Listening to Reflections: A Classroom Study
Owen, Lisa B., Teaching Children Mathematics
How often do we, as teachers, listen to what students are saying about mathematics? Listening to students solve problems can render insight into the diverse mathematical backgrounds from which they come. Listening reveals when, where, and how students make connections and helps us recognize their general understandings and misunderstandings.
Most students enter school with a basic understanding of some mathematical concepts (Resnick 1987). Many are able to add and count and use counting to solve various arithmetic problems. They model problems and invent procedures when computing answers (Carpenter 1985). Teachers should consider and value these problems and invented procedures and find ways to facilitate their ideas and incorporate them into instruction.
According to the Curriculum and Evaluation Standards (NCTM 1989), children must be given opportunities to "talk mathematics" to construct knowledge. Standard 2: Mathematics as Communication asserts that mathematics instruction should include opportunities for students to "reflect on and clarify their thinking about mathematical ideas and situations; [and] realize that representing, discussing, reading, writing, and listening to mathematics are a vital part of learning and using mathematics" (p. 26).
When children talk about mathematics, they have the opportunity to think out loud as well as listen to ways that others think. They compare and negotiate ideas as they clarify their thinking. They are engaged in what the author has labeled verbal reflection.
Teachers can create an arena for talking about mathematics; they can promote mathematical talk by framing questions or statements that will empower students to initiate learning. The lesson that follows describes the interactions in one classroom and exhibits characteristics similar to the assertions in Standard 2.
A Glimpse into One Classroom
A lesson on calculation methods had just begun. Ms. Bodi, a second-grade teacher, asked her students to decide which calculation method - calculator, paper and pencil, or mental mathematics would be used when solving various problems. Hanna was asked to share a problem for which a calculator would be appropriate. She suggested 150 plus 20. She chose large numbers that seemed to suggest a calculator. While Ms. Bodi wrote the problem on the chalkboard, Hanna watched and reacted. Her idea was clarified through verbal reflection when she said, "Right there, that is 170; that could be mental math."
During the lesson, Ms. Bodi asked questions and produced activities that allowed children to decide which method was appropriate for them. A consensus rarely occurred. Ms. Bodi hoped that students would agree that basic addition facts could readily be solved using mental mathematics. She gave one student a calculator and told another to use mental mathematics to solve the problem. The following discussion ensued.
Ms. Bodi: Okay, let's do this. I want Kim to use mental math, and Evan, you will use a calculator. We are going to see which one would be faster. Okay. Evan, are you ready? Kim, is your machine on? Where is your machine? (Kim pointed to her head.) Ready? Three plus two.
Ms. Bodi: Evan, what happened? (Everyone laughs.)
Ms. Bodi: Which way was the faster way to get the answer?
Evan: Mental math.
Ms. Bodi: So, when would you want to use a calculator and when might you want to use mental math? Sally?
Sally: Sometimes it takes longer to put all the numbers in the calculator and I can add faster in my head.
Evan: When you can't add real fast, use a calculator.
Ms. Bodi: Okay, sometimes it will take you longer to put the numbers in a calculator when you could have done the problem in your head, like Kim did.
Ms. Bodi was able to prove her point with an engaging activity; students could readily solve basic addition facts using mental mathematics. …