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This study examined the engagement of preservice elementary teachers (PSTs) with mathematics tasks that were open in various ways and to varying extents. In K-12 settings, practitioners who have implemented recently popular open tasks report increased engagement and from a wider variety of students. While these anecdotal reports and related theoretical and empirical research suggest the validity of these tasks, research has not yet systematically examined the impacts of these tasks. This study investigated the engagement of a specific population--elementary PSTs--with open mathematics tasks. Elementary PSTs were of interest because they were recently K-12 students themselves and elementary PSTs often require intervention in their narrow conceptions and sometimes negative perceptions of mathematics. I observed four task sessions in each of six sections of elementary mathematics methods classes across two semesters. The tasks in which PSTs engaged included "Notice and Wonder," "Which One Doesn't Belong?," and "How Many?" For purposes of comparison, I also observed PSTs during a task session revolving around a word problem of high cognitive demand. During observations, I video and/or audio-recorded (if possible) and took field notes while PSTs discussed the tasks as a whole group. PSTs also completed a brief semi-structured questionnaire. I collected identity webs of PSTs' identity as a mathematics learner from the classes that took place during the second semester. After initial analysis, from this larger data set, I selected one representative task session for each of the four tasks to analyze in more detail. In my analysis, I deductively coded for items related to a multi-dimensional understanding of engagement and inductively coded for evidence of influences on PSTs' engagement. I also considered the mathematical identities of PSTs who engaged in particular ways and were influenced in particular ways. Overall, PSTs demonstrated a similar quantity across all four tasks. However, there were qualitative differences in PSTs' engagement across the tasks, with PSTs demonstrating a wider range of cognitive engagement, greater self-initiation, and more intense affect in the most open tasks. The most consistent influences on PSTs' engagement were peers' thinking and the openness of the tasks. While not consistently named as an influence by PSTs, observations indicated the instructor's facilitation of the tasks clearly also played a role. This study has implications for mathematics teacher educators, mathematics education organizations, and mathematics education researchers. Implications are related to the validity of using the tasks with PSTs, the importance of considering openness and variations in ways tasks can be open, and the need to attend to multiple dimensions of engagement, including a greater range of cognitive engagement. [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/en-US/products/dissertations/individuals.shtml.]