As creating and critiquing arguments becomes more of a focus in mathematics classes, teachers need to develop their abilities to facilitate collective arguments. Many mathematics education researchers find Toulmin’s (1958/2003) model of argumentation to be useful in analyzing arguments, raising the question of whether mathematics teachers would find it useful as well. We introduced the model to prospective secondary mathematics teachers and asked them to analyze arguments using it. We found that the prospective teachers developed an appropriate understanding of what collective argumentation looks like in the classroom, and the model provided them a lens for analyzing teaching practice. This suggests the use of Toulmin’s model is a promising step in helping prospective teachers develop their conceptions of collective argumentation.

It can be difficult for teachers to make in-the-moment decisions about which solution strategies to cognitively challenging tasks should be included in the whole-class discussion (Stein, Engle, Smith, & Hughes, 2008). Teachers can purposefully select and sequence the solution strategies to help create a whole-class discussion that promotes the mathematical learning goal. An intervention was implemented in a middle school methods course that aimed to understand preservice teachers’ (PSTs’) competencies in formulating rationales for their selecting and sequencing choices. Results from the intervention suggest that PSTs’ sequencing rationales can be grouped into three categories.

We describe an activity designed to help preservice elementary school teachers develop mathematical knowledge for teaching in the domain of facilitating mathematical discussions. The activity involved preservice teachers writing task dialogues, imaginary conversations between a child and teacher about a problem-solving task, in which they practice responding to correct, partially correct, and incorrect student responses. Preservice teachers then implemented these same tasks with children in a field experience setting. We describe 2 different iterations of the activity and field experience in detail as well as the insights into preservice teacher knowledge each iteration afforded us.