In my last post, I talked about building on what students can do instead of focusing on what they can’t do. One way many of us try to do that is by using a variety of manipulatives in our classes.
I love using concrete objects with students of all ages. But I can remember several times when other teachers or administrators made fun of me for using “toys” with my 8th graders to teach algebra.
Manipulatives can be a great way to provide access into lessons for all students and when used well they can help build deep understanding.
But as two researchers, Stein and Bovalino (2001) pointed out: “manipulatives do not magically carry mathematical understanding.”
They identified two problems that can occur when using manipulatives.
1. They can sometimes be used in a way that only requires students to mindlessly follow what they see the teacher do. As a result, students mindlessly move around blocks just as they would mindlessly use a formula without meaning.
2. They can lead to unsystematic and non-productive exploration. If the mathematical goal is not clear and the activities are not carefully designed, the manipulative does not serve as a tool for developing the concept the teacher intends.
The researchers contend (and I would agree) that manipulatives need to be used in a way that links an activity a student can do to a new concept by allowing them to take in new information and give it meaning.
I’ll give an example. I recently worked with young students on how to find the area of rectangles. I gave out square tiles and asked them to find the area of a rectangular region. All of the students could complete the task by simply covering the region with the square tiles. What did they learn by doing this? Not much. At this point–they haven’t learned anything new.
It wasn’t until I asked them to how to find the number of squares without counting all the tiles that they began to develop new understandings. Some counted the number of rows and the number of tiles in one row and multiplied. Others counted the number of columns and the number of tiles in one column and multiplied. Others used repeated addition. As we worked on different sized regions and shared different strategies, they began to develop a rule for finding the area. Eventually, they were able to find the area when the dimensions were given without using any tiles.
Manipulatives allowed them to access the activity and have a concrete way to justify their answer but using the manipulatives alone did not lead to the new understanding developed.
If the student could do the activity before the lesson, he or she did not learn anything. Learning occurs when students use what they can do to develop a new understanding.
Or as the researcher Doug Clements (1999) says: “Students may require concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so.” Encouraging that reflection is a difficult task and one that I think often goes missing from lessons.
How do you use manipulatives? How do you encourage your students to reflect on their work with them? I’d love to know.
What to know more? Read what the research has to say:
Clements, D. H. (1999). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1, 45–60.
Stein, M. K., & Bovalino, J. W. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6, 356–359.