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Journal on Education

Volume 05, No. 02, Januari-Februari 2023, pp. 5313-5323

E-ISSN: 2654-5497, P-ISSN: 2655-1365

Website: http://jonedu.org/index.php/joe

Improving Students' Mathematical Problem-Solving Ability through the

Use of External Representations

Amirul Syah1, Harizahayu2, Gamar Al Haddar3, Annisah4, Emy Yunita Rahma Pratiwi5

1Universitas Muhammadiyah Sumatera Utara, Jl. Kapten Muchtar Basri No.3, Glugur Darat II, Kec. Medan Tim., Kota

Medan, Sumatera Utara

2Politeknik Negeri Medan, Jl. Almamater No.1, Padang Bulan, Kec. Medan Baru, Kota Medan, Sumatera Utara

3Universitas Widya Gama, Jl. Borobudur No.35, Mojolangu, Kec. Lowokwaru, Kota Malang, Jawa Timur

4SMPN 48 Surabaya, Jl. Bratang Wetan No.36, Ngagelrejo, Kec. Wonokromo, Kota SBY, Jawa Timur

5Universitas Hasyim Asy’ari Tebuireng Jombang, Jl. Irian Jaya No.55, Cukir, Kec. Diwek, Kabupaten Jombang, Jawa Timur

amirulsyah@umsi.ac.id

Abstract

This study has the objectives, namely first, to find out whether students' mathematical problem-solving abilities

can be improved by using external representations. Second, analyzing student activities in learning mathematics

by using external representations. Third, analyzing student responses to learning mathematics by using external

representations. This research uses classroom action research (CAR) or classroom action testing (PTK) methods.

According to O'Brien in Mulyatiningsih, "action research is carried out when a group of people (students)

identify the problem, then the researcher determines an action to overcome it." Classroom Action Research

(CAR) seeks to develop and reflect on a learning model with the aim of improving learning processes and

outcomes. Based on the results of the research, it can be concluded that: learning using external representations

can improve mathematical problem solving abilities; it can be seen that there is an increase in the number of

students who score above the KKM in cycle II, with the percentage of students who are declared complete

(reaching the KKM) at 71.4%, or as many as 20 students are higher than in cycle I, with a percentage of

53.57%, or as many as 15 students who are declared complete. This means that most students have achieved

learning mastery resulting from learning through the use of external representations.

Keywords: representation, classroom, learning, student.

Abstrak

Penelitian ini memiliki tujuan yaitu pertama, untuk mengetahui apakah kemampuan pemecahan masalah

matematis siswa dapat ditingkatkan dengan menggunakan representasi eksternal. Kedua, menganalisis aktivitas

siswa dalam pembelajaran matematika dengan menggunakan representasi eksternal. Ketiga, menganalisis

respon siswa terhadap pembelajaran matematika dengan menggunakan representasi eksternal. Penelitian ini

menggunakan metode penelitian tindakan kelas (PTK) atau pengujian tindakan kelas (PTK). Menurut O’Brie n

dalam Mulyatiningsih, “penelitian tindakan dilakukan ketika sekelompok orang (siswa) mengidentifikasi

masalah, kemudian peneliti menentukan suatu tindakan untuk mengatasinya.” Penelitian Tindakan Kelas (PTK)

berupaya mengembangkan dan merefleksi suatu model pembelajaran dengan tujuan untuk meningkatkan proses

dan hasil pembelajaran. Berdasarkan hasil penelitian dapat disimpulkan bahwa: pembelajaran dengan

menggunakan representasi eksternal dapat meningkatkan kemampuan pemecahan masalah matematis; terlihat

adanya peningkatan jumlah siswa yang mendapat nilai di atas KKM pada siklus II, dengan persentase siswa

yang dinyatakan tuntas (mencapai KKM) sebesar 71,4%, atau sebanyak 20 siswa lebih tinggi dari pada siklus I

dengan persentase 53,57% atau sebanyak 15 siswa yang dinyatakan tuntas. Ini berarti bahwa sebagian besar

siswa telah mencapai ketuntasan belajar yang dihasilkan dari pembelajaran melalui penggunaan representasi

eksternal.

Kata Kunci: representasi, ruang kelas pembelajaran, siswa.

Copyright (c) 2023 Amirul Syah, Harizahayu, Gamar Al Haddar, Annisah, Emy Yunita Rahma Pratiwi

Corresponding author: Amirul Syah

Email Address: amirulsyah@umsi.ac.id (Jl. Kapten Muchtar Basri No.3, Glugur Darat II, Kec. Medan Tim.,

Kota Medan, Sumatera Utara)

Received 20 January 2023, Accepted 26 January 2023, Published 29 January 2023

INTRODUCTION

Education is the main factor in the progress of a nation. Good education produces the best

graduates, who are expected to become implementers of national development. Along with the times,

5314 Journal on Education, Volume 05, No. 02 Januari-Februari 2023, hal. 5313-5323

the problem of education has received a lot of attention from the community. To support the

development of human resources (HR) who are competitive in responding to the challenges of an

ever-changing era, quality education is required. This is in accordance with the educational objectives

stated in Law No. 20 of 2003, Article 3, concerning the National Education System, namely:

"National education functions to develop capabilities and form dignified character and national

civilization in order to educate the nation's life. It aims to develop the potential of students to become

human beings who believe and fear God Almighty, have noble character, are knowledgeable, capable,

independent creatives, and become democratic and responsible citizens."

One way to develop the competitiveness of human resources (HR) is through education

conducted in educational institutions that organize learning processes in schools. One part of

education in schools that can make an important contribution to students' development of human

resource competitiveness is learning mathematics. Aspects of knowledge that are developed in

learning mathematics include the ability to solve problems. As stated in the attachment to

Permendiknas No. 22 of 2006 Concerning Content Standards (Ministry of National Education, 2006),

mathematics subjects aim for students to have the following abilities: (1) understanding mathematical

concepts; (2) reasoning; (3) solving problems; and (4) mathematical communication.

From the explanation above, it is clear that to increase the competitiveness of human

resources (HR), education is carried out by institutions that provide education. One part of education

that develops students' competitiveness is learning mathematics because learning mathematics can

improve students' problem-solving abilities, so students must do it. This can be seen from the results

of research conducted by the Trend in International Mathematics and Science Study (TIMSS) on class

VIII students in 2011, which were still low. In 2011, Indonesia was ranked 38th in the field of

mathematics, with a score of 386 out of 42 countries. The questions raised in TIMSS are not only at a

low cognitive level such as remembering, understanding, and applying but at a high level, namely

reasoning, which includes the ability to analyze, generalize, synthesize, assess, and solve non-routine

problems.

The subject matter contained in the TIMSS questions is divided into several more specific

topics, such as: Recall, recognize, compute, retrieve, measure, and classify or sort are all examples of

knowledge. Applying includes selecting, representing, modeling, implementing, and solving routine

problems. Analyzing, generalizing or specifying, integrating or synthesizing, justifying, and solving

non-routine problems are all examples of reasoning.

The conclusion from research conducted by TIMSS is not much different from research

conducted by PISA in 2015, which found that Indonesia is ranked 64th in the field of mathematics

with a score of 397 out of 70 countries. Specifically at level 2 (score 420–482), students can interpret

and recognize situations in contexts that require no more than direct inference, extract relevant

information from a single source, and make use of a single representational mode; at level 3 (score

482–545), students can execute clearly described procedures, including those that require sequential

Improving Students' Mathematical Problem-Solving Ability through the Use of External Representations, Amirul Syah,

Harizahayu, Gamar Al Haddar, Annisah, Emy Yunita Rahma Pratiwi 5315

decisions; and at level 4 (score 545–607), students can work effectively with explicit models on

complex, concrete situations that may involve constraints or call for making assumptions. According

to data from the 2015 PISA results, it shows that around 54% of students are proficient at level 3 or

more (proficient at levels 3,4,5, or 6), around 29% of students are at level 4.5 or 6, around 10.7% of

students are at level 5 or 6, and only 2.3% of students are at level 6. So students' mathematical

abilities are still at a low level. Because the PISA assessment process involves skills in

communication, mathematization, representation, reasoning, and argumentation, determining

strategies to solve problems, and using symbolic language, formal language, and technical language as

mathematical tools.

The use of symbols in solving mathematical problems includes the embodiment of

representation. Representation is an important competency for students to have in learning

mathematics. This can be seen in the expected goals for learning mathematics set by the National

Council of Teachers of Mathematics (NCTM). NCTM (2000) defines five standards of mathematical

ability that must be possessed by students, namely problem-solving abilities, communication skills,

connection skills, reasoning abilities, and representation abilities.

This statement is reinforced by Brenner's statement, which says that a successful problem-

solving process depends on problem representation skills such as constructing and using mathematical

representations in words, graphs, tables, equations, solving, and symbol manipulation. This means that

creating words, graphs, tables, equations, and manipulating symbols is part of the external

representation. External representations are embodiments of what students, teachers, and

mathematicians do internally.

In line with the characteristics of mathematics as a science that is systematic or has links

between its materials, it shows that there is a necessity to master the prerequisite material, which

forms the basis of new mathematical material. So that each student feels ownership of the external

representation in order to improve students' problem-solving skills toward higher-order thinking.

Based on observations made by researchers when carrying out observations on integrated

teaching profession practice activities, in fact, the teacher is still teacher-centered in learning and,

when explaining a material to students, immediately gives the mathematical formula, without first

associating it with the students' previous knowledge or with their daily lives. As a result, when the

teacher asks questions, students can only work on questions that are routine in nature, while students

cannot work on questions that are different or non-routine in nature, because students are not involved

in learning and concept discovery, so students' reasoning does not work. Besides that, in working on

the questions, students only solve them in the form of symbolic representations, so that the other

representations are not honed. Likewise, with the results of the research conducted by Rista Ayu, it

can be seen that students' external representation is lacking; students who have medium and low levels

of representation ability are still not able to solve problems related to their external representation

5316 Journal on Education, Volume 05, No. 02 Januari-Februari 2023, hal. 5313-5323

abilities properly. Students also still have a lot of confusion when rewriting the questions given in the

form of pictures or tables.

This study has the objectives, namely first, to find out whether students' mathematical

problem-solving abilities can be improved by using external representations. Second, analyzing

student activities in learning mathematics by using external representations. Third, analyzing student

responses to learning mathematics by using external representations.

METHOD

This research was conducted at MTs Darussalam Perigi Baru, which is located at H. Rasam

Street, RT.003/02 Perigi Baru Village, Pondok Aren District, and South Tangerang City. This

research was conducted in the class VII even semester of the 2021–2022 academic year. The research

was carried out in the even semester, from March 27 to May 3, 2022. The subjects of this research

were class VII students at MTs Darussalam Perigi Baru, totaling 28 people in the 2022 school year.

One person who acted as an observer in this study was a class VII mathematics teacher who was an

observer of the course of the research. During the implementation of the action, the observer helps the

researcher observe student responses during the learning process, writes down things that are not in

accordance with the plan, and together with the researcher reflects on the results of the action's

implementation.

This research uses classroom action research (CAR) or classroom action testing (PTK)

methods. According to O'Brien in Mulyatiningsih, "action research is carried out when a group of

people (students) identify the problem, then the researcher determines an action to overcome it."

Classroom Action Research (CAR) seeks to develop and reflect on a learning model with the aim of

improving learning processes and outcomes. In full, classroom action research aims to: (1) improve

the quality of content, processes, and learning outcomes in the classroom; (2) improve teachers'

professional abilities and attitudes; and (3) cultivate an academic culture so as to create a proactive

attitude in improving the quality of learning. Classroom action research is carried out in several cycles

until the desired results are achieved. The cycle is a round of successive activities that returns to its

original step. One class action research cycle includes four stages: (1) planning, (2) action, (3)

observation, and (4) reflection. Cycle I is carried out if the indicator of success has not been reached,

then cycle II is carried out, but if cycle I has achieved the indicator of success, cycle II is still carried

out to ensure the success of the treatment. However, if in cycle II the indicators of success have not

been achieved, cycle III is carried out using the results of reflection in cycle II as a reference in

preparing cycle III plans. When cycle I and cycle II indicators of success have been achieved, the

research is stopped.

HASIL DAN DISKUSI

Students' Mathematical Problem-Solving Ability

Improving Students' Mathematical Problem-Solving Ability through the Use of External Representations, Amirul Syah,

Harizahayu, Gamar Al Haddar, Annisah, Emy Yunita Rahma Pratiwi 5317

At the end of each cycle, a test is given that measures students' mathematical problem solving

abilities. The test given contains six questions that contain indicators of problem solving, namely

understanding the problem, making a model plan, completing the model plan, and interpreting the

results or solutions. The test results of the students' problem-solving abilities were as follows: there

was an increase in the average score between cycles I and II on each indicator of mathematical

problem solving. Of all the existing indicators, only the indicator for making a model plan

experienced a decrease in the average value, which was 0.18 points. The findings of this indicator's

analysis, namely, that the interpretation process was not carried out extensively in cycle I, but the

questions given in the final test in cycle I were in the moderate category with a difficulty level of 0.32

points, made it easier for students to answer these questions. Low student interpretation in cycle I

demands positive changes in cycle II. In cycle II, more interpretation processes were carried out,

especially interpreting an illustration into image form. With so many interpretation activities in cycle

II, students are getting used to interpreting mathematical problems from illustrations to pictures or

vice versa. In the final test of cycle II, the interpretation questions given were a little more difficult,

causing students to not be optimal in answering these questions, which had an impact on the decrease

in students' interpretation of the indicator test results.

The ability to complete the model plan, which was still low in cycle I at 45.36 points, made

researchers make learning changes in cycle II. One of the efforts made is to increase students' time in

carrying out the discussion process to understand problems and orders. In the process of discussion,

students are faced with problems that are able to develop their problem-solving abilities. Then, in

group learning, students are accustomed to expressing their opinions to one another in problem

solving rather than waiting for the teacher to explain first. This resulted in an increase in the indicator

of completing the student model plan by 39.28 points.

The ability of students to interpret low results or solutions in cycle I, which was equal to 28.04

points, made researchers carry out additional directions and provide lots of examples in cycle II in

order to get maximum results. One way to do this is to provide problems based on everyday life. One

of them lies in the solution stage. At the last meeting, students were asked to understand the problems

in word problems related to real life and then solve these problems and interpret the final results

obtained from the calculations. In cycle II, the ability to interpret the results or solutions reached a

score of 73.75. This shows a significant increase between cycle I and cycle II, namely achieving a

score of 45.71.

Student Activity

In comparison to the other aspects, the visual activities aspect has seen the least improvement.

An increase of 6.67% in cycle II was due to the fact that in cycle II, the students' focus on paying

attention to the explanations from the teacher and their friends had increased. Increasing student

activity in the aspect of visual activities also plays a role in improving other aspects, especially

5318 Journal on Education, Volume 05, No. 02 Januari-Februari 2023, hal. 5313-5323

aspects of oral activities.

The aspect of oral activities that focused on measuring student activity in expressing opinions,

asking questions, and answering questions posed by friends in discussions experienced a very

significant increase of 11.67%. The increase in this aspect was due to the fact that the discussion

process that took place in cycle II was more active than in cycle I. Students who were still passive

during the implementation of cycle I began to be able to express opinions in the discussion process in

cycle II. There are rules in discussion cycle II that require students to be able to understand the results

of their group discussions so that the material presented during the discussion is more optimal than in

cycle I.

The writing activities aspect experienced an increase of 38.33%. This aspect emphasizes the

focus of observation on student activity in drawing conclusions about the final results of problem

solving. In cycle II, students must be able to draw conclusions with their group mates so that they are

always ready when asked later by the researcher. These requirements and demands cause students to

be more active in carrying out activities in this aspect compared to the implementation of cycle I.

The next aspect is drawing activities, which have increased by 13.33% in cycle II. There are

several factors that cause this increase, one of which is the activeness of students in making or

describing flat shapes according to their types on buildings in real life. In cycle II, students are able to

describe flat shapes according to orders. Motor activities, which measure student activity in carrying

out direct motoric processes in learning, have increased by 5%. This increase occurred because in

cycle II students were given more space to be directly involved in the process of searching for data

and measuring objects in everyday life.

The mental activity aspect increased by 26.67%. In this aspect, students' activities in

remembering and using previous knowledge to solve problems become the focus of observation. The

mental activity aspect increased by 8.33%. In this aspect, students' activities in remembering and

using previous knowledge to solve problems become the focus of observation. The increase occurred

because there were more discussions compared to cycle I, which made students repeat information

that had been obtained repeatedly and made it easier for students to remember previous learning

material.

The emotional activities aspect experienced an increase of 18.33%. Students are more

enthusiastic and enjoy the learning process in cycle II because the material is more related to everyday

life and students are more active in the learning process. This is the main factor in increasing the

average percentage in this aspect.

Student Response

Students' daily journals were used to collect data on their reactions to learning about external

representation. Based on the data that has been obtained, there has been an increase in students'

positive responses from learning cycles I to II.

Improving Students' Mathematical Problem-Solving Ability through the Use of External Representations, Amirul Syah,

Harizahayu, Gamar Al Haddar, Annisah, Emy Yunita Rahma Pratiwi 5319

All researchers hope that carrying out classroom action research will lead to an increase in

positive student responses in each cycle and a decrease in negative and neutral responses from

students. The increase in students' positive responses was 17.85%, and respectively, there was a

decrease of 8.33% and 9.53% in students' neutral and negative responses to the external representation

learning that was carried out during cycles I and II.

There are many factors that influence student responses, but the most prominent are related to

the ongoing learning process and the material being studied. Students will feel happy if the learning

process is fun and conducive so that they can absorb the material being taught. But the material being

studied is also another factor. If students are involved in making learning fun, but the material being

studied is difficult, students become less enthusiastic and lose focus, making the learning difficult to

understand.

Discussion

The research findings revealed that students' mathematical problem-solving abilities (KPMM)

increased in cycle I by 53.21 and in cycle II by 71.39. The achievement of the KPMM score in cycle

II was 71.39, indicating that as many as 20 students, or 71.4%, had achieved the KKM. This means

that the KPMM achievements have met the success criteria, namely that 70% of students get a score

above the KKM. The increase in students' KPMM was supported by improvements in the learning

process summarized in the reflection of cycle I. Several improvements were made, including

improvements in writing down known and asked information in understanding problems,

improvements in planning models and discussion processes in completing model plans, and

improvements to the interpretation of the results or solutions made by students in solving problems.

The findings of this study differ slightly from the findings of Santia's (2013) research, which

took respondents from high school students and looked at the aspect of learning style, and reported

that representation of high school students in solving problems is the optimum value for students with

a fild independent cognitive style by understanding the information and what is asked by making

drawings, making a solution plan by making mathematical equations, manipulating numbers, and

manipulating words. While the subject solves problems from the field using an independent cognitive

style, he does so by understanding information and what is asked by writing mathematical equations

using formal symbols, making plans for solving by making mathematical equations, manipulating

these equations, and using trial and error, without re-checking the final results that are obtained.

Santia's findings also report that the methods used by independent and field-dependent fields are

almost the same as learning external representations.

External representation learning with each of its interrelated stages, namely translation,

integration, and solution, where the three stages focus more on the discussion process in forming

understanding of new concepts so that students understand problem solving, make plans, complete

plans, and finally interpret results or solutions, so that students with average problem solving ability

5320 Journal on Education, Volume 05, No. 02 Januari-Februari 2023, hal. 5313-5323

levels can benefit from the stages of representation. external. The findings in this study are in line

with previous findings by Roza Leikin et al. (2013). The results of the study indicate that external

representations have a very important effect on the process of solving mathematical problems for

students who have ordinary abilities. Student activity during learning by using external representation

is one of the focuses of this research. The research findings revealed that student activity increased by

61.43% in the first cycle and 80.00% in the second cycle. Achievements The percentage of student

activity in cycle II was 80.00%, which indicated that the percentage of learning activities during

learning using external representations met the expectations of researchers, namely 75%.

Learning with external representation is no longer teacher-centered but instead provides

opportunities for students to construct and develop their understanding independently, making them

more active. This is in line with research findings by Jarnawi (2011), who reported that some students

were able to develop forms of representation by using mathematical logic processes. Students begin to

formulate representations using known premises, arrange tables, make conjectures, and then arrange

formal representations.

Based on students' daily journals, the positive response of students during learning by using

external representations from cycle I to cycle II increased by 17.85%, with a breakdown of the

percentages for each cycle, namely 64.29% and 82.14%. The average percentage of students' positive

responses was 82.14% in cycle II, which is an indicator of the success of this study, namely that the

average percentage of students' positive responses during learning reached at least 80%. This shows

that learning with external representations can increase students' positive responses. Students who

give a positive response to external representational learning because learning by way of discussion is

different from learning usually give more freedom for students to process existing information, and

LKS-assisted learning can make it easier for students to understand the material provided. Students'

negative and neutral responses are more influenced by the level of difficulty of the material being

studied.

External representational learning in this study provides more learning that is able to assist

students in understanding problems and solving problems from their mathematical ideas, which are

then communicated through external representations. This finding is in line with the opinion of Goldin

and Shteingold, who stated that mathematical ideas are communicated through external

representations whose forms include spoken language, images, concrete, and written symbols.

Number systems, algebraic expressions, mathematical formulas, geometric shapes, and graphs are

models of representational forms.

CONCLUSION

Based on the results of the research, it can be concluded that: learning using external

representations can improve mathematical problem solving abilities; it can be seen that there is an

increase in the number of students who score above the KKM in cycle II, with the percentage of

Harizahayu, Gamar Al Haddar, Annisah, Emy Yunita Rahma Pratiwi 5321

students who are declared complete (reaching the KKM) at 71.4%, or as many as 20 students are

higher than in cycle I, with a percentage of 53.57%, or as many as 15 students who are declared

complete (reaching the KKM). This means that most students have achieved learning mastery

resulting from learning through the use of external representations. The use of external representations

in mathematics learning can increase student learning activities. This increase can be seen in aspects

of student learning activities, namely in the visual aspects of activities where students are able to pay

attention to the explanations of fellow group members without having to wait for the researcher to

explain, aspects of emotional activities, aspects of oral activities where students are able to actively

give opinions in discussions, Aspects of writing activities students are able to write down the final

conclusions from solving problems with their groups.

In aspects of drawing activities, students are able to describe shapes according to orders and

describe shapes according to their types on buildings in real life. Students are directly involved in the

process of searching for data and measuring objects in everyday life; the mental aspects of students'

activities are able to remember and use previous knowledge to solve the problems given; and the

emotional aspects of students' activities are more enthusiastic in the learning process with material

related to everyday life. In general, students' positive responses indicated an increase in learning

through the use of external representations.

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