Here is something I wrote about insight/grace when solving maths problems.
It includes some nice problems to have a go at, such as these two below.
Comments would be gratefully received.
Here is something I wrote about insight/grace when solving maths problems. It includes some nice problems to have a go at, such as these two below. Comments would be gratefully received.
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My calculator is not working properly. Only the add and reciprocal buttons work, as well as the numbers 09. Can you work out how to halve any number? Now the subtract button has started working. I think this means I can now square any number  can you work out how? Now, I think I can also calculate the product of any number. How? Are there any other operations I can do with only add, subtract and reciprocal?  I would love to hear your experiences when solving this problem, in particular with regards to getting involved. Consider the following problem: Four distinct lines A, B, C and D are given in the plane. A and B are parallel, and C and D are parallel. Find the locus of the point moving so that the sum of its perpendicular distances from the four lines is constant. You might like to try it before reading on.  What determines whether we become 'involved' in solving a problem, or not? First, and possibly foremost for students, is whether one 'has' to solve the problem  whether at the behest of the teacher or an examiner. Suppose then we have decided to try the problem. How does one then become 'involved in it'? For me, there is a critical period when I am not sure whether the problem is of genuine interest, or not. At this stage, I am not 'invested' yet  that is, I have not given myself to the problem  I remain relatively neutral. I am ascertaining whether the problem is at the 'right' level of challenge. This is perhaps dependent on whether there are any approaches, insights and conjectures available, alongside something intriguing. There may be a tension that is demanding to be resolved. All of these factors have the effect of mobilising the energy required to get involved in the problem. If they are not present, I might decide not to continue working on the problem, or to give up.  Should the energy be available to continue working on the problem, it becomes objectified through trying examples, sketching diagrams, making conjectures. When choosing what to explore, I pick examples that contains some of the complexity of the problem, but not too much, with the option to increase or reduce the complexity and/or generality of examples as the need arises. Often it might be illuminative to explore a special case, if there is one. Your sketch here might have included two pairs of parallel lines of different 'widths' and orientation, forming a parallelogram. Or perhaps you drew the more special case of a rectangle or square? Did you start by exploring the movement of the point that is equidistant from all four lines?  We gradually become more involved in the problem. Releasing vague thoughts/ideas/conjectures from the mind into the world (on paper, in words) may make it easier to regard to whole problem, thus mobilising further energy. As we get more involved, it becomes more difficult to maintain the initial neutral state. We may get attached to particular ideas, enabling and disabling emotions might surface. Should we become too attached, it may be beneficial to return to the contemplative state, the essential problem solving state. It provides a lightness through which we are able to maintain the whole in its complexity without grasping, thus improving our chances of receiving insight.  Hopefully at some point insight appears. You may believe that you have solved the problem. But how do you know? Have you reached the full solution? [Spoiler alert below!] You might have conjectured that the solution to this problem is the area between the parallel lines... but is this the whole story? In his book The Science of Education, Gattegno describes various ways of knowing. Try solving the problem below, paying particular attention to the moments that you know you are getting closer to the answer, especially the precise moment that you know you have the solution. How exactly do you know, at these moments, that you know? What are the feelings associated with knowing? Can you sense a mobilisation of energy in these moments? P is the centre of the square constructed on the hypotenuse AC of the rightangled triangle ABC. Prove that BP bisects angle ABC. Can you recollect 'giving up' while solving a maths problem, or any problem, recently? What, exactly, led you to make the decision to give up? Some possible reasons for giving up might include (a combination of):
Can you think of others? Have you shelved a problem that you intend to tackle later, when you are 'ready'? What might being ready mean? Can you recall any problems that you shelved a long time ago? Supposing that we are able to comprehend a given problem, it seems to me that there is no way of knowing whether we will reach a solution or not. If we agree that we cannot know whether a given problem is within our reach or not, what is it that leads us to give up? ExercisePlease read both problems below: Now answer the following questions as honestly as possible: 1. How interested are you in solving each problem? 2. How confident are you that you would be able to solve each one? Before reading on, I invite you to try the problem you are *least* interested in solving. ReflectionDid you manage to mobilise the energy to get started on, or remain involved with, the problem you chose? I am interested in what happens when asked/forced to solve problems that do not hold our 'interest'. I chose to work on problem 15. I became aware my lack of interest might be because I found it difficult to comprehend. It was not the sort of problem I usually try. I felt reasonably unconfident that I would be able to solve it (in the time I was prepared to give it). This resulted in finding it difficult to mobilise the energy to keep working on it. Is there a correlation between our ability to become involved in a problem, and our perceived ability to be able to solve it? ConjecturesA colleague once used the name 'sitters' to describe those students who require considerable motivation to get started on, or remain involved with, a problem. Are 'sitters' sitting because they are not 'interested', or because the problem is not at the right level for them, or perhaps due to a dialectic between the two? What does it mean to say we are not interested? In some cases it may be rooted in a difficulty in mobilising energy due to a feeling that the challenge is beyond our limitations. It may be the absence of a spark of insight, or recognition. One way of coping with a challenge that feels beyond our limitations is to avoid it. This may take the form of disinterest or dislike. We become able to meet this kind of challenge but not that one, we become this kind of person and not that. Repeat this a hundred times and we may understand why sitters are sitting. Here is a pdf about tension, coping and imagining when solving maths problems, following on from this post. References/links are: [1] Gattegno newsletter Problems and Solutions. [2] Transcript of Gattegno seminar on Awareness. [3] John Mason's 26 years of Problem Posing. [4] Gattegno's The Mind Teaches the Brain. Comments very welcome. 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. 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:
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'. Following recent conversations about whether it is possible to 'teach' problem solving, I wrote this. Comments very welcome. In today's meeting for worship I was reminded of something said by a friend, on his looking forward to visiting another friend: 'I'm looking forward to being in his presence again.' My thoughts turned to meeting, being in the presence of the others there, and a growing sense of unity. Today was the first time my daughter had attended meeting; she stayed for a short time and then left, but I remained very aware of her presence in the house. We have a number of children who come to the meeting; they bring a restless energy and joy. Sophie currently has a strong desire to be in the presence of us, her parents. She often just comes and sits next to us, or plays in the same room, and is currently waking up in the night to find us. Recently I was lying in bed and my cat jumped up and gently rested himself on my arm, just next to me. I was grateful for comfort, the light pressing of his weight.  And then, the meeting for worship, and the presence  or perhaps the absence  of God. I am currently not ready to worship what is called God, and currently think in terms of something akin to Bennett's higher energies  or perhaps Eckhart's God beyond God. This is perhaps the ground of what we might call presence, not accessible to common sense. Hence meeting in silence. Something similar might be felt in the presence of nature, of surrendering oneself to its vastness and plenitude, of teeming life and rocks, all in flux, of which we are a part. I was walking along the cliffs recently and held my hands in the air, in prostration to the sun. On another occasion recently, the words 'I understand' came out of my mouth, bringing to mind George Herbert's prayer, of 'something understood'.  Thoughts turn away from those present now, to being in the presence of friends; I have two friends coming to visit on Wednesday, and most of all I am just looking forward to being in their presence again. Since moving, I am grateful for others with whom I have a sense of presence through other forms of communication. My mind casts back to knowing my partner as a friend before starting our relationship, and the feeling we had of wanting to be in each other's presence more often. And there are others whose presence we are no longer able to be with, at least in the present moment, now. We are to be content with our memories of being present with them, suggesting that being in the presence of others, now, is what matters most, and my mind returns back to what is, here and now. 
