56840 EN the role of mathematical representation

ISSN: 2088-687X

11

THE ROLE OF MATHEMATICAL REPRESENTATION AND DISPOSITION IN
IMPROVING STUDENTS' MATHEMATICAL POWER
Imam Kusmaryono1, Hardi Suyitno2 and Dwijanto3
Sultan Agung Islamic University, Semarang, Indonesia
2,3
Mathematics Education Department, Semarang State University, Indonesia
kusmaryono@unissula.ac.id,
1

ABSTRACT
The success of students in problem solving is highly dependent on the ability of students
representing each developmental problems and a higher representation influenced by other
representations. The aim of research to uncover and describe the capabilities and dispositions
mathematical representation based on the students’ mathematical power. The subjects were
students (prospective teachers of mathematics) academic year 2015/2016 Mathematics Education
Department, Sultan Agung Islamic University. The numbers of students in this study were 29
students. The data collection technique using a mathematical representation tests, mathematical

disposition questionnaires and interviews. The study concluded that the representation is affected
by the disposition of the mathematical aspects of a person, through a positive disposition will help
increase students' understanding of mathematical concepts. Furthermore, the mathematical
representation will improve communication skills, perform conjecture and troubleshooting. In
general disposition and mathematical representation was instrumental in enhancing the competence
of the students’ mathematical power.
Keywords: disposition, representations, mathematical power

ABSTRAK
Keberhasilan siswa dalam pemecahan masalah sangat tergantung pada kemampuan
merepresentasikan masalah dan setiap perkembangan representasi yang lebih tinggi dipengaruhi
oleh representasi lain. Tujuan penelitian untuk mengungkap dan menggambarkan kemampuan
representasi dan disposisi matematis berdasarkan daya matematika siswa. Subjek penelitian ini
adalah mahasiswa calon guru matematika tahun akademik 2015/2016, Prodi Pendidikan
Matematika Universitas Islam Sultan Agung. Jumlah mahasiswa dalam penelitian ini adalah 29
orang. Teknik pengumpulan data menggunakan tes representasi matematika, kuesioner disposisi
matematika dan wawancara. Studi ini menyimpulkan bahwa representasi dipengaruhi oleh aspek
disposisi matematika dari seseorang, melalui disposisi positif akan membantu meningkatkan
pemahaman siswa tentang konsep-konsep matematika. Selanjutnya, representasi matematis akan
meningkatkan kemampuan komunikasi, melakukan dugaan dan pemecahan masalah. Secara umum

disposisi dan representasi matematis berperan penting dalam peningkatan kompetensi daya
matematika siswa.
Kata kunci: disposisi, representasi, daya matematika

Introduction

aspects of reasoning. On learning of

Learning math is basically learning about

mathematics in schools students are

the reason. Learning math is a learning

trained to do reasoning, meaning that

activity or activity that emphasizes the

learning should involve students actively

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ISSN: 2088-687X

to reason, strengthen their understanding

Brenner (Neria & Amit, 2004)

of mathematical concepts (Stacey, 2006).

states that the success of students in

In the process of reasoning students will

problem solving is dependent upon the

construct their own knowledge -stage

ability of students to represent a problem,

thinking students.

such

as

constructing

and

using

Learning math is not just convey

mathematical representations in the form

information, showing the formula and

of graphs, words, equations, tables and

insists on the procedure of processing a

images or manipulation of symbols

matter of course, but teachers act as

mathematical more. Hwang, et.al (2007),

mediators and facilitators and assist

in a study entitled influence the ability of

students

of

multiple representations and creativity to

conducive learning so that students

solving mathematical problems using

actively and continuously construct his

multimedia whiteboard system. Research

own knowledge to reason. Students not a

results showed that scores of students

photocopy of adults, what to think or

who use the formula better representation

rationalized by adults (teachers) can not

of

directly forcibly transferred from the

representations and graphic images or

teacher to the student. Each student has a

symbols.

through

the

creation

different way in the process of reasoning

the

student

NCTM

(2000)

using

and

verbal

Sumarmo

to construct knowledge, in other words

(2012) states that the representations are

the students had different levels of

expressions of ideas or the ideas of

mathematical

representation,

between

mathematics that displayed the students

students

teachers

between

in their efforts to seek a solution of the

students and other students. With regard

problems being faced as a result of the

this, teachers are expected to reduce

interpretation of the mind. The idea of a

plantings

mathematics.

mathematical representation in Indonesia

Problem-solving solutions no longer have

has been included in goal of learning

to follow the procedure and should be

mathematics in schools in Regulation No.

done as exemplified by the teacher,

23 of 2006 (Depdiknas, 2007). The

because it would be possible settlement

survey

of problem solving can be through a

Mathematics and Science Study (TIMSS)

variety of mathematical representations

in 2011 showed that the ability of

that can bring critical attitude and

students' mathematical representation of

creative students.

Indonesia is still low. Indonesia is ranked

and

doctrines

in

and

The Role … (Imam Kusmaryono)

results

Trends

International

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ISSN: 2088-687X

13

38 of 42 countries surveyed. This is

and higher level thinking and forming

because students in Indonesia are less

student

accustomed to solve problems with such

application of mathematics. Ability and

characteristics

TIMSS.

skill mathematical representation is also

Learning mathematics at school is not

indispensable for students and teachers in

giving students the opportunity to present

the learning of mathematics. Lack of

their own representation. Students are

ability and mathematical skills of the

only used to do the questions are routine

students

and imitate the teacher in solving the

mathematics

problem, so the ability of students to

decrease.

questions

in

attitudes

can

and skills in

lead

students

disposition

the

toward

will

also

develop their ideas and express them in

In mathematics, the disposition is

various forms of representation are

one component that is essential for

underdeveloped. As a result, the ability of

students (college students) for students

students' mathematical representation is

accustomed to getting issues that require

low.

and

a positive attitude, desire, passion, and

conventional as it only focused on low-

persistence as well as challenges to

level thinking skills.

complete. Without a good mathematical

Learning

monotonous

Conventional learning like this,

disposition then students cannot achieve

certainly not in accordance with the

competence or mathematical prowess as

objectives set curriculum of mathematics

expected. Disposition is defined as a

education in primary and secondary

tendency

of

schools.

students

in

The

purpose

of

learning

students

or

mathematical

individual
looked

mathematics in primary education and

positively or negatively (Kilpatric, Findel

secondary

&

education

is

to

prepare

Swaford,

2001).

In

simple,

students to be able to face changing

mathematical disposition can be regarded

circumstances in life and in the world is

as the attitude, interest and motivation

always evolving, through the practice of

toward mathematics. A large study has

acting on the basis of thinking logically,

proven that the disposition has a strong

rationally,

positive

critically,

careful,

honest,

association

with

cognitive

efficient and effective. In addition,

ability. Hudiono (2005) in his research on

students are expected to use mathematics

mathematics learning in junior high

and mindset of mathematics in everyday

school teacher concluded that the lack of

life, and in studying various scientific

knowledge and habits students learn in

emphasizes on the structuring of reason

the classroom in a conventional way has

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ISSN: 2088-687X

not been possible to develop student

Baki,2010). Dimensional mathematical

representation power optimally. Asikin

representation

and Junaedi (2013), found the disposition

mathematics can be presented as shown

has a strong positive relationship with

below.

solving

ability

power

of

In Figure 1 below, it is understood,

primary level. Likewise, the effect of

that the expected product are obtained

handling the disposition has a strong

when students utilize their knowledge of

relationship with the mathematical skills

mathematics with math skills together

of students of junior high schools

within the framework of the specified

(Syaban, 2009).

contents are indicators of Mathematical

addition

to

at

the

the

In

mathematics

of

low-level

Power.

mathematical thinking and high, students
also need to be trained to think advanced,
the students are trained to construct and
create your own picture mathematical
definition. Through constructing and find
definitions or concepts in mathematics
students are expected to develop other
mathematical abilities, in terms NCTM
the capability in question was referred to
as a mathematical power process.
Basically, every student has the
ability and potential in him. Including the

Figure 1. Mathematical Representation
of Mathematical Power Dimensions (MP)

mathematical power, but mathematical
power levels that each student is different

Do math including integrated and

(Kusmaryono, 2015). After reviewing the

dynamic activities, such as discovery,

relevant literature, in this study the

exploration, conjecture, and understand

mathematical power is defined as "the

the evidence. In this case it is clear that

belief of individuals to use knowledge of

this attitude, the general target primary

the conceptual and operational within the

and

framework of the content specified in a

related

situation

throughout the world is to develop

reasoning,
connection

to

solve

problems

using

communication
together"

(Mandaci

The Role … (Imam Kusmaryono)

and

secondary
to

education
learning

programs

mathematics

Mathematical Power.

&
AdMathEdu | Vol.6 No.1 | Juni 2016

ISSN: 2088-687X
NCTM

15

(2000)

states,

the

communication,

and

the

notion

of

mathematical power include the ability to

context. Mathematical power also served

explore, build conjecture; and give

to spur the success of the appreciation of

logical reasons; the ability to solve the

the complexity of the students in the

problem of non-routine; communicate

interdisciplinary

ideas about mathematics and using

Lourdes, 2005). Each individual involves

mathematics as a communication tool;

the

connect ideas in mathematics, between

(NCTM, 1989; Baroody, 2000). In the

mathematics

principles

activities.

and
As

other

an

intellectual

implication,

the

study

development

and

mathematics

of

(Mueller

personal

standards

(NCTM),

of

and

trust

school

namely

the

mathematical power is an ability that

principle

needs to be owned by students who

Principles), emphasizing the students'

studied mathematics at each school level,

activity to build new knowledge from

as has been recommended by the NCTM

experience

(Sharon L., Charlene. E. & Denisse, R.

students must learn to math with a true

1997).

mathematics

understanding. Bodner (1986) states that

should be implemented to cultivate the

knowledge is constructed by the learner

students'

with

Therefore

how

mathematical

power

(Kusmaryono, 2016).

of

learning

(Learning

and knowledge, so

trying

to

regulate

that

existing

experience in previous mental structure.

The results of the study Dharma et.

Thus, mathematics is a process of

al. (2013) showed that the realistic

acquiring knowledge that is created or

mathematics education more effective in

performed by the students themselves

improving understanding of the concept

through the transformation of individual

and students’ mathematical power. It is

experience.

important to note that in each of their

We know that, in mathematical

learning (students) need to be encouraged

knowledge that leads to mathematical

to discuss their processes in order to

power, requires the ability to use

improve understanding, gain new insights

information to think creatively and to

and

ideas

formulate, solve, and reflect critically on

(Thompson, 2008). The idea is based on

issues (NCTM.2000). Consistent with

the fact that mathematics is more than a

constructivist theories and supporting

collection of concepts and skills that must

evidence, NCTM (1989; 1991) has

be mastered. This includes methods of

recommended shifting from traditional

investigation

instructional approaches toward a better

can

communicate

and

their

reasoning,

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ISSN: 2088-687X

approach

encourages

math

The study was conducted with

power. Aspects of capabilities contained

survey approach and descriptive method.

in the mathematical power is part of

The purpose of study was to look for

mathematical high-level thinking skills.

information in a way to reveal and

Therefore develop students' mathematical

describe the capabilities and dispositions

power starts from the level of young

mathematical representation based on the

people has become an important goal of

power of mathematics students. The

the present study of mathematics (Philips

research subject chosen was a student

&

(prospective teachers of mathematics)

Anderson,

1993;

children

NCTM,

1989;

Diezman, 2005).

odd semester academic year 2015/2016

The assessment of students with a

on Mathematics Education Department,

mathematical measure of how much

Sultan Agung Islamic University. The

information they have to enter their level

numbers of students in this study were 29

of

ability and

implement,

willingness

and

to

use,

students. The data collection technique

communicate

that

was

performed

under

measurement

information (Signe, 2005). Marjolijn.P,

techniques in the form of mathematical

et.al. (2009), confirmed that the suspect

representation ability tests, questionnaires

with the true reason is the mathematical

regarding the disposition of mathematical

thinking process using Mathematical

and

Power. The assessment of mathematics

results

should examine the extent to which

mathematical representation is expressed

students have been integrated and make

in the form of scores and classified based

informed, whether they can apply them to

on the level of students. Here is presented

situations that require reasoning, and they

a mathematical representation indicator.

semi-structured
of

tests

the

interviews.
ability

The
of

can use math to communicate ideas

Table 1. Indicators Mathematical

(NCTM, 2000). On this basis, research

Representation

a

must be conducted to determine the role
of representation and disposition of
mathematics
students’

students

mathematical

to

improve

power

on

mathematics problems solving.
Research Methods

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17

Acquisition of data to measure the

consisting of: 12 positive statements and

ability of a mathematical representation,

12 negative statement following the

then performed the scoring as follows:

disposition indicator mathematically as
follows:

Table 2. Rubric Scoring Ability

(a)

Confidence

mathematical

Mathematical Representation

in

solving

problems;

(b)

Communicating mathematical ideas and
try an alternative method of resolving the
conflict;

(c)

persistent

in

math

homework; (d) Interested, curious and
creative in mathematics activities; (e) to
appreciate the role of mathematical tools
and languages; (f) Share your opinion
with other people .
This

section

questionnaire

presents

scoring

mathematical

dispositions

the
criteria

that

are

classified following the table below:
Furthermore, for the purpose of
clarifying

the

Table 4. Criteria Level Mathematical
Dispositions

mathematical

representation of the quality of students,
the score changed in terms of percentage
and is categorized as follows:
Table 3. Quality of Mathematical

Range of Scores
(%)

Criteria Disposition
Mathematically

76 to 100
51 to 75
26 to 50
0 to 25

High
Enough
Low
Very Low

Representation
Score Value
(In percentage )
90% ≤ Value ≤ 100%
75% ≤ Value < 90%
55% ≤ Value < 75%
40% ≤ Value < 55%
Value < 40%

After the students take the test the

Quality
Ability
Representation
very good
Good
Enough
Less
Very less

ability of mathematical representations,
interviews were conducted with the aim
to obtain in-depth information about the
mathematical representation, disposition
and power of mathematics and the

the

Questionnaires given to students at

difficulties experienced by students in the

beginning

test item .

of

the

study,

a

questionnaire technique is used to obtain

Result and Discussion

data on student mathematical disposition,
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The results of tests the ability of a

ability of representation in terms of the

mathematical

ability of

representation

of

each

students

the

results

are

student can be classified by aspects of

different. Students with a high capacity

mathematical representation (enactive,

amounted to 5 students, mathematical

iconic, symbolic), the level of students'

representation of test results on the

abilities and inclinations mathematical

student group is obtained 1 students were

power students are presented in the

in the category of iconic high and 4

following table.

students were in the category of very high

Table 5. Results of Student Mathematical

symbolic. While the average power of

Representation

mathematics (mathematical power) is at a
high level.
In the group of top level mathematical
power reaches a high level, this seems on
the ability of students in terms of
preparing a conjecture, investigation and

Achievement

indicators

mathematical

disposition through a questionnaire given
to students, obtained the following data.
Table 6. Disposition Mathematical

evidence. On top of this group indicates
that the disposition of high mathematical
achievement that indicator amounted to
75 % has an important role in the

Achievement Indicators
Ability
student
Top group
( high )
Central
Group
( medium )
Bottom
group (low )
Total Score

exploration and come to understand the

Total
Score
364

Percentage

Criteria

75%

High

1.182

77%

High

583

76%

High

2129

76%

High

mathematical representation. Based on
the results of interviews with students,
disposition

and

student

high

mathematical representation can help
increase power mathematical ability of
students.
Students with moderate ability

Based on the test results in Table

dispositions totaling 16 students have

4.1 and 4.2, the achievement indicators

high mathematical achievement indicator

questionnaire

result

with a score of 77%. The results of tests

disposition mathematical research, it is

on a mathematical representation of this

known that all student ability level low,

middle group of students obtained 10

medium and high overall had an average

students were in the category of iconic

high mathematical disposition . The

high and 6 students were in the category

and

analysis

The Role … (Imam Kusmaryono)

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19

of high symbolic. While the average

productive

power of mathematics (mathematical

Interestingly, when the students were

power) is at a high level. Students in the

asked to recall the experience of the

group of top and middle (highly skilled

importance of mathematics in life, they

and are) no longer depends on thinking

consistently choose the experience that

enactive but is in conformity with the

happened when they were in elementary

cognitive development that is iconic and

school. His interest in mathematics in

symbolic thinking.

primary school classrooms may serve to

Students with low ability were 8
students

have

disposition

a

with

high
a

mathematical

score

of

76%

than

David

and

Siti.

reinforce the idea that mathematical
disposition formed at the beginning of
one's school career. While David and Siti,

achievement indicator. The test results on

saw

student

representation

disposition, especially as a way to create

under the group acquired two students

a bond between friends experiences for

were in the category enactive, two

sharing and discussing, so that adds to the

students have low iconic representation

spirit of learning mathematics. It can be

and two students were in the category of

said that the experience of the individual

symbolic medium. For the average ability

person will establish a mathematical

of mathematics (mathematical power) are

disposition someone (Feldhaus, 2014).

mathematical

mathematics

as

a

positive

at moderate levels. At the bottom of this

The tendency of a mathematical

group, two students have engaged in

representation of students in this study is

representing the shape of the iconic and

a mathematical representation of iconic

symbolic.

and symbolic aspects of the most widely

development

Students’
is

still

cognitive
dependent

on

selected students in a math problem.

something real (iconic). Of course, at this

Based

stage the student must immediately

mathematical disposition questionnaire

release the ways of thinking that are

both at the top, middle and bottom have a

iconic, because science is accepted at the

high mathematical disposition. Students

college level are more iconic and

are required to be able to analyze the

symbolic.

problem,

on

the

collect

results

the

of

student

appropriate

Student (prospective teachers of

information and connect with their ideas,

mathematics education) at the time of the

and then present their thoughts in the

interview revealed that Bony and Nurul

form

has a mathematical disposition more

expressions,

AdMathEdu | Vol.6 No.1 | Juni 2016

of

images

or

and

finally

mathematical
found

the

The Role … (Imam Kusmaryono)

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ISSN: 2088-687X

solution of the given problem. Such

understanding of mathematical concepts.

activities

Furthermore,

can

improve

students'

mathematical representation.

the

representation

Based on the results of interviews

mathematical

will

improve

communication skills, perform conjecture

with students, disposition and student

and

high mathematical representation can

disposition

and

help increase power mathematical ability

representation

was

of students. This is consistent with the

enhancing

statement Hudiono (2005) states that "In

mathematical

view of Bruner, enactive, iconic and

Therefore, teachers or lecturers need to

symbolic,

mental

find the right way to be a member of

development of a person, and any

space to expand the representation of

developments

students in mathematics learning by

related

to

higher

the

representation

troubleshooting.

the

In

general

mathematical
instrumental

in

competence

power

of

of

students.

influenced by other representations."

interactive learning and problem-based.

Conclusion

Reference

Based on the test results of a
mathematical representation, disposition
questionnaires
student

and

teachers

interviews
of

with

mathematics

education, disposition and student high
mathematical representation can help
increase power mathematical ability of
students. This parallels the view Bruner

Asikin, Mohammad. Dan Junaedi, Iwan.
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Komunikasi

Matematika Siswa SMP Dalam
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a

symbolic)
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Representation

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