Developmental Analysis of Understanding Language About Quantities and of Solving Problems

We present information-processing models of different levels of knowledge for understanding the language used in texts of arithmetic word Problems, for forming semantic models of the situations that the texts describe, and for making the inferences needed to answer the questions in the problems. In the simplest cognitive models, inferences are limited to properties of sets that exist in a semantic model. In more complex cognitive models, relations between sets are represented internally and support more complex reasoning. Performance on three sets of problems by kindergarten through third-grade students was used to test the models. Global tests provided support for the models. These included measures of scalability and frequencies of individual children's patterns of solutions that agreed with predictions of the models. Performance on problems involving combinations and changes of sets was explained better by the cognitive models than performance on problems involving comparisons. Comparisons may require more advanced understanding of numbers as values of operators rather than only as cardinalities of sets.