This paper presents a novel task-oriented, user-centered, multi-method evaluation (TUME) technique and shows how it is useful in providing a more complete, practical and solution-oriented assessment of the accessibility and usability of Learning Management Systems (LMS) for blind and visually impaired (BVI) students. Novel components of TUME include a purposeful integration of a multi-theoretic foundation and multiple methods to accurately identify users' accessibility and usability problems in Web interaction and identify design problems and solutions to en-sure technical feasibility of recommendations. The problems identified by TUME remain hidden from extant evaluation methods--therefore, these problems remain in Web-based applications. As a result, evaluation of Web-based applications remains confounded by users' Web interaction challenges; their utility for specific user types remains unclear. Without appropriate evaluation of users' problems and challenges in using Web-based applications, we cannot begin to solve these problems and challenges. This paper demonstrates how TUME can be used to identify the unique problems and challenges of specific user types in using Web-based applications and suggests potential solutions. The outcome is an accurate understanding of specific design elements that present roadblocks and challenges for the user in interacting with the Web-based application and feasible design modifications to potentially improve the utility of these applications for specific user types. We illustrate the application of TUME to test the utility of LMS for BVI students. We use a popular LMS and an on-line exam as a representative task. We present results from three assessments that are essential components of TUME. Assessment I uses verbal protocol analysis to identify where and why BVI participants face LMS interaction problems in completing an exam. Assessment II performs text analysis of accessibility and usability design standards to identify design errors in the LMS exam environment. Assessment III employs recursive abstraction analysis of Web developers' comments to understand why these problems occur and how they may be addressed feasibly. Together, these assessments show how TUME can identify design errors and their consequent problems for BVI users, the source and nature of these problems, as well as potential remedies for these solutions. TUME provides researchers in education, human-computer interaction, and related fields a novel technique to understand the unique accessibility and usability needs of BVI users. Educators and education technologists may use this technique to accurately evaluate LMS for BVI students. With further validation and generalization across more problems, findings can be used to develop design principles for accessible, usable, and useful LMS for BVI students. Such design principles provide the basis to improve the efficacy of existing accessibility and usability standards for Web applications. TUME leads to improved accessibility and usability evaluation for non-visual interaction needed in mobile and ubiquitous computing environments and other multi-tasking situations.