Language – What Do the Rods Say?
The title of chapter 3 of Gattegno's Textbook 1 is Literal Work. Most people use the word literal when they mean something more like actually or as an intensifier like really. When Gattegno uses the word I think he means something like a literal translation; word-for-word. In this chapter, we are helping the students get command of the language of mathematics. We are translating ideas from our common language to the language of math and from the language of math back into our common language.
This will be intimidating for many parents. And if all you do is thumb through the textbooks, your eyes will glaze over. Yet, this is where I fell in love with Gattegno. He is very careful to take a student from awareness to awareness. Therefore, if the parent follows along with the exercises, pulls out the blocks and works them alongside the student, Gattegno will take the parent from awareness to awareness as well - you will learn math along with your students. He doesn't leave much for you to figure out, but you must trust him in the journey. That can be a little unnerving for some parents and teachers. But Gattegno is a trustworthy fellow and more than capable of getting you to your destination.
Gattegno pulls the most out of each session. He doesn't explain much. He provides only the language necessary for communicating ideas and then it is up to the student to explore those ideas. For example, last week we learned that o = e + w means that the orange is bigger than the blue by a white rod. He then asks the student what we learn from additional examples. It doesn't matter which examples are used, as the point of the exercise is that the student learns to use the language.
Like last week, we're going to build some staircases, only this time our common difference is a red rod. Once we have completed this exercise with the red rod, we will do it again with the light green rod.
You'll notice that Gattegno doesn't intend to correct a student if the student has erred when building of his staircase. Rather, he asks the student to take out a red rod and correct himself. Gattegno expects the student to become responsible for her own education.
We will make two staircases, one with a white rod as the beginning rod, and one with the red as the beginning rod. The student will fill in the gaps with a red rod to check his work. He'll read the rods going up and then down, and then with eyes closed while trying to visualize the rods.
The Flexible Use of Math Language
After we've made these staircases, you'll notice something very curious. You'll totally miss this if you are just skimming through the book. Gattegno uses plus and minus to express the rod's location in the staircase. You don't see kindergarten or 1st-grade students working with this kind of understanding. But my pupils, even one with Down Syndrome, does not have trouble with it. Also, you'll notice that in the expression of addition he says, "and the red rod is the common difference."
If you are a parent, unsure of yourself when it comes to math, using the word difference in the context of addition is going to be a bit confusing. It will be just as confusing if you are a student who has grown dependent on keyword strategies to solve word problems. Using the word difference will also cause confusion if your student is used to thinking of addition in terms of some plus some more only.
As I noted last week, we are seeking to understand math as an expression of relationships. We are still using addition and subtraction to express relationships of bigger and smaller. Gattegno states:
We use p = r + r to mean that the purple rod comes after the red rod when the common difference is red. How do we express that the dark green follows purple? How do we express that the brown follows the dark green?
We use b = o - r to express that brown comes after the orange [when reading the staircase down] in this staircase and that the common difference is red. How do we express that the dark green comes after the brown? What about purple comes after dark green.
Just as above, Gattegno isn't hand holding here. He provides an example of the language and then provides a challenge for the student. In this case, the challenge is to use the language of math to express similar relationships.
Our students are now able to express that mathematical statements involving addition and subtraction describe relationships of bigger and smaller. They learned this when they successfully told you, for example, that b = k - w means that the black rod is smaller than the brown by a white rod. And they can also take a relationship: the dark green comes after the purple when the difference is red, and express that in mathematical terms.
You will have to pardon my excitement as we go through the rest of the chapter. But I have looked through hundreds of things for sale on TpT and read a ton of math blogs. I have yet to see a curriculum that can do what Gattegno does. I want to grab people by the ears and scream, "Let me show you this!!!!" This program is not hard to do. The thing that is the hardest to overcome is parental and teacher attitudes. But once the shift in thinking happens, the rest is pretty easy. In the next few blog posts, our little k/1st kiddos will be successfully using parentheses, multiplying variables, and using fractions with deep understanding. We do this without pain. Without tears. Without frustration.
And for the homeschool parent, it doesn't get much cheaper. Less than $100 gets you everything you need to teach your student k-5/6-grade student. You can go slow and add things from the Moebius Noodles site, or Denise Gaskins Let's Play Math and throw in some geometry explorations like Stacey Speyer's polyhedra. If you did that, oh my, your child would not just know math, but enjoy math. Love math. And see the beauty of math. And that is what I am after. But I digress and I have to get back on track now.
The staircases are doing much more than just providing a tool for learning to express relationships and learning the language. We will be putting this to use later in our mathematical journey in the areas of skip counting, making addition and subtraction of large numbers easy peasy, and in understanding algebra.
A Little Bit of Housekeeping
This blog post is part of a series of posts on Gattegno Mathematics Textbook 1. Gattegno was the popularizer of Cuisenaire Rods in the 60's. He developed the program Numbers in Color or Visible and Tangible Math. You can purchase this book and the rest of the books in the series at Educational Solutions. If you want to read the other posts in this series, you can look in the categories sidebar and click on Gattegno Textbook 1.
Disclaimer:I have no financial interest in any of the books or tutors I recommend. If you see it listed on this site, it is because I use or have used the product myself. In the case of tutors, these are all people I know enough to trust them with math.