like learning a language…

Some methods, especially those that rely on memory and de-contextualised vocabulary and intense grammar are awful. They start with lists of words and complexities of grammar that generally mean something to those who have studied grammar in their own language but feel so fragmented and distant from speaking that one senses its going to be very hard going. The best (in my opinion) way of learning a language is through a live course using ‘The Silent Way’, which was created by Caleb Gattegno as he learnt how people learn. Some other methods however such as the Michel Thomas courses are very good for learning at home. I personally am learning arabic using this method and find in some ways it is like learning maths in the style of the subordination of teaching to learning. I could not find a course on arabic using ‘The Silent Way’.

One immediately feels useful learning is taking place and a certain hopefulness and confidence to continue takes over. It feels optimistic and fruitful. Arabic is a bit tricky because of the written language, the unfamiliar sounds and the unfamiliar words having in general no Latin roots. Most words are completely different to what an english speaker knows and the word order and other things are also unfamiliar.

OK so this sounds a bit like coming across mathematical ideas when you are little.

The knowing use of algebraic ideas in simple conversations and question and answer ‘sessions’ using the rods is similarly freeing and powerful. You can just see it so easily in the children’s eyes and actions. There must be no pressure at all. Certainly no pressure to write anything down, unless it comes from the children, and even then do not force them to ‘write it down correctly’. Do not ‘be a teacher’ in that sense. Bide your time.

Look at Caleb Gattegno’s  ‘Mathematics with Numbers in Colour’ Book 1, part II, ‘Qualitative Work with the Rods.’  ONLINE…

Don’t take it as a ‘course’, just read it through three times:

1. As if you were reading a newspaper

2. As if you were reading it out aloud to another person and

3. Try and fathom the general flow and gist of the chapter.



chronic boredom and irrelevance

please excuse this rant

a diatribe concerning the captive consumers of our woe, the children

They are the future. We create the future through them. They do not choose to be there. They do not choose to study the available curriculum. It has been chosen for them. They do not choose the structures of schooling including the structure of the day, with whom they are taught, their timetables nor the time for which individual subjects are taught. They do not choose just what is to be taught from the infinities of possibilities, nor why. They do not choose to divide knowledge into the separate subjects. They do not choose their teachers, the teachers’ personalities and the teachers’ particular teaching styles and the theories of learning on which they are (presumably) based. They rarely choose their own work. They are marked and assessed according to criteria they do not choose, and are categorized accordingly.  They are embedded in a system, a technological machine, which I imagine they have to tacitly assume is the best possible type of machine that society can construct, through much deep thought, which will enable them to live happy fulfilled lives and fully prepare them for the future. They accept it because it is the way things are. The status quo. The embedding in this prevailing school machine technology has inevitable consequences. Many like it. After all it is a vast social flux, they make friends, often for life, they meet people all the time. It is busy and involving. No one likes their work all the time after all, unless they are very lucky, and for much of the time, it is ‘OK’. It can lead to qualifications which are basically passports to further qualifications and restricted better jobs, even vocations and ‘professional’ activities and lifestyles. Children see this and it causes a certain degree of concentration (which is a kind of motivation imitating intrinsic motivation), even great concentration in some who can achieve excellent results in their exams. These achieve because they are born with the capacities that enable them to relatively easily fit in. It ‘suits’ them, as it ‘suited’ me up to a point. Everyone is pleased and their self-esteem becomes high. It is a good feeling. All you have to do is work hard and learn things and do well in exams, though exams are ‘not what anyone in their right mind would actually choose’. Still, they get on with it. If they make it to A levels or university, similar structures and processes prevail. If they decide to work hard and play hard, they generally have a good time. After all, they have the advantage of choosing (to some extent) their general areas of study. Sometimes, even specifics.  For others the story is not quite so rosy. They perceive, early on that the work is hard. That it is not very interesting. They look around and see others doing it easily and being rewarded in various ways. They get rewarded too, but as they get older they cannot help comparing themselves to those who will obviously ‘do better’ in the long run. As the work is ‘hard’ to understand and hence it is not easy to ‘do well’ it begins to be chronically wearing. Sooner or later most students experience this. Even for those who have inclinations and innate abilities praised and needed by the system, the relentless learning of new things, often not seeming to have much if any direct relevance to them and their needs, except for the need to pass exams, can cause chronic weariness can set in. It happens at university too. “Mum, why can it seem so irrelevant and dull? Why does it have to be done, apart from the exam angle? Why is it all considered to be necessary? Why is it all compulsory?  What is it all actually for?” Sometimes, they have ‘learning difficulties’. If they are told, or perceive that they are in fact in a different ‘category’ to most, they might easily come to ‘believe it’ hence totally altering their view of themselves, often for life. They might have some real special differences that are obvious, such as for instance, blindness, deafness or cerebral palsy which undoubtedly provide challenges to themselves and the providers of experience, their teachers. Sometimes, they have ‘learning difficulties’ that are less tangible, but which are still ‘seen’ to cause ‘difficulties’ for the ‘normal’ processes of the technological machine system. It is rarely seen as a product of the system itself. The system is generally not sensitively reflexive enough for that. It is the fault of the others – parents, society in general, the children in particular. Sometimes through individual teachers’ enthusiasms, and sometimes because the children like the teachers as people, students can get carried along down even very abstract and esoteric routes and actually enjoy the processes and products of learning along the way getting satisfaction through the sense of a job well done. There is nothing wrong with that, but it doesn’t always happen. In fact, it usually doesn’t happen. Some work is just too dull, or presented in too dull a style, in a flat, matter of fact, monotonous manner, ‘because its good for you’. It can be perceived as irrelevant by the students. Some teachers just cannot hold it all together. Other issues can emerge. ‘Disruption’ of the ‘lesson’ can easily occur due to others who have already given up the idea of pursuing the lessons’ aims and deciding to ‘have a bit of fun’ instead. It is true that it takes only one very disruptive person to destroy a teaching environment. Almost nothing can productively proceed in an atmosphere of noise and confusion. Threats and rewards can be issued and will ‘work’ with some at least temporarily. Machine gun nests work a treat. The lid can be kept on through externally imposed discipline. The repressive communist regime of Tito’s Yugoslavia managed this quite well. More ‘discipline’, including clearly visible, increasingly painful sanctions, parent/carer contracts and now quite common well established systems of ‘assertive discipline’ are acclaimed, and can ‘hold it all together’. Guilt and fear also work a treat. ‘Order’, real and apparent, is re-establishable. The normal processes of the status quo school technological machine are then re-established and proceed to the apparent satisfaction and indeed praise of all, often including the students. Certainly to most staff, the general population and government, who are seen to be improving standards. Indeed, at one level they are. Life is once again a bed of roses……but is it? Symptoms have been suppressed. What about the causes? Who bothers digging here? The causes are more insidious, more chronic, more pathological. They are potential viruses threatening the system itself. Leave them alone. In order to dig where it is dark one requires an active conscience, an unflagging, instinctual need for the development of self and professional knowledge. The digger requires courage, because what one might find in the dug hole might backfire on ones own beliefs and inner, tacitly held convictions. The digger might archaeologically expose himself and be taken away as a deviant. That is what societies do to maintain the deeply caused and little sensed status quo. It creates ‘objects of deviance’ which are cast out. Suck on these ideas and see for yourselves if there may be any substance in them. Do any cause resonances within you about teaching and learning. I am just one agent in this flux, self-constructed by my labours and experiences. Chronic boredom and irrelevance infect present educational systems, at all levels, but especially for the those compelled to be captive audiences. Dig where it is dark. Most people aren’t interested in this esoteric activity, because not only is it difficult, it is disturbing. It is also, transformative. Who wants that?




48 6x8 2x3x2x2x2


The LHS shows 6×8

The RHS shows 2x3X2X2X2  dust ( the 8 is 2x2x2 and the 6 is 2×3 )

As far as the number 48  is concerned the order of rods in the tower is irrelevant, but this needs ‘proving’. Take my word for it at the moment.

48 6x8 2x2x2x2x3


So long as the tower is constructed using the rods on the right, the order is irrelevant.

So, as 2x2x2x2x3 is the dust, this means we combine these a pair at a time in any order:

try it yourself..that’s best…but

here’s my mind at work for example:

start with 2, double it double it double it, that’s 16, times 3 is 48 (2 4 8 16 48)

2 threes are six, double it, double it, double it, that’s 48  (6 12 24 48)

2 twos are 4, two fours are 8, three of them is 24, double it, 48

and so on…..


(In general, the present school arrangements almost totally inhibit this…)

ps 6×8 is one piece of your ‘tables’, using the dust you see and get the ‘feel’ for 6×8, 8×6, 3×16, 16×3, 2×24, 24×2, 4×12, 12×4, never mind ‘half of 48 is 24’, ‘half of 12 is 6’, ‘half of 48 multiplied by 2 is 48’, ‘a quarter or fourth of 48 is 12’, ‘an eighth of 48 is a half of 12’…and so on till the cows come home…

yap yap yap…






there are no ‘objective’ events…

If you agree with the above then the consequences are the acceptance of diversity and complexity as grounding assumptions of any learning space you can devise or operate within…

There are no ‘objective events’. (See Poppy Pilgrim below). This is an illusion just as absolute space, absolute time are illusions of common sense and Newtonian physics. The illusions ‘work’ as a general rule and in the normal course of life up to a point, but the concepts are non subtle and simplistic. Listen to Einstein:

‘Considered logically, they (space, time and event) are free creations of the human intelligence, tools of thought, which are to serve the purpose of bringing experiences into relation with each other, so that in this way they can be better surveyed. The attempt to become conscious of the empirical sources of these fundamental concepts should show to what extent we are actually bound to these concepts. In this way we become aware of our freedom, of which, in case of necessity, it is always a difficult matter to make sensible use.’

Appendix V, Relativity and the Problem of Space, Relativity, the Special and General Theory, 1920. Appendix added 1952 My parentheses.

Bringing this through into the domain of teaching events, which are more complex than the domain of inanimate matter we could postulate that though it appears ‘obvious’ that for example a ‘teaching session’ (event) is taking place, the event as an absolute object is located nowhere but in the minds of the participants. Furthermore, all will perceive ‘it’ differently, according to their perspectives, capabilities and state. Hence it is clear that:

we must give up the idea that unique events exist

i.e. in our case, we must give up the idea that it is even possible to design and construct a teaching space that objectively exists separate from all participants and their perceptions, and that by following certain procedures, definite outcomes predicted beforehand will occur. The real situation is far more subtle and complex.

Hence a more subtle and illuminating relativistic concept, similar to the new conceptualisation of space, time, matter and event in Einstein’s general relativity is that:

there exists an event field which is potentially infinite, out of which diverse, multiple and unpredictable consequences will inevitably ensue

Events are located nowhere but the minds of the participants and, far from assuming uniformity and homogeneity (and striving pointlessly to achieve them by partitioning of various kinds).

we should assume diversity and complexity

This should be the grounding basis for proceeding with the enhancement of teaching, learning, planning, assessment and reflective and reflexive practices. All problems which then may proceed as consequences should be taken as food for thought for the analysis and transformation of current practices so that labelling, specialisation and marginalisation are inhibited and so that diversity and richness may flourish in their place in holonomic classroom worlds.

We must give up the generality still quite common in teaching and amongst the general population including politicians of all parties that the teacher teaches and the children learn unique objects, and that further more, if they do not learn these unique objects it is somehow the fault of the child.

By giving up the generality this does not preclude us from saying that in clearly obvious and simplistic learning situations, the ancient and limited view expounded above pertaining to the ‘objective’ illusory world of common sense, may in fact ‘work’, just as Newtonian mechanics is quite adequate to describe the motion of planets in their orbits of our central star, the regular predictable appearance of comets, the precession of the equinoxes and the ebb and flow of tides. Planets and comets are simple objects in comparison to the interpretive nets of human consciousness. Sir Isaac Newton was, as Descartes remarked, probably the smartest guy who ever walked the surface of this planet. His achievements marked the beginning of the last three and a half centuries of human scientific endeavor. However to quote Albert Einstein again:

‘Newton himself and his most critical contemporaries felt it to be disturbing that one had to ascribe physical reality both to space itself as well as to its state of motion; but there was at that time no other alternative, if one wished to ascribe to mechanics a clear meaning.’

Same source, p135

cover sets 1 how to sort the gold from the dross

1 cow + 1 cow makes 2 cows.

1 sheep + 1 sheep makes 2 sheep.

What’s one cow + one sheep? Mmmm…you cant do this unless you make a new category, called ‘animals’. Then:

1 cow + 1 sheep makes 2 animals



‘ANIMALS’ is a COVER SET and exists at the n+1 level.

‘COWS’ and ‘SHEEP’ are COVERED by the category above.

In this little diagram there are 2 animals at the nth level.

The DIMENSION of the nth level is 2.

If you added in meerkats and duck-billed platipuses (there appears to be no collective noun for platypus), the DIMENSION would be 4.

This is how mathematicians define DIMENSION.

Cover sets and dimension are going to come in useful when we start to create and criticise CURRICULAE… ahead…

You see you don’t want CATEGORY CONFUSION. That sounds like it could make you ill. However there is a lot of IT about. For example, you probably heard quite a lot of politicians with diametrically opposite views saying the same thing such as;

‘We want the best possible education for the children in our society’

Yes, true, we do, but the problem is they are relying on CATEGORY CONFUSION at the n+1 level to misle you into believing they are doing the RIGHT THING. Everybody of every persuasion agrees at that level but the point is what lies ‘beneath’ at other levels, what dimension they are referring to and what SPECIFICALLY do they mean.

I saw a T shirt once: ‘CAN’T YOU BE A BIT MORE F****** SPECIFIC’, excuse the french.

COVER SETS and DIMENSION are two of the tools that will sort out the idiots from the smart guys.

against too much prescribed content and the learning of undesirable outcomes

National curriculae have strong tendencies. A National curriculum might have as one of its aims to improve the access of all children to what is seen as a desirable amount of knowledge. That is fine, BUT…

National curriculae are mostly generated by advice from specialists tempered by ideologies of the prevailing political system. The first ensures an immense amount of material to be covered, because specialists live within their created worlds and love them so they can think of lots to do and the second ensures that a) the lists sound similar to what they did at school and b) great emphasis is given to processes, procedures and organisations for ‘learning’ that these power-possessing beings have had experience of. So, kids have a curriculum that is 1000 miles wide and an inch deep (maybe a foot) fragmented by Victorian attitudes and organisations for learning. The Victorian attitudes include the fragmentation of the world into a few subject areas, the fragmentation of the child’s time into standard units such as hours, and the fragmentation of the people of the world into ghettos they call groups or streams. The children quickly learn from these ancient environments who is smart or dumb, who is destined to be a king or a pauper, who will live to serve and who will serve, etc.



There are better ways…