and teachers and inventors and engineers. They are the potential leaders of their communities. Without the ethical core in our professional life which leads to this kind of outreach, we risk entering the new millennium in a rudderless and disconnected state which is a danger to the work of our profession and its purposes. We risk a rootless society which can become, in the words of Octavio Paz, "a wilderness of mirrors".

On the other hand, there is the potential for momentous change, change that depends on the comprehensive definition of teaching, as we work both with our students and in outreach to offer to ever larger numbers, especially those who until recently have been denied equal access, a place where they can become true discoverers and creators.