This paper presents seven basic principles or criteria that mathematics programs for mathematically gifted elementary students should meet. A rationale for each criterion is offered. Grouping these children together in one regular class is strongly encouraged so that they can be in one advanced group, as part of the traditional three groups in elementary classrooms. In low enrollment schools, grouping of mathematically gifted students from several grades together in one class is suggested. The seven program principles are that the mathematics program should: (1) be part of the regular mathematics curriculum; (2) have a rigorous identification procedure; (3) provide daily instruction; (4) include placement and interaction with peers who are also mathematically gifted; (5) involve faster pacing; (6) provide challenges at the student's level using advanced strategies; and (7) offer special training for teachers. (12 references) (DB)