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High school student’s mathematical modeling skills in open-ended
learning
To cite this article: U Damayanti et al 2020 J. Phys.: Conf. Ser. 1480 012042

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

IOP Publishing
doi:10.1088/1742-6596/1480/1/012042

High school student’s mathematical modeling skills in openended learning
U Damayanti1, Y Hartono1, C Hiltrimartin1, and Indaryanti1*
1

Mathematics Education Department, Universitas Sriwijaya, Palembang, South
Sumatra, Indonesia
*

Corresponding author’s email: indaryanti@fkip.unsri.ac.id

Abstract. This research is a descriptive study aims to determine the mathematical modeling
skills of students after using the open-ended learning to algebraic arithmetic operations in class
VII SMP Negeri 17 Palembang. The subjects of this study were grade VII.6 students of SMP
Negeri 17 Palembang with 32 students. The learning process is adapted to the stages of the openended learning. On the student worksheet, the problem given is an open-ended problem. The
researcher also uses problem-solving questions during the evaluation test. Data collected using
tests. The results of this study indicate that the mathematical modeling skill of students in openended learning in class VII.6 in SMP Negeri 17 Palembang is categorized good with an average
of 78.54%. Students who are still lacking in modeling mathematical problems because they still
incorrectly understand the problem and draw up a problem-solving plan.

1. Introduction
Based on Permendiknas No. 22 of 2006, one focus or goal in learning mathematics is to solve problems
that include the ability to understand problems, make mathematical models, solve problems, and
interpret solutions [1]. Mathematical abilities that students must be able to master through mathematics
learning, namely: (1) communication; (2) reasoning; (3) connection; (4) representation; (5) problemsolving [2]. Problem-solving is very important in mathematics, not only for those who will later learn or
explore mathematics, but also to be applied to other fields of study and in everyday life [3]. One of the
stages in problem-solving that can help students in solving problems is understanding problems and
planning problem solving that leads students to make mathematical models [4]. The cycles of model
construction, evaluation, and revision are valued in the light of befitting the professional practices of
mathematicians and scientist as engineering [5]. Mathematical modelling as problem-solving from a
modelling perspective does not insist that students get the correct answer [6]. The problem-solving
ability of students in Indonesia is still relatively low, this is based on the results of the Program for
International Student Assessment (PISA) test conducted by the OECD in 2015 which was conducted in
69 countries including Indonesia, which scored an average of 386 out of 500 grades average OECD and
ranked 62nd out of 69 countries [7]. Also, based on the 2015 TIMSS (Trends International Mathematics
and Science Study) study conducted in 50 countries, Indonesia received 397 out of 500 TIMSS average
scores. The characteristics of the questions given by TIMSS to measure the ability of reasoning and
problem solving, namely on the topics Number (number), Algebra (algebra), Geometry (geometry), and
Data and Chance (data and opportunity), this shows that the ability of solving student problems in
Indonesia is still low [8]. Traditionally, mathematical modelling tends to be associated with higher
school pure and applied mathematics (geometry, algebra, calculus, etc.) to solve real-world problems
[6]. Based on data on the results of the national junior high school exam the percentage of students who

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

IOP Publishing
doi:10.1088/1742-6596/1480/1/012042

were able to answer correctly on algebra material was only 41.88%. The national examination results
state that many students make mistakes in answering questions on algebraic material. In algebraic
material, students need problem-solving skills and understanding of concepts to model mathematical
problems to solve these problems [9].
In general, mathematics learning in schools is still characterized by teacher-centeredness, when the
learning process of the teacher is more dominant than the students and also the questions given for the
exercise are more routine which makes the student's reasoning ability in problem solving still lacking
and the ability to think only at low levels [10]. The teacher has a role to improve the ability to solve
problems with mathematical modeling in students can use the methods used, or by evaluating the form
of making questions that support [11]. One of the lessons that can improve students' ability to solve
mathematical problems with modeling is open-ended learning. Open-ended learning can provide
opportunities for students to gain knowledge, experience to find, recognize and solve mathematical
problems with a variety of strategies [12]. Open-ended learning can improve the ability to solve
mathematical problems by modeling a problem because with this learning students are not required to
memorize facts, but students are encouraged to be able to construct knowledge in students' own minds
[13]. The problem given in this open-ended approach is non-routine open-ended problems [13 - 15]. The
purpose of this study was to determine the mathematical modeling skills of students in open-ended
learning on the arithmetic operations material in the form of algebra in class VII SMPN 17 Palembang.
2. Method
This research is a descriptive study that aims to determine the ability of modeling mathematical problems
of students after using open-ended learning in class VII.6 SMPN17 Palembang. The subjects of this
study were students of class VII.6 SMPN 17 Palembang, amounting to 32 people. This study consisted
of 3 stages, namely the preparation phase, the implementation phase, and the data analysis stage. The
instruments used in this study were students' worksheets using open-ended or open-ended problems, test
questions, and lesson plans based on the stages of open-ended learning. Data collection techniques in
this study were obtained from the test data. The test is used to see students' mathematical modeling
abilities in solving problems after using open-ended learning. The test is done after two learning
meetings.
3. Result and Discussion
This research was conducted 3 meetings, 2 meetings in the form of a learning process using student
worksheets, and 1 meeting in the form of tests. Researchers conduct open-ended learning activities
consisting of 5 stages, namely (1) presenting the problem, (2) exploring the problem, (3) recording
student responses, (4) discussing student responses, and (5) guiding students to summarize the lesson.
In the implementation of learning the first meeting, and the second begins with an introduction, namely
greeting, praying, asking for news, and checking the presence of students, conveying the learning
objectives to be achieved after learning, and providing motivation in the form of the importance of
learning algebraic arithmetic operations, and presents material regarding the arithmetic operations of
algebraic forms. Then in the stage of presenting the problem, students are divided into 8 study groups
consisting of 4 people. Researchers distribute students' worksheets that contain open problems that can
be solved with various solutions or various answers, and also with the stages of problem-solving. In the
stage of exploring a problem, researchers provide direction and explanation to students about supporting
information in a brief, as well as activities or steps to solve the problems contained in the worksheets of
students. Students are asked to understand the problem and ask if they still don't understand the
information on the student worksheet. Then, students discuss and carry out various activities to solve
the problem. At the stage of recording student responses, each representative from each group was asked
to present the results of the completion obtained. Researchers observe and pay attention, the response of
each group and who provide responses to answers or ways of resolution obtained by other groups. At
the stage of discussing student responses, the researcher together with students discusses the answers
and methods of resolution obtained by each group. At the stage of summarizing the lesson, the researcher
guides the students to make conclusions about the count operations material on the algebraic form that
has been learned. After that, the researcher closes the lesson by asking students to learn the next material,
2

National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

IOP Publishing
doi:10.1088/1742-6596/1480/1/012042

and say hello. Overall in each meeting, the learning stages are carried out the same, which differs only
in the material being studied. At the first meeting studying the material operations of addition and
subtraction on the algebraic form. Meanwhile, at the second meeting studying the material multiplication
and division operations on algebraic forms.
In the implementation of learning, to be able to solve the problems given on the student worksheets
students can use the stages of problem-solving that are understanding the problem, planning problem
solving, solving problems, and re-checking the results obtained. The following answers from students
regarding the problems on the worksheets of students provided:

Figure 1. Answer the worksheets of students who are already appropriate.
From Figure 1, it can be seen that students can understand the problem and draw up a plan for solving
it by making a mathematical model of a good and appropriate problem. Based on the existing problems
students consider the first number with the letter a, and the second number with the letter b. Then, the
information that is known is modelled into a mathematical form 3𝑎 + 𝑏 = 60 and 𝑎 × 𝑏 = 225.
This shows that overall students have been able to change mathematical problem-solving problems in
the form of modeling.
However, there are still some students who have not been able to model a problem into mathematical
form. Here are the answers of students who are still mistaken in modeling mathematical problems:

Figure 2. Answer worksheets of students who are still wrong.
From Figure 2 proficiency level, it can be seen that students mistakenly changed or modelled the
problem in mathematical form. Students assume the first number and the second number with the same
variable, x. Then, students immediately add up the algebraic form obtained and get the x value. This can
occur because students are still mistaken in understanding the problem given and preparing a problemsolving plan. It has been acknowledged that students’ initial conceptions developed during the modelling
process could be characterised by unwarranted assumptions or the imposing of inappropriate constraints
[16].
After the test is carried out, the test result data that has been obtained are analysed to find out the
mathematical modeling abilities of students. The test results obtained are then examined and given a
score based on the scoring guidelines that have been made by researchers. Next, the researcher
determines the category of mathematical modeling abilities possessed by students which are very good,
good, satisfactory, poor, or very poor. Table 1 shows categories of students' mathematical modeling
skills.
3

National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

IOP Publishing
doi:10.1088/1742-6596/1480/1/012042

Table 1. Categories of mathematical modeling skills of students.
Score Interval

Category

81-100
61-80
41-60
21-40
0-20

Very good
Good
Satisfactory
Poor
Very poor

Based on the answers to the test questions written by students, by modeling the problem in the form
of mathematics, it seems that overall students can understand and plan the solution to the problem given.
The following answers to the test questions obtained by students:

Figure 3. Answers to the test questions student’s model mathematical problems.
From Figure 3, it can be seen that students correctly identify and write down what is known from the
questions given. Then, students can also identify and write what is asked from the problem correctly.
Assumptions made by students are clearly defined and supportive in making mathematical models of
these problems. Furthermore, the variables identified by students are clear, and the models made by
students are appropriate and appropriate. This shows that students can model problems into
mathematical form. Thus, the answers and final solutions obtained by students are correct because
students understand the problem and draw up a plan for solving it by modeling the problem into a good
and appropriate mathematical form.

Figure 4. Answers to student test questions that are still incomplete in modeling the problem.
Based on Figure 4, the student with the right to write what is known and asked of the matter. The
assumptions made and the variables used by students not defined and clearly identified, but according

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

IOP Publishing
doi:10.1088/1742-6596/1480/1/012042

to the given problem. However, students directly substitute values of b and c of the mathematical
modeling that the students in the problem solving process.
Based on the results of the final test, the average student’s mathematical modeling skills capability
can be categorized either VII.6 i.e. 78.54. Students are considered to have very good modeling
capabilities as many as 14 students with a percentage of 43.75%. Students are considered to have good
modeling capabilities as much as 9 students with a percentage of 28.13%. Students are categorized
modeling capabilities satisfactory as much as 7 students with a percentage of 21.87%. As well, students
are considered to have poor modeling capabilities as much as 2 students with a percentage of 6.25%.
The table below illustrates the percentage of students' mathematical modeling capability tests on the
material form of algebra:
Table 2. Percentage test students’ mathematical modelling capabilities
Frequency
14
9
7
2
Average

Percentage
43.75%
28.13%
21.87%
6.25%
78.54%

Category
Very good
Good
Satisfactory
Poor
Good

Based on Table 2, it can be said that most of the students have been able to model mathematical
problem since learning of open-ended at the stage of exploring the issue of students accustomed to
resolving problems by first writing down what is known and asked of the information provided and then
make a mathematical model of the problem, but there are still some students who are confused and have
trouble making a mathematical model due to not understand the intent of a given problem. In accordance
with [17] that said open-ended learning can improve students' problem-solving skills including modeling
the mathematical problem because with this approach makes students become actively involved in
solving the problem, from the research data obtained 60% of students who have the ability to model
mathematical problems in solving categorized problems both in learning with an open-ended learning.
Mathematical modeling should be and could be started early, when students already have basic
competencies then modeling can be developed [18, 19]. Mathematical modeling should be taught in
every level of mathematics education [20].
4. Conclusion
Based on research data obtained from VII.6 class at Junior High School 17 Palembang, it can be
concluded that the mathematical modeling capabilities of students in open-ended problem solving to
solve problems considered good. Students are used to model a problem into mathematical form due to
the learning process students are given the open questions that require students to understand the
problem and devise a plan that directs students' problem solving to model the mathematical problem.
5. References
[1] Depdiknas 2006 Permendiknas No 22 Tahun 2006 tentang Standar Isi (Jakarta: Depdiknas)
[2] NCTM 2000 Curriculum and Evaluation Standars for School Mathematics (USA: The National
Council of Teachers of Mathematics Inc)
[3] Ruseffendi H 2006 Pengantar kepada Membantu Guru Mengembangkan Kompetensinya dalam
Pengajaran Matematika untuk Meningkatkan CBSA (Bandung: Tarsito)
[4] Destiani L 2018 Kemampuan Pemodelan Siswa Kelas X SMA Negeri 11 Palembang dalam
Menyelesaikan Soal Pemecahan Masalah Matematika Bachelor Thesis (Universitas
Sriwijaya)
[5] Lesh R and Doerr H M 2003 Foundations of A Models and Modeling Perspective in Mathematics
Teaching and Learning Beyond Constructivism: Models and Modeling Perspectives on

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012042

[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]

[18]

[19]

[20]

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doi:10.1088/1742-6596/1480/1/012042

Mathematics Problem-solving, Learning and Teaching (Mahwah, New Jersey: Lawrence
Erlbaum Associates)
Chan C M E 2008 Thingking Classroom 9 37
OECD 2016 Programme for international student assessment (PISA) result from PISA 2015
online: https://www.oecd.org/pisa/
Mulis M O 2016 TIMSS 2015 International Results In Mathematics (Chestnut Hill: TIMSS &
PIRLS International Study Center)
Yahya A 2019 IJES 1 57
Husna 2013 Jurnal Peluang 1 81
Shimada S and Becker J P 1997 The Open-Ended Approach: A New Proposal for Teaching
Mathematics (Virginia: National Council of Teachers of Mathematics)
Sumartini T 2016 Mosharafa: Jurnal Pendidikan Matematika 5 148
Gordah E 2012 Jurnal Pendidikan dan Kebudayaan 18 264
Afandi A 2013 Pythagoras: Jurnal Pendidikan Matematika 8 1
Wahyuni D, Ariani N M and Syahbana A 2013 Edumatica 3 35
English L D and Watters J J 2004 Mathematical Modelling with Young Children Mathematical
Modelling: A Way of Life (Cambridge: Horwood Publishing Limited)
Sadevi N W S 2017 Efektivitas Pendekatan Open-Ended ditinjau dari Kemapuan Pemecahan
Masalah
Matematis
Siswa
online:
http://jurnal.fkip.unila.ac.id/index.php/MTK/article/download/11698/8344
Diezman C J J, Watters and English L D 2002 Teacher Behaviours that Influence Young
Children’s Reasoning Proc of the 26th Internationl PME Conference (Norwich: University of
East Anglia) p 289
Lehrer R and Schauble L 2003 Origins and Evolutions of Model-based Reasoning in Mathematics
and Science Beyond constructivism: Models and modeling perspectives on mathematics
problem-solving, learning, and teaching (Mahwah, New Jersey: Lawrence Erlbaum)
Bliss K 2016 GAIMME: Guidelines for Assessment & Instruction in Mathematical modeling
Education (United State America: COMAP & SIAM)

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