observing oneself…

splat-image

I wanted to do a ‘piece of research’ in which I attempted to ‘observe myself doing it.’ I have been interested in patterns in nature for many a year and thought I would drop ink droplets onto paper to see what patterns might emerge, if any. The following are 3 grabbed images of this research. I have used images here because it is necessary to maintain the complex formatting: read the first section, upper left. This shows how the original tape recording was transcribed and then thought about and re-thought about so to speak…

splat-1

a further piece some way in:

splat2

When I had finished the whole experiment I made a series of knowledge claims which were quite extensive. Here’s a small example:

splat-3

Well, that’s what I did…

challenge: observe yourself making a cup of tea

(or similar)

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gattegno buddhism and critical theory..ah well…

Gattegno maintained there were 4 stages in learning:

1. there is something to be explored

2. the exploration of this ‘field’

3. practice in this field leading to mastery

4. transfer of this knowledge to wider fields

The Four Noble Truths of Gotama Buddha, the cattari ariyasaccani are:

1. an unsatisfactory state exists

2. how this ‘suffering’ arises

3. how to end this ‘suffering’

4. the means by which this ‘suffering’ can be eliminated

In his book ‘The Idea of a Critical Theory’, Raymond Geuss describes a typical critical theory as being ‘ a transition from an initial state of bondage, delusion and frustration to a final state of freedom, knowledge and satisfaction.’ Further, he defines a typical critical theory as consisting of three parts:

1. the belief that a transition from the initial state to the final state is in fact possible

2. the demonstration that this is in fact necessary

3. this transition can come about if one follows the particular tenets of that critical theory.

Well, at least, Gattegno, the Buddha and Raymond Geuss have in common the idea that…

‘EMANCIPATION IS POSSIBLE’

but it is clearly not going to be a push-over

huatou: is it possible to change?

some important people…

Caleb Gattegno

Most of his works are available to read on line for free here:

calebgattegno.org

Roslyn Young – a modern exponent of Gattegno’s ideas

try this: On awareness and awarenesses

Read this book, co-authored with Piers Messum,

‘How We Learn and How We Should be Taught’,

Vol 1: Duo Flumina Ltd, 2011, ISBN 978 0 9568755 0 1

Caroline Ainsworth

An example of a ‘research process in all its complexity’

a case study of one teacher’s professional development journey

Madeleine Goutard

the wonderful exponent of STL*

Talks for Primary School Teachers

Educational Explorers Company, 1963

Mathematics and Children – a reappraisal of our attitude

Same series.

*The Subordination of Teaching to Learning – begin here

the ultimate zen assessment – pseudo object oriented reality…

Warning – the analogy to object oriented programming is FALSE

(if you want to see the point of this post go here)

Everything you and I perceive takes place in the mind. Your senses provide information to your mind. Your mind becomes aware of what you call reality due to information supplied by your senses which is then subsequently processed according to your established interpretative neural networks. These networks are evolved through awareness and analysis of these perceived conditions during your evolution as a perceiving, conceiving being… i.e. whatever you have experienced and then ‘made of’ these experiences.

Clearly, this involves your total historical and psychic environment. It is then clear that ‘this certain something’ that ‘you’ perceived was an interpretation of an interpretation, and was only an image of ‘the real thing’ whatever that was. This certain ‘real thing’ cannot be known absolutely. That is why great masters such as HUANG PO pointed out the ‘error’ of conceptual thought processes:

“There is no  “self”, no “other”. There is no “wrong desire,” no “anger,” no “love,” no “victory,” no “failure.” Only renounce the error of conceptual thought processes and your nature will exhibit its pristine purity-for this alone is the way to enlightenment.” HUANG PO, Wan Ling Record 24, p.86.

This ‘error’, is merely the knowing that what is perceived as ‘the truth, the Absolute Real Reality,’ even in its brute external form, as Searle would describe it, is NOT IT ITSELF. It is only ‘one interpretation of it’, and this is all we CAN KNOW. We cannot know ‘IT ITSELF’, because for us there is no ‘it itself.’ All we can know is what we perceive and then conceive through interpretation. This is relative reality and is different for all beings. This is ‘our world’… This is ‘my world’, this is ‘your world.’ This is why it is said that we ‘create the world.’ This is why there are as many cities you live in as there are perceivers of the city. There is not ‘a city’. There are no unique events.

Clearly there are ‘events’ at some level. There are earthquakes, there are floods. There are divorces. There is love. You will be hurt by the master’s stick. Yet you are the perceiver. You ‘create’ your specific take.  You create ‘your’ world. The external, the unknowable, is the CLASS which is knowable to us only as a fragment or ‘taste’. The specific, our individual  realities, are the INSTANCES.

Welcome to OBJECT ORIENTED REALITY…

huatou: I am the world

what makes a good adder?

When I was professor of wooden sticks at the university of Pokelsaltz I did a little experiment with some students. I gave them some little ‘sums’ and asked them to do them, taking note of how they did them and what actually they did and what thoughts occurred to them…

2 + 2 

recognised the symbol 2

recognised the + sign

was aware that an operation is actually possible

an awareness that these marks carry  meaning but that the marks are an immense abstraction from the external reality that was their origin

immediate knowledge of the answer – did nothing – just knew it was 4

didn’t scan it – didn’t read it – just knew

it trips a picture of equal groups

it looks like a symmetrical pattern to me – two halves of four

2 + 8

pairs that make ten – complements

didn’t do anything- just knew

recognised these as a way of making 10

it just looks like ten to me

I saw it and my eyes gravitated to the 8. I knew it was ten

you’d be crazy to count on eight

I think I’d do it backwards if I didn’t know it made 10

I have an image of 10 being a lot of other things – 10 is the destination – it isn’t one thing

10 has inner structures- I suppose all numbers do

you have to know the commutative rule

if you’re little it would be best if you had the confidence to scan it first rather than do it bit by bit from the 2. You’d save yourself a lot of trouble

12 + 3

I know that I can do the 2 plus 3 bit and basically that’s it  but I imagine that’s not obvious to little children

I see 5 and I see it in the teens

I have a picture of it living above 10 but before 20

I just know the answer

you could count on from 12

the 1 feels completely separate from the 2 – I know it s to do with place value – some people might not have that feeling

when I see “12” I see 10 and 2 – 12 doesn’t look like “one squiggle”, one thing. I see it as something to do with ten and some more

3 + 14

I know I have to add the 3 and the 4 and I know straight away that that’s 7

the 3 and the 4 belong to the same family – they live in the same room

I know you can ignore the 1,  because its different

you have to know the commutative rule or you’re in trouble

it’s best to scan the whole thing again first

it looks worse than the previous one – there the 2 and the 3 were together, now they’re separated by a 1 and the 1 is not really a 1, it’s a ten

place value is really important  but I bet it’s not easy to see it and understand it at first

we’re a long way away from a beginner aren’t  we? I suppose it’s like that with reading

In a way this is reading for meaning isn’t it?

15 + 6

I can see five’s in there

There’s two five’s and a spare 1

In my head I can hold the  first ten, see two five’s making another one, so that’s twenty, then add on the 1 to make 21

I see fifteen, a five and a one, so I know its 21

it’s a kind of making up to twenty

you have to know the five family

you have to know the way that five’s march on

when you look at the 15 it sort of sets your mind into knowing you’re looking for 5 more and so when you perceive the six, you see the 5 and a 1 in it

its to do with knowing complements again but its complements of five this time

I imagined before we did this that only complements of ten would be useful

I have an image of ten in two equal parts

I sort of see five’s and zeros going on and on and I feel the importance of the 10, 20, 30, 40 series is important somehow – they’re like barriers or boundaries

I see it as a sort of hole filling exercise – from the six I see 5 of them “falling” on top of the 15 and filling it up to 20, and then the one is all alone

6 + 17

I don’t like 6 add 7, I like 7 add 6, perhaps because 7 is nearer a ten boundary ?

it doesn’t matter which one you do – that’s the commutative rule again

6 and 7 look like 13 to me

I make the 17 up to twenty by using 3 of the six, then the 3 left  goes on top of the twenty

there’s a lot of manipulating going off in here

first  you scan it, you make a decision to make the 17 up to 20, you decide you need 3, you look at the six and take three off it, then you can forget about the first 3 that you’ve just taken off, but  you have to hold the 3 that’s left in your mind  and stick it onto the 20 you’ve just built – wow, there’s a lot of things in there – a lot of decisions, memory and pictures

useful complements to boundary  numbers like 20 and 60 are  the same as the complements to 10

that might not be obvious to little children

23 + 21

place value is fundamental

you have to see the 2’s being in a different sort of group to the 3 and the 1

I scanned it and then I felt like adding 20 on to the 23, which made 40, then I added the last 1 on to make 44

I sort of added the two sets of numbers simultaneously but my mind had sorted them out into the place value set, that’s the twos, and the “bits left over” set which is the  3 and the 1

so there were two parallel columns in my mind even though its written out horizontally

by just scanning it I know that this is an “easy” sort of problem because neither sets of numbers are going to cross a tens boundary so I know I haven’t got to hold much in my head

23 + 28

this looks worse than the last one because I know I’ve got to cross a boundary

I see 8 and 3 make 11, then I  added 40 to get 51

really there’s a lot in here again if you take it all to pieces

I scanned it and I suppose I was comparing what I perceived with my memory banks of tricks  and concepts trying to match something up.  I recognised the 3 and the 8 as being 11, though I think I actually “saw” 8 and 3, not 3 and 8. I knew that I could keep certain groups of numbers separate, the place value thing. Then I was floundering a bit because it didn’t come to me immediately to add on 40, there was a delay, and I was a bit worried about finding the path to the answer. I nearly added on 2 to the 11 and then two more, to make 15  but I knew this was wrong. Still, I nearly did it and the image of 15 momentarily popped into my mind. I was quite shocked

I noticed that though there’s quite a lot of separate things going on before we get to the answer, we only seem to be able to do them two at a time

the way we do things can be sort of binary and linear, but in other ways what we do is like “deal in wholes”

this scanning thing seems very important to me

I think the scanning process is connected with wholeness and a kind of “holistic  sensing” and this “informs” the analytic mind of “the way” to follow to get the answer. This “way” is then put into practice using the analytic binary “mind” that begins operating on the figures or images of the figures

sometimes though, one “sees” the answer immediately without seeming to have “done anything” and even when the “binary mind” is set  working, the “sub answers” still might “pop in” holistically

so I suppose the analytic and the holistic work together in there

some people seem to have a good “feel” for number and operations with numbers

perhaps this is connected with this “holistic sensing, scanning and tapping in to the body of previous experience and insight”

the scanning causes “resonances” with past insights

so in order to become good at adding we should make provision for them to have rich experiences and insights of all these matters we have been talking about – at least

56 + 29

I have to write the numbers down underneath each other

I scanned it again and decided to add 30 to the 56, which was easy and then take 1 away to get 85

I looked at it and just knew that 9 and 6 were 15, then I added 70

I took one off the 6 and gave it to the 29 to make it 30, keeping the 55 in my head, then I knew the answer was 85

I  saw 70, another 10  and a 5, but if you take that to pieces I suppose it was quite complicated really

29 + 59

well, when I see this I can see that its best to call it 30 and 60 and then take 2 off

some people might think that’s not “fair” but it gets the right answer

if you’re taking things off from a ten boundary then that’s complements again isn’t it, but this time you’re coming down rather than going up

it’s still complements that make ten

you’d have to know that if you add something on to change a number to an “easier” one, you’ve got to take whatever it was off again at the end

I suppose that might not be obvious to some children, or at least it might need a lot of practise

it’s an abstraction isn’t it ?

123 + 148

it’s not really any harder if you know all about place value

all the same kinds of thoughts apply

I suppose you really need to know that the 1’s are actually standing for 100’s

yes, its a matter of deciphering meaning again

there are more things to hold in your mind as the numbers get bigger, so I suppose people feel more of a need to write things down

yes, I suppose the individual reaches a point when it  becomes necessary  or at least useful to start writing things down, though I suppose some people can hold more things in their minds than others

I wonder if this is innate or whether it can be improved by practise ?

is there a best way  of writing the working out down?

well, no, because if we help them to scan “sums” and help them to have a good “feel” for number, this implies a variety of ways might be appropriate, not  just one way

I know but I was taught to do adding up in one particular way and it works for me

well yes it might work, but is that all maths means to you then, learning a few techniques so you can use these skills as tool s in other areas like science, or for doing your tax returns?

you’d probably use a calculator anyway

maths can be seen as a creative  vehicle  in its own right can’t it, like painting?

why can’t you see that individuality can be allowed a place in there?

most people don’t like maths and it’s not surprising is it?

it’s our job to try and let people see it as something good to do in its own right

if we want to be good teachers its no good just re-inforcing our own programming is it?

what was good enough for me is not good enough for the children I want to teach

if we work at it and try to see all the issues inside even the simplest looking thing like

2 + 2, and so on like we have been doing, and then provide loads of rich experiences so they can practise the things we think are important, and so get the children to have lots of success and so on at their own level, then that’s got to be good for the children and good for the image of maths hasn’t it?

see this for the results

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?’