This study investigated how problem-solving videos can be used in video-mediated professional learning to support secondary preservice mathematics teachers (PMTs) in developing teacher knowledge for noticing student thinking in the context of the derivative concept in calculus. A model of the trajectory of PMTs' noticing was constructed as six PMTs viewed and analyzed videos of students' problem solving. At the same time, the nature of video-mediated interactions that were found to be productive in supporting this knowledge development was examined. A design experiment was used as the research methodology. Data was collected from video recordings of eight semi-structured teaching episodes for each of the three pairs of PMTs and analyzed through a grounded theory approach. Considering that the knowledge was video-mediated, developed collaboratively, and assessed in action, the study was grounded in situated and sociocultural perspectives, and the conceptual framework of professional teacher noticing guided the analysis. The constructed model of noticing development that emerged from the experiment entails the following four processes: "describing," "interpreting," "comparing and contrasting," and "responding." This model represents an approach to an ambitious professional vision that can support calculus educators in iteratively improving their practice through readily accessible, video-mediated professional learning. The study also identified interactions that were found to mediate the development of PMTs' noticing. The social interactions that supported PMTs' noticing of student thinking are "highlighting" (moments worth noticing) and "prompting" (PMTs to notice what was highlighted). Prompts posed to the PMTs generated centers of focus (Lobato et al., 2013) and offered opportunities for their describing, interpreting, and responding to student thinking. The material interactions refer to features of problem-solving videos that were found to support the PMTs' learning to notice: (1) The videos offer images of students' problem solving that could serve as an opportunity for decentering; (2) They are effectively curated; (3) They can be paused and re-viewed repeatedly, and (4) They provide a comprehensive and appropriate scope and sequence of derivative topics. These findings contribute to research in calculus education, with implications for the design of learning experiences in mathematics teacher preparation. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]