Already at a very young age, children experience the wide applicability and intrinsic simplicity of linear/proportional relations. In primary and secondary school mathematics education, moreover, extensive attention is paid to this type of relations. In the long run, students develop the misbelief that each relation can be quantified as proportional, called the ''illusion of linearity''. The best-known misconception originating from such a ''synthetic model of linearity'' is that if a geometrical figure enlarges "k" times, its area and/or volume become "k" times larger too. This article reports and discusses a teaching experiment aimed at remedying this misconception in 8th graders. Ten experimental lessons were developed in order to obtain a conceptual change in these students. The learning results were tested by means of a pretest--post-test--retention test design with an experimental and control group. The problem-solving behaviour of control group students did not change. In the experimental group, the intervention was successful: students' automatic use of proportional strategies for solving non-proportional geometry problems drastically decreased. Nevertheless, some students continued to reason proportionally for all types of problems, while others suddenly started to apply non-proportional strategies to proportional problems too. The linearity illusion was broken in most students, but this did not always result in a deep conceptual understanding of proportional and non-proportional situations and relations.