A Dusty Discipline

Michael J. Barany and Donald MacKenzie

How does one see a mathematical idea? Can it be heard, touched, or smelled? If you spend enough time around mathematicians in the heat of research, you tend to believe more and more that when it comes to mathematics, the materials matter. To many, mathematical ideas look and sound and feel and smell a lot like a stick of chalk slapping and then gliding along a blackboard, kicking up plumes of dust as it traces formulas, diagrams, and other mathematical tokens.

Chalk and blackboards first made their mark in higher education at elite military schools, such as the École Polytechnique in France and West Point in the United States, at the start of the nineteenth century. Decades of war and geopolitical turmoil, combined with sweeping changes to the scale and social organization of governments, put a new premium on training large corps of elite civil and military engineers. Mathematics was their essential tool, and would also become a gateway subject for efficiently sorting the best and brightest. Blackboards offered instructors a way of working quickly and visibly in front of the large groups of students who would now need to know mathematics to a greater degree than ever before. They also furnished settings of discipline, both literal and figurative, allowing those instructors to examine and correct the work of many students at once or in succession as they solved problems at the board.

In the two intervening centuries, the importance of chalk and blackboards for advanced mathematics grew and grew. Blackboards reigned as the dominant medium of teaching and lecturing for most of the twentieth century, and continue to be an iconic presence in countless settings where mathematics is learned, challenged, and developed anew. As, indeed, it routinely is. While schoolbook mathematics seems as though it has been settled since time immemorial (it has not been, but that is another story), the mathematics taking place in universities and research institutes is changing at a faster rate than ever before. New theorems and results emerge across the world at such a dizzying pace that even the brightest mathematicians sometimes struggle to keep up with breakthroughs in their own and nearby fields of study. Long gone are the days when a single mathematician could even pretend to have a command of the latest ideas of the entire discipline.

The problem of keeping up might seem to lead one toward high technology, but to a surprising extent it leads back to the blackboard. When Barany followed the day-to-day activities of a group of university mathematicians, he found the blackboard was most prominent in their weekly seminar, when they gather after lunch to hear a local or invited colleague's hour-long presentation on the fruits and conundrums of recent and ongoing work. But blackboards are also present in offices, and even the departmental tea room. Their most frequent use came when mathematicians return to the seminar or other rooms to teach mathematics to students, just as their predecessors did 200 years ago. Wherever they are, blackboards serve as stages for learning, sharing, and discussing mathematics.

Blackboards are still more pervasive when one searches for them in unexpected places. The archetypal blackboard is a large rectangular slab of dark gray slate mounted on a wall, but over their history most blackboards have been made of other materials, some of which are not even black. (This includes the dark green boards in the seminar room of Barany's subjects.) Characteristics of blackboard writing can be found in the pen-and-paper notes researchers scribble for themselves or jot for colleagues. Gestures and ways of referring to ideas in front of a board translate readily to other locations. The blackboard is one part writing surface and two parts state of mind. To understand the blackboard, we realized, is to understand far more about mathematics than just seminars, lectures, and the occasional chalk marking in an office.

Even as blank slates, blackboards are laden with meaning. Because they are large and mostly immobile, they greatly affect how other features of offices or seminar rooms can be arranged. Entering a seminar room, one knows where to sit and look even when the speaker has not yet arrived, and the same principle holds for the different kinds of situations present in offices. We noted that when one arrangement of desks and chairs did not quite work it was the desks and chairs, rather than the blackboard, that were rearranged. Staring blankly at its potential users, a blackboard promises a space for writing and discussion. Depending on the context, too much writing on the board prompted users to use an eraser long before anyone intended to use...