The purpose of this multiple case study was to explain how mathematically proficient homeschool graduates acquired their proficiency. Bronfenbrenner's (1979, 1999) ecological and Process-Person-Context-Time (PPCT) models were used as theoretical frameworks to analyze the development of the graduates in the homeschool environment. The National Research Council's (NRC) Five Strands of Mathematical Proficiency (Kilpatrick, Swafford, & Findell, 2001) was used as a conceptual framework. I collected four types of data: (a) surveys; (b) National Assessment of Educational Progress; (c) Mathematical Self-Efficacy Scale scores; and (d) interviews. These data items were analyzed with explanation building, a special case of pattern-matching logic. Many themes emerged from this research: strong nuclear families, direct teaching, self-study or self-directed education, mastery learning, purposive conversations, and challenging curriculum (Saxon). As they matured, these graduates built upon the foundation set by their parent educators and took ownership of their studies. [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.]