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. The focus of this report is a dynamic, multidimensional, multidirectional model for the theory of the growth of understanding within a specific individual on a specific topic. Background to this theory is provided in the companion document, entitled "A Dynamic Theory of Mathematical Understanding: Some Features and Implications." Included is a discussion of various classroom situations surrounding the topic of quadratic equations that illustrate the eight embedded levels of understanding within the model, consisting of: (1) primitive knowing; (2) image making; (3) image having; (4) property noticing; (5) formalizing; (6) observing; (7) structuring; and (8) inventizing. These situational differences illustrate the various teacher-student interactions that guide the student through the eight embedded levels of understanding through the following specific processes: image doing, image reviewing, image seeing, image saying, property predicting, and property recording. (JJK)