The Effects of Different Problem Presentations and Formulations on the Illusion of Linearity in Secondary School Students

Abstract

This investigation focuses on the overgeneralization of the linear model-the so-called illusion of linearity. Previous studies by De Bock, Verschaffel, and Janssens (1998) generated strong empirical evidence for the strength of this phenomenon in 12- to 13-year-old and 15- to 16-year-old students working on applied geometrical problems about the relation between the linear measurements and the area of similar plane figures. In this article, we report on two follow-up studies wherein we investigated the effects of problem presentation and formulation on the strength of the illusion of linearity in the same age groups. In the first follow-up study, we altered the problem presentation by including metacognitive and visual scaffolds aimed at arousing students' doubts about the appropriateness of the linear model and at helping them to find the appropriate mathematical model. In the second follow-up study, we changed the problem formulation by transforming the problems, which were originally formulated in a missing-value format, into comparison problems. Both experimental manipulations had a positive effect on students' ability to resist the linearity illusion and to discover the nonproportional character of the problem situations, but these effects remained rather small, indicating that students' tendency toward linear modeling is indeed very strong, deep-rooted, and resistant to change.