Abstracts - Earli

pseudo-proportional problems and impossible ones. The examined grades were deliberatelychosen because they belong to two different educational levels in Cyprus with differentapproaches in the teaching of geometry. Confirmatory Factor Analysis was used for the analysis ofthe data in order to explore the structural organization of the various dimensions of geometricalproblem solving in each age group. This statistical technique was employed as the application ofother analyses (MANOVA) did not show a variation in students’ mean performance in pseudoproportionaltasks, with respect to grade level. Therefore, a more comprehensive analysis wasnecessary to further illuminate the phenomenon of pseudo-proportionality, based on theconjunctions of students’ handling the pseudo-proportional problems and the problems of differentreasoning requirements on the same content. Results suggest the existence of two differentstructural models - one for each age group - for the interpretation of students’ geometrical problemsolving behaviour. The students of both grades did not approach the three types of problems in thesame way but used different reasoning processes. For the younger students the pseudoproportionalproblems were of a similar nature as the usual problems and therefore, composed acommon factor. On the other hand, the pseudo-proportional problems formed a factor of their ownin the case of the older students, making obvious a different reasoning approach compared to theusual ones. This is indicative of the weaker impact of the linear model on 15-year old students’reasoning compared to younger students’ thinking.The linearity prototype in pupils’ and teachers’ perspectives on graphsConstantia Hadjidemetriou, Intercollege, CyprusJulian Williams, University of Manchester, United KingdomThis paper describes pupils’ and teachers’ performance on a task designed to diagnose theLinearity Prototype (LP). The ‘Charity’ item required pupils to draw a graph showing that after aCharity event ‘The more people help, the sooner we finish tidying up’. The whole test, consistingof 29 items, was administered to 425 pupils and their 12 teachers. Results showed 80% of 14-15-year olds exhibiting the LP in the Charity item with no significant differences among year 9 and10 pupils. Pupils’ responses were confirmed and enriched through group interviews in order toanalyse the thinking process behind their inappropriate linear reasoning. 18 pupils wereinterviewed. The results indicated some mismatch between pupils’ reasoning and their graphs withthe linearity answer being ‘conceptually’ but not ‘realistically’ correct for some of them. Pupils’responses also confirmed that, under test conditions, they answer questions superficially withoutengaging in deep mental processes, and that they fall into the ‘linearity trap’ because theyinappropriately apply the methods they used to draw linear graphs to unsuitable situations. Theteachers were asked to rate the difficulty of these items on a five-point scale, answer the questionsand predict possible difficulties of their pupils. Teachers’ ratings were analysed using the InversePartial Credit Model. Teacher’s difficulty estimates were compared to pupils’ actual difficultyestimates and discrepancies were detected. Although teachers accurately predicted pupils’difficulty in most of the items of the test, their prediction for this particular item was significantlyunderestimated. Semi-structured interviews with the teachers indicated that some carry the LPthemselves. They were generally not aware that children tend to exhibit this prototype, whichexplains their inaccurate rating of the items difficulty for the students.– 606 –