Mathematics belief impact on metacognition in solving geometry: Middle school students

Journal of Education and Learning (EduLearn)
Vol. 18, No. 2, May 2024, pp. 286~295
ISSN: 2089-9823 DOI: 10.11591/edulearn.v18i2.21110  286
Journal homepage: http://edulearn.intelektual.org
Mathematics belief impact on metacognition in solving
geometry: Middle school students
Mega Suliani1
, Dwi Juniati2
, Agung Lukito2
1
Department of Mathematics Education, Faculty of Mathematics and Natural Sciences, Universitas Negeri Surabaya, Surabaya,
Indonesia
2
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Surabaya, Surabaya, Indonesia
Article Info ABSTRACT
Article history:
Received Jul 14, 2023
Revised Oct 30, 2023
Accepted Nov 8, 2023
Mathematical beliefs and metacognitive knowledge play significant roles in
solving mathematical problems; thus, this study aims to investigate the
influence of middle school students' beliefs on their metacognitive
knowledge when solving geometry problems. This study utilizes both
quantitative and qualitative research methods. A linear regression test was
used to determine the effect of middle school students' beliefs on their
metacognitive knowledge. The results of the quantitative research analysis
were followed up with a qualitative research approach to describe the
metacognitive knowledge of students who have high and low confidence in
solving geometric problems. This research involved 352 middle school
students in the Tarakan area. Based on the results of linear regression, it is
known that the beliefs of middle school students have a positive effect on
their metacognitive knowledge when solving geometric problems. In
addition, it was found that students with different beliefs could solve a given
geometry problem, but the approach to solving it varied among subjects.
Middle school students have diverse beliefs, but these variations do not
affect their capacity to apply their metacognitive knowledge at every stage
of solving mathematical problems.
Keywords:
Geometry problems
Mathematics belief
Metacognition
Self-knowledge
Task knowledge
This is an open access article under the CC BY-SA license.
Corresponding Author:
Dwi Juniati
Department of Mathematics, Faculty of Mathematics and Natural Sciences
Universitas Negeri Surabaya
Ketintang St., Surabaya, Jawa Timur 60231, Indonesia
Email: dwijuniati@unesa.ac.id
1. INTRODUCTION
One of the main goals in the process of learning mathematics is for students to be able to solve a
given problem. However, there are still many students who find it difficult and feel anxious when learning
mathematics or solving math problems [1]. Many factors can influence students’ ability to solve problems,
such as working memory skills, cognitive awareness, beliefs and anxiety about mathematics [2]–[4].
Cognitive awareness and understanding of the problems faced by students are necessary in the process of
solving mathematical problems. Through awareness, students can improve their abilities [5]. However, the
results of a study conducted by Setyawati and Indrasari [6] show that students have not maximized their
awareness when solving mathematical problems even though it is essential. The principles of metacognition
involve being aware of knowledge and knowing how to apply it to solve problems. The results of the
research by Sutama et al. [7] shows that there are differences in the metacognitive activity of junior high
school students with field independent and field dependent cognitive styles in solving mathematical

J Edu & Learn ISSN: 2089-9823 
Mathematics belief impact on metacognition in solving geometry: Middle school students (Mega Suliani)
287
problems. In addition, the results of research conducted by Suliani et al. [8] state that both female and male
junior high school students were able to utilize their metacognition in solving math problems.
The concept of metacognition was introduced to describe and explain how a person can control his
thinking during learning and problem-solving, especially when a person experiences cognitive failure and
encounter difficulties in information processing and problem-solving [9]–[11]. The metacognitive aspect is
related to students' ability to organize their own thoughts. Lioe et al. [12] also stated that metacognition is
one of the main components of solving math problems. This emphasizes students' ability to monitor their
own thinking. This is in line with the concept of metacognition developed by Flavell [13]. Therefore,
students who excel at problem-solving always monitor their thinking process and evaluate the results they
achieve. These students know when to employ an effective strategy and when to change this strategy to make
a decision that aligns with a certain goal. There are three categories of metacognition: metacognitive
knowledge, metacognitive skills and metacognitive experiences [13], [14].
In this study, the researchers focused solely on aspects of metacognitive knowledge. This is based
on the idea that students who utilize their metacognition can understand how to solve tasks or problems.
Metacognitive knowledge can influence someone in solving mathematical problems, particularly those
related to geometry. According to Zulyanty [15] metacognitive knowledge helps individuals recognize the
truth and identify mistakes made when solving problems. In addition, it can help identify where the error lies
in problem-solving [13]. Based on this, metacognitive knowledge is an understanding of the process of
thinking about what to think about, how, and when to approach certain tasks.
Metacognitive knowledge serves as the foundation for utilizing cognitive and metacognitive
strategies in problem-solving process. It is developed through metacognitive skills, which involve managing
cognitive processes to achieve cognitive goals during problem-solving. Additionally, emotions play a role in
differentiating between metacognitive knowledge and metacognitive experiences. The presence of feeling
implies that the metacognitive experience involves a sense that the current subjective experience is a result of
the cognitive activity that occurred during the cognitive process. Metacognitive knowledge involves three
components: the learning process and one's beliefs about how they learn and others learn; learning tasks and
how to process information effectively; and determining which strategies to use and when to utilize it [13].
Apart from the cognitive aspect, students solving math problems must also pay attention to the
affective aspects, namely aspects that influence students' tendencies to solve problems [10], [11]. One of the
affective aspects that students must possess when learning mathematics is a belief in the subject and problem-
solving. The results of research conducted by Ozturk and Guven [16] concluded that beliefs not only affect
the process of problem-solving but also influence personal factors such as life experiences.
Solving problems related to affective aspects (beliefs) leads to the conclusion that students who
struggle to solve problem often feel frustrated. The results of research conducted by McLeod [17] show that
students who are unable to solve problems often panic. The same thing was done by Schoenfeld [18] who
showed that there was a strong relationship between the mathematics test results expected by students and
students' beliefs related to their abilities. Furthermore, belief can be divided into two categories: belief in
mathematics [19] and belief in solving mathematical problems [20]. Furthermore, Ozturk and Guven [16]
classify beliefs into two categories: high beliefs and low beliefs.
Geometry itself is a challenging subject for learners, particularly when it is being learned remotely.
In addition to space and form, geometry encompasses the concepts of distance, scale and relative position of
figures. Moreover, numerous occupations, such as architects, mechanical engineers, technicians and draughts
men utilize geometry. Due to that, geometry is an essential branch of mathematics. The majority of learners
find geometry difficult to study and have no desire to do so. This is due to the fact that learners frequently
feel unsure of themselves about what they have learned, experience anxiety when studying it and are unable
to use geometric theory to solve their problems [21].
So far, experts have conducted research in order to provide solutions to problems related to
metacognitive activity. For example, the role of metacognition is used to explain the relationship between
initial difficulties, students' understanding of reading, and the process of conjecturing [22], [23]. However,
research related to the relationship between metacognitive knowledge and junior high school students' beliefs
in solving geometric problems is still very limited in general [24], [25]. Even though there is a very important
essence when it involves students' beliefs in solving geometric problems to identify the metacognitive
knowledge of junior high school students, it is important to carry out further research, namely to investigate
and explore the metacognitive knowledge of junior high school students who have high and low belief in
solving geometric problems.

 ISSN: 2089-9823
J Edu & Learn, Vol. 18, No. 2, May 2024: 286-295
288
2. METHOD
Explaining this study aims to provide a comprehensive understanding of the metacognitive
knowledge of students who have high and low confidence in solving geometric problems, as well as their
relationships. The research approaches used are descriptive quantitative and qualitative. The design refers to
the collection, analysis, and integration of both qualitative and quantitative data at multiple stages of a
research [26]. The quantitative approach was intended to examine the correlation between metacognition and
students' beliefs in solving mathematical problems using the t-test statistical test. On the other hand, the
descriptive qualitative approach aimed to explore and investigate further regarding the metacognitive
knowledge of research subjects while solving geometric problems.
The hypothesis formulated in this study was that there is a functional relationship between
metacognitive knowledge and junior high school students' beliefs about solving geometric problems. The
population in this study included all students of a state middle school in Tarakan. The sample for this study
included 352 students of the middle school who were selected using a simple random sampling technique that
considered the homogeneity of the population. This is in line with Roscoe [27] who stated that an appropriate
sample size in research is between 30 and 500. The time for conducting research was in the even semester of
the 2021-2022 academic year.
Data regarding the students’ beliefs was collected using the Indiana mathematics belief (IMB) scale
questionnaire instrument, and the metacognitive knowledge questionnaire was used to collect the students’
metacognitive knowledge. After that, the correlation between metacognitive knowledge and students' beliefs
in solving mathematical problems was analyzed. Math tests were administered to assess the subjects’
mathematical ability. In the interview stage, a geometry problem solving assignment was given to the
selected subjects to evaluate their metacognitive knowledge. Furthermore, the IMB scale questionnaire
consisted of 30 statement items [20] the metacognitive knowledge questionnaire consisted of 14 statement
items [14], [28] related to middle school students' metacognitive abilities in solving mathematical problems,
they were found to be valid and reliable (Cronbach′
s Alpha = 0.934); the mathematics test consisted of 5
questions covering various subjects that students have studied. The problem to be solved by the selected
subjects for interviews was as follows: it is known that Firman will make a square photo frame with an outer
diagonal of the frame measuring 80√2 cm. Subjects were asked to calculate the total length of wood that
Firman would use and the minimum cost that Firman would incur if the price per meter of wood was IDR
40,000.
Data analysis techniques in this study used descriptive statistics, data reduction, data presentation,
triangulation, analysis and conclusion. To test the correlation between metacognition and students' beliefs in
solving mathematical problems, the study utilized the statistical t-test and to analyze the subject's
metacognitive knowledge in solving geometric problems, Polya stages [29] consisting of understanding the
problem, making plans, carrying out problem solving and evaluating for each stage, were utilized. The
subject's metacognitive knowledge was analyzed by looking at how the subject carried out metacognitive
knowledge activities that involve his knowledge of strategies which affect the direction and results of his
cognitive endeavors.
3. RESULTS AND DISCUSSION
A total of 352 middle school students participated in this study. All respondents completed a series
of tests, namely basic mathematics tests and then a questionnaire probing their mathematical belief in solving
geometric problems. After that, the respondents were asked to complete a metacognition knowledge
questionnaire. Respondents who took part in the study consisted of 167 male respondents and 185 female
respondents, with an age range of 12 to 14 years. Furthermore, 145 respondents had high belief in solving
geometric problems, consisting of 60 male respondents and 85 female respondents and 207 respondents had
low belief in solving geometric problems, consisting of 107 male respondents and 100 female respondents.
Description of the test results is shown in Table 1.
Table 1. High/low belief and metacognitive knowledge
High/low beliefs Metacognitive knowledge
Mean 86.91 42.25
Standard deviation 7.23 7.36
Min 61.40 19.71
Max 106.17 62.35

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From the Table 1, it can be seen that the average student's belief is 86.91 and the average
metacognitive knowledge is 42.25. This shows that students have a low level of mathematical belief in
solving geometric problems, which has an impact on their metacognitive knowledge. The researchers used
the SPSS application to test the normality of metacognitive knowledge data and students' beliefs about
solving geometric problems. The data used were scores of students' beliefs in solving math problems and
scores of students' metacognitive knowledge. Residual regression must follow a normal distribution and this
condition can be achieved by using the normal predicted probability (PP) plot, as shown in Figure 1.
Figure 1. Normal prediction probability plot (left) and scatterplot of residuals (right)
Figure 1 (left), shows a plot of metacognitive knowledge score points that correspond to the
normality diagonal line. This indicates that the normal conditions are met. Furthermore, homoscedasticity
refers to whether these residuals are evenly distributed and this condition can be checked by the distribution
of the residuals. The scatter plot of the residuals in Figure 1 (right) shows no particular pattern and the points
are evenly distributed above and below zero on the X axis. Furthermore, the points are also distributed evenly
to the left and right of zero on the Y axis. Therefore, it can be concluded that the homoscedasticity
requirements are met. The residuals are normally distributed and homoscedastic, so the student's belief
variable in the regression has a linear relationship with the student's metacognitive knowledge variable in
solving geometric problems. Output regression analysis of students’ metacognition and a belief in solving a
geometry problem displayed in Table 2.
The results of the summary results using SPSS from the regression analysis on students'
metacognitive beliefs and knowledge in solving geometry problems yield an R-squared value of 0.208 as
shown in Table 2, which indicates that the effect of the independent variable (belief) on the dependent
variable (metacognitive knowledge) is 20.8%, while other variables explain the rest. The next step is to
determine the regression model, test the suitability of the model and investigate the variables that affect
metacognitive knowledge. Table 3 and Table 4 show the coefficients and significance values of linear
regression analysis.
Table 2. Regression analysis of students’ metacognition and belief in solving a geometry problem
Model R R Square Adjusted R Square Std. Error of the Estimate
1 0.457a
0.208 0.206 6.56
Note: a. Predictor: (Constant), Total belief
Table 3. Analysis of variance (ANOVA) from the regression analysis out students’ metacognition and belief
in solving a geometry problem
Model Sum of squares df Mean square F Nilai p sig
Regression 3,965.769 1 3,965.769 92.181 0.000
Residual 15,057.514 350 43.021
Total 19,023.283 351

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Table 4. The output coefficients of regression analysis out students’ metacognition and belief in solving a
geometry problem
Model
Unstandardized coefficients Standardized coefficients
t p sig. value
B Std. Error Beta
Constant 1.833 4.224 0.434 0.665
High/low belief 0.465 0.048 0.457 9.601 0.000
Note: Dependent variable; metacognitive knowledge; Tolerance: 1.000; VIF: 1.000
This is shown from the results of the ANOVA table test. It was found that F = 92.181 with p value
Sig. = 0.000 < 0.05. Therefore, the overall regression model fits the data. The regression model is:
y = 1.833 + 0.465x, where the variable y is defined as the dependent variable, namely students' metacognitive
knowledge in solving geometric problems, while the variable x is defined as the independent variable,
namely students' beliefs in solving mathematical problems. Thus, middle school students' beliefs affect their
metacognitive knowledge when solving geometric problems. This means if there are students who have high
confidence in solving geometric problems, it is predicted that these students will have high metacognitive
knowledge of solving geometric problems. This is in accordance with the research of Setyawati and Indrasari
[30] which shows that students who have more belief in their mathematical abilities use better metacognitive
strategies. It can be said that the determinants of students' success in solving mathematical problems do not
only depend on their perception of thought processes but also on their beliefs about solving mathematical
problems. When students have good beliefs, they can improve their cognitive skills [16], [18].
To find out the effect of mathematical beliefs on metacognitive knowledge, a qualitative study was
conducted by giving assignments to 57 respondents. Selected participants consisted of 19 subjects with high
confidence in solving geometric problems and 38 subjects with low confidence in solving geometric
problems. Based on the results of the answers given by the participants, two students were selected,
consisting of one student with high belief and one student with low belief, who were identified through the
random sampling technique as research subjects who had unique problem-solving strategies and were able to
communicate their ideas when solving geometry problems.
Subjects who have high belief in solving geometry problems with an IMB score of 114 are labeled
with high mathematical belief (HMB), and subjects who have low belief in solving geometry problems with
an IMB score of 77 are labeled with low mathematical belief (LMB). The two subjects were female students
and had relatively balanced mathematical abilities, as shown in the TPMM score; the subject with high belief
earned a score of 100, and the subject with low belief earned a score of 95. The researchers also conducted
interviews with the two subjects, then presented in full the results of the analysis of knowledge data.
metacognitive subject in solving geometry problems.
In general, both subjects can solve geometry problems correctly, but there were differences in how
they answer the given geometry problem. Both HMB and LMB utilized metacognitive knowledge when
solving a mathematical problem. The two subjects reconsidered their own understanding on the task at hand
and decided on an effective strategy for solving the problem. Both subjects were able to assess their
respective abilities by mentioning their cognitive weaknesses and strengths when facing a task. Both subjects
were also aware of the steps that can be taken when faced with a particular task and can decide which
strategy to use. HMB was aware of her own knowledge in understanding the problem, specifically the
knowledge of how to select information to find important sentences in the problem. This is in line with the
results of a research conducted by Margono et al. [31] which stated that subjects who answered consistently
at the stage of understanding the problem had knowledge of themselves, knowledge of cognitive tasks and
knowledge of strategies.
On the other hand, LMB was aware of her own knowledge when facing certain cognitive tasks by
knowing the formula to use. Although LMB realized that understanding the problem was difficult, this
realization was further reinforced when LMB successfully solved a mathematical problem. She demonstrated
her understanding by knowing the purpose of the given problem. This is in line with the opinion of Tobias
and Everson [32], which states that students who have a lower understanding of what they know and do not
know may have greater difficulty retrieving previous lessons. In addition, LMB realized that solving a
problem required accuracy and the ability to understand the problem.
HMB and LMB systematically completed tasks based on the stages of Polya's problem-solving
process. At the stage of understanding the problem, both subjects were aware of their own knowledge
regarding cognitive abilities, the tasks that must be carried out, and the cognitive strategies that should be
employed to comprehend the problem. The two subjects, in order to understand the problem, first read and
identified every word or sentence in the problem, marked it as known, and asked for information in the
problem. The two subjects also decided on the formula to be used by utilizing their initial knowledge. This

291
knowledge was obtained from the learning experiences of the two subjects. Over time, it has been proven to
change students' beliefs regarding solving problems [33]–[36]. Thus, students can utilize their metacognitive
knowledge to solve a mathematical problem.
When understanding a problem, HMB could understand how to select information to find important
sentences in the problem. In addition, steps taken to understand the problem were reading, understanding,
and remembering the topic being discussed. HMB was also aware of deciding on effective strategies to
understand problems based on his learning experience. This finding is in line with the results of research
conducted by Riani et al. [37] which states that junior high school students can recall whether they have
solved problems like this before, think about whether previous knowledge can help solve problems, and
relate what is known and asked about problems with previous knowledge. On the other hand, LMB, in
understanding the problem, utilized the following steps: to record or mark important things in geometry
problems, determine the formulas used, and be aware of strategies that can be used by making sketches of
drawings and setting formulas to answer these problems.
When preparing the problem-solving plan, HMB was aware of her own knowledge regarding his
ability to identify and select important sentences and determine topics that were appropriate to the problem.
In addition, HMB was also aware of her abilities in terms of being able to show the keywords in the
questions and knowing the topics being discussed. The steps that HMB took were to re-read the problem,
mark important information, write down important information in their own language, determine the formula
and determine the most effective strategy to be able to plan problem solving based on their learning
experience. Meanwhile, LMB did not make plans to solve geometric problems, so LMB metacognition in
preparing problem-solving plans could not be described.
At the stage of applying or carrying out the problem-solving plan, both HMB and LMB knew how
to properly carry out the formula that had been established based on their understanding. Furthermore, the
steps used by HMB and LMB are carried out in accordance with the important information and questions
provided. The two subjects also applied algebraic principles and operations in a systematic manner. This is in
line with the results of research conducted by Nicolaou and Philippou [38] which states that students' beliefs
can improve their problem-solving and problem-posing skills in just a few weeks.
HMB reconsidered the knowledge required to implement its formulated plans, specifically by
understanding how to apply relevant formulas to the topic under discussion. In addition, HMB was also
aware of her ability apply the formula based on the topic being discussed. The steps that HMB took were to
complete them sequentially according to the questions on the problem and determine the most effective
strategy to apply the formula that had been determined based on the learning experience shown in Figure 2.
This is in line with the opinion of Alzahrani [39] who stated that metacognitive awareness can improve
students' learning process by rethinking the knowledge they have. Mokos and Kafoussi [40] describe a
metacognitive activity that is often used in solving open problems as the meta of procedural knowledge.
Figure 2. Results of HMB (subject with high belief) written work
On the other hand, LMB utilized her metacognitive knowledge by realizing the ability to apply
things that are known to the formula to use. This was done with the ability the subject had. Dunlosky and
Bjork [41] explained that metacognition is the mind's ability to monitor and regulate itself, or, in other words,
the ability be aware of one's own knowledge. Further steps that were taken by LMB were to apply the
information contained in the problem to the formula to be used and to be aware of the strategies that can be
𝟏: 𝟏: 𝟐
𝟏: 𝟐 = 𝒂:𝒄
𝟏
𝟐
=
𝒂
𝟖𝟎 𝟐
𝟏 × 𝟖𝟎 𝟐 = 𝟐 × 𝒂
𝒂 =
𝟖𝟎 𝟐
𝟐
𝒂 = 𝟖𝟎
𝒂 = 𝒃
𝟖𝟎 = 𝟖𝟎
length of wood = 𝟒 × 𝒂 = 𝟒 × 𝟖𝟎 = 𝟑𝟐𝟎
= 𝟑𝟐𝟎 × 𝟎. 𝟎𝟏 = 𝟑. 𝟐 m
charges = IDR 40,000 × 3.2 m = IDR 128,000

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292
implemented using the principles and operations of algebraic calculations, as illustrated in the example
shown in Figure 3.
Figure 3. Results of LMB (subject with low belief) written work
At the re-examining stage, both subjects recognized the appropriateness of their actions based on
their knowledge. The two subjects used different alternative methods to ensure that the calculations carried
out were appropriate for the problems they were facing. HMB utilized her metacognitive knowledge when re-
examining the solutions obtained, namely by realizing her ability to see the results obtained and knowledge
related to the topic being discussed. The steps taken were to adjust the results obtained and if the solutions
obtained were uncertain, it would be recalculated in a different way. In line with the findings of a study by
Juniati and Budayasa [42] which states that students feel more confident and competent when they can use
the theory they have learned effectively.
On the other hand, LMB thought differently in utilizing her metacognitive knowledge, namely by
realizing her ability to find out in advance the sides and angles of a square if the diagonal that divides the
square was known. The subject’s awareness at each stage of problem-solving could increase the subject’s
ability to solve problems properly and correctly. This is supported by the findings of a study Kozikoğlu [43]
which states that metacognition has a close relationship with mathematical problem-solving abilities. In
addition, Vissariou and Desli [28] argues that when students involve their metacognition in solving a
problem, they are able to represent and solve mathematical problems correctly, evaluate the effectiveness of
strategies, and recognize mistakes they have made. In addition, involving metacognitive knowledge in
solving problems can increase students' confidence so that they do not feel anxious when facing similar
problems at a later time. The following briefly presents the differences in HMB and LMB metacognition in
solving geometry problems, as show in Table 5 (see Appendix).
4. CONCLUSION
In conclusion, students' mathematical beliefs significantly impact their metacognitive knowledge in
solving geometric problems. This means if there are students who have high mathematical belief, it is
predicted that these students will also have high scores in metacognitive knowledge when solving geometric
problems. Furthermore, the metacognitive knowledge of the subject with high belief in their understanding
mathematical problems involves being aware of their own knowledge about the skills required to solve
problems and the ability to identify problems by specifying the known and requested information.
Furthermore, the subject is aware of the steps that can be taken to solve the problem. These steps include
reading the problem repeatedly, selecting the relevant information and connecting it with prior knowledge.
This allows them to determine the appropriate concept or formula to use in order to solve the problem. Every
problem-solving exercise undertaken by the subject with high belief always utilizes the metacognitive
knowledge. On the other hand, the subject with low belief in utilizing their metacognitive knowledge
experienced differences in their cognitive tasks. They lack awareness of their own knowledge and struggle to
identify the appropriate formula to us, although the subject realized the difficulty of understand the problem.
Is Know: Diagonal = 𝟖𝟎 𝟐 cm, Price of
wood per meter IDR 40,000.
Asked: Total length of timber and minimum costs
incurred?
Answer: 𝒃𝟐
= 𝒄𝟐
− 𝒂𝟐
𝒙𝟐
= (𝟖𝟎 𝟐)𝟐
− 𝒙𝟐
(𝒙𝟐
+ 𝒙𝟐) = 𝟏𝟐, 𝟖𝟎𝟎
𝟐𝒙𝟐
= 𝟏𝟐, 𝟖𝟎𝟎
𝒙𝟐
=
𝟏𝟐,𝟖𝟎𝟎
𝟐
= 𝟔, 𝟒𝟎𝟎
𝒙 = 𝟔, 𝟒𝟎𝟎 = 𝟖𝟎 cm
If the length of one side of the wood is 80cm, then the length of the
four sides of the wood is = 𝟖𝟎 × 𝟒 = 𝟑𝟐𝟎 cm.
320 cm = …. converted to meters
= 3.2 m
minimum costs incurred:
= 3.2 × Price of wood per meter
= 3.2 × IDR 40,000 = IDR 128,000

293
In scientific terms, this research findings can serve as a reference for educators and teachers in designing
more effective learning strategies to enhance students’ mathematical belief and metacognitive knowledge in
mathematics thereby assisting in improving their learning achievements.
APPENDIX
Table 5. Differences and similarities of HMB and LMB metacognitive knowledge in solving geometry
problems
Polya’s problem
solving stages
HMB (subject with high belief) LMB (subject with low belief)
Identify known
information,
information being
asked, check the
adequacy of
information.
− Understanding the problem by reading and
identifying every word or sentence in the problem
and marking it as known and asked information in
the problem.
− Deciding the formula to be used from the subject’s
prior knowledge.
− Realizing the knowledge in understanding the
problem regarding how to select information to find
important sentences in the problem.
− Be aware of the steps that can be taken in
understanding the problem by first reading,
understanding, and remembering the topic being
discussed.
− Realizing to decide on an effective strategy to
understand the problem based on the learning
experience it has.
− Understanding the problem by reading and
identifying every word or sentence in the problem
and marking it as known and asked information in
the problem.
− Deciding the formula to be used from the subject’s
prior knowledge.
− Knowing the steps to be used, namely
noting/marking important things in the problem
and determining the formula used.
− Making decision by sketching a picture and setting
a formula to answer the given problem.
Identifying operations
and strategies in
designing a problem-
solving plan at hand.
− Able to identify and select important sentences and
determine topics that are appropriate to the
problem.
− Planning problem solving, namely in the form of
the ability to show the keywords in the problem and
know the topic being discussed.
− Identifying the steps that can be taken in planning
problem solving by re-reading the problem,
marking important information, writing down
important information in their own language, and
determining formulas and establishing the most
effective strategies to be able to plan problem
solving based on their learning experience.
Finding a solution,
checking every step of
the strategy that has
been set to prove the
strategy chosen is
correct.
− Recognizing and knowing how to apply formulas
that are appropriate to the topic being discussed.
− Realizing his ability to operate the formula
according to the topic being discussed.
− Completing sequentially according to the questions
on the problem and establishing the most effective
strategy to be able to apply the formula he has set
based on his learning experience.
− Realizing the ability to apply the things that are
known and the formulas to be used.
− Knowing the steps that can be taken by applying
the information contained in the problem to the
formula to be used.
− Be aware of making decisions using algebraic
principles and arithmetic operations.
Check the overall
effectiveness of the
problem approach and
assess the solutions
obtained
− Looking at the results obtained and knowledge
related to the topic being discussed.
− Checking the suitability of the solution.
− Be aware of the steps that had been taken by
adjusting the results obtained and re-calculate in an
alternative way if necessary.
− Using different alternatives to ensure that the
calculations carried out were in accordance with the
problem at hand.
− Be aware of the ability to find out in advance the
sides and angles of a square.
− Checking the suitability of the solution.
− Knowing the steps to be taken by matching the
results obtained with the information listed in the
problem.
− Using different alternatives to ensure that the
calculations carried out were in accordance with
the problem at hand.
REFERENCES
[1] D. Juniati and I. K. Budayasa, “Working Memory Capacity and Mathematics Anxiety of Mathematics Undergraduate Students
and Its Effect on Mathematics Achievement,” Journal for the Education of Gifted Young Scientists, vol. 8, no. 1, pp. 271–290,
Mar. 2020, doi: 10.17478/jegys.653518.
[2] N. X. Fincham, Metacognitive knowledge development and language learning in the context of web-based distance language
learning: A multiple-case study of adult EFL learners in China. Michigan State University. Educational Psychology and
Educational Technology, 2015.
[3] N. Fatmanissa and N. Qomaria, “Beliefs on Realism of Word Problems: A Case of Indonesian Prospective Mathematics
Teachers.,” Mathematics Teaching Research Journal, vol. 13, no. 4, pp. 221–241, 2021.

 ISSN: 2089-9823
294
[4] D. Juniati and I. K. Budayasa, “The mathematics anxiety: Do prospective math teachers also experience it?,” Journal of Physics:
Conference Series, vol. 1663, no. 1, p. 012032, Oct. 2020, doi: 10.1088/1742-6596/1663/1/012032.
[5] A. In’am, Reveal the solution of math problems. Aditya Media, 2015.
[6] J. I. Setyawati and S. Y. Indrasari, “Mathematics Belief and The Use of Metacognitive Strategy in Arithmetics Word Problem
Completion Among 3rd Elementary School Students,” 2018, doi: 10.2991/uipsur-17.2018.29.
[7] S. Sutama, et al., “Metacognition of Junior High School Students in Mathematics Problem Solving Based on Cognitive Style,”
Asian Journal of University Education, vol. 17, no. 1, p. 134, Mar. 2021, doi: 10.24191/ajue.v17i1.12604.
[8] M. Suliani, D. Juniati, and A. Lukito, “Analysis of students’ metacognition in solving mathematics problem,” 2022, p. 020064,
doi: 10.1063/5.0096032.
[9] A. Efklides and G. D. Sideridis, “Assessing Cognitive Failures,” European Journal of Psychological Assessment, vol. 25, no. 2,
pp. 69–72, Jan. 2009, doi: 10.1027/1015-5759.25.2.69.
[10] R. Charles, F. Lester, and P. O’Daffer, “How to evaluate progress in problem solving. The national council of teachers of
mathematics,” Inc: Reston, VA, USA, 1987.
[11] A. J. Baroody and R. T. Coslick, Problem solving, reasoning, and communicating, K-8: Helping children think mathematically.
Merrill, 1993.
[12] L. T. Lioe, H. K. Fai, and J. G. Hedberg, Students’ metacognitive problem-solving strategies in solving open-ended problems in
pairs. Sense Publishers, 2006.
[13] J. H. Flavell, “Metacognition and cognitive monitoring: A new area of cognitive–developmental inquiry.,” American
Psychologist, vol. 34, no. 10, pp. 906–911, Oct. 1979, doi: 10.1037/0003-066X.34.10.906.
[14] G. Schraw and R. S. Dennison, “Assessing Metacognitive Awareness,” Contemporary Educational Psychology, vol. 19, no. 4,
pp. 460–475, Oct. 1994, doi: 10.1006/ceps.1994.1033.
[15] M. Zulyanty, “Metacognitive Knowledge of High School Students in Solving Math Problems,” Journal of Education in
Mathematics, Science, and Technology, vol. 1, no. 1, pp. 1–8, 2018.
[16] T. Ozturk and B. Guven, “Evaluating Students’ Beliefs in Problem Solving Process: A Case Study,” EURASIA Journal of
Mathematics, Science and Technology Education, vol. 12, no. 3, Jul. 2016, doi: 10.12973/eurasia.2016.1208a.
[17] D. B. McLeod, Research on affect in mathematics education: A reconceptualization, vol. 1. Macmillan, 1992.
[18] A. H. Schoenfeld, “Explorations of Students’ Mathematical Beliefs and Behavior,” Journal for Research in Mathematics
Education, vol. 20, no. 4, p. 338, Jul. 1989, doi: 10.2307/749440.
[19] P. O. Eynde, E. de Corte, and L. Verschaffel, Beliefs and metacognition: An analysis of junior high students’ mathematics-related
beliefs, Metacognit. New York, NY, USA: Nova Science, 2006.
[20] P. Kloosterman and F. K. Stage, “Measuring Beliefs About Mathematical Problem Solving,” School Science and Mathematics,
vol. 92, no. 3, pp. 109–115, Mar. 1992, doi: 10.1111/j.1949-8594.1992.tb12154.x.
[21] D. Juniati and I. Ketut, “The Influence of Cognitive and Affective Factors on the Performance of Prospective Mathematics
Teachers,” European Journal of Educational Research, vol. 11, no. 3, pp. 1379–1391, Jul. 2022, doi: 10.12973/eu-jer.11.3.1379.
[22] A. M. Ferrara and C. C. Panlilio, “The role of metacognition in explaining the relationship between early adversity and reading
comprehension,” Children and Youth Services Review, vol. 112, p. 104884, May 2020, doi: 10.1016/j.childyouth.2020.104884.
[23] Sutarto, I. Dwi Hastuti, D. Fuster-Guillén, J. P. Palacios Garay, R. M. Hernández, and E. Namaziandost, “The Effect of Problem-
Based Learning on Metacognitive Ability in the Conjecturing Process of Junior High School Students,” Education Research
International, vol. 2022, pp. 1–10, Jan. 2022, doi: 10.1155/2022/2313448.
[24] M. Güven and cS Dilek Belet, “Primary School Teacher Trainees’ Opinions on Epistemological Beliefs and Metacognition,”
Elementary Education Online, vol. 9, no. 1, 2010.
[25] R. A. Tarmizi and M. A. A. Tarmizi, “Analysis of mathematical beliefs of Malaysian secondary school students,” Procedia -
Social and Behavioral Sciences, vol. 2, no. 2, pp. 4702–4706, 2010, doi: 10.1016/j.sbspro.2010.03.753.
[26] J. W. Creswell, Research design: Qualitative, quantitative, and mixed methods approaches(3rd ed.). Thousand Oaks, CA: Sage
Publications, Inc, 2009.
[27] Roscoe, Research Methods for Business. New York: Mc Graw Hill, 1982.
[28] A. Vissariou and D. Desli, “Metacognition in non-routine problem solving process of year 6 children,” in Eleventh Congress of
the European Society for Research in Mathematics Education, 2019, vol. TWG02, no. 25, [Online]. Available:
https://hal.science/hal-02401125.
[29] G. Polya, How to solve it: A new aspect of mathematical method. Princeton university press, 2004.
[30] J. I. Setyawati and S. Y. Indrasari, “Mathematical belief and the use of metacognitive strategies in solving word problems in
arithmetic in 3rd grade elementary school students,” Advances in Social Science, Education and Humanities Research, p. 139,
2018.
[31] A. Margono, Mardiyana, and H. E. Chrisnawati, “Analysis of Using Students’ Metacognitive Knowledge in Solving Contextual
Problems Based on Polya’s Stages,” Jurnal Pendidikan Matematika dan Matematika Solusi, vol. 2, no. 6, pp. 471–484, 2018.
[32] S. Tobias and H. T. Everson, “Knowing what you know and what you don’t know: further research on metacognitive knowledge
surveillance,” College Entrance Examination Board, 2002.
[33] E. E. Arikan and H. Ünal, “An Investigation of Eighth Grade Students’ Problem Posing Skills (Turkey Sample).,” Online
Submission, vol. 1, no. 1, pp. 23–30, 2015.
[34] L. Chen, W. Van Dooren, Q. Chen, and L. Verschaffel, “AN INVESTIGATION ON CHINESE TEACHERS’ REALISTIC
PROBLEM POSING AND PROBLEM SOLVING ABILITY AND BELIEFS,” International Journal of Science and
Mathematics Education, vol. 9, no. 4, pp. 919–948, Aug. 2011, doi: 10.1007/s10763-010-9259-7.
[35] M. Nicolaidou and G. N. Philippou, “Attitudes towards mathematics, self-efficacy and achievement in problem solving,” in
European Research in Mathematics III: Proceedings of the Third Conference of the European Society for Research in
mathematics, 2003, pp. 1–11.
[36] F. Pajares and J. Kranzler, “Self-Efficacy Beliefs and General Mental Ability in Mathematical Problem-Solving,” Contemporary
Educational Psychology, vol. 20, no. 4, pp. 426–443, Oct. 1995, doi: 10.1006/ceps.1995.1029.
[37] R. Riani, A. Asyril, and Z. Untu, “Metakognisi Siswa dalam Memecahkan Masalah Matematika,” Primatika : Jurnal Pendidikan
Matematika, vol. 11, no. 1, pp. 51–60, Jun. 2022, doi: 10.30872/primatika.v11i1.1064.
[38] A. A. Nicolaou and G. N. Philippou, “Efficacy beliefs, problem posing, and mathematics achievement,” in Proceedings of the V
Congress of the European Society for Research in Mathematics Education, 2007, pp. 308–317.
[39] K. S. Alzahrani, “Metacognition and Its Role in Mathematics Learning: an Exploration of the Perceptions of a Teacher and
Students in a Secondary School,” International Electronic Journal of Mathematics Education, vol. 12, no. 3, pp. 521–537, Jul.
2017, doi: 10.29333/iejme/629.

295
[40] E. Mokos and S. Kafoussi, “Elementary Student’ Spontaneous Metacognitive Functions in Different Types of Mathematical
Problems,” Journal of Research in Mathematics Education, vol. 2, no. 2, pp. 242–267, Jun. 2013, doi: 10.4471/redimat.2013.29.
[41] J. Dunlosky and R. A. Bjork, “The integrated nature of metamemory and memory,” Handbook of metamemory and memory, pp.
11–28, 2008.
[42] D. Juniati and I. K. Budayasa, “Geometry learning strategies with optimized technology to improve the performance of
undergraduate mathematics students,” World Transactions on Engineering and Technology Education, vol. 21, no. 1, pp. 26–31,
2023.
[43] I. Kozikoglu, “Investigating critical thinking in prospective teachers: Metacognitive skills, problem solving skills and academic
self-efficacy,” Journal of Social Studies Education Research, vol. 10, no. 2, pp. 111–130, 2019, [Online]. Available:
https://jsser.org/index.php/jsser/article/view/362.
BIOGRAPHIES OF AUTHORS
Mega Suliani is a Ph.D. Candidate, Department of Mathematics Education,
Graduate School, Universitas Negeri Surabaya, Ketintang St., Surabaya, Jawa Timur 60231,
Indonesia. Her research focuses mathematics education, especially in metacognition student in
problem solving, mathematical representation and student competence in learning. She can be
contacted at email: mega.19003@mhs.unesa.ac.id, megasuliani94@gmail.com.
Dwi Juniati graduated her doctoral program from Universite de Provence,
Marseille – France in 2002. She is a professor and senior lecturer at mathematics
undergraduate and doctoral program of mathematics education at Universitas Negeri Surabaya
(State University of Surabaya). Her research interest is mathematics education, cognitive in
learning and mathematics (Topology, Fractal and Fuzzy). Affiliation: Mathematics
Department, Universitas Negeri Surabaya, Indonesia. She can be contacted at email:
dwijuniati@unesa.ac.id.
Agung Lukito is an Assoc. Prof and senior lecturer at Universitas Negeri
Surabaya (State University of Surabaya), Indonesia. His research is focused on Mathematics
education, Coding theory, and Algebra. Affiliation: Mathematics Department, Universitas
Negeri Surabaya, Indonesia. He can be contacted at email: agunglukito@unesa.ac.id.

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