How many apples make a quarter? The challenge of discrete proportional formats

Proportional judgments are easier for children in continuous formats rather than discretized ones (e.g., liquid in a beaker vs. in a beaker with unit markings). Continuous formats tap a basic sense of approximation magnitude, whereas discretized formats evoke erroneous counting strategies. On this account, truly discrete formats with separated objects should be even harder. This study (N = 565 7- to 12-year-old children) investigated that prediction. It also examined whether the format effects vary with children's fraction knowledge (FK; part-whole relations, computation, and fraction number line estimation). As found previously, discretized formats were more challenging than continuous ones; as predicted, discrete formats were yet harder. The format effect interacted with FK. Low-FK children were above chance only with continuous formats, medium-FK children struggled with discrete formats only, and high-FK children did well with all three formats.