The main concern of this paper is on the psychological processes of how students understand and solve mathematical word problems, and on how this knowledge can be applied to computer-based tutoring. It is argued that only a better understanding of the psychological requirements for understanding and solving those problems will lead to instructional improvement. Introduced first are the major points of a psychological theory of strategic and goal-directed understanding and solving mathematical situation problems derived from work on discourse processing and on work on cognitive simulation related to word problems. Secondly, the psychological and didactic principles, and the techniques used for the development of a computerized tutor designed at helping students to solve complex, verbally stated, mathematical problems are discussed. Developing partly intelligent tutoring systems may not only serve as a means to provide children with the kind of interactive problem solving and learning environments and instructional tools which have been described as didactic models of scaffolding and coaching, reflective learning, or of cognitive apprenticeship, respectively, but also serve as a means to the more fundamental educational goal of becoming an autonomous or self-directed learner. The computer tutoring program, called HERON, uses an iconic form of problem representation called solution trees or conceptual planning trees, a graphical network format of semantic concepts and mathematical operations, intended to capture the hierarchical structure of solution plans of multi-step mathematical situation problems. A narrative and graphical description of the program is included. (KR)