48

 

 

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…..

IF YOU HAVE THE TIME AND THE SPACE IN SCHOOL TO DO THIS TILL THE COWS COME HOME AND YOU ARE LITTLE, AND YOU START SLOWLY WITH THE NUMBERS UP TO 10 AT FIRST, STUDYING THE NUMBERS ONE BY ONE FOR A DAY OR TWO EACH FOR EXAMPLE WITHOUT STRESS, YOU WILL ‘GET A FEEL’ FOR THE NUMBER YOU ARE STUDYING WHICH WILL BE VERY POWERFUL IN YOUR FUTURE STUDIES OF THE NUMBER SYSTEM AND OPERATIONS YOU WILL NO DOUBT BE REQUESTED TO CARRY OUT…

(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…

TRY IT

 

 

 

 

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dust lies on top of tables…

SONY DSC

Here’s 8 with its factors: ‘two fours’ and ‘four twos’ which you see to the right.

Remember if you can find rods of the same colour which are the same length as another rod, as in the picture to the left, they are called factors of that number.

At the extreme right is the DUST of 8, ‘two times two times two’, 2x2x2

This is the ATOMIC STRUCTURE OF 8 in terms of multiplication.

Why is it useful and very very good indeed?

Because from the dust, 2x2x2, you can, if you feel like it:

Build ALL combinations of factors of a product

THIS BEATS ‘tables’

DUST EATS ‘tables’

DUST IS ABOVE ‘tables’

DUST BEATS ‘tables’

DUST LIES ON TOP OF ‘tables’ AS WE KNOW ONLY TOO WELL!

ps if you keep saying ‘tables’ it sounds weird too…

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

THE ANIMAL CATEGORY BELONGS TO A HIGHER LEVEL, it COVERS those beneath like this:

coverset1

‘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…..fun 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.

UNSATISFACTORY

This is the DARK SIDE of WORLD CREATION

There are better ways…

the content of every curriculum is relative and loaded

Awareness and attitude beat content. More later…

There are an infinite number of lists of content. Every curriculum that has ever existed or will ever exist or any that can be thought about in the future will have different sets of content. As now, when the government sees fit to allow those with no real knowledge or insight into education and learning to control what happens with content (and process), these lists will be enormous. What’s more they will be created by the prevailing mental structures of the beings involved.

A dog would create content relating to doggish items such as food, smell, territory etc.

Those who have been exposed to certain conditionings will often, though not necessarily, include aspects of these in their ‘curriculae’. George Monbiot recently wrote about this aspect. Check out his other writings on the reasons behind the present ‘revisions’…George Monbiot…’A Capitalist Command Economy’.

Gradgrind, from Dickens’ ‘Hard Times’ was keen on facts, facts, facts…

The half truth here is that of course everyone needs facts to operate in the world, but one needs much more than facts to be a creative, engaging, imaginative, happy productive human being.

The present ‘Revised National Curriculum’ for England and Wales is an example of ‘the Gradgrind Way’ created by those who do not understand but have the power to enforce. There are more sinister aspects to this however including the forcing or bribing of state schools to become so called academies by making them attempt to achieve the dull impossible. Then ultimately the profitable ones can be dealt out between the ‘unelected oligarchs’. This is truly a crime.

This revised curriculum is 1000 miles wide and 1 inch deep.

Every CURRICULUM contains CONTENT that is RELATIVE and LOADED.

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.

I CANNOT STRESS ENOUGH THAT IF YOU CHOOSE THIS CUI ZEN WAY OVER THE TRA WAY OF PERCEIVING NUMBERS YOU WILL ENTER ANOTHER RICHER, FAR MORE MULTI-DIMENSIONAL, CREATIVE WORLD AND WHAT’S MORE, YOU WILL HAVE NO FEAR.

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.