refutation of previous post…bugger

An instance of a class is maybe too like the class itself and each instance has the same behaviour other than by variation in its member variables. A sub-class in an OO hierarchy takes the behaviour of its parent and adds characteristics. These additional layers are maybe what our own selves add to what it really is. So this analogy is better:

Traversing the class hierarchy back to the root class

with many thanks to Myrthis Lathrop Jnr.

huatou: what is my root class?

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one last time concerning ‘tables’ with a little question at the end…

I can’t do any more of this ‘tables’ stuff, but I am asked about the issue so much…

Look, here’s a ‘tables’ square, slightly adjusted:

tablessquare1

I’ve removed the 1x and 10x sectors. 1x is trivial, 10x needs special treatment.

If you fully appreciate the flip rule you can forget the grey airbrushed section.

The square numbers in the blue-green squares are a special beautiful group, well worth studying. Many patterns and much al-jebr live here…

The rest are in 8 columns, from 1 to 8. Add up the numbers from 1 to 8 in your head and it’s 36. [I did 8×9 and halved it).

The orange numbers can be found by doubling from single digit numbers. (Conversely by halving from the products).

The blue by treblings from single digit numbers.

There are 4 spaces left, 5×6, 5×7, 6×7, 6×9.

double 15, the two primes, 5 and 7 make 35, double 21 and double 27.

Study George Cuisenaire’s original product wall chart:

Product-Wallchart

You can see all the doublings.  Look carefully.

These products, the numbers in black, form

MILESTONES in the unknown territory to 100

By studying them as numbers, as we have previously discussed, their inner structures will become apparent, thus lessening the awful stress on memory so prevalent in today’s schools.

In addition by briefly studying the Primes to 100 it will become apparent to the little yellow belts that many numbers are rich in factors and many are not. They will generate a ‘feel’ for the rich numbers, learn their inner structures by familiarity and learn, as an aside, the so called ‘tables’ relevant to that number.

This however takes

A CHANGE OF PROGRAMMING

even in the minds of  teachers, never mind the administrators

THIS IS PROBLEMATIC…

because even you and I dear reader

IMAGINE OURSELVES TO BE FREE

huatou: ‘Am I free?’

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

 

 

 

 

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.