Developing A New Eye For Mathematics Classrooms

Introduction

In the past few decades, we in the mathematics education community have come to see learning as the process of thinking through ideas —often puzzling, difficult ideas. We have come to see teaching as providing guidance and support for such thinking. And we have come to see that debate and discussion should be regular features of mathematics classrooms, even extending on occasion beyond class time and classroom walls. These changes constitute a significant departure from established perspectives on mathematics learning and teaching, and from established images of mathematics classrooms, in particular, the perspectives and images held by those of us who supervise teachers. In that supervisory role, we must make sense of what transpires in mathematics classrooms and have a positive impact on the teaching and learning that occur there. Therefore, it is important to ask: How should teacher supervision shift in order to support the kinds of teaching and learning of mathematics we have come to value?

This article addresses the reshaping of teacher supervision in this era of reform in mathematics education. It is based on a body of work with elementary-level administrators around issues of reform in mathematics education, carried out by Education Development Center (EDC). In the course of their work, the administrators experienced changes in several key areas of their thinking about supervisory practice. The following two quotes from participating administrators, give the flavor of some key shifts in thinking about the content observed and about the supervisory relationship:

Both comments were made after the administrators had been in the project for a while, and neither is representative of the thinking in the group at the start of the project. In this article, we discuss the kinds of relearning and change in beliefs the administrators experienced related to teacher supervision, with particular emphasis on how supervisors:

—judge quality in the content of mathematics lessons

—approach the supervisory relationship.

As part of the Mathematics for Tomorrow project, EDC staff facilitated a yearlong seminar on classroom observation and teacher supervision with three purposes: (1) to help administrators develop a sense of what is important to pay attention to when observing reformed/reforming mathematics classrooms; (2) to think with them about the pedagogy of supervision; and (3) to explore the characteristics of school culture that tend to hinder or support the supervisory process. The seminar, which consisted of ten monthly sessions, included twelve elementary-school principals, four district-level elementary mathematics supervisors, and two assistant superintendents of curriculum and instruction—all from the Boston, Massachusetts area, including Boston Public Schools. At each session administrators watched a short videotape of a classroom in which a teacher was in the process of transforming his/her mathematics instruction in accord with reform recommendations. They did the mathematics that would be presented in the lesson, and discussed it. They then viewed the videotape twice. After the first viewing they were asked, "What did you see," with special emphasis on the mathematics and pedagogy in the lesson. After the second viewing they discussed an assigned focus question designed to call their attention to a particular aspect of the pedagogy of the supervisor-teacher relationship.

The seminar allowed participants to encounter new ideas about mathematics, learning, and teaching, at the same time challenging older ideas. Central to the design is a distinctive perspective on knowledge, which is seen as the dynamic product of individuals working in intellectual communities, not a fixed body of immutable facts and procedures. Like many teachers, a large proportion of administrators were educated at a time when mathematics was viewed as a collection of facts and procedures, learning as the process of absorbing new information and practicing new skills, and teaching as the transmission of accumulated knowledge. Inevitably, these views influence the way administrators, in their supervisory roles, look at mathematics lessons and try to make sense of what they see.

The Supervisory Eye

Also influential when most administrators were being educated was a tradition of research on teaching—the so-called process-product program of research—which guided both training programs in teacher supervision and the design of instruments for use in classroom observation (Darling-Hammond & Sclan, 1992). Rooted in behavioral psychology, this tradition emphasized the decomposition of complex teaching tasks into their components and teachers were assessed on their mastery of the components and retrained if necessary (Shulman, 1986). Of particular interest were teacher behaviors associated through research studies with positive student outcomes, such as pacing of instruction (including wait time after questions); the structuring of lessons(incorporating an overview, transitions to and summaries of subparts, and a review of main ideas at the end); frequency of praise or criticism; use of lower- or higher-order questions; and so on (Brophy & Good, 1986).

The supervisory practice that has developed out of this tradition relies heavily on the use of checklists of observable behaviors. Accordingly, classroom observers go into classrooms planning to note the presence or absence of certain categorical items that have been shown to have statistical significance. During class, they pay attention to overtly observable teacher and student behaviors.

Today, a focus on behaviors alone is ill-suited for the needs of teacher supervision. Even a more contemporary list including elements such as "use of manipulatives," "small group work," and so on, will not suffice. Rather, a shift in the supervisory "eye," looking more deeply into the fabric of teaching and learning, is needed.
(See figure 1.). In their observations, supervisors should be attending to how well observed behaviors reveal student mathematical ideas, and the overarching observation strategy ought to focus on how well teachers’ decisions support the development of mathematical ideas.

Over the course of the seminar year, the administrators did change their perspectives on the content of observation, in particular, how they thought about:

Figure 2 shows a sampling of their comments on different features of the fifth-grade teacher and her students. We note the several cases where the before- and after- commnets are made by the same person.

The sample of comments represents some significant and potentially influential shifts in administrators making sense of what it means to know, learn, be engaged by, and teach mathematics. Apparently propelled by a growing recognition of the importance of mulling, exploration, discussion, and struggle in the learning of mathematics, the administrators were reshaping their beliefs about such things as i) the role of direct instruction which, they started to see, can be balanced with exploration; ii) the definition of "closure," since students use time during, as well as between, classes to discuss and make sense of mathematics; and iii) the assessment of understanding, a more complex task when understanding is no longer seen as a two-state phenomenon—either they got it or they did not —but is seen as developmental in nature. In addition, there were shifts in how the administrators thought about the role of classroom questions: "They might earlier have thought it adequate to observe that the teacher asked questions of all students in the class, evenhandedly; over time they began to wonder whether the teachers’ questions provided students with good opportunities to explore important mathematical ideas" (Nelson, Davidson & Sassi, 1998, p. 18).

The Supervisory Relationship

Toward the end of the seminar, as the administrators were attaching new meaning to mathematics, learning, and teaching, they began to explore implications for changing the way they approached the supervisory relationship. Typically, as teacher supervisors, administrators see their roles as critics or advice givers, a perspective on supervision that fits with a view of teaching as the transmission of knowledge and with a view of learning as the absorption of information. However, when views of teaching and learning shift, toward a greater appreciation of the value in the mathematics classroom of the exploration of ideas, of intellectual risk-taking, and of the collegial search for understanding, then this perspective on supervision may not fit.

The administrators’ yearlong seminar was an ongoing inquiry group, which encouraged the participants to consider the value of inquiry as an overarching strategy in educational change. This appreciation led the administrators to consider the possibility that the supervisor might engage in inquiry with the teacher about the events of the classroom, rather than just give advice. Nolan and Francis (1992) describe some of the key features of such an inquiry-based approach to supervision:

• data collection is transformed from a mechanism for documenting behavior to a mechanism for collecting information to help in generating knowledge about learning and teaching

• data are collected and interpreted from multiple sources (e.g., observation, student work, and tests) and over time, in order to capture a more complete picture of teacher and student thinking.

With this kind of inquiry frame around the supervisory relationship, advice giving can become far more constructive to the teacher’s professional learning and growth than in the kind of relationship based primarily on judging the congruence between the teacher’s classroom behavior and generic research on teaching. Advice has a role in the supervisory relationship, but it needs to be based on the supervisor’s perceptions about what is important. Often, in more complex mathematics learning and teaching—especially those that extend from one class to the next—what is important may not be evident. To reach a determination, the supervisor may need to collaborate with the teacher to inquire and gather more data. Together, they may need to (in the words of the administrator quoted at the beginning of this article) "think the question that’s raised that weighs on somebody’s mind is a wonderful thing." Then, following this collaborative sense-making, the supervisor may be in just the right position to advise the teacher "this is how you can use what you know" (Sassi, 1998, p. 17).

Conclusion

The NCTM Assessment Standards (NCTM, 1995) pointed to the centrality, across a variety of purposes of assessment in school mathematics, of the same cyclical process, portrayed in the following way:

For mathematics assessment, the NCTM document left a set of clear messages for rethinking: what counts as evidence; how evidence is interpreted; and how evidence is used. Similarly, for the process of teacher supervision, itself a form of assessment, we have been arguing that, in order for the ideas embedded in the mathematics education reform movement to take root in schools, there need to be fundamental shifts in what supervisors attend to as relevant evidence in classroom observations, what sense they make of the evidence, and how they use the evidence in their supervisory relationships.

The data from the administrators’ seminar suggest that it is possible to effect a shift in what supervisors attend to as relevant evidence in mathematics classrooms and the sense they attach to it. The vehicle described was an inquiry-based seminar, in which administrators had the opportunity to do mathematics together; see mathematics as more than facts and procedures; appreciate the complexity of children’s mathematical thinking; and appreciate approaches to teaching that capitalize on children’s mathematical thinking. They also had the opportunity to experience something of what it would be like to be in a reformed

mathematics classroom and learned, for themselves, some of the behaviors necessary for such a classroom to work—respect for others’ ideas; willingness to expose one’s own, often tentative, ideas to the scrutiny of others; and subjecting ideas to critical examination.

References

Brophy, J. & Good, T.L. (1986). Teacher behavior and student achievement. In Merlin C.Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 328-375). New York: Macmillan Publishing.

Darling-Hammond, L. & Sclan, E. (1992). Policy and supervision. In C. Glickman Supervision in Transition: The 1992 ASCD Yearbook (pp. 7-29). Reston, VA: Association for Supervision of Curriculum and Instruction.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

Nelson, B.S. (1998). Lenses on learning: Administrators’ views on reform and the professional development of teachers. Journal of Mathematics Teacher Education 1(2). Dordrecht, The Netherlands: Kluwer.

Nelson, B.S. & Sassi, A. (1998, April). Cultivating administrators’ professional judgment in support of mathematics education reform: The case of teacher supervision. Paper presented to the annual meeting at the American Educational Research Association, San Diego, CA.

Nelson, B.S., Davidson, E.M. & Sassi, A. (1998, April). Building new knowledge by thinking: How administrators can learn what they need to know about mathematics education reform. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.

Nolan, J. & Francis, P. (1992). Changing perspectives in curriculum and instruction. In Glickman, C.D. (Ed.) Supervision in Transition: The 1992 ASCD Yearbook (pp.44-60). Reston, VA: Association for Supervision of Curriculum and Instruction.

Sassi, A. (1998). Cultivating perception: Reframing the nature of advice-giving in a mathematics professional development seminar.

Shulman, L.S. (1986). Paradigms and research programs in the study of teaching: A contemporary perspective. In Merlin C.Wittrock (Ed.) Handbook of research on teaching (3rd ed., pp. 3-36). New York: Macmillan Publishing.

This article is adapted from Cultivating administrators’ professional judgment in support of mathematics education reform: The case of teacher supervision, a paper presented by Barbara Scott Nelson and Annette Sassi to the annual meeting of the American Educational Research Association, San Diego, CA. April, 1998.

The work described in this article was supported by a grant from the National Science Foundation (ESI-9254479).



Barbara Scott Nelson is Director of the Center for the Development of Teaching at EDC and Principal Investigator of the NSF- and Pew-funded project, Administrators Working for Change: Materials to support administrators’ learning about mathematics education reform.

Annette Sassi is research associate in the Center for the Development of Teaching at EDC and has done research on both teachers’ and administrators’ professional judgement.

Mark Driscoll is Co-Director of the Center for Professional Communities in Education at EDC, where he has managed a range of mathematics professional development and leadership projects.