Structure, Cuisenaire and Gattegno
We are going to take a break from our scheduled post on Gattegno Mathematics Textbook 1 for a small detour into how Gattegno uses Cuisenaire Rods to assist a student with understanding the structure of math.
In the last post, I mentioned that the value of making patterns for the green rod is not only that the child acquires the fact: 1 + 2 = 3. Yes, we want the child to acquire that fact. But we're also interested in developing the student's intuition and knowledge about the structure of mathematics. What did I mean by that?
The Structure of Math vs. A Formal Math Structure
There is a technical definition for a math structure. While I didn't mean that exactly, the ideas are related, but it's too much for this post to cover. Madeliene Goutard mentions math structures in her book Mathematics and Children. She doesn't offer a definition; she just assumes you know. That isn't helpful to the parent, but I'm not sure it's important that we wrap our minds around it right now anyway. Just know that something called a math structure exists and Gattegno has the study of math structures in mind as he leads you through his curriculum.
The whole idea of the structure of math (again, different than a formal math structure, but related) has been eating it me since one of my FB Group members asked why we don't mark the units on the rods. I've been looking for a way to talk about this and answer a lot of related questions at once. This proved way more challenging than I had imagined. I've written and rewritten this blog post so many times I've lost count.
Finally, I pulled out my trusty American Heritage Dictionary and looked up structure; it was only mildly helpful. Webster and Oxford were next: still not good. Google came to the rescue with an appropriate definition:
structure: the arrangement of and relations between the parts or elements of something complex
When I was talking about structure, I meant the entire body of arithmetic rules and concepts and how they are related to one another; in other words algebra. Usually, we wait until the late middle school years or high school before we generalize arithmetic. In chapter 3, we spent a lot of time working with letters, which is what most people think of when they remember their algebra experience. But as we continue to move ahead, Gattegno is still developing the student's understanding of the concepts and rules behind all of arithmetic. His focus is always on understanding the structure that holds all of this stuff together.
Structure And The White Rod
Below, we have the partitions for the light green rod. If the white rod = 1 then we have 1 + 2 = 3. We would also have 1 + 1 + 1 = 3 and 3 + 0 = 3. But what if we change this up and we say that w = any natural number. Now we have a situation where any number w added to twice w is equivalent to 3w. With this understanding, we can begin to understand the relationships between 1 and the other two numbers 2 and 3, and how those relationships are related to other groups of numbers that fit the same pattern. We could rewrite the following in terms of difference (subtraction), but I want to stick with the addition for right now.
We can notice every other sum is even and those sums can be divided in half. Every 4th sum can be divided into fourths and every 5th sum into fifths. But check this out: every other one can be divided into sixths but every 7th into sevenths. How curious is that? And why does that happen? Can we figure that out? That question isn't for your pre-k/k student to find, but it is certainly for a second grader.
Changing The Reference Rod
Above we wrote our statements in terms of the relationships to the white rod. What if we wanted to write our statements in terms of the relationships to the red rod. What would that look like?
The whole numbers that fit this pattern are still the same as we had above. 1 + 2 = 3, 2 + 4 = 6, 3 + 6 = 9 and so on. But this is a different way of thinking about these numbers. We haven't even added subtraction or multiplication to this yet. But we can easily do so. We could say 3 (1/2r) = 1 1/2 r. What subtraction statements could we write for the above image? What division statements could we write?
We can extend this using not just whole numbers but fractions as well. We don't have to start off with fractions. You will notice above that in the w + 2w = 3w example that all the 2w numbers are even. What if r equals an odd number like 3? What will our statements look like then? What if r is a fraction like 1/2 and g = 3/4; which rod is equivalent to 1?
Should we extend the activities like this the first time through? Probably not, but that depends on the child. The important part is that you realize that every time we work with the same basic structures -- trains, patterns, mats, staircases, rectangles, and towers -- the student will start making the connections about how all of this mathy stuff comes together. If you aren't sure how to take your student from 1 + 2 = 3 to the rest of it, don't worry. Gattegno gets you there. Just follow along.
Using the Light Green Rod as the Reference Rod
Let's get back to the math. In the first example, we changed the numbers. In the second, we changed the reference rod or the unit. What if we change the reference rod again? We can write the partitions in terms of the light green rod.
We will find that the numbers that fit this pattern fit the above patterns as well. What we have done is connect a few of the concepts and rules for arithmetic and generalized them. We have also looked at multiple ways in which the numbers one, two and three are related. These are not the only ways; there is more to discover in the partitions we have made. If allowed the time to think and investigate, the student could make a systemized study of this particular math structure, which is what Gattegno intended and Goutard alludes to.
Marking the Units Tends To Hide the Structure
By now, it should be obvious why we don't mark the units on the rods. The rods are pseudo-numbers. Numbers are abstract concepts, and we don't want the student to think that numbers are objects. Even if they can imagine the red rod is a 5, labeling the rods shapes a student's intuition and how they think about the way numbers work. When the focus is on the numbers, it tends to hide the underlying structure. The structures are abstract, marking the rods makes everything more concrete. There is much more to say on this, but this is my short answer. If you want to read an excellent treatise on the danger of making math concrete, I would suggest you pick up Mathematics and Children by Madeleine Goutard.
What Questions Should I Ask
One question I get a lot is, "How do I know which questions to ask?" I think a better question is, "How do I learn to quit talking?" But that is not what parents, who are math phobic, want to hear. But it is still true. Other than talk less, my advice depends on who you are.
If you are math-phobic or had a poor math education, the best thing you can do is trust that Gattegno provides you with the questions you need and he will get you there. Don't push your child and don't skip the stuff you think is too easy. Keep a math notebook and work alongside your child. The benefit of doing this is that you stop talking while you are writing and you will be giving your student time to think while you are thinking. You will start developing an intuition for math. And finally, don't be too eager to jump into this chapter on numbers. Stay in Chapter 3 as long as you can and suck as much understanding from those exercises as your student will allow.
If you are down this road a bit and starting to figure out how the concepts and rules work together, the above advice to be silent probably applies to you more. Now that your puzzle is nearly together, you want your student to put theirs together today (I'm not the only one like this, you know who you are). You need to give your kids the space to think and to think carefully about what they are doing. If you are asking questions that are not from Gattegno's book, those questions should mostly fall under two categories: helping the student order their thinking and big idea questions. Big idea questions are structure questions rather than fact finding questions.
You should probably rethink your questioning strategy if the kinds of questions you ask the most are fact finding questions. Those kinds of questions produce an answer like 1 + 2 = 3. Now before someone gets on my case about not learning our basic math facts, I never said we don't want to. We do. But those facts are not isolated facts taken out of the context of the whole structure of mathematics. We are not pitting intuition and understanding against getting correct answers. Rather, our goal should be to have an understanding of and a deep intuition for correct things.
The purpose of this blog post is to convince you to NOT mark off the units on your Cuisenaire Rods and to not skip these basic exercises. There is more going on here than meets the eye. I know from my own experience that working through the exercises works out the kinks in my understanding. Kinks I didn't know I had. Give your children the time they need to work out theirs, or even better, not develop the kinks in the first place.