Learning The Permutation Concept Through Role-Playing

Journal of Education and Learning (EduLearn)
Vol.12, No.3, August 2018, pp. 422~431
ISSN: 2089-9823 DOI: 10.11591/edulearn.v12i3.9241  422
Journal homepage: http://journal.uad.ac.id/index.php/EduLearn
Learning The Permutation Concept Through Role-Playing
E.I. Pusta Siligar, Somakim, Hapizah
Department of Education Mathematics, Sriwijaya University, Palembang, Indonesia
Article Info ABSTRACT
Article history:
Received Mar 12, 2018
Revised Apr 21, 2018
Accepted May 25, 2018
This research aimed to produce a learning trajectory which could help
students to understand the concept of permutations through role-playing
activity in the election of chairman and vice-chairman of Intra School
Students Organization. This research was a design research method through
three stage, 1) preliminary design/preparing for the experiment were used to
design the Hypothetical Learning Trajectory, 2) design experiment was a
Hypothetical Learning Trajectory test phase consisting of pilot and teaching
experiment, 3) retrospective analysis. It was Indonesia’s Realistic
Mathematics Education Approach at tenth grade of Senior High School No
15 Palembang. Data collecting technique were obtained from observations,
recorded videos, photos, students’ sheet activities, and field documentation.
The result of this research was a learning trajectory which has three activities
that could help student to understand the concept of permutations, there are:
1) the students could determine the form of administrative committee and
calculate the number of formed groups formation in the election from 2, 3, 4
and 5 nominated candidates, 2) Students could determine the number of
groups’ formation convert into the closest form of the multiplication and the
factorial, 3) by using the students understanding of factorial, students found
the concept of permutations form: nPr =
–
Keywords:
Indonesia’s Realistic
Mathematics Education
Approach
Intra School Students
Organization
Permutations
Role-playing
Copyright © 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Somakim,
Departement of Education Mathematics,
SriwijayaUniversity,
Jl. Srijaya Negara, Bukit Besar, Palembang, 30139 - Indonesia.
Email: somakim_math@yahoo.com
1. INTRODUCTION
The students’ ability in understanding the problems of permutations study is still considered very
low [1]. The cause of the students’ difficulties in solving the permutation problems is that they are confused
to use permutations formulas or combinations of formula in solving the problem of enumeration rules,
Thereby, the presentation of the material from the mathematics learning books always made the students to
follow the examples of problem-solving that exist. Typically, when the question is changed, it will make
students feel confused in determining the concept of problem-solving [2]. The students’ low achievement in
learning mathematics was caused by some factors, such as an uninteresting presented learning material and
the lack of examples of application in everyday life, as well as during the learning process, the implemented
teaching method was mainly focused on teacher-centered activity only throughout class time [3]. In the
process of learning mathematics, the students were generally directed at the process of memorization rather
than in understanding the learning concept of mathematics [4].
As stated in the Standards of Graduate Competency of Permendiknas No. 21 the Year 2016,
permutations study is an important learning material to be learned, that students must be able to solve the
contextual problems related to permutations [5]. The importance of studying the concept of permutations
study is to learn the bases of one kind of combinatorial analysis [6], and it could be applied in daily life, for
example in scheduling procedure [7].

EduLearn ISSN: 2089-9823 
Learning The Permutation Concept Through Role-Playing (E.I. Pusta Siligar)
423
Fundamentally, an appropriate alternative learning is needed to improve students’ understanding
toward the concept of permutations study, as well as the importance of these permutations materials in the
learning process and students’ life in every day. One alternative could be used in studying permutations
materials is that students need real-life problems in a more relevant context, which can stimulate the students’
interest in learning and maintain their prior knowledge [8].
One of the learning approaches that could be used by teachers of mathematics in developing
students’ ability to think, understand, communicate, and solve the problems in both the learning process and
daily life is that by using Indonesia’s Realistic Mathematics Education approach [9]. It was developed as
Indonesian version since 2001. It was taken from Realistic Mathematics Education developed by Freudental
Institute in the Netherlands, and its concept was adapted to Indonesian culture and based on the theory of
Realistic Mathematics Education [10]. There are 3 principles of Indonesian Realistic Mathematics Education
in the learning process; (1) guided reinvention and progressive mathematizing, (2) didactical
phenomenology, and (3) self-developed models [11].
Furthermore, the recent research has a positive result, that the use of Indonesia’s Realistic
Mathematics Education approach with the barcode context could improve students’ understanding of the
concept of enumeration rules [12]. In line with this, Indonesia’s Realistic Mathematics Education could lead
students in understanding the mathematical concepts by constructing their background knowledge related to
everyday life in such a “real” situation [13].
In addition, the concept of playing the game could create situations related to the students’ real
experiences as well as phenomena and events in the environment, which could be used as starting
points/opening activities/series of activities for learning [14]. Permutations study could be solved by using
playing game [15], and it could be taught through exchange-places game [16].
One kind of playing game activities that could be played by students in learning permutations study
is role-playing activity in the election of chairman and vice-chairman of Intra School Students Organization.
It is the best way to develop skills, initiative, communication, problem-solving, self-awareness and
cooperative work in groups [17]. Intra School Student Organization is one of the organizations that exist in
Junior High School and Senior High School in Indonesia, which used to determine the chairman through the
conducted election followed by all students and teachers. The period of management of the Intra School
Student Organization would be held in every year; thereby it should be a change of leadership. It is hoped
that through role-playing activity in the election of chairman and vice-chairman of Intra School Student
Organization could raise possibilities and a large number of formed groups as expected to help the students to
understand the concept of permutations study.
2. RESEARCH METHOD
The kind of this research was a ”Design Research”, which designed to produce a learning trajectory
on permutations study by using the context of role-playing activity in the election of chairman and vice-
chairman of Intra School Students Organization. It is a form of research that aimed at developing Local
Instructional Theory with the cooperation between the researcher and the teachers to improve the quality of
learning [18]. In this research, the developed design was Hypothetical Learning Trajectory, which contained
a series of students’ learning activities. There are three stages in this research design: preparing for the
experiment, design experimental, and retrospective analysis [19]. This article will describe the results of
research at the design experimental stage (teaching experimental group). The research stages are shown in
Figure 1 below.
Figure 1.The Process of Designing Research [20]

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EduLearn Vol. 12, No. 3, August 2018 : 422 – 431
424
Data collecting technique in this research were obtained from recorded videos at the study process,
students’, and activity photos, and the result of student’s activity in students’ sheet activities. The stage of
presentation of learning video result data by transcribing into the written form.
3. RESULTS AND ANALYSIS
This research was designed for produced learning trajectory which can help students to understand
the concept of permutation through role-playing of election chairman and vice-chairman of Intra School
Student Organization. It was implemented at tenth grade of Senior High School No 15 Palembang with six
students at pilot experiment step and 36 students at teaching experiment step. The following is the result of
the study:
3.1. Preparing for the Experiment
First step, researcher reviewed all literature to produce Hypothetical Learning Trajectory of
permutations learning with the through role-playing of the election chairman and vice-chairman of Intra
School Student Organization, which is: reviewed standard content based on curriculum K13 with basic
competencies understanding the concept of permutation, reviewed Indonesia’s Realistic Mathematics
Education Approach, and design research as a method of that used in this research. The Following
Hypothetical Learning Trajectory has been designed which can be seen in the following figure 2:
Figure 2. Hypothetical Learning Trajectory
3.2. Design Experiment
The second stage, design experimental that consisted of two cycles; cycle 1as the experimental pilot
and cycle 2 as the experimental treatment. In cycle 1, the researcher acted as model teacher and tested the
Hypothetical Learning Trajectory to the six students apart from the students who became the subject of the
research at the time of conducting the research cycle 2. Those six students were selected based on the level of
ability; 2 high ability students, 2 students with medium ability, and 2 students with low ability. Meanwhile, in
cycle 2 of experimental treatment, the researcher acted as an observer. In this cycle, the teaching and learning
process was under the mentoring of their own teacher as the model teacher to take control in this class, and
there are 36 involved students of Senior High School Number 15 Palembang in the learning process.
Following explained the result of the research at the teaching experiment of cycle 2:
Students can determine the form of group formation that may occur in the election of the
chairman of Intra School Students Organization from 2, 3, 4, and 5 candidates by using role-playing
activity
In the first activity, the students were asked to play “Role Playing” game in solving the election
problems of the management of Intra School Students Organization, and they were asked to find out how
likely the chairman and vice-chairman that might be formed from 2, 3, 4, and 5 nominated candidates in the
Intra School Students Organization election. At the beginning of this activity, the teacher has divided the
students' groups into six (6) groups that consisted of six students in each group, then the students have chosen
one of their members to be the candidate of chairman and vice-chairman in the Intra School Students
Organization election. The chosen students played their role as the chairman and vice-chairman candidates
while the other member students participated to vote the chosen one.

425
In order to determine the two (2) candidates to be nominated at the first activity, the students have
written the name of candidates students on the Students’ Activity Sheet and put it into a provided box. After
that, the election of chairman and vice-chairman of the Intra School Student Organization were decided by
group members who served as observers by looking at the possible form of arrangement from the 2
candidates. The transcript of the conversation can be seen as follows.
Transcript 1
Teacher : Please...your group explain the answer to the problem no 1. Who will be the candidates
from your group?
Student : The candidates of our group are Khairul and Inayah Salsabilah.
Teacher : Then, determine the administrative arrangement that may occur in the chairman and
vice chairman election of Intra School Students Organization?
Student : The chairman candidate is Khairul Ihsan and the vice-chairman candidate is Inayah
Salsabilah (Both of students hold the paper with the names of each corresponding
position directly)
Teacher : Are you sure if they are as the candidates from your group?
Students: We have voted them Ma’am.
Teacher : So, how many possible forms of arrangement from these 2 candidates?
Students : Only one form of arrangement.
Teacher : Then? Is there any form of arrangement?
Students : Hmmm (Thinking...)
Teacher : Please... practice one more time?
Student : Oh... there is another possible arrangement, Inayah salsabilah can be as a chairman and
her vice-chairman is Khoirul Ihsan (Both of them hold the paper with the names of each
corresponding position directly)
Teacher : What do you think if there are two possible candidates?
Student : There are two formed arrangements. In the first arrangement, Khairul Ihsan can be as
the chairman and his vice-chairman is Inayah Salsabila, Meanwhile, in the second
arrangement Inayah Salsabilah can be as the chairman and KhairulIhsanas her vice-
chairman.
Teacher : So, they exchange their position one another?
Students : Yes, they do.
Teacher : Is there another formed arrangement beside those two arrangements?
Students : no...nothing.
In the election, each group has different ways to give their votes, one of the ways is by giving the
most votes of group members to the non-candidates of chairman of Intra School Students Organization, the
student who got the most votes, he/she acted as the elected chairman by holding a paper with the name as a
chairman. Meanwhile, the student who got the least votes, he/she acted as a vice-chairman by holding the
name as vice-chairman. Then, the teacher guided the students to see if there are any possible group
formations that can be formed. To determine other arrangements, the student exchanged his/her position and
roles. For example, if A is chosen as the chairman then B will become the vice-chairman, meanwhile, if B is
chosen as the chairman then A will become as co-chairman. In other words, that the arrangement of
permutations could be AB≠ BA.
Furthermore, the results of students’ answers to this first activity showed that by using role-playing
activity, the students are able to determine the possible group formations in the election of chairman and
vice-chairman of Intra School Students Organization. Further, there are no serious obstacles experienced by
students while carrying out this activity. If there are 2 candidates of the board then the number of formations
that can be formed is 2 forms of arrangement, if there are 3 candidates then the number of formations that can
be formed is 6 forms of arrangement, if there are 4 candidates then the number of formations that can be
formed is 12 forms of arrangement, and if there are 5 candidates then the number of formations that can be
formed is 20 forms of arrangement. Some students' group answer in solving the election problems if there are
2, 3, 4, and 5 possible candidates can be seen in Figure 3 below.

 ISSN: 2089-9823
426
Figure 3.The result of the student’s answer in determining the possible groups’formation in the election of
chairman and vice-chairman of Intra School Students Organization
Students can determine the number of groups’ formation in the election of chairman and vice-
chairman of Intra School Students Organization from 2, 3, 4, and 5 candidates, and convert them into
the closest form of the multiplication and the factorial concept.
In the second activity, the students were asked to calculate the number of formed groups formation
in Student Intra School Organizations election from 2, 3, 4 and 5 nominated candidates. The students’
success in determining the number of the groups’ formation was based on the results of their first activity.
For example, if there are 2 candidates, then there are 2 formations exist, if there are 3 candidates, then there
are 6 formations exist, if there are 4 candidates, then there are 12 formations exist, and if there are 5
candidates, then there are 20 formations exist. Moreover, in this activity, the students were asked to convert
the number of groups that they have found into the closest form of the multiplication. Further, there is no
constraint experienced by the students in converting the number of formations into the closest form of the
multiplication. The transcript of the conversation between the teacher and students can be seen as follows.
Transcript 2
Teacher : Try to explain if there are 20 arrangements converted into the closest form of
multiplication, and how do you form its shape?
Student : It is 4 x 5 or 5 x 4
Teacher : How do you know
Student : It’s about the closest multiplication form Ma’am, After number 4, it is followed by
number 5, and if 4 x 5 or 5 x 4 the result is 20.
In this activity, the students can explain the result of their answers that they have found; for 20
formations, it could be converted into the closest multiplication with 4 x 5 or 5 x 4, for 2 formations, it could
be converted into the closest multiplication with 2 x 1 or 1 x 2, for 3 formations, it could be converted into
the closest multiplication with 2 x 3 or 3 x 2, and for 12 formations, it could be converted into the closest
multiplication with 4 x 3 or 3 x 4. The aim of this activity is to facilitate the student to find the concept of
factorial because in the next activity the students would be asked to convert the form of the closest
multiplication into the form of the factorial concept. The result of students answer in factorial form can be
seen in the following Figure 4.

427
Figure 4. The results of student answers in the factorial form
Based on the result of students’ answers, it showed that students are able to convert the number of
group formations that they have found into the closest form of the multiplication, there is no constraint
experienced by students in converting the number of formations into the closest form of the multiplication. In
other hands, the result of multiplication 5 x 4 could be converted into factorial form with , the result of
multiplication 2 x 1 could be converted into factorial form with or , the result of multiplication 3 x 2 could
be converted into factorial form with or , and as the result of multiplication 4 x 3, it could be converted
into factorial form with .
Students can find the concept of permutations form: nPr =
–
The third activity, to make it easier for students in determining the general concept of permutations,
the students were required to summarize all the results of activity 1 and activity 2 into an available table on
the Students’ Activity Sheet. In this activity, the teacher guided the students to find out the general concept or
the permutation formula by converting the form of the factorial pattern to the general concept of permutations
by utilizing the number that exists on the most candidates and the number that exist on many positions object.
The transcript of the conversation between the teacher and the students in finding the general concept of
permutations and/or its formula was presented as follows.
Transcript 3
Student : There are only 2 nominated candidates at the beginning.
Teacher : How many chosen positions object?
Student : Only 2 positions
Teacher : What?
Student : There are two corresponding positions, namely the chairman and vice-chairman of Intra
School Students Organization.
Teacher : Ok, how many formed groups arrangement we have....?
Students: 2
Teacher : Are the form still in the same kind of that multiplication?
Students: Just move its position.
Students : Where is it?
Student : 2 multiplied by 1 (2 x 1), it is in form of factorial.
Students: Two factorials per nol
Student : How about in form of permutations?
Teacher : Try to write again those two factorials, per?
Students: Nol
Teacher: How do you get nol?
Students : (Thinking)
Student : By multiplying 1 and 0 (it’s the result of 1 x 0)

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Teacher : In this point of view (look at the number of the most nominated candidates in the column
and the number of positions of the object in the table?)
Students : “from” (thinking)
Student : Yups... 2 reduced by 2 (2 - 2)
Teacher : Okay, please write it now?
In this activity, the student succeeded in finding the general concept of permutations still in form of
the numbers, the result of student’s answer in the form of factorial concept was converted to by
changing 0! to (2-2)!, it was obtained from 2 numbers in the column of many candidates which was reduced
by 2 numbers in the column of many position objects. For the next permutation pattern, students initially
wrote in the factorial concept column, but after the student subtracts the 3 numbers in the column of many
candidates with 2 numbers in the column of many objects, then the result was 1, thereby the students wrote ,
then it was converted to permutations of form . Moreover, in the next steps the students converted to
, and to
Furthermore, the teacher guided the students to find out the general concept or formula of
permutations that were already in the form of symbols. The following conversation is the discussion result
between the teacher and the students in determining the general concept of symbolized permutations.
Transcript 4
Teacher : Okay, please look at the number! There are 2, 3, 4, and how do you symbolize it?
Students : Eeehm those (Thinking), “n”.
Teacher : Alright, then try to write what n it is?
Students : “n” factorial. This is, “per” inside the bracket “n” min “r” for the closed bracket
factorial (
–
).
The result of the students’ answers in the determining the concept of permutations is shown
in Figure 5.
Figure 5. The students’answer in determining the permutations concept
From the result of the student’s answer, it showed that the students are able to understand the
concept of permutations by using role-playing activity with the model teachers’ instruction, and finally, the
students could find out the permutations formula nPr=
–
.
3.3. Retrospective Analysis of Teaching Experiment
In this study, the students’ exchanged roles and positions as elected chairman and vice-chairman of
the Intra School Students Organization could generate a sequence of management formations in the election.
The students’ activity is in line with the definition of, that permutations study is an arrangement of sorted
objects [21]. In the subsequent activity, the students found out the number of formations of board

429
management that can occur in the election of chairman and vice-chairman of Intra School Students
Organization from 2, 3, 4, and 5 nominated candidates. This shows that students’ was very enthusiastic to do
role-playing activity and shows that permutation could be learned through role-playing. In this study, to find
out the general concept of permutations study, the students determine the number of board management that
can occur from the “r” object in many needed positions by paying attention to “n” objects of all existing
candidates. Herein, the students could find the general concept or the formula of permutations
–
. Based
on the result of the retrospective analysis in the activity 1, activity 2, and activity 3 of cycle 2 (teaching
experiment), that the learning procedure was in accordance with the designed Hypothetical Learning
Trajectory. It can be concluded that the students have understood the concept of permutations study via the
role-playing activity in the chairman and vice-chairman election of Intra School Students Organization by
determining the possible board management and the number of formed groups.
4. DISCUSSION
In this research, learning trajectory that has been designed and done by the researcher to understand
the concept of permutations was through role-playing to the election the chairman and vice-chairman of Intra
School Student Organization that has three learning activities that has been done at the process of students'
learning. This learning process reflected three principles of Indonesia’s Realistic Mathematics Education.
Three principles of Realistic Indonesia’s Mathematics Education are guided reinvention and progressive
mathematizing, didactical phenomenology, and self-developed models [11]. Based on the guided reinvention
principle in this permutations learning process, the students were given equal opportunity to play the role of
determining the form of chairman and vice-chairman of Intra School Students Organization in their own way
through the direction and guidance of the teacher. Hereby, those students have their own learning
experience activities.
The second principle is didactical phenomenology. This principle stated that the situation with the
phenomenon used as material and application area in the teaching of mathematics must depend on the real
context of the students’ background before they can reach the level of mathematics formally, it means that
realistic mathematics teaching emphasizes the importance of the contextual problems to introduce the
mathematical topic to the students. The use of the context of chairman and vice-chairman election in Intra
School Students Organizations is a contextual problem that existed in the students’ life as a phenomenon in
the concept of permutations study because students can have the experience in the election of the board
management of Intra School Students Organization.
The third principle is self-developed models in this principle; the developed model by the students is
used to serve as a bridge of informal knowledge to formal mathematics. It can be seen from students’ activity
in determining the board management that may occur, such as determining the number of group
management, converting the formations into the form of the closest multiplication, in turn, it is converted to
the form of the factorial concept, and at last, the students were able to find out the permutations formula as a
result of students’ understanding of the permutations study concept.
5. CONCLUSION
Based on the findings of this research and its discussion, it can be concluded that the encountered
learning trajectory consisted of 3 activities: Activity 1, the students could determine the form of
administrative committee and calculate the number of formed groups formation in the election from 2, 3, 4
and 5 nominated candidates. Activity 2, students could determine the number of groups’ formation convert
into the closest form of the multiplication and the factorial. Activity 3, by using the students understanding of
factorial, students found the concept of permutations form: nPr =
–
. Moreover, the findings showed that
by using role-playing in the election of the chairman and vice-chairman of Intra School Students
Organization could help students to understand the concept of permutations study.
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BIOGRAPHIES OF AUTHORS
E.I. PustaSiligar, S.Pd.
Postgraduate Students of Mathematics Education
Sriwijaya University
Dr. Somakim, M.Pd.
Lecturer of Mathematics Education
Dr. Hapizah, S.Pd., M.T.
Lecturer of Mathematics Education

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