Gattegno suggests we may educate ourselves in the workings of the self through becoming sensitive to movements of what he calls ‘free energy’:
…free energy displays a temporary structuration involving minute amounts of energy coagulated by the slightest expression of the self: a thought, a mood, an emotion, an activation of a functioning.
The Mind Teaches the Brain
I am interested in how we might become aware of such movements of energy whilst working on maths problems, in an attempt to understand how we might enable ourselves and others to become better equipped to solve them.
I invite you to work on this problem - taken from John Mason’s 26 years of problem posing - whilst attending to the following, and before reading the rest of the post:
- What are you doing when you are not performing some kind of mathematical functioning?
- Are there times when you feel you are in a relatively ‘neutral’ state? What precipitates this state?
- Are there times when you feel more and less able to solve the problem?
- Are you in any way aware of ‘temporary structurations involving minute amounts of coagulated energy’?
The first noticeable shift in energy I felt was a desire to start working immediately on a couple of possibly fruitful ideas, but then tempered with experience that working without a period of contemplation is rarely successful for me.
This is an example of the constant tension and resolving of tension that I experience when working on maths problems, that I will describe here as the mobilisation and dissipation of energy.
Regulation of energy by the self through mobilisation and dissipation results in a return to a neutral, contemplative state. In this state, intuition - considering the whole, maintaining complexity - is the dominant way of knowing.
From the contemplative state, the self gathers information, integrating the past with the present. What do I recognise? What might be relevant? This may lead to a question or conjecture, mobilising further energy.
Writing of symbols - algebraic manipulation, diagrams, tables - is a means of objectifying this energy, increasing the possibility of something being noticed, whilst freeing the self to turn its energy in other directions.
Possible futures are available in the present, through imagining. What do I envision might happen if I follow this approach? The future is also brought to the present via affectivity.
Energy may be mobilised (and objectified) through discussing the problem with others. I must be careful that my ego does not become obstructive, and I may need to remove myself from ‘aggressions’ over which I have no control.
All the time, the self continues to regulate the flow of energy, returning to the contemplative state, in preparation for, and in hope of, insight - ‘grace’ - the revelation of as yet unreached awarenesses, leading to further conjectures and mobilisations of energy, and hopefully a solution.
It may be that sufficient energy is not available to be mobilised in the present. Sleeping on it may allow the self to work on the problem without the inhibitions of the waking state.
Giving up is a viable option, although it may well be the case that we learn most about ourselves in moments of wanting to give up; it often happens that I solve problems following such moments, giving it 'one last try'.
