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.



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


what makes a good adder?


2 + 2 

recognised the symbol 2

recognised the + sign

was aware that an operation is actually possible



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

it trips a picture of equal groups


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


you’d be crazy to count on eight



10 has inner structures- I suppose all numbers do

you have to know the commutative rule


12 + 3


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



3 + 14



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




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



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




I have an image of ten in two equal parts



6 + 17



6 and 7 look like 13 to me


there’s a lot of manipulating going off in here



that might not be obvious to little children

23 + 21

place value is fundamental






23 + 28







this scanning thing seems very important to me



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



the scanning causes “resonances” with past insights


56 + 29

I have to write the numbers down underneath each other





29 + 59




it’s still complements that make ten



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


yes, its a matter of deciphering meaning again




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




you’d probably use a calculator anyway



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






see this for the results




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…..


(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…






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’


ps if you keep saying ‘tables’ it sounds weird too…

dust – your first view of the number 6

Here is  a six rod, with a mat pattern beneath.


Each line of the pattern can only have rods of the same colour. Only ‘same-coloured’ rods that fit the 6 are allowed this time and for this reason:

These same- coloured rods are FACTORS of 6… (2 and 3 are factors of 6)

6 is called the PRODUCT

The towers to the right say ‘three twos’ and ‘two threes’. They are ‘crossed rods’ and we read them as multiplications. They are equivalent to the line of red rods and the line of green rods to the left.

They are DUST

They are FACTOR TOWERS in their simplest form

They show the ATOMS of 6, if you like…


6 can be re-constructed from this DUST in EVERY (multiplicative) WAY 


(in the multiplicative domain)