Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Dr. Julie A. Washington

Second Advisor

Dr. Lee Branum-Martin

Third Advisor

Dr. Lynn Fuchs

Fourth Advisor

Dr. Rebecca Williamson


Leading theories of arithmetic cognition take a variety of positions regarding item formatting and its possible effects on encoding, retrieval, and calculation. The extent to which formats might require processing from domains other than mathematics (e.g., a language domain and/or an executive functioning domain) is unclear and an area in need of additional research. The purpose of the current study is to evaluate several leading theories of arithmetic cognition with attention to possible systematic measurement error associated with instrument formatting (method effects) and possible contributions of cognitive domains other than a quantitative domain that is specialized for numeric processing (trait effects). In order to simultaneously examine measurement methods and cognitive abilities, this research is approached from a multi-trait, multi-method factor analytic framework.

A sample of 1959 3rd grade students (age M=103.24 months, SD=5.41 months) were selected for the current study from the baseline time points of a larger, longitudinal study conducted in southeastern metropolitan school districts. Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, and the Exact versus Approximate Calculations Hypothesis (a specification of Triple Code Theory) were evaluated with confirmatory factor analysis, 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 from this study provided support for both Triple Code Theory and Encoding Complex Theory, and to some extent, Exact Versus Approximate Calculations Theory is also supported. 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. The relationship between problems that required exact calculations (across formats) also provided support for Exact Versus Approximate Calculations Theory’s stipulation that exact calculation problems may draw from the same cognitive processes. As predicted by Encoding Complex Theory, executive function was a direct predictor of all arithmetic outcomes. Language was not a direct predictor of arithmetic outcomes; however, the relationship between language and executive function suggested that language may play a facilitative role in reasoning during numeric processing, particularly for language-formatted problems.