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.
He is quick to remind the student that 2p means p + p and the student can substitute p + p if she wants or needs to do so.
Something happens when we get to the end of this activity. Gattegno gives us a statement like this: d+p = 2 (g + r) and asks the student if this is true. I don't know about you but I didn't see anything like this in k/1 when I was a kid. There is no explanation for what to do with the brackets. Gattegno expects the child to take what he learned from the activities on brackets and apply it to this new situation that includes multiplication. If your student struggles with this, it is a sign to go back and make sure she understands what brackets mean. But don't be too quick to go back. Let her struggle and think before stepping in to help.
There are 3 types of activities covered in the rest of this section.
Finding Equivalent Forms for Multiplication
When we begin to work with equivalent forms, Gattegno has us start with 3 rods of a single color for which there are 4 forms. You can sit down and work this out systematically if you want. That is what Gattegno is doing here. I expect that Dr. Hajar and Dr. Powell would use this exercise as a way to get the student to start thinking about how to organize his thoughts so that he is able to work through these exercises systematically. This is what P. is working on now. We've been using equivalent forms for about 6 months, but he wasn't aware of the term. He was aware that we could take any pattern in a mat, break it up and make it in a different way. When I brought this to P. to work on, I said something like, "We know that if we have 4 of the red rods, I could say 4 red or I could say 1 red plus 3 reds, right? That is called an 'equivalent form'. It means that 4 red rods are equivalent to 1 red rod plus 3 red rods. How can we find all the equivalent forms for 4 red rods? And how will you know you have them all when you are done?" This has proved more difficult for P. than I expected. We've been substituting for ages now and I thought he would get it right away. He has come up with 1 and 3 and flipping to get 3 and 1, but he didn't have a good plan after 30 minutes and lost focus. When I bring it back up again, I will use 3 rods instead of 4.[youtube id="NxVH6gSSiLg"]
Reading Multiplication Expressions
This is pretty straightforward: 8r is read, "eight red." 8e in our house is read, "eight blue." Gattegno has the student practice these methodically. Evidently, he thinks this is fairly important so we didn't skip it.
Finding The Difference
In these exercises, Gattegno combines multiplication with subtraction. Instead of making up your own problems, I would use his at first. He gradually increases the degree of difficulty. There's a lot of multiplication and bracket combining so I wouldn't push through these exercises. We're not using this as a traditional textbook. Do some work in this section, work on something else and come back to a few days later.
Note that Gattegno does not have us use the word minus. Thus, 5r - 3w is read the difference between '5 red rods and 3 white rods'. When the student finds the difference she should read the rods back, "The difference between 5 red rods and 3 white rods is a black rod," or "Black is the difference between 5 red and 3 white."
If you write these problems on a whiteboard ,the students are also going to get experience translating written work back into the rods. At first, we worked with the rods and read them, then we made oral expressions and created rod patterns. Our students may or may not be writing at this point, but we want them to be able to see written expressions and create rod patterns for them.
Is It True?
Just as above, the student is taking a written mathematical expression and converting it to rods. This time, however, the student needs to determine if these statements are true or false. There are a couple sections of these. I would do his first and then make up your own. Gattegno sets the pattern for increasing difficulty that should be followed for most students, you'll want to become familiar with it.
None of what we are doing in this chapter is new to the students. We did nearly all of it in the last chapter, but we didn't have the language to describe what we were doing. Remember our focus is not right answers, though the students need to get to right answers, but on the language. Our students should be able to read the rods, to write down what the rods say, and to create rod patterns from written and oral mathematical expressions. If your student is not writing they can dictate or use math cards.
Do You Want More Support?
If you are looking for more support, join our Facebook Group for Base Ten Blocks. There you'll find a bunch of other parents who are using base ten blocks to teach their children math. Most of us had a horrible math education and are learning right along with our kids. Some of us are actually really good at teaching our children math, we're grateful for those folks, they give us a lot of advice.