# Multiple representations in mathematical problem solving Exploring

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```					Multiple representations in
mathematical problem solving:
Exploring sex differences

Department of Education
University of Cyprus

Barcelona, January 2007
Introduction
The focus of the study

Pictures           Number line

solving
Theoretical considerations
 This study focuses on one-step change problems
(measure-transformation-measure).

b
a               c

 Change problems include a total of six situations.
 The placement of the unknown in the problems
influences students’ performance (e.g. Adetula, 1989).
Representations used in additive problem solving

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

Schematic drawings,
Picture of a                                  a triadic diagram of
particular situation                                   relations
(Duval, 2005; Theodoulou,                           (Willis & Fuson, 1988;
Gagatsis & Theodoulou, 2004)                    Vergnaud, 1982; Marshall, 1995)

Number line
(Shiakalli & Gagatsis, 2006)
The informational picture

more cakes in the

?      morning.

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

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     numbers
Points on the line                                    depicted on the
can be numbered                                       line correspond
to vectors

The simultaneous presence of these two
conceptualizations   may      limit     the
effectiveness of number line and thus
hinder the performance of learners in
Panaoura, 2003).
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
 To examine the sex differences in the structure of
the processes involved in the solution of additive
problems with multiple representations.
Method

Participants: Primary school students
6 to 9 years of age

Girls   249       243       223     715
Boys    252       243       281     776

Total   501       486       504     1491
The test
b
a            c
18 one-step change problems
(measure-transformation-measure)

Type of the relation
9 join situation (J)                   9 separate situation (S)
The placement of
the unknown
start.amount(a) transf.(b) fin.amount(c) start.amount(a) transf.(b) fin.amount(c)

V   P    L V     P     L V     P    L     V    P    L V     P   L    V    P    L

Representation
V= verbal, P= informational picture, L = number line

An example of the symbolization of the variables:
VJb= a verbal problem of a join situation having the unknown in the transformation
Results
Sex effect
The results of multivariate analysis of           Group
variance (MANOVA) showed that
   the effect exerted by sex was not                 Girls             1.481
significant {F (1,1473) = 0.588,
p=0.443, η2=0.000} on students’                   Boys              1.499

Boys and girls performed
equally well.

   Similar results were obtained in each

Girls       1.096           1.595      1.751

Boys        1.180           1.597      1.720
Sex and representations
Group     Verbal     Picture    Number line
problems   problems   problems
Girls     1.569      1.379      1.495

Boys      1.602      1.379      1.516

 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.
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
The first, second and third
.70 .71 .70
VSa6
coefficient of each factor stand                    Verbal,     .65 .63 .60
for the application of the model on                unk. a, b    .79 .78 .80    VJb15
the performance of the whole                                    .67 .67 .68    VSb12
sample, girls and boys respectively.
PJa9
1.01 1.02 1.00                     .74 .74 .74    PSa18
Picture,     .56 .55 .57
unk. a, b    .73 .74 .73    PJb17
.71 .71 .71     PSb8
.93 .95 .91
Problem-
solving ability                                                 LJa11
.74 .75 .73    LSa16
1.02 1.02 1.03                     .69 .70 .68
Number line,                   LJb7
unk. a, b     .73 .76 .72
.68 .69 .66     LSb5
.94 .93 .94
VJc10
.54 .54 .55
VSc1
Whole sample: Χ2(131)=544.716, CFI=                              .63 .62 .64
Unk.c       .50 .49 .50
0.965, RMSEA=0.046                                               .60 .62 .58
PJc4

Whole sample, girls, boys:                                       .52 .53 .52   PSc13
Χ2(276)=741.621, CFI = 0.961, RMSEA =                           .72 .70 .70     LJc2
0.048                                                                          LSc14
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.
Figure 2: The CFA model for the role of the representations and
the positions of the unknown on additive problem solving by first
VJa3
.62 .64
Verbal,                 VSa6
.63 .55
unk. a, b    .73 .78    VJb15
.52 .55    VSb12
PJa9
1.01 1.01                       .69 .69    PSa18
Picture,     .54 .58
unk. a, b    .73 .70    PJb17
.68 .68     PSb8
.96 .92
Problem-
solving ability                                         LJa11
.63 .66    LSa16
1.02 1.04                       .61 .62
Number line,               LJb7
unk. a, b     .69 .66
.52 .54    LSb5
.84 .93
VJc10
.51 .54
VSc1
Χ2(276)=449.815, CFI=0.942,                              .51 .59
Unk.c      .47 .55
RMSEA=0.050                                              .58 .59
PJc4

The fit of the model                                 .46 .51    PSc13
.59 .70
was good.                                                   LJc2
LSc14
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

VJa3
.69 .66
Verbal,                  VSa6
.55 .48
unk. a, b    .80 .75     VJb15
.68 .64     VSb12
PJa9
1.04 1.02                       .64 .66     PSa18
Picture,     .47 .46
unk. a, b    .66 .68     PJb17
.64 .63      PSb8
.87 .82
Problem-
solving ability                                          LJa11
.71 .62     LSa16
1.02 1.07                       .62 .53
Number line,                LJb7
unk. a, b     .72 .65
.70 .59     LSb5
.99 .91
VJc10
.47 .48
.64 .58     VSc1
Χ2(276)=519.138, CFI=0.920,                   Unk.c       .46 .40     PJc4
RMSEA=0.060                                               .57 .47
.45 .38    PSc13
The fit of the model                                 .71 .65      LJc2
was acceptable.                                                LSc14

 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
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
Concluding remarks
Common remarks between boys and girls across the

 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.
Sex and age

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.

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