Mathematics education research has shown that teachers possess and continually refine a particular type of mathematical knowledge specific to teaching mathematics (Hill, Ball, & Schilling, 2008). Thus, professional development should begin with an emphasis on mathematical content. Teachers need to be comfortable with the mathematics embedded in the Problems of the curriculum. In addition, they need to form a more complete picture of how the mathematical ideas build on previous ideas and how those ideas in turn provide the foundation for the mathematics in subsequent Units and grades. Teachers also need to make connections between multiple solution strategies and between the five representations often used to communicate mathematics (Preston and Garner, 2003). See Five Star of Classroom for a detailed description of using the five representations.
We place mathematical knowledge first because good instructional decisions and practice rely on deep understanding of the mathematics embedded in the Problems. Emphasizing pedagogy too soon may interfere with teachers' focus on the mathematics. However, effective instructional practices should be modeled as mathematics content is being learned.
Teaching and Learning
Once teachers have developed a strong understanding of the mathematics within the CMP curriculum, the focus of the professional development can shift to teaching and learning. When the focus shifts to pedagogy, teachers continue to develop their own understanding of the mathematics through conversations about student work and student understanding of the mathematics.
Teachers need to experience inquiry-based pedagogy in their professional development so that it will serve as a model for their own teaching. They also need to be involved in conversations about how to teach problem-centered materials. Because CMP requires a major shift from demonstrating mathematical procedures to facilitating classroom discussions around important mathematical ideas, professional development on pedagogy might begin with strategies for facilitating such discussions, such as Prompt, Wait Time, Revoice, Restate, and Apply Reasoning, which were discussed on page 81 of The CMP Classroom. One professional development resource for facilitating effective and mathematically productive classroom discussions is the Mathematics Discourse in Secondary Classrooms (MDISC) project (Herbel-Eisenman, Steele, and Cirillo, in press).
After teachers have taught several Units, they will need more in-depth work on instruction. The student books and teacher support can help teachers implement the curriculum within their own classrooms, but teachers also need time away from their classrooms to talk with peers and to fully investigate the potential of the curriculum. A resource for this ongoing professional development is the Five Practices for Orchestrating Productive Classroom Discussions (Smith and Stein, 2011). These Five Practices are:
- Anticipating students solution strategies
- Monitoring students' work
- Selecting students' solution strategies to share
- Sequencing students' solution strategies to highlight important mathematics
- Connectig and summarizing students' solution strategies
The CMP Classroom, explains how to use the Five Practices to enhance instruction.
Once teachers have begun using inquiry-based instruction in CMP, it becomes clear that traditional forms of assessment are insufficient to gauge student learning. CMP offers a variety of assessment to support teachers, including embedded assessment, which may be unfamiliar to teachers and require them to develop new skills.
Concerns about how to assess student learning and how to grade assessments tend to arise later in the implementation process. Therefore, professional development involving assessment should occur after the teachers have experienced some of the curriculum and after the focus on teaching and learning. Professional development should also address the role of assessments in teachers' reflections and decisions.
For example, with support and experience, teachers begin to see both formative and summative assessment as data they can use to drive instructional decisions. Furthermore, formative assessment is the very essence of teaching. Throughout a lesson, teachers are questioning, observing, and listening to students and adjusting their strategies as needed to accommodate and enhance learning. Homework, Check-ups, Quizzes and Unit tests provide additional assessment that also informs teachers' subsequent planning and teaching. For more information on assessment see Assessment.