Students with mathematical learning disabilities (MLDs) experience persistent challenges learning even the most elementary mathematics. While prior research on MLDs has classified students on the basis of test scores and documented performance differences between groups, this dissertation focuses on the qualitative differences in individual students with MLDs as each student attempted to learn. This study extends the understanding of MLDs by (1) focusing on the mathematical domain of fractions, (2) selecting for study students whose difficulties with mathematics are clearly due to an MLD and not other factors, (3) conducting detailed diagnostic analyses of video-taped tutoring sessions, and (4) contrasting the difficulties that the students with MLDs experience to data collected with five typically achieving fifth grade control students. The two case study students, "Lisa" (19-year-old community college student) and "Emily" (18-year-old high school student) each participated in a pretest, four weekly tutoring sessions, and a posttest focused on fraction concepts. Achievement test scores, interviews, and videotaped tutoring session data were used to establish that Lisa and Emily met classic MLD qualifications, and more stringent response-to-intervention criteria. Both students demonstrated unexplained persistent low math achievement and neither student benefited from a tutoring protocol that had been effective for typically achieving fifth grade students. The tutoring sessions were analyzed using microgenetic methods. Analysis indicated that each student relied upon a unique collection of atypical understandings, which reoccurred across the sessions, were resistant to standard instructional approaches, and proved to be highly consequential for the student's ability to understand more complex fraction concepts. A cross case analysis revealed surprising similarities in the atypical understandings displayed by both case study students. These atypical understandings stemmed from and contributed to the student's inability to conceptualize and manipulate representations of fractional quantity. These atypical understandings were not similarly problematic for the fifth grade control students, but did appear in one additional student with an MLD. This suggests that there are qualitative differences in the difficulties experienced by students with MLDs and that it may be possible to design screening measures to identify these indicators of atypicality. In addition, remediation approaches should take into account and specifically target these atypicalities. [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.]