Indicator C.3.2. Understand and Recognize Students’ Engagement in Mathematical Practices

Because doing mathematics is at the core of learning mathematics, well-prepared beginners recognize that students will present various approaches to problems, representing them, justifying solutions, and critiquing the reasoning of others. As such, they are inclined to anticipate, analyze, and respond to students’ diverse solution strategies. Sets of mathematical practices have been usefully elaborated in standards and research syntheses over the past few decades to help teachers know what mathematical practices to develop in their students (cf. NCTM, 2000; National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010; NRC, 2001a). Although accomplished teachers have greater breadth of knowledge of students' engagement in mathematical practices across many mathematical content domains, well-prepared beginners recognize challenges that students may face when engaging in certain mathematical practices. For example, they know that students may accept arguments that have weak mathematical foundations, especially from peers who are their friends or someone they think typically responds correctly.

Acknowledging that they are only beginning their careers, well-prepared beginners have an emerging understanding of how to nurture development of mathematical processes and practices. They anticipate how students’ use of mathematical practices will look and sound within specific grade-band mathematical topics, knowing that over years of experience, their knowledge of students’ ways of using mathematical practices will expand to more mathematical topics. For instance, well-prepared middle level teachers can describe how middle level students might reason abstractly and quantitatively about algebra, geometry, and statistics. Well-prepared beginners use their own understanding of mathematical practices responsively when they interact with students who may engage in those practices differently than they do or are just beginning to develop aspects of those practices.