Number – Introduction With Cuisenaire
We have finally entered the world of number with Cuisenaire Rods. We are in chapter 4 of Gattegno Mathematics Textbook 1, covering the first four activities. If you have stumbled upon my blog at this post, you will want to go back and read the other posts in this series. You can find them on the right-hand side under the category Gattegno Textbook 1.
In our house, we spent well over a year on chapters 2 and 3. We've finally started dipping our toes into chapter 4. Most of what we do is still centered around patterns and mat work with letters. Since P. is very comfortable with the letters and manipulating the symbols, his work has become increasingly more sophisticated. This has come about since the introduction of brackets. But I also think this happened because he was ready for brackets, and because he no longer has to think about the symbols and the rods, he can explore his ideas with confidence.
In the first three chapters, we learned to manipulate the rods, make observations, verbalize those observations, apply mathematical language to those observations and some students began writing those observations down or dictating them. It is after the student develops fluency with operations and fractions that we introduce number. When Gattegno introduces numbers, he doesn't do it as a set of problems the student must work through using the rods as a calculating tool. Rather, he approaches it as a systematic study of each number. The study numbers two and three begins with addition and subtraction. With the number 4, we add on brackets. Numbers 5 and beyond we add multiplication and fractions, and we'll backtrack a bit to numbers 2-4.
Chapter 4 is the study of numbers up to 10. The first exercise is to measure the rods using the white rod as our unit of measure. We could just as easily use the red rod or the yellow rod as the unit of measure. It is important that the student does not get the idea that a red is 2. A red rod is no more two than the figure '2' is two. The rods only have the meaning we give them in a particular situation.
After the student has measured a rod, they will indicate how many white rods it takes to equal that rod. For instance, the white rod is 1, the red rod is 2, the light green rod is 3. The student continues in this manner until they have expressed the value of each rod when measured with the white rod. The student may do this either by writing the figures or by dictation.
After we have measured the rods and given them number names, according to the white rod, we move to the study of numbers. Gattegno first deals with addition and subtraction with the red and green rods. He avoids multiplication and fractions. But P. was ready to tackle those immediately. I didn't stop him. However, Gattegno hits the concepts of multiplication and fractions from a slightly different perspective than he did in the last chapter. Since I don't want to discourage P.'s exploration and observations, I will still cover the fraction and multiplication exercises as Gattegno has laid them out when we come across them. I don't have to put a damper on his enthusiasms as I know he needs lots of time to internalize what is happening. Good grief, I need a lot of time to internalize what is happening.
How do we work on addition and subtraction? We do exactly what we did before. We build patterns for the rods.
Patterns for the Number 2
How many ones does it take to build the pattern for the red rod? If we take one away what is left against the red rod?
When we place the rods end to end, we will use the word 'plus' just as we have been.
What is 1 + 1 if you use the color name for the rod? What is 1+ 1 if we use a number name for the rod?
What is 1+ 1 if we use a number name for the rod?
Other statements that can be made with 1 + 1 = 2 configuration:
One is shorter than two by one. Two is larger than one by one. One is the difference between two and one.
Patterns for the Number 3
Now we're going to take the light green rod and make its pattern. My son made all the permutations for the light green when he did this.
After making the patterns, the student should either dictate each line or write each line using the plus sign. If your student didn't make all the permutations for the light green rod, you should ask the student to write the red line another way. If she wrote 1 + 2, she would need to write it 2 + 1.
Gattegno introduces us to another pattern for the green rod. We place a white rod on top of the light green rod. Now we will use this pattern if we want to know the difference between 3 and 1. The student should find the rod that fits the space that is left and name that rod with a figure. After completing the task, the student is invited to read the rods. This is the first time we come across the word minus. This pattern is read 'three minus one equals two'. Students are now expected to read the following 3 - 1 = 2 and 2 - 1 = 1.
Notice the progression in this lesson. The student makes the patterns. The students writes or dictates the patterns. Then the student reads the written notation. It is important that the student is able to do all of these. It shouldn't be an issue if sufficient time was spent in the previous chapter working on reading and writing mathematical statements.
We repeat this whole process with the red rod placing it on top of the light green rod on either end. It doesn't matter which end the students places the rod. The student should make the subtraction pattern, find the missing rod, and then read or write the pattern. Then the student should read both of the following: 3 - 2 = 1 and 1 = 3 - 2.
The last part of exercise 4 is a group of problems the student should complete in writing if he is writing. If not, it is a fine time to play, "What's Under The Cup". That just means the teacher can write the addition and subtraction statement or use rods and hide the missing part under a cup for the student to figure out. Another variation is placing the missing part in a bag or under a piece of cloth.
We didn't stop at these exercises. I asked P. to create his own exercises with a symbol missing and I figured out what was missing. We practiced reading statements both ways: 1 + 1 + 1 = 3 and 3 = 1 + 1 + 1; 3 = 2 + 1 and 2 + 1 = 3. And we practiced without looking at the rods or the symbols.
The purpose of these exercises is not just to learn that 1 + 2 = 3. If that is all you are after. skip all this and get the Preschool Prep videos, your students can have the basic math facts memorized in a week. However, if done properly, the Cuisenaire rods provide an opportunity for the concrete working out with the rods to become a labratory where the student is able to discover the structure of addition, subtraction, multiplication, division, and fractions.
If you want to find out more about the Gattegno Method of Mathematics you can check out the resources section of this blog. You can join our Facebook Group where you will find all kinds of base ten block users not just Cuisenaire. And if you are really ready to go full Gattegno, you can click on the Conference tab at the top of the page. You can join our private FB group with conference video and comments from our training last August at the Bronx Charter School for Better Learning. The focus of most of the week was the Gattegno method of teaching math. There is a little bit about Gattegno's Word's In Color, which is his language program.