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

Toshihiko Shindo, Keiichi Magara

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1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)320-329
Number of pages10
JournalJapanese Journal of Educational Psychology
Issue number3
Publication statusPublished - 2011



  • Illusion of linearity
  • Substantiating the results of mental operations
  • Undergraduates

ASJC Scopus subject areas

  • Developmental and Educational Psychology
  • Education

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