This post contains fragments of my mathsconf7 session ‘A golden age’, about maths education writing from 1960 - 2000.
The number of 'matches' is 60, but how did you get to that answer? As Dave Hewitt describes:
There is a difference between counting and watching yourself counting
It would appear that not everyone counts as you might expect.
Sophie's first words
Sophie, my daughter, is 21 months old. I describe a brief account of her development. One of her first words was ‘more’, Walkerdine’s more, as opposed to no more, and not as opposed to less. Then ‘two’, in the sense of more than one of the same. She is bi-lingual, so ‘two’ sometimes becomes ‘dos’. Then random sequences, followed by truncated sequences: ‘cinqo, seis, siete, ocho, nueve…’, but often ‘five, six, seven, eight, ten’ in English. Where is 9?
We read this, from Caleb Gattegno:
You all know that it takes a little time for little children to say: one, two, three, four, five, ... They say: one, seven, three, two, four. Don't they sometimes say this? And after a while they say: one, two , three, ... And you are very, very happy, that they say one, two, three, like you. But you didn't ask yourselves: why do they say one, seven, three, two? Why?
It happens so often; so many people know that their own children do that. But they don't stop to ask the question. Once the children get it right - which means they do what everybody does - then we are satisfied. It doesn't strike anyone that they do it, that they have spent some time sorting things out, by themselves. Doesn't it make you wonder, how do they work, these children? How do they use their minds in order to get what we are going to build our arithmetic on
I have decided to remove the tables, and moved the chairs close together so we can hear each other read, and can talk more easily.
There are questions: Are we too quick to correct? How would children, our students, learn differently if we responded differently? What are children doing when ‘counting’ in this way? Do they ascend from the abstract to the concrete, or perhaps move between them?
Two of the audience are familiar with the work of Gattegno. A lady (I regret I did not ask her name, or take her contact details) recalls that her father knew Gattegno, that she learned to read through colour charts, and mathematics through cuisinaire rods. We talked at the end of the session, she is going to go home and read Gattegno rather than do her year 10 marking. Sometimes we need to transcend the every day.
Teacher as researcher
Another question from Gattegno:
Which exercises do children give themselves to be able to grasp objects so that they can hold them?
Here are some of the exercises Sophie gives herself:
- Walking, running, chasing, hide and seek, …
- Touching - play, repetition, grasping, trial and error, cuddling, putting together, pulling apart, pushing through, …
- ‘Using’ sound: naming, joining words together, animal noises, commands, noticing sounds, …
- Asking me to: make (play-doh), read, …
- Putting stuffed animals to bed - different sizes for different beds, …
- Spontaneous counting (stairs, ants, aeroplanes, woodlice..)
What might we learn by watching our children, or those we teach, more carefully? Gattegno implores that we (as teachers) must research our own practice:
One of the instruments, that you all have, is to ask questions… It's a discipline you have to give yourself - a spiritual discipline. This is part of the education of teachers... We have to study learning as a matter of course and all the time: not only in books, but where it takes place: in front of us, in the classroom…
Teaching as a spiritual discipline....
Awareness of awareness
Before we start the next reading, I ask: Where is our attention when reading? When teaching? What are our students attending to? Can we become aware of our awareness, and in so doing, the awarenesses of others?
Dick Tahta describes how we might come into contact with our own awareness:
As you read these words, let yourself become conscious of your reading. Do your eyes flicker? Do you take in chunks at a time? What images do you invoke? Are you with the reading? Are you now with the self that was with the reading?
It is helpful to be quite pedestrian about this, to try to avoid thinking of awareness as some special form of insight or ‘raised consciousness’. Try at first to note [simply] what happens...
With this in mind, we read this from John Mason:
To abstract means to draw away.. It is easy to sympathise with the student’s sense of abstract as removed from reality... But perhaps this sense of being out of contact arises because there has been little or no participation in the process of abstraction, in the movement of drawing away. Perhaps all the students are aware of is the having been drawn away, rather than the drawing itself.
Articulation of a seeing of generality, first in words and pictures, and then in increasingly tight and economically succinct expressions, using symbols and perhaps diagrams, is a pinnacle of achievement, often only achieved after a great struggle.
The expression acts as a signal to recall salient associations. Just as there is a huge difference between drawing your own figure in order to stabilise your mental imagery and to extend your thinking power, so there is a huge difference between expressing your own generality and doing someone else’s algebra.
The groups discuss what resonates for them, while I resist the urge to intervene.
Who was present with the reading? Some found it hard; it is easier for some to be present when listening. I describe how I have been working at listening, about being present-to others, instead of thinking about what I want to say or interrupting before others have finished speaking.
We talk about an awareness of ourselves when teaching. It seems that a significant number of us have had ‘out-of-body’ experiences whilst teaching. I am reminded of Mason’s two birds:
I read some common themes I have discovered in the literature that may be relevant for our work as teachers:
[It is] the process of making objective to oneself alone what one was allowing into one’s awareness and then, in a secondary process, making it objective [explicit] for others... We must offer ourselves new exercises to continue our growth… by being present in one’s exercises. (Caleb Gattegno)
It arises spontaneously of itself… Noticing is not something we do, it is something that happens to us. Noticing not only brings us into the world, but brings the world also into being. (JG Bennett)
The point is that [an action] is not necessarily good or bad in itself, but rather whether I am choosing to do it, or whether it chooses me… Finding yourself doing something is easy; catching yourself about to do something and choosing to act differently in a more informed manner, is much harder. (John Mason)
(The process of) abstraction
There is a huge difference between expressing your own generality and doing someone else’s algebra. In abstraction there is an element of change in stressing ignoring, an alteration of foreground and background. (John Mason)
Just what is meant by this word attention? Is it something which I control? I have struggled with this question on and off for some time, and have been unable to answer it. (John Mason)
Teaching is in substantial part about directing attention. (David Pimm)
A teacher cannot directly manipulate a student’s awareness nor give a student awareness. It is the student’s work to educate their own awareness. A teacher can, however, use techniques to help the student carry out this work, to direct attention. (Dave Hewitt)
By being explicit, by focussing attention on the brief but important moments of abstraction movement, more students could be helped to appreciate mathematics… (John Mason)
Working with children’s ‘human’ powers
Children spontaneously stay with problems. And they stay for as long as is required. Children spontaneously transform... they live close to their powers of transformation and to their mental dynamics. (Caleb Gattegno)
The question is not whether one should use a mix of [teaching] methods (of that I have no doubt), but precisely how that blend should be achieved. Perhaps the chief demand on the teacher is to learn to live consciously with the creative tension which exists between exploration and instruction - in deciding, for example, whether to tell or not to tell about a particular matter. (Alan Wigley)
[There are] many who would be far better served by a ruthlessly economic set of principles and minimal list of required awarenesses. Something more economical, yet more fertile; something that is based on repeatable experiment and presented without imperative exhortation. (Dick Tahta)
Caring is a way of being in relation: in its most basic form, a connection or encounter between two human beings. The state of consciousness of the carer is characterised by engrossment and motivational displacement. By engrossment I mean an open, non-selective receptivity to the cared-for. Motivational displacement is the sense that our motive energy is flowing towards others and their projects. We are seized by the needs of another. (Nel Noddings)
When you give a child range from which to make choices… it’s your decision… it’s something you are responsible for... If it’s a decision to let him alone you are just as responsible for it as if it’s a decision to intervene. (David Hawkins)
Listening [is] an activity of openness, humility, caution and trust. The listener must always be oriented toward gaining a fuller understanding, always vigilant to the fallibility of interpretation. This is why listening cannot be silent; it is itself a kind of speaking. Listening is not a solitary act, it is a reciprocal engagement. (Brent Davis)
There are other issues around authority and community that I didn’t get time to discuss. My edited research is here, with more complete excerpts here.
Our process of abstraction
I ended the session by describing how this theory is informing our department practice.
Our (tentative) purposes are to educate awareness (of mathematics, self, other, of awareness), and to create equitable communities.
Our values are responsibility, receptivity, attentiveness, decentration, and perhaps something about presence that we have yet to explore.
We act along three dimensions of action regarding distribution of authority: dialogue, time and space.
The question is not whether we should ‘tell’ or let students do, it is both/and... or rather: when and how to direct attention.
Each of these values and purposes are imbued with meaning for us, sought through our process of abstraction. I would encourage any teacher to conduct their own process of abstraction.
Next year we will analyse to what extent our practice adheres to these purposes and values through a video project.
My current thinking is about energies (as described by Gattegno and JG Bennett) and the idea of mathematical vulnerability, as described by Linda Brandau:
The issues of control, resistance and vulnerability are ones dominating my life these days. What does really knowing someone, or something, mean anyway... risk taking, vulnerability, intimacy, responsibility, freedom, resistance, and insecurity.
It is difficult to be honest, not only with others, but with one's self.
To grow we must take risks and place ourselves in vulnerable positions. We must admit to the artificial separation of emotion, cognition, subject matter, professional and private lives. By keeping these areas separate we are concerned with controlling them. By loosening the boundaries, perhaps even cutting them, we become open to enriching and learning about ourselves and others...
Thank you to those who were present at the session. I would love to hear from you.