What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number based prior knowledge, certain intermediate states of understanding emerge. This paper presents a study of Greek and Flemish 9th grade students who solved a test about the infinity of numbers in an interval. The Flemish students outperformed the Greek ones. More importantly, the intermediate levels of understanding—where the type of the interval endpoints (i.e., natural numbers, decimals, or fractions) affects students’ judgments—were very similar in both groups. These results point to specific conceptual difficulties involved in the shift from natural to rational numbers and raise some questions regarding instruction in both countries.

► Greek and Flemish students face similar difficulties with rational numbers.
► The idea of discreteness is transferred from natural to rational numbers.
► Natural numbers, decimals, and fractions are treated as unrelated number sets.
► Instruction in both countries neglects these conceptual difficulties.