ISSN 2087-8885
E-ISSN 2407-0610
Journal on Mathematics Education
Volume 10, No. 3, September 2019, pp. 425-444
425
SENIOR HIGH SCHOOL MATHEMATICS LEARNING THROUGH
MATHEMATICS MODELING APPROACH
Bambang Riyanto, Zulkardi, Ratu Ilma Indra Putri, Darmawijoyo
Universitas Sriwijaya, Jalan Padang Selasa 522, Palembang, Indonesia
Email: [email protected]
Abstract
By modeling learning students enjoy learning and doing mathematics in new ways. This study aimed firsly to
produce senior high school mathematics modeling tasks, lesson plan, and student worksheet for valid
mathematical learning; secondly, to produce senior high school mathematics modeling, lesson plan, and student
worksheet for practical mathematics learning; lastly, to produce senior high school mathematics modeling tasks,
lesson plan, and student worksheet for potentially effective mathematics learning. This study used method of
development research that consisting of 3 steps, i.e., analysis, design, and evaluation. In the analysis stage,
researcher did student analysis, curriculum, and mathematical modeling. Second stage are to design and product.
Finally, researchers applied a design of formative evaluation consists of self-evaluation, one-to-one, experts
review, small group, and field test. Based on experts review, one-to-one, small groups, and field test were
obtained valid, practical, and potentially effective, i.e. mathematical modeling tasks, lesson plan, student
worksheet to teach mathematical modeling in senior high school and Mathematical modeling tasks and student
worksheets to learn mathematical modeling in senior high school.
Keywords: learning, senior high school mathematics, modeling approach
Abstrak
Dengan pembelajaran pemodelan siswa menyenangi belajar dan doing mathematics di dalam cara baru. Tujuan
Penelitian ini adalah, pertama, menghasilkan tugas pemodelan matematika sekolah menengah atas, lembar kerja
peserta didik (LKPD), dan rencana pelaksanaan pembelajaran (RPP) untuk pembelajaran matematika yang
valid, kedua, menghasilkan tugas pemodelan matematika sekolah menengah atas, ), lembar kerja peserta didik
(LKPD), dan rencana pelaksanaan pembelajaran (RPP) untuk pembelajaran matematika yang praktis, terakhir,
menghasilkan tugas pemodelan matematika sekolah menengah atas, lembar kerja peserta didik (LKPD), dan
rencana pelaksanaan pembelajaran (RPP) untuk pembelajaran matematika mempunyai efek potensial. Studi ini
menggunakan metode penelitian pengembangan yang memiliki 3 langkah, yaitu analisis, desain kemudian
evaluasi. Langkah pertama dilakukan analisis siswa, kurikulum, dan pemodelan matematika. Langkah kedua
adalah desain dan product. Kemudian langkah terakhir, peneliti menggunakan desain evaluasi formatif, yaitu
dilakukan self-evaluation, one-to-one, review ahli, small group, dan field test. Berdasarkan experts review, one-
to-one, small group, dan field test diperoleh, yaitu soal pemodelan matematika, lembar kerja peserta didik
(LKPD), dan rencana pelaksanaan pembelajaran (RPP), dan yang valid, praktis dan memiliki efek potensial
untuk mengajar pemodelan matematika di sekolah menengah atas dan tugas pemodelan dan lembar kerja peserta
didik untuk belajar pemodelan matematika di sekolah menengah atas.
Kata kunci: pembelajaran, matematika sekolah menengah atas, pendekatan pemodelan
How to Cite: Riyanto, B., Zulkardi, Putri, R.I.I., & Darmawijoyo. (2019). Senior High School Mathematics
Learning through Mathematics Modeling Approach. Journal on Mathematics Education, 10(3), 425-444.
https://doi.org/10.22342/jme.10.3.8746.425-444.
Modeling learning makes Mathematics more meaningful and interest for students in this approach
(Arseven, 2015). Traditional teaching and separate teaching from problem solving are not enough to
prepare students for the 21st century. Bliss et al. (2016) stated that the preparing of 21st century
students can be achieved by mathematical modeling because modeling and competency modeling
have important contributions to make whether learns able learn and how they must learn in
mathematics education. In traditional learning, students are conditioned by "results orientation"
426 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
(Kantowski, 1981). The traditional approach also aimed that students produce correct answers
validated by the teacher. Students are considered achievers if their actions are the same as the
teacher's expectations. The teacher prioritizes the purpose of the following specific procedure
instructions to be effective (Cobb et al., 1992). It is very contrary to the modeling learning.
Kilbane and Milman (2014) state that learning in the 21st century is characterized by the use of
information and thinking to solve problems. In Problem-Based Learning models, students do both of
these, resulting in their development of 21st century skills. It is very compatible with mathematical
modeling learning that modeling learning is learning matching the characteristics of 21st century
learning, and modeling can access 21st century students' skills (Bliss et al., 2016). Research
conducted by Maa et al. (2018) show that with learning modeling the teachers explained the positive
changes in general motivation for mathematics, and stated that their students decided to study
mathematics or science, or technology because they enjoyed learning and doing mathematics in new
ways. This shows the teacher is very happy to teach mathematics in new ways and students enjoy
learning mathematics in new ways.
Yanagimoto (2005) states that traditional mathematics currently taught in schools does not
contain subject areas under development (everything is already complete and finished). There is a
little examples of modeling in the practice of learning mathematics in many countries. The cause for
the abyss between educational programs and facts is that mathematical modeling is not easy for
teachers (Arseven, 2015). The looking for of student's problems when modeling is the aerly step
before giving teacher intervention or a feedback (Ferri, 2018). This shows that mathematical modeling
aiming to diagnose student weaknesses in mathematics and teachers having to have content and
pedagogical knowledge to do the mathematical modeling can be done by focusing on modeling tasks.
Mathematical modeling taught in school mathematics must be introduced together with strengthening
new content (Kawasaki, 2012).
Mathematical modeling methods make learners understand better the relationship between real-
world problem and mathematics (Bonotto, 2007; Blum, 2002). Then, studies have shown that
mathematical modeling is a approach that teachers do not know well (Frejd, 2012). Quite similarly,
introducing modeling into curriculum of mathematics resulted on teachers creating their strategy of
learning, causing their learning of mathematics more related to real-world problems (Martinez-
Luacles, 2005). Consequently, models have an essential influence in creating mathematics real to
learns (Anderson, 2010).
There were three research questions in this Study. Firstly, how senior high school mathematics
modeling tasks, student worksheet, and lesson plan were designed valid. Secondly, how senior high
school mathematics modeling tasks, student worksheet,and lesson plan were designed practical.
Lastly, how senior high school mathematics modeling tasks, student worksheet, lesson plan were
designed potentially effective.
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 427
METHOD
This study applied method of development research. This method consists of 3 steps, i.e.,
Analysis, design, then evaluation (Akker, et. al., 2006). On the analysis stage, researcher did student
analysis, curriculum, and mathematical modeling. The second stage, researcher did design and
produce mathematical modeling tasks, LKPD and RPP. Finally, this research applied a design of
formative evaluation (Figure 1). On this stage, researcher did self-evaluation, one-to-one, experts
review, small group, and field tests (Tessmer, 1993, Zulkardi, 2006).
Figure 1. Design of Formative Evaluation.
The criteria of success of this research used the form of mathematical modeling asks, student
worksheet , and lesson plan in Senior High School that was valid, practical and potentially effective.
The validity was got from the comments of experts of mathematics, RME, mathematics education and
mathematical modeling. The practicality was got from the students' comments since working
mathematical modeling tasks via observations of the one-to-one, small group by video and interview,
and the effective was obtained from the field test. Practicality implies simple to implement. It can be
interpreted, and is not double meaning.
The collecting of data techniques were, firstly, walkthrough. It based on the comments
of experts to produce a valid mathematical modeling tasks, student worksheet, lesson plan on
language, content, construct and contexts aspects, secondly, interview and student’s solution,
it produced from on-to-one, small group and field test to get the practicality and potentially
effective of the mathematics modeling tasks, student worksheet, lesson plan. The obtained
data were analyzed by implementing method of descriptive analysis, firstly, walkthrough,
walkthrough by sheet analysis based on the comments of validation of experts to produce
valid mathematical modeling tasks, student worksheet, and lesson plan, secondly, interview
and student’s solution, it analyze the outcomes of one-to-one and small group to produce
practicality and potentially effective of the mathematics modeling tasks, student worksheet,
and lesson plan.
Self
Evaluation
Expert
review
One-to-One
Revise
Revise Small
Group
Revise Field
Test
Low resistance of revision High resistance of revision
428 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
RESULT AND DISCUSSION
Preliminary research, i.e. the first cycle of research was done at SMP Negeri 6 Kayuagung
with 3 students Grade VIII.2 students as research subjects, namely K. F. P., M. R. A. P. and O. A.
In this study the valid and practical mathematical modeling tasks were produced using UBER
contexts. Figure 2 shows that the students' commented that the task of mathematical modeling with
UBER context was very good and made them think, and reason, and need revisions to terms in the
process of modeling that they were not yet familiar with it. It is because the mathematical modeling
was new for them.
Suggestion:
In my opinion, the material is very good, because it can make children to think reasoning, but the
language that used too high for yunior high school, therefore, children do not understand to the tasks
that is given.
Figure 2. Students' comments on the modeling task
The first cycle of the study also indicated that learners could create identification of the
problems, make assumptions, work mathematically to get results, and provide recommendations even
though they were not able to generalize. Figure 3 is a solution for students of UBER context modeling
tasks.
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 429
Do the math: to get solution
car 250
8
people 200
x
000.250.125000.50x
Analyze and assess the model and the solutions
The model is suitable
Iterate as needed to refine and extend the model
No
Implement the model and report the results
Conclusion
If there are 250 people on the trip, 25 car will be
rented at cost 1.250.000 rupiah
Figure 3. Students Solution
The students' commented that the tasks of mathematical modeling with nutritional context was
very good and made them imaginative, but it was quite difficult. This shows that the modeling task is
good and needs revision. The comments of student can be seen in Figure 4.
In my opinion, this task is good. It can invite to imagine. But, some fact and
problem above out of sync and not clear what’s the meaning.
Figure 4. Student comment’s on mathematical tasks
The second cycle of the study indicates that students could only identify and simplify the tasks but could not
to complete the modeling tasks according to the modeling process. This resulted the fact that mathematical
modeling was new to them. Figure 5 shows the solution for students in the task of modeling nutrition contexts.
Identify and specify the problem to be solved
How to solve the problem of malnutrition in
Indonesian society? whether by increasing
consumption of fruits and vegetables and
reducing the consumption of foods that contain
high fat and cholesterol
Do the math: get a solution
In one serving size
White rice: 129 0,285 27,94 2,664
Corn : 132 1,62 29,29 4,966
Wheat bread: 67 1,07 12,269 2,376
Pearled barley: 123 0,44 26,229 2,263
Figure 5. Student’s Solution
430 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
The Third Cycle of the study took 3 subjects as the subjects of SDIT Bina Insani Kayuagung,
namely C. A. M., A., and F. The students' commented that they learn interesting mathematics with
mathematical modeling learning. Figure 6 is a student's comment on a mathematical modeling task.
This learning mathematics is interesting
Figure 6. Students' comments on modeling tasks
In this third cycle of the study, students could only identify and specify problems and make
assumptions. They were not able to do a mathematical process to get a solution and cannot evaluate
and generalize mathematical models (Figure 7).
Identify and specify the problem to be solved
home, parking lot and playground
Do the math: get a solution
Land size 15 x 40 meter
Figure 7. Student Solution to the Modeling Task
The Fourth Cycle took 6 students as research subjects of SDIT Bina Insani Kayuagung,
namely S. A., K. N. I., F. A., M. S. A., M. F. A. H. and D. R. F. The students commented that the
mathematics learning became interesting and made them think with mathematical modeling learning.
Figure 8 explains a student's comment on a mathematical modeling task. This fourth cycle study
used the context of “Jumat Sejahtera”.
The problem is good and interesting and
makes us think
Figure 8. Students' comments on modeling tasks
In this fourth cycle, students could only identify and specify problems and make assumptions,
determine important variables, perform mathematical processes to get solutions, produce
mathematical models, but they were not able to generalize and do iterations that can be seen in Figure
9.
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 431
Create a mathematical model/formula to
calculate only the cost of food from "Prosperous
Friday" using the variables (numbers / values /
prices) that you have created!
Food per box = 3 x 3000 = 9000
Drink per box = 1 x 1000 = 1000
= 10.000 rupiah
Total of students = 21 people
Total of teacher = 2 teachers
23 orang
23 orang x 10.000 = 23.000 rupiah
Create a mathematical model/formula to
calculate how long to save from "Prosperous
Friday" using the variables (numbers / values /
prices) that you have created by assuming how
much money you save per day!
Total of students = 21
The price of donut = 5000
The money that save in one day = 5000
5000 x 21 = 105.000 rupiah
105/5 = 21 hari
Figure 9. Student Solution to the Modeling Task
This study was conducted at SMA Negeri 1 Palembang. In this study, modeling learning used
the context of Musi 2 Bridge, Toll Fee, Online and Conventional taxi, Car Speed on Toll Road, and
Water subsciption fee of PDAM. These mathematical modeling tasks and students worksheet was
conducted experts review, i.e. Prof. Hendra Gunawan, Prof. Edi Cahyono, Al Jupri, S.Pd., M.Sc.,
Ph.D., and Dr. Rusdy A. Siroj. Figure 10 show the comment of expert of mathematical modeling, i.e.
Edi Cahyono (Figure 10).
Expenditure per month on the payment
receipt above is almost the same. Thus, it
would be difficult to formulate significantly
different recommendations. Data cost per m
^ 3 difference is needed, for usage groups.
Data can be reproduced, for example, receipt
for 1 year
Figure 10. Expert’ comment to the Modeling Tasks on PDAM contexts
After experts review, this study conducted one-to-one. The subjects of one-to-one were the
learners of Grade XI IPA 1 and 3 of SMA Negeri 1 Palembang as many as 15 students, namely A. K.,
A. A., A. K. A. I., A. A. R., A. Y. N., D. T. T., H. K., J. A, M. A. A., M. B. H., M. D. S. N., M. A., N.
K., N. S. A, and R. R. A.
432 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
One of the comments of students when one-to-one students, named A. K. state that this
modeling task was interesting and he never studied modeling tasks like this pervously and this task
needed a strong and careful analysis. A. K.'s comments is shown in the Figure 11.
This problem is interesting, because before I never solved a math problem this complicated. To do
this problem requires a strong and careful analysis.
Figure 11. Student comment’s on one-to-one
Judging from the solution of students, the task of modeling with the context of the Toll Road Fee and Musi
2 Bridge, the students were not able to bring up mathematical models even though they did the mathematical
processes. But for the other three contexts, the students were able to make mathematical models even though they
could not validate and evaluate and do iterations to improve the mathematical models obtained. This is due to new
mathematical modeling for them. Figure 12 is a solution for students who work mathematically.
charging
nth money
on the toll
card (e-
money)
The total
amount of
money
spent on
toll card
filling
(Rp)
The total
amount of
administrative
costs (Rp)
The
amount of
money on
the toll
card (Rp)
the amount
of money
paid for
the nth trip
(Rp)
The
amount of
money on
the toll
card on the
nth trip
(Rp)
Money on
the card
filled in
for the
number of
trips?
Remaining
money (e-
money) on
the toll
card on the
nth trip
(Rp)
1 Not filling 1.500 48.500 20.000 45.200 2 25.200
2 Not filling 1.500 48.500 20.000 53.200 2 33.200
3 50.000 1.500 48.500 20.000 33.200 1 13.200
4 Not filling 1.500 48.500 20.000 61.700 3 41.700
5 Not filling 1.500 48.500 20.000 41.700 3 31.700
6 Not filling 1.500 48.500 20.000 58.700 2 38.700
7 50.000 1.500 48.500 20.000 38.700 1 18.700
Figure 12. The results of doing mathematics students.
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 433
After conducting one-to-one, the small group was also conducted using the context of "Musi 2 Bridge",
"Toll Road Fee," Online and Conventional Taxis”, "Car Speed on Palindra Toll Road ", and "Water subscription
fee of PDAM". The result of small group was that students could solve mathematical modeling problems in the
context of "PDAM". The solution to the mathematical model is shown in Figure 13.
Make a mathematical model to calculate the cost and
cost of clean water per cubic meter using the variable
you created. Determine the monthly cost and water
price per cubic meter!
Mathematical model
October: 13 a + b = 37.300
September: 12a + b = 35.200
We get the values of a and b as follows
a = Rp. 2.600 cost of water per cubic meter
b = Rp 4000 fixed costs per month
Figure 13. Context Mathematical Model PDAM
The comments of students after small group were that they were happy and the material was
new and interesting. Student comments are shown in Figure 14.
I enjoy this material and this material is new, and the formulas are unknown to me. Pretty interesting
Figure 14. Student’s comment on small group
After that, the study continued to conducting a field test to Grade XI IPA 3 using modeling
learning with the context of the PDAM. Then, field test was conducted to Grade XI IPA 3 and IPA 1
with modeling learning of the context of the musi 2 bridge and the toll roads fee. Then, it was
conducted to Grade XI IPA 3 using the context of PDAM Fee. Then, it was conducted to Grade XI
IPA 5 using the modeling learning the context of Musi 2 Bridge. Then, on Tuesday, April 9, 2019, it
was carried out to Grade XI IPA 3 with the modeling learning of the context of the Toll Road Fee.
For the field test, students commented that this modeling task was interesting and they never
studied modeling tasks like this before and this task required strong and careful analysis. Figure 15 is
the student’s comment after the learning process, i.e.
This problem is interesting, because I have never done math problems this complicated before. To do
this problem requires a strong and careful analysis.
Figure 15. Students comment’s on one-to-one and small group
434 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
The modeling learning activities indicate that students are very enthusiastic about learning
mathematics which can be seen in the videos and observations. Student solutions in the modeling
process also show that students could to create mathematical models for the five contexts of modeling
learning that are done even though students have not been able to evaluate the mathematical models
they have found. Figure 16 shows the solutions of students for modeling tasks with the context of the
PDAM.
Make a mathematical model to calculate the cost and cost
of clean water per cubic meter using the variable you
created. Determine the monthly cost and water price per
cubic meter!
Cost per cubic meter = x, Fixed costs per month = y.
13x + y = 37.800 12x + y = 35.200
12x + y = 35.200 12(2.600)+ y = 35.200
x = 2.600 31.200 + y = 35.200
y = 4000
Make a mathematical model to estimate the cost
that must be incurred by the household in one
year based on the cost of water costs and costs
for one year!
Usage: a = amount per cubic meter, x = price per
meter cubic, y = expense per month
bill amount in one month (no expense) = (ax)
bill amount in one year (no expense) = (ax + y)12
Figure 16. Mathematical model of fee PDAM in one year
The same results also show the context of the Car Speed on the toll road. The students could to
create mathematical models of problems given. Figure 17 is a student solution.
Make a mathematical model for the speed of a car that
crosses the motorway using the variable you have
created! The average speed of all cars and their
mathematical models.
Car peed:
Sa = Sb va . ta = vb. tb
...(1)
.v
v
b
a
a
b
t
t
...(2)
.v
v
a
b
b
a
t
t
Average speed: ..
....v
v
a
rata
cba
vv
cb
.
v
v
rata
n
n = many cars, and v = car speed
Figure 17. Mathematical model of the “car speed on toll road” context
For the context of Musi 2 Bridge, the students were able to produce two models, namely
assuming the arch of the Musi’s bridge 2 is a quadratic function and a trigonometric function. Figure
18 is a solution to the mathematical model they found.
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 435
Create a mathematical model to
calculate the height of the big and
small arches of the musi bridge 2
using the variables you have created!
y = -0,35(x – 0)(x – 10)
y = -0,35 x(x – 10)
y = -0,35x
2
– 10x
yn = nth string height, y = tinggi maksimum
x = distance end to end horizontally
The bridge arch is a function of sin
Sin graph with a hill shape has angle intervals 0x. So, for this bridge
has a difference in the angles 15
for each adjacent rope (to determine height) and a small arc.
Create a mathematical model to calculate the height of large and small
arches of the musi bridge 2 using the variables you have created!
Large arch height:
15
.sin.10
ny for its breaking point
The value of 1
15
.sin
n , so,
2
sin
15
.sin
n
So, n = 15/2. Function for small arches, i.e.
8
.sin.10
ny
n
Figure 18. Mathematical model for “Musi 2 Bridge” context
The context of online and conventional taxi show that students could produce mathematical
model derived from the given problem. Figure 19 is the mathematical model found by the student.
Make a mathematical model to calculate the cost of
online and conventional taxis!
Online taxis (Jawa, Sumatera, Bali):
a = 0, b = 3.500, Un + a + nb
Kalimantan, NTT: The highest: Un = n . 3.700
The lowest: Un = n. 6.500
Bluebird conventional: Un = 6500 + n . 4100
Execitive: The highest: Un = 13.000 + n. 7500
The lowest: Un = 17.000 + n . 9000
Express: Un = 6500 + n. 3800
Figure 19. Mathematical model of online and conventional taxi
For the context of the online and conventional taxi fee also shows that students could make
mathematical models of the problems given. Figure 20 is a mathematical model found by students.
436 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
Make a mathematical model for the costs that
have to be incurred by toll road users for one
year and sal do on the toll card at the last use
of the toll road at the end of the year using
the variable you have made!
Charging Fee = 50.000 rupiah
Un = a + (n – 1)b
Un = 50.000 + (n – 1) 50.000
Un = 50.000n
Figure 20. Mathematical model for “Toll Road Fee” context
After doing field test potentially effect was looked. This can be seen from the comments of the
model teacher, the observer teacher and the students themselves as research subjects stating that
modeling learning is good to apply in the future because it is interesting, challenging, and meaningful
for students and makes students motivated to learn. The learning process of mathematical modeling
tasks can produce good mathematical models as expected and there is a validation process as in the
modeling cycle. The mathematical model that has not been generated is in the context of the Musi 2
Bridge. The quadratic function model that still has parameters appears. Whereas for trigonometric
functions mathematical models have emerged but there are no arguments or validations from the
students. Whereas for the contexts of conventional online and taxi taxis, the toll road fee, PDAM and
car speed on toll road produced the correct model. Therefore, results of the study shows the learning
process on field test is good in accordance with the expectations in accordance with the characteristics
of the task and the modeling learning process.
This study produced mathematical tasks, lesson plan, and student worksheets for senior high
school mathematics modeling. As a results, the valid, practical and potentially effective senior high
school mathematics task of mathematics modeling, lesson plan, and student worksheet were designed
valid, practical, and potentially effective to be implemented for more interesting and meaningful
mathematics learning to better the quality of learning and mathematical achievement.
The first research problem, how senior high school mathematics modeling tasks, lesson plan,
and student worksheet were designed valid?. This research conducted experts review from
mathematicians Prof. Hendra Gunawan, experts of mathematical modeling Prof. Edi Cahyono, expert
of realistic mathematics education Al Jupri, S.Pd., M.Sc., Ph.D and expert of mathematics education
Dr. Rusdy A. Siroj. They state that mathematical modeling tasks in this reseach is good and need
revision. After experts review, one-to-one was conducted. Students say that this tasks was interesting
and make student to think. Student could make mathematical model in one-to-one. This show that this
tasks was valid.
In this research of designing modeling tasks based on the characteristics of the modeling task,
the problem comes from the real world. Sullivan et al. (2015) state that designing tasks should
consider three aspects of pedagogy, namely, student motivation, task recognition to students, which
relates to the teacher wanting learners to have the ability to interpret the demands of the task, and
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 437
accessing assignments by all students, on which this issue causes the teacher to think of student
motivation, the level of initial knowledge to push with assignments, the current mathematics culture
class, expanding tasks that could distinguished to allow all learners to push appropriately. To
implement this designed task the teacher can make modeling tasks as an effective way of learning.
This modeling tasks can motivate students; with mathematical modeling tasks the students will carry
out the process of interpreting assignments and modeling tasks which are valid for all students who
are weak, moderate and high in their mathematical abilities.
In desiging tasks of mathematical modeling learning, teachers must make use of everyday
contexts. The function of a context is very essential in mathematical modeling because the modeling
needs a context for problem frame, and mathematical applications (Lingefjard, 2006; Mousoulides,
2007). A focus characteristic of traditional mathematics learning in many countries is the activity of
'solving story problems' (Schoenfeld, 1992; Mousoulides, 2007).
Saxena et al. (2016) state that sometimes they find it difficult to solve problems and learn
formulas and prove theorems. Although a lot of effort and research were carried out in this field but
there was a lot of work done at the basic level. However, the level of difficulty gets high at the middle
and high levels because we have a lot of content in grades 11 and 12 as well as many new things for
students and many formulas, theorem relationships and a few parts of the application. In research it
shows that it is very essential to understand the contexta in modeling.
The second research problem is how senior high school mathematics modeling tasks, lesson
plan, and student worksheet were designed practical?. This research was conducted small group. In
this small group students very active in learning. Students could learn mathematical modeling to get
mathematical model. Students also state that this learning is very interesting. This show that
mathematical modeling tasks and student worksheet is practical.
It is not easy to implement mathematical modeling lessons in schools due to time constraints
(Hino, 2007). Research in mathematics education emphasizes the essential of teacher knowledge
about learner thinking (Ball & Cohen, 1999; Ball et al., 2008; Even & Tirosh, 1995; Stillman et al.,
2010). But teacher knowledge about students’ thinking is not easy matter because it needs them to
see, hear, understand, and interpret learners' thinking when driven by mathematics tasks (Ball, 1997).
Teachers must be able to estimate difficulties, lack of understand, and predict whether students are
appealing in choosing examples (Ball et al., 2008; Hill et al., 2008). It shows the importance of the
task of modeling school mathematics. Modeling learning place new challenges on the teacher (Doerr,
2006).
The study conducted by Riyanto et al. (2017) indicates that junior high school learners are very
extracted in learning mathematical modeling. Likewise, in the study conducted by Riyanto et al.
(2018), students are also interested in learning mathematical modeling, and the research in elementary
schools shows the same thing that elementary school pupils can produce good mathematical models
and are interested in learning mathematical modeling (Riyanto et al., 2019). Based on the Eric's
438 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
(2010) research results, he recommend that future studies be required in the area of mathematical
modeling to link theory and practice, to fit between contemporary and approaches of traditional, and
role of the teacher as a facilitator. The study conducted by Anthony and Hunter (2010) show that
mathematical modeling and application tasks that are discussed in group settings to give a solid basis
to learner development of argumentation because they are inherently social experiences with support
of effective teacher and communication, collaboration, teamwork, and reflection effectively.
The results of the study conducted by Biccard (2010) show that modeling competencies can not
develop by implementing traditional approach and essential teaching to develop mathematics as their
own competence. Development of modeling competencies requires support by wise task selection,
heterogeneous group formation, and effective teacher knowledge about modeling. The study findings
of Wong (2008) can have implications to training of teacher on modeling tasks to improve quality of
mathematical achievement. The suppression on mathematical modeling in school level is to focus on
the process rather than on the product (Balakrishnan et al., 2010).
The third research problem is how senior high school mathematics modeling tasks, lesson plan,
and student worksheet were designed potentially effective?. This study conducted field tests for
PDAM, Fee Tol, Musi 2 Bridge, Online and Conventional taxi, and Car speed on Tol road. From field
tests, students was very enthusiastic and enjoyed learning mathematics with mathematical modeling
and students could make mathematical models using their own methods. From 36 students only one
student didn’t like this tasks. On field tests also students could make mathematical model but students
couldn’t validate the mathematical model, so that couldn’t do iteration in modeling process. It is
caused that learners are rarely to learn mathematical modeling.
In this study, students studied mathematical modeling to solve real-world problem with emply
topic in mathematics. In learning mathematical capital, students must make assumptions, make
selections, make variables, mathematization, working mathematically and validating the obtained
results both mathematically and in the real world. Similar, the study results of Stillman and Brown
(2014) show that the inability to mathematize in the classroom is connected to the inability to apply
relevant mathematical skill in the context of modeling than lack of relevant mathematical skill per
person, or an orientation view of mathematical applications or persistence on tasks.
Creativity is needed to face an unknown future and the education should complement learners
for this purpose (Wessels, 2014). Several skills, such as creativity, spatial, and problem solving, are
generally associated with art, but it is also a basic part of mathematics, technology, economics and
politics - forming an integral part of a person's daily life (Robinson, 1999; Hendroanto et al., 2018;
Ahamad et al., 2018). In mathematical modeling, it is very important for creativity, fluency,
flexibility, and novelty. It is very important in facing an increasingly complex life. Furthermore,
Saxena et al. (2016) state that the students' performance has not been at the desired level, the students
did not solve mathematical problems interestingly. The importance of mathematical modeling in
schools is at all levels (Mousoulides, 2007). In this study, the creativity of students in learning
Riyanto, Zulkardi, Putri, & Darmawijoyo, Senior High School Mathematics Learning … 439
mathematical modeling rises. In addition, metacognitive development and critical thinking skills are
seen as importance and must help learners reflect on their answers to modeling tasks (Garofalo &
Lester, 1985; Schoenfeld, 1992; Mousoulides, 2007).
In the study conducted by Eric (2013) it concludes that encouraging students in mathematical
modeling opens up a strategy to teachers to redesign strategies in which leaners be able solve
problems in real-world problems. Modeling tasks of "travel plan" context take a dimension for
students to look at agreed phenomena for mathematical thinking, asking them to explain the purpose
of modeling situations, evaluating these amounts and variables leads relevance to work with
manipulating, developing, and interpreting the model in many strategies to get solutions (Eric, 2013).
In mathematical modeling approach, students should be able to use the many stages in
modeling cycle to many open-ended tasks (Haines & Crouch, 2010; Knott, 2014). This approach can
be applied to solve other problems successfully and to develop a critical, reflective individual (Knott,
2014). The teachers in his study comment that it is essential for learners to learn mathematics is
helpful in extra-mathematical situations (real-world) and one approach to make students learn
mathematics with this pupose in thinking must be integrate more modeling activities in mathematics
learning (Frejd, 2012; Muhtadi et al., 2017; Mumu et al., 2018).
Good modeling tasks invite learners to encounter the whole modeling process (Balakrishnan et
al., 2010). In this study, student was given the entire modeling process. In this study, it can be seen
from experts review, student’s opinion and student’s solution of mathematical modeling tasks. The
same results occur in the study using the contexts of Musi 2 Bridge, Toll Road Fee, Car Speed on Toll
Roads, online and conventional taxis, and the context of the PDAM whose contexts originating from
the real world situation in which the learning uses authentic, opened-ended problems, students
assuming, doing election. In this learning, the students produced good mathematical models, active,
effective learning processes and they were interested in learning mathematics. The results supported
to several research that shown students are interested in realistic material use real-time daily
applications for them (Zulkardi, 2002; Saleh et al., 2018; Nuari et al., 2019).
Zulkardi and Putri (2019) said that some new components of the 2013 curriculum have the
same characteristics as PMRI. PMRI and Learning mathematical modeling are not much different
(Zulkardi, 2017). It indicates the importance of modeling learning in the 2013 curriculum.
Consequently, teachers should design learning tasks based on the characteristics of mathematical
modeling tasks. This statement also correspondends to Blum et al. (2019) stating that the design of
activities in mathematical didactics can be in the form of design of tasks, lessons (lesson plan),
sequences of teaching, textbooks for mathematical leaning, curriculum, assessments, and ICT-based
teaching material or programs for teacher education and must be done by teachers, educators, book
writers, curriculum and assessment developers, designer of ICT, and researchers. This statement is in
accordance with this study, namely designing modeling tasks, student worksheet, and lesson plan.
440 Journal on Mathematics Education, Volume 10, No. 3, September 2019, pp. 425-444
CONCLUSION
This study produces the valid and practical high school mathematics modeling tasks, lesson plan, and
student worksheet. The products alaso have potentially effective. The findings in this study were that
students were very enthusiastic and enjoyed learning mathematics with mathematical modeling and
students could make mathematical models using their own methods. Student can apply modeling
process, so this promoting student’s modeling literacy. Mathematics learning should implement
mathematics modeling learning starting from elementary school to college level. Then students
individually and in groups solve this mathematical modeling problem by using suitable mathematical
topics. There needs to be a change in curriculum that uses real-world problems in mathematics
learning. There needs to be a training for teachers in designing modeling tasks, implementing
mathematics learning, and evaluating mathematical modeling learning in schools and colleges.
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