Proceedings of the Annual Meeting of the Cognitive Science Society

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.