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Al-Jabar: Jurnal Pendidikan Matematika
Volume 15, Number 02, 2024, Pages 559 – 576
DOI: 10.24042/ajpm.v15i2.24527
559
Reversible thinking in solving mathematics
problems in terms of cognitive style
Hakmi Rais Fauzan*, Erry Hidayanto, Tjang Daniel Chandra
Universitas Negeri Malang, Indonesia
raisfauzanhakmi@gmail.com*
Abstract
Article Information
Submitted Nov 05, 2024
Accepted Dec 12, 2024
Published Dec 22, 2024
Keywords
Cognitive Style;
Field-Dependent;
Field-Independent;
Problem-Solving;
Reversible Thinking.
Background: Reversible thinking, the ability to think bidirectionally, is a
crucial component of mathematical problem-solving. Differences in
cognitive styles, particularly field-dependent and field-independent
characteristics, play a significant role in students' reversible thinking,
necessitating a deeper exploration of these relationships.
Aim: This study aims to describe students' reversible thinking processes in
solving mathematical problems based on their cognitive styles, focusing on
field-dependent and field-independent traits.
Method: A qualitative descriptive approach was applied to 32 eighth-grade
students from a junior high school in Malang City, Indonesia. Data were
collected using the Group Embedded Figures Test (GEFT), a reversible
thinking test, and semi-structured interviews. Students were categorized
into field-dependent and field-independent groups using GEFT before
undertaking a reversible thinking test. Semi-structured interviews were
conducted to gain deeper insights into their problem-solving approaches.
Results: The findings indicate that students with field-independent
cognitive styles exhibit better performance in the aspects of negation and
reciprocity. They carefully apply problem-solving strategies, consistently
reverting to initial values after achieving correct solutions. Conversely,
students with field-dependent cognitive styles are more prone to errors,
particularly in changing operation signs and applying the concept of
reciprocal equivalence.
Conclusion: This study highlights significant differences in reversible
thinking between students with field-dependent and field-independent
cognitive styles. The results suggest the need for tailored teaching methods
to enhance reversible thinking based on cognitive styles. Further research is
recommended to explore barriers and additional factors influencing
reversible thinking.
INTRODUCTION
As a fundamental part of education, mathematics shapes the way individuals think and
approach problems in their daily lives. Mathematics significantly influences various
aspects of human life, driving advancements in technology, science, and business
(Siregar & Nasution, 2019). Its contribution extends to sharpening logical reasoning and
systematic thinking, enabling students to better understand complex concepts. Through
mathematics education, students are trained in diverse ways of thinking that prepare them
to tackle real-world challenges (Darwanto, 2019). One prominent form of such thinking
is reversible thinking, which enhances analytical skills and equips students with the
ability to approach problems from different perspectives.
How to cite Fauzan, H. R., Hidayanto, E., & Chandra, T. D. (2024). Reversible thinking in solving mathematics problems
in terms of cognitive style. Al-Jabar: Pendidikan Matematika, 15(2), 559-576.
E-ISSN 2540-7562
Published by Mathematics Education Department, UIN Raden Intan Lampung
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
560
Mathematics cultivates various types of thinking skills, one of which is reversible
thinking, a crucial aspect of problem-solving in mathematical contexts. This cognitive
ability allows students to mentally retrace or reverse their thought processes. Inhelder &
Piaget (1958) defined reversibility as the capacity to mentally reverse a process to return
to its starting point. He identified two main forms: negation and reciprocity. Negation
refers to an operation having an inverse that neutralizes the original action, such as
division reversing multiplication or exponential functions reversing logarithms.
Reciprocity, on the other hand, involves relationships characterized by equivalence or
mutual compensation. Krutetskii (1976) further explored reversibility, describing it as a
process of shifting from forward to backward thinking. For example, if a student learns
a sequence of steps (A, B, C, D, E, F), they master the progression from A to F. By
reversing the order—from F back to A—they gain the ability to restructure their
reasoning process. Krutetskii’s framework highlights that reversibility encompasses
reversing operations, recognizing reciprocal relationships, and applying both theorems
and their inverses (Muzaini et al., 2021). This cognitive skill underscores the importance
of flexible thinking, equipping students to approach mathematical problems with a
broader perspective and greater adaptability.
Reversible thinking has been widely examined due to its significance in enhancing
students' problem-solving abilities in mathematics. Pebrianti et al. (2023) observed that
while many students demonstrated forward-thinking proficiency, some encountered
challenges when required to reverse their reasoning to construct answers. This difficulty
was linked to a lack of contextual understanding during the initial stages of learning,
highlighting the importance of developing more robust conceptual foundations.
Similarly, Purwaningrum & Sutiarso (2022) emphasized that strengthening reversible
thinking involves nurturing students' mathematical attitudes and reasoning skills through
varied instructional approaches, strategies, and models. Moreover, Balingga et al. (2016)
identified that students with lower reversible thinking capabilities often struggled with
solving unfamiliar problems, leading to confusion and difficulty in adapting to new
challenges. These insights underscore the critical role of reversible thinking in equipping
students with the flexibility and adaptability required for effective problem-solving.
Reversible thinking offers significant potential in enhancing students' ability to
approach and solve mathematical problems effectively. Maf’ulah & Juniati (2020)
highlighted that this ability aids in the development of new understandings within a
student’s cognitive framework, resulting in more meaningful learning experiences. This
is particularly valuable when students encounter problems that differ from examples
provided by their teachers. Similarly, Sutiarso (2020) emphasized that reversible thinking
plays a crucial role in stimulating students' mental knowledge and experiences, ultimately
reinforcing their problem-solving confidence. Additionally, Maf’Ulah et al. (2019) noted
that reversible thinking enables students to examine problems from both forward and
backward perspectives. As a result, students who develop strong reversible thinking
skills are better equipped to overcome challenges in mathematics with greater ease and
adaptability.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
561
The variation in students' reversible thinking abilities highlights the complexity of
this cognitive skill in the context of mathematics learning. Amalia et al. (2024)
emphasized that differences in these skills are evident among students. Supporting this,
Purwaningrum & Sutiarso (2022) found that among 31 students tested, only 20.96%
demonstrated reversible thinking skills, while 79.04% lacked them. Similarly, Sutiarso
(2020) revealed that of 40 students tested, 42.5% possessed reversible thinking abilities,
whereas 57.5% did not. Amalia (2024) further explained that these differences are
primarily influenced by how students comprehend and internalize material during the
learning process, which is closely tied to their cognitive styles. Such findings underscore
the importance of recognizing individual differences in cognitive processes to enhance
learning outcomes.
Cognitive style plays a crucial role in shaping how students approach and solve
mathematical problems. Ulya et al. (2014) described cognitive style as a unique way
individuals respond to, process, organize, and utilize ideas when facing various situations
or phenomena. Ngilawajan (2013) emphasized that each student has distinct potentials,
strategies, and thinking styles for addressing mathematical challenges. These differences
in cognitive processes are influenced by environmental factors and past educational
experiences, which categorize cognitive styles into field dependent and field independent
(Siregar & Nasution, 2019). Field independent students tend to adopt a more analytical
approach to problem-solving, while field dependent students often struggle with
identifying the core elements of a problem (Wulan & Anggraini, 2019). Observations
during the preliminary study revealed similar patterns, where some students
demonstrated ease in understanding and solving problems, while others faced significant
difficulties in interpreting the problem context and executing solutions. Sellah et al.
(2017) explained that variations in cognitive style directly influence how students think
and complete their tasks. Understanding these differences is crucial for developing
teaching strategies that cater to diverse cognitive styles and enhance overall learning
outcomes.
Previous research has discussed students' reversible thinking ability at the school
level (Balingga et al., 2016; Muzaini et al., 2021; Purwaningrum & Sutiarso, 2022) and
college (Sutiarso, 2020). Maf’ulah & Juniati (2020) explored the reversible thinking
ability of prospective teacher students through algebraic problem solving tasks, while
Ikram et al. (2018) examined students' reversible reasoning in composition function
problems. In addition, some studies have also focused on improving reversible thinking
skills through learning media/strategies (Bharata et al., 2022). However, this study is
different because it seeks to identify how students process the information provided in
mathematical problems. This is important, considering that differences in the way
students process information can affect their reversible thinking ability (Amalia et al.,
2024). The findings from this study are expected to make a significant contribution to
mathematics education theory by expanding the understanding of the relationship
between information processing and reversible thinking ability. Practically, this research
can serve as a basis for designing learning strategies that are more adaptive to the
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
562
different ways students understand and process lessons so as to improve overall
mathematics learning outcomes.
Considering the differences in how students process information and the influence
of cognitive styles, further research is essential to understand how reversible thinking
interacts with field-dependent and field-independent characteristics. Reversible thinking,
as a fundamental mathematical skill, plays a pivotal role in enhancing problem-solving
abilities. By addressing these cognitive styles, this study seeks to fill the gap in
understanding how students' information processing impacts their reversible thinking.
The findings are expected to contribute significantly to mathematics education theory
and practice, particularly in designing adaptive instructional strategies that align with
students' diverse cognitive profiles.
METHODS
Design
In this study, researchers used a qualitative approach that aims to describe the phenomena
experienced by research subjects, such as perceptions, behavior, actions, and motivation,
which are described narratively in natural situations (Waruwu, 2023). The qualitative
approach was chosen because it allows researchers to explore in depth the reversible
thinking process of students, which is complex and difficult to measure quantitatively.
As stated by Rusandi & Rusli (2021), descriptive qualitative research is a research
strategy that directs researchers systematically in investigating events, phenomena, and
various facts related to the research subject. This research is descriptive with a qualitative
approach to provide a detailed description of the reversible thinking ability of students in
solving mathematical problems in terms of field dependent and field independent
cognitive styles. Descriptive design was chosen because it is relevant to understanding
and revealing differences in students' thinking patterns based on their cognitive styles,
thus providing a comprehensive picture that supports the development of more effective
learning strategies.
Participants and Instruments
This study used GEFT (Group Embedded Figure Test) adapted from Witkin's theory to
determine the type of cognitive style of students, including field dependent cognitive
style and field independent cognitive style. The instrument developed by Witkin has been
tested for validity and reliability before, so the GEFT can be used directly without
validation (Davis, 2006). Meanwhile, the reversible thinking test used was first validated
by an expert in the field of mathematics education. Based on the validation process that
has been carried out, the validator states that the reversible thinking test instrument is
valid and can be used for research purposes.
The research subjects of this study were eighth-grade students in one of the junior
high schools in Malang, Indonesia. The research subjects were based on the results of the
Group Embedded Figure Test (GEFT) and the results of solving the reversible thinking
test. In the selection of research subjects, the first step is the grouping of cognitive styles
with GEFT. This cognitive style test was conducted to categorize students into the type
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
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of cognitive style field dependent (FD) and cognitive style field independent (FI). In the
next step, students were given a reversible thinking test. Based on the results of these two
tests selected 2 students with field dependent cognitive style and 2 students with field
independent cognitive style.
Data Analysis
The data collected in this research is qualitative. The analysis was carried out from the
beginning to the end of the research. The following are the stages: (1) Data reduction,
the data reduced in this study is in the form of GEFT test results and reversible thinking
tests, which are used to determine research subjects. At the same time, the interview data
is reduced by listening to the recorded interviews conducted, writing a summary of the
interview results, and discarding unnecessary data. (2) Data presentation, the reduced
data will provide a specific picture of students' reversible thinking in solving mathematics
problems. The results of the description of students' reversible thinking are presented in
narrative form to faciliatate concluding. (3). Making conclusions, on the results of the
presentation of data that has been reduced and presented previously. This stage is done
to discover how reversible thinking students in solve mathematical problems regarding
field dependent and field independent cognitive styles.
RESULTS AND DISCUSSION
Result
This section describes the data found in the field. Before being presented, the collected
data were selected according to the research needs. The data were selected based on their
relevance to the research objectives and their involvement in supporting other findings
in this study. Meanwhile, unnecessary data will be excluded and not presented in this
study.
Subjek field independent SFI1
Issue 1
Figure 1. SFI1’s first problem
SFI1’s problem-solving activity begins with understanding the problem. In this
activity, SFI1 understands important data and information, according to SFI1’s answer in
Figure 1. SFI1 wrote that the area of the first plot is , the area of the second
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
564
plot is , and the difference between the two plots is . Then, the
question asks about the area of each plot and the difference between the two plots.
The next step is SFI1 developing a solution plan. This activity is shown by making
an equation for the difference between the two plots. SFI1 did the calculation between the
two known plots, which SFI1 knew from the problem that the first plot was wider than
the second plot, . After that, SFI1 made an equation from the
results of the previous calculation with the known difference.
Furthemore, SFI1 continued to solve problems that gave rise to reversible thinking
indicators. SFI1 brought up the negation indicator when solving the problem, which can
be seen when changing the equation to the equation . As is
known, the negation aspect appears when students use inversion of related operations;
this can be seen when students cancel the subtraction of with . In addition, there
is also the aspect of reciprocity, this can be seen when SFI1 changed the form of the
equation to
. At the completion, as shown by SFI1, it can be concluded
that the reversible thinking indicators that appear are negation and reciprocity.
In the next activity, SFI1 returned to the initial equation by substituting into
each land plot area. Then, SFI1 was verified by recalculating the difference between the
two plots of land. This result strengthened SFI1’s belief that the final value he obtained
was correct. From this stage, it can be concluded that SFI1 raises an indicator of reversible
thinking, namely returning to the initial value after obtaining the result.
Issues 2
Figure 2. SFI1’s second problem answer
SFI1’s problem-solving activity begins with understanding the problem. In this
activity, SFI1 could identify the important data in the problem, as shown in SFI1’s answer
in Figure 2. SFI1 wrote the perimeter of the rectangle as 64 cm, length (3x+7) cm, and
width . Then, the problem asked for the value of length and width and
recalculated the perimeter of the rectangle. SFI1 could understand the purpose of the
problem well.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
565
The next step is developing a solution plan. This activity is shown by making the
formula for the perimeter of the rectangle. SFI1 used the formula for the perimeter of a
rectangle that had been learned previously. After that, he entered important data obtained
from the problem, such as length and width. SFI1 can develop a solution plan carefully
and accurately.
Furthermore, SFI1 succeeded in bringing up reversible thinking indicators, namely
negation and reciprocity. This can be seen when SFI1 changed the form
to . As is known, the negation aspect appears when students use inversion
of related operations, which, in this case, SFI1 cancels with . In addition, there
is also a reciprocity aspect; this can be seen when SFI1 performs an equivalent reciprocal
relationship where both segments are equally divided by ten so that the equation is
. If the student is involved a reciprocal relationship, then the student provides the same
treatment on both sides. From the stages of solving this problem, SFI1 brought up the
reversible thinking indicators of negation and reciprocity.
In the next activity, SFI1 recalculated the perimeter of the rectangle by substituting
in the length and width equations. This result strengthened SFI1’s belief that the
result obtained was correct. SFI1 can verify the perimeter of the rectangle again. From
this, it can be seen that the reversible thinking aspect emerges as the ability to return to
the initial value. This is indicated by the suitability of the answer obtained by SFI1 with
the information known from the problem.
Subjek filed independent SFI2
Issue 1
Figure 3. SFI2’s first problem answer
SFI2’s problem solving activity begins with understanding the problem. In this
activity SFI2 obtained information contained in the problem, according to SFI2’s answer
in Figure 3. SFI2 wrote that the area of the first plot is , the area of the
second plot is , and the difference between the two plots. SFI2 can
understand the purpose of the problem well, so SFI2 managed to write down all the
important information known and asked in the problem.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
566
The next step is SFI2 development of a solution plan. This activity is shown by
making an equation for the difference between the two plots. SFI2 did the calculation
between the two known plots, which SFI2 knew from the problem that the first plot was
wider than the second plot. After that, SFI2 made an equation from the results of the
previous calculation with the known difference. SFI2 can develop a solution plan well
and correctly.
Furthermore, SFI2 solved the problem by displaying indicators of reversible
thinking, where SFI2 used aspects of negation and reciprocity in solving the problem.
This can be seen when changing the equation to the equation
. The negation aspect appears when students use inversion of related operations, where
SFI2 cancels the substraction of with . In addition, there is also the aspect of
reciprocal relationship, which can be seen when SFI2 changed the form to
and obtained the value of . SFI2 can apply the step plan or problem solving
strategy well. It can be concluded that the reversible thinking indicators that appear are
negation and reciprocity.
In the next activity, SFI2 returned to the initial equation by substituting the value of
into each land plot area. Then, SFI2 was verified by recalculating the difference
between the two plots of land. This result strengthened SFI2’s belief that the results
obtained were correct. SFI2 can verify the value of the difference between the two plots
of land again. From this, it can be seen that the reversible thinking aspect emerges as the
ability to return to the initial value. This is indicated by the suitability of the answer
obtained by SFI2 with what is known from the problem.
Issue 2
Figure 4. SFI2’s second problem answer
SFI2’s problem solving activity begins with understanding the problem. In this
activity, SFI2 got important information from the problem, according to SFI2’s answer in
Figure 4. SFI2 wrote the perimeter of the rectangle, which is , length , and
width . Then, the question asks SFI2 to determine the length and width and
recalculate the perimeter of the rectangle. SFI2 understood the purpose of the problem
well and managed to explain the information known and asked in the problem.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
567
The next step is SFI2’s development of a solution plan. This activity is shown by
making the formula for the perimeter of the rectangle. SFI2 used the formula for the
perimeter of a rectangle that had been learnt previously. After that, he entered important
information from the problem, such as length and width. SFI2 can develop a solution plan
well and correctly.
In solving the problem, SFI2 managed to indentify the indicators of reversible
thinking, namely negation and reciprocity. This can be seen when SFI2 changes the form
to . As is known, the negation aspect appears when
students use inversion of related operations, which, in this case, SFI2 cancels with
. In addition, there is also a reciprocity aspect, this can be seen when SFI2 performs
a reciprocal relationship when changing the form of the equation from to
and obtaining the value . SFI2 can apply the step plan or problem solving strategy
well. At the completion stage, SFI2 brought up reversible thinking indicators: negation
and reciprocity.
In the next activity, SFI2 recalculated the value of the perimeter of the rectangle by
substituting the values in the length and width equation and immediately
recalculated the perimeter of the rectangle. This result strengthened SFI2‘s belief that the
result obtained was correct. SFI2 can verify the value of the perimeter of the rectangle
again. From this, it can be seen that the aspect of reversible thinking emerges as the
ability to return to the initial value. This is characterized by the suitability of the answer
obtained with the information known from the problem.
Subjek field dependent SFD1
Issue 1
Figure 5. SFD1’s first problem answer
SFD1‘s problem solving activity begins with understanding the problem. In this
activity, SFD1 can get important information contained in the problem, according to
SFD1‘s answer in Figure 5. SFD1 wrote the area of the first plot as , the
area of the second plot as , and the difference between the two plots as
. Then, SFD1 wrote down what was asked in the problem, namely the area of each
plot and the difference between the two plots. SFD1 understood the meaning of the
problem well, and SFD1 explained the important information known and asked in the
problem.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
568
The next step is SFD1 developing a solution plan. This activity is shown by making
an equation for the difference between the two areas of the plots. SFD1 made the equation
for the difference between the two known plots, with the first plot’s area minus the
second plot’s area. SFD1 was able to plan the solution well and correctly.
Furthermore, SFD1 performed algebraic operations by involving reversible
thinking indicators. SFD1 used aspects of negation and reciprocity to solve the problem.
This can be seen when SFD1 changed the form to
. In addition, there is also a reciprocity aspect, this can be seen when SFD1
changes the form to
. In the process of solving, SFD1 made mistakes, so
this affected the results at the next stage of the work. The mistake SFD1 made occurred
when SFD1 was wrong in grouping the terms of the equation. As explained earlier, the
reversible thinking indicators, namely negation and reciprocity, can be seen in the
working process.
In the next activity, SFD1 returned to the initial equation by substituting each land
plot area. Then, SFD1 was verified by recalculating the difference between the two plots
of land. SFD1 verified the difference between the value of the two plots. From this, it can
be seen that the reversible thinking aspect emerges as the ability to return to the initial
value. However, due to the completion process, there was an error in the previous stage,
thus making the recalculation of the difference value of the two plots of land wrong.
SFD1 also found an error in solving the problem he was working on due to the difference
between the final difference value he obtained and the difference known from the
problem.
Issue 2
Figure 6. SFD1’s second problem answer
SFD1 ‘s problem solving activity begins with understanding the problem. In this
activity, SFD1 got important information about the problem, according to SFD1‘s answer
in Figure 6. SFD1 wrote the perimeter of the rectangle as , length as ,
and width as , and the problem asked in the problem is the value of length,
width, and re-verification of the perimeter of the rectangle. SFD1 can understand the
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
569
purpose of the problem well, where SFD1 managed to explain the important information
known and asked in the problem.
The next step is SFD1’s development of a solution plan. This activity is shown by
making the equation for the perimeter of the rectangle. SFD1 used the formula for the
perimeter of a rectangle that he had learned before. After that, he included important
information that was known from the problem, such as the length, width, and perimeter
values of the rectangle. SFD1 could plan the solution well and correctly.
Furthermore, SFD1 performed algebraic operations involving reversible thinking
indicators. However, SFD1 made mistakes in the negation aspect, where SFD1 did not
succeed in using the inversion of the related operation, namely canceling the
multiplication of 2. Further errors were also seen at the problem solving stage, where
SFD1 also failed to use inversion of related operations on to . At the same time,
the reciprocity aspect is seen when SFD1 does the equivalent reciprocal relationship,
namely when
. At the completion stage, SFD1 only succeeded in using the
reciprocity indicator, while for the negation indicator, there were still errors, so further
reinforcement was needed in the future negation.
In the next activity, SFD1 recalculated the value of the perimeter of the rectangle
by substituting in the length and width equations, and SFD1 immediately
recalculated the perimeter of the rectangle. SFD1 can verify the value of the perimeter of
the rectangle again. From this, it can be seen that the aspect of reversible thinking
emerges as the ability to return to the initial value. However, an error in the previous
work made SFD1 less confident about the results of his work.
Subjek field dependent SFD2
Issue 1
Figure 7. SFD2’s first problem answer
SFD2‘s problem solving activity begins with understanding the problem. In this
activity, SFD2 obtained important information about the problem, according to SFD2‘s
answer in Figure 7. SFD2 wrote the area of the first plot is , the area of the
second plot is , and the difference between the two plots is . Then,
write down what is asked in the problem: determine the area of each plot and recalculate
the difference between the two plots. SFD2 understood the meaning of the problem well,
where SFD2 explained what information was known and asked in the problem.
The next step is SFD2 developing a solution plan This activity is shown by making
an equation for the difference of the two plots. SFD2 made the equation for the difference
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
570
of the two known plots, with the first plot’s area minus the second plot’s area. SFD2 can
develop a solution plan.
Furthermore, SFD2 performs algebraic operations by involving reversible thinking
indicators, but there are errors in the appearent aspect of reciprocity. This error occurred
when SFD2 changed the equation from to . In the next
stage, SFD2 succeeded in using an aspect reciprocity, namely when SFD2 changed the
equation to
. At the completion stage, SFD2 made a mistake, which
affected the next stage of work.
In the next activity, SFD2 returned to the initial equation by substituting
into each land plot area. Then, SFD2 was verified by recalculating the difference between
the two plots of land. SFD2 can verify the value of the difference between the two plots
of land. From this, it can be seen that the reversible thinking aspect emerges as the ability
to return to the initial value. However, due to the completion process, there was an error
in the previous stage, making the recalculation of the difference value of the two plots of
land wrong.
Issue 2
Figure 8. SFD2’s second problem answer
SFD2‘s problem solving activity begins with understanding the problem. In this
activity, SFD2 got important information on the problem, according to SFD2‘s answer in
Figure 8. SFD2 wrote the perimeter of the rectangle as , length as , and
width as . Then, what is asked in the problem is to determine the length and
width of the rectangle and recalculate the perimeter of the rectangle. SFD2 can understand
the meaning of the problem well, whereas SFD2 explains what information is known and
asked about the problem.
The next step is SFD2 developing a solution plan. This activity is shown by making
the equation for the perimeter of the rectangle. SFD2 used the formula for the perimeter
of the rectangle obtained previously. After that, he included important information that
was known about the problem. Furthermore, SFD2 performed algebraic operations by
involving reversible thinking indicators, but there was an error in using equivalent
reciprocal relationships, where when SFD2 changed the equation from
to . At a later stage, SFD2 used equivalent reciprocal relationships,
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
571
namely when SFD2 changed the equation to
. SFD2 was confused
about moving segments. It can be concluded that at the problem solving stage SFD2
only uses the reciprocity indicator, but there are still errors.
In the next activity, SFD2 recalculated the value of the perimeter of the rectangle
by substituting in the length and width equation, and SFD2 immediately
recalculated the perimeter of the rectangle. SFD2 can verify the value of the perimeter of
the rectangle again. From this, it can be seen that the aspect of reversible thinking
emerges as the ability to return to the initial value. However, an error in the previous
work made SFD2 less confident about the results of his work.
Discussion
Field independent subjects solve problems by involving aspects of reversible thinking.
The aspects of reversible thinking that appear in the problem solving process of field
independent subjects are negation and reciprocity. The negation aspect appears when
students use inversion of related operations, while the reciprocity aspect appears when
the subject performs compensation or equal reciprocity by giving the same treatment on
both sides (Maf’ulah & Juniati, 2020).
In the first problem, the negation aspect appears when the field independent subject
cancels the subtraction of with . This can be seen from the answer of the field
independent subject, namely when the field independent subject changes the form of the
equation from to . While the reciprocity aspect appears
when the field independent subject divides the two segments by . We can see this when
the field independent subject changes the form of the equation from to
so that the final value of is obtained. Based on this, it can be said that SFI1 and
SFI2 fulfill the indicators of reversible thinking, namely negation and reciprocity. This is
in line with research (Maf’ulah et al., 2023), which states that students with good
mathematical ability development will solve problems correctly. In line with that, his
research (Hasan, 2020) revealed that students with field independent cognitive styles
have a meticulous nature in explaining the problem and can organize the information
obtained so that in the process of solving the problem, they will get the right end.
The next aspect of reversible thinking appears to be a return to the initial value after
obtaining the results. This can be seen from the field independent subject's answer when
recalculating the difference between the two plots. The field independent subject first
calculates each plot area; after finding the area of each plot, the field independent subject
immediately calculates the difference. In the second problem, the activity of rechecking
the answer results was carried out by recalculating the formula for the perimeter of a
rectangle. This can be seen from the answer of the field independent subject, who
recalculates the perimeter of the rectangle by changing the value that has been obtained.
Field independent subjects can believe in the truth of the results they have obtained due
to the suitability of the final value obtained with the information known from the
problem. Therefore, it can be concluded that the field independent subject fulfills all
indicators of reversible thinking and solves the problem correctly. This is in line with
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
572
research Sutiarso (2020), which reveals that reversible thinking is very influential for
someone in solving mathematical problems. In line with that, in her research stated that
in solving mathematical problems, students who have the ability to think reversibly will
find it easier to deal with it (Purwaningrum & Sutiarso, 2022).
Field dependent subjects perform problem solving according to the plan that has
been formulated at the previous stage. However, SFD1 and SFD2 failed to solve the
problem according to the plan, where the field dependent subject could not perform the
solution steps properly. Students often make mistakes in solving problems due to a lack
of care or thoroughness when dealing with problems (Ramadhani & Roesdiana, 2023).
Errors made by field dependent subjects occur when changing the sign of operations and
when bringing up aspects of reversible thinking.
In the first problem, SFD1 made a mistake in changing the sign of the operation,
where when multiplying the operation with , SFD1 still wrote the result
. While in the reversible thinking aspect, SFD1 managed to bring up the negation
aspect and the reciprocity. The negation aspect occurs when SFD1 inversion of
related operations, where SFD1 cancels the addition operation and with
and . Likewise, for the reciprocity aspect, this can be seen when changing
the form of the equation from to
. On the other hand, SFD2 made a
mistake when changing the form of the equation to .
Here, SFD2 is also wrong in the equivalent reciprocal relationship but is correct when
using the negation aspect, which cancels the operation with .
In the second problem, SFD1 made mistakes in bringing up aspects of reversible
thinking. The first mistake made by SFD1 was when canceling the multiplication
operation of , where SFD1 was wrong in changing the form of the equation from
to . The second mistake made by SFD1 was when
canceling the operation of , where SFD1 mistakenly changed the form of the equation
from to . However, at a later stage, SFD1 managed to come
up with an equivalent reciprocal relationship, namely dividing both segments by . This
can be seen from SFD1's answer, which changes the form of the equation from
to
. Similarly, the mistake made by SFD2 failed to bring up the reversible aspect
of thinking. The error occurred when SFD2 was wrong in the equivalent reciprocal
relationship but was correct when using the negation aspect, which canceled the
operation with . The mistakes made by SFD2 in the second problem are the
same as the mistakes he made in the first problem. From these errors, we can see that the
field dependent subject is less careful when performing mathematical operations, so the
final value obtained is incorrect. This is in line with his research (Suraji et al., 2018),
which states that there are students who understand the problem but are less careful when
performing calculations. This is also in line with research (Alifah & Aripin, 2018), which
states that students with field dependent cognitive styles tend not to show a coherent line
of thinking, make inappropriate steps, and skip some important steps. As a result, the
final value obtained is not based on strong arguments, so it can be said that they failed in
solving mathematical problems.
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
573
In the final stage, the field dependent subject checked the answers that had been
obtained. The aspect of reversible thinking that appears at this stage is to return to the
initial value after obtaining the results. However, at this stage, the field dependent subject
realized that there was an error in his work, wherein, in the first problem, there was a
difference in the final value obtained with the information in the problem. SFD1 obtained
the final difference result of , and SFD2 obtained the final difference result of ,
while the difference known from the problem is . Likewise, with the second problem,
the field dependent subject realized the mistakes made, because there was a difference in
the circumference value obtained with the circumference known from the problem. SFD1
obtained the final perimeter result of , and SFD2 obtained the final perimeter result
, while the perimeter value known from the problem is . From this, it can be
concluded that the field dependent subject has not mastered the ability of reversible
thinking well enough to solve mathematical problems. This is in line with research
Maf’ulah et al. (2017), which states that some students still fail to bring up reversible
thinking skills in solving mathematical problems. In line with that, in research (Nurlatifah
& Hakim, 2024), stated that students who do not have good abilities in reversible thinking
will have difficulty solving problems correctly.
The theoretical difference between field independent and field dependent cognitive
styles significantly affects their reversible thinking ability. Field independent students'
analytical and problem-solving abilities enable them to utilize reversible thinking
effectively, ensuring accurate solutions and logical consistency. In contrast, field
dependent students' reliance on external guidance and difficulty in separating relevant
details often lead to errors in mathematical operations and incomplete problem solving.
While field independent students verified their results systematically, field dependent
students found that they made mistakes in the solution process due to the difference
between the final result and the initial known value.
The results of this study are in line with research findings by Pebrianti et al. (2023)
and Amalia et al. (2024), who stated that students with well-developed reversible
thinking ability can solve problems more effectively, while those without this ability face
significant challenges. By understanding this difference, educators can design various
approaches, strategies or learning models targeted at developing reversible thinking
ability, such as visual-media-based learning, which will potentially improve their
mathematics problem-solving performance.
Limitation and Suggestion for Further Research
This study has several limitations that may affect the results. This study only focuses on
reversible thinking and cognitive style, with subjects limited to certain groups. This may
affect the diversity of the data and limit the generalizability of the results. In addition,
data collection using the GEFT and reversible thinking test also has the potential for bias.
For example, reversible thinking tests can be influenced by the way students understand
the questions or external factors such as test anxiety, potentially reducing the accuracy
of the results. To overcome these limitations, it is recommended that future studies
expand the scope of subjects, including students from different levels of education and
Hakmi Rais Fauzan, Erry Hidayanto, Tjang Daniel Chandra
574
socio-economic backgrounds, so that the results of the study are more varied and can be
better generalised. In addition, future research could also involve more varied
instruments to reduce potential bias, such as a combination of performance-based tests
with in-depth interviews. The identification of other factors, such as barriers to reversible
thinking, logical-mathematical intelligence, maths anxiety or personality type, could also
enrich the understanding of the factors that influence reversible thinking in solving
mathematics problems.
CONCLUSIONS
There are significant differences in reversible thinking between students with field
independent and field dependent cognitive styles. Students with field independent
cognitive styles successfully used aspects of negation and reciprocal relationships in
problem solving. In contrast, students with field dependent cognitive styles had more
difficulty in using aspects of reversible thinking, especially in using inversion of related
operations. In the final stage of completion, both students with field independent and
field dependent cognitive styles attempted to return to the initial value after obtaining the
result. Students with field independent cognitive styles do it in a structured and
systematic of the answer results with the information in the problem is obtained. While
students with field dependent cognitive styles get the difference in the final answer with
the information in the problem, they are unsure of the answers they get. From this, it
shows that the ability to think reversibly students with cognitive style field independent
have a better tendency in solving math problems than students with cognitive style field
dependent.
AUTHOR CONTRIBUTIONS STATEMENT
All authors contributed to this research. HRF followed the guidance of each supervisor,
such as compiling research instruments, conducting data collection, reporting routine
activities, and compiling the final report. EH coordinates and is responsible for the
research process, as well as guiding and directing research both technically and
substantively. TDC assists the lead researcher, checks research instruments, and guides
students in data collection activities, data analysis processes, reporting research results,
and publication processes.
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