This blog post is part of a series of posts on the Gattegno Mathematics Textbook 1. If you are interested in the previous posts, you can find them here. We’re in chapter 3, and we’ve come to the section on equations which covers activities 10-12.
Gattegno defines equation as any of the writings we can put down for a pattern that has two equivalent lengths. Notice here that writings is plural, which means that any given pattern with two equivalent lengths has more than one equation that will go with it. Even if we are only using the operation of addition, we can write three equations from a pattern of one rod that is equivalent to two rods by removing one. Gattegno uses a square to indicate that one of the rods has been removed from the equation.
Gattegno Cuisenaire Conference
Hey, everyone!!! I am so excited to announce the Gattegno Cuisenaire Conference for Homeschooling Families. Click here to get all the details. You don’t have to homeschool to attend. The conference will be excellent for teachers and tutors who want to learn more about Gattegno and who are interested in using his methods in the classroom. It’s just that my circle is mostly homeschoolers.
Check our dedicated Facebook page to get the latest announcements about the conference. And it’s a good place to post any questions you have. See you at the BBL!
We are still working our way through Gattegno Mathematics Textbook 1. We’re on chapter 3, activities eight and nine which cover reading and building equivalencies. The gist of these two exercises is to have the student learn to read addition and subtraction statements, check those statements for accuracy by building them with the rods, and lastly, students will build patterns and create their own equivalencies based on what they can see in the rods. Reading math symbols and creating the patterns based on those symbols is the next step to fluency.
Reading The Symbols
The first activity in this section provides us with numerous math expressions related to addition and subtraction. Each expression contains 2 rods which are equal to a single rod. The student will be reading those expressions using the correct words for the signs.
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.
This blog post covers exercises 3 & 4 in Gattegno Mathematics Textbook 1, chapter 3. This is where Gattegno first introduces the student to the word ‘relation’. This concept is critical not just for these particular exercises, but for all of math. If you’ve paid attention to anything I’ve posted in the last year or so, you know that ‘relationships’ is one of my favorite words. It’s not really, but in the context of teaching children mathematics, they won’t get anything unless they understand this one thing. Oh, they might learn some facts, but they won’t get the big picture of mathematics because ‘relationships’ are the big picture.
Activities 3 and 4 are exercises in building staircases. We did this in chapter two. The last time we did it, the students became aware that there is something called a staircase, that the rods can be ordered smallest to largest or largest to smallest, and that there is an order to the rods. If they were astute, they might have noticed that each successive rod is larger by the same length. If they didn’t, that’s fine because we are going to measure that distance with a rod. Then we will ask the students to express the relation between two successive rods.
By ‘successive’ we mean things that follow or things in a row; consecutive. By ‘relation’ we mean how one thing or idea is connected to another thing or idea. When we say ‘express,’ we mean how we talk about or how we write about the relationships. Please note that where I quote Gattegno directly I will use his letter patterns for the rods. Otherwise, I will use the one we use at home. To see what we do, you can look here.
Understanding Addition and Subtraction As Relationships
When we make a staircase using all the colors, one of the things that bring these rods into a relationship is that each successive rod is one white rod larger than the rod before it. Those relationships can be verbally expressed by saying that a white plus a white is equivalent to a red, and a red plus a white is equivalent to a green. To express those relationships in written form we write w + w = r and r + w = g.
The last few weeks I’ve come across a lot of stuff about young children and their confusion with the equal sign. I’ve tutored quite a few students who come to me thinking that the equal sign means “Write the answer here.” I’ve been very careful about what I say and how I present equivalencies since I discovered how confusing this is for kids.
Given my heightened sensitivity to this subject, it came as a shock that my 5 year old was confused about what the equal sign means. How this got twisted in his head is forever a reminder that we are dealing with young children who don’t have a firm grasp of the language they speak. I can think back to my own math education and how much confusion has been cleared up in the last 6 years. I didn’t develop a relational understanding of math until I discovered base ten blocks almost 2 years ago.
If you are working with kids, they need time. They need time to play with the ideas and wrap their heads around the concepts. And concepts need to be approached from multiple directions. Does doing math this way guarentee no confusion? Nope. But it does help minimize it. And if we are educating a child’s awareness, much of the areas of confusion will eventually sort themselves out.
So my discovery began when Lisa Cranston @lisacran tweeted this:
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.
We made it through chapter 2 of Gattegno Mathematics Textbook 1! If you followed this series, then the foundation for teaching Gattegno’s mathematics with Cuisenaire has been laid. We ‘ll be adding a few new things here and there, but for the most part, we’re just adding more details to what we’ve already put down. You will find the entire series by clicking Gattegno Textbook 1 in the sidebar under categories.
The Bricks of Our Foundation: Free Play, Trains, Patterns, Mats and Staircases
The end goal of book one is mastery of the 4 basic operations and fractions as operators in numbers up to 10. In this first section, the big picture goal is that the student becomes familiar with the rods and the activities, is capable of manipulating the rods, and begins to develop the language to communicate what s/he is seeing and doing.
Free Play: Don’t skip this. It is the temptation of most parents to skip this part and get to the work of “real math”. However, free play does teach real math. Beyond that, children develop an intuition about how math works, develop spatial reasoning, plus free play encourages problem-solving skills and much more. Intuition is not something you can teach didactically. If you want to skip free play in lesson time, give your students free access to the blocks and encourage your students to use them outside of math time.
This is the last post on chapter 2 of Gattegno Mathematics Textbook 1. I did the happy dance. I did. We’re covering activities 37-39, which are all about odd and even numbers. The next post will be a recap, and I’ll share some of the mistakes I’ve made and what I’ve learned since I started.
Odd and Even Numbers
One of the first math concepts I taught my son was odd and even. He could tell me if a number was odd or even when he was two. We played lots of games with odd and even because knowing if a number is odd or even is probably one of the most important things you can know about a number.
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.