Posted by: Alexandre Borovik | September 24, 2008

Psychophysiology of Blackboard Teaching

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 BeingRafael 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.

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Responses

  1. Automated “possibly related posts” function of WordPress instantly produced a link to

    + Gestures Enhance Learning!, in http://dyslexia.wordpress.com/2007/08/24/gestures-enhance-learning/, an article which originally appeared in The Washington Post, 6 Aug 2007,
    http://www.washingtonpost.com/wp-dyn/content/article/2007/08/05/AR2007080501179.html

    Quote:

    “Susan Wagner Cook uses hand gestures to teach math to third grade students. The students look at an unfinished equation printed on the whiteboard:

    4 + 3 + 6 = ___ + 6

    Dr. Cook says, “I want to make one side [sweeping her left hand under the left side of the equation] equal to the other side [as she sweeps her right hand under the right side of the equation].” [...]

    “Teachers who use gestures as they explain a concept — such as the hand sweeps that Cook uses to emphasize an equation’s symmetry — are more successful at getting their ideas across, research has shown. And students who spontaneously gesture as they work through new ideas tend to remember them longer than those who do not move their hands.”

  2. It seems to me that you are creating a false dichotomy. It is not blackboard vs powerpoint but blackboard vs something else.

    I would agree that a passive (pre-composed) powerpoint may not give what you want. However consider some of these possibilities:

    use powerpoint with a “smartboard” then you can present some stuff (perhaps a real life example of the use of the bit of maths as an image or video?) followed by writing it on the smart board into a previously blank slide. This has the advantage that when the student has copied it down wrong, lost their notes or whatever they can see the whole thing as it was originally presented (you could even record what you are saying as a voice over as you go along – though comments, questions, answers etc from the floor may be inaudable).

    Or, how about doing the same thing using a tablet PC then you can write as you would on paper and it appears on the screen, with the advantage that you are facing the students all the time instead of facing the board half the time. Again, what you have written is recorded for later study.

    Of course, with a blackboard the students can use their mobile phones to photograph it if they want, but the image might be not very good.

    I am not saying that these would necesarily work for you, but for some (and especially those who dont like to put their back to the audience or who have messy handwriting on a blackboard) these might be a great help.

    It also means that students can concentrate on what you are saying and writing without having to take everything you write down if they want access to it later.

  3. Dear Alexandre,

    Here’s my small contribution to the collective wisdom of mathematicians on this subject (since you asked for it). I agree with much of what you’re saying, but I completely disagree regarding whiteboards vs blackboards. Personally I much prefer whiteboards. The disadvantages of blackboards and chalk are well explained in the post by Phil Beadly. And I don’t see why it should be that hard to speak loudly while using a whiteboard; especially compared to trying to speak at all when surrounded by a cloud of chalk dust that gets in your throat…

    But I guess it’s a matter of personality too. One of my calculus students had great difficulties with the whiteboard. He would try to write something and then complain that the marker was out of ink, but when I tried it, it worked perfectly. It turned out that his way of writing was too “violent”; he moved the marker too fast for the ink to have time to stick to the board. For me with blackboards it’s the other way around (my writing is hardly legible since I can’t push hard enough with the chalk).

    PS. Off-topic remark: the third blackboard from the left in the photo says “Daddy has a pomegranate. I don’t have a pomegranate.” :)

  4. Part of the problem, well understood now in the Communications Theory community, is that use of MS Powerpoint promotes linear thinking. For people who think in straight lines, from A to B, Powerpoint is fine. For those, in contrast, who think creatively or laterally, or wish to demonstrate creative or innovative thinking, it is a disaster.

    It is certainly ironic that in the same decade that the western world embraced hyperlink technology (in the WWW), we also adopted the non-linking, ultra-linear, anti-creative technology that is MS Powerpoint.

  5. Microsoft is a ghastly monopoly that shoves their ghastly software down everybody’s throat. It has done a tremendous damage to the intellectual health of mankind. Take Excel, for example, it makes you think like an accountant.

  6. I teach at my smartboard pretty much all the time, but for the at least half of that time I use it just as a whiteboard, one on which it’s a little harder to write fast and neatly but on which one can leaf back and forwards through multiple pages of today’s work – or yesterday’s, or last year’s. For almost all of the rest of the time, I use it with pre-printed “notes with gaps” – my page on the board matches the page the students have, and we fill them in together. Just because it can run powerpoint doesn’t mean that it must.

    I suspect that if I were still teaching classes of 100+ in large theatres, though, I would still want a blackboard. In the smaller rooms, whiteboards are superior.

  7. I have to say I completely agree with this; as an older student learning math, I found that I prefer professors who write on blackboards. I’m able to follow the flow of logic, and am more apt at understanding what is being written in real time. And the gestures do help when the professors are making a point, and as they are particular to each professor, I find them charming (i.e., point understood). I find that in my classes with slide presentations, I understand less of what is being taught – it is too passive. And those laser pointers! They are the bane of my learning career. All I do is follow the jittery light of whatever is being circled, not the actual text itself.

  8. We run an interdisciplinary institute. For what it’s worth here is our experience:

    1. Powerpoint is only good for presentations (‘shows’ – not serious lectures) and forces a linear argument as indicated above. What about ‘sidebar’ proofs on multiple boards? Or long proofs on ‘continuous’ blackboards. As an undergraduate, we could have 3 continuous blackboards running at any time – with different information on each and the lecturer moving easily through the information in 3×2 directions. So PPT is totally useless for the development of any serious mathematical derivation – in any mathematically intensive subject. Of course PPT has other strengths – but is hopelessly weak in this particular area.
    2. We have tried smartboards for workshops (in all fields). No-one uses them – they seem to prefer whiteboards or flipcharts (blackboards are not available yet – we are just putting them in) and then take photos with their mobile phone. Smartboards (or tablets) have poor resolution, time lag, poor tip/mark alignment, so don’t allow in-between filling, annotating, noting up etc easily. It may be a good upcoming technology but IMO is ‘not quite there’ yet.
    3. Whiteboards in public spaces – the pens are always dry, there is no cleaner, the board doesn’t clean well so e.g. if you are cancelling out or correcting errors, they always leave a ghost.
    4. Filling in. Has anyone tested whether students ‘writing things down’ helps them remember the material? Maybe it’s a generational thing, but I find that when studying anything (or listening to a seminar), the physical process of writing helps me remember what I am reading and improves my concentration – even if I never look at my notes again. In a time when students are becoming increasingly mathematically incompetant, maybe the distribution of preprinted notes and other staff-generated material needs to be rethought (heresy!) and the students made more dependent on self-generated material which requires some engagement from their ‘wet ware’.

  9. [...] without speaking. The advantages & disadvantages can be discussed at some length — read this blog post of Alexandre Borovik for a strong defence of the medium and the accompanying habits of presentation [...]


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