Developing A Sense of Place Value: Gattegno Textbook 2
Instead of taking a brief stab at all the concepts I covered in my last post regarding Gattegno's Mathematics Textbook 2, I will dedicate a blog post to each concept and show you where Gattegno is going with the material. I can't show the entire route, otherwise we'd be covering the entire curriculum. Even if I can't give you the entire route, I can at least show you where the next turn is. This post I am covering place value. So much for the 2 post overview.
Place Value and Number Building
Early last spring, I introduced P. to place value using fingers. I would actually do that exercise somewhere near the end of book one and the beginning of book 2. Gattegno writes about it in the first chapter of the Common Sense Teaching of Mathematics.
There is a lot of focus in the Gattegno Textbooks on number building and place value. Place value is more than just saying the names of numbers and being able to read and write them. It's about knowing how numbers are built and manipulating those numbers to our advantage. It is all about algebra.
Place Value: Addition
When Gattegno introduces addition, he has students write out problems in the expanded form. For example, 356 + 245 would be written 3 x 100 + 5 x 10 + 6 + 2 x 100 + 4 x 10 + 5. You will likely say something like: Good grief, that is confusing! Why not just show students the standard algorithm for addition and be done with it, it's faster and more efficient.
My answer back would be: Faster and more efficient for what?
What is our goal? Is our goal to get our students to the right answer or is our goal to create students who know and understand mathematics, who also get the right answers?
Before you panic, problems are not written this way forever. They are only written this way in the beginning so that students understand what it is they are doing when they are adding and subtracting and multiplying. We are also learning about the Associative and Commutative Properties of Addition, this understanding will come in handy when we get to formal algebra. Below, I created a video that shows how Gattegno students are introduced to adding numbers.
You'll notice that students are adding in a line rather than vertically in the beginning. Later in book 2, he uses the vertical arrangement for problems and then he asks the student which way of writing problems is easier for them. This fits with the theme that there is no "right" way, only the way that works best for them and the way they think.
Adding On The Gattegno Place Value Chart
The Gattegno Place Value or Tens Chart is useful for developing a students awareness of reading, writing and ordering numbers, and the concepts of addition and subtraction using place value. The transition areas - tens, hundreds and thousands - are easily explored on the chart. We can also use the chart for activities that develop awareness of powers of ten, fractions and percentages. Place value is at the center of student's awareness.
In the video below, I start with counting. Counting on the chart helps a student understand how numbers are ordered. You'll want to start with first naming numbers, and then by counting by ones and moving on to skip counting by various numbers. I also show you how we do addition on the Gattegno Chart.
Alf Coles has written a great post over at NRICH about activities on the Gattegno Chart, you'll be surprised how much even young children can abstract from the chart when place value is laid out before them. I will be posting additional video of how we use the chart in future blog posts.
You can get the Gattegno Chart over at Learning Well At Home here. We use ours a lot, so I laminated mine.
Place Value: Beyond Addition
Place value isn't just about addition and subtraction. It is at the heart of our understanding of mathematics. I found this recently on the MTBoS (Math Twitter Blog-o-Sphere).
I had questions about this tweet, a lot of them. How were we getting to one hundred without a concept of place value? How do students know what one hundred is? Have students ever skip counted by 10's? If students don't have a concept of place value yet, why are we asking them which ones they would rather have? That question assumes an understanding of place value.
Place value isn't only about knowing that 245 is the same as 200 + 40 + 5. It's not just about the position and value of numbers. It's also about decomposing numbers and knowing their relationship to the other numbers in our base 10 number system. Without this understanding, math becomes a mystery and the confusion carries over into multiplication and division.
In the twitter example above, a student would need to know that 16 tens = 100 + 60. But first they must understand that 15 is the same as 1 x 10 + 5. The student must then make the leap from 1 x 10 + 6 to (1 x 10 + 6) 10. So sure, kids can count and regurgitate numbers back to you. And you can even have them adding/subtracting multi-digit numbers if you drill the algorithm into them. But parroting back numbers and memorizing a procedure does not mean a student knows what's going on.
So when you look at Gattegno's Textbook 2, and you see that he has numbers broken down over and over, he is hammering home the idea of place value. He has students write numbers in different ways: 5000 can be written 500 x 10 or five hundred of the 10's. It can be written as fifty of the 100's or 50 x 100 or it can be written as 5000 x 1 or five thousand of the ones.
Why does Gattegno teach the way he does? One of the reasons is to keep teachers from creating these kinds of problems in the first place.