Megan L. Franke

Capturing Teachers’ Generative Change: A Follow-Up Study of Professional Development in Mathematics

This study documents how teachers who participated in a professional development program on understanding the development of students’ mathematical thinking continued to implement the principles of the program 4 years after it ended. Twenty-two teachers participated in follow-up interviews and classroom observations. All 22 teachers maintained some use of children’s thinking and 10 teachers continued learning in noticeable ways. The 10 teachers engaged in generative growth (a) viewed children’s thinking as central, (b)possessed detailed knowledge about children’s thinking, (c) discussed frameworks for characterizing the development of children’s mathematical thinking, (d) perceived themselves as creating and elaborating their own knowledge about children’s thinking, and (e) sought colleagues who also possessed knowledge about children’s thinking for support. The follow-up revealed insights about generative growth, sustainability of changed practice and professional development.

A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction.

This 3-year longitudinal study investigated the development of 82 childrens understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms. An understanding of most fundamental mathematics concepts and skills develops over an extended period of time. Although cross-sectional studies can provide snapshots of the development of these concepts at particular points in time, longitudinal studies provide a more complete perspective; however, relatively few studies have traced the development of fundamental mathematics concepts in children over more than a single year. In this paper we report the results of a 3-year longitudinal study of the growth of childrens understanding of addition and subtraction involving multidigit numbers. We focus particularly on childrens construction of invented strategies for adding and subtracting multidigit numbers. The overarching goal of the study was to investigate the role that invented strategies may play in developing an understanding of multidigit addition and subtraction concepts and procedures. We trace the development and use of invented addition and subtraction strategies and examine the relation of these strategies to the development of fundamental knowledge of base-ten number concepts and the use of standard addition and subtraction algorithms. Finally, we consider what the use of invented strategies affords by way of avoiding systematic errors and extending knowledge of basic multidigit operations to new problem situations.

Using Children’s Mathematical Knowledge in Instruction

This article describes how knowledge of children’s thinking in mathematics, derived by using a cognitive science research paradigm, was used by a first-grade teacher to make instructional decisions. Children in the classroom learned mathematics to a level that exceeds what is recommended by the NCTM Standards (NCTM, 1989). The study is situated in the Cognitively Guided Instruction (CGI) project chain of inquiry. We describe the teacher’s knowledge, beliefs, and method for using research-based knowledge of children’s thinking in addition and subtraction in her classroom. We describe her mathematics curriculum, the expectations she had of children, and the way her classroom was structured to enable her to continually assess children’s thinking and knowledge. The article includes statements of the teacher that indicate the importance of the research-based knowledge to her as she taught. The article ends with some speculation about how knowledge of children’s thinking can influence curriculum reform.

Keeping It Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching

We analyze a particular pedagogy for learning to interact productively with students and subject matter, which we call “rehearsal.” Our goal is to specify a way in which teacher educators (TEs) and novice teachers (NTs) can interact around teaching that is both embedded in practice and amenable to analysis. We address two main research questions: (a) What do TEs and NTs do together during the kind of rehearsals we have developed to prepare novices for the complex, interactive work of teaching? and (b) Where, in what they do, are there opportunities for NTs to learn to enact the principles, practices, and knowledge entailed in ambitious teaching? We detail what happens in rehearsals using quantitative and qualitative methods. We begin with the results of our quantitative analyses to characterize how typical rehearsals were structured and what was worked on. We then show how NTs and TEs worked together to enable novices to study principled practice through qualitative analyses of a particularly salient aspect of ambitious teaching, namely, eliciting and responding to students’ performance.

A Longitudinal Study of Gender Differences in Young Children’s Mathematical Thinking

One area in which gender differences in mathematics have been studied minimally deals with strategies used to solve mathematical problems. The limited evidence available suggests that there may be some gender differences in problem-solving strategies. Differences have been found in grades 1-3, with girls tending to use observable strategies (such as counting) and boys tending to use mental strategies (Carr & Jessup, 1997). Gallagher and DeLisi (1994) studied high-ability secondary school students and reported that while there were no overall differences in the number of selected SAT items answered

Using Designed Instructional Activities to Enable Novices to Manage Ambitious Mathematics Teaching

If teacher education is to prepare novices to engage successfully in the complex work of ambitious instruction, it must somehow prepare them to teach within the continuity of the challenging moment-by-moment interactions with students and content over time. With Leinhardt, we would argue that teaching novices to do routines that structure teacher–student–content relationships over time to accomplish ambitious goals could both maintain and reduce the complexity of what they need to learn to do to carry out this work successfully. These routines would embody the regular “participation structures” that specify what teachers and students do with one another and with the mathematical content. But teaching routines are not practiced by ambitious teachers in a vacuum and they cannot be learned by novices in a vacuum. In Lampert’s classroom, the use of exchange routines occurred inside of instructional activities with particular mathematical learning goals like successive approximation of the quotient in a long division problem, charting and graphing functions, and drawing arrays to represent multi-digit multiplications. To imagine how instructional activities using exchange routines could be designed as tools for mathematics teacher education, we have drawn on two models from outside of mathematics education. One is a teacher education program for language teachers in Rome and the other is a program that prepares elementary school teachers at the University of Chicago. Both programs use instructional activities built around routines as the focus of a practice-oriented approach to teacher preparation.

Teacher Questioning to Elicit Students’ Mathematical Thinking in Elementary School Classrooms:

Cognitively Guided Instruction (CGI) researchers have found that while teachers readily ask initial questions to elicit students’ mathematical thinking, they struggle with how to follow up on student ideas. This study examines the classrooms of three teachers who had engaged in algebraic reasoning CGI professional development. We detail teachers’ questions and how they relate to students’ making explicit their complete and correct explanations. We found that after the initial “How did you get that?” question, a great deal of variability existed among teachers’ questions and students’ responses.

Learning to Teach Mathematics: Focus on Student Thinking

Megan Loef Franke is associate professor of education at the University of California, Los Angeles; Elham Kazemi is assistant professor of education at the University of Washington, Seattle. T FIELD OF MATHEMATICS EDUCATION has made great strides in developing theories and research-based evidence about how to teach elementary school mathematics in a way that develops students’ mathematical understanding. Much of this progress has grown out of research projects that engage teachers in learning to teach mathematics. These projects have not only shown what is possible for teachers and students but have also provided insight into how to support teachers in their own learning. We use one particular research and development project, Cognitively Guided Instruction (CGI), as an illustration of how theory and research inform the teaching and learning of mathematics. The key to CGI has been an explicit, consistent focus on the development of children’s mathematical thinking (Carpenter, Fennema, & Franke, 1996). This focus serves as the guide for understanding CGI’s contributions as well as its evolution. Initiated over 15 years ago by Carpenter, Fennema, and Peterson, CGI sought to bring together research on the development of children’s mathematical thinking and research on teaching (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). This research project engaged first-grade teachers with the research-based knowledge about the development of children’s mathematical thinking. We studied the teachers’ use of this knowledge within their classroom practice and examined the changes in teachers’ beliefs and knowledge. This work created a rich context for our own learning as well as for the teachers. We learned a great deal about teachers’ use of children’s thinking, teacher learning, and professional development. Since the initial project, our understandings and our work with teachers and students have continued to evolve. We now work with K-5 teachers across a number of different content areas (Franke & Kazemi, in press; Kazemi, 1999). Our learning reflects much of the learning occurring in the field. In this article, we divide our learning about CGI into two sections. These sections represent an ongoing shift in our thinking from a consistent cognitive paradigm to a more situated paradigm. These different notions of learning have influenced our views of student and teacher learning and how to maximize that learning. We tell this story by describing our initial CGI work and what we learned about students and teachers. We then elaborate by describing and characterizing our current work and how that has influenced our learning about creating a focus on students’ mathematical thinking within professional development.

Algebra in elementary school: Developing relational thinking

We have characterized what we callrelational thinking to include looking at expressions and equations in their entirety rather than as procedures to be carried out step by step. For the last 8 years, we have been studying how to provide opportunities for students to engage in relational thinking in elementary classrooms and how to use relational thinking to learn arithmetic. In this article, we present interviews with two third-grade students from classrooms that foster the use of relational thinking. In both cases, we focus on the distributive property. The first example illustrates how a teacher scaffolds a sequence of number sentences to help a student begin to relate multiplication number facts using the distributive property. The second example shows another student who is already using the distributive property and the extent of his knowledge.

Teachers’ shifting assessment practices in the context of educational reform in mathematics

Abstract This paper presents a study of primary and secondary mathematics teachers’ changing assessment practices in the context of policy, stakeholder, and personal presses for change. Using survey and interviews, we collected teachers’ reports of their uses of three forms of assessment, one linked to traditional practice (exercises), and two linked to reforms in mathematics education (open-ended problems and rubrics). Findings revealed several trajectories of change in the interplay between assessment forms and the functions that they serve. Teachers may implement new assessment forms in ways that serve ‘old’ functions; teachers may re-purpose ‘old’ assessment forms in ways that reveal students’ mathematical thinking. Our developmental framework provides a way to understand the dynamics of teacher development in relation to ongoing educational reforms.