Numerical cognition: On the convergence of componential and psychometric models

Abstract

The relationship between the elementary operations underlying the processing of numerical information and performance on psychometrically derived ability measures defining the Numerical Facility, Perceptual Speed, General Reasoning, and Memory Span factors was examined for a sample of 112 Air Force recruits. The process variables were derived based on a componential model for arithmetic and an additional measure that indexed the ability to allocate attention while executing arithmetical operations in working memory. All numerical facility, general reasoning, and memory-span tests required the processing of numbers, but only the numerical facility and general reasoning measures required arithmetic. Analyses of the pattern of relationship between the process variables and the ability measures indicated that the rate of executing the operations of arithmetic fact retrieval and carrying contributed to individual differences on the measures defining both the Numerical Facility and General Reasoning factors, but was unrelated to performance on the memory-span tests. Moreover, the attentional allocation aspect of working-memory capacity directly contributed to individual differences on only the general reasoning and memory-span measures. The results demonstrated a pattern of convergent and discriminant relationships for parameters reflecting the processing of numerical information and performance on theoretically similar and subtly dissimilar ability measures. Implications for the representation of individual differences in human abilities were discussed.

Counting on working memory in arithmetic problem solving

“…As noted earlier, our own view is of a multicomponent system; some authors view working memory as a single flexible system providing temporary storage, processing resources, and the allocation of attention (e.g., Engle, Cantor, & Carullo, 1992;Geary & Widaman, 1992;. This latter form of working memory appears to correspond roughly to the central executive component in the Baddeley and Hitch (1974;Baddeley, 1986) model of working memory.…”

“…This latter form of working memory appears to correspond roughly to the central executive component in the Baddeley and Hitch (1974;Baddeley, 1986) model of working memory. In a study ofindividual differences, Geary and Widaman (1992) have suggested that the measure of working memory used in their studies is related to arithmetic performance. This is entirely consistent with our own view that the central executive component of working memory serves an important function in these tasks.…”

“…In the case of arithmetic problems that cannot be tackled in this way by most people (e.g., 234+429 or 23 X47), more complex models have been developed that specify stages in the calculation. However, there is a continuing debate among the authors of the various models as to the nature of the cognitive processing that might be involved (e.g., Campbell & Clark, 1988;Dehaene, 1992;Geary & Widaman, 1992;McCloskey, 1992;McCloskey et aI., 1991;McCloskey et aI., 1986;Sokol, GoodmanSchulman, & McCloskey, 1989;Sokol et aI., 1991;Widaman et aI., 1989) or how the necessary skills are acquired and applied by young children (Gallistel & GeIman, 1992;Graham, 1987;Hitch, Cundick, Haughey, Pugh, & Wright, 1987;Siegler, 1987). Despite the debate, most researchers in this area agree that there is a requirement for temporary storage of information during calculation in addition to any other processes that might be involved.…”

Attention and intelligence: The validity of the Star Counting Test.

“…During this century, correlational studies have provided considerable insight into the structure of intelligence (e.g., Carroll, 1993). More recently, experimental researchers have identified some of the mechanisms underlying performance on specific intelligence tests (Butterfield, Nielsen, Tangen, & Richardson, 1985;Carpenter, Just, & Shell, 1990;Geary & Widaman, 1992;Steraberg, 1985). In contrast, in the field of attention there is still a gap between what Cronbach (1957) called the "two disciplines of psychology" (p. 671), the experimental and the correlational approach.…”

Mathematical development: the role of broad cognitive processes

“…Baddeley and Hitch (1974) conceptualized working memory as a multicomponent cognitive device comprised of three systems: the central executive (the "central processor," which carries out the operations), the phonological loop (the "auditory processor," which organizes verbal information and feeds it to the central processor), and the visual-spatial sketchpad (the "visual processor," which organizes visual information and feeds it to the central processor). Calculation skills and problem solving have been theorized to depend, in part, on the working memory central executive, the phonological loop, and the visual-spatial sketchpad (Geary & Widaman, 1992;Hitch, 1978;Swanson, Cooney, & Brock, 1993). It is reasonable to hypothesize that the working memory central executive is engaged during translation of word problem sentences into equations, and in executing arithmetic steps.…”