In order for teachers to support students' mathematical thinking, Battista (2004) believed they must identify core mathematical concepts, recognize conceptual frameworks for understanding children's thinking, and use appropriate assessment tasks. In his view, learning trajectories provide teachers with information on children's cognitive abilities as well as a structure for assessment. The present study investigated the ways in which preservice teachers used, and reflected on their use of, learning trajectories to assess, plan, and instruct during a one-on-one tutoring project focused on geometric shapes. In addition, preservice teachers were asked to reflect on the ways in which they might use learning trajectories during small-group and whole-group instruction. This study employed collective case study methodology as a qualitative research design methodology. The goal of the research was to understand how three preservice teachers interpreted their experience. Participants were preservice teachers seeking K-8 teaching certification with a minor in mathematics education. Throughout the twelve-week study, digital recordings of fieldwork and interviews were collected, along with journal entries, lesson plans, and fieldnotes. The data analysis strategy followed Stake's (2006) methodology for collective case study analysis. Trustworthiness was accomplished through thick description, triangulation of data, and member checks. Supported by constructivist learning theory as the theoretical framework guiding the research, the study found that during the assessment phase, learning trajectories gave preservice teachers flexibility in identifying their students' level of mathematical thinking. While planning, preservice teachers created lesson plans that encouraged active learning and were within their students' zone of proximal development. And during instruction, learning trajectories were used as a tool for formative assessment. When asked to reflect on how learning trajectories might be used during small-group and whole-group instruction, preservice teachers surmised that learning trajectories could be used to create hetero-homogeneous groupings and used to ask questions that increase in sophistication, respectively. [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.]