Learning Fractions with Cuisenaire Rods
I'm posting my way though Gattegno's Mathematics Textbook 1, we're in chapter 4 studying the number 5. Students are coming into contact with addition, subtraction, multiplication, and fractions as operators. Our lessons often get derailed. If you've read my blog at all, you know I think this is a good thing. Obviously, today's lesson got derailed, so you won't find this in the Gattegno textbook. But we are learning fractions and that's what's important.
Learning Fractions by Naming
We were supposed to move onto the study of 1/4 today. But when we reviewed 1/3, P. had a few glitches in his understanding. He was giving correct answers, but I could tell that while 1/3 of the light green rod was clear, his awareness of fractions and number names still isn't solid. What I decided to do is ask P. to show 1/3 of various rod lengths. We used the Cuisenaire Rod color names the first time around. This was no trouble for him at all.
Next, I asked P. to say the names of the rod using number names. This proved to be much more difficult. He actually said, "Can we be done? This is getting harder by the minute?" I smiled and replied that he was more than capable of meeting the challenge, he just needed to slow down and think. I was correct, and he was surprised.
Learning Fractions as Proportion
After naming each fraction set, we arranged the sets into a staircase. If you don't know, I happen to really like staircase work. It is a valuable tool for showing relationships. What we learned from this is that 1 is to 3, as 2 is to 6, as 3 is to 9, as 4 is to 12. Each of these are a x 3 relationship, a 1/3 relationship, and a 1 to 3 relationship. It depends on how you want to talk about it.
Learning Fractions By Renaming
The last thing we did was make another staircase; did you know I like staircases? I really like staircases. Our staircase is flipped on it's side with the smallest rod on the bottom. This time we played a renaming game. White is what of red? White is what of light green? White is what of purple? White is what of dark green? Then we renamed the rods with their number names and started again. One is what of two? One is what of three? Two is what of one? Two is what of six? Two is what of 10? And so on.
This became P's math composition for today. When we started the lessons he was still struggling with fractions when he used with number names for the rods. By the end of the lesson, he was all, "Oh come on Mom, make it harder!" That is another way I know that he has this down.
P. came away much more confident using number names and fractions. We spent almost a year and a half on chapter 3 of Gattegno's Textbook 1. The transition to numbers names from letter names has been bumpy. As far as I can tell, this is a memory and language issue not a comprehension issue.
As he is learning fractions, he is gathering all kinds of awareness in addition to what is required for fractions. One doesn't have to worry so much about cross-teaching. It just shows up all by itself: a little change here, a renaming there, and a bit of rearranging the rods and all kinds of relationships present themselves to you.
Learning Fractions by Developing Awareness
In my last blog post, I wrote about using Cuisenaire with older students. I always tell parents to start at the beginning because you don't know where their gaps in awareness are located. The student can't tell you because the student isn't aware either. You can't ask because you don't know where your gaps happen to be.
In that post, I gave a list of strategies I use for adapting the exercises in the textbook and making them more interesting for older students. I used three of those strategies in this post and added an additional one.
1. What Can Be Changed?
Gattegno was working with 1/3 inside 5, but I'm not under any obligation to keep the student at 5. It's not like if we go outside of 5 someone will ground me or send me to bed without supper. We're not ruled by the curriculum, and Gattegno wouldn't want you to be either. There is no reason just use 1/3 as the fraction. You can use any fractions, including odd ones like 5/12.
2. What Can Be Compared, Contrasted, and Constrained?
Instead of just leaving it at 1/3 of three, we decided to look at 1/3 in multiple contexts and compare them. We did this by rearranging the rods into staircases. This helps us get a fuller picture of 1/3.
3. What Can Be Renamed?
In the last image, we renamed. We do this a lot. The white rod is not the number one any more than a single cow in a field is one. Numbers aren't objects and we can call one anything. Renaming reinforces this idea. But it also develops flexible thinking when it comes to mathematics. There is not a singular image for one-third; and there is no magic that turns something into one-third. One third describes a relationship of one quantity to another.
4. Does This Remind You Of Anything You've Seen Before?
I've just started asking P this question mostly because he has started to notice relationships in all kinds of places. He is coming to realize that characters in stories follow types. And the same types show up over and over again. Fractions and ratios look very similar and so do fractions and multiplication. I've started asking it in the hopes that he will start asking himself this question as a strategy for noticing what is in front of him.
What if Math Were the Easiest and Most Fun Part of Your Day?
Are you ready to change the way your students learn math? Are you ready to make it both fun and rigorous?
Get Hands-On Learning With Gattegno - My Gift To You