How We Use What We Learn in Math: An Integrative Account of the Development of Commutativity

One crucial issue in mathematics development is how children come to spontaneously apply arithmetical principles (e.g. commutativity). According to expertise research, well-integrated conceptual and procedural knowledge is required. Here, we report a method composed of two independent tasks that assessed in an unobtrusive manner the spontaneous use of procedural and conceptual knowledge about commutativity. This allowed us to ask (1) in which grade students spontaneously apply this principle in different task formats and (2) in which grade they start to possess an integrated concept of the commutativity. Procedural and conceptual knowledge of 8 to 9 year olds (163 second and 180 third graders) as well as 46 adult students was assessed independently and without any hint concerning commutativity. Results indicated procedural as well as conceptual knowledge about commutativity for second graders. However, their procedural and conceptual knowledge was unrelated. An integrated relation between the two measures first emerged with some of the third graders and was further strengthened for adult students.