In this paper we propose a developmental framework that makes distinctions and links among schemes—conceptual structures and operations children construct to reason in multiplicative situations. We provide a set of tasks (problem situations) to promote construction of such schemes. Elaborating on Steffe et al.’s (Steffe & Cobb, 1998) seminal work, this framework synthesizes findings of our teaching experimentsi with over 20 children who have disabilities or difficulties in mathematics. This empirically grounded framework contributes to articulating and promoting multiplicative reasoning—a key developmental understanding (Simon, 2006) that presents a formidable conceptual leap from additive reasoning for students and teachers (Harel & Confrey, 1994; Simon & Blume, 1994). In place of pedagogies that focus primarily on multiplication procedures, our framework can inform teaching for and studying of children’s conceptual understandings. Such understandings provide a basis not only for promoting multiplication and division concepts and procedures but also for reasoning in place-value number systems, and in fractional, proportional, and algebraic situations (Thompson & Saldnha, 2003; Xin, 2008). We contrast our stance on children’s cognitive change and teaching that promotes it with the Cognitively Guided Instruction (CGI) approach (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998)). CGI grew out of research on children’s solutions to addition and subtraction tasks. By asserting that “children’s solution processes directly modeled the action or relationships described in the problem” (Carpenter, Hiebert, & Moser, 1983, p. 55), CGI researchers seemed to equate children’s cognitive processes with tasks. In contrast, we argue for explicitly distinguishing between task features as adults conceive of them and schemes children bring forth for solving tasks. Consider a Join task such as, “We had 7 toys and got 4 more; how many toys we then had in all?”
A child may solve such a task by counting-all 1s (1-2-3-…10-11), by counting-on (7; 8-9-10-11), or by using a through-ten strategy (7+3=10; 10+1=11). The latter two indicate the child understands number as a composite unit, hence preparedness for multiplicative reasoning, whereas the first does not. We concur with CGI’s premise of the need to use children’s ways of thinking in teaching. However, we disagree that the structure of a task as seen by an adult determines, in and of itself, the way a child makes sense of and acts to solve it. The next section presents the conceptual framework that underlies our synthesis.