Improving Instruction by Listening to Children
Chambers, Donald L., Teaching Children Mathematics
In the vignette that follows, identify the ways in which the teacher's behavior is or is not consistent with the NCTM's Standards. The first-grade teacher presents the following problem to her students:
"I made eight pies for my husband and he ate two of them. How many were left? Everyone get out your counters and we will do it together." She pauses briefly. "Lay down eight counters." She pauses again. "Picture them as pies. My husband eats two of them. How many pies do I have left? Write your answer beneath the line."
After making sure that each child has used his or her counters to model the eight pies, removed two of the counters, counted the remainder, and written 6 in the appropriate spot, the teacher continues, "Nan has five candy hearts. Dan gave her six more. How many candy hearts does she have now?" She asks Jim to show the class how he solved the problem. Jim demonstrates his solution on the overhead projector. First he counts out five hearts, then he counts out six into another pile. Then he counts all the hearts to get eleven. The teacher points out that Jim should have started with the larger pile and added on to get the answer. She says that they should always start with the larger pile when adding on.
Some features of this vignette seem consistent with the recommendations of the NCTM's Professional Standards for Teaching Mathematics (1991). The children are engaged in a problem-solving activity. They are using manipulative materials to help them solve the problem. They have an opportunity to describe how they solved the problem, and the manipulative materials serve as tools for discourse.
But a closer look reveals less desirable features. The teacher seems unreceptive to a student's solution unless it conforms to principles that she has formulated. In this example, the student must start with the larger pile. Furthermore, a student's communicating to negotiate good mathematics is not valued.
In traditional mathematics teaching, communication by teachers consists largely of explanations; teachers explain and demonstrate mathematical procedures to the children. Students watch, listen, and practice. Even when teachers emphasize understanding by explaining procedures, the pattern of communication remains the same. The teacher talks and the students listen.
During the last decade, the understanding of cognition in learning mathematics has increased, and curriculum developers and teachers are being encouraged to use this knowledge. Studies indicate that the behavior of teachers changes as their understanding of children's thinking increases and that this change in behavior results in changes in patterns of classroom communication and in better mathematics learning by their students.
Indeed, this teacher did change. Consider the ways in which a subsequent episode, recorded in the same teacher's classroom one year later, reflects a changed pattern of communication:
Teacher: Alice collects stamps. She had twenty-five stamps and she wants to have forty stamps. How many more does she need?
This example is a change-unknown problem, which is more complex than the simple result-unknown problem in the first episode. The numbers are also much larger. The teacher wanders around the room looking at individual papers. One child has written 46 rather than 40, so she prompts the student to correct the number by repeating, "She's got 25 and needs 40." She notices that Morgan is puzzled, so she suggests that he use counters. She restates the problem for Tom and says, "How are we going to find the answer?" She then calls on various children to explain how they arrived at a solution.
Tom: 20 plus 20 is 40, so I just took 5 away.
Teacher: Why did you take away 57 (Long wait time)
Tom: Because he had 25 so that would be 15.
Judy: I counted by nickels.
Teacher: I didn't think of it that way. Ben, would you show us on the overhead projector how you solved it? …