A HIERARCHICAL ORDERING OF AREA SKILLS BASED ON RULES, REPRESENTATIONS, AND SHAPES

Abstract

A hierarchy of skills in the measurement topic of area was validated on three-hundred and six students between grades six and nine. The hierarchy of skills was based on the rules underlying the individual skills. When a rule for one skill was considered a component of a rule for another skill, then the two skills were hypothesized to be hierarchically ordered. In addition, if a simple rule for a particular skill was replaced by a more complex rule, resulting in a different skill, then these two skills were hypothesized to be hierarchically ordered. The physical representations of the area tasks, as well as the shapes of the area figures were hypothesized as influencing the skill orderings. The use of Latent-class analysis revealed that seven of the nine skill orderings analyzed were hierarchically ordered based on difficulty level and not prerequisiteness. The other two skill orderings indicated equaprobable partial mastery classes. In addition to Latent-class analysis, the incorrect processes used by the students were coded and tabulated. The results revealed that (1) nonstandard shaped area problems were the most difficult for this sample, (2) the most frequent process associated with incorrect responses involved the addition of numbers shown in area problem figures, (3) the second most frequent process involved some form of multiplication, without regard to the area concepts inherent in the task, and (4) students beyond the sixth grade made more errors involving multiplication processes than errors involving addition processes. The study revealed that the use of rules, representations and shapes as the basis for a hierarchy does appear to have merit. In addition, process analysis revealed that students respond in a large variety of ways when they do not know the correct process for area tasks.