Journal of Mathematics Education
Volume 9, No. 1, February 2020

p-ISSN 2089-6867
e ISSN 2460-9285

https://doi.org/10.22460/infinity.v9i1.p111-132

THE ROLE OF CONSTRUCTIVISM-BASED LEARNING
IN IMPROVING MATHEMATICAL HIGH ORDER
THINKING SKILLS OF INDONESIAN STUDENTS
Ani Minarni*1, E. Elvis Napitupulu2
1,2Universitas Negeri Medan

Article Info

ABSTRACT

Article history:

To make students actively involved in learning to grasp mathematical higherorder thinking skills (MHOTS) is not easy. Meanwhile, the ability is so
important for students to master for it takes place when students continue their
studies to a higher level as well as work within a variety of professions,
especially in the era of the industrial revolution such nowadays. Many factors
affect students' thinking abilities, including learning factors. This study, which
implemented constructivism-based learning, aims to investigate the role and
contribution of constructivism-based learning approaches as well as
mathematical prior knowledge (MPK) to the achievement of MHOTS of
middle secondary school students. The data tested through Multivariate
Analysis at the 0.05 significance level. In general, this study found that: (1) In
the experimental class, the learning approach plays an important role in the
way it increased students' MHOTS significantly. (2) The average contribution
of constructivism-based learning to MHOTS was at the range of 18% to 57%.
(3) Student activity in learning increased significantly. (4) In some cases, there
is an effect of interaction between learning factors and MPK towards the
achievement of MHOTS. The study recommended the teachers to have
courageous in implementing constructivism-based teaching and learning to
i
e de
MHOTS.

Received Nov 4, 2019
Revised Feb 10, 2020
Accepted Feb 16, 2020
Keywords:
Constructivism-based learning,
MHOTS,
MPK

Copyright © 2020 IKIP Siliwangi.
All rights reserved.

Corresponding Author:
Ani Minarni,
Departement of Mathematics Education,
Universitas Negeri Medan
Jl. William Iskandar Ps. V, Kenangan Baru, Deli Serdang, North Sumatera 20221, Indonesia.
Email: animinarni10@gmail.com
How to Cite:
Minarni, A., & Napitupulu, E. E. (2020). The role of constructivism-based learning in improving
mathematical high order thinking skills of Indonesian students. Infinity, 9(1), 111-132.

1.

INTRODUCTION

Parents keep some will for their children, for example they want their children to be
useful people for themselves, family & society, and to serve their parents, country and
religion. God has the will of His creatures; God wants His creatures to be on a straight path
in terms of ways that can make His creatures feel peace in living. The teachers want their
students to have thinking competencies, life skills and useful characters. The teacher wants
students to have the habit of lifelong learning and succeed in learning according to what
NCTM document emphasized (NCTM, 2000).

111

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

Many factors influence student achievement, but the point is there are two, namely
factors within themselves and external factors (Brown, 1990). What can be considered as
internal factors include intellectual level (intelligence level), learning ability, learning
motivation, learning independence (self-regulated learning), attitudes, feelings, interests,
psychological conditions, and due to socio-culture. Meanwhile, external factors include the
attitude of parents and teachers to students, learning factors applied in schools, curriculum,
school discipline, teachers themselves, learning facilities, grouping students, social systems,
student social status, teacher and student interaction, political economy conditions,
circumstances time and place or climate.
By paying attention to a series of influential factors, one of the things teachers can
try in schools is to improve learning outcomes through the implementation of appropriate
learning. Learning should be empowering students to think and construct their knowledge,
arise students' interest in learning and make students understand the topics they learn.
Learning that has characteristics like this is constructivism-based learning (Resnick, 1987),
such as Problem-based learning, Discovery learning, Cooperative learning (Arends, 2012;
Ronis, 2008), even contextual learning and Open-ended approaches, and mathematical
realistic approach.
The learning achievement will be even better if parents help encouraging their
children to understand what is learned in school through often asking what they learned,
whether they understand the subject matter learned today, or trigger the student to do
homework. If students understand they learned and can construct their knowledge for
themselves, they will have the opportunity to gain understanding skills at the HOTS (high
order thinking skills) level, the level that reaches the ability to apply knowledge to solve
problems. In other words, the understanding ability will bring up the ability of problem
solving. On the other side, problem-solving ability will build up HOTS as well.
HOTS always plays an important role from time to time, especially in the present era
that has entered the era of industrial revolution 4.0. HOTS, moreover MHOTS, has proven
to be the basis/foundation in the development of that era (Formaggia, 2017). To achieve
MHOTS is not enough just to rely on learning factors, but the mathematical prior knowledge
(MPK) also needs to consider since its also holds an important role in problem solving
process. This is because according to the results of the research, the MPK factor contributes
to the achievement of mathematical problem solving abilities (Minarni, 2017). Reasoning
ability, connection ability & mathematical representation, even the interactions between the
fac
ca a
affec
de
ea i g
c e . The ef e, i i i e e ting to
investigate how is the role, especially the contribution, of constructivism-based learning
approaches to the achievement of mathematical thinking skills of middle secondary school
students. The findings of this study could be triggered the teachers to grasp the courage in
implementing constructivism-based learning approaches that they considered difficult to
implement.
Mathematical High Order Thinking Skills (MHOTS)
High Order Thinking Skills (HOTS) is the concept of education reform based on the
taxonomy of learning objectives from Bloom and its revisions in Marzano & Kendall (2007).
The idea is that some types of learning not only require higher-level thinking skills but also
require ways to teach it differently from other types of learning so there is the term HOTS.
In Bloom's taxonomy, for example, skills that involve analysis, evaluation and synthesis (the
creation of new knowledge) are considered abilities at the highest level that require learning
and teaching methods different from learning or teaching methods that require students to

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113

master facts and concepts (Anderson et al., 2001). Whereas in Marzano & Kendall (2007)
the level of high-order thinking is self-system & metacognition.
HOTS generally consist of include critical, logical, reflective, metacognitive
thinking, problem solving, and creative thinking. The ability to think has the potential to
develop and increase when a person faces a problem that is not familiar to him, uncertain,
raises a dilemma or invites questions. HOTS that runs successfully produces explanations,
decisions, a series of performance, and products that are valid in the context of existing
experience and knowledge and it fosters the growth of other intellectual skills in a sustainable
manner. HOTS is rooted in the skills of simply applying and analyzing knowledge and
cognitive strategies that intertwined with prior knowledge. The appropriate learning strategy
and learning environment is a facility for the growth of HOTS along with the growth of
accuracy, self-supervision, openness, and flexibility in students. This explaination is in line
i h he he
e a ed
h
HOTS i ea ed a d de e ed i
de
c g ii e
structure (Kulm, 1990).
One kind of the HOTS is critical thinking that has many different definitions. Scriven
& Paul (1987) stated that critical thinking is the intellectually disciplined process of actively
and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating
information gathered from, or generated by, observation, experience, reflection, reasoning,
or communication, as a guide to belief and action. In its exemplary form, it is based on
universal intellectual values that transcend subject matter divisions: clarity, accuracy,
precision, consistency, relevance, sound evidence, good reasons, depth, breadth, and
fairness. Therefore, a good critical thinker generally needs to be able to both analyze and
synthesize information. Another definition of critical thinking comes from Cottrell (2005)
that stated critical thinking is a cognitive activity that involves mental processes such as
attention, categorization, selection, and judgment.
Thus, based on the definition of critical thinking, it can be said that mathematical
creative thinking is the abilities that require elements to test, question, connect, evaluate, all
aspects that exist in mathematical problems. All of the areas of mathematics requires critical
thinking, including algebra and geometry. Algebra work requires analysis, which is the
ability to break apart the pieces of a problem to solve while doing geometry involves more
synthesis than analysis, in that we take all the elements of geometry and combine them to
solve problems (do geometric proofs).
The term of mathematical problem solving skills (MPSS) referred to the definition
of problem solving defined by Anderson et al. (2001), which is the process of applying
mathematical knowledge in new and unfamiliar situations (problems). In the process of
i g a he a ica
be
e i g h gh P a f
e (P a, 2004) which
include understanding the problem, device a plan, carry out the plan and looking back.
Understand the problem is the ability to represent the problem in any other form that makes
one easier to attain the solution. Understanding skills also enable one to demonstrate
mathematical connection skills (the ability to connect among mathematical
knowledge/ideas/ procedure/concepts) (NCTM, 2000). The stage of looking back or
reflection can be interpreted as draw conclusions for the solutions.
In solving mathematical problems, one should have creativity. The ability to think
creatively in math problem solving is the ability to solve mathematical problems flexibly,
involving convergent thinking and divergent thinking. Mathematical creative thinking
enable one to make connections between problems under consideration, mathematical
knowledge, variety of strategies for possible solutions, variety solutions, models and related
questions, evaluate the problem solving process and not just at the end, communicate with
peers, teachers, and other interested adults while working on the problem as well as
following its solution (Jensen, 1976).

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Torrance (1960) had studied for long time to reveal an understanding of students
mathematical creative thinking skills. Just like other skills, mathematical creative thinking
can be learned through the teaching and learning process that required students to construct
their knowledge via problem solving. In addition, Sheffield (2013) stated that encouraging
and supporting the development of mathematical creativity have the added benefit to
increasing students enjoyment of and engagement in mathematical reasoning, sense making,
problem solving, and problem posing. Mathematical creative thinking skills is consisted the
aspects of to think flexibly, original (proposed new idea), fluency, detailed and depth
explanation to the solution.
Constructivism-based learning
For a long time, the learning approach teachers used in Indonesian schools is
dominated by direct learning or direct instructions and demonstrations to resolve routine
problems with communication tend to be one-way from teacher to student (Minarni,
Napitupulu, & Husein, 2016). This learning approach can indeed foster mathematical
problem solving abilities but the problem that can be solved is the problem that exists in the
textbook which is often not related to real life problems (Silver, 2013), does not have the
characteristics of a problem that serves to increase mathematical high order thinking skills
(HOTS) such as problem solving (Ronis, 2008).
The learning approach aimed at reaching HOTS requires clarity of communication
to avoid ambiguity and confusion and to increase students' positive attitudes towards tasks
that require them to think and as a way out to address the diverse needs of students. This
kind of learning needs scaffolding techniques, i.e., the support and assistance as needed in
students at the beginning of their problem solving. Scaffolding should gradually reduce until
finally the students are left to work independently. Excessive or too little scaffolding can
hinder a student's development or progress in reaching HOTS.
At present, learning that is expected to improve HOTS and MHOTS is
constructivism-based learning because this learning carries the principle that knowledge is
the result of human construction (Widodo, 2004), knowledge is the result of social
construction (Vygotsky, 1980). Social interaction participates in giving an important role in
the process of knowledge construction (Phillips, 1997). Knowledge is constructed in a
particular social context and influenced by a variety of 'strengths', including ideology,
religion, politics, economics, human interest, and group dynamics. Therefore, individuals
must construct their own knowledge due to knowledge cannot be simply transferred directly
from the teacher to students or from the book to the readers.
Constructivism-based learning such as problem-based learning (PBL), discovery
learning (DL), cooperative learning (Arends, 2012), realistic mathematics education (RME)
(Gravemeijer & Doorman, 1999), contextual teaching-learning (CTL), and the Open-ended
approach (Becker & Shimada, 1997) is designed by considering the factors that influence
the learning outcomes. For example, PBL carries interdisciplinary learning, considers local
culture and places emphasis on social interaction to foster students' problem solving skills
and social skills (Arends, 2012). Meanwhile, social skills allow the improvement of
academic achievement (Minarni, 2013).
Instead of starting the learning process by presenting content for students to
memorize and understand, PBL emphasizes the process of how humans learn naturally, that
is, learning occurs when there are problems (Hmelo-Silver, 2004). To obtain a solution to
the problem, people will be motivated to learn the skills and knowledge related to the
problems they face, learn or recall the contextual knowledge related to the problem. PBL
relies on problems that integrate useful knowledge for students in their personal lives or in

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115

facing their professional careers later on. Problems are designed to be authentic,
unstructured, and sufficiently challenging students to become active and reliable problem
solvers. It can be inferred from Savin-Baden & Major (2004) that the goal of PBL is to guide
students to construct meaning rather than gathering facts and to become collaborative
learner. Those characteristics open the opportunities for the achievement of HOTS.
Another learning approach based on constructivism is Realistic Mathematics
Education (RME). RME is a teaching and learning theory in mathematics education that was
first introduced and developed by the Freudenthal Institute in the Netherlands (de Lange,
1996). RME is an approach that insisted mathematics should be connected to reality and
human activity, close to children and be relevant to everyday life situations (Gravemeijer &
Doorman, 1999). H e e , he
d ea i ic , efe
j
he c ec i
i h he
real-world, but also refers to problem situations which real in students' mind. The context of
the problems presented to the students can be a real-world one but this is not always
necessary. De Lange (1996) stated that problem situations can also be seen as applications
or modeling.
There are two types of mathematization in RME formulated explicitly in an
educational context (Treffers, 1991). These are horizontal and vertical mathematization. In
horizontal mathematization, the students come up with mathematical tools that can help to
organize and solve a problem located in a real-life situation. Examples of horizontal
mathematization: identifying or describing the specific mathematics in a general context,
schematizing, formulating and visualizing a problem in different ways, discovering relations,
discovering regularities, recognizing isomorphic aspect in different problems, transferring a
real world problem to a mathematical problem, and transferring a real world problem to a
known mathematical problem. On the other hand, vertical mathematization is the process of
reorganization within the mathematical system itself. Examples of vertical mathematization:
representing a relation in a formula, proving regularities, refining and adjusting models,
using different models, combining and integrating models, formulating a mathematical
model, and generalizing (Gravemeijer, 1994).
The learning process starts from contextual problems. Using activities in the
horizontal mathematization, for instance, the student gains an informal or a formal
mathematical model. By implementing activities such as solving, comparing and discussing,
the student deals with vertical mathematization and ends up with the mathematical solution.
Then, the student interprets the solution as well as the strategy used to another contextual
problem.
RME is closely related to socio-constructivism (de Lange, 1996; Gravenmeijer,
1994). In both approaches, students are offered opportunities to share their experiences with
others. In addition, de Lange (1996) stated that the compatibilities of socio-constructivist
and RME are based on a large part or similar characterizations of mathematics and
mathematics learning. Those are: (1) both struggle with the idea that mathematics is a
creative human activity; (2) that mathematical learning occurs as students develop effective
ways to solve problems (Streefland, 1991; Treffers, 1991); and (3) both aim at mathematical
actions that are transformed into mathematical objects (Freudenthal, 2006).
Like wise RME, discovery learning (DL) is also based on constructivism. DL is also
referred to problem-based learning, experiential learning and 21st century learning.
Discovery learning is the work of learning theorists and psychologists Jean Piaget, Jerome
Bruner, and Seymour Papert (Arends, 2012). Jerome Bruner is often credited to the origin
of DL in the 1960s, but his ideas are very similar to those of earlier writers such as John
Dewey. Bruner argues "Practice in discovering for oneself teaches one to acquire
information in a way that makes that information more readily viable in problem solving".

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This philosophy later became the discovery learning movement of the 1960s. This
philosophical movement suggests that people should "learn by doing".
The label of DL can cover a variety of instructional techniques. A discovery-learning
task can range from implicit pattern detection, to the elicitation of explanations and working
through manuals to conducting simulations. DL can occur whenever they do not provide the
student with an exact answer but rather the materials in order to find the answer. Discovery
learning takes place in problem solving situations where the learner draws on his own
experience and prior knowledge and is a method of instruction through which students
interact with their environment by exploring and manipulating objects, wrestling with
questions and controversies, or performing experiments.
It has been suggested that effective teaching using discovery techniques requires
teachers to do one or more of the following: 1) Provide guided tasks leveraging a variety of
instructional techniques, 2) Students should explain their own ideas and teachers should
assess the accuracy of the idea and provide feedback, 3) Teachers should provide examples
of how to complete the tasks. A critical success factor to discovery learning is the teacher
assistance. On the other hand, DL potentially make the students feel confused and frustrated.
Silver (2013) argued that pure unassisted discovery should be eliminated due to the lack of
evidence that it improves learning outcomes. Bruner (1961) who was one of the early
pioneers of discovery learning cautioned that discovery could not happen without some basic
knowledge (mathematical prior knowledge).
I
a , he eache
e i di c e
ea i g i c i ica
he cce
f
learning outcomes. Students must build foundational knowledge through examples, practice,
and feedback. This can provide a foundation for students to integrate additional information
and build upon problem solving and critical thinking skills. Early research demonstrated that
guided discovery had positive effects on retention of information at six weeks after
instruction versus that of traditional direct instruction. It is believed that the outcome of
discovery-based learning is the development of inquiring minds and the potential for lifelong learning. Discovery learning promotes student exploration and collaboration with
teachers and peers to solve problems. Children are also able to direct their own inquiry and
be actively involved in the learning process with the support of sufficient motivation (Reid,
2007).
The next one is contextual teaching and learning (CTL). CTL involves making
learning meaningful to students by connecting to the real world (Johnson, 2002). It draws
de
di e e ki , i e e , e e ie ce , a d c
e a d i eg a e he e i
what and how students learn and how they are assessed. In other words, contextual teaching
situates learning and learning activities in real-life and vocational contexts to which students
ca e a e, i c
ai g
c e , he ha , f ea i g b he ea
h ha
learning is important.
Some examples of contextual teaching and learning are interdisciplinary activities
across content areas, classrooms, and grade levels; or among students, classrooms, and
communities. Problem-based learning strategies, for instance, can situate student learning in
he c e
f
de
c
i ie . Ma
ki
ea ed a a
f c e a ea i g
activities are transferable skills, can be used not only for successful completion of a current
project but also in other content areas to prepare a student for success in later vocational
endeavors. Contextual learning, then, engages students in meaningful, interactive, and
collaborative activities that support them in becoming self-regulated learners. Additionally,
these learning experiences foster interdependence among students and their learning groups.
Complementary outcomes assessments for contextual student learning are authentic
assessment strategies, i.e., the assessment is not only limited to the results of the written test
but also based on the students' performance in doing the assignments.

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Another constructivism-based learning model or approach is cooperative learning,
an educational approach aimed to organize classroom activities into academic and social
learning experiences (Arends, 2012). Cooperative learning is actually not merely arranging
students into groups; it is characterized as "structuring positive interdependence." Students
must work in groups to complete tasks collectively toward academic goals. Unlike individual
learning, which can be competitive in nature, students learning cooperatively can capitalize
on one another's resources and skills (asking one another for information, evaluating one
another's ideas, monitoring one another's work, etc.). Furthermore, the teacher's role changes
from giving information to facilitating students' learning. Everyone succeeds when the group
succeeds. Ross & Smyth (1995) describe successful cooperative learning tasks as
intellectually demanding, creative, open-ended, and involve higher order thinking tasks.
Cooperative learning has also been linked to increase levels of student satisfaction. In
cooperative and individualistic learning, student efforts are evaluated on a criteria-referenced
base while in competitive learning teachers grade in a norm-referenced base.
Five essential elements are identified for the successful incorporation of cooperative
learning in the classroom i.e:
i. Positive interdependence
ii. Individual and group accountability
iii. Promotes interaction (face to face)
iv. Teaching the students the required interpersonal and small group skills
v. Group processing.
Students in cooperative learning settings compared to those in individualistic or
competitive learning settings, achieve more, reason better, and gain higher self-esteem. The
next constructivism based learning approach is Open-ended approach. Open-ended approach
provides students with experience in finding something new in the process of open problem
solving (Becker & Shimada, 1997), while open problem solving is based on open-ended
problems. It can be concluded that Open-ended problems used in mathematics lessons from
elementary through high school grades. These problems proposed have several or many
correct answer, and several ways to get the correct answer. There are five advantages of
Open-ended approach:
a. Students participate more actively in lessons and express their ideas more frequently
because Open-ended approach provides free, responsive, and supportive learning
environment. The problem has many different correct solutions, so each student has
opportunities to get his own unique answer. Hence, students are curious about other
solutions and they can compare on and discuss their solutions. Those activities bring a
lot of interesting conversation to the classroom.
b. Students have more opportunities to make comprehensive use of their mathematical
knowledge and skills. Since there are many different solutions, students can choose their
favorite ways toward the answer and create their unique solution. Activities can be the
opportunities to make comprehensive use of their mathematical knowledge and skills.
c. Every student can respond to the problem in some significant ways of his/her own.
Therefore, it is very important for every student to be involved into the classroom
activities, and the lessons should be understandable for every student. The open-ended
problems provide every student with the opportunities to find his/her own answer.
d. The lesson can provide students with a reasoning experience. Through comparing and
discussing in the classroom, students are intrinsically motivated to give reasons of their

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

solutions to other students. It is a great opportunity for students to develop their
mathematical thinking.
There are rich experiences for students to have the pleasure of discovery and to
receive approval from fellow students. Since every student has each solution based on each
i e hi ki g, e e
de i i e e ed i fe
de
utions. There are also some
disadvantages of the Open-ended approach (Sawada, 1997), such as the difficulty of posing
problems successfully, the difficulty of developing meaningful problem situations, and the
difficulty of summarizing the lesson.
Besides learning approach, ICT also influences learning outcomes (Agyei & Voogt,
2011). Moreover, affective aspects also affect learning achievement (Minarni, Napitupulu,
Lubis, & Annajmi, 2018). Therefore, this paper focuses more on the description of the
contribution of learning approach as well as the influence of interactions between the
learning approach and the MPK on the achievement of MHOTS. However, the MPK factor
plays an important role as well since it is needed to be recalled previously learned or provide
the results of a calculation, which were considered lower cognitive questions in previous
studies, played key stages at introducing new mathematical content as well as in the stage of
solving mathematical problems.
The study first aimed to seek the answer on how the contribution degree of the
constructivism-ba ed ea i g
he achie e e
i
e e
f de
MHOTS i ,
and second to reveal if there exists an interaction between the learning approach and the
mathematical prior knowledge on the achievement of MHOTS.
2.

METHOD

The population of this study was junior high school (PJHS) students in Medan, Deli
Serdang, Binjai, and Padang Sidempuan in the Province of North Sumatera, and Banda Aceh
in the Province of Nanggroe Aceh Darussalam. Because the school does not allow students
to take randomly from each class, samples are taken per class. Classes are taken through
simple random sampling because the students at all classes assumed homogenous
mathematical prior ability, two classes from each district. One class is used as the
experimental class, the other one is the control class. The experimental class applied
constructivism-based learning, while the control class applies conventional learning. This
research runs in the odd semester in the 2017/2018 and 2018/2019 academic years.
The instrument used in this study was an essay test of high-order mathematical
thinking skills (MHOTS), which included tests of mathematical problem solving skills
(MPSS). The MPSS indicators used in this study are modifications of Polya (2004) and
NCTM (2000), including:
1. The ability of mathematical understanding shown by external representations and
connections between ideas/facts/concepts/mathematical procedures.
2. The ability to propose problem solving strategies that are demonstrated by the existence
of techniques/methods of problem solving in student worksheets, whether in the form
of mathematical models, graphs, tables, diagrams, or others.
3. The ability to execute the proposed problem solving strategy, shown by calculations and
mathematical manipulations to obtain a solution.
4. Summing up the solution obtained by following the initial problem.
Problem 1 below is an example of a question for developing mathematical creative
thinking skills (Maharani, 2014) Grade VIII junior high school students.

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Problem 1
Look at the quadrilateral model below.

10 cm

i. Draw a quadrilateral that has same area with the image above.
ii. Create at least two different questions related to square and solve it.
The following problem is an example of a problem that teacher proposed in the OpenEnded class (Problem 2).
Problem 2
The average of mathematics score of students from a junior high school is 65. What
is the additional score if the average score of the exam becomes 68. Write down the
steps you do to get a solution.
Implementation of Constructivism-based Learning
After the learning tools are validated, the next stages of research are as follows:
i. Conduct the test of mathematical prior knowledge (MPK).
ii. Implement constructivism-based learning.
iii. Organizing post-tests
iv. Analyzing research data
v. Discuss the results of the research
vi. Conclude
As long as the learning program took place, students are directed to solve
mathematical questions contained in student worksheets (SWS). Each SWS, which consists
of three to four problems, is designed based on the aspects of MHOTS the students must
achieve. The teacher directs students to work cooperatively and collaboratively in groups to
solve them. In general, this is how the learning process takes place in the classroom,
whatever type of constructivism-based learning approach used. Of course, there are syntaxes
differences between one learning approaches to the other, for example in the Open-ended
approach, the questions contained in the SWS are open, that is, have various ways to get the
solution and diverse solutions. In the RME approach, the questions are required to be
c e a , a d i he di c e ea i g a
ach, he e i
e ie e
gh eache
guidance to enable the student in getting the solution. In general, the questions contained in
the SWS are designed based on the MPSA indicators modified from Polya and NCTM as
mentioned in the introduction to this article.
In the control classroom, the teacher implemented direct teaching to the whole class.
In this case, the teacher is considered as an essential role model and is expected to be an
expert or learned figure. A good grasp of the subject matter is more important and serves as
a prerequisite for this kind of pedagogy.

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

Data Analysis
Univariate and multivariate analysis is used as a statistical tool to analyze the
contribution of treatment towards mathematical high order thinking skills (MHOTS)
achievement, while t-Students is used as a statistical tool to determine the significant
improvement of MHOTS (Glass & Hopkins, 1996). All analyses use a 0.05 level of
significance. The role of the learning approach is elaborated through linking MHOTS
achievements with the steps of the learning approach applied in the classroom based on the
output of the regression analysis.
3.

RESULTS AND DISCUSSION

The research took place in six different schools in the provinces of North Sumatra
and Nanggroe Aceh Darussalam continuously from 2017 to 2019. The results of the study
are presented in the following order.
3.1. Mathematical Communication achievement in PBL Classroom
The first study was conducted at public junior high school (PJHS) Muara Batu, Aceh.
The purpose of this study is to improve mathematical communication skills (MCS) as one
of the mathematical high-order thinking skills (MHOTS). For the sake of this matter, we
implemented instructional materials that integrated Acehnese cultural context to problembased learning (PBL). Instructional materials based on PBL is designed so that they meet
valid, practical and effective criteria. Table 1 show the data on mathematical communication
skills at trial I and II as a result of the research.
Table 1. Students MCS achievement at PBL classroom
Category
Highest
Lowest
Average

Trial I
87.5
50.0
74.3

Trial II
95.8
68.8
80.3

Every aspect of average MCS scores in experiments I as well as experiment II is
presented in Table 2. There was an increase in mathematical communication skills after PBL
implementation. This supports the results of previous studies that PBL can improve the
ability of mathematical high-order thinking skills (MHOTS), where mathematical
communication is one of the MHOTS. The implementation of PBL also allows the
development of social skills (Arends, 2012) where one of the benefits of social skills is
increased academic achievement (Minarni, 2013). Whether academic achievements in the
field of mathematics or social fields, this requires separate research.
Table 2. Average score of students MCS at each aspect
Aspect
Explain the idea or situation of an image
in his own words
Describe a situation in image
Describe the situation in mathematical
equation

Trial I

Trial II

10.3

11.2

13.1

13.9

12.2

13.0

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121

Overall, instructional materials based on PBL that integrate Aceh culture have
fulfilled the criteria valid, practical and effective in accordance with the objectives of this
study. The meaning of these criteria is:
a. Validity and Practicality:
1) The average validity of RPP, Student Book, and Students Work Sheets (SWS) given
by five validators is 4.60.
2) In trial I, this instructional material only requires a slight revision. In trial II, the
validators stated that this instructional material was valid.
3) Based on the interview and questionnaire, the teachers and the students stated that
there were no obstacles in using this instructional material.
b. Effectivity:
1) More than 75% of students involved in this study have achieved minimum learning
completeness requirements, namely achieving test scores more than 65 (in
accordance with what was agreed by the ministry of education) (Table 1 and Table
2).
2) Time provided is sufficient for learning implementation.
3) Both the teachers and the students respond positively to the instructional materials.
4) The mathematical communication skills of the students in the experimental
classroom increased with average N-Gain 0.61 (calculated based on Table 1).
These findings show that integrated PBL in the instructional materials affects
significantly the achievement of MCS. The learning approach gives contribution to students
learning outcomes.The study implies that if the teacher has the opportunity to design
appropriate instructional materials based on the constructivism learning approach for
developing MHOTS, then teacher's desire to improve students HOTS will be viable.
Furthermore, Indonesia is a country with rich types of local culture; it should be easier to
enrich the repertoire of cultural-based mathematical knowledge. Another idea conveyed
based on this research is that schools can ask the government to provoke the implementation
of research results such as learning materials developed based on constructivism. In addition,
based on an interview the students give positive responses to the implementation of Problembased learning, so it makes sense that PBL gives contribution to the improvement of students'
MCS.
3.2. Mathematical Understanding Achievement in Cooperative Learning Classroom
The second study was carried out to investigate the effect of cooperative learning
assisted mapping concept and Microsoft Visio (CLMV) towards mathematical
understanding concepts (MUC). The sample consisted of 34 students of eighth-grade AlUlum Islamic Middle Secondary School, Medan, in the Academic Year 2017/2018. CLMV
was used in the experiment classroom, while HOTS to be developed was MUC. Previous
research (Arslan & Altun, 2007) revealed mathematical prior knowledge (MPK) does not
affect the achievement of mathematics learning outcomes, but many other researchers
confirmed that it is influential. MUC test scores from the experimental class and the control
class are shown in Table 3.

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

Table 3. MUC score test of the students
MPK
High
Medium
Low

Learning Approach
CLMV
Conventional
82
56
62
42
63
44

Table 3 shows that there is a difference in the MUC test score between the
experimental class and the control class. This suggests that there is a significant contribution
of CLMV to the achievements of the MUC. Thus, we used ANOVA to test the contribution.
The result of the test is presented in Table 4.
Table 4. Contribution of learning approach to MUC
Source
Corrected Model
Intercept
MPK
Learning Approach
Learning App.*MPK
Error
Total
Corrected Total

Sum of
Squares
8065.8a
108817.9
960.3
4303.3
416.2
11997.0
223107.0
20062.8

Df
5
1
2
1
2
63
69
68

Mean
Square
1613.2
108817.9
480.2
4303.3
208.1
190.4

F

Sig.

8.5
571.4
2.5
22.6
1.1

0.000
0.000
0.088
0.000
0.342

a. R Squared = 0.40 (Adjusted R Squared = 0.36)

The test results in Table 4 interpreted as follows:
a. There is an effect of the learning model on the ability of MUC.
b. The contribution of learning factors to the achievement of MUC is 40%.
c. There is no effect of interaction between the learning approach and MPK factors on
MUC achievement.
d. MUC of the students taught through cooperative learning assisted mapping concepts
and Microsoft Visio software is better than MUC of the students taught through direct
instruction.
Indeed, the involvement of software as part of ICT undeniably gives a positive impact
on student learning outcomes (Agyei & Voogt, 2011). Furthermore, Indonesia is a country
that is responsive to the development of ICT. Almost all PJHS students have smartphones
that can make it easier for them to download software or other applications needed in the
learning process. Therefore, the readiness of teachers is needed to integrate ICT in
mathematics learning.
Besides, the results of the observation indicate that learning in the experimental class
is in line with the stages specified in the cooperative learning approach. Activities to solve
problems in the class that is done cooperatively give results in the form of increasing student
MUC achievements. This is the important role of Cooperative learning in improving
mathematical high order thinking of the students. This research is in line with the theory of
cooperative learning, which states that learning through small groups enables increased

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123

learning achievement because in cooperative learning; tasks are designed so that they meet
intellectually demanding, creative, open-ended, and involve higher-order thinking tasks
(Ross & Smyth, 1995) which allow the growth of MUC as one of MHOTS.
The weakness found in this study is that the teacher is a little excessive in assisting
because some students experience dead ends in solving problems. This needs to get the
attention of policymakers so that teachers do not give up in applying this innovative learning.
3.3. The Achievement of MPSS in Contextual and Cooperative Learning Classroom
Subsequent research was carried out at Medan Budi Agung Middle Secondary
School. Mathematical high order thinking skills (MHOTS) is investigated is mathematical
problem solving skill (MPSS). Through the implementation of Cooperative learning and
Contextual Teaching Learning (CTL) Geogebra-assisted, this study aimed to investigate the
diffe e ce be ee
de
MPSS i C
e a i e ea i g c a
(E e i e a c a
I) and contextual classroom (Experimental class II). In both experimental classes, learning
was implemented with the help of Geogebra software. Student MPSS test scores are
presented in Table 5.
Table 5. Statistic of students MPSS score
Learning Approach
Cooperative
CTL

Statistic
Average
SD
52.79
17.059
53.45
15.999

Table 5 shows that there is a difference in the MPSS score test between the two
experimental classrooms. Geogebra-assisted contextual learning (CTL-G) is superior in
improving student MPSS compared to Geogebra-assisted cooperative learning. The
difference in MPSS achievements shows that there is an influence or contribution of the
learning approach to the MPSS. Thus, we do the test of difference achievement of the MPSS
through two-way ANOVA at 0.05 significance level. The test result is presented in Table 6.
Table 6. Test of the Effect of Learning Approach to MPSS
Source
Corrected Model
Intercept
MPK
Learning Approach
Learning App. *
MPK
Error
Total
Corrected Total

Sum of
Squares
1206.4a
326446.6
178.2
571.4

F

Sig.

3
1
1
1

Mean
Square
402.1
326446.6
178.2
571.4

4.2
3.4E3
1.9
5.9

0.009
0.000
0.177
0.018

402.2

1

402.2

4.2

0.045

5340.2
332448.0
6546.6

56
60
59

95.4

a. R Squared = 0.18 (Adjusted R Squared = 0.14)

Df

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

Based on data in Table 6, it can be concluded that the test result is significant, this
means that:
a. There is the MPSS difference between students taught by Geogebra-assisted cooperative
learning and students taught Geogebra-assisted CTL. This means, there is an effect of
ea i g fac
he
de
MPSS achie e e . The effec i
ea ed b he
degree of contribution. The contribution of learning factors to MPSS achievements is
around 18%. Both Geogebra-assisted CTL and Geogebra-assisted Cooperative learning
plays a substantial role in achieving MPSS.
b. There is an interaction effect between the learning approach and MPK factors on MPSS
achievement. It means the students from low and medium MPK get benefit from this
kind of learning approach.
The advantages of contextual learning that make it possible to play an important role
in improving the MPSS of the students are characteristics of problems that are designed to
connect with the context of students' daily lives. Based on interviews, students acknowledge
that the problems given by the teacher are quite interesting and easier to understand because
they are familiar with the theme of the problem. Understanding the problem is the first and
foremost thing in solving problems and according to the CTL theory, making learning
meaningful to students by connecting to the real world is the core element in CTL (Johnson,
2002).
Meanwhile, the weakness of this study mainly lies in the weakness of student MPK,
in line with the results of other studies (Minarni et al., 2016), such that the teacher is forced
to remind students of mathematical knowledge that is not well stored in the cognitive
structure of students. Based on the results of this study, it is suggestions that:
a. The teacher is advised to use Cooperative learning and CTL to enhance student
achievement in mathematical problem solving skills.
b. In implementing constructivism-based learning such as Cooperative learning and CTL,
the teacher is advised to involve information and computer technology (ICT) such as
Ge geb a f a e, e ecia f ge e a i g de
i ee i
d i g ge e .
Because through the help of the software, the display of geometric forms can be
visualized more accurately and more 'eye-catching' which increases students'
enthusiasm for learning and challenging them to explore other problems related to
geometry problems. From this activity, it is hoped that the student's perseverance and
life-long learning will be grown.
c. The teacher is advised to strengthen students' comprehension of mathematical
knowledge and mathematical concepts, as well as MPK.
If MPK becomes an obstacle in achieving MPSS then the implementation of
innovative learning such as CTL and cooperative learning becomes increasingly important
because the main advantage provided by these two constructivism-based learning is that
learners will be able to store knowledge in long-term memory to guarantee the availability
of MPK. This again shows that self-constructed knowledge can make a person firm in storing
his knowledge. The following explanation is an example of the problem used in research and
alternative solutions.

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125

Problem:
"A cube-shaped aquarium with a length of 85 cm is filled with water. If a decorative stone
with a volume of 125 litres is put into the tank, determine the volume of water left in the
tank."
This problem is closely related to other disciplines, namely Physics. Moreover,
this question is also related to the context of students' daily life where students are very
familiar with aquariums as a container for keeping fish that require ornamental stones to
mimic original fish habitat.
Solution:
𝑉

𝑉

s × s × s cm
85 × 85 × 85 cm
614.125 cm
614.125 dm
614.125 liter
𝑉 –𝑉

614.125 – 125.000
489.125 liter.

To solve the problem, students should execute four steps, i.e.:
a. Calculate the initial volume of water in the aquarium.
b. Convert water volume units (from cm3 to liters) to equal the volume units of ornamental
stones.
c. Find the reduction of the initial volume because of the insistence of ornamental stones.
d. Conclude the volume of water left in the aquarium after ornamental stones press some
water out.
Thus, this mathematical problem has characteristics as a good question, that is
contextual, interesting, related to other disciplines, and requires multi-step to get a solution.
All of these characteristics are in line with HOTS proposed by Resnick (1987).
3.4. MPSS Achievement in Realistic Mathematics Education Classroom
The fourth study was carried out at PJHS 2 Beringin, District of Deli Serdang. This
research is an effort to improve students' mathematical problem solving skills (MPSS)
through a realistic mathematical education (RME) approach assisted by Autograph. The
MPSS score test of the students is presented in Table 7.
Table 7. Statistic of students MPSS
Learning Approach
RME
Conventional

MPK
High
Medium
Low
High
Medium
Low

Statistic
Average
SD
19.02
2.452
18.02
2,706
17.20
2,680
17.03
1.150
15.90
2.850
12.80
2.030

Data from the research was analyzed through two-way ANOVA. The result of the
analysis is presented in Table 8. It can be seen in Table 8 that there are differences in MPSS
scores between students in the experimental class and students in the control class. The

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

difference in achievement is quite large, this means that there is an influence or contribution
of the treatment to MPSS achievements (Glass & Hopkins, 1996). The significance level test
of the contributions is carried out through the Analysis of Variance. The output of the test is
displayed in Table 8.
Table 8. The Contribution of RME towards MPSS
Source
Corrected Model
Intercept
Learning Approach b)
MPK
Learning App. *MPK
Error
Total
Corrected Total

Sum of Squares
175.9a)
12886.6
59.5
98.5
2.2
625.9
18292.0
801.9

Df
5
1
1
2
2
58
64
63

Mean Square
35.2
12886.6
59.5
49.3
1.1
10.8

F
3.3
1193.9
5.5
4.6
0.1

Sig.
0.012
0.000
0.022
0.014
0.902

a) R squared = 0.22 (Adjusted R Squared = 0.15)
b) Learning Approach: RME

Based on the result of the analysis in Table 8, the research findings are:
a. The enhancement of the student MPSS in the experiment classroom is higher than the
in the conventional classroom.
b. The learning factor has a significant influence on the achievement of MPSS.
c. The adjusted R squared is 0.15. This means the contribution of the learning approach to
MPSS is 22%.
d. There is no interaction effect between the learning approach and MPK factor on the
MPSS achievement.
e. The process of solving mathematical problems shown by students in the experimental
classroom is better (in terms of more MPSS indicators that are met, systematic and
directed).
In this study, the teacher has implement RME properly, that is, the learning is
conducted so that the students construct knowledge by themselves, in line with the socioconstructivism as one principle of RME (de Lange, 1996; Gravemeijer, 1994) and students
are offered opportunities to share their experiences with others. Besides, de Lange (1996)
stated that the compatibilities of socio-constructivist and RME are based on a large part or
similar characterizations of mathematics and mathematics learning because they are
struggling with the idea that mathematics is a creative human activity and mathematical
learning occurs as students develop effective ways to solve problems (Streefland, 1991;
Treffers, 1991). So, if this research is not successful enough in developing student MPSS, it
is shown by the low MPSS score and the low contribution of RME to MPSS achievements
(only 15%), then that becomes a problem that we must think of a solution.
The most striking obstacle in this study is the weakness of students in representing
problems into various forms of representation such that they have trouble at the stage of
de f
a he a ica h i
a
age. A h gh
de
ha e
b e i he
h i
a a he a ica age a d e ica a he a ica i he RME,
he
de f a d
de f
age ake de fee he ed i
i g
b e . I ee
ha hi gives a
role in increasing student MPSS.

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127

One thing could be suggested from the study is time allocation. As in other studies,
the time available is always insufficient to conduct the learning process that aims in
enhancing high-order thinking skills. For this reason, it is recommended that the teacher
prevent the debate or a prolonged argument among students. The teacher must immediately
take over and decide firmly which correct solution is for a certain problem, and which
solution still have shortcomings or mistakes.
3.5. Mathematical Creative Thinking Achievement in Open-Ended Classroom
The fifth study took place in the Public Middle Secondary School Number 2 at
Padangsidempuan. The constructivism-based learning applied here is the Open-ended
approach. The research objectives are:
a. To investigate the influence of the Open-ended approach integrated Batak Angkola
culture (OEBC) towards students' mathematical creative thinking skills (MCTS).
b. To investigate the effect of the interactions on MCTS.
The average score of MCTS of the students is presented in Table 9.
Table 9. Statistic of students MCTS
Learning
Approach
OEBC

Conventional

MPK
Low
Medium
High
Low
Medium
High

Statistic
Average
SD
61.99
8.39
69,70
16.30
90.70
7.00
32.30
5.40
48.70
13.00
68.15
8.40

Data from the research was analyzed through two-way ANOVA. The ANOVA
output is presented in Table 10.
Table 10. Contribution of open-ended approach towards MCTS
Source
Corrected Model
Intercept
MPK
Learning App.
Learning App.*MPK
Error
Total

Sum of
Squares
15436.534a
195925.534
9121.630
5829.018
499.494
11665.216
258944.000

Df
5
1
2
1
2
58
64

Mean
Square
3087.307
195925.534
4560.815
5829.018
249.747
201.124

F

Sig.

15.350
974.151
22.677
28.982
1.242

0.000
0.000
0.019
0.012
0.296

a) R Squared = 0.57 (Adjusted R Squared = 0.53)

The results of the study indicate that: (a) There is an influence of the Open-ended
approach integrated Batak Angkola culture towards the achievement of students'
mathematical creative thinking skills (MCTS); (b) R Squared = 0.57. This means the

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

contribution of learning factor to the achievement of MCTS is 57%; (c) There is no common
influence (interaction) between the Open-ended approach integrated Batak Angkola culture
and MPK towards the achievement of students' mathematical creative thinking skills.
The MCTS score test of the students at the experiment classroom is high enough, so
does the contribution of learning approach towards MCTS. It is possible since the Openended approach implemented in the classroom provides students with experience in finding
something new in the process of open problem solving (Becker & Shimada, 1997), while
open problem solving is based on open-ended problems. These characteristics enable the
students to participate more actively in lessons and express their ideas more frequently; have
opportunities to get a unique answer; develop curiosity about other solutions and they can
compare on and discuss their solutions; make comprehensive use of their mathematical
knowledge and skills. Since there are many different solutions, students can choose their
favorite ways toward the answer and create their unique solution. The study also shows that
the students involved in the classroom activities and enable the students to have reasoning
experience and build intrinsic motivation to give reasons for their solutions to other students.
It is a great opportunity for students to develop their mathematical thinking. All of these
characteristics meet the demands of the Open-ended approach.
The weakness of this study is the students are less courageous in conveying ideas.
This may be due to Indonesian culture that children are generally educated not to argue with
their parents or other family members. They are usually educated to obedient Children.
Overall, based on the results of this study, the following suggestions are offered.
a. The teacher is suggested to be creative in creating a learning atmosphere that allows
students to express mathematical ideas in their language such that the students have selfconfidence, creativity, and courage to argue with their classmates.
b. The teacher is suggested to provide a variety of mathematical problems that are in line
with the context of the local culture and lure students to relate them to the subject matter
or other mathematical problems. If this is done, it will build students' perception that
mathematics is useful in their daily lives.
c. The teacher should allocate a more accurate time so that this constructivist-based
learning activity can run smoothly.
Preparation of discussion groups is only carried out once at the beginning of learning
such that there is more time available for group discussion activities in the next session.
3.6. MPSS Achievement in Discovery Learning Classroom
The fi a e ea ch a c d c ed de e
de
achie e e i Ma he a ica
Problem Solving Skills (MPSS) through the implementation of instructional materials based
on discovery learning approach. There are 40 students included in the experiment class and
40 students in the control class. All of the students are from Public Junior High School
(PJHS) 17 Medan. An essay test is used to c ec da a f he
de
MPSS. S de '
MPSS score test is presented in Table 11.
Table 11. Statistic of students MPSS
Learning Approach
Discovery
Conventional
Note: Ideal score = 20

Statistic
Average
SD
15.03
2.282
10.33
1.493

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129

Based on Table 11, the MPSS achievement of students in the experimental class
(Discovery learning class) is better than the conventional class. This shows that there is an
influence/contribution of learning to student MPSS achievements. The results of this study
support the theory that through the discovery learning approach, students' trust in efforts to
solve problems increases because they are accustomed to conducting investigations to find
information needed to solve problems. According to Dewey's opinion in finding such
knowledge a person unknowingly stores information in ways that make information easier
to use in solving new problems. (Arends, 2012). Thus, the achievement of problem solving
skills become significant. The time limitation to implement discovery learning is a major
obstacle in the completion of complete learning. The enhancement of MPSS performance
was presented in Table 12.
Table 12. The enhancement of MPSS
Eq. var.

F
0.003

T
3.557

Df
78

Sig.
0.001

Mean Diff.
3.800

Lower
1.673

Upper
5.927

This significant increase in MPSS encouraged researchers to statistically test the
contribution of discovery learning to MPSS achievements. Test results are presented in Table
13.
Table 13. Test of learning approach effect towards MPSS
Source
1. Corrected model
2. Intercept
3. App
4. Error
5. Total

Type III Sum of Squares
308.112
11640.313
308.112
1210.575
13159.000

Df
1
1
1
78
80

Mean Square
F
308.112
19.852
11640.313 750.011
308.112
19.852
15.520

Sig.
0.000
0.000
0.000

R Squared = 0.203

Results of the research: (a) The e ha ce e
f de
MPSS i ig ifica
ih
a mean difference of 3.80 (MPSS ideal total score was 20) (Table 12); (b) The contribution
of the developed instructional materials based on discovery learning towards MPSS is 20.3%
(Table 13).
The results of this study indicate a high increase in MPSS (3.8 points), but this
research found that the increase is not due to the contribution of the learning approach
because of its small contribution, which is only 20%. There may be other factors contribute
more significantly to students' mathematical problem solving abilities. It is common in the
world of education that there are indeed many factors that contribute to student learning
outcomes in matthematics, including teacher factors, school environment, friends, and
affective aspects such as mathematical disposition, social skills, self-confidence, selfregulated learning, and others (Minarni et al., 2018), and others. Finding out the dominant
factors contribute to learning outcomes is the attention and interest of educational
researchers.
4.

CONCLUSION

Based on the result of the research then the conclusion are, firstly, constructivismbased learning can improve mathematical high order thinking skills (MHOTS) such as
mathematical connection, mathematical understanding, mathematical problem solving, and
mathematical creative thinking skills/ability. In the experimental classroom, students'

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

MHOTS increased significantly. The contribution of constructivism-based learning to
MHOTS is in the range of 18% to 57%. Secondly, based on the results of observations made
by the observer, the activity of students in the learning process increases significantly. Third,
in some cases, there is an influence of interaction between the learning approach and
students' mathematical prior knowledge towards the achievement of MHOTS. Fourth, based
on observations and interview results, the integration of ICT to the learning approach
increases students' enthusiasm in solving mathematical problems. Maybe there are other
factors besides learning that contributes more to higher-order mathematical thinking skills
(MHOTS), for example, affective factors such as mathematical disposition, social skills, and
learning motivation. To approve this allegation, of course, requires special research.
Some suggestions can be drawn from the study are (1) We suggest that the teacher
dares to implement constructivism-based teaching-learning approach to improve HOTS; (2)
The teachers are advised to have the willingness to integrate local culture in the learning
ce
i
e de
i e e i olving mathematical problems; (3) The teachers are
advised to integrate ICT in explaining subject matter and describing the solution the student
gets so that the explanation is easier for students to understand.
REFERENCES
Agyei, D. D., & Voogt, J. (2011). ICT use in the teaching of mathematics: Implications for
professional development of pre-service teachers in Ghana. Education and
information technologies, 16(4), 423-439. https://doi.org/10.1007/s10639-0109141-9
Anderson, L. W., Krathwohl, D. R., Airasian, P. W., Cruikshank, K. A., Mayer, R. E.,
Pintrich, P. R., & Raths, R. (2001). M. C Wittrock. A Taxonomy for Learning,
Teaching and Assessing. New York: David McKay Company.
Arends, R. I. (2012). Learning to teach. McGraw-Hill Companies.
Arslan, Ç., & Altun, M. (2007). Learning to solve non-routine mathematical
problems. Elementary Education Online, 6(1), 50-61.
Becker, J. P., & Shimada, S. (1997). The Open-Ended Approach: A New Proposal for
Teaching Mathematics. National Council of Teachers of Mathematics, 1906
Association Drive, Reston, VA 20191-1593.
Brown, A. L. (1990). D ai
ecific
i ci e
children. Cognitive science, 14(1), 107-133.

affec

ea i g a d

a fe

i

Bruner, J. S. (1961). The act of discovery. Harvard educational review, 31, 21-32.
Cottrell, S. (2005). Critical Thinking Skills: Developing Effective Analysis and
Argument. New York: Palgrave Macmillan.
De Lange, J. (1996). Using and applying mathematics in education. In International
handbook
of
mathematics
education,
49-97.
Springer,
Dordrecht.
https://doi.org/10.1007/978-94-009-1465-0_3
Freudenthal, H. (2006). Revisiting mathematics education: China lectures (Vol. 9). Springer
Science & Business Media.
Formaggia, L. (2017). Mathematics and Industry 4.0. International CAE Conference,
Vicenza, November 6-7, 2017.

Glass,

G., & Hopkins, K. (1996).
psychology. Psyccritiques, 41(12).

Volume 9, No 1, February 2020, pp. 111-132

131

Statistical

and

methods

in

education

Gravemeijer, K. (1994). Developing Realistic Mathematics Education. Utrecht: Freudenthal
Institute.
Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics
education: A calculus course as an example. Educational studies in
mathematics, 39(1-3), 111-129. https://doi.org/10.1023/A:1003749919816
Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students
learn?. Educational
psychology
review, 16(3),
235-266.
https://doi.org/10.1023/B:EDPR.0000034022.16470.f3
Jensen, I., R. (1976). Five-point Progress for the Gifted. Poster Presentation. International
Congress on Mathematical Education, Karlsruhe, Germany.
Johnson, E. B. (2002). Contextual teaching and learning: What it is and why it's here to stay.
California: Corwin Press.
Kulm, G. (1990). Assessing Higher Order Thinking in Mathematics. American Association
for the Advancement of Science Books.
Maharani, H. R. (2014). Creative thinking in mathematics: Are we able to solve
mathematical problems in a variety of way. In International Conference on
Mathematics, Science, and Education, 120-125.
Marzano, R. J., & Kendall, J. S. (2007). The new taxonomy of educational objectives. Corwin
Press.
Minarni, A. (2013). Pengaruh pembelajaran berbasis masalah terhadap kemampuan
pemahaman matematis dan keterampilan sosial siswa SMP Negeri di Kota
Bandung. Jurnal Paradikma, 6(02), 162-174.
Minarni, A., Napitupulu, E., & Husein, R. (2016). Mathematical understanding and
representation ability of public junior high school in north sumatra. Journal on
Mathematics Education, 7(1), 43-56. https://doi.org/10.22342/jme.7.1.2816.43-56
Minarni, A. (2017). On Eight Grade Students Understanding in Solving Mathematical
Problems. Asian
Social
Science, 13(12),
86-96.
https://doi.org/10.5539/ass.v13n12p86
Minarni, A., Napitupulu, E. E., Lubis, S. D., & Annajmi (2018). Kemampuan Berpikir
Matematis dan Aspek Afektif Siswa. Medan: Harapan Cerdas Publisher.
NCTM (2000). Principle and Standards for School Mathematics. Reston: VA.
Phillips, D. C. (1997). How, why, what, when, and where: Perspectives on constructivism in
psychology and education. Issues in Education, 3(2), 151-194.
Polya, G. (2004). How to solve it: A new aspect of mathematical method (No. 246).
Princeton university press.
Reid, G. (2007). Motivating learners in the classroom: ideas and strategies. Sage.
Resnick, L. B. (1987). Education and learning to think. Washington DC: National
Academies.
Ronis, D. L. (2008). Problem-based learning for math & science: Integrating inquiry and
the internet. Corwin Press.

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Minarni, & Napitupulu, The role of constructivism-based learning in improving mathematical …

Ross, J. A., & Smyth, E. (1995). Differentiating cooperative learning to meet the needs of
gifted learners: A case for transformational leadership. Talents and Gifts, 19(1), 6382.
Savin-Baden, M., & Major, C. H. (2004). Foundations of problem-based learning. McGrawHill Education (UK).
Sawada, T. (1997). Developing Lesson Plans. In: J. Becker, & S. Shimada (Eds.). The OpenEnded Approach: A New Proposal for Teaching Mathematics (pp. 1-9). Reston, VA:
National Council of Teachers of Mathematics.
Scriven, M., & Paul, R. (1987). Defining critical thinking. In 8th Annual International
Conference on Critical Thinking and Education Reform, Summer.
Sheffield, L. J. (2013). Creativity and school mathematics: Some modest
observations. ZDM, 45(2), 325-332. https://doi.org/10.1007/s11858-013-0484-8
Silver, E. A. (2013). Teaching and learning mathematical problem solving: Multiple
research perspectives. Routledge.
Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of
developmental research (Vol. 8). Springer Science & Business Media.
Torrance, E., P. (1960). Educational Achievement of the Highly Intelligent and Highly
Creative: Eight Partial Replications of the Getzels and Jackson Study. Research
Memorandum. BER 60-68, Bureau of Educational Research: University of
Minnesota.
Treffers, A. (1991). Realistic mathematics education in the Netherlands 19801990. Realistic mathematics education in primary school, 11-20.
Widodo, A. (2004). Constructivist oriented lessons: The learning environments and the
teaching sequences. Frankfurt: Peter Lang GmbH.
Vygotsky, L. S. (1980). Mind in society: The development of higher psychological
processes. Harvard university press.