like learning a language…

Some methods, especially those that rely on memory and de-contextualised vocabulary and intense grammar are awful. They start with lists of words and complexities of grammar that generally mean something to those who have studied grammar in their own language but feel so fragmented and distant from speaking that one senses its going to be very hard going. The best (in my opinion) way of learning a language is through a live course using ‘The Silent Way’, which was created by Caleb Gattegno as he learnt how people learn. Some other methods however such as the Michel Thomas courses are very good for learning at home. I personally am learning arabic using this method and find in some ways it is like learning maths in the style of the subordination of teaching to learning. I could not find a course on arabic using ‘The Silent Way’.

One immediately feels useful learning is taking place and a certain hopefulness and confidence to continue takes over. It feels optimistic and fruitful. Arabic is a bit tricky because of the written language, the unfamiliar sounds and the unfamiliar words having in general no Latin roots. Most words are completely different to what an english speaker knows and the word order and other things are also unfamiliar.

OK so this sounds a bit like coming across mathematical ideas when you are little.

The knowing use of algebraic ideas in simple conversations and question and answer ‘sessions’ using the rods is similarly freeing and powerful. You can just see it so easily in the children’s eyes and actions. There must be no pressure at all. Certainly no pressure to write anything down, unless it comes from the children, and even then do not force them to ‘write it down correctly’. Do not ‘be a teacher’ in that sense. Bide your time.

Look at Caleb Gattegno’s  ‘Mathematics with Numbers in Colour’ Book 1, part II, ‘Qualitative Work with the Rods.’  ONLINE…

Don’t take it as a ‘course’, just read it through three times:

1. As if you were reading a newspaper

2. As if you were reading it out aloud to another person and

3. Try and fathom the general flow and gist of the chapter.

WARNING: DEFINITELY DO NOT USE IT RIGIDLY AS A ‘COURSE’

awareness 2 some examples in rod pre-number…

some algebra before arithmetic pre-number awarenesses

examples of free play awarenesses:

rods are good things to play with

we can make pictures

we can build models and buildings

we can play games

we can invent games

we can share our ideas

we can learn from each other

rod colours are in families

rods can be packed away in family colour order if we like

rods can be packed away in other ways

rods have a definite regular order of size

I can recognise rods by how they feel

rods can be named by colour and in other ways

directed informal awarenesses:

rods of same colour are same length

rods of same length are same colour

groups can be made of the same colour

groups can be made of the same lengths

language (word concept) awarenesses:

trains can be made in a variety of ways

rods can be swapped for others

 staircases can be made in a variety of ways

 mats can be made

mats can be made with rows of rods of the same colour

we can answer questions by making patterns of rods

awarenesses relating to adding and its inverse:

I can find one rod to fit two others

I can find two or more rods to fit one

I can find rods to fit long trains

I can find the difference between two rods in different ways

I can find the difference between two trains of any length

awarenesses relating to equivalence, multiplication and factors:

I can find one rod which fits two or more rods of the same colour

I can find rods of the same colour to fit some other longer rods

I can find rods of the same colour to fit some, but not all trains

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:

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:

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?’

the flip law will transform the learning of ‘tables’ if you insist

wait a little and I’ll show you how not to be concerned with 63 of the products you have to learn up to 100 so you’ll only have to become familiar with 37, but I hope you don’t just ‘learn your tables’ without thought, that’s antiquated..here’s two quick tips to be going on with when you are considering two factors to be multiplied together:

don’t learn both forms, learn one

learn the one in which the smallest number comes first

e.g. 6×8, 8×6

forget the 8×6

that’s halved the problem nicht wahr?

slight problem: you’ll really have to work on the flip law until it’s second nature

the flip law 2

it works for multiplication and addition but not subtraction and division.

in multiplication it works for fractions as operators too

play yourself…you must own flip it as second nature

ps the normal name for flip it is the commutative rule

pps dressing is not commutative – you don’t put your socks on over your shoes

the flip law 1

This rule is profound and will change many things, illustrated here with a few rods:

language and rod domain (with a bit of number):

two threes is just as long as three twos, and also they have the same volume

number domain with some signs:

2 x 3 = 3 x 2 = 6

al-jebr domain with signs:

a x b = b x a = c

more and more abstract, more and more general

disambiguation blurb: in the ‘real’ world, two green rods is not the same as three reds. This is why some people object to agreeing that 2 x 3 is the same as 3 x 2. They are correct. However, in the number domain, the answer, which is a pure number, is not affected by the order. The product as a number is invariant to the transformation. If you want another example, it is as invariant as taking a homotopy group functor on the category of topological spaces. You probably don’t need this information. As most people working on calculations are looking for ‘the answer’, one can say that for all intents and purposes, calculationally speaking,

the order of operations in multiplication is irrelevant

further more, if you wish to mention it, and which also makes no difference to the product, the number sentences, transformations or equations, whatever you want to call them, contain the sign ‘x’. This sign is called an operator and it has to be attached to something. It has to be attached either to the first numeral or the second in this case. If it is attached to the first numeral, like this ‘2x’, this ‘whole’ is again called an operator, in this case a ‘doubling’ operator. In language it says ‘two lots’ or ‘two groups’, so this, 2x  3 says

two lots of three or two threes

2x  3

with little children, the easiest and most meaningful form is by using this choice in the attaching of the operator

because:

a) just like in reading, one reads the first numeral first and

b) one doesn’t have to hold the first number in the mind to the same degree as the form below whilst reading the second number. (Saying ‘two threes’ seems somehow less complicated for little learning minds than saying ‘two multiplied by three’)

c) one doesn’t even have to mention the ‘x’ in language, merely recognise it ….two threes

nevertheless the product is still invariant to order

if the operator is attached to the 3, we get 2  x3 which says:

two multiplied by three

2  x3

most teachers call this the ‘correct’ way, but it is just one way

the x3 becomes a ‘trebling’ operator

so, in summary, and for the benefit of little children:

THE ORDER IS IRRELEVANT

and this 2 x 3 with the operator ‘x’ in the middle, at first means ‘two threes’ to little people

later, with much practice, it looks like ‘two threes and three twos’ at the same time

(ps you can introduce all the other ways of saying it whenever you feel it’s appropriate)

JUST UNDERSTAND WHAT’S GOING ON…

48

 

 

 

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.

 

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…

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…