Dynamic geometry software can help teachers highlight mathematical relationships in ways not possible with static diagrams. However, these opportunities are mediated by teachers’ abilities to construct sketches that focus users’ attention on the desired variant or invariant relationships. The study described in this paper looked at two cohorts of preservice secondary mathematics teachers and their attempts to build dynamic geometry sketches that highlighted the trigonometric relationship between the angle and slope of a line on the coordinate plane. The authors identified common challenges in the construction of these sketches and present examples for readers to interact with that highlight these issues. They then discuss ways that mathematics teacher educators can help beginning teachers understand common pitfalls in the building of dynamic geometry sketches, which can cause sketches not to operate as intended.
A student, Stuart, related perimeter to pixels and the professor, Beth, moved back and forth between reserved believing and reserved doubting and doubting teacher actions (Elbow, 1986; Harkness & Noblitt, 2017) while assessing the merit of his conjecture in the moment. Video allowed the researchers to rewatch the episode multiple times after the moment and to attempt to believe (Elbow, 1986; 2006), or find merit or strength, in Stuart’s conjecture and then explore the mathematics that he suggested. Within this paper the researchers “restory” (Creswell, 2012) chronologically what transpired in the moment in the classroom, their later conversations, and their after-the-moment mathematical explorations of Stuart’s conjecture. Video can, perhaps, allow teacher educators to help preservice teachers and classroom teachers notice and reflect on missed opportunities for believing. Video also has the potential to empower teachers to explore the mathematics suggested by students after the moment and then use what they learn in future lessons.
This study examines whether preservice teachers’ experiences with video analyses during teacher preparation have long-lasting effects on their practices once they enter the profession. Specifically, the authors examined whether teachers who had opportunities to analyze student thinking and learning during teacher preparation continued to do so when they reflected on their teaching effectiveness as full-time teachers. A group of elementary school teachers who attended a video-enhanced mathematics methods course were compared to a control group at the end of their first year of full-time teaching. Teachers were asked to assess two lessons they had just taught by describing lesson learning goals and providing a rating of lesson effectiveness and a rationale for their evaluation. Teachers who attended the video-enhanced course during teacher preparation outperformed their counterparts in both the quality of evidence they drew upon and their attention to individual or subgroups of learners. Study limitations and future directions are discussed.
In this article the authors described their exploration of a particular design element they labeled “video in the middle.” As part of the video in the middle design, the viewing of carefully selected video clips from teachers’ classrooms is sandwiched between pre- and postviewing activities that are expected to support teachers’ engagement in and learning from the video. These three elements (prevideo, video, and postvideo), taken together, comprise a videocase. Videocases can then be further sequenced to create a specified professional development (PD) curriculum. Purposeful selection of each video clip allows for coherence between the prevideo, video viewing, and postvideo activities, which in turn, supports the link between a given videocase and identified teacher learning goals. Incorporating a video in the middle design within a video-based mathematics PD environment can promote a detailed and focused examination of complex mathematical content, the relationship between pedagogical decisions and practices, and an unpacking of students’ mathematical thinking. It is essential to underscore the major role that facilitators play in video-based PD, and that the effective application of the video in the middle design is, in large part, dependent on skillful facilitation. The video in the middle design can be useful across different content areas and teacher education settings.