Contexts for PD
The activities of a strong professional development program for teachers implementing CMP emphasize four areas:
Experiencing
Planning
Teaching
Reflecting
Experiencing
First and foremost, to implement a problem- centered curriculum like CMP, teachers need deep understandings of the key mathematical ideas and ways of reasoning that are embedded in solving the problems within a unit. In addition, they need to see how understanding of these ideas develops over time and connects to content in other units. Thus, during the workshops (or whenever teachers are being introduced to a new unit), teachers should experience the curriculum in a way that is similar to what their students will experience. This does not mean they need as much time for each problem, nor that they must do every problem. Problems for the workshops should be chosen to highlight the development of key mathematical ideas. The supporting problems can be more quickly examined so that the flow of development is clear, but the main focus is on the key idea.
Professional development leaders should model good teaching; they should set a context for teacher learning, encourage teachers to investigate, and help teachers make their conclusions explicit. This allows teachers to focus on making sense of the mathematics needed to solve the problems posed. By setting the context as, "How do you think your students might solve this problem?", the workshop leader can shift the focus to students' understanding. Teachers should be encouraged to make a good faith effort not to superimpose their own store of remembered knowledge on to each problem. The goal is not so much to find the answer to the problem as it is to ask, "What would my students bring to this problem? What solution strategies might they try? Which seem productive and rich in mathematical ideas? What are some of the misunderstandings that students might evince, and how can I best use discussions around these misunderstandings to help everyone learn more?"
Some teachers may think that the problems, or the mathematical ideas, are too difficult for students. A powerful strategy for helping teachers with the mathematics and showing what students can learn is to use examples of student work. This can also help teachers gain a deeper knowledge of the mathematics that underpins the problem and the unit. It allows teachers to ask questions they might be reluctant to ask. In such an environment teachers can and do learn the mathematics of a unit. Positioning the mathematics and the teaching through the lens of the student helps provide a comfortable environment for discussions of teaching and learning of the mathematics.
Through follow-up discussions of the problems, the mathematical potential of the problems, the reasoning that students employ, and the connections that can be made become more explicit. Through such interactions teachers begin to value such questions as, "What is the mathematics? At what stage are we in the development of understanding of the key idea? What do students need to bring to the problem? To what do these ideas connect in a student's future study of mathematics?"
While effective teaching strategies are modeled and occasionally discussed during the study of a unit by the participants, it is most effective if attention to teaching becomes explicit in the professional development. Teachers will need help with the teaching model. Knowing how to launch a problem, how to assist and guide all students during the exploration, and how to summarize student understandings and strategies are very crucial to the development of the mathematics. A good stimulus for discussions on teaching is observation of good teaching, either live in a classroom or on video. Analyzing students' strategies can lead to conversations on how the classroom environment/discussion may have affected learning. This is also a time for teachers to collaborate with colleagues to make sense of the evidence for good teaching. Questions like, "What aspects of the launch were effective? What aspects of the summary were effective? What is my evidence? How would I have addressed that student question?" prepare teachers to become reflective practitioners.
Planning
Planning is key to success with any problem- centered curriculum such as CMP. Professional development activities should include opportunities to plan collaboratively. Occasionally teachers should be asked to plan together to teach a problem, asking, "What is the mathematics? What difficulties will students encounter? What mathematical discoveries might they make? How will I launch this?" It is crucial that administrators recognize that, while the planning load reduces somewhat after the initial implementation stage, there will always be a need for teachers to plan lessons and reflect on what students learned from the lessons, and for administrators to help find time for these planning and reflecting activities. This is a way to optimize and continue professional development.
For each class session it is important for teachers to identify the mathematical concepts or strategies, their stages of development, and the time needed to develop these understandings. The power of CMP does not lie in any one activity or any one unit. Important ideas are studied in depth within a unit and further developed and used in subsequent units. It is both the depth of understanding within units and the careful building and connecting of the units that allow students to develop to their fullest mathematical potential.
Initial planning can occur in the first summer, prior to the implementation of the CMP. However, teachers also need time during the year to plan, particularly with their colleagues. Planning sessions allow teachers to share problems they have experienced, learn new ideas from their colleagues, probe the mathematics more deeply, look for connections, and plan upcoming class sessions.
Once teachers are comfortable with the mathematics and inquiry-based instructional model, they are ready to look more closely at assessment and how to use assessment to evaluate students' knowledge and to inform their teaching. Fuller discussions on assessment are appropriate during the second year of teaching CMP and continued professional development. However, examining student work with a colleague is valuable. Asking questions about what the students' work shows not only deepens teachers' knowledge, but it can also serve as a guide to planning effective teaching strategies. Planning also allows teachers time to discuss and share management and grading strategies as well as ways to address the needs of diverse student populations.
See suggestions for planning to teach a unit.
Teaching
Teachers need to think critically about creating a classroom environment that fosters students' expectation that they will work together to solve problems, reason about possibilities, justify their ideas, and solutions, and look for connections. Posing problems that provide a challenge for the students, allowing students to explore the problem and guiding class discussion on the solution of the problems requires the teacher to play many roles at the same time.
Teachers need help in learning how to ask effective questions that can guide and probe students' understanding, and at the same time they need to learn to listen carefully to their students. These are not skills that teachers, even those with many years of experience, have traditionally practiced. District administrators who take the time to become knowledgeable about inquiry- based learning are better able to support teachers directly. They can help set the expectation that teachers will collaborate and learn from and with each other as the curriculum is enacted.
Setting and achieving high expectations for understanding, problem solving, representing, and communicating for all students is a task that confronts teachers on a daily basis. Reflecting on one's practice with a lens on student understanding is important for teachers to make progress. Establishing the kind of environment in a school where administrators support and expect teachers to collaborate with each other can change the whole school's daily focus to such teacher questions as, "What evidence do I have that my students learned something? What did they learn?"
Reflecting
Professional development activities should model reflective practices. It is through reflection on their teaching and their students' understanding that teachers continue to grow in their capacity to build powerful mathematical experiences for all students. Planning with a colleague, peer coaching, observing a peer, or sharing with colleagues are some ways to encourage reflection.
Videotapes of lessons can serve as a catalyst for reflections. Caution must be exercised if videos are used. If a teacher within a building agrees to be video taped, then the focus should be students' learning rather than critique of the teacher. Centering conversations on student learning is a way to help the teacher think about his or her practice. "What is there about my students' ways of approaching the problem that I like?" Why do I think this is effective? What should I do to encourage more of this? What aspects of my students' actions are not productive? Why is this? What can I do to redirect my students?" Finding the fine line between trying to help the students be successful with the problem and allowing the students freedom to explore a more open problem will take reflection and growth over time.
Similarly, using rich collections of student work in professional development activities can focus teachers' attention on the role and importance of the summary phase of the lesson. When teachers study a collection of student work, some say that all of the solutions are acceptable, or they correct those that are not, and go on to the next lesson. But it is the analysis and comparison of the collection of student work that can bring the important mathematics to the forefront. Student work can also be the way to center discussions and reflections on students' understandings.
A variety of assessment tools can be used including mathematical reflections, quizzes, unit tests, projects, and district wide instruments. To be effective, discussions on student learning should go hand in hand with discussions on teaching. A focus on student learning leads naturally to looking at the development of ideas over time. Talking and planning with colleagues in different grade levels provides the opportunity for teachers to build and share a coherent curriculum vision. Collaboration and reflection are key elements in creating a community of teachers and administrators within the school that can support improvement in teaching and learning over time.