Abstracts - Earli

correctly with lower magnitudes than the latter one. Nevertheless, the curve of their estimationsexhibited for both linear and quadratic growth a linear shape. Combining these finding with thoseof earlier research in this domain, it might be hypothesized that the concept of non-linearitydevelops in several phases. First, no differentiation between linear and non-linear processes willtake place. Thereafter, estimations of non-linear processes will become higher than those of linearprocesses but, nevertheless, will continue exhibiting a linear shape. Later, these estimations willshow a non-linear shape, but only a part of the true exponent of the underlying non-linear functionis taken into account. Finally, non-linear processes will be estimated with the appropriatemagnitudes. However, it was shown that even adolescents and adults achieve this last phase onlywith simple tasks, whereas in a variety of other tasks this sample grossly underestimated nonlinearprocesses. The development of the concept of non-linearity will be discussed also withregard to people’s ability to estimate linear processes.Pupils’ over-use of proportionality on missing-value problems: How numbers may changesolutionsWim Van Dooren, Catholic University of Leuven, BelgiumDirk De Bock, European University College Brussels, BelgiumLieven Verschaffel, Catholic University of Leuven, BelgiumPrevious research showed that primary school pupils over-use proportional methods especiallywhen solving non-proportional missing-value word problems®. The current study examineswhether the numbers appearing in these word problems partly explain this phenomenon. In mostprevious studies, the numbers in the problems formed integer ratios (i.e., the outcome could beobtained by making an integer multiplicative jump). This may have stimulated pupils to useproportional methods, also in cases where these are inappropriate. A test containing proportionaland non-proportional word problems was given to 508 4th to 6th graders. Numbers in theseproblems were experimentally manipulated so that the ratios were sometimes integer andsometimes not. For example, a non-proportional problem with integer ratios was: Ellen and Kimare running around a track. They run equally fast, but Ellen started later. When Ellen has run 16laps, Kim has run 32 laps. When Ellen has run 48 laps, how many has Kim run? while the versionwith non-integer ratios was Ellen and Kim are running around a track. They run equally fast, butEllen started later. When Ellen has run 16 laps, Kim has run 24 laps. When Ellen has run 36 laps,how many has Kim run? Correct (additive) reasoning is comparably easy for both versions, butproportional reasoning is far less evident (though still possible, of course) for the version with nonintegerratios. As expected, problems with integer ratios elicited much more (inappropriate)proportional methods in pupils than non-integer ratios. This effect was particularly strong in 4thgrade, dropped in 5th grade to disappear in 6th grade. Theoretical, methodological, and practicalimplications of these findings are discussed.The illusion of linearity in geometrical problem solvingModestina Modestou, University of Cyprus, CyprusIliada Elia, University of Cyprus, CyprusAthanasios Gagatsis, University of Cyprus, CyprusGiorgos Spanoudes, University of Cyprus, CyprusThis study explores the different dimensions of students’ abilities in geometrical problem solvingconcerning area and volume, with special emphasis on students’ behaviour while handling pseudoproportionalproblems and on alteration of this behaviour with students’ age. Students in 9th and10th gradewere given a test involving three types of problems: usual computation problems,– 605 –