Abstract
One goal for teacher preparation programs is to develop preservice teachers' ability to plan student-centered lessons that include meaningful mathematical discourse. The five practices for orchestrating productive mathematical discussions provides one framework for planning this critical component of mathematics instruction. This chapter discusses several strategies the authors implemented in undergraduate mathematics education courses. Mathematics teacher educators can use these strategies to support preservice teachers as they develop their understanding of mathematical discussions and their mathematical knowledge for teaching. The culminating activity, a three-part lesson planning sequence, prompts preservice teachers to apply knowledge of the five practices to plan responsive lessons in which student thinking is centered.
TopBackground
Five Practices for Orchestrating Productive Mathematical Discussions (Stein et al., 2008) offers a popular framework that includes actions that occur during planning as well as during classroom implementation of a lesson. Stein et al. suggest that meaningful mathematical discussions based on students’ ideas can be attained through this process of five steps, each step depending on the previous one. Teachers must anticipate students’ responses to a task that will be taught, monitor students’ thinking as they are engaged in the task, purposefully select students to present based on their choice of representations, sequence those representations in a purposeful way, and then make connections among the representations so that students are able to understand key mathematical concepts. To facilitate a student-centered discussion, teachers predict how students may engage in the task, plan key questions, consider scaffolds students may need, and think about how big ideas will be summarized at the end of the lesson. Without careful planning, teachers who use rich tasks can feel out of control of the learning, reducing teacher efficacy (Schoenfeld, 1998; Smith, 1996). Stein et al. (2008) noted that while some expert teachers may be able to successfully improvise in the classroom, most cannot. As such, making lesson planning a public and reflective practice between MTEs and PSTs allows instructors to reinforce the idea that intentional planning versus improvisation is the norm for well-designed lessons that center students’ thinking. The Five Practices provide a process for PSTs to use when planning to implement rich mathematical tasks that value students’ ideas and strategies.
Preparing PSTs to plan lessons in which they will facilitate meaningful mathematical discussions requires MTEs to carefully consider the knowledge and skills required to do so. Ball et al. (2008) provided a framework that distinguishes among several types of Mathematical Knowledge for Teaching. In this framework, both mathematical content knowledge and pedagogical knowledge include three subsets of math teacher knowledge. In addition to common content knowledge, teachers use other types of knowledge including specialized content knowledge, knowledge of content and students, and knowledge of content and teaching. These distinctions help conceptualize what PSTs need to know and be able to do, while also supporting math teacher educators in considering how to develop these areas of knowledge.
Key Terms in this Chapter
Monitoring: The second step in Five Practices (Smith, et al., 2008 AU11: The in-text citation "Smith, et al., 2008" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ), monitoring is a practice where teachers observe students’ thinking and solutions as they are solving rich math tasks .
Connecting: The last step in Five Practices (Smith, et al., 2008 AU10: The in-text citation "Smith, et al., 2008" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ). Teachers pose questions that will support students in making connections to important mathematical ideas during a class discussion.
Jigsaw: An instructional strategy that splits up a large text or group of texts among students. Students thoroughly read their assigned text then teach it to other students who have not read that text.
Anticipating: The first step in Five Practices for Orchestrating Productive Mathematical Discussions (Smith et al., 2008 AU9: The in-text citation "Smith et al., 2008" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ) . Anticipating refers to considering the variety of strategies students will use to solve a rich task, including correct, partially correct, and incorrect solutions.
Sequencing: The fourth step in Five Practices (Smith, et al., 2008 AU13: The in-text citation "Smith, et al., 2008" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ), sequencing is the practice of choosing the order in which selected students’ solutions are shared during a class discussion with the purpose of illuminating important mathematical ideas.
Equipartitioning: Splitting a set or region into equal-sized groups or pieces.
Selecting: The third step in Five Practices (Smith, et al., 2008 AU12: The in-text citation "Smith, et al., 2008" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ), selecting refers to intentionally choosing students to share their thinking during a discussion to illustrate an important mathematical idea.
Mathematical Knowledge for Teaching: A framework for the domains of knowledge of mathematics teachers ( Ball et al., 2008 ). It includes three subtypes of content knowledge: common content knowledge, horizon content knowledge, and specialized content knowledge. It also includes three subtypes of pedagogical content knowledge: knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum.
Iterating: A fraction concept in which a fractional piece is repeated or copied.