Well-prepared beginning teachers of mathematics have foundational understandings of students’ mathematical knowledge, skills, and dispositions. They also know how these understandings can contribute to effective teaching and are committed to expanding and deepening their knowledge of students as learners of mathematics.
Middle level students have their own unique understandings from their elementary schooling, life experiences, and personal preferences. The complex mathematical ideas at the middle level can almost always be approached in a variety of ways. Therefore, teachers of mathematics at the middle level must prioritize individual student reasoning as part of their planning and teaching.
Well-prepared beginning teachers of mathematics at the middle level support emerging mathematical practices of middle level learners. [Elaboration of C.3.2]
Early adolescents benefit from opportunities to explore meaningful and authentic tasks that relate to their interests and backgrounds (AMLE, 2012). Such authentic contexts provide environments from which middle level learners can further their abilities to use mathematical practices and processes. As noted earlier in this chapter, middle level mathematics is more abstract and symbolic than elementary school mathematics. Middle level learners must have regular opportunities to engage in mathematical practices and processes in a more sophisticated manner than they may have demonstrated in earlier grades. As noted by Gojak (2013), the middle grades are a time to get “messy” with mathematics. Therefore, well-prepared beginners must be able to engage their students in representing and explaining their mathematical thinking and making mathematical arguments. For example, they must be able to understand the various approaches students might use to solve problems and create environments in which strategies are discussed, critiqued, and compared.
Well-prepared beginners understand that particular teaching moves support (or inhibit) student development of mathematical practices and processes. For example, as discussed in Activities 2 and 3 in Vignette 6.1, posing questions can help to develop students’ abilities to analyze problem situations, select appropriate strategies, and reason quantitatively. Well-prepared beginners also recognize that early adolescents may be self-conscious and therefore have a variety of ways for students to demonstrate their unique thinking. For example, a well-prepared beginner might collect work and project particular solutions anonymously to highlight reasoning or particular representations. Well-prepared beginners reflect on ways their own actions affect the ongoing development of student thinking.