Quantitative and qualitative analyses of 37 first-graders' solutions to addition problems were conducted to re-examine inconsistencies in children's progress from counting-all to counting-on. The authors applied the constructivist's participatory-anticipatory stage distinction as a tool for fine-grained assessment. Among solutions given by case-representing children during 20-minute clinical interviews, the authors distinguish both stages within each of 2 known schemes that constitute counting-on, "Anticipating Start From" and "Anticipating Stop At" (with double counting). The prompt-dependent, Zone of Proximal Development-correlated participatory stage provides a lens for explaining children's regress to counting-all and concurrent use of various solution strategies. The study advances a novel, theoretical coordination among 3 frameworks: constructivist scheme theory with a focus on the notion of prompt, Vygotsky's sociocultural approach, and Siegler's Overlapping Waves Model, and provides a corresponding, threefold stance on mathematical tasks. (Contains 4 tables, 2 figures and 1 footnote.)