The role of visual representations in mathematical cognition and understanding mathematics

Abstract

The problem of visual representations in mathematics is analyzed within the framework of activity approach. Visual representations of mathematical concepts reflect visual material operation schemes rather than visual material objective characteristics that are independent of the recipient. A fundamental distinction is made between internal and external means of visualization in mathematics: an internal image is included in the schemes according to which it is perceived and used. Common roots of sensible and non-sensible intuition in mathematics are identified: intuition allows to grasp an operation scheme with external sign models. It is proposed to treat the mathematical concept as a coordination of various operation schemes with sign-symbol models: spatial, verbal and algebraic. Educational consequences of the presented approach to visual representations are discussed