K-12 standards and curricula are beginning to include engineering design as a key practice within Science Technology Engineering and Mathematics (STEM) education. However, there is little research on how the youngest students engage in engineering design within the elementary classroom. This dissertation focuses on iteration as an essential aspect of engineering design, and because research at the college and professional level suggests iteration improves the designer's understanding of problems and the quality of design solutions. My research presents qualitative case studies of students in kindergarten and third-grade as they engage in classroom engineering design challenges which integrate with traditional curricula standards in mathematics, science, and literature. I discuss my results through the lens of activity theory, emphasizing practices, goals, and mediating resources. Through three chapters, I provide insight into how early-elementary students iterate upon their designs by characterizing the ways in which lesson design impacts testing and revision, by analyzing the plan-driven and experimentation-driven approaches that student groups use when solving engineering design challenges, and by investigating how students attend to constraints within the challenge. I connect these findings to teacher practices and curriculum design in order to suggest methods of promoting iteration within open-ended, classroom-based engineering design challenges. This dissertation contributes to the field of engineering education by providing evidence of productive engineering practices in young students and support for the value of engineering design challenges in developing students' participation and agency in these practices. [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://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]