Measuring Arithmetic: A Psychometric Approach to Understanding Formatting Effects and Domain Specificity

Using multitrait, multimethod data, and confirmatory factor analysis, the current study examined the effects of arithmetic item formatting and the possibility that across formats, abilities other than arithmetic may contribute to children's answers. Measurement hypotheses were guided by several leading theories of arithmetic cognition. With a sample of 1,314 third grade students (age M = 103.24 months, SD = 5.41 months), Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, and the Exact versus Approximate Calculations Hypothesis were evaluated, using 11 measures of arithmetic with symbolic problem formats (e.g., Arabic numeral and language-based formats) and various problem demands (e.g., requiring both exact and approximate calculations). In general, results provided support for both Triple Code Theory and Encoding Complex Theory. As predicted by Triple Code Theory, arithmetic outcomes with language formatting, Arabic numeral formatting, and estimation demands (across formats) were related but distinct from one another. As predicted by Encoding Complex Theory, executive attention was a direct predictor of all arithmetic outcomes. Language was no longer a direct predictor of arithmetic outcomes when executive attention was accounted for in the model; however, a strong and enduring relationship between language and executive attention suggested that language may play a facilitative role in reasoning during numeric processing. These findings have important implications for assessing arithmetic in educational settings and suggest that in addition to arithmetic-focused interventions, interventions targeting executive attention, language, and/or the interplay between them (i.e., internal speech during problem-solving) may be a promising avenues of mathematical problem-solving intervention. Educational Impact and Implications Statement Symbolic formats (e.g., Arabic numerals, spoken language, written language) are usually used for teaching and testing arithmetic ability in formal educational settings; however, research has suggested that different symbolic formats may lead to different sorts of arithmetic problem-solving. Using a large sample of elementary school-aged children, this study explored the possibility that the manner in which problems are conveyed during testing may be an important factor for understanding arithmetic cognition and achievement. Findings suggested that (1) different types of symbolically-formatted arithmetic problems meas