We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts.
Tzur, R., Johnson, H. L., McClintock, E., Xin, Y. P., Si, L., Woodward, J., Hord, C., & Jin, X. (2013). Distinguishing schemes and tasks in children's development of multiplicative reasoning. PNA, 7(3), 85-101.