require vast numbers of mathematically trained professionals, and there was sufficient mathematical talent and motivation to survive any pedagogy, this filtering system was considered benign. Many even considered it desirable.

The emergence of a highly competitive and technological world economy has fundamentally enlarged the demands on mathematics education. We now seek, for the broad work force, levels of scientific and technical competence and literacy that approach what was formerly deemed appropriate for only a select and specialized student population. These same changes make increased demands of technical literacy for responsible and informed participation in our modern democratic society. These pressures give an added practical edge to the traditional argument for the cultural enrichment and intellectual empowerment that mathematical ideas and thinking can confer. When large numbers of students fail and/or leave mathematical study, which is the gateway to such competence and literacy, this is judged now to be the failure not of the students but rather of the educational system. Moreover, the students lost are disproportionately from the minority and female populations that constitute the major influx into the work force.

The time has come for mathematical scientists to reconsider their role as educators. We constitute a profession that prides itself on professionalism, on an ethos of quality performance and rigorous accountability. Yet academic mathematical scientists, who typically spend at least half of their professional lives teaching, receive virtually no professional preparation or development as educators, apart from the role models of their mentors. Imagine learning to sing arias simply by attending operas; learning to cook by eating; learning to write by reading. Much of the art of teaching—the thinking, the dynamic observations, and the judgments of an accomplished teacher—is invisible to the outside observer. And, in any case, most academic mathematical scientists rarely have occasion to observe really good undergraduate teaching.

While one does not learn good cooking by eating, neither does one learn it just by reading cook books or listening to lectures. Cooking is best learned by cooking, with the mentorship of an accomplished cook, that is, by an apprenticeship model. In fact, teacher education also is designed with a mixture of didactic and apprenticeship instruction. Professional development of academic mathematical scientists as teachers should perhaps be similarly modeled on learning in the context of practice, with only relatively small doses of more formalized styles of learning with which we are most familiar. Good designs for doing this in a systematic way are not now common. Education professionals can help us in creating and experimenting with such designs.

Effective teaching requires that teachers know their students, to be able not only to explain things to them, but also to listen to them, closely, and with understanding. And knowing something for oneself, or for communication to an expert colleague, is not the same as knowing it for explanation to a student. Further, the experience of a mathematical scientist as a learner may not be the best model for the learning of his/her student. These are the kinds of skills and awarenesses that professional development can help cultivate.

Of course there have always been in our professional ranks some very effective, even inspiring, teachers. They have become so through a combination of talent, personal commitment, hard work, and practice—and without recourse to professional educators. But do these isolated individuals constitute a model for the educational responsibility of our profession? Are we—and the public we serve—to be content with the condition that some few among our ranks have chosen to take the individual initiative to develop their teaching skills? Imagine, by contrast, abandoning our disciplined education in rigorous mathematics for future researchers to a laissez-



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