Multiple representations in mathematical problem solving: Exploring sex differences Iliada Elia Department of Education University of Cyprus Barcelona, January 2007

The focus of the study PicturesNumber line Additive problem solvingIntroduction

This study focuses on one-step change problems (measure-transformation-measure). ca b Theoretical considerations Change problems include a total of six situations. The placement of the unknown in the problems influences students’ performance (e.g. Adetula, 1989). Additive change problems

Verbal description (DeCorte & Verschaffel, 1987; Carpenter, 1985) Representations used in additive problem solving Number line (Shiakalli & Gagatsis, 2006) Schematic drawings, a triadic diagram of relations (Willis & Fuson, 1988; Vergnaud, 1982; Marshall, 1995) Picture of a particular situation (Duval, 2005; Theodoulou, Gagatsis & Theodoulou, 2004)

The informational picture

Geometric dimension Arithmetic dimension The numbers depicted on the line correspond to vectors Points on the line can be numbered The simultaneous presence of these two conceptualizations may limit the effectiveness of number line and thus hinder the performance of learners in arithmetical tasks (Gagatsis, Shiakalli, & Panaoura, 2003) The number line

Purpose To explore the effects of the informational picture, the number line and the verbal description (text) on the solution of one-step change problems. To investigate the possible interaction of the various representations with the mathematical structure and more specifically with the placement of the unknown on students’ ability to provide a solution to additive change problems. To examine the sex differences in the structure of the processes involved in the solution of additive problems with multiple representations.

Method Grade 1Grade 2Grade 3Total Girls Boys Total Participants: Primary school students 6 to 9 years of age

The test 18 one-step change problems 18 one-step change problems (measure-transformation-measure) 9 join situation (J) 9 join situation (J) 9 separate situation (S) 9 separate situation (S) V= verbal, P= informational picture, L = number line V P L start.amount(a) transf.(b) fin.amount(c) Type of the relation The placement of the unknown Representation start.amount(a) transf.(b) fin.amount(c) V P L V P L V P L V P L An example of the symbolization of the variables: VJb= a verbal problem of a join situation having the unknown in the transformation ca b

Results The results of multivariate analysis of variance (MANOVA) showed that the effect exerted by sex was not significant {F (1,1473) = 0.588, p=0.443, η 2 =0.000} on students’ additive problem solving performance. Similar results were obtained in each grade separately. Group Girls1.481 Boys1.499 Boys and girls performed equally well. GroupGrade 1Grade 2Grade 3 Girls Boys Sex effect

S ex and representations  Boys and girls exhibited similar problem solving performance in each type of representation.  They both encountered greater difficulty in the solution of problems represented as informational pictures compared to the other types of problems. GroupVerbal problems Picture problems Number line problems Girls Boys

Whole sample: Χ 2 (131)= , CFI= 0.965, RMSEA=0.046 Whole sample, girls, boys: Χ 2 (276)= , CFI = 0.961, RMSEA = Figure 1: The confirmatory factor analysis (CFA) model for the role of the representations and the positions of the unknown on additive problem solving by the whole sample and by girls and boys, separately VJa3 VSa6 VJb15 VSb12 PSa18 PSb8 LJb7 LJa11 LSa16 PJb17 PJa9 LSb5 VSc1 VJc10 PJc4 PSc13 Verbal, unk. a, b Unk.c Number line, unk. a, b Picture, unk. a, b LSc14 LJc2 Problem- solving ability The first, second and third coefficient of each factor stand for the application of the model on the performance of the whole sample, girls and boys respectively.

Remarks on the role of representations in problem solving The findings revealed that students (boys and girls) dealt flexibly and similarly with problems of a simple structure regardless of the mode of representation. However, when they confronted problems of a complex structure they activated distinct cognitive processes in their solutions with reference to the mode of representation. Apart from the structure of the problem, the different modes of representation do have an effect on additive problem solving. There is an important interaction between the mathematical structure and the mode of representation in problem solving.

Χ 2 (276)= , CFI=0.942, RMSEA=0.050 Figure 2: The CFA model for the role of the representations and the positions of the unknown on additive problem solving by first grade girls and boys, separately The fit of the model was good VJa3 VSa6 VJb15 VSb12 PSa18 PSb8 LJb7 LJa11 LSa16 PJb17 PJa9 LSb5 VSc1 VJc10 PJc4 PSc13 Verbal, unk. a, b Unk.c Number line, unk. a, b Picture, unk. a, b LSc14 LJc2 Problem- solving ability

Χ 2 (276)= , CFI=0.920, RMSEA=0.060 Figure 3: The model for the role of the representations and the positions of the unknown on additive problem solving by second grade girls and boys, separately The fit of the model was acceptable VJa3 VSa6 VJb15 VSb12 PSa18 PSb8 LJb7 LJa11 LSa16 PJb17 PJa9 LSb5 VSc1 VJc10 PJc4 PSc13 Verbal, unk. a, b Unk.c Number line, unk. a, b Picture, unk. a, b LSc14 LJc2 Problem- solving ability

The model in third grade The application of the model in third grade students as a whole was acceptable [Χ 2 (131)= , CFI=0.931, RMSEA=0.056], but the relations among the abilities involved (factor loadings) were weaker compared to the younger students’. This indicates that the dependence of the older students’ solution processes on the mode of representation and the placement of the unknown was different from the younger students. The fit of the model on boys and girls of third grade was poor [Χ 2 (276)= , CFI=0.877, RMSEA=0.074]. The model seemed to apply to the boys of the particular grade (after some minor modifications), but not to the girls. The particular structure was not sufficient to describe the solution of the additive problems by third grade girls.

Concluding remarks The results provided a strong case for the role of different modes of representation in combination with the placement of the unknown in additive problem solving. Informational pictures may have a rather complex role in problem solving compared to the use of the other modes of representation. the very interpretation of the informational picture requires extra and perhaps more complex mental processes relative to the verbal mode of representation. That is, the thinker needs to draw information from different sources of representation and connect them. Boys and girls in the whole sample and in each grade exhibited similar levels of performance both in general and at each representational type of problems. Common remarks between boys and girls across the three grades

Sex and age Boys and girls in first and second grade made sense of additive problems in multiple representations by using similar processes. This phenomenon was stronger among the younger students. Third grade boys and girls, despite their similar performance, were found to activate different processes in problem solving with multiple representations. Third graders used processes that were less dependent on the mode of representation and thus on its interaction with the placement of the unknown compared to younger students. Older students could be able to recognize the common mathematical structure not only of the simple problems (model), but also of the complex problems in different representations and deal more flexibly with them than younger students (Gagatsis & Elia, 2004).

Concluding remarks Implications for future research Development generates general problem-solving strategies that are increasingly independent of representational facilitators (Gagatsis & Elia, 2004). This study indicates that girls probably begin to develop or employ explicitly and systematically these strategies earlier than boys. It would be theoretically interesting and practically useful if this inference was further examined in a future study. This would require a longitudinal study combining quantitative and qualitative approaches to map the processes activated by boys and girls at different stages of the particular age span.