The teacher finishes the introduction with the following directions. “After you have finished your sculpture, you will draw a picture of the object as if it is lying flat on the paper. Then you will look at each layer, from top to bottom, count each layer and write a number sentence. Of course, the number sentence will add up to ten.”
The children get up from the rug and take their seats. In the middle of each table is a bin of the connecting cubes. Each student takes ten. Some take all of one color. Some take five each of two colors, and some just take a random assortment of colors. Immediately they start putting the cubes together to create their objects. As they attach the cubes they talk quietly, unconsciously narrating both their intentions and progress to date. “This is the goal post and here is the kicker.” “My object can move into different shapes.” “My colors are for Halloween.” “My object can transform from robot to truck.”
As they complete the objects, the students stand them up or lay them down in front of the drawing paper at their place on the table. They begin drawing the object carefully observing the location of each cube. These are free hand drawings, but the size of each cube is remarkably similar. You can see their heads bobbing up and down as they complete drawing one cube and look up to their object to see the next one. Some students count the number of squares on their drawing, touching each square with their finger and then doing the same with the object, checking for accuracy.
As they finish they call the teacher over to see what they have done. The teacher offers observations about the intricacy of the object and the drawing. “I see the pattern of colors.” The teacher might point out errors and observe corrections being made. “Is the bottom of your object drawn on the paper?”
The teacher reminds the students to finish by writing the “number sentence” on the drawing, and to count layer by layer. Almost instantly the hands of the students move to count the top layers horizontally. They write the number, followed by a plus sign, and then continue to the next layer. One student’s NFL goal post is now represented by the number sentence 2+2+4+1+1. Another student’s laser is represented by 5+2+2+1.
To conclude the lesson the teacher moves to the white board and asks the students to read their number sentences. She writes them on the board and asks, of course, “What do all these sentences add up to?” The chorus of students responds loudly, “Ten!” The teacher, refining and memorializing the fundamental concept, asks, “Is there only one number sentence that adds up to ten?” “Are there many ways to make the number ten?” Again the chorus responds enthusiastically, “There are many ways to make ten!”
As I start to leave the classroom I ask the students to tell me what they had made. Here is the list:
- Pyramid
- camera
- ghost
- medicine holder
- candy corn, stairs
- the letter L
- hand stretcher
- pullout couch
- video camera
- and thig-a-ma-gig
Finally I asked the students what they were going to say when their parents asked them, “What did you do in school today?” I said, “Of course, you will probably say, ‘Nothing.’ But if you were to say something, what would it be?” One eager and broadly smiling second grader stated, “We used connecting cubes to build sculptures and then made number sentences that added to ten.”
I was smiling as I left the classroom, thinking to myself, “It is always good to know that students do indeed understand what they are doing!”
I hope my description has helped you understand how much cognitive development was going on in this lesson. Fundamentally, intelligence is the ability to create abstract symbolic language that described physical reality. In this activity, the students first manipulated physical objects to create a symbol of a real object. Then they created a pictorial representation, and then a numerical representation. Each representation was more and more abstract, further and further away from the first hand experience of the real object. Is this not analogous to the design constraints any engineer and/or scientist must endure as she/he creates a tool, machine or physical system? Our students were guided through a learning process that both engaged the imagination and fostered the development of facility with abstract symbols. They are not just memorizing some meaningless procedure. Our students not only know how to add numbers in a flexible and fluid manner, but they understand how those numbers connect to the physical world.
They are thinking like scientists and engineers.
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