Jackson and I…

The other one, the one called Jackson has worked in education since 1970, in many schools and a university. He has been privileged to work with a very wide range of people, from 4 to 80 years, many children, many adults, many courses… I read in the sunlight on my summerhouse porch then tend vegetables. I know of him from the letters and emails he receives. I like playing with my grandchildren, constructing images and reading Borges. I chase froth on the beach with a large camera. He shares these preferences, but in a vain way that turns them into the attributes of an actor. I am quite aware of his custom of treating things he thinks of as precious. All things long to persist in their being. Years ago I tried to free myself from him and travelled in distant lands but all that is far way, playing those games with time and infinity. I shall now imagine other things. Thus my life, as all, is a flight to oblivion or to him.

with great apologies to Jorge Luis Borges and ‘Labyrinths’

Here lie a few of his previous activities. The highlighted are basically maths related:


Now he specialises in rods and the subordination of teaching to learning or as some might say the elevation of learning above teaching…and also, the creation and operation of rich, motivating learning environments.

communication is possible….contact: drjax at ntlworld.com

some important people…

Caleb Gattegno

Most of his works are available to read on line for free here:


Roslyn Young – a modern exponent of Gattegno’s ideas

try this: On awareness and awarenesses

Read this book, co-authored with Piers Messum,

‘How We Learn and How We Should be Taught’,

Vol 1: Duo Flumina Ltd, 2011, ISBN 978 0 9568755 0 1

Caroline Ainsworth

An example of a ‘research process in all its complexity’

a case study of one teacher’s professional development journey

Madeleine Goutard

the wonderful exponent of STL*

Talks for Primary School Teachers

Educational Explorers Company, 1963

Mathematics and Children – a reappraisal of our attitude

Same series.

*The Subordination of Teaching to Learning – begin here

Numbers, what are they?

Everybody thinks they know what numbers are, and at a common functional level, most believe they do, and what they do with them tends to work well enough. As to what they actually are, this is much more problematic. Bertrand Russell defined them a few decades ago. This is what he said, followed by what his, (the best) definition of a number looks like:

(I’m leaving this out for now)

Mmmmm!   Best left alone. One thing you need to know is that they are not as obviously simple as you probably thought. There are a few things to make clear. What are numerals? These are signs that are made to represent numbers. They are not numbers. These signs can take many forms. Different languages have different sounds for numerals and different squiggles. The important thing is that they Represent numbers. I will say it again, the noises and squiggles that are made ARE NOT NUMBERS. They are indicators which point to the idea of number. Number is an idea. The way you imagine a number has IMMENSE consequences for what you do with them. The zen way of Cui is a billion light years distant from the way they tend to be imagined in the curriculae of most school systems.

When a Cui zen master sees a mark on a piece of paper known as a numeral, he senses an immense set of possibilities. He senses a whole world of relationships. He does not sense one position on a number line. He feels ‘worlds’. He senses holistic wholes floating in a sea of infinite soup. He is ready to collapse this infinite whole into one of its possibilities according to the best solution appertaining to the present moment. He sees the whole but has the power to collapse this ‘wave function’ into a specific form. If there are several marks on the paper which have been interpreted by the perceiver as being a ‘written problem’, the Cui zen master will select the ideal form of the number which he senses (as a whole) in order to solve the problem effectively, often immediately. He collapses the ‘wave function’ of the number into its job-specific eigenvalues. He chooses a specific form of the number in order to solve the problem effectively and aesthetically. I am using the metaphor of quantum electrodynamics here because it is (a little bit) isomorphic, or of similar form. The master will be aware that there are many different ways of solving the particular problem and he will take responsibility for choosing that particular way. He will not be upset if someone demonstrates another way, he will be grateful for this fresh perception, having had his stock of possibilities increased.

This is not just true concerning the perception of number, it is true in relation to the whole of the beautiful, creative, imaginary world of maths.


It is like learning to play an instrument properly and being able to write music yourself rather than learning a few simple tunes from memory. It is like learning to be a painter rather than messing about. It is about being creative rather than mechanical. That’s all., but IT IS ALL.

Enjoy the world rather than be fearful of it.