Profiles in understanding the density of rational numbers among primary and secondary school students

Abstract

The present cross-sectional study investigated 953 fifth to tenth grade students’ understanding of the dense structure of rational numbers. After an inductive analysis, coding the answers based on three types of items on density, a TwoStep Cluster Analysis revealed different intermediate profiles in the understanding of density along grades. The analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our profiles showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a natural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers.