Effects of substantiation of the results of mental operations on understanding the ratio of the areas of two similar figures

Prior research demonstrated that students, even at the university undergraduate level, thought that when the length of the sides of a rectangle was enlarged k times, the rectangle's area was also enlarged k times. It was hypothesized that this "illusion of linearity" was the result of inappropriate mental operation of the formula for the area of a rectangle. The aim of Experiment 1 was to increase learners' understanding of the following rule: "If length×k, then area × k ²". Participants (159 undergraduates) were divided into 3 groups. As a demonstration of incorrect linearity, materials were prepared to substantiate the participants' incorrect mental operation. Group A was presented with 2 rectangular photos of a face, one of which was the original photograph, the other, the same photograph enlarged 3 times by enlarging only the width of the original, so that the enlarged version was an oblong version of the original. Group B was presented with 2 rectangles that were the same shape as the photographs used in Group A. Group C was presented no such materials. The performance of the students in Group A was best on the post-test. Experiments 2 (233 undergraduates) and 3 (109 undergraduates) demonstrated that the oblong was effective in changing the students' understanding only when it gave them a sense of incongruity in contrast to actual objects.