Cuisenaire Rods – One Can Be Anything

Gattegno's Textbook 1

The study of 1-10 is coming to a close and so is Chapter 4 of Gattegno Textbook 1. This last section, we cover the Cuisenaire Rod mantra - One Can Be Anything. The rods are not marked for a reason. You can get a free copy of the book here.

Next blog post, I will not be doing chapter 5 as it is word problems and self-explanatory. 

I will pick it back up in chapter 6 with the study of numbers 11-20.

Cuisenaire Rods - One Can Be Anything

Let's talk about this "one can be anything" thing. When Gattengo talks about concrete representations, he means the use of buttons or beans or those little counting bears. A concrete representation is anything where we say this is 1 and this is 2.  Mortensen blocks and Math-U-See blocks do the same thing. Jerry Mortensen would argue for the the concrete - he would say that students should move from concrete to the abstract. Charlotte Mason expressed this sentiment as well.

Gattegno and Goutard were not in favor of representing numbers in this fashion. "Numbers aren't objects," says Gattegno and we don't want to give students the idea that numbers are concrete as mathematics is an activity of the mind - an activity of both discovering and imposing relationships on the world around us. Those relationships are mental not physical. Gattegno referred to Cuisenaire Rods as pseudo numbers and semi-concrete. What he means by this is that Cuisenaire Rods convey relationships not the numbers themselves.

Algebra, just as arithmetic, has a thoroughly perceptive basis. In fact, C. Gattegno's suggestion that algebra should be taught before arithmetic is psychologically sound. Perception relies largely on relations rather than absolute values, and generalities precede particulars in sensory experience. The colored Cuisenaire sticks represent relations among quantities; their absolute length is irrelevant and readily transposable.                                           Rudolph Arnheim, Visual Thinking

Below is one of my favorite videos about this concept - I never get sick of watching it. However, "one can be anything" goes much deeper than this video does. 

It is here that Gattegno seems to be more consistent with Charlotte Mason's philosophy of education than she is, at least when it comes to mathematics. Charlotte Mason comments in School Education that our nature craves after unity. It is this very unity in mathematics that Gattegno is getting at with the Cuisenaire Rods. It is this relational unity Gattegno seeks to uncover in all of his work.

Well, you might say, "Children are not able to perceive that unity, that is why we wait to show it to them until late middle school or early high school." Gattegno would heartily disagree with you on that point, and actually so does Charlotte Mason, at least in all other areas of education but mathematics. That is why I heed Mason's advice in most other areas but not mathematics. It is also why I believe Gattegno was more consistent with Mason's own philosophies about education than she was when it comes to math.

The Cuisenaire Rods, and Gattegno's way of using them, provide a visual for how some relationships are similar to other relationships and how they are different. If anything can be called one, then the student is able to perceive similar relationships even though the numbers change. Not only that, the student is able to perceive what is a similar relationship and what is not.

Cuisenaire Rods - One Can Be Anything Staircase

One Can Be Anything Staircase

The staircase above counts +1 only if white equals 1. If white equals 1, then red equals 2, and light green equals 3. 

What happens if red equals 1? What kind of staircase do we have? We have a staircase that counts + 1/2 . That means that w = 1/2, red = 1, light green = 1 and 1/2. What does the blue equal? 

What if yellow = 1? What kind of staircase do we have? We have a staircase that counts + 1/5. What is light green in a staircase that counts +1/5? What is black? 

What if black = 1? What kind of staircase do we have? What is white when black is one? What is orange when black is 1? 

What does the 10th step equal in a staircase that counts by +1/10?  

What does the 4th step equal in a staircase that counts +1/3? 

What is the 7th step if brown equals one?

Here are some questions to think about:

How is 1/7 and 6/7 related to 1 and 6? How is 1/2 and 6/2 related to 1 and 6? 

What is the relation between 6/4 and 6/2? How does work on this staircase help a child understand fractions?

How is 1/2 to 1 similar to 1 to 2  in a way that 1/3 to 1 is not, but 1/3 to 2/3 is?

What would happen if we changed this to a staircase with a common difference of red starting with a white? If white = 1 what is black? What happens if black = 1? What are the other rods in the staircase? What do you notice? What do you wonder?

Are You Smelling All This Relational Goodness?

Here we have a real study of relations. But this is just staircases - there are other structures full of relational goodness. And there is so much more in staircase studies than renaming rods - a lot more!

But that is for another day. If you want to learn more download  Hands-On Learning with Gattegno (it's free) and I'll share with you how you can join a Gattegno Study Group where we play with this kind of thing. Yes, I do have friends that want to play with little colored Sticks. It's not just me, and yes, they are adults. And my friends are so weird that they say things like, "Ohhh, that is so cool!" and "Oh my gosh, I've never seen that before!" and "This makes it so much easier!".

If that seems a lot like overwhelm and kinda geeky for you, no problem. You can join my Facebook support group for base-ten block users and get some support from tutors, teachers, and other parents who are just like you. Some of us know what we are doing. Most of us are winging it. A bunch of us are scared of math - like really scared. We take everyone and there is no judgement and a lot of help.  

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