Multiple Multiple representations in representations in mathematical problem mathematical problem solving: Exploring sex solving: Exploring sex differencesdifferences

Iliada EliaIliada EliaDepartment of Education

University of Cyprus

Barcelona, January Barcelona, January 20020077

The focus of the studyThe focus of the study

Pictures Number line

Additive problem solving

IntroductionIntroduction

This study focuses on one-step change problems (measure-transformation-measure).

ccaabb

Theoretical considerationsTheoretical 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 problemsAdditive change problems

Verbal description(DeCorte & Verschaffel, 1987; Carpenter, 1985)

Representations used in additive problem Representations used in additive problem solvingsolving

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

?

The cakes I have had The cakes I have had ready since yesterday.ready since yesterday.

I made some I made some more cakes in more cakes in the morning.the morning.

These are the cakes I have These are the cakes I have now. How many cakes did I now. How many cakes did I make in the morning?make in the morning?

Geometric d

imensio

nArithmetic dim

ension

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).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

The number line

PurposePurpose

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.

MethodMethod

Grade 1

Grade 2

Grade 3

Total

Girls 249 243 223 715

Boys 252 243 281 776

Total 501 486 504 1491

Participants: Primary school students 6 to 9 years of age

The test

18 18 one-step change problems one-step change problems (measure-transformation-

measure)

9 9 join situationjoin situation ( (JJ)) 9 9 separate situationseparate situation ( (S)S)

VV= = verbalverbal, , PP= = informational pictureinformational picture, , LL = = number linenumber line

VV P P L L

start.amount(a) transf.(b) fin.amount(c)

Type of the relationType of the relation

The placement of The placement of the unknownthe unknown

RepresentationRepresentation

start.amount(a) transf.(b) fin.amount(c)

VV P P L L

VV P P L L

VV P P L L

VV P L P L

VV P P L L

An example of the symbolization of the variables:

VJb= a verbal problem of a join situation having the unknown in the transformation

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ResultsResults

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, η2=0.000} on students’ additive problem solving performance.

Similar results were obtained in each grade separately.

Group

Girls 1.481

Boys 1.499

Boys and girls Boys and girls performed equally well. performed equally well.

Group Grade 1

Grade 2 Grade 3

Girls 1.096 1.595 1.751

Boys 1.180 1.597 1.720

Sex effect

Sex and representations

Boys and girls exhibited similar problem solving Boys and girls exhibited similar problem solving performance in each type of representation.performance in each type of representation. They both encountered greater difficulty in the They both encountered greater difficulty in the solution of problems represented as informational solution of problems represented as informational pictures compared to the other types of problems.pictures compared to the other types of problems.

Group Verbal problems

Picture problems

Number line problems

Girls 1.569 1.379 1.495

Boys 1.602 1.379 1.516

Whole sample: Χ2(131)=544.716, CFI= 0.965, RMSEA=0.046

Whole sample, girls, boys: Χ2(276)=741.621, CFI = 0.961, RMSEA = 0.048

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

.70 .71 .70

.94 .93 .94

1.02 1.02 1.03

.93 .95 .91

1.01 1.02 1.00

.72 .70 .70 .52 .53 .52 .60 .62 .58 .50 .49 .50 .63 .62 .64 .54 .54 .55

.68 .69 .66

.73 .76 .72

.69 .70 .68

.71 .71 .71

.73 .74 .73

.56 .55 .57

.74 .74 .74

.67 .67 .68

.79 .78 .80

.65 .63 .60

.74 .75 .73

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 Remarks on the role of representations in problem solvingrepresentations 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.

Χ2(276)=449.815, 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.

.62 .64

.84 .93

1.02 1.04

.96 .92

1.01 1.01

.59 .70 .46 .51 .58 .59 .47 .55 .51 .59 .51 .54

.52 .54

.69 .66

.61 .62

.68 .68

.73 .70

.54 .58

.69 .69

.52 .55

.73 .78

.63 .55

.63 .66

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

Χ2(276)=519.138, 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.

.69 .66

.99 .91

1.02 1.07

.87 .82

1.04 1.02

.71 .65 .45 .38 .57 .47 .46 .40 .64 .58 .47 .48

.70 .59

.72 .65

.62 .53

.64 .63

.66 .68

.47 .46

.64 .66

.68 .64

.80 .75

.55 .48

.71 .62

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 gradeThe model in third grade

The application of the model in third grade students as a whole was acceptable [Χ2(131)=334.744, 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 [Χ2(276)=658.382, 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 remarksConcluding 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 remarksConcluding remarksImplications for future researchImplications 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.