The goal of this study was to document the characteristics of students' dispositions towards mathematics when they engaged in the exploration of parts of unsolved problems: Graceful Tree Conjecture and Collatz Conjecture. Ten students, Grades 4 and 5, from an after-school program in the Midwest participated in the study. I focused on the cognitive, affective, and conative aspects of their mathematical dispositions as they participated in 7 problem-solving sessions and two interviews. With regard to cognitive aspects of the students' dispositions, I focused on the students attempts to identify and justify patterns for labeling graphs. Overall, the unsolved problems were accessible to the students and they found patterns that enabled them to gracefully label specific classes of graphs for the Graceful Tree Conjecture. With regard to affective aspects of students' dispositions, I found five themes that characterized their beliefs about the nature of mathematics. Also, students exhibited a variety of emotions throughout the study. The two emotions exhibited most frequently were frustration and joy. The third type of disposition that students exhibited was the conative construct of perseverance. This was related to the interplay of frustration and joy and characterized the productive struggle that students experienced throughout the study. To examine students' dispositions in greater depth, I conducted a case study analysis of the positional identities of two students, which I report in a detailed narrative. [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.]