conditioning 1 – tendencies

Being human and having been exposed to innumerable influences of a mathematico-teaching-learning nature you may have innate tendencies to:




WARNING: These items and many others like them will get liquidised and turned into very nice tasty paste if you follow the path of becoming a Cui Master.


conditioning 2 – you, maths, space and time, the government etc

Whatever you have been exposed to in general society, whatever you have exposed yourself to through conversations, books, courses and every other transaction you have made with the world, others, the environment and your self, will have CONDITIONED you and INFLUENCED you. Whilst this is obvious to a gibbon, people in general have strong tendencies to believe that the manner in which they view and respond to the world is THE ONE TRUE PATH. Clearly this is the extreme view and your path is clearly ONE PATH, But…these paths are deeply entrenched. They have a tendency to become YOUR TRUTH.

These ‘paths’ can be considered to be ‘interpretative networks’ through which the ‘real world’ is perceived and then conceived. These interpretative networks are filters through which ‘a world’ is conceived. As such these networks ‘create’ the ‘world’ which you ‘see’. You do not see the ‘real world’ because you cannot. All you can ever see is your version.

That is not to say that if you are not sitting properly in Silent Illumination, the overseeing zen priest will not hurt you when he whacks you on the shoulder with his whacking stick. It will hurt. The stick is ‘real enough’…

The ‘brute reality’ exists at a level that will effect you. Even when discussing the nature of this ‘brute reality’ however, never mind the mental realms (where our main interest lies), the colour we perceive as red, for example, DOES NOT EXIST ‘out there’. It is a construct of our perceptual apparatus. There is no red out there. There might well be frequencies of what we call electromagnetic radiation that can be considered to be isomorphic with this ‘red’, but red itself does not exist except in our minds.

We do not see things as they are but how we are enabled to see them.

If we are considering the ‘outer world’ in general for a mo’, consider this then: that which we perceive as space and time, also like red,  DO NOT EXIST ‘out there’, they are categories of perception that come into existence when we as humans do. I.e. when we are born (or maybe in the womb). Kant figured that one out. However, even though these categories are overwhelmingly powerful, controlling and utterly ‘obvious’ to us in our general day to day life, they do not really EXIST ‘out there’. They certainly exist for us, we have cornered the market in them, but they are created categories.

How much more created then are our beliefs about MATHS?

What you have experienced throughout your school lives and everything you have been exposed to since has undoubtedly conditioned you into believing that what you understand by ‘maths’ IS maths.

IT IS OBVIOUSLY NOT THE ONLY WAY of conceiving this game that can be called mathematics.

You have to accept that even though there is a vast government approved WAY of thinking and speaking about and then of teaching about this thing called maths, it is only ONE way, not THE way.

Unfortunately, the government way is a thousand miles wide and one inch deep.

There is too much in the curriculum, and it is done too quickly and in a shallow manner, resulting in fear and confusion for too many.

It does not, except in the rarest of individuals who can get through without too much damage, lead to a creative approach to the subject.

The Cuisenaire Way is a creative way of viewing maths as something which can be created, composed and improvised with, to use a musical metaphor. One can learn to write music, not play a few restricted tunes.

The Cui Way is a light year wide and bottomless.

You will need to peer into your own conditioning and reconsider…

Unfortunately you will be beaten with sticks if you do not follow the government line, and the government line is inadequate. The government and its representatives are not in the business of being truly open, honest and willing to learn, of reconsidering their conditioning.

You however, probably are.

Good on ya sport!

Read on…

disambiguation 4 – domain specific signs or not?

What signs to use in specific domains is problematic. For example, the addition sign, the ‘plus’ sign originated in the number domain, the arithmetic domain. If you are going to use numerals as signs, the appropriate signs that indicate ‘operations’ are the familiar ones:  +  –   X   and ‘divide’ which is generally not easily accessed in a normal font set, as in this case.

The issue is that if at the moment we are in the ROD MARK MAKING DOMAIN and manipulating real rods in order to see things which we then want to ‘write down’, should we have for example a special sign which means, ‘put the rods end to end’? It would be easy to say ‘What’s the problem, just go ahead and use the normal signs.’ I have done this, and to be honest I am not sure that this is a bad idea. It seems easy and obvious, mainly because people in general including teachers don’t know of any other domain-specific signs. (THESE DO EXIST HOWEVER)

It can certainly be construed as being ambiguous, particularly by some modern rod proponents and almost undoubtedly philosophers of maths. They say that DOMAINS should keep their MARKS specific and separate.  What to do, what to do?

More of this later…

making alphabet marks on flat things

The ROD written language MARK-MAKING DOMAIN is another. It is different to the rod language (speech) domain because now marks that represent rods are going to be made on paper (or at least on flat things such as paper or computer screens, white boards etc.) Rods might be drawn and coloured in previously, and then real letters can be used if the children know how to form at least some of them. STRESS: you do not have to move to this particular abstract domain IN ANY HURRY, quite the opposite.

Caleb Gattegno used the following letters to represent the Cuisenaire rods:

w r g p y d b t B o

White, red, green, purple, yellow, dark green, black, tan, blue and orange.

Others have used other letters. If you want to use Gattegno’s books, it might be best to stick to these, but you don’t have too, but you do need to be consistent. As a slight aside, it is useful to provide help in forming the letters properly, because here like everywhere else, a little good direct tuition is very useful. For example, form all those letters starting at the top. This goes for the great majority of the letter and number formation in the English language. Bad habits are persistently difficult to break.

You notice that black, brown and blue start with the same letter, as do green and (dark) green. Hence the alternatives. You get a tan if you stay in the sun for a while. They mostly see that straight away, and they accept that blue could be represented by a capital letter to avoid confusion. Familiarity soon takes over. Colour blind children can apparently distinguish the colours by shades, but I have never come across a totally colour-blind person.

This ROD written language MARK-MAKING DOMAIN is where ‘other’ SIGNS might be met. So long as the child totally understands the signs all might be well, but THIS WILL NOT BE THE CASE AT FIRST. However, even if you decide to make ‘mathematical’ signs as well as the written alphabet signs there are issues you should be aware of…….for example, do you think that signs are ‘domain specific’ or not? if you claim to be a master, you should consider this.

disambiguation 3 – more on domains

OK, so we can see straight away that if we take a general overview of this domain lark, there is an obvious ROD DOMAIN, an ARITHMETIC DOMAIN and an ALGEBRAIC DOMAIN. Each of these have associated ‘mark making’ activities on flat pieces of paper. So, at first glance, there are THREE main domains where we can be active plus their associated mark-making activities. Not surprising children (and adults) get confused is it? Inside the ROD DOMAIN we have various sub domains as described previously. Same with the others. Maybe we are getting a bit too pernickety with these domains. Maybe we’re getting a bit too philosophically mathematical. On the other hand, maybe not.

Looking a bit more carefully at the rod domain, you can see that there is a PURE ROD DOMAIN where there is no mark-making on paper (unless the children decide to of their own accord and in their own way). This is where children can play. When I say play, I mean, totally pure, unadulterated play with no adult interference at all. You, as a teacher are not attempting to get them to do anything or see anything in particular, just let them play.

You might decide it is time to start saying something. In this case you will have entered the ROD SPEECH DOMAIN. You can start saying things like, “I bet you can’t make a chrono-synclastic-infundibulum?”, and so on…  (Kurt Vonnegut-The Sirens of Titan. A CSI is a ‘kind of wormhole through space and time where all kinds of truth fit together….” Just what we really need, nicht war?

You might decide to say, “Can you show me half of an orange rod?” Or, “Which one looks most like a strawberry?” Or, “Could you make a monkey using one rod of each colour?”

An awful lot of very useful ideas which are purely algebraic can be accessed in this rod speech domain totally without writing anything down. In fact practically everything that is useful in arithmetic can be first seen here. THIS ALONE IS ABSOLUTELY AMAZING, JUST THINK ABOUT IT. MOST IF NOT ALL OF THE USEFUL ALGEBRA CAN BE ACCESSED IN THIS ROD SPEECH DOMAIN……………

The things seen don’t have to be perfectly understood before moving across to other domains, it just means that when they are seen in other domains there will be some memory trace of the ideas already present at least in the subconscious, maybe more. In addition it must be said that it is a VERY GOOD THING to make a very strong decision to do this TILL THE COWS COME HOME. I.e. do a great deal of it. Do not be in a hurry. This is another big issue that becomes an immense problem. If you move too rapidly to other domains there will be trouble. On the other hand, activities can proceed in other domains at the same time, but only if the child is ready… This is an area where skillful teaching decisions have to be made.

disambiguation 2 the ‘rod’ domain and the ‘marks on 2D paper’ domain

If you are keen to work with Cuisenaire rods and arithmetic, you should be aware that you will be working in several different areas, or domains. If you are using real rods, this is the ROD DOMAIN. It exist in the familiar world of 3D objects all around us. What we are going to be looking at are “Parallel worlds”, or “Similar worlds”. We are going to gradually (I can’t stress that word enough) move over into other worlds, which are similar, but not the same. Same means, “another one or an identical thingy”.

Similar means “somewhat like….” So we start with the rod domain.

We might start making marks on paper. This is not the rod domain, we have switched over to the flat 2D world of “marks on paper”. This is a flat, more abstract world where we make marks that represent real things. You can make marks of different natures. You could make ‘drawing’ marks, ‘letter’ marks or even, ‘numeral’ marks. These each have different characteristics. Imagine you are a child. They are not real things in the sense of the real 3D things you are used to. Your teachers might think they are a bit removed. You might think it seems a lot removed from what you are used to and are familiar with. In fact if you are not really ready to operate in this world, it will become a very confusing world, a world where you can feel inadequate, stupid and incapable of seeing precisely what you are being asked. You will have entered a FEARFUL SPACE OF HORROR. You will not want to come here again. You will dread it and just like a snail who’s touched something it doesn’t like, you will RETRACT YOUR FEELERS. Remember this. It’s not where I meant to go, but as it is so important I went there to emphasise it anyway. I was becoming paragraphically ambiguous. However, if the above does happen, IT IS A CRIME AGAINST HUMANITY, really, a criminal offence that should be punished. I’m not joking. It happens all time because teachers in particular either don’t know what they are doing and/or don’t have enough time to do this domain switching properly and gradually. Lets start again with this domain disambiguation……already we have seen, the pure rod domain, the rod plus speech domain, the drawings on paper domain, the written language domain, the numeral domain………..You see, it is a bit complicated for a little child, as it is for you maybe…..

disambiguation 1

My mum used to send me to the butchers. Sometimes she’d say “Go and get me some meat.” This was very upsetting for a shy young boy. Imagine all the women (in those days it was just women) in the butchers listening to me saying that and getting asked by the butcher, “Yes young man…..what precisely do you f****** mean?” Nightmare. My son once had a T shirt that read “Can’t you be a big more f****** precise?

You see, in the imaginary world of mathematics it is necessary, if you really want to get anywhere, to be quite precise. Or rather, totally precise. Exactly and unambiguously precise. Disambiguation means to make things absolutely precise and utterly clear. Then, you can be sure that Arthur Pilbeam, who lives on Mars, will know exactly what you mean when you ask him a question.

This problem of not being quite precise can get you into all kinds of trouble. The are many areas that need “sorting out”. One of them is to be absolutely clear about the particular area, or DOMAIN, in which one is operating. Please read on………

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.

emptiness 1 and 2

It’s all a game….maths is a game played in the mind. All of maths is perfectly pure, it has only an isomorphic relationship with the real world. Isomorphic means of similar form. The real world applications are reduced and imperfect versions of the pure ideas. Sometimes the ideas that can be thought of have no corollary in the ‘real’ world. An example of this is the square root of minus one. It is actually very useful in certain calculations and the results however certainly can be of practical significance.

What you see when you make a pattern of rods or put any rod down is up to you. If you are relatively unconditioned you are free to see more possibilities. This is an important attribute of the perceiver. Part of the teacher’s role is to allow this to be. To allow the perceiver to be free to perceive and then to conceive.