A Dynamic Theory of Mathematical Understanding: Some Features and Implications.

Given the current and widespread practical interest in mathematical understanding, particularly with respect to higher order thinking skills, curriculum reform advocates in many countries cite the need for teaching mathematics with understanding. However, the characterization of understanding in ways that highlight its growth, as well as the identification of pedagogical actions that sponsor understanding, represent continuing problem areas. This paper presents the theoretical basis for a companion document, entitled "The Characteristics of the Growth of Mathematical Understanding." It shows a theory of mathematical understanding that is based on the consideration of understanding as a whole, dynamic, levelled but non-linear process of growth. A model for the theory is represented by eight embedded rings, each representing a level of understanding activity attainable for any particular topic by any specific person. The levels are illustrated by tracing the building of an understanding of the concept of fractions. Beginning with the initial cognitive stage, the levels are called: (1) primitive knowing; (2) image making; (3) image having; (4) property noticing; (5) formalizing; (6) observing; (7) structuring; and (8) inventizing. The application of the theory to teaching is discussed. (MDH)