A prolegomenon--a "critical discussion to introduce and interpret an extended work"--exactly describes the purpose of this essay, since the Appalachian Collaborative Center for Learning, Assessment, and Instruction in Mathematics (ACCLAIM) is very much an extended work. This essay develops a tentative theoretical framework to help represent intended purposes, commitments, and arguments of ACCLAIM's extended work. ACCLAIM's mission is the cultivation of indigenous leadership capacity for the improvement of school mathematics in rural places. It aims to understand the rural context as it pertains to learning/teaching mathematics, and to articulate in scholarly works, including empirical research, the meaning and utility of that learning and teaching for and by rural people. Elements of the framework developed in this essay include the nature of mathematical knowledge, the view of the context of education as being as important as technical concerns of curriculum and instruction, the rural context versus cosmopolitan norms, and mathematical knowledge in rural context and vice versa. Four principles are offered for rural mathematics education research: it should describe the salient relationships between mathematical knowledge and rural context, examine rural schools as they serve or subvert the gatekeeping functions of school mathematics, examine hypotheses about the place occupied by mathematics knowledge in and for rural communities, and elaborate theories of "pedagogy of place" for mathematics education in rural schools. Appendix discusses exemplary rural mathematics programs. (Contains 62 references.) (SV)