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.