The role of structural and semantic factors in the solution of algebra speed problems

A model for the solution of speed problems was proposed which is based on their structure, and its component schemata of INFER DURATION RELATION and INFER LENGTH underlie the representation and modelling of these problems. The acquisition of the INFER RELATION schemata and their use were studied by testing 178 ninth to eleventh grade students. Tests included three isomorphic problems of six problem types. The findings are: the inference of the Time Nondirect Relation is a prerequisite for the solution. The INFER DURATION RELATION schema was used correctly in three fifths of the solutions, but was acquired by one tenth of the students. Students did not prefer to model any relation within and across problem types but the success rates depend on both the relation modelled and the requirement for evidence for the inferences. The success rates of the problem types partly match the hypothesized cognitive complexity, and are proportional to the number of equations produced in the test. Students neither recognized nor capitalized on the problem isomorphism. The results generally substantiate the validity of the analysis, and possible reasons for the deviations from the model's predictions are proposed, and instructional implications that the experiment may have for students and teachers are outlined.