Modeling is an important element of discovery and design processes because it can help individuals to comprehend and facilitate solutions to problems, mediate among mental and external representations, and off-load cognitive demands. However, engaging in model generation, comprehension, and transformation requires the orchestration of domain…

This article reports a validity study of History Assessments of Thinking (HATs), which are short, constructed-response assessments of historical thinking. In particular, this study focuses on aspects of cognitive validity, which is an examination of whether assessments tap the intended constructs. Think-aloud interviews with 26 high school…

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs,…

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In…

Clinical interviewing is viewed here as a social interactional pattern in order to examine the nature and limits of the technique as a means of scientific data acquisition. I defend the technique against criticisms that it is ecologically suspect and prone to systematic biases, mainly due to influence of the interviewer on the interviewee or to…

In this article we examine students' perspectives on the customary, public work of proving in American high school geometry classes. We analyze transcripts from 29 interviews in which 16 students commented on various problems and the likelihood that their teachers would use those problems to engage students in proving. We use their responses to…

This study was designed to identify the types of understandings, skills, and beliefs that affect pre-service teachers' evaluations of students' mathematical arguments in classroom contexts. Thirty-four pre-service teachers read a transcript of a third grade lesson in which the students were expected to prove a generalization. To investigate…