Profiles in understanding operations with rational numbers

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively different profiles regarding students’ understanding is scarce. The current study investigated the different ways students reasoned in arithmetic operation items with fractions and decimals. A cross-sectional study with 1,262 participants from 5th to 10th grade was performed. A TwoStep Cluster Analysis revealed eight different student reasoning profiles. We found that the natural number bias is first overcome in addition and subtraction, and later in multiplication and division. Moreover, differences regarding representation were only found in addition and subtraction items, indicating that natural numbers interfered more strongly in fractions than in decimal numbers. Finally, results showed that some students’ difficulties with rational number multiplications and divisions had other explanations than the natural number bias.This research was carried out with the support of the project PROMETEO/2017/135 from Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain), and the predoctoral grant from the University of Alicante (UAFPU2018-035), and the postdoctoral grant (I-PI 69-20)