Transformations with Gattegno

We are nearing the home stretch for chapter 2; our topic is transformations. This post is my second to the last post on this chapter of Gattegno Mathematics Textbook 1. Once again, if you have landed here out of the blue, you can find the whole series by clicking Gattegno Textbook 1 in the categories sidebar. When we are finished with chapter 2,  I’ll do a recap post and then move on to chapter 3.


We practice transformations starting with two trains. A train of one rod and a train made up of three blocks in two colors. We will place the trains side by side. In the train of three cars, we will start with the odd colored car on the outside.

The textbook uses the example a train consisting of a single brown car and a train of a red and two greens. For clarity, I will refer to the single car train as a train and the train used for transformations as the t-train. We want the student to say that both trains are the same length. We will then move the red rod to the center and ask if the trains are still the same length. Then we will remove the red rod and slide the green ones together. Which rod do we need to make the trains the same length? If you have a young child, they may not know that it is the red rod; they may need to think about it or even measure.

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The textbook makes suggestions for colors of the t-train. Again, those colors are arbitrary. We’re going to be using rods over five units long for the train. For the t-train, we’ll practice with different rods using either a rod of one color and a two rods of the same color or three rods of different colors.

Progression for Transformations

Our progression is as follows:

  1. Start with a complete t-train. Remove one rod, slide the other two together. Ask the student to find the rod that will make both trains the same length.
  2. Make a t-train of two rods with a space left. The student will find the rod that fits the space/makes the trains the same length.
  3. Make a t-train of two rods with a space left. Without touching the rods, the student tells which rod fits in the space.
  4. Place a train of one rod on the table. Tell the student which two rods make part of the t-train. Without touching the rods, the student shares which rod completes the t-train.

Transformations are part of algebra and geometry. Using Cuisenaire we create an awareness of transformations from the beginning. Learn to teach with Cuisenaire.

I mentioned in other blog posts that we aren’t out to master this in a week. Gattegno doesn’t give teachers instruction how to accomplish mastery either. He did write in one of his books that there is enough material in the second chapter to last through kindergarten, which is an indication that mastery comes with continued use of the blocks over time. We’ve been on chapters 2 and 3 for a long time. Our approach has morphed into working with activities in a mat and the moving away from the mat to explore concepts more narrowly like the exercises above and then moving back to mat work.

What Are We Learning?

In this set of exercises, we are dealing with transformations. What do we want the children to learn?

  • The empty spaces can only be filled by one rod. The length of the space and the lengths of the rods are connected.
  • We can transform a rod of one length into a set of multiple smaller rods. But there are rules that must be followed.
  • It doesn’t matter if the empty space is on the left, right or in the middle, the size of the space dictates the rod that will fill it.
  • You cannot make a t-train with 3 random rods. Each choice of rod sets limits on which subsequent rods can be chosen.
  • Students are also learning to visually discriminate distance and the relative lengths of the rods.

We still work on some of these exercises. My son cannot visually discriminate all of the rods by the empty space, nor is he able to tell me which rod fills a space using words only. He still needs the visual clues. We have a tendency to do mat work for a day or two and then move away from that to work with a couple of trains and then back to mats again.

Transformations are part of algebra and geometry. Using Cuisenaire we create an awareness of transformations from the beginning. Learn to teach with Cuisenaire.

To reinforce transformations I will have him build a mat for one rod with the constraint that each train must contain 3 rods.  Then I will have him walk away from the table. I will remove one rod from each train and he has to fill the spaces up as quickly as possible. Sometimes this works, and sometimes we end up with blocks all over the table. But that just means he gets to rebuild the mat. Only this time, he will usually choose different combinations.

Remember there is no hurry. Play along with your children. Let them make transformations and you guess which rod is missing. Have fun. We are still playing games.

The following is an excellent video I came across on the Math In Unexpected Places blog.  I trust this will help you understand why we need to repeat these activities over and over, even if they look easy for us, and even if it appears they have “got it”. He, like Gattegno, tells you to play close attention to the student. He also tells you what you are looking for, which is helpful.

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