That’s enough about domains for now. You can see what I’m talking about. Now its time for you to explore this idea yourself. The next time you start ‘doing’ some maths consider which domain(s) you may be working in and try to think of anything that could be thought about…….that’s all
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…
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.
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…..