Welcome the 101st Math Teachers At Play Blog Carnival (MTaP)! If this is your first experience with the blog carnival, a little background information is in order. MTaP was started by Denise Gaskins over at Let’s Play Math. It’s a collection of blog posts from teachers around the internet. Notice that these teachers are at play. This is supposed to be the fun side of teaching math. The last carnival was hosted at Three J’s Learning. You can find the current edition of our partner carnival, edition 138, at the Carnival of Mathematics.
We are at the end of chapter 3 of Gattegno’s Textbook 1. I’m pretty excited about this section of the material. My oldest son struggled with fractions and we didn’t introduce them until 2nd grade. Even then, it was pretty simple stuff. We’ve been doing fractions for almost a year with P., who has a very good grasp of what’s going on. There was a time I dreaded math lessons in our house, not anymore. This last tool in the box makes mat work and other activities interesting and fun.
If your student has made the multiplication connection they are probably ready for fractions. Fractions are something else I introduced very early. Like Gattegno, we started with a half. I think half is pretty easy because most kids know what a half is, especially if they have a sibling. I did it before we decided to purposefully use the Gattegno books. You can read my first post on fractions and preschoolers here. A couple weeks after that post, I began to read Gattegno’s work seriously and switched from another base ten block system to exclusively using Cuisenaire Rods.
We are nearing the end of chapter 3 of Caleb Gattegno’s Mathematics Textbook 1. Next week, we cover fractions and then we move onto chapter 4 and the introduction of numbers. If you’re new to this blog, please see the other posts in this series by clicking here. We are working with multiplication which covers activities 15 – 19 of chapter 3.
In our house, the idea of multiplication came within a session or two of introducing trains of one color in chapter two. If students are required to read the rods, it is very tedious to read a train of more than 3 rods. The student needs multiplication to make the oral communication of mathematics easier and they will bring up the idea of replacing w + w + w + w + w with 5w. Your job is to provide your student with the arbitrary information of how we say it and how we write it. They know about multiplication because they already understand the nature of things. They know that we say, ‘I have 3 balls’ not ‘I have a ball plus a ball plus a ball.’
In this chapter, students are learning to speak and write mathematical ideas. Gattegno expects that the teacher’s math language will be clear and precise. Everyone should understand what we mean by what we say. This is no different than learning to use any language. We are given 3 ways we can say 2(x) a rod; the student is expected to practice all three.
If you are still with us after my little hiatus that included life and death, taking a newborn into our home, and attending conferences – – you know, that stuff we call life – – we are plugging along in Chapter 3 of Gattegno Mathematics Textbook 1. You can find the rest of the posts in the sidebar under the category by the same name. This post covers the topic of brackets which are introduced in activities 13 and 14. For those of you who are new to this series, I recommend starting at the beginning and reading the series alongside Gattegno’s Book, which you can purchase here.
I’ve said before that this is not a standard curriculum. There are activities which need not be done in order. For example, we skipped all activities involving the use of brackets as my son wasn’t writing yet. His interest in brackets was piqued about six weeks ago.
Post Gattegno Cuisenaire Conference wrap-up, I can’t get it done in one post. Instead, I’ll break down the salient points in a series of posts. Arbitrary and necessary is where we’re starting. This idea has rocked my world. It was worth the price of admission for this idea alone.
Day One Morning Session
Monday morning started out kind of slow. We were given a set of problems and asked what we would teach children about those problems. We were then separated into groups and told to pick one problem, discuss what we would do, and write it on large sheets of paper. Then each group was to pick a spokesperson that could explain our reasoning to the rest of the group. I’ve been around Gattegno enough that I thought I had this all wrapped up. The equal sign situation should have brought me up short but I’m a slow learner, what can I say?
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.