Introduction to Mathematical Concepts

Chapter 3 of Gattegno Mathematics Textbook 1 starts with the introduction of math language and the exploration of mathematical concepts related to that language. I want to note that there is no clear transition between chapters 2 and 3. Your student need not master every exercise in chapter 2 before you move to chapter 3. When your student has a good grasp of the exercises in chapter 2, can manipulate the rods when asked and has started to memorize some of the patterns you can move ahead.

If you have found yourself here by accident, you can click Gattegno Textbook 1 in the categories sidebar where you will find all the posts in this series.

Beginning written mathematical compositions with Gattegno Cuisenaire.

Teaching Goals of This Stage

In chapter 2, the students were discovering the characteristics of the rods. Now we want to make them aware of and gain insight into beginning mathematical concepts. We want the children to have a clear understanding of addition, subtraction, and multiplication. In this chapter, they will also gain their first experiences with fractions. The students will know the meaning of terms like plus, minus, multiply and equal.

In chapter 2 the activities were informal. We were creating awareness in our students. We were assisting them in their discoveries. But now we expect that they will give some expression for what they are doing. We are embarking on a more intensive study of the rods and rod patterns. As they learn the language that will allow them to express what they are doing, we want them to be able to:

  • Read what the rod patterns say.
  • Write what the rod patterns say.

Our big triumph at the end of this chapter will be that the student can express the mathematical concepts that she creates AND can understand oral mathematical statements well enough to create rod patterns for them.

If you follow the textbooks alone, you will feel pressed to introduce written symbols at this point. However, that was not the practice of Gattegno teachers. If your student is already writing and enjoys writing, it might be fine to introduce the written symbols. However, if your student is an emerging writer, he will be better served by sticking with oral exercises right now. We have been at this for quite awhile, and my son, age 5, still isn’t writing. I have a student, a year younger, who is writing, and one who is almost nine that just started writing. I wouldn’t force written work. In fact, both Chambers and Goutard suggest that you should never rush written work.  Madeliene Goutard has an entire section on this in her book Mathematics and Children. I will cover this more in a later blog post.

Replace the Rod Colors With Letters

At some point, as we transition to chapter 3, we need to switch from using the colors names for the rods to letter names. The books use a pattern, but you are free to make your own. I don’t like using upper and lower case letters, so we made our own. In the beginning, I made a small poster for reference which was a huge help. Since I introduced the letter names immediately, there was no transition period for us. We easily go back and forth between letter names and color names of the rods.

Working through Gattegno Textbook 1. Introducing letter names for the rods.

Introducing the Correct Mathematical Terms

If we have spent enough time with the blocks in the last chapter and we’ve used phrases like end to end, equivalent to or is the same length,  adding another piece of information will not be difficult. In the first two exercises we find five mathematical terms Gattegno wants us to learn: compare, difference, plus, minus, and equivalent.

Compare, Difference, Minus, and Equal

When we ask the children to compare the rods we want them to note which one is smaller and which is larger.



If we compare a blue rod and a purple rod we see that the blue rod is bigger than the purple rod. Which rod, when placed end to end with the purple rod, will make a train equivalent to the blue length. We will call the rod that fills that space the difference between the length of the blue rod and the length of the purple rod. If we want to say that quickly, we say e minus p equals y. When we say equals it is a fast way of saying is equivalent to. When we say minus, it means we have two rods, and we want to measure the difference between them.


To reinforce the idea, we want to practice. Ask the student to find two rods that are not the same color. Place them side by side with the larger one on top. Ask him to find the difference between the two rods. Once he has done this, ask him to read his rods. Repeat this activity with different rods until the student has no trouble with expressing subtraction constructions.


Take a dark-green rod and a green rod and place them end to end. Which rod is equivalent in length in length to the dark-green rod and the green rod? If we want to say this quickly, we say d plus g equals b. When we say plus it means we have two or more rods that are placed end to end. Practice making more trains and finding the equivalent lengths. Have your student read what she has created with her blocks.

plus 1

A Short List of Don’ts

  • Please do not replace the word equals with makes. We want the children using correct mathematical terms from the beginning. There is nothing inherently difficult with the word equals that makes it harder to understand than the word makes. It is better for the student to start using the correct terms from the beginning.
  • Do not use take-away as a replacement for minus or subtraction.
  • When presenting a challenge to the student such as, “Can you find the rod that is equivalent to a red plus a green?”, do not intervene unless it is necessary. And then, ask questions instead of giving answers. It is good for the student to struggle and gather more information. If the student grabs the wrong rods, don’t say, “No.” Just keep quiet and wait.

If you are interested in learning more about base ten blocks to teach math, and/or would like support from other parents and tutors, you can join our FB Group here. You will find users of Gattegno, Mortensen Math, Miquon, Education Unboxed, and others. Some of them just use the blocks to model problems in their regular curriculum. We’re a pretty friendly bunch. No one ever bites.
















Arithmophobia No More