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This article is the first of two articles to appear in the Arithmetic Teacher reporting the third-grade and seventh-grade results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP). The secondary school data appear in companion articles in the Mathematics Teacher (Brown et al. 1988a, 1988b). This article summarizes the major results of the performance on number, operations, and word-problem items. Some examples of the data are given to support conclusions about what students in general are and are not learning. The specific items reported are altered items that are parallel to the actual items and are illustrative of the results. The actual items are not included because they may be used in future assessments. Interpretations were made from analysis of the actual items. Not all items in the assessment were given to both third-grade and seventh-grade students. Some were administered only when appropriate for the given grade level. Results for eleventh-grade students are included when such information aids in understanding the performance of the third-grade or seventh-grade students.

One hundred twenty-eight children in Grades 1, 2, and 3 were given two multiplication and four division word problems that differed in semantic structure. The children's solution strategies were classified by degree of abstraction into direct representation, double counting, transitional counting, additive or subtractive, or recalled number fact. Strategies were also classified by use of physical objects: use as representations of individual elements, use as tallies or repeated references, or no use. Analyses of the solution strategies indicated the existence of a two-step intuitive model for multiplication, two common intuitive models for partitive division and measurement division each of which was related to an intuitive model for subtraction, and a third model for partitive division.

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