Mastering Addition and Multiplication Concepts with Cuisenaire Rods
In this post, I am continuing the work on Gattegno's Textbook 1, chapter 4. We are partway through our study of the number 5. We've parked ourselves in the middle of a long section that is dealing with multiplication. If you've been following, you know that my 6-year old is highly skilled in the art of distraction and rerouting our math lesson. I put my foot down, well sort of.
This is what happened: I presented him with a clear task. I wrote a statement on the board; he was to make the same statement in a different way. Well, that’s not what happened.
I’m starting to catch on to this whole teaching and asking the right questions thing. Instead of repeating myself and pointing to my statement, I decided to ask a question. If you read my last post, you should have watched the "notice and wonder" video at the end. If not, go watch that video now and come back. That video has revolutionized all of my teaching. I can use notice and wonder as a strategy my students can use when approaching a problem. But it is also a strategy for teaching me to teach. And it fits with everything Gattegno is doing AND it is easy for me to remember. Am I stuck and don't know what to do? I notice and wonder about my student.
So here I have a situation where my son has given me the wrong answer and since this the the 3rd time we've come at this, I know it is on purpose. It is time to pull out my notice and wonder strategies.
“I’m curious, can you tell me more about 8 – 3 = 5? I noticed that there are no 1’s or 2’s in your statement.”
“That statement is boring,” was his response. OK then. That is all well and good but there are ideas that Gattegno wants to cover in these lessons and I’m not keen on skipping them.
This is how it goes in most homeschools. In a public school, the child plays along because there is peer pressure. Children don’t want to stand out, they don’t want to feel different. There is some degree of compliance just because of the situation. When a child is at home, and schooling, there is no need for compliance. There are no peers to cause embarrassment. Mom has doesn't have many options.
As I see it, I have two choices.
1. Force him to do the work, that he finds boring, by way of coercion or some other manipulation.
2. Engage him where he is and attempt to work the concepts Gattegno is covering into our session.
I'm working hard to steer away from education by force. I can force a 6-year old to read through tears. I've done that. I can force a child to do math. I've done that. This blog is my penance. Parents and teachers can keep doing those things and console themselves that their students should just comply because well, they're bigger. That didn't work out so well for me or my oldest child the first time around. I'm working on becoming a better teacher. I picked option two.
What is Gattegno covering in this section? We are working with multiplication and addition and he is making sure the student understands how multiplication and addition are related but different. Since he commits a huge section of this chapter to addition and multiplication, we can assume that it is important and also, it’s safe to assume that many children get hung up here. If you read through the material, Gattegno approaches this forward, backward and sideways. He doesn’t move in a straight line either. He moves between rods, written work, and mental work. It isn't a seamless line. That probably means this material should be spread out over the course of a few weeks. I attempt to work on the book 2 days a week and we are free to do our own thing 3 days. Plus there is a lot of informal math learning happening all the time.
Since exercise 19 covers fractions, I’m only going to look at exercises 14-18 and leave the rest for the next post.
Mastering the Concepts of Addition and Multiplication
- Students can rewrite repeated addition as multiplication and multiplication as repeated addition.
- Determine, using the rods, which of 3 given statements, that include similar addition and multiplication statement, is larger. Such as 2 x 3, 3 + 2 and 2 + 3.
- Determine, without using the rods, which of 2 given statements, that include similar addition and multiplication statements, is smaller.
- Without using the rods, write the answer to statements containing addition and multiplication, with a little bit of subtraction thrown in for fun. These include statements like: 1 x 2 + 1 x 4 = and (4 – 2) x 2= .
Given P.'s distain for what he calls "baby problems", I wanted to create problems that include the concepts, but not the problems, that Gattegno is stressing. I wanted to know whether P. has mastered the idea of addition and multiplication. Since I've made assumptions in the past, like here, about what he knows and doesn't know, I'm more inclined to push him so that I know for sure.
I allowed him to make some trains for reference. I did not give him clues or hints. I required that he rewrite his statements to include addition, subtraction or both. He ended up rewriting my statement in the end anyway. And we rewrote it several ways. Plus, he did his. What was interesting to me is that we didn't run into trouble until we hit 2 x 3, 3 x 2, and 3 + 2.
Victory On the Multiplication Concept Front
He chose to rewrite 8 as 6 + 2 to meet my constraint of using addition and multiplication. He wasn't able to use the rods. He told me that 3 + 2 + 2 was 8. I told him I wasn't sure I agreed and that maybe we should verify that statement using the rods. You can see his check in the image below. This is what sparked a wonderful conversation about what is happening with multiplication. What does 2 x 3 mean? We've gone over it before. But obviously, it isn't solid for him yet. What does 3 + 2 mean, which he knew. We went back and forth between 2g and 2 of light greens, and 3r and 3 of the red. Then we changed the rod names to numbers number names, with white = 1. P. was asked to connect the knowledge he gained from letters to our new situation with numbers. Then we changed the rod names again, with w = 2, just to show that the concept of multiplication doesn't change even when we change the name of the unit.
We started a game we call "Guess My Train" but sometimes it is "Guess My Rod". We take turns making verbal a statement about a train or a rod and the other person has to guess what it is.
- My rod can be made with 3 x 1 + a red.
- My rod can be made with 1 + 1 + 1 plus 2 x 2.
- My train has a pair of twins and it equals 5. One of the rods is white.
- My rod has 2 odd rods and one even and it equals 8.
- My rod can be made with three times one plus one.
This game helps P. create clues based on his level of comfort. Since he wants me to get stuck, he has to be creative and work hard. I am able to determine his level of understanding. He is able to work at a higher level with the rods in front of him. Since we like this game, we're doing both. We play the verbal games with easy clues while driving in the car or while we are doing chores. The other is played as part of math class or just for fun.
We'll get to that fraction work in the next blog post.