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Analysis of students’ understanding of mathematical concepts in the Faraid calculation using modulo arithmetic theory

Authors:
  • Universitas Ibrahimy, Situbondo, Indonesia
  • Universitas Ibrahimy, Situbondo

Abstract and Figures

This study aims to determine students' mathematical understanding of the modulo arithmetic concept, the Faraid concept, and the modulo arithmetic application in Faraid calculations. This study employs a combination research method (mixed methods) with Sequential Explanatory Design model in which the quantitative data analysis is supported by qualitative data analysis. Data collection techniques consist of test questions and interviews for sixth semester students of the Mathematics Education Study Program, the Faculty of Tarbiyah, Universitas Ibrahimy, Situbondo, Indonesia. The initial data analysis consists of the Normality Test and the Paired Samples Test. The second data analysis entails research subject selection, data reduction, data presentation, and conclusion. The findings show that: (1) the average value of understanding the modulo arithmetic concept is 84,100, while the average value of understanding Faraid concept is 83,500; (2) the statistical results show the significance of Sig. (2-tailed) is 0.016<0.05 and 0.014<0.05; it can be concluded that there is a significant difference in the results of the mathematical concept understanding test between the concept of modulo arithmetic theory and the application of the modulo arithmetic concept to Faraid calculations, and the Faraid concept with the application of the modulo arithmetic concept to Faraid calculations; (3) the second analysis is the interview, which is divided into three categories: high, medium, and low. The result of the interview is students' understanding of mathematical concepts through the use of modulo arithmetic theory as a new alternative method of solving Faraid calculations.
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AIP Conference Proceedings 2633, 030005 (2022); https://doi.org/10.1063/5.0102211 2633, 030005
© 2022 Author(s).
Analysis of students’ understanding of
mathematical concepts in the Faraid
calculation using modulo arithmetic theory
Cite as: AIP Conference Proceedings 2633, 030005 (2022); https://doi.org/10.1063/5.0102211
Published Online: 14 September 2022
Mohammad Tohir, Muzayyanatun Munawwarah, Saiful, et al.
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Analysis of Students' Understanding of Mathematical
Concepts in The Faraid Calculation Using Modulo
Arithmetic Theory
Mohammad Tohir1, a), Muzayyanatun Munawwarah1, Saiful2, Abd. Muqit3, Khoirul
Anwar2, Kandiri2, Asmuki2
1Study Program Student of Mathematics Education, Faculty of Education, Universitas Ibrahimy, Situbondo,
Indonesia
2Universitas Ibrahimy, Situbondo, Indonesia
3Universitas Islam Negeri (UIN) Sunan Ampel Surabaya, Surabaya, Indonesia
a) Corresponding author: matematohir@ibrahimy.ac.id
Abstract. This study aims to determine students' mathematical understanding of the modulo arithmetic concept,
the Faraid concept, and the modulo arithmetic application in Faraid calculations. This study employs a
combination research method (mixed methods) with Sequential Explanatory Design model in which the
quantitative data analysis is supported by qualitative data analysis. Data collection techniques consist of test
questions and interviews for sixth semester students of the Mathematics Education Study Program, the Faculty
of Tarbiyah, Universitas Ibrahimy, Situbondo, Indonesia. The initial data analysis consists of the Normality
Test and the Paired Samples Test. The second data analysis entails research subject selection, data reduction,
data presentation, and conclusion. The findings show that: (1) the average value of understanding the modulo
arithmetic concept is 84,100, while the average value of understanding Faraid concept is 83,500; (2) the
statistical results show the significance of Sig. (2-tailed) is 0.016 < 0.05 and 0.014 < 0.05; it can be concluded
that there is a significant difference in the results of the mathematical concept understanding test between the
concept of modulo arithmetic theory and the application of the modulo arithmetic concept to Faraid
calculations, and the Faraid concept with the application of the modulo arithmetic concept to Faraid
calculations; (3) the second analysis is the interview, which is divided into three categories: high, medium, and
low. The result of the interview is students' understanding of mathematical concepts through the use of modulo
arithmetic theory as a new alternative method of solving Faraid calculations.
INTRODUCTION
Mathematics is a subject that demands students to focus their minds in order to recall and recognize previously learned
material, so they must be able to understand the material's concept accurately. Students' accomplishment in
understanding the fundamental concepts of mathematics paves the way for the delivery of mathematical concepts in
the following material. Therefore, students must establish a connection between new and prior knowledge to
understand a mathematical concept. According to Sutadi, understanding a concept is the ability to capture and master
more than several facts related to particular meanings [1]. According to Dewanti, understanding concepts for students
will impact student learning outcomes, namely in the form of learning achievement, so it is critical for educators or
lecturers to improve students' understanding to develop problem-solving abilities [2]. Eggen states that understanding
concepts can demand a variety of cognitive tasks such as remembering, explaining, identifying facts, providing
examples, generalizing, applying, analogizing, and presenting new concepts in another way [3]. Thus, understanding
mathematical concepts refers to a person's ability to construct a mathematical concept based on the basic knowledge
they already possess using their own words and forming connections with the new knowledge gained in solving a
Mathematics Education and Learning
AIP Conf. Proc. 2633, 030005-1–030005-9; https://doi.org/10.1063/5.0102211
Published by AIP Publishing. 978-0-7354-4376-1/$30.00
030005-1
particular case or problem. However, the fact that occurs in the field is different from what is expected, as many
students still do not fully understand the material that has been taught. According to Chairunnisa, Apriliaswati, &
Rosnija, the first variable that causes inefficient education is passive students who do not enjoy or are not interested
in the teaching materials provided [4]. Armianti, Wildan, Robiansyah, Trissiana, & Prahmana also adds Silver and
Turmudi's opinion that forcing students to memorize all the material in a course is not good for them [5]. The best
thing to do is to ensure that students understand the material being taught because one's enthusiasm for something also
significantly impact one's understanding.
Students' difficulties in understanding the material often occur in subjects that require high perseverance and
accuracy, such as Faraid. Therefore, there is a need for innovations that can increase students' enthusiasm in learning
about Faraid. As the legal science that determines the procedure for distributing inheritance according to Islamic law,
Faraid is a challenging discipline for most students to learn. The reason for this is that, in addition to containing the
theory of the inheritance distribution according to Islamic law, Faraid also involves mathematical aspects of the
inheritance distribution to the eligible heirs [6]. Tohir explains that Faraid is closely related to mathematics, and the
two cannot be separated because mathematics is a science of arithmetic that helps Muslims easily distribute the
inheritance left by the deceased based on Islamic law contained in the Al-Quran and Hadith [7].
Due to their lack of understanding of Faraid, students are afraid to practice inheritance measures that should have
been implemented under the Shari'a. Consequently, Faraid is rarely practiced by some people and has been replaced
with a letter of testamentary in which the testator has decided the heirs' share of each property. Many scholars who
have mastered this field have passed away, and next generation has little interest in learning it since Faraid is a slightly
complicated branch of science in which each part of the section is related to other sections [8]. For instance, to know
the heir's share of the inheritance, students must also understand the concept of Faraid itself and to understand the
calculation, Faraid requires mathematical understanding. The complexity of determining the distribution and
calculation in Faraid resulted in some students having a very poor understanding of it.
Faraid is a branch of science that studies the rules for living inheritance distribution, including assets and legal
rights according to Islamic law [9]. According to Ula, Meliyana, Ilahiyah, & Tohir, Faraid discusses the issue of
inheritance rights (tirkah) for the deceased owner's heirs [10]. Junaidi also defines that Faraid or inheritance law is a
law that regulates the transfer of ownership rights of the inheritance (tirkah) to the heir, determining who are eligible
to become the heirs and how many shares each of them gets [11]. The property left by the deceased is the object of
Faraid, and the calculation of the share of the deceased's heirs must adhere to what has been stated in the Shari'a's
nash. According to Tohir, Faraid is closely related to mathematics and cannot be separated, considering that
mathematics serves as an arithmetic science that helps Muslims in distributing the inheritance left by the deceased
based on Islamic law contained in the Al-Quran and Hadith [7].
In mathematics, modulus operation is an operation that results in the remainder from dividing one number by
another. The word 'modulo' is usually abbreviated to (mod). Modulo arithmetic has been widely applied in various
fields, including ISBN (International Book Serial Number), hash functions (mathematical functions that can convert
numeric input values into compressed input), and cryptography (a science that studies techniques in mathematics
related to information security). According to Munir in his book entitled Discrete mathematics, modulo arithmetic is
a sub-chapter of algorithms and integers [12]. According to Masnur, Sukirwan, & Asih, Modulo arithmetic (modular
arithmetic) is a method in mathematics that expresses the remainder of an integer after being divided by another
integer, or it can also be said that modulo is a number operation that produces the division remainder of an integer
number to other numbers; usually, the word 'modulo' is shortened as (mod) [13]. Therefore, there are numerous
fundamental concepts in this course that students must understand.
The application of modulo arithmetic theory to the Faraid calculation process is an innovation in integrating the
two concepts. With the innovation in the Faraid calculation process, students are expected to understand the concept
of Faraid calculation more easily and gain a better understanding of the modulo arithmetic theory. Knowing the heirs'
inheritance share, students can immediately apply the concept of the modulo arithmetic theory with the basic formula,
namely "a mod m = r or a = mq + r, assuming a is the origin of the problem, m is the heirs' inheritance share (only the
denominator), q is the result of a and m, and r is the remainder of the distribution (if a is divided by m).
Research done by Lestari & Surya reveals that the Realistic Mathematics Education approach is effective towards
the ability to understand students’ mathematical concepts [14]. Findings shown by Ula et al. showed that the ability
of basic and mathematical fraction operations play a fundamental role in the distribution of inheritance rights. The
selection of the suitable problem-solving strategy is helpful in the distribution of inheritance rights, and therefore the
role of mathematics in that case could minimize controversy in the society [10]. The modulo arithmetic theory has
been widely used by previous researchers, including the use of it to test the validity of ISBN [15], determine marriage
date [16], test the validity of ISSN [17], and is indirectly used in Primbon Jawa by the Javanese people [18].
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Therefore, it is necessary to analyze students’ understanding towards the concepts of Faraid and modulo
arithmetic. This is aimed to make innovation in the Faraid science using the concepts of modulo arithmetic. Doing
so, the advantages of this study are aligned with the purpose of the study. That is to describe the understanding of the
Faraid concepts using modulo arithmetic theory in sixth semester students of Mathematics Education batch 2018 at
Universitas Ibrahimy Sukorejo.
METHOD
Research Design
This study uses mixed method as the research design. It combines qualitative and quantitative methods in order to
obtain more comprehensive, valid, reliable, and objective data [19]. According to Creswell, the concurrent embedded
strategy is a mixed methods strategy that applies the stages of collecting quantitative and qualitative data
simultaneously [20]. Qualitative descriptive approach is used to describe the creative thinking process of students that
is expected to solve mathematical problems based on the stages of the Wallas model. Saryono states that qualitative
research is used to investigate, describe, explain, and find the qualities or features of social influences that cannot be
explained, measured or described through a quantitative approach [21]. On the other hand, the descriptive quantitative
approach is used to analyze data obtained from the final semester exam scores and the results of the creative thinking
process test. According to Creswell, quantitative research is a type of educational research in which researchers decide
what to research, formulate specific questions, limit questions, collect measurable data from participants, analyze
numbers using statistics, and conduct impartial investigations objectively [20].
Participants
The subjects of this study are twenty-five students in the sixth semester of Mathematics Education study program on
the Faculty of Tarbiyah, at Universitas Ibrahimy. The indicators of the type of understanding are based on the
mathematical concepts of understanding by Anderson, L.W. & Krathwohl (2010) revised Bloom's Taxonomy. The
taxonomy includes remembering (C1), understanding (C2), applying (C3), analyzing (C4), evaluating (C5) and
creating (C6). There are also seven indicators that are developed at the cognitive process in the category of
understanding (C2). This includes interpreting, exemplifying, classifying, summarizing, inferring, comparing, and
explaining. Furthermore, the cognitive process in the category of applying (C3) consists of two kinds of cognitive
processes, namely executing familiar tasks and implementing unfamiliar tasks [22]. This study uses cognitive levels
C2 and C3 with reference to the indicators of the concept of understanding according to Skemp. Some of these
indicators indicate the level of student understanding. The indicators of relational understanding above, according to
Skemp, refer to indicators of the understanding concept by Kilpatrick, Swafford, & Findell (2002) [23]. Those are: (a)
the ability to restate the concepts that have been learned; (b) the ability to clarify objects based on whether or not the
requirements that make up the concept are met; (c) the ability to apply concepts algorithmically; (c) the ability to
provide examples of the concepts learned; (d) the ability to present concepts in the form of mathematical
representations; and (f) the ability to relate various concepts (internal and external mathematics).
Data Collection
The data collection techniques used in this study are tests of conceptual understanding, documentation, observation,
and interviews. The questions on the test of conceptual understanding that had been validated by three expert lecturers
using Aiken's Value were included in the moderate criteria because the V index value is > 0.4 and < 0.8 and is proven
to be reliable with the Cronbach Alpha value of the arithmetic aspect modulo 0.865, the Faraid science 0.619, and the
application of arithmetic modulo to the calculation of Faraid science 0.681. The guidelines and interview questions
had also passed the validity test with three expert lecturers. The test items that are provided in this research are: (1)
Faraid science test questions, (2) Modulo arithmetic test questions, and (3) Modulo arithmetic test questions for
calculating Faraid science. The instruments were made in the form of 10 test questions based on the indicators of
Skemp’s concept of understanding which refers to the opinion of Kilpatrick and Findell. These are a grid of concept
understanding test questions, and concept understanding test questions and answer keys. The answers of twenty-five
students were then evaluated according to the scoring guidelines or assessment rubric that had been made.
Data Analysis
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The first stage of the data analysis technique is to determine students' understanding of the modulo arithmetic theory,
Faraid science, and whether or not there is a difference in understanding the concept after the application of Faraid
calculations with modulo arithmetic theory. Thus, before testing the data hypothesis, it is necessary to test the
normality of the data to find out whether the data is normally distributed or not with Chi Square formula which later
proven using SPSS version 26. Once done, a Paired Samples Test was carried out to find out whether there is a
significant mean difference between (1) modulo arithmetic by applying modulo arithmetic to Faraid calculations and
(2) Faraid science by applying modulo arithmetic to Faraid calculations. SPSS version 26 was used in the Paired
Sample Tests. At last, to test that the data is homogeneous, homogeneity test was carried out using the F test formula.
The second stage of data analysis technique is to analyze the results of interviews and documentation obtained
during the study. The interviews were conducted with three students representing three categories of high, middle,
and low groups. This were later analyzed using data reduction techniques, data presentation, and conclusions. Lastly,
the validity of the data was checked using a credibility test with triangulation activities. The source triangulation
checking was carried out to check the validity of research data by comparing data from the outside. Comparing the
test result data with interviews is a way to see and check the degree of trustworthiness of the information that had been
obtained.
RESULTS AND DISCUSSION
The data that had been obtained during the research are the results of a written test containing test questions for
understanding the concepts of modulo arithmetic, Faraid science, application questions in Faraid science calculations,
and interviews. The interviews were conducted to strengthen the results that had been analyzed in the previous stage.
Before using the research instruments, validity and reliability tests were carried out. The results of the validation of
the test questions by three expert lecturers using Aiken's Value are included in the moderate criteria because the index
value of V> 0.4 and < 0.8. The following table is Rater Agreement table to facilitate the arrangement of the score
results:
TABLE 1. Rater Agreement
No.
V
1
V
2
V
3
S1
S2
S3
s
CVI
Criteria
1
3
4
3
2
3
2
7
0,58
Middle
2
3
3
4
2
2
3
7
0,58
Middle
3
3
3
3
2
2
2
6
0,5
Middle
4
3
3
3
2
2
2
6
0,5
Middle
5
3
3
3
2
2
2
6
0,5
Middle
Note:
V : Validator
S : Subject
The results of Coefficient V are further interpreted according to the rules of < 0,4 = low, 0,4 - 0,8 = medium, and
> 0,8 = high. In conclusion the instrument validity rate indicated that the instrument in this study is valid with medium
validity level. After the instrument validity test is conducted, the instrument is further tested to fsict semester students
to prove its reliability. The result of reliability test for every aspect towards Cronbach Alpha value using SPSS 26 are
as the following: (1) The instrument aspect modulo arithmetic is proven to be reliable with Cronbach Alpha value >
0,6 resulted in 0,865; (2) The instrument aspect Faraid science is also proven to be reliable with Cronbach Alpha
value 0,619; and (3) The instrument aspect application of modulo arithmetic is proven reliable with Cronbach Alpha
value 0,681.
Concept understanding test is conducted on Wednesday, September 1, 2021 towards twenty-five students in the
sixth semester of Tadris Mathematics study program on the Faculty of Tarbiyah, at Universitas Ibrahimy. Tabel 2 in
the following are the result of concept understanding tests for the three aspects which are (1) faraid science; (2) modulo
arithmetic theory; and (3) application of modulo arithmetic theory on Faraid science calculation.
TABLE 2. Test Result of Students’ Concept Understanding
Aspect
N
Minimum
Maximum
Mean
Std.
Deviation
Varianc
e
Kurtosis
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Statistic
Statistic
Statistic
Statisti
c
Std.
Error
Statistic
Statistic
Statistic
Std.
Error
Modulo
Arithmetic
25
65.
00
100
84.100
1.934
9.
679
93.687
-
0.
687
0.902
Faraid Science
25
65.00
100
83.500
1.909
9.547
91.146
-0.620
0.902
Application
25
60.00
100
77.400
2.293
11.467
131.500
-0.023
0.902
The Table 2 above presents that the average score of students’ conceptual understanding of modulo arithmetic,
Faraid science concept, and the application of arithmetic modulo on Faraid science calculation are in the range of
84,100; 83.500; and 77,400. The minimum and maximum score on each aspect is 65,00 and 100. The data shows that
the students’ understanding of mathematical concepts on the three aspects is considered good and the third concept
on the application of modulo arithmetic on Faraid science needs to be paid attention to. The test results also show that
students’ understanding of the three concepts needs to be deepened for them to be better. The study shows that
students’ understanding on the application of modulo arithmetic concept on Faraid science can be a solution to
enhance their understanding on Faraid science, thus a better Faraid science learning is needed.
The mathematical concept understanding test result on students is further analysed to discover the students’
understanding on the modulo arithmetic theory, Faraid science, and the application of modulo theory in Faraid
science calculation. The analysis is done to find out whether; (1) there is any significant difference on modulo
arithmetic and the application of modulo arithmetic theory on Faraid science; and (2) there is any significant
difference on Faraid science and the application of modulo arithmetic on Faraid science.
Kolmogorov-Smirnov statistical test were deployed as a prerequisite test to find whether there are any significant
differences on Faraid science concept, modulo arithmetic concept, and the application of modulo arithmetic concept
on Faraid science calculation. Normality data test using the Kolmogorov-Smirnov statistics resulted in a significance
value of 0,200. Due to the significance value 0,200 > 0,05, then the residual value is distributed at normal level. After
the Kolmogorov-Smirnov test, statistical parametric test is deployed to find whether there are any significant
differences on Faraid science concept, modulo arithmetic concept, and the application of modulo arithmetic concept
on Faraid science calculation. The results on the Paired Samples Test are shown on Table 3 below.
TABLE 3. Paired Samples Test
Aspect
Paired Differences
t
df
Sig. (2-
tailed)
Mean
Std.
Deviation
Std.
Error
Mean
95% Confidence
Interval of the
Difference
Lower
Upper
Modulo Arithmetic -
Application
6.700
12.
985
2.
597
1.340
12
.
060
2.580
24
0.016
Faraid Science -
Application
6.100
11.
571
2.
314
1.324
10
.
876
2.636
24
0.014
Decision Making Basis
1. If value of sig. (2-tailed) < 0,05, there are a significant difference on the test result of mathematical understanding
concept between the understanding of modulo arithmetic theory concepts and the application of modulo arithmetic
on Faraid science calculation, as well as Faraid science concept understanding and the application of modulo
arithmetic on Faraid science calculation.
2. If the value of sig. (2-tailed) > 0,05, there are not significant difference on the test result of mathematical
understanding concept between the understanding of modulo arithmetic theory concepts and the application of
modulo arithmetic on Faraid science calculation, as well as Faraid science concept understanding and the
application of modulo arithmetic on Faraid science calculation
Based on the mathematical concepts understanding test results on the three aspects, the Paired Sample test presents
the value of sig. (2-tailed) 0,016 < 0,05 and 0,014 <0,05, thus the results show that there is a significant difference of
students’ ability in understanding modulo arithmetic concepts and the application of modulo arithmetic on Faraid
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science calculation, as well as Faraid science concept understanding and the application of modulo arithmetic on
Faraid science calculation.
Figure 1 below shows the high-achieving student’s (Subject A) answer on the understanding of mathematical
concepts on the calculation of Faraid science using the modulo arithmetic.
Based on the Subject A’s answer sheet as displayed on Figure 1, Subject A student shows a good understanding
of the concepts by answering the questions in a detailed manner. Students are also capable of determining the parts
received by the inheritance, using the conventional mode. Aside from that, Subject A student understands (C2) the
root problem of the Least Common Multiple (LCM) on the inheritance variable. Subject A student also owns a good
understanding (C2) of the basis theory of modulo arithmetic and is capable of applying (C3) the theory on Faraid
science calculation. Based on the interview, Subject A student shows a very good understanding of the concepts, and
the indicator of conceptual understanding has been thoroughly achieved according to the Klipatrick and Findell’s
indicator of conceptual understanding.
Subject A is capable of explaining the concept and making determination on parts received by the inheritance as
well as remembering the modulo arithmetic formula. Subject A is also capable of applying the concepts and presenting
examples of mathematical representation while calculating in a correct manner using the conventional mode. The
understanding that Subject A possesses helps them to find connection between the internal and external concepts,
namely figuring out root problems by using LCM on inheritance variables. Subject A also possesses a good
understanding of the basis theory of modulo arithmetic and is capable of applying it on Faraid science calculation. In
accordance with previous studies done by [24], Tohir, Maswar, Atikurrahman, Saiful, & Pradita, and Tohir, Abidin,
Dafik, & Hobri, students with high ability in preparation step will be able to identify problems in a well manner and
they will possess the both the useful and unuseful knowledge to solve the problem correctly [25] [26]. Munawwarah,
Laili, & Tohir further argued that students’ thinking process will move correctly as teachers expected if the information
FIGURE 1. Subject A answer sheet
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processor theory component from the stimulus up to long-term memory on students are well-functioning [27]. The
results of research by Lestari & Surya showed that Realistic Mathematics Education approach is effective on the
ability of students’ mathematical concept understanding [14].
The results of Subject B’s answers show that Subject B’s conceptual understanding is already great because the
student knows and memorizes the concept, however the student still has not understood the materials in detail, so the
student is still not skilful enough in applying the concept. The conceptual understanding towards the Faraid aspect is
already great, but it is still not enough to clarify the parts of the inheritors. However, in calculating the inheritance
conventionally, it is already correct. In the modulo arithmetic aspect, Subject B is already capable of understanding
and solving the provided questions correctly in this research. It can be said that Subject B is already capable of
correctly correlating the application of the modulo arithmetic in the Faraid calculation. The following interview results
towards Subject B show that Subject B’s understanding of the concept is already great and subject B has already
achieved most of the indicator of conceptual understanding according to Kilpatrick and Findell’s indicator of
conceptual understanding. The ability to restate the concept is already great, but there is a lack in the ability to clarify
the object which is deciding the parts of the inheritance that the inheritors will receive. Moreover, the ability to
memorize the formula of modulo arithmetic is already great. In applying the concept, Subject B is capable of providing
examples and coming out with mathematical representation which show that Subject B is able to calculate properly
and correctly according to the conventional method or the theory of modulo arithmetic. This strengthens the results of
Tohir, Atikurrahman, et al. research which indicates students with the ability are digging information, identifying the
questioned problems properly, but the consistency of choosing between the necessary and unnecessary information is
lacking in solving the problems [28]. This is in line with the statement from Tohir et al. which states that in solving
the problems, students will experience the process of realizing and arranging an approach towards the problems [29].
Furthermore, students will also choose the strategy and ask themselves regarding the problems.
Lastly, students with low conceptual understanding can get the big picture of the concept and answer the questions
by using their instincts and their prior learning experiences, but as they are not able to theoretically analyze their
results, the low achiever students cannot elaborate the steps of their results during the interviews. That is most possibly
because the students know the theory, but they do not understand it. The results of this study are in line with the results
of research by Ibrahim & Bahri on the use of modulo arithmetic showed that modulo arithmetic was useful for
determining test characters (check digits) in the ISBN [15]. These results strengthen the research result by Tohir,
Maswar, et al. which states that students with low interests in mathematics cannot solve the problems properly and
correctly [21]. According to Muttaqin, Susanto, Hobri, & Tohir, this is because the low achiever subjects do not
regularly memorize or often forget all the acquired information and they do not regularly repeat their prior language,
re-learn, and practice by doing similar questions at home [30]. In line with Tohir which indicates that someone can
forget the acquired information because that person fails to change the short-term memory into long-term memory as
they do not regularly study over the knowledge again and they are unable to categorize or unite their acquired
information [31].
In that sense, it can be concluded that high achiever students do not only know and memorize a concept or a theory,
but they know how to operate it into the other problems with similar characters. On the other hand, students with low
conceptual understanding know and memorize the concept, but they do not understand the theory in detail, so they are
not that skilful when they have to implement the theory. Lastly, students with low conceptual understanding can get
the big picture of the concept and sometimes they can get the right answer by guessing and using their instincts.
However, as they are not able to theoretically analyze their results, the low achiever students cannot elaborate the
steps of their results during the interviews. That happens because the students know the theory, but they do not
understand it. Students’ new perspective towards the Faraid with the modulo theory is however unable to significantly
increase students’ conceptual understanding. However, researchers hope students’ motivation to explore more Islamic
knowledge such as the Faraid science or the other knowledge that is related to mathematics and to understand that
every scientific knowledge is based on a mathematical concept.
CONCLUSION
In conclusion, according to research results and discussions, this study shows that: (1) Students’ understanding
towards the modulo arithmetic theory has the average of 84,100. There are 8 students (32%) within the high conceptual
understanding category, 12 students (48%) within the moderate conceptual understanding category, and 5 students
(20%) within the low conceptual understanding category. The analysis results of interviews towards conceptual
understanding indicate that: the high category is able to achieve the indicator of conceptual understanding, the
moderate category is able to achieve half of the indicator of conceptual understanding, and the low category is still
not able to achieve the indicator of conceptual understanding; (2) Students’ understanding towards the Faraid science
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has the average of 83,500. There are 7 students (28%) within the high conceptual understanding category, 12 students
(48%) within the moderate conceptual understanding category, and 6 students (24%) within the low conceptual
understanding category. The analysis results of interviews towards conceptual understanding indicate that the high
category is able to achieve the indicator of conceptual understanding, but students under the moderate category and
the low category have the same difficulty which is the difficulty to clarify the parts that the inheritors will receive; and
(3) The application of the modulo arithmetic theory towards the calculation of the Faraid has the average of 77,400.
There are 3 students (12%) within the high conceptual understanding category, 13 students (52%) within the moderate
conceptual understanding category, and 9 students (36%) within the low conceptual understanding category. The
analysis results of interviews towards conceptual understanding show that: students within the high category are able
to implement the concepts as their understanding towards the two concepts are already great, students under the
moderate category are also able to implement the concepts despite the difficulty in clarifying the parts that the
inheritors will receive, whereas students under the low category will have to understand more the concepts of modulo
arithmetic and Faraid science, so there will be no further mistakes.
The Wilcoxon test in SPSS version 26 showed the results of Asymp. Sig. = 0,034 for modulo arithmetic with the
implementation of modulo arithmetic towards Faraid science and 0,036 for Faraid science implementation of modulo
arithmetic towards Faraid science. Because both Asymp. Sig. < 0,05 thus the hypothesis is accepted which means
there is a significant difference in implementing the theory of modulo arithmetic in Faraid science calculation in sixth
semester students of the Mathematics Education Study Program, the faculty of Tarbiyah, Universitas Ibrahimy,
Situbondo, Indonesia. Students’ new perspectives towards Faraid science guide to new challenges to keep exploring
knowledge solutions in mathematical way.
ACKNOWLEDGMENTS
We would like to thanks for the support from Faculty of Education, Ibrahimy University, Situbondo, Indonesia, 2021.
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