Making connections is essential for deep understanding in mathematics. Teachers have often expressed frustration about the fact that they can use manipulatives to teach, but when they work with the numbers students are not successful. I believe one of the issues is that students are not connecting the concrete experience with the abstract. In this two part blog series I’m going to show you a simple way to guide students to make these connections.
I’m going to use the concept of decomposing and composing as we journey from the concrete to the abstract. Decomposing and composing is new language in the common core, but the idea has been around in mathematics for a long time. In fact in Table 1 of the Common Core Standards they use the language of put together and take apart as one of three addition and subtraction situations. (If you want more info on those situations, check out my blog series on Teaching Addition and Subtraction.)
In this series, I will show you an example of a series of tasks for kindergarten and/or 1st grade. My goal is for these ideas to intrigue you and maybe cause you to come up with a different creative series of tasks that follow the same path from concrete to abstract. In part one, we are going to look at the concrete and representational. In part two, we will look at the representational and abstract. Each stage will take some time depending on your students understanding. Don’t rush through, the concrete stage may last the longest because it is important during the representational stage that they remember their experience in the concrete stage.
1. Start by having students show you numbers using their fingers. Use a small number that they can find success with like three (in kindergarten) or five (in 1st grade). Have all the students use only one hand to show you the number.
2. Now explain that you are going to decompose that number. Have students show you the same number using two hands, and ask different students. How many fingers are you holding up on your left hand? . . . right hand? How many fingers do you have altogether?
3. Have the students hold up their left hands and explain that this is one part. Then have them hold up their right hands and explain this is their second part. When you have two parts and put them together you have a new whole.
It is a big developmental step for students to see each part and the whole and be able to communicate about it. When I have observed kindergarteners and 1st graders, they often answer addition and subtraction with only using the strategy of one by one counting. They would count one hand to verify it was the correct number. Then they would count the second hand to verify it was the correct number. Last they would count all the fingers. This is an appropriate way to get an answer, but we want to see students develop more sophisticated strategies using subitizing, counting on, and decomposing and composing numbers.
Use an Ellison Die Cut Machine to make a lot of hands. Choose a number, and using two hand cut-outs students show you different ways they can decompose the number. They can glue the hands on a piece of construction paper.
Students don’t have to show you all the ways you can decompose a number in the beginning. Their ability to decompose the number in more than way will happen over time.
Next weeks blog post I will be taking the journey from the Representational Stage to the Abstract. Also, in part two I’ll have some work mats your students can use to demonstrate their ability to decompose using their fingers.