What Can We Do To Improve Maths Education?

This depends on who we are.

In our compartmentalized system, it is hard for a single organization or individual to do very much to affect the overall system directly. But addressing the local situation will indirectly influence the global situation. I will address the question from the point of view of college and university mathematicians.

First, college and university mathematics departments should develop courses which can give students a fresh chance in mathematics. Remedial courses are widespread, but their success is limited: going over the same material one more time is tedious and boring, whether you understood it the first time or you didn’t. There is a built in handicap to enthusiasm and spontaneity. Instead, there should be more courses available to exploit some of the breadth of mathematics, to permit starting near the ground level without a lot of repetition of topics that students have already heard. For instance, elementary courses in topology, number theory, symmetry and group theory, probability, finite mathematics, algebraic geometry, dynamical systems (chaos), computer graphics and linear algebra, projective geometry and perspective drawing, hyperbolic geometry, and mathematical logic can meet this criterion.

Second, we should work to create better channels of communication between the compartments of the educational system. We need to find devices so that the educational accomplishments of professors are visible within the profession, not merely within the classroom or within the department. We need to find vehicles for exchange of interesting ideas between different departments. We should visit each other’s classrooms. We need more talks and special sessions related to education at our professional meetings, and more prizes for educational accomplishments.

Even more important and more difficult is the creation of channels for communication between the strata: most important for colleges and universities is communication between high school, college and university mathematics departments. This communication must be two-way: college and university professors can learn a lot about how to teach from school teachers. The state mathematics coalitions that they have helped to stimulate, but what we need is a much more massive exchange. How can the senior professors, who are at the top of a system which is clearly not doing such a great job, presume to teach their juniors how to do better? The graduate students and the junior faculty often do a better job at teaching mathematics than the senior faculty, who have sometimes become resigned to the dismal situation, settled into a routine, and given up on trying any new initiatives. Even when they do a pretty good job in their own classrooms, against the odds, they do not usually get involved in improving the overall system.

Often other professors are suspicious of the professor who does take an interest in education. They tend to assume that research is the only activity which really matters and that turning to education is a sign of failure in research. Senior professors sometimes explicitly advise junior faculty not to waste too much energy on teaching, or they will never be promoted.

We must recognize that there are many different ways that we can make important contributions to society and to our institutions. It is dumb to measure mathematicians against the single scale of research. Education is an important and challenging endeavor, which many people engage in by choice, not necessity. We should judge them by what they accomplish, not by what they might have accomplished if they spent their time and energy elsewhere. What urgently needs to change is the system of professional rewards. We need something better than the current situation within university mathematics departments where there is lip service to the importance of teaching, but, when it comes to the crunch of hiring and tenure decisions, teaching and service count only in the marginal cases where the candidates cannot be differentiated by the quality of research.

People are socially motivated. As we discuss education with each other, we put more energy into it, and it becomes more important to us. The academic culture can change, and it has changed. The process of change is mostly an informal one (what you talk about at lunch), not controlled by organizational decisions. But when the time is ripe, as I believe it is now for mathematical education, a little nudging by organizations can help stimulate a huge change. The needed reforms will take place through collegial, cooperative efforts. Good mathematical ideas spread very rapidly through informal channels in the mathematical community. As we turn more of our attention to education, good educational ideas will also spread rapidly.