Fractions: The Inverse of Multiplication
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.
Fractions as a Description of Relationship
To introduce fractions, Gattegno takes us back to working with a train of one color, specifically exercise 19 from last week. It doesn't matter what the problem is, but exercise 19 has multiple examples to use. We'll use Gattegno's example of e = 3g. This statement tells us that a blue rod is equivalent to 3 light green rods. That's the expression most of us would use when talking about multiplication with our student. Gattegno has another statement for us, "Blue is 3 times as big as a light green rod or as long as a light green rod." In these examples, we are talking about the blue rod and it's relationship to the light green rod. Now we are going to do the opposite. We are going to talk about the light green rod and it's relationship to the blue rod. When we do that, we say that the green rod is 1/3 as big as blue or 1/3 of blue.
C.E. Chambers states that Gattegno saw fractions, not division, as the inverse of multiplication. I didn't really understand what he meant by that as I saw fractions and division as basically the same thing. What I've come to realize, after carefully reading his work, is that Gattegno is careful to use math language to always describe the relationship, not the operation. Math operations are the action taken upon the number to determine the relationship. This is what I think Ben, from Crewton Ramone's House of Math, means when he says "Arithmetic is not the math, it's how we do the math."
When we say 4 = 10 - 6 we read that as '4 is equivalent to the difference between 10 and 6' not '4 equals 10 minus 6'. We read the relationship, not the action performed on the number. So it is with fractions and multiplication. If we are describing relationships, then fractions are the inverse of multiplication. If e is 3 times as long as g, then g is 1/3 as big as e. If you have 12 balls and I have 3, you have 4 times more balls than I do, and I have 1/4 as many balls as you do. To find out how many times more, I would have to perform the operation of division but division does not tell me about the relationship, fractions do.
If we take the time to help the students discover the ways in which the 4 basic operations can be used to describe relationships, they won't be forced to rely on keywords later. Next week, I will describe a game we play to help P. use operations to describe relationships using addition, subtraction, multiplication and fractions. Notice, in this chapter, there is no division work.
Verbalizing Relationships in Terms of Fractions
In exercise 20, Gattegno gives a multiplication problem and it's fractional inverse. He then provides the model for saying the relationships and expects the student to use the model to say them. He covers fractions 1/2 through 1/10 in this way.
It is very important that students get this down. When I was at the BBL this summer, I watched adults struggle to get this right. When I struggled, in a different area, I noticed that I didn't have clarity in my own mind. Repeating the verbiage helped solidify my understanding of what was happening with the math.
In exercise 21, Gattegno gives a list of equations and asks the student to determine whether the equations are true. Expressions like: 3 ( w + r ) = 1/3 x e + d and o - (1/2 x n + 1/5 x y) = 1/2 x 0. Students are expected to do this mentally. This is k/1 material, we don't want to push this kind of work on them the first go around, which is why we need to spend a lot of time learning the rods and not skipping over the mental work in chapter 2 and earlier in chapter 3. But don't assume your students can't do it either. This kind of work is easily within the grasp of a k/1 child.
While we haven't done this yet, we've done some pretty serious work with fractions and multiplication but our work with brackets has lagged behind. Behind what? Only the book. Gattegno didn't expect anyone to use this as a textbook in which you work through all of this in order. Below is some of P.'s work from over 6 months ago. He was 5 at the time.
The next post is a wrap-up of what we've covered in this chapter. I will also cover how I use the mats and trains to tie all these lessons together and how easy it is to cover multiple concepts at once. Each time you cover the material, there is an opportunity for the student to cover it with deeper understanding. Given that there are only 21 activities, you will be hard pressed to make this chapter last a couple months let alone a year if you go over the activities only one time. What we need is a way to bring this whole chapter together.
If you would like to read the rest of the posts in this series you can find them here.