Contexts for Professional Growth
A strong professional development program for teachers implementing CMP should emphasize the five areas summarized in the chart.
First and foremost, teachers need a deep understanding of the key mathematical ideas and reasoning that are embedded in the Problems. In addition, they need to see how student understanding of these ideas develops over time and connects to content in other Units. Thus, 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 does it mean 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 not to use their own knowledge to solve each Problem. The goal is not so much to find a solution 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 hard. A powerful strategy for helping teachers with the mathematics and showing what students can learn is to use examples of student work. This alleviates the anxiety of teachers who have never learned or understood the mathematics in the Problem or have no confidence in their ability to do the mathematics. It allows teachers to ask questions they might be reluctant to ask. Looking at the mathematics from a student’s point of view provides a comfortable environment for discussions of teaching and learning of the mathematics.
Follow-up discussions of the Problems help teachers better understand the mathematical potential of the Problems, the reasoning that students employ, and the connections that can be made. 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?”
During the portion of professional development that focuses on mathematical knowledge, effective teaching strategies are modeled and occasionally discussed. However, explicit attention to teaching needs to follow. 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 crucial to the development of the mathematics. A good stimulus for a discussion on teaching is the observation of good teaching, either live in a classroom or on video. Analyzing students’ strategies can lead to conversations on how the classroom environment and discussion may have affected learning. This is also a time for teachers to collaborate and make sense of evidence of good teaching.
Developing the habit of asking, “What aspects of the launch were effective? What aspects of the summary were effective? How would I address that student question?” prepares teachers to make the same demands of themselves.
Planning is key to success with any problem-centered curriculum such as CMP. During professional development, teachers plan together to teach a Problem, asking, “What are the mathematical ideas? What difficulties will students encounter? What mathematical discoveries might they make? How will I launch this Problem?” 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. Administrators need to help teachers find time for planning and reflecting. This will help teachers optimize their professional development time.
For each class session, it is important that teachers 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 in 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 over the 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 the inquiry-based instructional model, they can look more closely at assessment and determine how to use assessments to evaluate students’ knowledge and to inform their teaching. More thorough 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. During planning time, teachers can also discuss management and grading strategies as well as ways to address the needs of diverse student populations.
We have taken the stand that curriculum and instruction are not distinct—“what to teach” and “how to teach” are inextricably linked. The circumstances in which students learn affect what is learned. (Lappan & Phillips 1998. p. 84)
Teachers need to create a classroom environment that fosters collaboration between students as they reason through and solve problems, justify their ideas and solutions, and look for connections between mathematical ideas. Teachers must pose a Problem that provides a challenge for students, allow them to explore the Problem, and guide a class discussion on solutions to the Problem, all of which require the teacher to play many roles at the same time.
Teachers need to learn how to ask questions that can effectively probe students’ understanding. They also need to learn to listen carefully to their students. These may be new skills for many teachers, even those with many years of experience. 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 with each other as the curriculum is enacted.
The CMP curriculum requires a paradigm shift for both teachers and students. The old paradigm, in which the teacher demonstrates procedures while students listen, take notes, and imitate the teacher, is based on the idea that the teacher knows all and that students come in as blank slates. Under the new paradigm, the teacher poses a problem and facilitates student learning by listening, questioning, and orchestrating class discussions. It is based on the idea that the teacher knows the necessary mathematics and teaching strategies and continues to learn more while students build on their prior experience and knowledge of mathematics to solve problems. This paradigm shift requires that teachers continually expand their knowledge of teaching and learning mathematics. This is best achieved through collaboratively planning, enacting, assessing, and reflecting on student learning.
Teachers must set and achieve high expectations for understanding, problem solving, representing, and communicating for all students on a daily basis. In order to make progress, they also need to reflect on their practice, focusing on student understanding. By establishing a school environment in which administrators support collaboration and expect teachers to work together, teachers throughout the whole school can begin to focus on questions such as, “What evidence do I have that my students learned something? What did they learn?”
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 and peer coaching are some ways to encourage reflection.
Videos of lessons can serve as catalysts for reflections. However, caution must be exercised when analyzing videos: the focus should be on student learning rather than a critique of the teacher. Centering conversations on student learning can help teachers think about their practice. "What do I like about my students" ways of approaching the problem? "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 in solving a 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 help teachers reflect 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. Others correct those that are not acceptable 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.
To be effective, discussions on student learning should go hand in hand with discussions on teaching. A focus on student learning leads naturally to a discussion of 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.
Encouraging teachers to share their successes is one way for schools and districts to promote teacher ownership of the curriculum. Sharing can be done within the school by mentoring new teachers or through shared planning times, through district newsletters, through online discussion groups, or by volunteering to speak at local or state meetings. Networking, attending professional meetings, and joining mathematics teachers' associations are all ways to continue to grow in mathematical knowledge and in pedagogical strategies.
Often overlooked is the problem of teacher turnover, which occurs in many middle schools. It is critical to develop a plan to provide professional development for new hires. It is equally important to develop collaborative relationships with experienced mathematics teachers. Such relationships are mutually beneficial to the new and experienced teachers and, in many instances, result in lowering the rate of attrition of new teachers.
Going for More
As teachers become comfortable with CMP, it becomes a natural part of the fabric of the school. A sense of complacency, a "We've done it!" feeling, sets in. This is the time to take a more exacting look at the potential of the curriculum and ask, "Can we do better?" This is the time when teachers should look more deeply at the mathematics embedded in the Problems and find ways to better promote student understanding. These more advanced professional development experiences can re-energize teachers and result in improved student learning. Moving to this level of implementation is a crucial step and is very often overlooked. Professional development should not be a one-time or a brief experience; rather, it should help teachers stay fresh, enthusiastic, and highly effective over time.
The Power of Collaboration
A growing body of research suggests that one avenue to improve mathematics teaching and learning is to engage practicing and future teachers in collaborative professional development involving teacher inquiry which is focused on teachers’ professional concerns and situated in their schools and classrooms (Atweh, 2004; Loucks-Horsley, Stiles, Mundry, Love, & Hewson, 2010). For over a decade, education researchers have advocated for the use of lesson study, a form of collaborative teacher inquiry focused on student learning, as a strategy to improve teaching and learning in the United States (Fernandez & Chokshi, 2002; Fernandez & Yoshida, 2004; Hiebert & Stigler, 2000; Lewis, 2005; Richardson, 2004; Stepanek, 2001; Stigler & Hiebert, 1999; Yoshida, 2005). In lesson study, teachers discuss the mathematics of the lesson and study how students learn that mathematics. Teachers then collaborate to plan the lesson, with a focus on how they anticipate students will interact with the lesson content, and how they as teachers will respond to student questions and answers. Teachers then enact the lesson, often with other teachers observing the lesson specifically focusing on how and what students learn from the lesson. Lastly teachers reflect upon the lesson enactment and collaboratively debrief the lesson, and plan how the lesson might be improved. There are other forms of collaborative teacher inquiry, such as the collaborative analysis of student work on mathematics problems. However, to be effective, all teacher collaboration situated in the professional work and knowledge required for teaching mathematics, such as: (a) the examination of the mathematics and how students learn mathematics, (b) planning lessons, (c) the teaching of a lesson and observing each other teach, and (e) reflecting on teaching the lesson or the lesson observation. These essential four contexts for effective professional development are discussed in more depth later in this section.
It is through the collaborative process of sharing ideas; planning, examining student work; looking for gaps; and finding ways to make even bigger gains in student understanding, reasoning, and communication that teachers continue to move forward. Collaboration might focus on student understanding, perceived mathematical weaknesses as evidenced on local and state testing; teacher strategies; reports from teachers who have attended state or national conferences; preparing presentations for administrators, parents, meetings at the state or national level; and effective use of technology. In the early stages of implementation, the community may include the entire staff of mathematics teachers, but as implementation continues, it is likely that teachers will rely heavily on their grade-level colleagues for support, ideas, and guidance. Professional development opportunities are needed to ensure that these collaborations are able to continue throughout the implementation, even after the curriculum appears to be institutionalized at the school.
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