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’

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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 1

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

SONY DSC

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…

dave eats a 1 rod and britney spears

davesniffsrods

That’s a bit annoying Dave… we can’t see all the complements to 10 now, we’ll have to imagine that one above the 9, as well as the missing one above the 5…

SONY DSC

what can you see?

complements to 10 and the sum of all the numbers to 10

a nice rectangle 5 x10 + 5 = 55

the old devil al-jebr would see:

 halfnaddn

a half of any number multiplied by the same number plus a half of that particular number is;

the sum of all numbers up to that number.

so, add all the numbers up to 100:  50 x 100  + 50  makes 5050   nice…

the power of al-jebr

by the way, here’s the same infamous Dave eating Britney:

video

( thanks to MMc and J )

video: add up all the numbers from 1 to a million in your head

cuisenaire rods the way of zen 130 add from 1 to a million in your head

the power of PATTERN PATTERN PATTERN PATTERN PATTERN

I said:

PATTERN

The music is me singing a long version of ‘Dance to your Daddy’ I constructed