I move up my post of 25 August 2006 from now defunct old incarnation of this blog. I am prompted by Phil Beadle’s paper on blackboards, You can’t dance in front of an interactive button, in The Guardian, 28 October 2008. I offer to my readers a wonderful quote from Beal:

If I am modelling sentence construction or the semicolon, drawing a map illustrating colonialism in Africa, or scribing arrows outlining connections between ideas, I want to be able to do it quickly: as quick as I think; as quick as I talk. I want to be able to teach with my whole body, use gesture, employ pause to illustrate nuance, become as one with the board; become, in those rare moments of flow, both dancer and dance. Now the board dictates that, rather than pirouette, twist and enthuse, I click a frigid button.

This picture is a shot from a wonderful film by Samira Makhmalbaf which won Grand Prix du Jury at the Cannes Film Festival 2000. The film is a tale of a band of teachers, who, with blackboards on their backs, wander in search of students from village to village in remote mountains of Iranian Kurdistan. See a larger picture; you may notice that the blackboard of the leftmost teacher is covered with mathematical or physical formulae.

It is relatively easy to explain why a projector-only lecturing is unsuitable for mathematics. A mathematics teacher is not just conveying information, he or she teaches to think mathematically, and teaches by example, in real time. It is crucially important to be in full control of timing and tempo of the narrative. If a lecture involves calculations (and they are inevitable in most mathematical disciplines), it is crucially important to let students feel the subtle play of rhythms, emphasize switches and branch points in the procedure, highlight recursion and reduction to simpler cases.

Next, one should not forget gesture. One of my colleagues (an exuberant but highly popular lecturer) loves to say that “one is doing mathematics with one’s entire body”. Notice that mathematics is full of motion and action metaphors applied even in the situations where there is no motion or action:

we say that an asymptote of the hyperbola xy=1 approaches the x-axis;

that a variable t runs through the set of real numbers;

that the sequence {1/n : n = 1,2,3,…} converges to zero as n tends to infinity,

and the list of examples can be easily expanded.

Recent findings of cognitive psychologists point to an important role of gesture in subconscious processing of mathematics. In their book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Rafael Nunez and George Lakoff put the gesture/motion metaphor at the center of cognitive understanding of mathematics. Nunez’s paper (R. Nunez, Do real numbers really move? Language, thought and gesture: the embodied cognitive foundations of mathematics. In: 18 Unconventional Essays on the Nature of Mathematics (R. Hersh, ed.). Springer, 2005, pp. 160-181) roots mathematical concepts into unconscious mechanisms linking speech and gesture:

the internal mental image of the concept of “convergence” is the shared component of the meaning of the spoken phrase which describes convergence, and a semi-conscious or unconscious gesture which accompanies the phrase.

At a first glance, the idea appears to be far-fetched, until one checks the list of experimental evidence showing intimate relation of speech and gesture. I quote Nunez’s paper, where the list is supported by detailed bibliographic references.

Speech accompanying gesture is universal. This phenomenon is manifested in all cultures around the world.

Gestures as less monitored than speech, and they are, to a great extent, unconscious. Speakers are often unaware that they are gesturing at all.

Gestures show an astonishing synchronicity with speech. They are manifested in a millisecond-precise synchronicity, in patterns which are specific to a given language.

Gestures can be produced without presence of interlocutors. Studies of people gesturing while talking on the telephone, or in monologues, and studies of conversations among congenitally blind subjects have shown that there is no need of visible interlocutors for people to gesture.

Gestures are co-processed with speech. Studies show that stutterers stutter in gesture too, and that impending hand gestures interrupts speech production. (I have once witnessed how a skilled speech therapist made a severely stuttering child to speak flawlessly from the first seconds of the very first speech therapy session: she held the child’s hand to feel his spasms, and spoke to him, leaving prompts and pauses for his responses at precision timed moments when the hand was relaxed. For observers – including the child’s mother – this looked like a miracle.)

Hand signs are affected by the same neurological damage as speech.

Gesture and speech develop closely linked. Studies in language acquisition and and child development show that speech and gesture develop in parallel.

Gesture provides complimentary content to speech content. Studies show that speakers synthesize and subsequently cannot distinguish information taken from the two channels.

Gestures are co-produced with abstract metaphorical thinking. Linguistic metaphorical mappings are paralleled systematically in gesture.

At risk of committing a mortal sin of using introspection as a source of empirical evidence, I have to say that I am very sympathetic to Nunez’s ideas: they appear to reflect my everyday experiences in mathematics and my observations of other people thinking and talking about mathematics. (I do not agree with everything that Nunez proposes in respect of mathematics teaching, but I will perhaps return to that on another occasion.)

We can now return to our blackboards. Well, psychophysiology of gesture explains why, in teaching mathematics, a blackboard is better than a Powerpoint presentation, but why are blackboards preferable to whiteboards?

The following observation belongs to Israel Gelfand. Using chalk on a blackboard, we write by moving the entire arm. With felt markers on a whiteboard, we use smaller scale movements of fingers and the wrist. At a purely instinctive, physiological level, people tend to hold breath when they do small movements with their fingers. On the contrary, wider movements of arms fit naturally in the cycle of breathing and speaking. In mathematics teaching, we have to write on the blackboard and speak more or less simultaneously (although I take care to never speak “to the blackboard” and turn to face the audience every time I utter a word – but I do keep chalk pressed against the board, thus ensuring that I continue writing exactly from the spot where I stopped). In a lecture, the lecturer has to speak clearly, loudly and project his/her voice onto the audience, filling the whole room with the sound. Ever tried to do that while writing with a thin felt pen on a transparent roll of film (yet another horrible contraption found in many lecture rooms)?

For the first installment, I said enough in defence of blackboards — but I invite comments from my colleagues, I would really like to enmasse our collective wisdom on the subject.

I hope that my university colleagues who work in education studies forgive me if I bluntly state my credo:

Mathematics teaching is not a science. It is an art.

Moreover, it is a performance art, like drama or ballet, and should be treated as such. Unsurprisingly, ballet dancers are very fussy about the state of the stage floor: they need a surface with just right level of friction, support and spring. Normally, we are very fussy about the quality of the blackboard surface -I can tell a lot on that subject. Unfortunately, we are reduced to fighting for continuing existence of our old blackboards; we do not even dare to raise the issue of their quality.

Ddisclaimer: my posts do not necessarily constitute or represent the views of my University.