Grant: Discovery Grants Program - Individual

The development of arithmetic concepts is an essential aspect of children's arithmetic knowledge. In the area of children's mathematical cognition, much of the emphasis has been on the development of additive concepts rather than on multiplicative concepts and yet multiplicative concepts are more complex and difficult for children to understand. Further, research on how children's conceptual development occurs in conjunction with procedural and factual knowledge of arithmetic is critical to understanding children's mathematical knowledge as a whole. My research program aims to address both of these issues by investigating how both additive and multiplicative versions of the concepts of inversion and associativity develop across time in conjunction with procedural and factual knowledge. The concept of inversion, that pairs of operations are inversely related to each other, appears to be grasped, even before formal schooling when the additive aspect of the concept is assessed on problems such as a + b - b. However, when multiplicative problems such as d x e ÷ e are presented, children appear to have a much weaker understanding of the inversion concept. The concept of associativity, that any pair of numbers can be solved first in problems of the form a + b - c and d x e ÷ f, appears to be weak on both additive and multiplicative problem types. There is some evidence that understanding of one concept is related to understanding of the other. Further, there is some evidence that the understanding and application of the concepts is related to factual and procedural knowledge of the operations under investigation. The goal of this research program is to follow children's developing understanding of both concepts and to isolate and identify how concept development can be enhanced or promoted. Children's solid understanding of how the arithmetic operations relate to one another is key to their further and more complex understanding of the mathematic problems and concepts that they will be exposed to in their later years of formal schooling.