Cuisenaire Rods With Older Students

We've been working through Caleb Gattegno's Textbook 1 with younger students. The textbook is fit for 3-7 year olds depending on when they start. But what about starting Cuisenaire Rods with older students? What do you do with the student who's been working in Saxon 5/4 for the last year?

One of the things Gattegno had in mind was the study of structures. You can find my post on Gattegno and Structures here. If the student has a few years of math under their belt, and they don't want to go through chapter 2 of Textbook 1, holding rods behind their back (though it is a good exercise), don't make them. Student's will feel like they are going backwards. But that doesn't mean that the student has actually gained all the awareness Gattegno is leading the student to in chapter 2. How do we assist this student? 

The answer, I think, lies in studying the rod structures Gattegno presents in the first book intentionally, as structures. I can't go through all of them, so I'm going to take staircases as an example. Staircases allow a student to study sequencing and series in a visual manner; it turns out series are everywhere. In the first book, the student is just beginning to notice and experiment with how a staircase works. In the following video, Vi Hart is talking staircases but she's using a number line instead. What happens if the space between the steps is +2 or +6 or +10? Where does it take you?

What makes me pumped everyday is that I don't have to know all of this stuff. I am learning right along with P. We have already figured out a lot of the counting in steps, and have had a ton of fun manipulating the steps in all kinds of ways. 

When we play math or study the rods as structures there's a few things I am thinking about all the time:

1. How Can This Be Changed?

This one is the easiest. There are so many variables for a structure. In my math notebook, when I am working on something like a staircase, I write down all the ways I can imagine to take this particular scenario and make it a little different. I am also paying attention to what P notices so that some changes aren't arbitrary to him, but based on what he is already thinking about. In the image below, I made suggestions for the value of the white rod. But there is more...

  • We can talk about the rods as a fraction of the next rod: w = 1/2 r, r = 2/3 g, g = 3/4 y. Is there a pattern?
  • Instead of changing the step, we can change one of the rods. Dark green = 30 and orange = 50 what is the value of each step? What are the other rod values?
  • Ask the student to change one thing and make it hard for you to figure out.  
  • Ask the student to change something, but make it easy enough for a little brother or sister to figure out.
  • What happens if we make an identical staircase? How many ways can the two staircases be combined? 

​We can also change the space between each step. The step can be increased so that it is a difference of a red rod. If the staircase has a difference of red, the student will be working with odd and even numbers if w = 1. The step distance can also grow. In the image below, you can see two different kinds of growth. All of the above changes apply to these staircases as well.

2. What Can Be Compared, Contrasted and Constrained?

Now that we have a bunch of staircases to work with, and we've had a chance to work them, we can compare and contrast the staircases. We can talk about what is the same and what is different. We can do this between two completely different staircases. And we can do this when we make a change in a single staircases as in the first image. Once we start comparing and contrasting, we can create more staircases by adding constraints or by making it similar to something we've already done, but also different in some way:

  • A difference of one instead of +1
  • x 3 instead of + 3
  • Stair growth of +1, +3, +5, +7 
  • Stair growth +2, +4, + 6

Here is a wonderful post by Simon Gregg on patterns. Here is a question for you: How is what he is doing similar to staircases and how is it different? Can you do something like this at home? 

3. What Can Be Generalized? 

If the student is older, this will depend on how much older, can the student make generalizations from the information gained?  For instance, if I want to find any stair in the first staircase I could write the following generalizations:

    • Step + 1 = the next step going up the staircase or n + 1.
    • Step - 1 = the next step in the staircase going down or n - 1.

4. What Predictions Can Be Made?

What is going to be the 10th step in this staircase? What about the 21? We can then go back to generalize and determine if there is a way to determine any step in the staircas.

5. What Can Be Renamed?

Renaming helps us get out of a rut and it is exactly as Vi described in her video. 1 + 1 is 2. In any of the staircases above, I can rename any step and the distance between each step can also be renamed. This is particularly fun for a student to do to a teacher.  For instance, in the second image, first staircase, instead of writing 1, 2, 4, 8, 16,  we could write the sequence as  20-19, 2 x 1, 2 + 1/2 (4), 4 ^2.

This is not the all that can be done with a staircase. But this should get you thinking about how to change any structure into something for study. If you read the post I wrote on structures, you'll notice how I approached the structure of w + r = g using some of the same approaches I took in this post.

Lacy, from Play, Discover, Learn, recently posted about an exercise she did with her students. They are using some of the same approaches I use, even though we've never talked about it. She doesn't state it, but she is also doing staircase work made into a game. Her kids are ex-Saxon users. It would be interesting to read her post and see how many of the above strategies you can pick out. Are there some she uses that I left out?

If you have a favorite activity or strategy that helps you "work in a Gattegno kind of way" I'd love to hear about it. If you don't feel like you have enough ideas and want some more, you can join us over in our Facebook Group for parents and teachers who are using base ten blocks. Not all of us are using Gattegno. Most of us are learning to teach and we all pretty much had a poor math education. Folks are kind, generous with advice and the conversations are usually pretty good. 

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