On the difficulty of rational number formats

Learning rational numbers poses significant challenges on mathematics education. Recently, it was suggested to facilitate learning rational numbers by introducing them 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 responses) and problem types (e.g., visualization, mere calculations, or word problems). Yet, conclusive evidence on how responses 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 students completing more than 7,000,000 problems) from an intelligent tutoring system. We found large and significant differences in error rates within each rational number format, ranging from about 26\% for problems including visualizations and up to 77\% for word problems. We did not observe significant differences between multiple-choice vs. open responses. These findings suggest that problem types, i.e. the way rational numbers are included in mathematical problems, drive the difficulty of rational number formats considerably and more than rational number format itself. Therefore, curriculum design may consider the arrangement of problem types more compared to arrangements based on format. This means starting with problem using visualizations first, before gradually progressing to more difficult problem types such as word problems.