We contend that the primary role of an illustration or physical manipulable for teaching mathematics is to help the learner understand the language of the mathematics by providing the learner with a referential senr«ntics. Having taught this to subjects, w e address the question of how to assess their understanding. Problem-solving performance, w e show, is insufficient by itself. A n assessment of students' m e m o r y for the original problem statement, and their ability to use cues within the referential semantics is demonstrated as a potential method. Fourth graders (n=24) solved word algebra problem after (a) training with a designed referential semantics from a computer tutor called the Planner, (b) training with symbolic manipulatives, or (c) receiving no training (control). Although pretest-posttest gains were only moderately better for the Planner group than the symbol group, the former showed reliably better ability to reconstruct the problem statements after a 5-day delay. A particular advantage for recall of algebraic relations (as compared to assignments) was evident. Mental representation of relations has been singled out as a major obstacle to successful word problem solving. The support that a well-designed referential semantics plays in the formation and retrieval of appropriate mental structures for problem solving are discussed, as are methods for assessing problem comprehension and conceptual change.