A week or so ago I had the pleasure of meeting Dr. Les Steffe, whose research we have been learning from. As he was speaking, he made a statement that I think we notice, but often as an educational system, we tend to ignore. He said, “You can’t teach math.” Now, what did he mean by this? By no means is he implying that children should not come to know math, but just that it is not something pre-made that we can hand over to students for them to take. So, then what do we do?
If you are in the educational community, you have likely learned the importance of manipulatives in the mathematics classroom. We know that manipulatives can help students to understand mathematics conceptually, but it’s difficult to know when and how to use them. How do we develop lessons that purposefully implement manipulative use, and achieve the goal of abstracting out a mathematical concept? I believe Dr. Steffe’s research has begun to answer that question.
We know that students benefit from “seeing” what is going on with mathematics and our hope is that they can abstract that understanding into a mental operation. So we use manipulatives, relating them to a mathematical concept. Often times, this doesn’t seem to be enough. Dr. Steffe has added what I have come to find to be a critical transitional component to how students use manipulatives. He has students work with actual objects and then begins to cover the objects. Imagine counting seven visible objects and being told there are four more under a cloth – the objective being to count all items together. The students begin to imagine the objects, or use their fingers, pretending they are the objects. It is a step that helps them to take a physical object and bring a slight separation from the students needing to have the actual object in order to count. It is a step in between the student needing the manipulative, and forming a mental structure for the mathematical concept. Hiding the objects helps them to separate themselves from the physical representation, and to develop a purely mental representation, that is stripped of the physical attributes, yet allows the student to still operate as if they were working with objects.
This gives us some guidance when we attempt to make mathematics real to young children through the use of manipulatives. What if we asked ourselves, “How can I have the students work with the manipulatives physically, then after they do this successfully, have the objects present but hidden in order for them to begin to imagine them?” Eventually, they will no longer need anything in front of them, because they will have developed mental structures around the mathematical concept. The research is showing that we can use this process for students to come to know how to count, add, subtract, multiply, divide, and to develop fractional ways of thinking. I’ll share more in my next blog.