Chapter 3: Putting It All Together

Chapter 3 of Gattegno’s Mathematics curriculum has come to a close. If you are new to Gattegno or Cuisenaire Rods, you can find a growing list of resources here. If you would like to locate all the posts in this series you can find them under the category Gattegno Textbook 1.

exhausting the relationships

Chapter 3 and The Big Picture

Chapter 1, which is blank on purpose, was all about free play. There were no instructions because that sort of ruins the whole ‘play’ part.  In chapter 2, the student was discovering the attributes of the rods; those things that consist of what I like to call “the natural order of things”. In other words: rods of the same color are the same length; rods that are the same length are the same color; some rods can be made with an equivalent train of two colors and some cannot; some rods can be made from a train of one color that are equivalent in length; and some rods can only be made from an equivalent train of white rods.

In chapter 3, we moved on to more formal activities with the rods and the exercises required a great deal more mental activity. Gattegno is a fine guide and it becomes easy to understand why we spent so much time on the mechanics of manipulating the rods in the previous chapter. In chapter 3, the student’s mental energy is focused on:

  • learning the language of addition, subtraction, multiplication, and fractions;
  • learning the symbols for the various relationships;
  • combining these two abilities to express any given rod pattern in a correctly written or dictated mathematical statement;
  • correctly interpreting written or oral mathematical statements in terms of rod patterns.

We’ve observed that a student need not be writing to complete the exercises in chapter 3, neither does Goutard or Chambers recommend pushing a child to write. I’ve discussed this in more detail in previous posts so I won’t rehash it now. P. is 6 and he is an emerging writer (very slowly emerging, I might add). I just take dictation. One of the benefits of taking dictation is that the child must verbalize the mathematical statements and I am able to hear what P. is thinking. Sometimes, he has an idea of what he wants to do but isn’t clear on how to get there. Making him verbalize his expressions allows me the opportunity to know what kinds of questions to ask.

The stages of mastery at this level can be roughly divided into 3 categories of activity:

  1. manipulating the rods and making observations;
  2. manipulating the rods and reading what is seen in the rod patterns;
  3. manipulating the rods and writing what is seen in the rod patterns.



P’s great discovery after we began a serious study of brackets.

Combining Lessons for Greater Effectiveness

We intended to use Gattegno textbook 1 for at least 2 years, which seems astonishing given how small these textbooks are. But, we’ve been working on chapter 2 and chapter 3 for about a year and haven’t run out of things to do yet. While at the BBL this summer, Dr. Hajar agreed that it’s probably most effective to move between mat work and trains.

We might spend a day or two on one mat, making observations, writing expressions, and asking questions. We move from the mat to trains to work on a specific skill set, perhaps fractions. Then we move back to either the same mat and bring our new awareness to the situation, or we may build a new mat and then apply our newly acquired awareness to the new situation. Over time, junior begins to bring all of his mathematical skills together to work on mats.

If you have only one student, the parent will need to play partner with the child. Children feed off each other and spark ideas. Without another child to stir the intellect, the parent will need to fill that role. We don’t want to frustrate the child, but we also don’t want to make it super easy for them. When we do mat work and attempt to exhaust the relationships, I’ll often take turns with P. so it doesn’t become boring. Over time, we’ve both become more creative with our expressions. Which, of course, means that his understanding has grown deeper and so has mine.

We can compare an image from 6 months ago to his recent work. What I love about the work on the right is that in the last line P. suddenly discovered how to use fractions as a creative way to state zero. I had to put an end to the zeroing, I could tell he was on a roll and we were in for a long statement. Which is fine if he is writing, not so much if I am writing.

Combining Lessons, Chapter 3, Gattegno

Using Operations to Describe Relationships

In the last blog post, I talked about a game we play that forces P. to think about using addition, subtraction, multiplication and fractions to describe relationships. It’s possible P. would gather this ability naturally over time. But this is such a simple game that can be played anywhere, that I’m content to keep playing. Besides, it makes me exercise my brain cells as well.

I have joked that my oldest son scored post-high school on standardized tests in math since the 6th grade, but couldn’t accurately calculate the how much carpet I’d need to cover my living room. This is something I’ve been working hard to avoid. Part of that is knowing what operations to use when. That means understanding how math actually works, not just memorizing facts. (One more time, for the record, your children should know their math facts cold.)

Let’s say we are down at the beach collecting rocks. P. has a pile of just 10 very flat rocks to use for skipping. Z. is pretty indiscriminate. She collects everything and her pile is much larger. Our game is fairly easy. Let’s figure out how many math statements we can make to express the relationships between P.’s pile and Z’s pile:

  • p + z tells us how many stones we have all together.
  • z – p tells us how many more stones Z. has than P.
  • we can use a fraction to tell us what portion P. has compared to Z.
  • we can use multiplication to tell us how many times more rocks Z. has than P.

Notice here that we aren’t looking for correct answers or even figuring out the answer, but how can we use the operations to describe what is in front of us. My husband had surgery recently and was in the hospital for a week. We traveled the elevator 6 times a day. These are just a few things we observed:

  • We could use subtraction to tell us the difference between the total number of floors and the floors we pass on the elevator.
    • Do we count the floor we started on and the one we land on? Hmmm. Let’s think about that. What do we mean by pass?
  • If we know how many stairs between floors, we can use multiplication to find out how many total stairs.
  • We could use fractions to tell us what portion of stairs is between each floor.
  • We can use subtraction to say you must go down one floor from dad’s room to get to the gift shop.
  • We can use addition to state that you must go up two floors from the laboratory to get to dad’s room.

This concludes our work in chapter 3, I look forward to chapter 4 where we move to the study of numbers. If you want more information on Gattegno, please check out my resources page, I add new ones as I find them. If you want to connect with other base ten block users, you are welcome to join our Facebook group. We are mostly a group of parents, with poor math training, attempting to teach math to our kids. We have some folks who know what they are doing, we count on them to keep the rest of us in line.





  • Yuyan Zimmerman says:

    This is very well written and we can see the teaching is well delivered to the student. Thanks Sonya!

  • Arithmophobia No More