Improper use of linear reasoning: An indepth study of the nature and the irresistibility of secondary school students' errors
 Dirk De Bock,
 Wim Van Dooren,
 Dirk Janssens,
 Lieven Verschaffel
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Several recent ascertaining studies revealed a deeprooted and almost irresistible tendency among 12–16year 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 indepth investigation by means of individual semistandardised 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 wellspecified questions by the interviewer with respect to one single nonlinear 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.
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 Title
 Improper use of linear reasoning: An indepth study of the nature and the irresistibility of secondary school students' errors
 Journal

Educational Studies in Mathematics
Volume 50, Issue 3 , pp 311334
 Cover Date
 20020501
 DOI
 10.1023/A:1021205413749
 Print ISSN
 00131954
 Online ISSN
 15730816
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 illusion of linearity
 length and area
 misconception
 ratio and proportion
 similarity
 Authors

 Dirk De Bock ^{(1)} ^{(2)}
 Wim Van Dooren ^{(1)} ^{(3)}
 Dirk Janssens ^{(4)}
 Lieven Verschaffel ^{(1)}
 Author Affiliations

 1. Center for Instructional Psychology and Technology (CIP&T), University of Leuven, Vesaliusstraat 2, B3000, Leuven, Belgium
 2. Europese Hogeschool Brussel, (EHSAL), Belgium
 3. Research fellow of the of the Fund for Scientific Research, Flanders (Belgium) (F.WO.– Vlaanderen), Belgium
 4. Department of Mathematics, University of Leuven, Belgium