How we use what we learn in Math: An integrative account of the development of commutativity. - Haider, Hilde; Eichler, Alexandra; Hansen, Sonja; Vaterrodt, Bianca; Gaschler, Robert; Frensch, Peter (Frontline Learning Research, 2014)
One of the crucial issues in mathematics development is how children acquire mathematical concepts and procedures. Most researchers now agree that this knowledge develops iteratively (e.g., Resnick, 1992). However, little is known about how well this knowledge is integrated into a more abstract concept and how children come to spontaneously apply such concepts. Expertise research suggests that spontaneously spot and use a principle whenever it applies requires well-integrated conceptual and procedural knowledge. Here, we report a method allowing to asses procedural and conceptual knowledge about the commutative principle in an unobtrusive manner. In two different tasks, procedural and conceptual knowledge of second and third graders as well as adult students were assessed independently and without any hint concerning commutativity. Results show that, even though second graders according to our measures already possessed procedural and conceptual knowledge about commutativity, the knowledge assessed in these two tasks was unrelated. An integrated relation between the two measures first emerged with some of the third graders and was further strengthened for adult students.