TY - JOUR

T1 - Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults

AU - Fitousi, Daniel

AU - Noyman, Ran

N1 - Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2024/1

Y1 - 2024/1

N2 - Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate 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 rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration: (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants’ integration strategies 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 sub-optimal (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 or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

AB - Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate 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 rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration: (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants’ integration strategies 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 sub-optimal (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 or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

KW - Conjoint measurement

KW - Fractions

KW - Numerical cognition

UR - http://www.scopus.com/inward/record.url?scp=85177601876&partnerID=8YFLogxK

U2 - 10.1016/j.cognition.2023.105656

DO - 10.1016/j.cognition.2023.105656

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C2 - 37979219

AN - SCOPUS:85177601876

SN - 0010-0277

VL - 242

JO - Cognition

JF - Cognition

M1 - 105656

ER -