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## How Much Can We Accomplish? Elementary Mathematics Methods Revisited

by Megan Loef Franke
October 2000

Teaching elementary mathematics methods is extremely challenging. We know that prospective elementary school teachers must be prepared to teach everything from counting to algebraic thinking. They must understand the mathematics that they are to teach, as well as how to engage students in that content. For me as a methods teacher, this is not simple pedagogy. The pedagogy must be tied to the mathematics being taught.

Determining how to provide students with both the knowledge of the mathematics and the knowledge of how to teach that mathematics led me to Deborah Ball's work on knowledge-in-action. I no longer think about these two types of knowledge separately. Making instructional decisions that support student learning requires that teachers understand content beyond knowing what procedures to use, when to use them, and why they work.

Knowledge-in-action demands that teachers know how to use their knowledge of the content to enhance the understanding of their students. Teachers need to know what question to ask when a student tells them that 7 = 3 + 4 is not true. They need to know which problem to pose next and what a student could say to demonstrate understanding.

Ball's idea of knowledge-in-action helps me integrate content and pedagogy; however, it also forces me to engage preservice teachers in such a way that the methods course furnishes the basis for continued development of their knowledge.

Research on teacher learning and generative growth shows that generative teachers, teachers engaged in ongoing learning, share some common characteristics: they have detailed knowledge of students' mathematical thinking, a way of organizing that knowledge, and a view that this knowledge is their own to add to, challenge, and adapt. This conception of generativity led to focusing the methods course on detailing students' mathematical thinking and creating student-learning trajectories. With a focus on students' mathematical thinking, content and knowledge-in-action come together.

Throughout the course, my preservice teachers assess individual students. They bring examples of students' thinking to the class, they detail the thinking for their peers, they discuss how the mathematical thinking differs across examples, they consider which ideas are mathematically more sophisticated, and they talk about potential responses to the student. For instance, they may take a series of number sentences like 7 = 3 + 4 and ask students whether those sentences are true or false and why. In class, we detail each different student response and characterize the responses across grades. We discuss what is meant when a student says that it is false because the plus sign is in the wrong place or that the 7 has to come after the equals sign. We discuss different conceptions of the equals sign that students bring and how we as teachers can respond to them; we create new number sentences to challenge their different conceptions.

By having the preservice students articulate both in discussion and in writing what they know about student thinking and how they organize that knowledge for themselves, the preservice teachers move with their thinking from one class to the next, build on it, refine it, change it, and challenge it. I help them recognize that their knowledge can be developed through practice. Like me, they also learn that content and methods are not separate curricula; they must be interwoven.

Bibliography
Ball, Deborah Loewenberg. "Developing Mathematics Reform: What Don't We Know about Teacher Learning—but Would Make Good Working Hypotheses." In Reflecting on Our Work: NSF Teacher Enhancement in K–6 Mathematics, edited by Susan N. Friel and George W. Bright. Lanham, Md.: University Press of America, 1997.

Franke, Megan Loeb, Thomas P. Carpenter, Linda Levi, and Elizabeth Fennema. "Capturing Teachers' Generative Growth: A Follow-up Study of Professional Development in Mathematics." American Educational Research Journal. Forthcoming.

 Megan Loef Franke, mfranke@ucla.edu, is an associate professor in the Graduate School of Education and Information Studies at the University of California at Los Angeles, where she studies teacher and student learning in the context of mathematics professional development.

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