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.