Paper Series Abstracts

Detailed descriptions of classroom process are needed in order to ground discussion of the principles animating the mathematics education reform movement. In response to this need, teachers who were already engaged in transforming their mathematics instruction were invited to write reflective narratives about their evolving instructional practice. This paper describes the structure of that project and presents excerpts from some representative narratives. It also considers how writing these narratives contributed to the continuing development of their authors and discusses how reading them affected students in teacher education courses. Finally, it urges that such narratives be seen as a medium through which teachers can become centrally involved in the national conversation about mathematics education reform.

Paper 2:

Goldsmith, L. T. & Schifter, D. (1994) Characteristics of a Model for the Development of Mathematics Teaching. 15pp.

The current mathematics reform movement has recognized that new forms of mathematics teaching will be needed to support the proposed curricular changes. These new forms extend beyond the acquisition of new teaching techniques and strategies to the reconstitution of fundamental notions of teaching, learning, and the nature of mathematics as a discipline, and also to the creation of different classroom opportunities for learning. The means by which teachers effect this kind of transformation are, as yet, little understood. This paper describes a set of components of developmental models that can be used to guide efforts to build models of the process of teachers' development in mathematics practice. Drawing from theories of cognitive development, the paper focuses on three components of the change process: (1) qualitative reorganizations of understanding; (2) orderly progression of changes; and (3) the contexts and mechanisms by which transitions are effected; and suggests a fourth component-individual motivational and dispositional factors.

Paper 3:

Hammer, D. (1995) Epistemological Considerations in Teaching Introductory Physics. 20pp.

Epistemological beliefs are about knowledge and learning. In a physics class, for example, some students might believe learning consists of memorizing facts and formulas provided by the teacher, whereas others might believe it entails applying and modifying their conceptualizations of phenomena. This paper explores, in the context of a debate about velocity from the author's high school physics class, how an epistemological perspective on students' knowledge and reasoning might influence a teacher's perceptions and intentions.

Paper 4:

Hammer, D. (1995) Misconceptions or P-prims: How Might Alternative Perspectives of Cognitive Structure Influence Instructional Perceptions and Intentions? 21pp.

The notion that students come to science courses with misconceptions has become quite widely accepted by those who follow or participate in education research. DiSessa and his colleagues (diSessa, 1988, 1993; Smith, diSessa, & Roschelle, 1993) have challenged the theoretical and empirical validity of the misconceptions perspective and offered an alternative account of cognitive structure in "phenomenological primitives," or "p-prims." The purpose of this article is to further clarify and contrast the two accounts, in particular to consider their utility and generativity as conceptual tools for teachers; in other words, how might each perspective influence instructional perceptions and intentions? The article recounts a discussion about forces and motion from a high school physics class; analyzes how a teacher might perceive students' participation in that discussion from either perspective; and considers what, based on those perceptions, the teacher might see as tasks for instruction.

Paper 5: (Abstract for each chapter included)

Nelson, B. S. (Ed) (1995) Inquiry and the Development of Teaching: Issues in the Transformation of Mathematics Teaching. 65pp.Chapter 1: Nelson, B.S. "Introduction". pp 1-8.

Six years after the publication of the National Council of Teachers of Mathematics' (NCTM's) Curriculum and Evaluation Standards for School Mathematics, which set the course for a new era of mathematics education reform, professional development for mathematics teachers has moved to the center of the reform agenda. The argument has been successfully made that new curricula and educational policies alone will not adequately power the reforms (Cohen, 1990; Little, 1993; Lord, 1994). Rather, they depend on the transformation of teaching in the nation's many classrooms. Many teachers have embarked on the project of changing their teaching toward that envisioned in the Standards. Their work leads us to the following questions: Where are we in our understanding of the nature of this process? How can we help teachers in their efforts to invent a new form of teaching? and How can we continue to learn about what such invention entails?

Chapter 2: Russell, S.J., Schifter, D., Bastable, V., Yaffee, L., Lester, J.B., & Cohen, S. "Learning Mathematics while Teaching". pp 9-16.

This paper examines cases of elementary grade teachers learning mathematics in the context of their own teaching, as they explore mathematics content they are using with their students, consider student strategies and representations that are new to them, and try to understand how students are thinking about complex mathematical ideas. We consider what teachers must already understand in order to do this and discuss implications for teacher education.

Chapter 3: Schifter, D. "Teachers' Changing Conceptions of the Nature of Mathematics: Enactment in the Classroom". pp 17-26.

This paper distinguishes four conceptions of mathematics as enacted in the classrooms of teachers working to transform their instruction along lines urged by the reform movement. Drawing on teachers' accounts of their own teaching, it proposes these enacted conceptions as four stages of a typical developmental trajectory. Questions concerning the implications of this model for both a theory and practice of mathematics teacher development are raised.

Chapter 4: Goldsmith, L.T. & Davenport, L.R. "Affective Issues in Developing Mathematics Teaching Practice". pp 27-36.

This paper explores the role of teachers' emotions in the process of developing a mathematics practice predicated on constructivist principles of learning and teaching. While most teachers and teacher educators would recognize from a clinical perspective that emotional responses to change processes are important, there has been little interest in engaging in systematic exploration of the roles that emotions might play in influencing the process of change itself. Below we consider possible roles that affect may play in this process in order to begin examining ways to use reflections on emotions to help promote growth in teaching.

Chapter 5: Davenport, L.R. & Sassi, A. "Transforming Mathematics Teaching in Grades K-8: How Narrative Structures in Resource Materials Help Support Teacher Change". pp 37-46.

This study examined the role of material resources in supporting teachers attempting to transform their mathematics teaching practice. Teachers in a larger funded project with regular access to a wide range of resources were asked to identify and discuss resources of significance to them. Most of the resources identified as helpful conveyed detailed information about other teachers' classrooms or contained numerous examples of student work. This information was often conveyed through narrative form. These findings suggest that resources using narrative structures to provide concrete images of teachers and students exploring this "new way" of doing mathematics can be of great value to teachers.

Chapter 6: Hammerman, J. "Teacher Inquiry Groups: Collaborative Explorations of Changing Practice". pp 47-56.

This paper describes the collaborative inquiry group structure of the Mathematics for Tomorrow project, focusing primarily on the dynamic relationship between the community being formed and teachers' learning and growth, especially around issues of mathematics and pedagogy. It presents several vignettes of mathematical and pedagogical explorations in the inquiry group as well as teachers' own descriptions of the effects of participation in the group on their thinking and classroom practice. The paper also raises a variety of questions for further investigation and research.

Chapter 7: Nelson, B.S. "Epilogue". pp 57-62.

Each of the papers in this anthology has focused on a dimension of the process entailed for teachers as they embark on the project of moving their teaching toward that envisioned in the NCTM Standards-the impact of the nature of teachers' mathematical knowledge on their visions for teaching, the role of affect in the process of change; the essential characteristics of helpful materials, and the issues to be addressed in developing a teacher community that supports investigation into practice. The existence of the set of papers invites "conversation" about the relationships between these elements: What role does affect play in teachers' developing mathematical sophistication? How can materials help teachers become better mathematicians? Does the presence of a supportive culture enhance both the expression of affect and mathematical growth? and so on.

Paper 6:

Schifter, D. (1997) Learning Mathematics for Teaching: Lessons in/from the Domain of Fractions. 24pp.

It is widely recognized that developing a successful teaching practice, one that is grounded in the principles that guide the current effort to reform mathematics education, requires a qualitatively different and significantly richer understanding of mathematics than most teachers in the 1990s possess. However, it is not as clear how teachers' mathematical understandings develop and how those understandings affect instruction. This paper explores two avenues for K6 teachers' mathematical investigations‹inquiry into mathematics itself, and inquiry into children's mathematical thinking‹and illustrates how they arise in elementary teaching situations and how they can be explored in a professional development setting.

Paper 7:

Nelson, B.S. (1997) Lenses on Learning: How administrators' Ideas about Mathematics, Learning, and Teaching Influence Their Approaches to Action in an Era of Reform. 22pp.

If the intellectual norms and values embedded in the mathematics education reform movement are to move beyond individual classrooms and significantly influence entire schools and districts, school and district administrators will need to become centrally, rather than peripherally, involved. This paper discusses the way that administrators' ideas about the nature of mathematics, learning, teaching, and school culture affect their interpretations of the nature and intent of the mathematics reform movement and their thoughts about how they might support it. In particular, administrators' views of parents' concerns, professional development for teachers, and how new ideas move around in a school are discussed. I suggest that administrators have well-formed ideas about mathematics, learning, and teaching, and that these ideas influence their views of reform and how to provide support. These ideas need to be taken into account if administrators are to be central actors in reform.

Paper 8:

Rosebery, A. S. & Puttick, G. M. (1997) Teacher Professional Development as Situated Inquiry: A Case Study in Science Education. 26pp.

In the last decade, a literature of cases has been developed that provides various perspectives on the nature of expertise in teaching, highlighting, among other things, the importance of teachers' knowledge of both subject matter and pedagogy. Recently, these studies have begun to stress the challenges, dilemmas and uncertainties that teachers face daily, and to portray these moments, as well as moments of certainty, as opportunities for examining the nature of accomplished teaching. This case study explores the ways in which a beginning elementary classroom teacher gained a foothold in this complex terrain in the domain of science. The analyses includes episodes from the teacher's first three years of teaching while she was a participant in an educational research project that, among other things, investigated an inquiry-based approach to teacher professional development. We examine the particulars of her experiences in learning scientific content and practices, as well as the particulars of her initial struggles to bring her students' ideas into contact with standard scientific knowledge and ways of knowing.

Paper 9:

Warren, Beth & Ogonowski, Mark. (1998). From Knowledge to Knowing: An Inquiry into Teacher Learning in Science. 25 pp.

In this paper we elaborate the idea of pedagogical content knowledge through close examination of a teacher’s learning in science. We address a question derived from Shulman’s (1986) original work on teacher knowledge: What is learning for teaching? We locate our exploration in a view of pedagogical knowing as a practice of seeing into the subject matter through the eyes, hearts, and minds of learners, an image we adapt from Ball (in press). We present a case study of a second year, fifth grade teacher as she conducted an investigation of aquatic ecology over a period of several months, in the context of a four-year project in which teachers examined science, science learning, and teaching through their own and their students’ experience as learners. We analyze how this teacher came to see into the subject matter, her own learning, and her students‘ learning as she worked to understand aspects of the ecology of a local pond, and how her experience figured in her identify as a learner and her practice as a teacher.

Paper 10:

Hammer, David. (1999). Teacher Inquiry. 21 pp.

The progressive agenda of science education reform, particularly the goal of promoting student inquiry, places substantial intellectual demands on teachers. If this reform is to succeed, the education community must do more to appreciate and address its demands. This paper presents three examples of high school physics teachers’ conversations about "snippets" of each others’ work with students. The purposes are (1) to highlight the central role and intellectual demands of teacher inquiry, in particular teachers’ diagnoses of students’ strengths and needs; (2) to suggest that teachers often experience and express their diagnoses in terms of instructional strategies; and (3) to suggest that the value of education research for instruction should be understood primarily with respect to what it may contribute to teacher inquiry.

Paper 11:

Nelson, B. S. (1999). Building New Knowledge by Thinking: How Administrators Can Learn What They Need to Know About Mathematics Education Reform. 19 pp.

The mathematics education reform movement is built on very different ideas about the nature of learning, teaching, and, indeed, mathematics itself, than the views that have prevailed in American schooling for many years. This has implications for administrative practice, since an enterprise that exists to support rigorous thinking on the part of students requires different administrative supports than one that exists to transmit accepted knowledge from teacher or textbook to student. Understanding new ideas about mathematics, learning, and teaching, and exploring the implications of these ideas for administrative practice, requires conceptual change on the part of many administrators. This paper describes the pedagogical principles that underlie a program designed to provide opportunities for such conceptual change for administrators.

Paper 12:

Geist, P. K. & Remillard, J. T. (2000). What an Innovative Curriculum for Teachers Reveals About Supporting Teachers' Professional Learning. 20 pp.

Researchers agree that achieving the fundamental changes called for by current reforms in mathematics education requires new learning on the part of teachers. To meet this challenge, a tremendous variety of teacher-enhancement projects, representing a range of perspectives and approaches to supporting teachers’ learning, currently exists across the country. This paper presents a comparative analysis of three teacher educators using a curriculum, Developing Mathematical Ideas (DMI), designed to serve elementary teachers in an inquiry-group setting. The aim of the study was to examine the process and demands of supporting teachers’ learning and their efforts to reform their practices. Analyses revealed that the central demand of supporting teachers’ learning through inquiry involved navigating through what we have called openings in the curriculum. These openings took the form of unanticipated questions, challenges, observations, or actions by participating teachers and required facilitators to make on-the-spot judgments about how to guide the discourse. Examinations of the teacher educators’ processes for navigating these openings revealed that they used a set of three activities in determining how to respond. Analysis of facilitators’ activities further illuminates the work involved in supporting teachers’ learning and offers implications for the type of support needed by teacher educators engaged in this work.

Paper 13:

Davenport, Linda Ruiz & Morse, Amy (2001).Fostering a Stance of Inquiry among Teachers: Professional Development in Mathematics Education. 43 pp.

Smith, Margaret Schwan, Introduction, p 1

New elementary curricula offer considerable promise in meeting new goals for student learning. These curricula represent a dramatic departure from more traditional textbooks. These new curricula, however, represent only one component of what is needed. New forms of instruction will become broadly available to students only if a substantial portion of the current teaching force transforms its current pedagogical practice. This transformation will require teacher professional development and support.

Morse, Amy, Forging a Partnership: Intent, Decision making, and Curricula, pp 3-12

This paper explores how the successful enactment of a new, reform-based curriculum requires a significantly different relationship between the curriculum and the teacher. He or she must immerse herself in the mathematics content in a way she has not before, as a learner and a seeker of sense-making.

Davenport,Linda Ruiz, Elementary Mathematics Curricula as a Tool for Mathematics Education Reform: Challenges of Implementation and Implications for Professional Development, pp 13-41.

A number of standards-based elementary mathematics curricula have been created to serve as a tool for mathematics education reform. Although these curricula have much to offer teachers, they also pose serious challenges; in order to use these curricula as intended, teachers must shift how they think about mathematics, mathematics learning, and mathematics teaching. This paper provides two stories of teachers learning to work with an innovative elementary mathematics curriculum while they are participating in a year-long Developing Mathematical Ideas seminar. An examination of these stories shows how professional development that engages teachers in thinking deeply about the mathematics content of the elementary mathematics curriculum, and exploring how students think about that mathematics content, can help prepare teachers to use standards-based curricula as a tool for reforming their practice.

Paper 14

Schifter, Deborah & Hammer, David (2001). Practices of Inquiry in Teaching and Research. 29 pp.

There are three central purposes in this paper. The first is to gain insight into the inquiry of teaching, that is, the ongoing, everyday inquiry teachers conduct into their students' understanding and learning. The second is to explore the similarities and differences between practices of inquiry in teaching and research. The third is to consider what these practices of inquiry may offer each other. We pursue these objectives by examining (1) a conversation among a group of physics teachers about a discussion that took place in one of their classes, and (2) essays by two elementary school teachers about their first and second grade students' early reasoning about triangles.

Paper 15:

Sassi, Annette (2002). Cultivating Perception: Helping Teachers To Attend to the Salient Features of their Mathematics Classrooms. 43 pp.

Efforts to align mathematics instruction with NCTM’s (2000) Principles and Standards for School Mathematics have stressed the importance of teacher learning both as a goal of professional development and as an integral aspect of teacher practice itself. This paper explores the idea of teacher learning embodied in standards-based pedagogies and argues that at its core is the belief that teachers need to be intellectually curious and need to cultivate a particular type of attention toward and appreciation of their teaching situations. Using the theoretical framework of Aristotelian practical judgment as articulated by philosopher Martha Nussbaum and specified for educational contexts by education philosopher Shirley Pendlebury, the paper considers how a critical component of mathematics teaching is “perception” — the discernment of and responsiveness to the salient features of mathematics teaching and learning. It reframes Deborah Ball’s account of her teaching as an example of Aristotelian practical judgment, judgment that rests with perception rather than with general rules or formal procedures. It then uses a discussion from a teacher study group to illustrate how learning to deliberate about the actions one should take are inseparable from cultivating perception of the salient features of one’s situation. In this case, the salient features are the mathematical ideas behind the curriculum and to the ways children come to understand them. Using a teacher’s vignette written for the study group, it looks at how the writing and discussing of classroom stories with a focus on the mathematical ideas and the students’ thinking can help teachers cultivate and refine their attention to the salient features of their classrooms.