Introduction To Patterns
This post covers activities 3 and 4 of chapter 2, in Gattegno's Mathematics Textbook 1. Here Gattegno introduces patterns to us. A pattern is simply two or more trains of a given length placed side by side. We say that a white rod and a white rod are a pattern for the red rod if you are using Cuisenaire rods for base ten blocks.
Patterns are at the center of the work we'll be doing throughout Textbook 1. From here, students will learn to use all four basic operations and fractions as operators. When introducing patterns to the student, we will build a pattern first. Let's use the black rod. We'll make a pattern for the black rod and then say something like, "I just built a pattern for the black rod. Can you build a pattern for the yellow rod?"
If your student can create a pattern for the yellow rod, we will ask her to add more trains to her pattern. If the student was not able to complete the task, we want to find out why. We can not know why by telling the child no and doing it for him. Rather, we must learn to read our students and figure out how they are thinking. I would start by making a pattern for a yellow rod and then ask the child to add another train under the one just created.
When we first started making patterns, my son would rarely follow my directions and complete the task using the rod I suggested. He might choose a yellow and a dark green instead. Sometimes he chose completely different rods. At first, I saw it as defiance. Since rod choice is arbitrary, I decided to give him the option of which block to use. He frequently decides when he does his work, and what work we do. He becomes heavily motivated when he gets some control over the work he is doing. I have noticed that when he decides, he will often choose something I think is too difficult for him.
A group of patterns placed side by side is called a mat. A mat should not be confused with a complete pattern. A complete pattern is a mat that contains all the combinations available for a particular rod. This gets a bit tricky. A group of patterns can be considered complete if it contains all the combinations of trains with the rods in any order. Or the pattern may be complete if contains all the possible trains and all their variations.
For instance: a pattern for the yellow rod contains two red rods and a white rod. Those rods can be made into the following trains: red + red + white, red + white + red, and white + red + red. The mat in the next image could be considered a complete pattern for the yellow rod if you are are only concerned that a train consisting of a red, a red and white exist in the pattern. We do not call this a complete pattern. For us, a complete pattern includes all the possible trains and all their variations.
It is parental/teacher preference how you want to address complete patterns. Given my definition, however, the only complete patterns your students will make are for the red, green, purple and yellow rods. The number of patterns just becomes too many.
Once your student has a mat of patterns, Gattegno suggests that we ask the student if he has a particular pattern in his mat. We made a bit of a game out of this. Mom and the kids would all make mats, and then we would take turns reading one of our patterns to see if anyone else had the same pattern, "Do you have a red and a red and a white?" This became a competition to see who could make the most creative patterns. Anything with more than 2 rods qualifies as creative.
After much handling of the rods, we will ask the student to create patterns from memory. Maybe the student has a mat for the orange rod; we will then ask her to add to it by speaking the new pattern out loud without looking at the blocks. We have math masks for this kind of thing. Our math masks are pillow cases with faces drawn on them. This prevents little eyes from peeking, and my kids think they're hysterical to wear.
Once the student has experience with a few mats, we will say a pattern and ask the student to find the rod that makes that pattern. In the beginning, my son would need to make the train and look for the correct rod. After a couple of months, he could do it mentally.
So Why Are We Building Patterns, And Reading Them?
- Reinforces the idea of equal. Many children develop the idea that the equal sign means "write the answer here". It takes time to understand sameness/equivalence.
- Rods are made up of other rods and numbers are made up of other numbers.
- We are laying a strong foundation right now. If you've read my other blog posts, you can now look at the image of the mat and recognize addition, subtraction, multiplication, division and fractions.
- The children are playing around with combinations and sets. We are learning what can be included in a set, different ways things can be combined and still be considered part of a set.
- Exploring the associative and commutative properties of addition. It doesn't matter how the blocks are ordered or how they are grouped the sum is still the same.
- We are gradually training children's ears to hear and then translate what they hear into creating with the blocks; and also to look at the blocks and translate what they see into words and eventually into the written symbols of mathematics.
Chapter 2 of Gattegno Mathematics Textbook 1 has 39 activities in it. This doesn't mean 39 lessons, but 39 activities you can do over and over again with a pre-k/k student. Don't rush. We've been in chapters 2 and 3 for a long time. There is a lot to learn here so Play with your students and learn math with them.