Notice and Wonder for Educators| Chapter 2 of Gattegno Textbook 1
Notice and wonder for Chapter 2 of Gattegno Textbook 1 isn't just for the student. It is more than ever for the educator.
Sonya from Arithmophobia No More reminded me of this point. As one who only learned the traditional methods of teaching mathematics, I found a great need to relearn mathematics.
Notice and wonder wasn't a habit that came naturally to me. What was I suppose to notice?
I soon learned that there is only one answer to that question. That is, whatever I noticed was what I was suppose to notice.
With the practice of notice and wonder, I grew as an educator. I saw more opportunity with each time I played and looked at the rods.
An Educator's Math Journal
I talked about giving your child space to think and do dishes in the last post, but really, this is a great opportunity to spend time noticing and wondering about the rods yourself. You need to perceive math within the rods yourself. Why?
First, if the student sees something and tries to show you, there maybe a gap in understanding between you and the student. I hate to tell you but it will probably be your fault more than the student. If you already have a developed eye for the math within the rods, you will understand easier when the student tries to convey a math idea.
Second, there may be a concept the student has difficulty perceiving despite the tasks in Gattegno's textbook. If you have worked with the rods on your own, you will become aware of another way to present the concept with the rods. There are more ways than Gattegno could ever put down in the Gattegno textbooks or a handbook.
As you relearn mathematics in this gentle way, you will notice and wonder things that your student isn't ready for, which is why you keep a journal for yourself. When the student is ready, you will be ready too.
Now let's finish up Module 1. The first play mat is in the Math Task Cards for Module 1. The rest of the play mats featured here are included in the Interactive Math Notebook Bundle except the Notice and Wonder journal page which is found in the Staircase Extension Activity task cards.
Module 1.5 Complements
Complements begin to deepen a student's perception of the relationships between equivalency, addition and subtraction. They can be presented through staircases or missing rod exercises.
In either case, the task is to find a rod equivalent to a pair of rods, pairs of rods equivalent to a rod and the missing rod. After working thru all the exercises in module 1.5, a student may notice something like the following:
Educator: Can you tell me which rod is needed for the light green stair to reach the chick in the apple tree?
Student: The black rod.
Educator: *Continues working thru the play mat until all the missing rods are found.
Educator: What do you notice about the missing rods you found to rescue the chick?
Student: I see the blue stair needed a white rod and the white rod needed a blue stair.
Educator: I notice that the tan rod needed a red rod and the red rod needed a tan.
Student: It is like that for all the rods.
The last comment from students is a common one. You understand what they mean, but it isn't the clearest way to express it. There are two choices. You can ask them to show you what they mean with the rods. Then provide a clearer way to express their idea.
Or you also could assume what is meant and rephrase the statement in the form of a question. "Do you mean no matter the order of the rods they equal the same length, like purple plus a dark green equals dark green plus a purple?"
I would favor the first as the second is a bit leading. Sometimes a student will just say yes just to get on with it. They may not know what you meant.
The task card below is also a part of module 1.5. After working through a few of the blue missing rod cards, have the student look at the work they accomplished.
Educator: How many missing rods did you find for the blue rod?
Educator: What did you notice about the missing rods?
Student: I notice a purple rod, a yellow rod, a light green rod and a dark green rod.
Educator. I notice if I have a yellow rod, I need a purple rod to make the length of the blue rod.
Student: If you have a purple rod, you need a yellow rod.
Notice is a great opportunity to model a statement several ways. We can present this card as a statement of addition, difference, and missing rod. Students may notice transformations in these activities, but if they don't, no worries. There is a selection of activities in module 1.7 for that.
The educator might wonder how the statement might change if they add a white to each rod. "Is the missing rod still the same?" I wouldn't expect a young student to wonder this, but it might be an activity to extend the student's thinking if they are breezing through this task.
Module 1.6 Trains, Patterns, and Mats
Pretty much all the tasks in Gattegno's Textbook 1 and Sonya's Manual provide opportunity to talk of the same mathematical qualities of the rods over and over again. Varying the activities allow the concepts to appear fresh and interesting to the student.
Remember that after module 1.3, you should vary the activities by pulling from a different module every day. You may even considering layering by bring yesterday's task to the table for a few minutes as well as adding a new task from a different module.
The following activity is from the Number Building Play Mat that you can obtain for free when you subscribe. For the purpose of module 1, let's look at what a child can notice after working through it.
Educator: Do you notice any solid color trains?
Educator: Which colors?
Educator: Do you notice any more solid color trains?
Educator: I wonder if you can build another solid color train equal to yellow.
Of course, the student may give it a go. If not, move on. Notice and wonder is a quick activity in the beginning. Don't drag it on unless the student leads the notice and wonder.
Educator: I notice four whites fit into a purple rod. What do you notice?
Student: I notice the red mat has only two trains.
Educator: I notice it has two whites as the other train for the red mat.
Student: I notice a red and white train for the light green mat.
Educator: I notice a light green and a white for the purple mat.
Student: I notice two whites and a red in the purple mat.
Educator: I wonder if you can build another solid color train equal to purple.
Comparing trains offers more opportunity to notice differences. I enjoy using Building Equation paths as a fun alternative to building trains of rods.
The student builds the paths first. Then ask the student to compare the paths by making the paths straight and lining the paths side by side as if building trains.
In this dialogue, notice is more specific but not necessarily leading. There is enough available to the student to notice differences about each path that answers will vary depending on the interest of the student.
Educator: I notice one light green in the first path and two light green in the second path. What do you notice is different?
Student: I notice the first path has red rods and the second path does not.
Educator. I notice two whites in the first path and three white sin the second path.
Student: I notice the second path is longer.
Extend this activity by looking for the missing rod that makes the trains equivalent.
I am not going to cover the rod races as I feel Sonya's manual covered the kinds of opportunities available to notice and wonder.
Module 1.7 Transformations
Sonya says this about transformations.
At the heart of relationships is the understanding of how numbers, ideas, and situations can be transformed into similar ideas and situations that are related.
One of the first ideas students observe in Gattegno Textbook 1 is that the rearrangement (i.e. transformation) of rods create a new equivalent train.
One play mat that makes a great exercise for transformations is the Fraction Exploration Play Mats. Set up the activity first for the student by creating four piles of rods containing the exact same rods in each pile.
Don't tell the student exactly what you are doing. Just state simply that you are racing with just these rods. Then race with the student by randomly pulling one rod from each pile until the race is finished.
The student may create the same paths and that is why it is important that you build different paths from the student.
Then notice and wonder.
Educator: I notice the raccoon's path is white, yellow, yellow and white. What is the frog's path?
Student: A yellow, white, white and yellow
Educator. I notice the whites are on the outside of the yellow for the raccoon's path but on the frog's path, the whites are inside of the yellows.
Student: I notice the frog and fox ended their race with a yellow.
The student may not perceive the transformations at first, but with the other activities, in module 1.7, the student will deepen their understanding for transformations.
Module 1.8 Odd and Even
Per Sonya's manual:
Gattegno defines odd numbers as those rods whose length can be formed by a white rod and two rods of the same color, or a white rod followed by two more white rods. Even numbers have a length, which can be formed by two rods of the same color.
Sonya discusses the importance of odd and even in developing a student's insight in number behavior, and that is worth reading.
Gattegno's Textbooks are absent of these kinds of explanations. He assumes you are an educator familiar with the importance of such concepts. Sonya realizes that a lot of us are just relearning math and provides this important information in her manuals.
For the module 1, the goal for the student is to identify odd and even rods. With such a clear definition, students can verify whether a rod is odd or even with ease. You can use just about any play mat to practice identifying odd and even rods. Can you find a pair of the same color rods to cover that rod?
Another way to verify if a rod is odd or even is whether the rod can be made of all reds. Use a staircase play mat for this activity. Ask the students to build as many of the stairs with red rods. If there remains a space left for any stair, add a white rod.
Educator: What do you notice?
Student: A notice this stair is one red rod.
Educator. That is the second step. The first step is a white rod.
Student: I notice this step is all red rods, too.
Educator: That is the fourth step. I wonder if the next stair after the tallest stair will be all reds or all reds with one white.
The student may offer a guess to what they think the next stair will be. If not, you may call it a day or build it yourself to find out. Let math be curious to you and it will be curious to the student.
A side note: Any staircase work is an opportunity to reinforce the ordinal names of each step, so name the ordinal names of the stairs throughout the activity.
After students can well identify odd and even, they will enjoy playing with odd and even staircases. Odd and Even exploration play mats are a great introduction into playing with these staircases.
The above play mat is composed of 4 even stair cases plus a lone white rod. Curiously enough, you can notice that the picture is concentric odd squares, that is one squared, three squared, five squared, seven squared, nine squared and eleven squared.
A student won't notice this in module 1, but as the habit of observation is built, the student is sure to find this and other math curiosities. This is what you have to look forward too.
Last Notes on Notice and Wonder
Well, that concludes Notice and Wonder for Chapter 2 of Gattegno's Textbook 1. I didn't provide very long dialogues to encourage you to keep notice and wonder short for module 1.
These are a few examples to the million your student will discover. Be happy if all they see is a tornado. You are just building a habit of observation and curiosity for predicting patterns at this point. Keep modeling curiosity.
The most important thing you can do while waiting for the student's math-eye to develop is to develop and improve your own. If Gattegno was still discovering new insights from Cuisenaire rods after decades of play, you can too.