Posted by: Alexandre Borovik | September 26, 2010

## More on Blackboards

My regular commentator Peter left the following comment on my post Psychophysiology of blackboard teaching:

There are many disciplines where some crucial knowledge is communicated to new learners ONLY in the physical demonstration by a lecturer of some argument. Perhaps this is because it it not possible to communicate the knowledge in any other manner. In Economic game theory, for example, it is very hard to find an adequate written treatment of extensive form representations of games (ie, games as tree diagrams), because economists, to a person it seems, acquire their understanding of the semantics of these diagrams from one another in personal, face-to-face, contact, and nobody has bothered (or is able) to write this knowledge up in a textbook. The situation is similar with the use of so-called comparative statics in Economic Theory – the use of graphs in the XY-plane to compare one state of a complex adaptive system with another, later state or states. I don’t know of any adequate written treatment of this type of reasoning which could explain to a complete novice what is going on, without someone who is already knowledgeable drawing the diagrams in real-time in front of the beginner.

The same is true, I suspect, with commutative diagrams in category theory. Is there, anywhere, an adequate written explanation of the reasoning involved in chasing arrows around a commutative diagram which would explain the reasoning without having to resort to a person-to-person real-time drawing?

The same applies to all kinds of diagrammatic thinking. Once upon a time three my postgraduate students worked on three completely different problems: groups of finite Morley rank, symplectic matroids and probabilistic recognition of finite simple groups. I used to talk to each of them referring to the same picture that was sitting for a month or two on the blackboard in my office. It represented the root system of type $B_3$ – an eternal object. Could the same be done without a blackboard (or at least a whiteboard)? I doubt it.

Root system B_3. Picture by Maria Borovik

The picture inserted here was drawn by my daughter Maria. A couple of weeks ago she brought me as a gift a cofee mug with a chalk writable surface:

A coffee mug with chalk writable surface

The quality of the writing surface is fantastic — I can only dream of having a blackboard of the same texture.