This article describes the authors' three-component instructional sequence--a before-class activity, a during-class activity, and an after-class activity--which supports students in becoming self-regulated proof learners by actively developing class-based criteria for proof. All four authors implemented this sequence in their classrooms, and the…

The implementation of the Common Core's Standards for Mathematical Practice can pose a challenge to all teachers of mathematics but especially to preservice teachers. These standards require teaching in a way that often differs from what preservice teachers have experienced as learners. Standard 1--"Make sense of problems and persevere in…

Generalizing--along with conjecturing, representing, justifying, and refuting--are forms of mathematical reasoning important in all branches of mathematics (Lannin, Ellis, and Elliott 2011). Increasingly, however, generalizing is recognized as the essence of thinking in algebra (Mason, Graham, and Johnston-Wilder 2010; Kaput, Carraher, and Blanton…

While teaching a methods class for preservice secondary school mathematics teachers, Christopher E. Smith found that although all students could draw a reasonably close approximation of a circle, not all could provide an entirely accurate definition of a circle. A discussion with students led him to think about ways of reintroducing students to…

Many preservice teachers come into teacher education programs with a rule-based view of mathematics, a view that is to a large extent indicative of their previous school experiences. In teaching preservice elementary school teachers, this author has used his instrumental understanding as the starting point for rich discussions about the…

Mathematical letter writing can be a mutually beneficial partnership between high schools and universities. Secondary school preservice teachers in methods courses require authentic experiences in focusing on students' mathematical thinking. High school students benefit from increased opportunities to engage in individualized tasks with personal…

A focus on mathematics vocabulary must be part of teachers' instructional plans to develop students' understanding of key ideas. The author presents examples from work with preservice teachers regarding two vocabulary strategies and other related activities that can be used by middle and high school mathematics teachers. (Contains 8 figures.)

During a six-week methods course for preservice elementary school teachers, student pairs were asked to solve a problem and record their experiences in a dairy. Presented are the dairy entries of two students as they worked on determining the number of squares that can be formed on a five-peg geoboard. (MDH)