Dr. Alf Coles - In Pursuit of a Science of Education
Gattegno calls on teachers to become scientists of education in their classrooms by using the tool of
watchfulness. A problem, as Gattegno was aware, is that watching and seeing is more complex than it may appear. There is a problem with perception – which arises because seeing (or hearing, or any other sense) is an act of categorizing.
A parish magazine is delivered to my house every month. One month I noticed, as part of an article, our parish council had a logo, which is an arrangement of three standing stones. I reflected at the time that for this to be the parish council logo, these standing stones must be near where I live, yet I had never seen nor heard of them. The next day I was driving down a road outside our village that I must have gone down and back on several hundred times. Whilst traveling close to 60 mph, without conscious deliberation, I turned my head fully 90 degrees to the right and glimpsed for a fraction of a second, at a break in the wall by the road, the configuration of standing stones.
I do not believe this story is one of chance. I needed to see the stones in the magazine to recognize them in the many fleeting glimpses I must have unknowingly had along the road. Because I see in categories, I only see what I am familiar with – the problem then arises – how am I ever able to see anything ‘new’? If, to see something, I need to connect it to something I already know, how can I learn? What can I do to see the strange in the familiar? Foundation art-degree courses run lectures on ‘learning to see’ in a way that is perhaps not so widespread in initial teacher training.
Yet arguably it is as necessary; to become a teacher, and certainly to become a scientist of education, I need to use myself in a way that allows me to expand my categories of perception. Laurinda Brown puts it simply as, “seeing more, seeing differently,” or in Gattegno’s language making myself “vulnerable” to the new.
One technique for opening ourselves to the new is to stay with the detail of our observations – and this is what, in part, I read into Gattegno’s call for ‘watchfulness.’ John Mason distinguishes ‘accounts of ’ from ‘accounts for’ phenomena – the former containing the detail, the latter the interpretation. Of course all description no matter how detailed is interpretation, since all categories impose a splitting up of the world that could have been done differently. Yet there is a difference between saying (of the same imagined event): “I asked a question and no one put up their hand to answer,” and “The class seemed
Gattegno calls on teachers to become scientists of education in their classrooms by using the tool of watchfulness.
disengaged today.” If we begin with ‘accounts of,’ without judgment, we have the chance of noticing something we had not observed before – particularly if we are able to keep records over time and look back at similarities and differences. Someone else might see ‘thoughtful’ where I see ‘disengaged’ – holding off interpretation and judgment means alternative descriptions become possible. And what of the students – how can we work with them to ‘see more’ and ‘see differently’ in relation to mathematics? The same distinction between accounts ‘of ’ and ‘for’ is equally valid. Offer students something visible or tangible, and everyone can say what they see (an ‘account of ’). If the context is rich enough, then mathematics (moving to an agreed ‘account for’) can follow from what students notice.
To start with, then, I need a context in which students have something to say.
My two-year-old daughter was given a balance bicycle for Christmas – this is basically a bicycle with no pedals that you push and then freewheel. In a matter of weeks she was confidently freewheeling downhill and leaning in to go around corners. As I know from seeing her elder brother it will be a short while before she can use these awarenesses to ride a pedal bicycle. In contrast, children given pedal bicycles with stabilizers learn to lean out as they go around corners. This must be unlearnt, and the transition to riding without stabilizers can be hard.
I wonder how much of what we offer students is the equivalent of a stabilizer – fractions as pieces of cake; ‘a’ stands for ‘apple’; the separate treatment of addition and subtraction, or of any two inverse processes. And, more importantly, what are the balance-bike equivalents we can give students? Dick Tahta, who collaborated over many years with Gattegno, would perennially offer the challenge for someone to come up with the minimal set of images needed to cover the whole mathematics curriculum (up to, say, aged 16) – in Dick’s words, what are the canonical images of mathematics?
Gattegno was certain there could be no images for algebra, since algebra is an awareness of dynamics; which I find easier to think about as an awareness of process, or an awareness of similarity and difference.
1 -> 2
3 -> 4
10 -> 11
20 -> ??
This function game puzzle will be straightforward to most school students. It requires a stepping back from the simple giving of the answer, however, to recognize structure, and the operation being performed, to become aware of the dynamic or process – i.e., the shift from ‘it’s one more’ to ‘I am adding one to the number’ – and hence to providing the algebraic notation: n -> n+1.
As a teacher, a favorite activity was working with students on creating graphs of this and other more complex rules. A student might choose to draw graphs of these rules: n-> n+1; n-> 2*n+1; n->3*n+1, and notice they cross the y-axis at the same point. Awareness of this similarity can lead a student to conjecture about graphs of the form: n -> m*n+1, and further practice to test this idea. If algebra is awareness of dynamics, it must be around whenever we are doing mathematics – if we can hear it.
By starting with images (I would include a function game as an image – indeed a canonical one), and letting the algebra arise from the dynamics of such work, worries about ‘understanding’ can disappear. Consider learning about percentages using the Gattegno ‘tens’ chart below (surely another canonical image).
1000 2000 3000 4000 5000 6000 7000 8000 9000
100 200 300 400 500 600 700 800 900
10 20 30 40 50 60 70 80 90
1 2 3 4 5 6 7 8 9
I begin: “Ten percent of 1,000 is 100. What is 10% of 3,000?” [touching it with a stick]. You call out “300.” Ten percent of 600? 60. Ten percent of 320? 32, etc. Finding 10% of a number may initially become associated with ‘moving down one row.’ (And since inverse functions can always be treated together, moving from 10% to 100%, associated with moving up one row.) The action can be repeated, to give the relations between 1% and 100% as a move up or down two rows. I recognize the ‘me’ that began teaching in 1994 might have been worried that, in this approach, students do not really ‘understand’ what they are doing. Yet what else, really, is there to get? It is true there is no linking ‘metaphor’ (e.g., that ‘percent’ means ‘out of a hundred,’ or that percentages can be connected to shading squares on a 10 by 10 grid). The treatment is ‘metonymic’ - i.e., not about an alternative meaning, but a substitution within the same context. Instead of a metaphoric shift to some (usually unconnected) other domain, the substitution involved here means that finding 10% gets linked to an action students perform (moving down a row) and can see. In a reverse of my thinking when I began teaching, I now see the kind of ‘teaching metaphors’ that I used to offer students as ‘stabilizers.’ I would explain negative numbers as cold bricks, positives as hot bricks, altering the temperature of my pot, so taking out a cold brick, or subtracting a negative – abracadabra – makes it warmer. Such translations may support learners over an initial hurdle, but quickly become barriers – how do you multiply by a cold brick! Gattegno’s metonymic approach focuses attention on how to perform key operations (e.g., finding a third of a number by finding three identical Cuisenaire Rods that fit along it; finding the Sine of an angle by finding the perpendicular distance from the x-axis to a point on the circumference of a unit circle). Such operations are actions, real or virtual, that need to be practiced. However, by avoiding metaphor, the focus easily shifts to the transformations themselves, and hence to the inherent algebra, like finding a third of a third, or finding angles with the same Sine length. Gattegno provides the context, the operations (as actions), the notation, and often a challenge – he leaves to students the task of making sense of it all.
Dr. Alf Coles