Conceptually-Based Strategy Use Investigating Underlying Mechanisms and Development Across Adolescence and into Early Adulthood

Abstract

Researchers have used inversion and associativity problems (e.g., 2 × 8 ÷ 8, 3 + 19 − 17, respectively) to assess whether or not individuals have the conceptual understanding that addition and subtraction and multiplication and division are inverse operations (i.e., the inversion concept, Robinson & Ninowski, 2003; Starkey & Gelman, 1982) and whether or not they understand that numbers can be decomposed and recombined in various ways and still result in the same answer (i.e., the associativity concept, Canobi, Reeve, & Pattison, 1998; Robinson, Ninowski, & Gray, 2006). It is not known when the development of these two concepts reaches adult levels. Furthermore, it is not known whether the application of these concepts during problem solving requires individuals to interrupt the execution of well-practiced procedural knowledge (e.g., Siegler & Araya, 2005). In the present study, 40 adolescent participants per grade from Grades 7, 9, and 11 and 40 adult participants who had graduated from high school the previous academic year solved multiplication and division inversion and associativity problems. Also, participants completed a task that measured whether the execution of the inversion shortcut or associativity strategy interrupted the execution of computational strategies. The results suggest that inversion shortcut and associativity strategy use increase in Grade 9, that inversion shortcut use approaches adult levels before associativity strategy use, and that the execution of both conceptually-based strategies interrupts computational strategies. Therefore, the present study identifies adolescence as an important developmental period for inversion shortcut and associativity strategy use and provides the first evidence that applying conceptual mathematical knowledge to problem solving requires the interruption of procedural mathematical knowledge.