Children’s estimates of equivalent rational number magnitudes are not equal: evidence from whole numbers, percentages, decimals, and fractions

Fractions, decimals, and percentages are generally assumed to differ in difficulty based on the degree to which their structure aligns with, or differs from, that of whole numbers. Percentages are viewed as most similar to whole numbers with their fixed, unstated denominator of 100. Decimals are often assumed to be easier than fractions because their place-value structure is an extension of the base-ten system for whole numbers, unlike fractions, which have a bipartite structure (i.e., a/b). However, unlike whole numbers, a longer string-length for decimals and fractions does not always signify a larger magnitude. To assess understanding of the four notations, we measured number line estimation of equivalent fractions and decimals with shorter string-lengths (e.g., 8/10 and 0.8) and longer string-lengths (e.g., 80/100 and 0.80), percentages (e.g., 80%), and proportionally equivalent whole numbers on a 0-100 scale (e.g., 80.0). Middle school students (N = 65, 33 female) generally underestimated all numbers (Whole Numbers: 3%, Percentages: 2%, Decimals: 17%, and Fractions: 5% below the actual value). Shorter string-length decimals and fractions were estimated as smaller than equivalent longer string-length ones; and larger magnitude decimals and fractions were underestimated by greater amounts than smaller ones. Overall, percentages were estimated similarly to corresponding whole numbers, fractions had modest string-length effects, and decimals were the most underestimated, especially for single-digit decimals. These results highlight the strengths and weaknesses of children’s understanding of each notation’s magnitudes and challenge the assumption that decimals are always better understood than fractions.