Beta: Jurnal Tadris Matematika, 16(2) 2023: 151-171

DOI 10.20414/betajtm.v16i2.567

Research articles

Student’s ways of thinking and ways of understanding analysis in solving

mathematics problems in term of adversity quotient
Siti Lailiyah1, Fitria Anis Kurlillah1, Kusaeri1

Abstrak Terdapat dua kategori keterampilan berpikir yang saling mempengaruhi dalam pengetahuan
matematika yaitu proses berpikir yang disebut Ways of Thinking (WoT), dan cara pemahaman siswa
yang disebut dengan Ways of Understanding (WoU). Penelitian ini bertujuan untuk mendeskripsikan
Ways of Thinking (WoT) dan Ways of Understanding (WoU) dalam menyelesaikan masalah
matematika ditinjau dari Adversity Quotient (AQ). Penelitian ini adalah deskriptif kualitatif. Subjek
dipilih berdasarkan hasil tes Adversity Respons Profile menggunakan teknik purposive sampling,
sehingga didapatkan 3 siswa climbers, 3 campers dan 2 quitters di MTs Muhammadiyah 1 Taman di
Sidoarjo. Data tes tulis dan wawancara dianalisis sesuai indikator WoT dan WoU. Hasil penelitian
ini menunjukkan bahwa WoT siswa climber dalam menyelesaikan masalah matematika cenderung
memiliki satu strategi yang mengarah pada solusi benar, cara berpikir empiris, dan memiliki
keyakinan yang sangat baik terhadap konsep matematika. Sedangkan WoT siswa camper dan quitter
cenderung memiliki satu strategi yang mengarah pada solusi salah, dan cara berpikir out of the box.
Siswa camper memiliki keyakinan yang baik, sedangkan siswa quitter memiliki keyakinan yang
kurang terhadap konsep matematika. WoU siswa climber berkategori sangat baik, siswa camper
berkategori cukup. dan siswa quitter berkategori kurang. WoT dan WoU siswa climber lebih baik
dibandingkan siswa camper dan quitter.

Kata kunci Cara berpikir, Cara memahami, Adversity quotient

Abstract There are two categories of thinking skills which influence Mathematical knowledge called
Ways of Thinking (WoT) and Ways of Understanding (WoU). This study aims to describe students'
WoT and WoU in solving mathematics problems in terms of Adversity Quotient (AQ). Descriptive
Qualitative was implied to this study. The subjects were selected based on the results of the Adversity
Response Profile test using a purposive sampling technique. As a result, there are 3 climbers, 3
campers, and 2 quitter students from MTS in Sidoarjo. The data collection technique was gained
from written test and interview section which were analyzed based on WoT and WoU indicators.
The results of this study indicated that WoT of climbers’ students tended to have one strategy that
led to the correct solution, an empirical WoT, and a good belief in mathematical concepts.
Meanwhile, WoT of campers and quitters’ students tended to have one strategy that led to the wrong
solutions and they had beyond belief WoT. Further, campers’ students had good confidence while
quitter students have less confidence in mathematical concepts. WoU of climbers’ students were
good, campers’ students were enough, and quitters’ students were less. In conclusion, WoT and WoU
of climbers’ students were better than the camper and quitter students.

Keywords Ways of thinking, Ways of understanding, Adversity quotient

1
Department of Mathematics Education, Faculty of Teacher Training and Education, Universitas Islam Negeri Sunan
Ampel, Jl. Ahmad Yani 117 Surabaya, Indonesia, lailiyah@uinsby.ac.id

© Author(s), licensed under CC-BY-NC

Lailiyah et al.

Introduction
In the 21st century, humans are required not only to master technology but also to handle
and solve problems with a strong and resilient attitude in managing new ideas and being
responsive to changes (Fauziansyah et al., 2013; Marques, 2012; Sanabria & Arámburo-
Lizárraga, 2017). It leads to the idea of critical thinking. This critical thinking skills can be
formed and developed through mathematics (Kusaeri et al., 2022; Murawski, 2014). In learning
mathematics, students need to do various exercises. When they are accustomed to doing the
exercises on Math, they will realize that Math requires lots of practices to sharpen their abilities
in solving various mathematical problems.
Previous studies show that the level of mathematical thinking ability, especially geometric
thinking, for junior high school students in solving mathematical problems is in a low level
(Junining et al., 2022; Kurniati et al., 2016; Ma’rifah et al., 2019; Megawati et al., 2019; Rabu
& Badlishah, 2020). Likewise, there is a research on students' difficulties in solving geometry
problems (Arifendi & Wijaya, 2018; Fauzi & Arisetyawan, 2020; Indrayany & Lestari, 2019;
Maryanih et al., 2018; Sholihah & Afriansyah, 2017). One of the causes of students' lower
thinking ability lies in the students' mindset which only focused on one solution without trying,
analyzing and finding new ways and tended to use the same way or formula of solving problems
as what has been given by the teacher. This happens because the mathematics learning process
in class generally emphasizes students' mastery of calculation formulas rather than emphasize
students' thinking process abilities (Nurhasanah, 2019).
Learning mathematics is in line with problem-solving and it requires a problem (Wu, 2017).
According to Mujib (2015), solving mathematical problems get the students to use reason and
think creatively, so the problems created must be challenging which directs students to combine
all known concepts related to the problems they face and form a new concept so that the problems
given can be solved. Thus, learning mathematics can be used as an exercise for students in
building and developing their thinking skills (Toker & Baturay, 2021; Widyatiningtyas et al.,
2015). In addition, there are two categories of their thinking skills that influence each other in
mathematical knowledge which are a thinking process called Ways of Thinking (WoT) and
Ways of Understanding (WoU).
Several studies on WoT and WoU have been reviewed in recent years. Nurhasanah et al.,
(2021) examined the characteristics of WoT that are interconnected with students’ WoU in
vector material. Further, in her following research regarding the implementation of WoT,
students who have high, medium, and low cognitive abilities in geometry material can be
reference in developing mathematics teaching materials (Nirawati et al., 2022). As a result, the
difference between this study and previous ones is the material used which are lines and angles,
as well as the selection of subjects based on the adversity quotient. Therefore, this study aims to
describe and identify the WoT and WoU types of climbers, camper, and quitter students in
solving problems on lines and angles.

Prior research
According to Nurhasanah (2019), students had difficulty with the geometry because teacher
generally get students to master the calculation formula instead of having the ability of thinking
process in learning mathematics during the class. Sholihah & Afriansyah (2017) study also
shows that the achievement of students in the process of solving geometric problems based on
Van Hiele's thinking stages is in the stage of 0 (visualization) as much as 96.87%, while stage 1
(analysis) is 3.13%, stage 2 (informal deduction) and stage 3 (deduction) does not exist. The
same thing is done by Moses (2016), he states that the Van Hielle geometric thinking level of
class VII students generally only reaches level 1 which means that students can recognize shapes
based on their properties. According to Fauzi & Arisetyawan (2020), students experience
difficulties in basic geometry such as: (1) students have difficulty using concepts; (2) students

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get difficulty using principles; and (3) students have difficulty solving verbal problems. These
show that the thinking skills of Indonesian students in solving geometric problems are still
relatively low (Basri et al., 2019; Sandy et al., 2019).
Students had difficulty with the geometry of the cube and block material, and dynamic
geometry (Lingefjärd et al., 2012; Maryanih et al., 2018). In their research, it was also found
that this study also provides alternative solutions to students' learning difficulties, such as:
(a) Using computer applications or software (PowerPoint, Microsoft Word with SmartArt
Graphic) and software such as Cabri Geometry, The Geometer's Sketchpad (GSP),
Geometry Expert, Logo, GeoGebra, Wingeom, and Maple (Alabdulaziz et al., 2021;
Kilicman et al., 2010); (b) Activating prerequisite material about flat shapes that explain the
sides of the shapes; (c) Applying the guided discovery method using guided worksheets; and
(d) Doing more practices both contextual and non-contextual questions. By doing those
activities, hopefully teachers could design a various learning process which encourage
students to develop their thinking abilities (Chusni et al., 2021; Toker & Baturay, 2021).

Theoretical review
Ways of thinking (WoT) and ways of understanding (WoU)
One theory that can be the reference by teachers to see the process of learning mathematics
is Harel's theory in the principle of Duality, which explains that two categories of knowledge
influence each other in mathematical knowledge. They are thinking process called as Ways of
Thinking (WoT) and students understanding’s way which is called as Ways of Understanding
(WoU) (Harel, 2008b). WoU is a collection of structures consisting of certain axioms,
definitions, theorems, proofs, problems, and solutions. While WoT is all ways of thinking. It is
a characteristic of a mental act performed by a person, meanwhile the product is all ways of
understanding or WoU.
Students' WoT is divided into three interrelated actions, namely problem-solving
approaches, proof schemes, and beliefs in mathematics (Harel, 2008a). The problem-solving
approach is a way of thinking related to problem actions and is often referred to as a heuristic.
One of the well-known heuristic models is Krulick and Rudnick which is divided into several
stages of problem-solving (Krulick & Rudnick, 1996). These stages are: read and think, explore
and plan, select a strategy, find and answer, reflect and extend (Kusdinar et al., 2017).
The following action is the scheme of its evidence and a discussion of it is needed to find
out someone's way of thinking (Koichu et al., 2013). This evidence is implemented by someone
to confirm himself or to convince others that the statement is true, while the evidence scheme is
the collective cognitive characteristic of the proof produced by someone. The first proof scheme
consists of extra-belief proof schemes which consist of ritual and authority schemes. The second
is empirical which consists of proceptual and inductive schemes. The third is a deductive proof
scheme consisting of transformational and axiomatic proof schemes. From the evidence scheme
above, the empirical evidence scheme is a common and strong way of thinking among many
students. This is because students tend to do something based on the understanding on the
experiments results or examples given by the teacher.
In addition to the categories of problem-solving and proof schemes, the category of belief
in mathematics is also essential, but the result of teachers’ belief in Indonesia is low (Kusaeri &
Aditomo, 2019; Muhtarom et al., 2017). Beliefs about mathematics are one's view of
mathematics itself (Harel, 2021). Confidence is categorized into two, which are a belief in
learning mathematics and a belief in the problem-solving process (Muhtarom et al., 2017).
According to Nurhasanah's research, beliefs about mathematics show the extent to which
students are aware of the use and relation between concepts they have and problem-solving,
know the effectiveness of the methods/concepts chosen, and know the advantages of the many

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interpretations of concepts that are carried out (Nurhasanah, 2019). Furthermore, there is also
WoU which is a product of a way of thinking. It needs to be highlighted that mathematical
understanding is not just remembering concepts or following procedures, yet the results of
solving mathematical question is the main point. Thus, students are expected to deeply
understand the basic of mathematics, from facts to mathematical proofs.

Adversity quotient (AQ)

Ways of thinking and ways of understanding mathematics are closely related to
psychological factors, one of which is the personality that shows one's character. Everyone who
has a different character must have a different way of thinking (Zhao et al., 2021). Adversity
Quotient (AQ) is considered to play a role in students' thinking processes when completing
mathematics. According to Stoltz (2005), he defines AQ into three forms, namely (i) AQ is a
new conceptual framework for understanding and improving all aspects of success, (ii) AQ is a
measurement for knowing a person's response to difficulties, and (iii) AQ is a series of scientific
means to improve a person's response to adversity. Apart from that, AQ is defined as an ability
that exists within a person to overcome and process difficulties using the intelligence they have
so that it becomes a challenge to be resolved (Mustika et al., 2018). Therefore, from the
explanation above, researchers conclude that AQ is a person's effort to overcome the difficulties
they are experiencing.
AQ is categorized into three categories, which are climber, camper, and quitter (Hasanuddin
& Lutfianto, 2018). Quitters are a group of people who are lack of the desire to accept challenges
in their life. Whereas a camper is a group of people who already have the desire to try and face
existing problems and challenges, but they stop because they feel they can't stand to it anymore,
while climbers are a group of people who choose to continue to survive and struggle to face all
the problems, obstacles that hit them. To find out more about the characteristics of the quitter,
camper, and climber categories, see Table 1 below.

Table 1. Quitter, camper, and climber profiles

Profiles Characteristic
Quitter a. Refusing to climb any higher
b. Unpleasant lifestyle
c. Working is just enough for survival
d. Tends to shy away from tough challenges
e. Rarely have true friendships
Camper a. Keep climbing until you feel enough and stop at that place.
b. At a certain stage they feel satisfied
c. Still have initiative and a little enthusiasm to try.
d. Tends to build good relationships with other campers.
Climber a. They will continue to climb by thinking about the possibilities.
b. Their lives feel "complete" because they appreciate the small amount
of time and effort they have had.
c. Have high motivation and enthusiasm to continue to strive for the
best.
d. Not afraid to explore unlimited potential, willing to take high risks
and accept criticism.
e. Willing to accept any changes that push them in a positive direction.

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Methods
This type of research is qualitative descriptive research. The descriptive methods were used
to describe the students’ WoT and WoU in solving mathematical problems. This research was
conducted at MTs Muhammadiyah 1 Taman in Sidoarjo Regency, East Java in October 2022 for
5 days. The reason for choosing this school as a research location was because the students'
conditions were heterogeneous and this school was one of the best schools in Sidoarjo Regency.
This research procedure consists of 3 stages, namely the preparation stage, the implementation
stage, and the final stage. In the preparatory stage, the researcher compiled the instrument,
validated the instrument, made a research permittion letter to the research location school, asked
for permission, and made an agreement with the teacher in conducting the research. At the
implementation stage, the researcher gave ARP tests to all students via Google Forms. This ARP
test is to find out a person's AQ score. After being analyzed, it is then classified based on the
categories in Table 1 and selected several research subjects. In the next stage, this research
subject solves mathematics questions to explore the students' WoT and WoU. The next step is
each student was interviewed to explore more about the WoT and WoU processes. In the final
stage, the researcher analyzed the written test data and interview data.
Moreover, the subject-taking technique was used purposive sampling in which students
were selected including the Adversity Quotient (AQ) types (climbers, campers, and quitters).
The steps for taking the subject began with giving the Adversity Response Profiles (ARP) test
to 28 students in class VIII-A and 26 students in class VIII-B via Google form. The ARP used
in this study is adoption from (Indrawati, 2019). The ARP used was not validated again because
it had previously been validated and used by Indrawati. Then the ARP results which had been
filled in by 54 students were analyzed by giving a score and determining their AQ category. The
following AQ scores and categories are presented in Table 2.

Table 2. AQ category (Damayanti et al., 2020)

AQ category ARP value

Climber 166 - 200
The transition from camper to climber 135-165
Camper 95 - 134
The transition from quitter to camper 60 - 94
Quitter 0 - 59

After the ARP were analyzed, it was found that 37 students transition from camper to
climber (20 students from class VIII A and 17 students from class VIIIB), 5 climbers (2 students
from class VIII A and 3 students from class VIII B), 3 students of the transition from quitter to
camper (3 students from class VIIIA), 7 camper students (3 from class VIII A and 4 students
from class VIII B), 2 quitter students (2 students from class VIII B). The subjects in this study
were based on the results of the ARP and the recommendations of the mathematics teacher based
on representatives of the AQ score levels starting from low, medium, and high of AQ. As a
result, 8 AQ students were chosen as the subjects which consisting of 3 students in the climber
category, 3 students in the camper category, and 2 students in the quitter category. In the quitter
category, two students were selected because there were only two students who were included
in this part as the data above. The list of selected research subjects is presented in Table 3 below.

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Table 3. List of selected research subjects

No. Name Initials Total ARP Score Category Subject Code

1. DLP 184 climbers SC1
2. ANR 174 climbers SC2
3. IN 166 climbers SC3
4. NDH 126 campers SP1
5. MFN 106 campers SP2
6. IFS 104 campers SP3
7. DD 52 quitters SQ1
8. IDS 56 quitters SQ2

Data collection began with the subject being given a material problem or questions of lines
and angles, then students were asked to solve the exercise. The implementation of this test was
carried out offline at school. Then the interview process was carried out after completing the
problem-solving exercise. The interview process was used to find out more about students’ WoT
and WoU in solving math problems. The interview method used was semi-structured. This
interview was conducted in-depth until the data or information was obtained. The steps of this
research took the following stages: (1) preparing recording devices; mobile phones and writing
test instruments, (2) asking students to complete the questions that have been given, and (3)
conducting interviews regarding students' WoT and WoU that couldnot be detected by recording
devices. To get students’ WoT and WoU which are problem-solving online test and angle
material. The research instrument was in the form of one problem with 2 sub-questions and an
interview guide sheet. The indicator for the written test is "to determine the size of an angle if
the other angle is known as a result of two parallel lines cut by a transverse line". The written
test in this study is shown in Figure 1 below.
The two instruments were validated by four validators, which consisted of two validators
from mathematics education lecturers at UIN Sunan Ampel and two mathematics teachers from
SMPN 2 Taman and MTs 1 Muhammadiyah Taman. The first validator had to revise the use of
sentences in the instruction such as using sentences that can be understood by junior high school
students. Having it done, the instrument was validated again by the second validator. In the
second validating process, the instrument needed more improvement in the term of research
method, especially in the written test. The revision was the test should ask about how many
strategies could be used to solve this problem or questions. After being validated by the second
validator, the researcher revised it again by adding the statement 'Write it down if you have more
than one way or formula'. The following step was it checked by the third validator. The third
validator stated that the questions are good based on the material, but the questions need to relate
to junior high school students. Then, the researcher immediately changed the context of the
questions which were closer to the students. After that, the fourth validator had a look and
validated the instructions. There was another revision in the instructions. The revision was it
should state that the answers would not affect the value of the learning outcomes in the report
cards. The assessment of the four validators starting from the material aspect, the construction
of the questions, and the writing and languages obtained a B grade, which means it is feasible to
use with revision so that the instrument is feasible to use.

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 Students’ ways of thinking….

D C
𝑙

𝑘
A B
In the picture above, the line is parallel to line l, points A, B, and C, D respectively lie on the
̅̅̅̅ = 𝐷𝐴
lines k and l. If known 𝐶𝐷 ̅̅̅̅ = 𝐴𝐶
̅̅̅̅ , and ∠𝐷𝐵𝐴 = 40°. Define ∠𝐵𝐷𝐴, ∠𝐷𝐴𝐵, and ∠𝑎!
Three students get an assignment from the teacher to determine the size of the angles in the
picture above. Three students have different opinions about the size of the outer angle of the
triangle
a) Andi, a measure of ∠𝑎 is smaller than the measure of ∠𝐵𝐷𝐴 + ∠𝐷𝐴𝐵
b) Beni, a measure of ∠𝑎 equals the measure ∠𝐵𝐷𝐴 + ∠𝐷𝐴𝐵
c) Cici, a measure if ∠𝑎 is bigger than measure ∠𝐵𝐷𝐴 + ∠𝐷𝐴𝐵
In your opinion, which opinion is correct? Explain!
(Write it down if you have more than one way)

Figure 1. Written test instrument

The data analysis was gained through writing test and interview section. They werecarried
out by reducing data, presenting data, and drawing conclusions (Miles et al., 2020). Reducing
the data in this study could be done by transcribing the interview results. As for the stages of
presenting the data, it was presented in the form of a description by displaying the results of
interview transcripts and it was based on the WoT and WoU indicators for each subject. The
WoT indicators were presented in Table 3 and the WoU indicators were shown in Table 4 which
were modified from Nurhasanah's research (Nurhasanah, 2019). Modification of the WoT and
WoU indicators in this study was the proof scheme indicator, which in Nurhasanah's research
that wass divided into two aspects; Result Pattern Generalization (RPG) and Process Pattern
Generalization (PPG). While the evidence scheme is divided into three; deductive, empirical,
and beyond belief in this study.

Table 4. WoT indicators in solving problems

Action Category Indicator

Problem Very good Have a variety of strategies and use effective and efficient strategies
solving that lead to the right solution.
approach Good Create a problem-solving plan and allow it to lead to the correct
solution.
Enough Create a problem-solving plan and possibly lead to wrong solutions.
Less Don't have a plan.
Proof Deductive One's way of thinking in solving problems is based on a good
Scheme understanding of the concept, validating the process to produce a
true statement.
Empirical A person's way of thinking is based on the results of an experiment.
for example, substitution of answers or numbers, understanding the
concept, and the resulting statement is true or false.
Beyond The way of thinking is based on non-referential symbols.
belief

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Belief in Very good Consciously using and connecting between known concepts to
mathematics solve problems.
Recognize the advantages of using and relating the selected
concepts.
Knowing the effectiveness of the selected concept.
Good Consciously using and connecting between known concepts to
solve problems.
Recognize the advantages of using and relating the selected
concepts.
Do not know the effectiveness of the selected concept.
Enough Consciously using and connecting between known concepts to
solve problems.
Do not know the advantages of using and connecting the selected
concept.
Do not know the effectiveness of the selected concept.
Less Do not know the concept formula used in solving the given
problem.

Table 5. WoU indicator in solving problems

Category Indicator
Very good Explain the problem completely and correctly, select concepts/algorithms, explain
concepts verbally and in writing appropriately, and a link between concepts in
solving problems logically.
Good Explain the problem completely and correctly, choose concepts/algorithms,
explain concepts verbally and in writing appropriately, and a link between
concepts in solving problems logically, but the final answer is incorrect.
Enough Explains the problem as a whole, and explains concepts well, but is not precise in
choosing certain concepts, and is unable to link between concepts.
Less Unable to fully explain the problem, misinterpreting the problem, or unable to use
concepts in solving problems and not linking one concept to another.

As for the process of conclusion, the researchers first described and analyzed the data
according to the WoT and WoU indicators and based on checking the validity of the data using
triangulation, then categorized them based on WoT and WoU indicators.

Findings and Discussion

The source of data for this research was 8 selected subjects who were coded SC1, SC2, SC3,
SP1, SP2, SP3, SQ1, and SQ2. The answers from the results of problem-solving tests, interviews,
and observations were used by researchers to identify students' WoT and WoU in solving
problems on lines and angles.

WoT and WoU of climbers’ students

Subjects SC1, SC2, and SC3 tended to only write one problem-solving strategy with
different problem-solving approaches. In the problem-solving approach stage, SC1 used the
concepts of division, subtraction, the addition of angles, and the concept of rectifier angles, SC2
used the concept of the properties of angles and equilateral triangle properties, and SC3 used the
concepts of addition, equilateral triangles, the sum of triangular angles, and rectifier angles, as
shown in Figure 2.

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 Students’ ways of thinking….

Figure 2. The results of the climbers’ students’ work; SC1 (left image) and SC2 (right image)

At the proof scheme stage, SC1 wrote information in graphic form, SC2 wrote information
in written and graphic form, and SC3 wrote information in written form. When it came to
planning, SC1 started by looking for ∠BDA, ∠CAB, and ∠ used the concepts of division,
subtraction, and addition of angles and straight angles, SC2 used the concept of angle properties,
and equilateral triangles, while SC3 used the concept of addition, equilateral triangles, the sum
of angles, triangles and straight angles in solving problems. Further, at the stage of choosing a
strategy, SC1, SC2, and SC3 apply their strategies to find ∠BDA, ∠CAB, and ∠. So, SC1
strategy is not quite right, while the SC2 and SC3 strategies are correct and appropriate, as shown
in Figure 3.

Figure 3. The results of the climber student’s work; SC3

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The excerpts from the interview results in Table 6 and Table 7 show that at the answer-
finding stage, SC1 concluded that Andi's opinion was correct, SC2 concluded that Cici's opinion
was correct but changed to Beni's opinion was correct, while SC3 concluded that Beni's opinion
was correct. Then, SC1, SC2, and SC3 believed that their answers were correct. However, SC1,
SC2, and SC3 re-examined the strategies that had been worked out. SC1 could solve the problem
in 34 minutes, SC2 finished in 29 minutes, and SC3 completed in 24 minutes. Briefly to conclude
that at the stage of the proof scheme, SC1, SC2, and SC3 are included in the empirical way of
thinking.

Table 6. Transcript of subject climber's interview

Question Transcript of SC1 Transcript of SC2 Transcript of SC3

interview interview interview
What plans Find the angle Draw first, then write Write down what is
do you BDA, CAB, and BDA down any information known and what is
have in then add the angle that is known after that asked and then search
mind to DAB and then the search ∠ADB, ∠CAB for ∠CAB, ∠BDA equals
solve the angle . equal ∠. ∠
problem?
What The concept of The concept of the In addition, equilateral
concept did division, subtraction, properties of angles, and triangles, the sum of
you use in and sum of angles equilateral triangles. angles in triangles,
solving the equal supplementary equilateral triangles.
problem? angle sis.
Explain in I'm confused, sis... I First of all, I first drew the I am looking for ∠CAB
detail how drew the corner first... angles, Sis, and gave the Sis, ∠CAB it's the same
you did the while thinking about numbers. Then I write as 60⁰. For example, if I
problem! how to do it when I down any information that draw this line Sis (while
read the question is known about the drawing), then I will
again... um, something problem. After that I name the angles A, B, C,
went wrong, here's sis started looking ∠BDA ... D, E, F, G, and H. The
CD = DA = AC. which ∠BDA, I got this from angle that I mean is the
means that all the ∠ADC by subtracting angle opposite this line,
angles in the ADC ∠BDC. ∠ADC is equal to like angles C and D. For
triangle are equal to 60 60⁰. From the problem, it example, angle C is
degrees. Earlier I tried is known that AD = DC = equal to 60⁰. Angle C is
to find ∠BDA first, CA means that triangle the same as angle G, so
∠BDA if I see that it is ADC is an equilateral angle G is equal to 60⁰.
half of ∠CDA, so I triangle. Then ∠BDC is So, angle G, if the sum is
divide 60⁰ divided by the same as 40⁰ as big as equal to angle E, is equal
2 equals 30⁰. Then the ∠DBA. Follow the to 180, just like angle E
continue searching concept of angle is the same as angle D.
∠DAB, ∠DAB there is properties, but I forgot the So, angle E equals 180⁰
an angle that is not yet name. minus 60⁰ equals 120⁰. If
known, and I will look angle E has met, it
for ∠CAB this first I means that angle D
get from 180 minus 60 equals 180⁰ minus 120⁰
divided by 2 the result equals 60°.
is 60⁰.

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Table 7. Follow-up transcript of subject climber's interview

Transcript of SC1 Transcript of SC2 Transcript of SC3

Question
interview interview interview
Does this Yes, sis, there are three Yes, sis, I'll draw (draw Yes, sis, I'm looking for
information angles, then the DAC two corners cut by a ∠BDA, it's in the ABD
relate to angle in the ADC transverse line). This is triangle, then in this
your triangle, which means Sis, for example, ∠DBA triangle, all the angles
strategy? the DAC angle is equal was in the same position are known ∠DAB,
to 60⁰. Next, the as ∠B1. So, ∠B1 it's equal ∠ABD except for the
supplementary angle is to 40⁰, if ∠B1 it's 40⁰, it ∠BDA. So, I use the
equal to 180⁰, so I means ∠B2 it's equal to formula for the sum of
subtract the angle 180⁰ 140⁰ because it's straight. the angles of a triangle.
equals 60⁰ equal to ∠B2 it's equal to ∠A2, So, there's a relation.
120⁰ ... I divided it by because, the faces are the After searching ∠BDA, I
2 because there are same, so it's equal to 120⁰. entered the ∠DAC and
only 2 unknown Then ∠A2 it's equal to ∠CAB into ∠DAB. Next,
corners left. Wait a ∠A4, and it's also add up the ∠BDA and
minute... oh I added straightened out, so it's ∠DAB. After that I'm
∠BDA and ∠DAB ∠A4 equal to 40⁰. looking for ∠, 180⁰
together and the result Find ∠CAB using the same minus 40⁰ equals 140⁰,
was 150⁰. After that, I method as the properties I'm sorry Sis, this is the
searched ∠ using the of the angle earlier, it is correct one. After that,
supplementary angle ∠CAB equal to ∠DCA, compare the results of
for this angle (pointing ∠DCA = 60°. So, ∠CAB is the sum of the angles
at an angle of 40⁰), also equal to 60⁰. After with ∠.
180⁰ minus 40⁰ equals looking for the angle
120⁰. earlier, I immediately
looked for ∠. ∠ equals
180⁰ minus 40⁰ equals
140⁰. Lastly, ∠BDA added
∠DAB together equals
140⁰.
What ∠ bigger than In my opinion, Cici's Beni's opinion is correct,
conclusions ∠BDA + ∠DAB … uh opinion is correct because not Andi Sis because ∠
can you small sis… it means the sum of ∠DAB and equal to ∠BDA + ∠DAB.
draw? Andi is right ∠BDA is equal to ∠. uh So, Beni's opinion is
sorry wrong Sis … Beni I correct.
mean.
Are you Already Sir... Already Yes sir
sure about
your
answer?
Explain because the method I I have checked everything. Because earlier I
why do you use is correct Insha Allah, the method is counted again Sis and I
believe in by the concept given by checked the methods is
the results my teacher correct.
you found?

According to the tables above, at the stage of confidence in mathematics, SC1, SC2, and
SC3 were aware of using several mathematical concepts in solving problems. SC1 knew the

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advantages of the chosen concept so she was confident about the chosen strategy but had a less
effective strategy. While SC2 and SC3 knew the advantages of the chosen concept so they felt
confident about the chosen strategy and had an effective strategy. On the indicator of confidence
in mathematics, SC1 was in a good category, meanwhile SC2 and SC3 had a very good category.
Whereas WoU of SC1was in enough category because the SC1 could explain the problem as a
whole, and explain concepts well, but was not precise in choosing certain concepts and was
unable to make connections between concepts. WoU of SC2 and SC3 were in the very good
category because they could explain the problem as a whole and the strategy well, and the
concepts used were appropriate. Based on the description and data analysis of SC1, SC2, and
SC3 in the questions, the WoT of SC1, SC2, and SC3 in solving line and angle questions could
be seen in Table 8, while the WoU can be seen in Table 9 below:

Table 8. Ways of thinking (WoT) subject climber conclusion

WoT Indicator SC1 SC2 SC3 Conclusion

Problem solving approach Enough Good Good Good
Proof Scheme Beyond Empirical Empirical Empirical
belief
Belief in mathematics Good Very good Good Good

Table 9. Ways of understanding (WoU) subject climber conclusion

WoU Indicator SC1 SC2 SC3 Conclusion

Understanding problems, Enough Very Very good Very good
choosing concepts/algorithms, good
explaining concepts and linking
concepts in solving problems

The three climber students had high determination in which they understood the problem
well, could mention written and unwritten information on questions, plan strategies properly and
precisely, and solve problems well even though they only employ one solution strategy. In line
with the research of Yani et al., (2016), she states that climber students have assimilation
thinking processes that occur when planning problems and implementing problems, and can use
cognitive schemes well in solving problems.
Then, the approaches used by climber students were vary and the correct problem-solving
was obtained, this showed that in solving problems, climber students tried to do it to the fullest
and best. This is in line with the study conducted by Chabibah et al., (2019) who state that
climber students can carry out all stages of the thinking process in solving mathematical
problems with various problem-solving approaches. Climber students with the highest AQ level
will always try to solve every problem well (Husain et al., 2022). Climbers students do not
simply believe the truth of the results they get before they student does a re-examination
(Widyastuti, 2015). This shows that climber students have an awareness of using the concept
and know the effectiveness and advantages of the chosen concept.

WoT and WoU of campers’ students

SP1, SP2, and SP3 tended to only write one problem-solving strategy with different
problem-solving approaches. In the problem-solving approach stage, SP1, SP2, and SP3
employed the concept of adding and subtracting angles, as shown in Figure 4.

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 Students’ ways of thinking….

Figure 4. The work’s results of SP1 campers (left image) and SP2 campers (right image)

At the proof scheme stage, SP1 and SP2 did not write down information either in the form
of pictures or writing. When planning, SP1 started by looking for ∠BDA, ∠CAB, and ∠ used
the concept of division and sum of angles in solving problems, SP2 and SP3 used the concept of
sum and subtracting angles in solving problems. At the strategy selection stage, SP1, SP2, and
SP3 employed their strategies to find ∠BDA, ∠CAB, and ∠. SP1 and SP2 admitted that they got
the strategy from friends without understanding the meaning of the concept used (authority
scheme), while SP3 believed in a concept based on what the subject usually did when solving
problems without understanding the meaning of a concept (ritual scheme). So, the strategies of
SP1, SP2, and SP3 are not quite right, as shown in Figure 5.

Figure 5. The work’ result of SP3 camper

This could be seen in the excerpts of the interview results in Table 10. At the answer-finding
stage, SP1 concluded that Andi's opinion was correct, and SP2 concluded that ∠BDA, ∠CAB was
smaller than ∠, in this case, Andi's opinion was correct. While SP3 concluded that Beni's
opinion was correct. SP1, SP2, and SP3 believed that their answers were correct. SP1, SP2, and
SP3 subjects re-examined the strategies that had been worked out. SP1 could solve the problem

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in 44 minutes, SP2 could complete it in 39 minutes, and SP3 finished in 20 minutes. So, it was
concluded that at the stage of the proof scheme, SP1, SP2, and SP3 were included in the way of
thinking beyond belief.

Table 10. Transcript of interview with subject campers

Transcript of SP1 Transcript of SP2 Transcript of SP3

Question
interview interview interview
Describe in detail At first, I first divided I'm looking for ∠BDA, First of all, I'm
how you tackled 60⁰ by 2 to find BDA equals 60⁰ minus looking for ∠ Sis, by
this problem? ∠BDA. Then I find 40⁰ equals 20⁰. Then subtracting the angles
the angle ∠CAB, search ∠DAB, ∠DAB 180⁰ and 40⁰. Then I
∠CAB equal to 60⁰. equals 60⁰ plus 40⁰ try to find ∠BDA, by
Then ∠BDA I add equals 100⁰. Next, dividing the 60⁰ by 2.
∠DAB it equal to search ∠, ∠ equals After that, I'm looking
150⁰. Then I find ∠, 90⁰ plus 50⁰ equals for ∠CAB by adding
I subtract the 140⁰. ∠BDA is equal to the 60⁰ and 40⁰, then I
supplementary angle ∠DAB but is not yet divide by 2 equals 50
180⁰ with 40⁰ equals added together... ∠BDA degrees. After that, I
140⁰ plus ∠DAB is equal to added ∠BDA and
20⁰ plus 100⁰ equals ∠DAB, and the result
120⁰ so it ∠ is greater is 140 degrees. So,
than ∠BDA + ∠DAB. Beni's opinion is
correct.
Ok, where did you Me, at first, I was From me... earlier I just Then I tried to find
get the idea to confused and then tried to do as much as I ∠BDA by dividing the
search? Try to when I saw the could Sis, I didn't 60° angle by 2. I can
explain!∠BDA picture... it turned out understand a bit... see from your picture
∠BDA that this was because my teacher that the DB line cuts
half from ∠ADC. So, I only explained briefly the ADC angle in
divided by 2 equal 60⁰ half.
… It was shared by
my friend Sis, so I
couldn't answer this
one.
How did you From the question, Already Yes sis
know that triangle Sis. It is known that
ADC is the line CD = DA =
equilateral? AC
OK, based on the Andi ∠BDA + ∠DAB is Because ∠ it is equal
strategy you smaller to ∠ to ∠BDA + ∠DAB.
explained earlier, So, I think Beni's
what conclusions opinion is correct.
can you draw?
After listening to Yes Not sure Yes
your explanation,
are you sure about
your answer?
Explain why you Checked again Sis, the - I already checked
believe in the result is equal to in the
results you found? picture

164
 Students’ ways of thinking….

At the stage of confidence in mathematics, SP1, SP2, and SP3 were aware of using several
mathematical concepts in solving problems. SP1, SP2, and SP3 knew the advantages of the
chosen concept so they felt confident about the chosen strategy but had less effective strategy.
On the indicator of confidence in mathematics, SP1 was in a good category, SP2 was in enough
category, while SP3 was belong to good category as well. Whereas the students’ WoU, SP1,
SP2, and SP3 were in enough category because SP1 could explain the problem as a whole and
explain concepts well, but was not precise in choosing certain concepts and was unable to link
between concepts, while SP2 and SP3 could partially explain the problem. Based on the
description and data analysis of SP1, SP2, and SP3 in the questions, the WoT of SP1, SP2, and
SP3 in solving line and angle questions can be seen in Table 11, while the WoU can be seen in
Table 12.

Table 11. Ways of thinking (WoT) subject campers conclusion

WoT Indicator SP1 SP2 SP3 Conclusion

Problem solving approach Enough Enough Enough Enough
Proof Scheme Beyond Beyond Beyond Beyond belief
belief belief belief
Belief in mathematics Good Good Good Good

Table 12. Ways of understanding (WoU) subject campers conclusion

WoU Indicator SP1 SP2 SP3 Conclusion

Understanding problems, choosing Enough Less Enough Enough
concepts/algorithms, explaining concepts
and linking concepts in solving problems

The three campers’ students had moderate determination in which they understood the
problem sufficiently, could mention information in writing only and analyze some of the
information well. Campers’ students experienced difficulties in determining and using concepts
in designing settlement strategies, so they were not able to solve problems properly and
appropriately. This is in accordance with the research by Nurhasanah, (2019) that campers
students tend to have accommodation and semi-conceptual thinking processes. They have not
been able to use cognitive schemas properly in solving problems. Campers students can carry
out the stages of the thinking process up to the stage of implementing strategies for solving math
problems with limited abilities (Chabibah et al., 2019). This shows that campers students do not
have an awareness of using the concept and know the effectiveness of the chosen concept.

WoT and WoU of quitters’ students

SQ1 and SQ2 tended to only write one problem-solving strategy with different problem-
solving approaches, but they didn't complete the answer, as shown in Figure 6.

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Lailiyah et al.

Figure 6. Student work results in quitters SQ1 (left image) and SQ2 (right image)

At the proof scheme stage, SQ1 and SQ2 did not write down information either in the form
of pictures or writing. When planning, SQ1 started by writing down several angles to look for;
∠, ∠BDA, and ∠DAB, while SQ2 started by looking for arcs. At the stage of choosing a strategy,
SQ1 and SQ2 used their strategy to find ∠. SQ1 and SQ2 subjects didn't know what concept
used for the next plan. The subjects admitted that they got the plan from a friend (authority
scheme). So, the SQ1 and SQ2 subject strategies were not quite right.
This could be seen in the excerpts from the interview results in Table 13. At the stage of
finding answers, SQ1 had no conclusions, while SQ2 concluded that Beni's opinion was correct.
At the checking stage, SQ2 re-checked the results of the answers he obtained by measuring
again, but when he re-examined, the results obtained were different so the SQ2 subject was
unsure of the results of the answers. Regarding to the duration, SQ1 could solve the problem in
49 minutes, while SQ2 was in 12 minutes. So, at the stage of the proof scheme, SQ1 and SQ2
were included in the way of thinking beyond belief.

Table 13. Transcript of interview with subject quitters

Question SQ1 interview transcript SQ2 interview transcript

Describe in detail how I don't know sis; I can just Earlier I wanted to use a ruler arc
you tackled this look for ∠ it. Difficult Sis. Sis... because earlier there was no
problem? math class and my friends didn't
bring it either, so I looked for
something else and I found it with a
wire. I made this wire earlier and
then I pasted it like a corner. When
it's finished, I slide it to the BAD
corner, then I slide it to ADB.
Where did you get this - From the teacher…. He... it's not
idea from? right Sis, this is from me... the
teacher measured it using a ruler arc,
instead of a wire but there wasn't an
arc before... so I took the initiative
to use a wire.
Did you not use the - No Sis, I don't understand, it's better
information you know to to use the tool right away
solve this problem?
Is there any other way - There is, use the formula, but I can't
besides the way you do? memorize it... it's hard...

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 Students’ ways of thinking….

OK, so you're also Yes, Sis, I don't understand -

having trouble finding the material and I don't
∠DBA and ∠DAB? Why? know how

OK, based on the No sis Beni

strategy you explained
earlier, what conclusions
can you draw?
After listening to your - Not sure sis.
explanation, are you sure
about your answer?
Why? - I checked again...the answers have
changed but it seems that the answer
is that.

At the stage of confidence in mathematics, SQ1 and SQ2 were aware of using several
mathematical concepts in solving problems. SQ1 and SQ2 did not know the benefits of the
chosen concept and had fewer effective strategies. This happened because the results obtained
were inaccurate even though the subject could complete the test. On the indicators of confidence
in mathematics, SQ1 and SQ2 were belong to less category. Whereas WoU of SQ1 and SQ2
were in the less category because they could not explain the problem correctly, explain the
concepts verbally and choose concepts correctly, and they could not relate concepts in solving
problems logically. So, WoT and WoU of climbers’ students were better than camper students
and quitter students. Based on the description and data analysis of subjects SQ1, and SQ2 in the
questions, the WoT of subjects SQ1, and SQ2 in solving line and angle questions can be seen in
Table 14, while the WoU can be seen in Table 15 below:

Table 14. Ways of thinking (WoT) subject quitters conclusion

WoT Indicator SQ1 SQ2 Conclusion

Problem solving approach Enough Enough Enough
Proof Scheme Beyond Beyond Beyond belief
belief belief
Belief in mathematics Less Less Less

Table 15. Ways of understanding (WoU) subject quitters conclusion

WoU Indicator SQ1 SQ2 Conclusion
Understanding problems, choosing Less Less Less
concepts/algorithms, explaining concepts
and linking concepts in solving problems

The two quitters’ students had low determination and they were able to understand the
problem properly. Quitters’ students could only mention part of the information in writing and
were not able to analyze the information properly. Quitters’ students experienced difficulties in
determining settlement strategies so the final results obtained were not correct. This is in line
with research by Kusumawardani (2018) which states that quitter students have not been able to
use cognitive schemas properly in solving problems. Quitter students are only able to focus on
results and are unable to understand the meaning of the resulting concepts and processes
resulting in the wrong answers. The two quitters’ students also did not validate the results of
their answers and felt doubtful about the results obtained. This is in line with research by
Chabibah et al. (2019) which states that quitter students are only able to carry out the stages of
the thought process in reading and planning without solving the problem properly. This shows

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that quitter students do not have awareness of using the chosen concept and do not know how to
do it.
The characteristics of WoT and WoU are interconnected. The WoU that is incorrect and
incomplete results in a WoT that is illogical or wrong, while a good WoU will produce a WoT
that is systematic, logical, and effective (Mefiana & Herman, 2023; Nurhasanah et al., 2021).
The implication of WoT and WoU in education is that when teachers know the students' thinking
processes in both WoT and WoU. Therefore, teachers could design an appropriate learning that
considers the characteristics of their students' WoT and WoU. Apart from that, the difference
between students' WoT and WoU influences students' mathematical abilities and differences in
understanding a concept (Samosir & Herman, 2023). Therefore, to support students' WoT and
WoU, teachers must familiarize students with solving non-routine problems in various
mathematics learning contexts (Aiyub, 2023).

Conclusion
Based on the results of the research that has been done, it could be concluded that regarding
to the WoT of climbers’ students, they tended to have only one strategy that led to the correct
solution, had an empirical way of thinking, and had very good self-confidence in mathematical
concepts in terms of in problem-solving on lines and angles. While the WoU were very good.
Furthermore, the WoT of campers’ students in solving problems on lines and angles tended to
have only one strategy that led to wrong solutions, the way of thinking used by them was beyond
belief, and they had good confidence in mathematical concepts. Whereas WoU were in enough
category. The last but not least, the WoT of quitters’ students in solving problems on lines and
angles tended to have only one strategy that led to the wrong solution, the way of thinking used
by them was beyond belief, and they had less belief about mathematical concepts. Meanwhile,
the WoU were less category. In brief, WoT and WoU of climber students were better than the
camper and quitter students.

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