Good leaders look through multiple lenses and enter into partnerships with the teachers they work with.—A school administrator

Growing up, DonnaMarie—now an elementary school principal in an urban district for 13 years—was a good student, "even in math." As a child, she memorized mathematical procedures to get the right answers, but she had no understanding of what she was doing or why it worked. This strategy fell apart in high school when math became more abstract. She was eventually placed in a special trigonometry section and received a passing grade because she was a "good" student. DonnaMarie never took math again.

After college, DonnaMarie spent 21 years as an elementary school teacher. She prepared lessons that made it easier for children who struggled with mathematics to get correct answers. When she became a principal, she relied on what she viewed as her strengths and centered her supervisory work on language arts.

It bothered DonnaMarie that she knew so little about the math that was being taught in her school. She felt poorly equipped to judge the value of the teaching methods or to support any changes that were needed.

## Lenses on Learning

Administrators can profoundly influence teachers who are working to change the ways they teach such content subjects as mathematics. But how prepared are administrators to observe mathematics classes? Good elementary mathematics teachers not only explore new pedagogical approaches and deepen their own understandings of content, but they also develop their acuity for making sense of children's developing mathematical understandings.

How can supervisors best support teachers? Can they look beyond standards-based instruction to help teachers discern opportunities for helping children make sense of math?

The lenses through which administrators observe classrooms and the ways in which they frame discussions with teachers are central to good supervision. Content knowledge is important, too. When administrators and teachers talk about how children make sense of mathematical ideas, administrators need to bring depth and substance to the conversation.

In 1994, knowing that administrators need to explore mathematics, reflect on students' mathematical ideas, and consider the implications for their own professional practice, SummerMath for Teachers—an inservice teacher education program at Mount Holyoke College—began offering a series of monthly seminars for administrators. Although administrators learned a great deal in the program, they told SummerMath staff that they needed opportunities to continue learning and to engage in ongoing discussions. Last year, the staff of SummerMath for Teachers leapt at the chance to pilot modules of a new program—Lenses on Learning: A New Focus on Mathematics and School Leadership, developed at the Education Development Center with funding from the National Science Foundation and the Pew Charitable Trusts. The Lenses on Learning course provides a coherent program that builds seamlessly on conversations from one session to the next, from one module to the next.

- View and discuss videotapes of mathematics classes.
- Explore student thinking through an examination of student work.
- Read and discuss articles that provoke thinking about mathematics, teaching, and learning.
- Engage in mathematics explorations and discussions.
- Complete guided assignments in their schools related to the ideas explored in the course.

The Lenses on Learning seminar allows participants to explore mathematics for themselves. The problem-solving activities that they do help them appreciate the nature of mathematical understanding beyond algorithmic manipulation and allow them to redefine their personal attitudes toward mathematics. They begin to recognize their own power as mathematical thinkers and to shift their vision of what students are capable of as learners.

SummerMath for Teachers began the first pilot program by inviting principals, curriculum directors, associate superintendents, and teacher representatives from local school districts to a meeting to plan for professional development for administrators during the 1999–2000 academic year. Together, the group decided to work with the new Lenses on Learning program as the core of ongoing professional development for area administrators.

Attendees agreed that teamwork and monthly meetings were essential. Participants formed a core group of professionals that remained constant throughout the academic year. Each participant pledged to attend all of the monthly meetings and to complete homework in preparation for each one.

When the group met for the first time the next fall, it consisted of 14 administrators: 11 principals, two assistant or vice principals, and one curriculum director. An Eisenhower Professional Development grant funded this pilot project. The participants represented six very different school systems—urban, suburban, and rural—each with its own supports and limitations. All but two participants attended as part of a district team. DonnaMarie enrolled in the course with one other principal employed in her school district.

## A Community of Inquiry

What kind of change is possible when administrators have the opportunity to engage in intellectually challenging work with colleagues they trust? When DonnaMarie entered the room for the first meeting of our Lenses on Learning seminar, she apprehensively surveyed the group. Like several other administrators, she approached the course with trepidation. DonnaMarie knew that she needed to overcome the constraints that her limited math background placed on her work as a supervisor. She decided that this might be her chance to overcome her fear of math and to learn something, too.

DonnaMarie confided to the group her math phobia. Although during the first several sessions she sometimes felt anxious, she stuck with the seminar. She was motivated to improve her supervision in mathematics classes and to overcome her fears. She continued to push herself to attend each session until she became more comfortable and eventually looked forward to each meeting.

What transformed DonnaMarie's attitude? The course materials provided grounding for intellectual work. In light of new ideas, participants reexamined their long-held beliefs about learning and teaching mathematics and their own practice as instructional leaders. The community of inquiry that formed in the seminar played a central role in fostering that intellectual growth. Group members strove to understand the complexities of their respective situations, and they helped one another think through how to challenge ideas while still performing supervisory tasks.

Participants established a safe, engaging, intellectually vigorous environment for discussing their differing perspectives. They affirmed the importance of building a climate of inquiry in which questions are valued and people are encouraged to share half-formed ideas.

For example, at the beginning of one seminar session, participants shared stories about classroom observations. One principal told about observing a timed fact test. His anecdote generated a discussion that highlighted three different perspectives: getting the right answer quickly is important; getting the right answer is important, but the amount of time it takes to get that answer may not be; and both efficiency and correct responses are important, but how those answers are arrived at may not be. The participants were not surprised by the range of opinions; however, they were excited to recognize such discussion as an opportunity to articulate their current thinking, to understand that everyone's ideas are in process, and to realize the shared value of remaining open to new possibilities.

They discussed how complicated it is to attend to one's own needs as a learner while also paying attention to the group's needs and dynamics. They listened carefully to each idea, considering how it was the same or different from their own thinking. They worked to derive individual meaning from shared experiences and discussions.

In the fall of the school year, the group rethought ways in which children learn to solve two-digit subtraction problems by operating on the numbers themselves without the use of a pencil and paper. Our sample problem was

*53 – 38 = X.*Everyone agreed that the answer was*15*, but the surprise came in the number of strategies members of the group used to arrive at a solution.One administrator worked the problem as50 – 30 = 2020 – 8 = 1212 + 3 = 15

Another stated the solution as60 – 30 = 3030 – 8 = 2222 – 7 = 15

A third shared strategy was58 – 38 = 2020 – 5 = 15

After sharing strategies, participants tried to use each with a different set of numbers. Several administrators found this process unfamiliar and uncomfortable, but most participated and experimented with new ideas. Suddenly, DonnaMarie's hand flew into the air: "Is it wrong to be solving this the way I have always done so before?" Rather than treat DonnaMarie's question as an interruption, the other seminar participants used it as an opportunity to consider their own experiences with manipulating the numbers more flexibly. What is gained by such flexibility? What are the advantages of teaching everyone to compute the same way? What are the disadvantages?

As they reflected on the ways they had learned mathematics, several administrators talked about how those experiences influenced their perspectives when they observed teachers in mathematics classes. They realized that teaching mathematics should focus on the process rather than on the end product. Math instruction should develop a thinking process rather than teach rote computational skills.

## Growing as a Supervisor

A year after she nervously walked into her first Lenses on Learning meeting, DonnaMarie reports that her participation in the seminar has made a significant difference in how she sees her role as principal. Her work in supervision and evaluation has shifted. She is no longer terrorized by the thought that someone might ask her about mathematics. She has gained confidence in her ability to think about math and has been able to create opportunities for her staff to reflect more deeply about both mathematics and the teaching of mathematics. As part of the evaluation process this year, DonnaMarie asked each teacher in her school to allow her to observe a math lesson. Because she was able to focus her energy on understanding the ideas that children expressed during the classes she observed, she found that she had valuable advice to offer the teachers.

Later, DonnaMarie visited a 2nd grade classroom to observe a lesson introducing estimation. In a traditional sense, the teacher did a good job. She shared some practical applications for estimation skills and interacted well with the students. But when the teacher proceeded to tell the children how to estimate, DonnaMarie wondered whether the students could find other ways to solve the problems.

In the post-observation conference, DonnaMarie pointed out to the teacher that she had missed an opportunity to draw on the students' ideas and to learn more about what they already knew and understood. "What would have happened if you had asked the students if they could figure this out for themselves?" she posed. The teacher gave serious thought to DonnaMarie's query. As collaborative partners, DonnaMarie and the teacher talked about what had come before this lesson and what the next steps might be.

DonnaMarie has decided to take part in next year's seminar on classroom observation and teacher supervision in elementary mathematics. In this second yearlong course, with the continued support and challenge of her colleagues, she intends to further examine the implications of

*Principles and Standards for School Mathematics*(National Council of Teachers of Mathematics, 2000) for her work as a supervisor.## New Perspectives

DonnaMarie originally found it puzzling to be thinking about mathematics in new ways. She also found it liberating, because she learned she could draw on resources other than her memory. She could see that she, too, had important mathematical ideas. If she had these ideas, then students must also bring important understandings about mathematics with them to the classroom. This realization gave her fresh insights into what is important for teaching.

New ideas about teaching and learning mathematics have become central to DonnaMarie's reexamination of her work as a supervisor. She now works with a number of questions: What does a mathematics class that is based on the standards look like? What is the role of the teacher? What is the role of the students? How should the ideas of students be taken into account in planning and enacting lessons? What is the role of the supervisor? How can the supervisor's relationships with teachers be collaborative rather than critical and allow room for teachers to construct their own knowledge about mathematics, learning, and teaching?

For administrators to supervise effectively, they need ongoing opportunities to explore the links among subject matter, student learning, and teaching. The Lenses on Learning course is designed to enable administrators to redefine their relationships to mathematics, to reflect on children's mathematical thinking, and to extend this work to revise their ideas and consider the implications for their own professional practice. DonnaMarie's story provides us with rich images of ways in which deep, sustained professional development experiences can contribute to the gradual transformation of supervisors' work.