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?

Advertisements

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

 

 

 

 

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…

current material factors inhibiting robust learning

There is, in spite of everything, powerful learning proceeding within some classrooms, but the prevailing forces of coercion and shallow curriculae inhibit it intensely and prevent renewal.

The material factors inhibiting it, to be more precise include:

A shallow, extremely wide curriculum not emphasizing robust learning in depth

Fragmented mechanical coercive organisation of the teaching day

Fragmented subject areas containing fragmented progressions of ideas

Not enough time available for in depth exploration due to inappropriate immense demands of the shallow curriculum structure

Teachers having to teach for national summative testing

The dismantling of in-service and primary teacher education services for ideological reasons