On the difficulty of rational numbers

Learning rational numbers poses significant challenges in mathematics education. Recent work has suggested introducing rational numbers starting with the easiest format (i.e., fractions, decimals, or percentages). However, difficulty may not only be influenced by the format itself but also by other factors which drive difficulty within formats, such as responses types (e.g., multiple-choice vs. open-ended responses) and problem types (e.g., visualization, mere calculations, or word problems). Yet, conclusive evidence on how response types and problem types drive the difficulty within each of the three different rational number formats is largely missing. To address this gap, we analyzed a large-scale dataset (around 38,000 learners completing more than 7,000,000 problem steps) from an intelligent tutoring system. We found large differences in error rates within each rational number format, ranging from about 22% for problems including visualizations to 69% for word problems. These findings suggest that difficulty is not driven by rational number format alone but also varies substantially with response type and problem type within formats. Thus, instructional sequencing may benefit from considering not only the order in which fractions, decimals, and percentages are introduced, but also variation in problem difficulty within each format. Our findings provide an empirical basis for developing and experimentally testing sequences that progress from relatively easier to more difficult problems, thereby allowing researchers to identify sequencing approaches that are optimal for performance and learning.