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.
Well-prepared beginning teachers of mathematics at the high school level recognize that their students may not think about mathematics the same way they do. They need to see their roles as building on the thinking of their students rather than only transmitting their mathematical knowledge to their students [C.3.1]. For example, when given a quadratic equation to solve, students may use a guess-and-check strategy. Rather than counting this strategy as wrong because it is not deductive in nature, well-prepared beginners consider how that solution could become part of the discourse around how to solve quadratic equations. Acknowledging the thinking behind that approach while also noting its limitations in situations in which solutions are not integers will motivate the need for other approaches. In this example, students are naturally engaged in the doing of mathematics, developing mathematical processes and practices [C.3.2].
By the time they are in high school, many students may conclude that they are not good at mathematics and certainly do not like studying it; well-prepared beginning teachers need to be cognizant of strategies for increasing their students’ confidence in engaging mathematics as well as their appreciation for its usefulness [C.3.3].