Is counting a bad idea? Complex relations among children's fraction knowledge, eye movements, and performance in...

Understanding fraction magnitudes is crucial for mathematical development but is challenging for many children. Visualizations, such as tape diagrams, are thought to leverage children's early proportional reasoning skills. However, depending on children's prior knowledge, these visualizations may encourage various strategies. Children with lower fraction knowledge might rely on counting, leading to natural number bias and low performance, whereas those with higher knowledge might rely on more efficient strategies based on magnitude. This study explores the relationship between students' general fraction knowledge and their ability to visually compare fraction magnitudes represented with tape diagrams. A total of 67 children completed a fraction knowledge test and a set of comparison tasks with discretized and continuous tape diagrams while their eye movements, accuracy, and response times were recorded. Cluster analysis identified three groups. The first group, high-achieving and applying magnitude-based strategies, showed high accuracy and short response times, indicating efficiency. A second high-achieving group frequently used counting strategies, which was unexpected. This group achieved the highest accuracy but the longest response times, indicating less efficiency. The third group, low-achieving and rarely using counting strategies, had the lowest accuracy and short response times. These students tended to compare absolute sizes rather than relative sizes (i.e., showing a size bias). None of the groups exhibited a natural number bias. The study suggests that counting, although inefficient, does not necessarily lead to bias or low performance. Instead, biased reasoning with fraction visualizations can originate from reliance on absolute sizes.