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November 2000

Finding a Mathematical Voice: An Interview with Deborah Schifter

In her years of research and collaboration with teachers, Deborah Schifter knows how difficult it is to change the way you teach. It's particularly hard in mathematics, where prescriptive textbooks have provided a welcome crutch for many teachers.

"The cleanliness of traditional math teaching has worked for a lot of teachers," says Schifter. "It's a time of day when they can almost relax and rely on the lesson plan and the answers in the textbook." In her professional development program, Developing Mathematical Ideas (see box at right), Schifter urges teachers to move away from the textbook and concentrate on analyzing and responding to students' mathematical ideas.

"Now, everything is messy," Schifter observes. "They don't know what ideas children might come up with, what they will have to evaluate and react to." DMI provides teachers with a wealth of examples of what they might expect from students—and how to prepare for the unexpected.

For example, one of the DMI videotapes features a case study of a real teacher struggling to understand the thinking of a student who, in turn, is struggling with the assigned problem (see example). "The student comes up with this answer and the teacher isn't sure where he had gone wrong. So she made that a homework assignment. The class ended up spending two days figuring out where Thomas had gone wrong."

"This was a teacher who said she had always hated math," Schifter adds. "But doing this kind of math brought back to her why she had wanted to be a teacher. Instead of hiding behind a textbook, she could explore mathematics along with the kids."

Schifter understands the appeal of cleanliness in math. "There is something cleaner about mathematics than other subjects. That's part of what drew me to it. It wasn't political like social studies, where something controversial like abortion might come up, or writing class, where students might write personal essays about difficult issues in their life. In math, you can just focus on students' ideas and their thinking."

And yet Schifter sounds a little like a writing teacher when she discusses the power of mathematical thinking. "Children need to learn that they have ideas worth listening to, that they have a mathematical voice, and that is a powerful way to communicate. We want teachers to take their students and themselves seriously as mathematical thinkers."

Of course, Schifter also acknowledges that mathematics has become politicized-with a national debate ranging over textbooks, the role of the teacher, and standards. "We're not saying, as some critics of the reforms have charged, that any answer is okay. What we are saying is that the quality of the logic underlying the answer is important. Teachers need to be able to follow a mathematical line of thought and help students see which steps were good ones and which ones got them off in the wrong direction.

"It's not about just simply giving teachers a pep talk. We can all be cheerleaders, but in the end they really have to understand the mathematics."

Example: Delving Into Student Thinking

Teacher Liz Sweeney asked her fifth grade class to come up with methods for solving some two-digit multiplication problems. Jemea solved 29 X 12 correctly when she rounded the single factor, 29, to 30. She added twelve 30s to get 360 and then subtracted 12 to get 348.

Thomas used a rounding strategy also, but his led him astray. Setting out to simplify the problem, 36 X 17, he added 4 to the 36 to get 40 and 3 to the 17 to get 20. That left him with a multiplication problem he could solve easily: 40 X 20 = 800. Thomas then subtracted the 4 and the 3 from 800 to arrive at his answer: 793. (The correct answer is 612.)

Ms. Sweeney found Thomas's strategy intriguing and asked him to present it to the class. The students saw right away that Thomas had reached an incorrect answer, but they couldn't understand why. The class spent the next two sessions investigating multidigit multiplication as they searched for Thomas's error.

They came to see that when Thomas added 4 to 36 and 3 to 17, he changed the problem to (36 + 4) X (17 + 3), which equals: 36 X 17 + 4 X 17 + 3 X 36 + 4 X 3. Thomas had added in a lot more than the 4 and the 3 he later subtracted.

 

Measuring the Impact of DMI

Researchers Linda Davenport and Sophia Cohen of EDC have each conducted detailed studies to measure the impact of four on teacher learning and classroom practice. Data from the studies offer converging evidence that virtually all of the teachers came to believe that children can and do generate mathematical ideas and thoughts; that teachers themselves generate mathematical ideas and thoughts; and that these understandings can and should figure prominently in elementary classrooms. For all eight of the teachers for whom classroom observations and interviews were available, teaching practices moved toward increased student reasoning and making ideas explicit and more frequent teacher invitations to reflect both publicly and privately on these ideas. Davenport's study also indicates that teachers were better able to articulate mathematics learning goals for their students.

 

 

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Developing Mathematical Ideas

Developing Mathematical Ideas (DMI) is a set of commercially available materials designed for teachers' professional development. The materials grow out of a collaboration with TERC's Education Research Collaborative and SummerMath for Teachers at Mount Holyoke College.

DMI seminars are structured to help teachers think through the major ideas of K-6 mathematics and examine how children develop those ideas. At the heart of the materials are sets of classroom cases presenting student thinking as described by teachers. For each of five modules—Building a System of Tens; Making Meaning for Operations; Examining Features of Shape; Measuring Space in One, Two, and Three Dimensions; and Working with Data—a facilitator's guide lays out a plan for eight three-hour sessions including focus questions for case discussions, mathematics activities, and homework assignments. A videotape of the classroom cases accompanies each module. Cases are based on actual teacher writing gather during a prior EDC project, Teaching to the Big Ideas.

Developing Mathematical Ideas

Developing Mathematical Ideas (DMI) is a set of commercially available materials designed for teachers' professional development. The materials grow out of a collaboration with TERC's Education Research Collaborative and SummerMath for Teachers at Mount Holyoke College.

DMI seminars are structured to help teachers think through the major ideas of K-6 mathematics and examine how children develop those ideas. At the heart of the materials are sets of classroom cases presenting student thinking as described by teachers. For each of five modules—Building a System of Tens; Making Meaning for Operations; Examining Features of Shape; Measuring Space in One, Two, and Three Dimensions; and Working with Data—a facilitator's guide lays out a plan for eight three-hour sessions including focus questions for case discussions, mathematics activities, and homework assignments. A videotape of the classroom cases accompanies each module. Cases are based on actual teacher writing gather during a prior EDC project, Teaching to the Big Ideas.