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Title: The illusion of linearity: A decade of research in the field of geometry
Authors: De Bock, Dirk
Van Dooren, Wim
Janssens, Dirk
Verschaffel, Lieven #
Issue Date: Jul-2008
Conference: International Congress on Mathematical Education edition:11 location:Monterrey, Mexico date:July 2008
Abstract: Socrates drew a square and asked his servant, a boy named Meno, to create one that is twice as large. Meno responded by doubling the length of each side…

From a long way back, the linear (or proportional) relationship is a key concept in mathematics education. Both from a psychological and a mathematical point of view, the idea of linearity comes first. However, just like Meno, students tend to overrely on linearity, or as Freudenthal argued in his Didactical Phenomenology of Mathematical Structures “linearity is such a suggestive property of relations that one readily yields to the seduction to deal with each numerical relation as though it were linear” (p. 267). This phenomenon is often referred to as the illusion of linearity. Although the negative impact of this illusion of linearity has always been acknowledged, empirical research on this phenomenon started only about one decade ago.

On this poster, we present the major findings of a systematic line of research in the field of geometry, more specifically, with respect to problems about the relations between the linear measurements and the area or volume of similarly enlarged or reduced geometrical figures (and students’ belief that if a figure enlarges k times, the area and/or volume enlarge k times too). In this line of research, various methodological instruments were used: the administration of collective paper-and-pencil tests to large groups of students under different experimental conditions (each providing some form of help), the performance of in-depth semi-standardised individual interviews involving a series of cognitive conflicts, and the design, implementation and evaluation of an experimental lesson series.

This program has empirically demonstrated that the illusion of linearity is indeed a widespread and deep-rooted phenomenon affecting many students from a broad range of ages and educational settings. Even with considerable support (such as the request to make a drawing of the problem situation before solving the problem or the provision of ready-made drawings on plain or squared paper, or making the problems more authentic), only very few students made the shift to correct non-linear responses. Remarkably, once students discovered the non-linear character of certain situations as a consequence of the provided help, they almost immediately overgeneralised these new non-linear insights to ‘simple’ linear problems.

Our research revealed the psychological and educational factors that lie at the roots of the occurrence and persistence of the illusion of linearity. Explanatory factors were found in (1) the intuitive, heuristic nature of the linear model, (2) students’ experiences in the mathematics classroom, and (3) elements related to the mathematical particularities of the problem situation in which the linear error occurs. Our findings not only seem important to understand students’ mathematical thinking and learning, but can also guide the development of teaching/learning environments aimed at the development of students’ deep conceptual understanding of linearity, including the disposition to distinguish between situations that can and cannot be modelled linearly.
Description: see also http://icme11.org/node/700
Publication status: published
KU Leuven publication type: IMa
Appears in Collections:Leuven Academic Centre for Professional Development in Education
Mathematics - miscellaneous
Education and Training
Instructional Psychology and Technology
Research Centre for Educational Research & Development, Campus Brussels (-)
Faculty of Economics and Business (FEB) - miscellaneous
# (joint) last author

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