Why Fractions are Difficult? Modeling Optimal and Sub-optimal Integration Strategies of Numerators and Denominators by Educated Adults
Many children and educated adults are challenged by fractions, experiencing difficulties in understanding and manipulating these rather complex mathematical objects. In this study, we argue that a major cause of this challenge is rooted in the need to combine information from two separate informational sources (i.e., denominator and numerator) according to a normative arithmetic rule (i.e., division). We contend that in some tasks, the correct arithmetic operation on the constituent components, is replaced by an inadequate (suboptimal) psychological operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by harnessing two rigorous models of information integration techniques: (a) functional measurement (Experiments 1-3) and conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare empirical behaviour with that of an ideal-observer model. Functional measurement analyses on data from the number-to-line task, revealed that participants could represent the global magnitude of proper and improper fractions quite accurately and combine the fractions’ components according to an ideal-observer model. However, conjoint measurement analyses on data from the comparative judgment task, showed that most participants combined these fractions’ components according to a suboptimal (saturated) observer model, that is inconsistent with an ideal-observer (additive) model. These results support the view that educated adults are capable of extracting multiple types of representations of fractions depending on the task at-hand. These representations can be either accurate and conform with normative arithmetic (system one) or approximated and inconsistent with normative arithmetic (system two). The latter may lead to the observed difficulties people experience with fractions.