Modeling Quantitative and Covariational Reasoning

In the Piagetian tradition, researching reasoning regards modeling cognitive structures and their development. For quantitative and covariational reasoning, these are primarily descriptions of mental images and mental actions, as well as the tasks and observations used to make inferences about them. The chapter begins by contrasting the meanings of quantitative and covariational reasoning with their more common usages in applied statistics. Next described are Piaget’s constructs of gross, intensive, and extensive quantities, which he initially used to categorize stages in children’s reasoning about measurement. These constructs are also useful for distinguishing older students’ ways of thinking about quantities, including their images of how quantities vary (and covary). Three models for continuous variational reasoning (chunky, smooth, and scaling) are introduced and compared. Methodological considerations are discussed with regard to examples of students’ thinking, including contrasts between arithmetical and quantitative reasoning as well as figurative and operative thinking. The chapter closes by addressing the roles of mathematical modeling and problem-solving in promoting students’ quantitative and covariational reasoning.