Kind of warrants and proofs used by upper secondary students in intuitively approaching the density of Q in R

While research in Mathematics Education has mainly focused on the density of Q, little is known about the conceptualization of the density of Q in R. This study investigates upper-secondary school students’ warrants and proofs regarding the density of Q in R. A classroom activity focused on continued fractions, which provide a unified representation of rational and irrational numbers, was carried out. Through warrantand proof-analysis, this study reveals how different kinds of warrant and proof shape the students’ reasoning on the density of Q in R, providing insight into the level of mathematical depth reached and the reasoning effectiveness produced by students.