stoner

“He found his release and fulfilment in the classes in which he himself was a student. There he was able to recapture the sense of discovery he had felt that first day, when Archer Sloane had spoken to him in class and he had, in an instant, become someone other than who he had been. As his mind engaged itself with its subject, as it grappled with the power of the literature he studied and tried to understand its nature, he was aware of a constant change within himself; and as he was aware of that, he moved outward from himself into the world which contained him, so that he knew that the poem of Milton’s that he read or the essay of Bacon’s or the drama of Ben Jonson’s changed the world which was its subject, and changed it because of its dependence upon it.”

‘Stoner’ by John Williams

what drives exploration and investigation?

jo2-124

 

Surprisingly, it is not the intellect.

The intellect may well be utilised as a tool but it is not the driver.

How do you lay yourself open to inspiration?

There is a big difference between simile and metaphor: If you are asked to dance, quietly feel the difference between ‘dance like a bird’ and ‘you are a bird’. If your presence can be filled with images , thoughts and feelings of you as a bird, and you forget the other story of yourself, you will dance well. As soon as you become aware of dualities, for example between what you are being and what others think you are doing it will fail.

When you are the bird you feel your feet clutching the branches, the wind in your face, the hardness of your beak. When you slip out of this into dancing like a bird, the duality reappears and you become fragmented as usual.

This non dual deep belonging is not a mental visualisation. All activity is transformed. Spontaneity appears. The person you were, or you normally think you are is not present. That story has vanished.

There is no one left who dances, there is only the dance.

The actor does not act the part. Only when the actor as person is absent does the other temporarily emerge. The other is the creative mathematician, the scientist, the artist, the writer.

That’s all I can say about becoming an explorer.

The main problem is that people do not believe they can explore, they do not believe they are capable, they believe the story of themselves that is their normal story which has unfortunately been constructed over the years by inadequate experiences.So treat yourself how you would wish to become.

Do not fear the judgement of others.

Forget ‘can’t’, ‘don’t want to’, ‘don’t like’………….

Be imaginative, be explorers.

Find things that have never been seen by you before.

Begin.

Find out something you didn’t know before about a square

no wonder adding up is tricky…

If you are asked to do an addition you are really being asked to

add the coefficients of a polynomial equation

Look , here’s what 326 really means:

polynomial1

3 is the coefficient of the second power, 2 is the coefficient of the first power and 6 is the coefficient of the zeroth power.

THAT’S WHY ITS HARD FOR LITTLE KIDS TO UNDERSTAND AT FIRST

huatou: why polynomial?

good adder results

Here’s a list of the main ideas that are useful in becoming ‘a good adder’ generated from the previous post:

interpreting the meaning of numerals and operation symbols

to know the complements to and from 10 and to and from 5

immediate recognition of number pairs as complements 

the transference of the above to all other decades and patterns

familiarity with the inner structure of numbers

scanning problems first as a  fundamental activity

awareness of equivalent transformations after the scanning process

the power of  visual images of relationships and processes within memory

total awareness of the commutative rule 

total awareness of our number system including place value

transformation is OK and is allowed

generation of own recording styles

qed

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?’

chronic boredom and irrelevance

please excuse this rant

a diatribe concerning the captive consumers of our woe, the children

They are the future. We create the future through them. They do not choose to be there. They do not choose to study the available curriculum. It has been chosen for them. They do not choose the structures of schooling including the structure of the day, with whom they are taught, their timetables nor the time for which individual subjects are taught. They do not choose just what is to be taught from the infinities of possibilities, nor why. They do not choose to divide knowledge into the separate subjects. They do not choose their teachers, the teachers’ personalities and the teachers’ particular teaching styles and the theories of learning on which they are (presumably) based. They rarely choose their own work. They are marked and assessed according to criteria they do not choose, and are categorized accordingly.  They are embedded in a system, a technological machine, which I imagine they have to tacitly assume is the best possible type of machine that society can construct, through much deep thought, which will enable them to live happy fulfilled lives and fully prepare them for the future. They accept it because it is the way things are. The status quo. The embedding in this prevailing school machine technology has inevitable consequences. Many like it. After all it is a vast social flux, they make friends, often for life, they meet people all the time. It is busy and involving. No one likes their work all the time after all, unless they are very lucky, and for much of the time, it is ‘OK’. It can lead to qualifications which are basically passports to further qualifications and restricted better jobs, even vocations and ‘professional’ activities and lifestyles. Children see this and it causes a certain degree of concentration (which is a kind of motivation imitating intrinsic motivation), even great concentration in some who can achieve excellent results in their exams. These achieve because they are born with the capacities that enable them to relatively easily fit in. It ‘suits’ them, as it ‘suited’ me up to a point. Everyone is pleased and their self-esteem becomes high. It is a good feeling. All you have to do is work hard and learn things and do well in exams, though exams are ‘not what anyone in their right mind would actually choose’. Still, they get on with it. If they make it to A levels or university, similar structures and processes prevail. If they decide to work hard and play hard, they generally have a good time. After all, they have the advantage of choosing (to some extent) their general areas of study. Sometimes, even specifics.  For others the story is not quite so rosy. They perceive, early on that the work is hard. That it is not very interesting. They look around and see others doing it easily and being rewarded in various ways. They get rewarded too, but as they get older they cannot help comparing themselves to those who will obviously ‘do better’ in the long run. As the work is ‘hard’ to understand and hence it is not easy to ‘do well’ it begins to be chronically wearing. Sooner or later most students experience this. Even for those who have inclinations and innate abilities praised and needed by the system, the relentless learning of new things, often not seeming to have much if any direct relevance to them and their needs, except for the need to pass exams, can cause chronic weariness can set in. It happens at university too. “Mum, why can it seem so irrelevant and dull? Why does it have to be done, apart from the exam angle? Why is it all considered to be necessary? Why is it all compulsory?  What is it all actually for?” Sometimes, they have ‘learning difficulties’. If they are told, or perceive that they are in fact in a different ‘category’ to most, they might easily come to ‘believe it’ hence totally altering their view of themselves, often for life. They might have some real special differences that are obvious, such as for instance, blindness, deafness or cerebral palsy which undoubtedly provide challenges to themselves and the providers of experience, their teachers. Sometimes, they have ‘learning difficulties’ that are less tangible, but which are still ‘seen’ to cause ‘difficulties’ for the ‘normal’ processes of the technological machine system. It is rarely seen as a product of the system itself. The system is generally not sensitively reflexive enough for that. It is the fault of the others – parents, society in general, the children in particular. Sometimes through individual teachers’ enthusiasms, and sometimes because the children like the teachers as people, students can get carried along down even very abstract and esoteric routes and actually enjoy the processes and products of learning along the way getting satisfaction through the sense of a job well done. There is nothing wrong with that, but it doesn’t always happen. In fact, it usually doesn’t happen. Some work is just too dull, or presented in too dull a style, in a flat, matter of fact, monotonous manner, ‘because its good for you’. It can be perceived as irrelevant by the students. Some teachers just cannot hold it all together. Other issues can emerge. ‘Disruption’ of the ‘lesson’ can easily occur due to others who have already given up the idea of pursuing the lessons’ aims and deciding to ‘have a bit of fun’ instead. It is true that it takes only one very disruptive person to destroy a teaching environment. Almost nothing can productively proceed in an atmosphere of noise and confusion. Threats and rewards can be issued and will ‘work’ with some at least temporarily. Machine gun nests work a treat. The lid can be kept on through externally imposed discipline. The repressive communist regime of Tito’s Yugoslavia managed this quite well. More ‘discipline’, including clearly visible, increasingly painful sanctions, parent/carer contracts and now quite common well established systems of ‘assertive discipline’ are acclaimed, and can ‘hold it all together’. Guilt and fear also work a treat. ‘Order’, real and apparent, is re-establishable. The normal processes of the status quo school technological machine are then re-established and proceed to the apparent satisfaction and indeed praise of all, often including the students. Certainly to most staff, the general population and government, who are seen to be improving standards. Indeed, at one level they are. Life is once again a bed of roses……but is it? Symptoms have been suppressed. What about the causes? Who bothers digging here? The causes are more insidious, more chronic, more pathological. They are potential viruses threatening the system itself. Leave them alone. In order to dig where it is dark one requires an active conscience, an unflagging, instinctual need for the development of self and professional knowledge. The digger requires courage, because what one might find in the dug hole might backfire on ones own beliefs and inner, tacitly held convictions. The digger might archaeologically expose himself and be taken away as a deviant. That is what societies do to maintain the deeply caused and little sensed status quo. It creates ‘objects of deviance’ which are cast out. Suck on these ideas and see for yourselves if there may be any substance in them. Do any cause resonances within you about teaching and learning. I am just one agent in this flux, self-constructed by my labours and experiences. Chronic boredom and irrelevance infect present educational systems, at all levels, but especially for the those compelled to be captive audiences. Dig where it is dark. Most people aren’t interested in this esoteric activity, because not only is it difficult, it is disturbing. It is also, transformative. Who wants that?

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