It's Not Too Late to Conceptualize: Constructing a Generalized Subtraction Schema by Abstracting and Connecting Procedures

In this study, children were encouraged to abstract mathematical principles by making connections between procedures. Children from 2 sixth-grade classes (N = 58) were asked to solve and explain subtraction examples in 3 different number domains (whole numbers, fractions, and decimals), solve subtraction word problems, compare the procedures, and discuss subtraction principles. Forty percent of the children could identify procedural similarities and could abstract general subtraction principles. The rest of the children received more instruction. One group received individual abstraction and mapping instruction that encouraged them to generalize procedural steps and connect procedures. Another group received domain-specific instruction without connections between domains. The results show that following mapping instruction, children whose original instruction was mostly procedural could make connections and abstract principles, that is, construct a general subtraction schema. These children improved in conceptual knowledge and in solving transfer problems.