There has been much interest in mathematical understanding and a variety of approaches to study of it in the area of research on mathematics education. The aim of this paper is to study the transcendent recursive theory of understanding mathematics which has been developed and elaborated by S. Pirie and T. Kieren in their work over the past six years.
The theory and its model are examined thoroughly from view points of its foundamental ideas, features and usefulness as a theory of understanding mathematics in order to answer the following questions.
1) Why does this theory regard the growth of child' s understanding mathematics as a whole, dynamic, levelled but non-linear, transcendently recursive process?
2) What are features of this theory? In particular, what are properties of eight potential levels or distinct modes as basic elements of this theory?