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…

disambiguation 4 – domain specific signs or not?

What signs to use in specific domains is problematic. For example, the addition sign, the ‘plus’ sign originated in the number domain, the arithmetic domain. If you are going to use numerals as signs, the appropriate signs that indicate ‘operations’ are the familiar ones:  +  –   X   and ‘divide’ which is generally not easily accessed in a normal font set, as in this case.

The issue is that if at the moment we are in the ROD MARK MAKING DOMAIN and manipulating real rods in order to see things which we then want to ‘write down’, should we have for example a special sign which means, ‘put the rods end to end’? It would be easy to say ‘What’s the problem, just go ahead and use the normal signs.’ I have done this, and to be honest I am not sure that this is a bad idea. It seems easy and obvious, mainly because people in general including teachers don’t know of any other domain-specific signs. (THESE DO EXIST HOWEVER)

It can certainly be construed as being ambiguous, particularly by some modern rod proponents and almost undoubtedly philosophers of maths. They say that DOMAINS should keep their MARKS specific and separate.  What to do, what to do?

More of this later…

making alphabet marks on flat things

The ROD written language MARK-MAKING DOMAIN is another. It is different to the rod language (speech) domain because now marks that represent rods are going to be made on paper (or at least on flat things such as paper or computer screens, white boards etc.) Rods might be drawn and coloured in previously, and then real letters can be used if the children know how to form at least some of them. STRESS: you do not have to move to this particular abstract domain IN ANY HURRY, quite the opposite.

Caleb Gattegno used the following letters to represent the Cuisenaire rods:

w r g p y d b t B o

White, red, green, purple, yellow, dark green, black, tan, blue and orange.

Others have used other letters. If you want to use Gattegno’s books, it might be best to stick to these, but you don’t have too, but you do need to be consistent. As a slight aside, it is useful to provide help in forming the letters properly, because here like everywhere else, a little good direct tuition is very useful. For example, form all those letters starting at the top. This goes for the great majority of the letter and number formation in the English language. Bad habits are persistently difficult to break.

You notice that black, brown and blue start with the same letter, as do green and (dark) green. Hence the alternatives. You get a tan if you stay in the sun for a while. They mostly see that straight away, and they accept that blue could be represented by a capital letter to avoid confusion. Familiarity soon takes over. Colour blind children can apparently distinguish the colours by shades, but I have never come across a totally colour-blind person.

This ROD written language MARK-MAKING DOMAIN is where ‘other’ SIGNS might be met. So long as the child totally understands the signs all might be well, but THIS WILL NOT BE THE CASE AT FIRST. However, even if you decide to make ‘mathematical’ signs as well as the written alphabet signs there are issues you should be aware of…….for example, do you think that signs are ‘domain specific’ or not? if you claim to be a master, you should consider this.

disambiguation 3 – more on domains

OK, so we can see straight away that if we take a general overview of this domain lark, there is an obvious ROD DOMAIN, an ARITHMETIC DOMAIN and an ALGEBRAIC DOMAIN. Each of these have associated ‘mark making’ activities on flat pieces of paper. So, at first glance, there are THREE main domains where we can be active plus their associated mark-making activities. Not surprising children (and adults) get confused is it? Inside the ROD DOMAIN we have various sub domains as described previously. Same with the others. Maybe we are getting a bit too pernickety with these domains. Maybe we’re getting a bit too philosophically mathematical. On the other hand, maybe not.

Looking a bit more carefully at the rod domain, you can see that there is a PURE ROD DOMAIN where there is no mark-making on paper (unless the children decide to of their own accord and in their own way). This is where children can play. When I say play, I mean, totally pure, unadulterated play with no adult interference at all. You, as a teacher are not attempting to get them to do anything or see anything in particular, just let them play.

You might decide it is time to start saying something. In this case you will have entered the ROD SPEECH DOMAIN. You can start saying things like, “I bet you can’t make a chrono-synclastic-infundibulum?”, and so on…  (Kurt Vonnegut-The Sirens of Titan. A CSI is a ‘kind of wormhole through space and time where all kinds of truth fit together….” Just what we really need, nicht war?

You might decide to say, “Can you show me half of an orange rod?” Or, “Which one looks most like a strawberry?” Or, “Could you make a monkey using one rod of each colour?”

An awful lot of very useful ideas which are purely algebraic can be accessed in this rod speech domain totally without writing anything down. In fact practically everything that is useful in arithmetic can be first seen here. THIS ALONE IS ABSOLUTELY AMAZING, JUST THINK ABOUT IT. MOST IF NOT ALL OF THE USEFUL ALGEBRA CAN BE ACCESSED IN THIS ROD SPEECH DOMAIN……………

The things seen don’t have to be perfectly understood before moving across to other domains, it just means that when they are seen in other domains there will be some memory trace of the ideas already present at least in the subconscious, maybe more. In addition it must be said that it is a VERY GOOD THING to make a very strong decision to do this TILL THE COWS COME HOME. I.e. do a great deal of it. Do not be in a hurry. This is another big issue that becomes an immense problem. If you move too rapidly to other domains there will be trouble. On the other hand, activities can proceed in other domains at the same time, but only if the child is ready… This is an area where skillful teaching decisions have to be made.

disambiguation 2 the ‘rod’ domain and the ‘marks on 2D paper’ domain

If you are keen to work with Cuisenaire rods and arithmetic, you should be aware that you will be working in several different areas, or domains. If you are using real rods, this is the ROD DOMAIN. It exist in the familiar world of 3D objects all around us. What we are going to be looking at are “Parallel worlds”, or “Similar worlds”. We are going to gradually (I can’t stress that word enough) move over into other worlds, which are similar, but not the same. Same means, “another one or an identical thingy”.

Similar means “somewhat like….” So we start with the rod domain.

We might start making marks on paper. This is not the rod domain, we have switched over to the flat 2D world of “marks on paper”. This is a flat, more abstract world where we make marks that represent real things. You can make marks of different natures. You could make ‘drawing’ marks, ‘letter’ marks or even, ‘numeral’ marks. These each have different characteristics. Imagine you are a child. They are not real things in the sense of the real 3D things you are used to. Your teachers might think they are a bit removed. You might think it seems a lot removed from what you are used to and are familiar with. In fact if you are not really ready to operate in this world, it will become a very confusing world, a world where you can feel inadequate, stupid and incapable of seeing precisely what you are being asked. You will have entered a FEARFUL SPACE OF HORROR. You will not want to come here again. You will dread it and just like a snail who’s touched something it doesn’t like, you will RETRACT YOUR FEELERS. Remember this. It’s not where I meant to go, but as it is so important I went there to emphasise it anyway. I was becoming paragraphically ambiguous. However, if the above does happen, IT IS A CRIME AGAINST HUMANITY, really, a criminal offence that should be punished. I’m not joking. It happens all time because teachers in particular either don’t know what they are doing and/or don’t have enough time to do this domain switching properly and gradually. Lets start again with this domain disambiguation……already we have seen, the pure rod domain, the rod plus speech domain, the drawings on paper domain, the written language domain, the numeral domain………..You see, it is a bit complicated for a little child, as it is for you maybe…..