the symbol of infinity…

Norman Cantor notes the three roles Boniface played that made him “one of the truly outstanding creators of the first Europe, as the apostle of Germany, the reformer of the Frankish church, and the chief fomentor of the alliance between the papacy and the Caroligingan Family”. Well that’s as maybe , but people say that the first instance of the appearance of the sign for infinity was in St. Boniface’s cross…

I am doubtful.

here it is:


and again, somewhat purified:


I would say these are re-arrangements of arcs drawn by compass and not this:

Dekes-Techniques-103-infinity-symbolThis sign was first given its current mathematical meaning in “Arithmetica Infinitorum” (1655) by the British mathematician John Wallis (1616-1703).

Thanks John…

observing oneself…


I wanted to do a ‘piece of research’ in which I attempted to ‘observe myself doing it.’ I have been interested in patterns in nature for many a year and thought I would drop ink droplets onto paper to see what patterns might emerge, if any. The following are 3 grabbed images of this research. I have used images here because it is necessary to maintain the complex formatting: read the first section, upper left. This shows how the original tape recording was transcribed and then thought about and re-thought about so to speak…


a further piece some way in:


When I had finished the whole experiment I made a series of knowledge claims which were quite extensive. Here’s a small example:


Well, that’s what I did…

challenge: observe yourself making a cup of tea

(or similar)

gattegno buddhism and critical theory..ah well…

Gattegno maintained there were 4 stages in learning:

1. there is something to be explored

2. the exploration of this ‘field’

3. practice in this field leading to mastery

4. transfer of this knowledge to wider fields

The Four Noble Truths of Gotama Buddha, the cattari ariyasaccani are:

1. an unsatisfactory state exists

2. how this ‘suffering’ arises

3. how to end this ‘suffering’

4. the means by which this ‘suffering’ can be eliminated

In his book ‘The Idea of a Critical Theory’, Raymond Geuss describes a typical critical theory as being ‘ a transition from an initial state of bondage, delusion and frustration to a final state of freedom, knowledge and satisfaction.’ Further, he defines a typical critical theory as consisting of three parts:

1. the belief that a transition from the initial state to the final state is in fact possible

2. the demonstration that this is in fact necessary

3. this transition can come about if one follows the particular tenets of that critical theory.

Well, at least, Gattegno, the Buddha and Raymond Geuss have in common the idea that…


but it is clearly not going to be a push-over

huatou: is it possible to change?

we learn through awarenesses…

I was in my greenhouse this morning and thought I would plant some cucumber, melon and gherkin seeds in a root trainer:


I had to fill them with potting compost first. I started filling them but quickly realised I had to pack the compost down considerably into the individual trainers before they’d be suitable for planting. When I had filled them to my satisfaction the whole 32 section unit was very scruffy looking so I tried to brush the excess off. A soft brush was no good so I had to hunt down a hard hand brush. This did the job a treat. I was pleased. I now had a good idea of what was required. I had learnt something. Next time will be quicker…

Simply, I became aware of what was necessary to accomplish this (easy) task.

Look at my chilli plants:


Coming on, but the weather…

Coming back up the garden, feeling pleased, I was unaware of this rope:


and tripped…c’est la vie…


awareness 2 some examples in rod pre-number…

some algebra before arithmetic pre-number awarenesses

examples of free play awarenesses:

rods are good things to play with

we can make pictures

we can build models and buildings

we can play games

we can invent games

we can share our ideas

we can learn from each other

rod colours are in families

rods can be packed away in family colour order if we like

rods can be packed away in other ways

rods have a definite regular order of size

I can recognise rods by how they feel

rods can be named by colour and in other ways

directed informal awarenesses:

rods of same colour are same length

rods of same length are same colour

groups can be made of the same colour

groups can be made of the same lengths

language (word concept) awarenesses:

trains can be made in a variety of ways

rods can be swapped for others

 staircases can be made in a variety of ways

 mats can be made

mats can be made with rows of rods of the same colour

we can answer questions by making patterns of rods

awarenesses relating to adding and its inverse:

I can find one rod to fit two others

I can find two or more rods to fit one

I can find rods to fit long trains

I can find the difference between two rods in different ways

I can find the difference between two trains of any length

awarenesses relating to equivalence, multiplication and factors:

I can find one rod which fits two or more rods of the same colour

I can find rods of the same colour to fit some other longer rods

I can find rods of the same colour to fit some, but not all trains

awareness 1 an atom of education…

Caleb Gattegno never tired of saying that only awareness is educable. He said that all learning occurs due to one becoming aware of factors relating to the field of study one is engaged in. It may be one awareness or it may be a series of awarenesses.

Roslyn Young, in ‘How we Learn and How we should be Taught’, says:

‘This means that the awareness becomes the conceptual unit of learning. The awareness is to education what the atom is to chemistry.’

huatou: awareness is an atom of learning

nasruddin looks for his keys…


It was night time. The Mullah was on his knees searching for something under a lamp post in the street.  His friend Parviz came along. “What are you doing Nasruddin?” “Looking for my keys Parv.” “OK, I’ll help you then.” They carried on looking for a while, then Parviz said, “Are you sure you dropped them here?’ “Oh, no Parv, I dropped them in the middle of the street but there’s no light for looking there.”

Benny Thompson has lots of Nasruddin tales…

assessment as it is…


 The ultimate zen assessment:

You cannot know another you do not know yourself

…it is not possible. You cannot ‘know’ another in the fullest sense. I cannot know you. I ‘see’ your outer coverings, your skin, your body, your clothes. When I speak to you we discover more about each other. All the time we are making judgments and attempting to interpret each others’ responses and so on. Clearly we can get somewhere, but…

‘I’ can ‘know’ ‘you’, up to a point, but that is all. Ultimately I do not know myself, never mind you.

We are always assessing people in general though, through automatic responses, automatic generalisations. ‘I like the look of him.’ ‘People who have their trousers hanging low are stupid.’ ‘I hate people who smoke.’ and so on… These ‘automatisms’ are deeply ingrained. If we are unaware of them, well…what do you think?

As teachers we are required to assess others in more specific ways

For example if its my job to teach you addition it would be very useful for me to know what you already know. At least, that seems a reasonable supposition. How do I do it?

I can only attempt it by spending time and focusing attention on the specific areas I am interested in.

To start with I need to look at myself. Do I know what I am looking for? Do I actually know what I am talking about when I say ‘I want to teach addition?’ Most people don’t apply too much attention in these areas. It’s dark in here and the ground is hard for digging. Read this Sufi story.

Basically, if you know what you are doing and you look and listen openly whilst asking suitable probing questions that’s a start

You will have to spend time and effort and you should work on each child at a time

You may have noticed that this is very hard to achieve in a large class and when you are burdened with too many other demands

to create order, energy is required

in physics, to increase order is to decrease a measure known as entropy

(I like entropy that’s why I’ve put it in….it’s closely related to the most fundamental law of physics, the Second Law of Thermodynamics…)

Entropy tends to increase throughout the whole universe…

Your desk becomes dis-ordered in the course of a day

iron rusts

you age

if you wish to tidy (order) your desk you will have to expend energy

If you wish to assess another, to increase your ordered knowledge of them, you will have to


to expend energy in this sense you will need TIME…

ps For those interested in our most accurate representation of matter: There is no unique universal relation in quantum theory that could stand on equal footing with the position-momentum uncertainty relation. The energy operator is called H the Hamiltonian. There is no dual relationship between H and T.

Not yet, anyhow…

see sister post