An Analysis of the Relationships among Computation-Related Skills Using a Hierarchical-Clustering Technique

Abstract

A hierarchical-clustering technique was applied to data on the mastery of mathematical skills for the purpose of examining relationships among the skills. The data were obtained from two administrations of criterion-referenced tests to 715 pupils in Grades 1 to 4 in two Irish elememary schools toward the end of their school year. A clear hierarchical ordering of skills was obtained at each grade. An examination of groups of computation-related skills showing a high degree of consistency in their orderings across the two administrations suggested that factors considered in the formulation of the skills could account for some of the orderings obtained. The results highlight the value of cluster analysis as a tool for the preliminary analysis of relationships among mathematical skills.

Patterns of knowledge in children's addition.

“…Four main outcomes are noteworthy. First, consistent with previous findings (Canobi et al, 1998;Close & Murtagh, 1986;Langford, 1981), children noticed that addends were reordered before noticing that addends were decomposed and recombined. Second, although children's understanding that numbers can be added in different orders appears surprisingly abstract, there was a concrete-toabstract ordering in their knowledge of the ways in which addends can be decomposed and recombined.…”

“…The goals were to discover whether children were more likely to recognize commutativity relations than relations based on aspects of associativity and additive composition (cf. Canobi et al, 1998;Close & Murtagh, 1986;Langford, 1981) and whether children made more correct conceptual judgments in the context of objects than in the context of symbols, as might be expected if children's representations of addition concepts develop along a concrete-to-abstract dimension. Second, we investigated whether we could identify meaningful patterns of procedural knowledge based on children's problem-solving performance.…”

“…The aim of the first analysis was to examine the comparative difficulty of addition concepts because although Resnick (1992) argued that commutativity and associativity are not distinct in children's understanding, others (e.g., Canobi et al, 1998; Close & Murtagh, 1986; Langford, 1981) have argued that commutativity knowledge emerges prior to associativity knowledge. Therefore, we examined whether children were more accurate at judging and justifying their judgments for order problems than for composition problems.…”

“…With a few exceptions (e.g., Baroody, Ginsburg, & Waxman, 1983; Canobi, in press; Canobi et al, 1998, 2002; Hsieh & Baroody, 2000; Langford, 1981), children's knowledge of each addition principle has been examined separately, making it difficult to assess changes in the interrelations among them and, ipso facto, conceptual development (Bisanz & Lefevre, 1992). Indeed, although Resnick (1986) argued that commutativity and associativity are not distinct in children's understanding but arise from knowledge of additive composition, other researchers have argued that children acquire an understanding of commutativity prior to an understanding of associativity (Close & Murtagh, 1986; Langford, 1981). Therefore, further research into the comparative difficulty of recognizing different kinds of principle-based relations between addition problems is needed.…”

“…First, are children more likely to recognize conceptual relations in which addends are reordered than those in which addends are decomposed and recombined (cf. Canobi et al, 1998; Close & Murtagh, 1986; Langford, 1981)? Second, are children more likely to recognize conceptual relations in the context of objects than in the context of symbols, as might be expected if their representations develop along a concrete-to-abstract dimension?…”