This paper describes a model of student attainment of mathematical concepts and its development. In this model three types of activities (developmental, connecting, and abstract) are considered as an overlay on the three ways of representing mathematical concepts (physical/visual, oral, and symbolic). Each activity type involves some means of…

A discussion is presented concerning variables associated with children's representation of certain addition and subtraction problems. Problem components are classified as physical, pictorial, sensual, verbal-spoken, verbal-written, symbolic, or internal. Problems are discussed as abstract (symbolic) or nonsymbolic. Three problem components are…

The difficulties and misconceptions that many students experience when learning to visualize graphs of functions are discussed. Specifically, the inability to make accurate connections between symbolic and graphical representations, the lack of understanding about scale and scaling, and the confusion caused by transformations are examined with…