Educational Studies in Mathematics

, Volume 50, Issue 3, pp 311–334 | Cite as

Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors

  • Dirk De Bock
  • Wim Van Dooren
  • Dirk Janssens
  • Lieven Verschaffel


Several recent ascertaining studies revealed a deep-rooted and almost irresistible tendency among 12–16-year old students to improperly apply the linear or proportional model in word problems involving lengths, areas and volumes of similar plane figures and solids. While these previous studies showed to what extent students' improper use of linear reasoning is affected by different characteristics of the task, it remained largely unclear what aspects of their knowledge base are responsible for the occurrence and strength of this phenomenon and how these aspects relate to other more general misconceptions and buggy rules identified in the literature. This paper reports an in-depth investigation by means of individual semi-standardised interviews aimed at analysing the thinking process underlying students' improper linear reasoning and how this process is affected by their mathematical conceptions, beliefs and habits. During these interviews,students' solution processes were revealed through a number of well-specified questions by the interviewer with respect to one single non-linear application problem, as well as through their reactions to subsequent kinds of cognitive conflict. The interviews provided a lot of information about the actual process of problem solving from students falling into the ‘linearity trap’ and the mechanism behind it. Although some students seem to really ‘believe’ that quantities are always linked proportionally, their improper use of linearity often results from superficial and intuitive reasoning, influenced by specific mathematical conceptions, habits and beliefs leading to a deficient modelling process.

illusion of linearity length and area misconception ratio and proportion similarity 


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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Dirk De Bock
    • 1
    • 2
  • Wim Van Dooren
    • 1
    • 3
  • Dirk Janssens
    • 4
  • Lieven Verschaffel
    • 1
  1. 1.Center for Instructional Psychology and Technology (CIP&T)University of LeuvenLeuvenBelgium
  2. 2.Europese Hogeschool Brussel, (EHSAL)Belgium
  3. 3.Research fellow of the of the Fund for Scientific Research, Flanders (Belgium) (F.WO.– Vlaanderen)Belgium
  4. 4.Department of MathematicsUniversity of LeuvenBelgium

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