Interpreting the Standards: Translating Principles into Practice
Schifter, Deborah E., O'Brien, Deborah Carey, Teaching Children Mathematics
Since the publication of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991), such phrases as "mathematics should be taught for understanding," "teachers should facilitate the construction of mathematical concepts," and "classrooms should be student centered" have become identified with a reformed mathematics pedagogy. However, when issues remain framed in these very general and abstract terms, the significance for practice of this shared vocabulary of reform is difficult to assess: even a cursory survey of current reform initiatives shows that despite their prominence at conferences, workshops, and in the literature, the meaning of these phrases for day-to-day life in the classroom is hardly self-evident.
Today, around the country, real teachers, facing twenty or more real children each day, are discovering how the principles underlying the reform vision can be translated into a practice of mathematics instruction. By communicating detailed images of specific classroom events, such teachers can provide their colleagues with grounding for much-needed discussion of possible meanings - enacted meanings - of the vocabulary of reform. To begin a conversation about whether those of us who urge, say, that "teachers should facilitate student construction of their own mathematical understandings" really do share similar instructional goals, we can now ask, "You mean the sort of thing that happens in so-and-so's class when ...?"
In this article, Deborah O'Brien, a third-grade teacher and participant in an in-service project led by Deborah Schifter, draws just such a detailed image from her classroom. By examining how O'Brien translates the vocabulary of reform into pedagogical practice, we elucidate our understandings of the principles that underlie the vision of mathematics instruction proposed in the Standards documents.
One day last spring, O'Brien was working with her twenty-seven third graders on area and perimeter. The task given to the class was to find out in how many ways they could arrange twelve one-inch-square tiles into a rectangle and then find the perimeter of each resulting figure. O'Brien had laid out materials that her students might choose to use - tiles, rulers, paper. The children got to work. They knew the routine: first work individually, then share ideas with a classmate until their teacher calls them all together for whole-group discussion.
As her students worked, O'Brien circulated about the room, keeping track of emerging ideas and questions. Her attention was caught by two girls who were working together.
Lin: The perimeter is seven!
Tina: No. I'm sure it's ten.
Lin and Tina had obtained differing, and wrong, answers - and each seemed sure that she was right. They had reached an impasse, and it seemed that they were not going to find the correct answer, fourteen inches, without O'Brien's intervention [ILLUSTRATION FOR FIGURE 1 OMITTED].
O'Brien: What are you getting?
Each girl took her turn explaining her solution to the teacher. However, O'Brien was disturbed that they were not listening to each other:
"For the moment, I wanted them to talk to each other instead of to me, so I asked if they would try to see how they were getting different results. Lin showed her way again, and so I asked Tina to explain what Lin had done."
O'Brien's behavior will likely puzzle readers whose images of teaching derive from the mathematics classrooms in which they once sat as children. For many decades, mathematics has been taught in the same way: the teacher demonstrates procedures for getting correct answers and then monitors students as they reproduce those procedures. Why does O'Brien want these two children to listen to one another's faulty reasoning? And not only listen, but explain how her partner arrived at her - incorrect - answer? Such a teaching strategy seems inefficient, maybe even downright harmful - suppose that Tina later remembers Lin's way as the right way? …