This study aimed to identify elements of personal knowledge that were hypothesized to underlie within-person, across-context variations in students' appraisals of self-efficacy for coping with challenges encountered in Science, Technology, Engineering, and Mathematics (STEM) education. Freshman in a college of engineering completed assessments of (a) 4 elements of personal knowledge regarding themselves and their educational resources; (b) their subjective beliefs about links between these 4 personal and social qualities and each of 32 specific educational challenges; and (c) perceived self-efficacy for successfully handling each of the 32 challenges. Individual students' self-efficacy appraisals commonly varied substantially within-person, across contexts. This variability was predictable. Students displayed relatively high (low) self-efficacy within subsets of situations they subjectively linked to positively (negatively) valenced knowledge that they possessed, a finding consistent with the knowledge-and-appraisal model of personality architecture that guided the research. Additional analyses demonstrated that students with similar overall academic self-efficacy scores may display markedly different profiles of self-efficacy appraisal across context. Students' narrative accounts enriched understanding of these profile patterns.