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Students in the United States are not achieving in mathematics as indicated on the NAEP (2019) exams and other measurements of student achievement (NCES, 2019; OECD, 2019; O'Dwyer, et al., 2015). Mathematically gifted and promising students are especially impacted by this phenomenon, though it is not exactly known what factors contribute to successful teachers of these students. This phenomenological case study focused on the beliefs, instructional practices, and conceptual understanding of mathematics of five teachers in a public charter school for gifted students. Data sources collected included semi-structured interviews, classroom observations, and questionnaires based on Swan's (2006) practices and beliefs research with effective mathematics teachers. To better understand these phenomena, two conceptions of giftedness served as the lenses for this study: Renzulli's Three-Ring Model (1978) and Gagne's Differentiated Model of Giftedness and Talent (1985). Using an interpretive phenomenological analysis several themes emerged in response to each research question. Findings for instructional practices indicated that teachers used both student-centered and teacher-centered practices and consistently utilized differentiated groupings. Additionally, teacher participants believe that gifted students possess both positive traits and challenges, and specifically for math, believe that sense-making is key, and math is a subject students should enjoy. Teachers also noted that school and home environments impact access to an equitable mathematics education for children. Teachers' conceptual understanding of mathematics is guided by their ongoing practice, the curriculum, and math experiences prior to teaching. These findings indicate the importance of ongoing training and professional development in mathematics and gifted education, as well as the recruitment and retention of teachers who possess a strong conceptual understanding of mathematics, a passion for the subject, and a student-centered approach to teaching. Keywords: mathematically gifted and promising, instructional practices, beliefs, equitable access, teachers' conceptual understanding of mathematics. [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/en-US/products/dissertations/individuals.shtml.]