Characteristics of Mathematical Reversible Thinking in Junior High School Students

. Reversible thinking is a mathematical competency that influences students' success in solving problems. Problem-solving is the core of mathematics education. This research aims to identify the characteristics of students' thinking in solving problems that require reversible thinking ability. A qualitative method with a case study approach was used in this research. The study used tests and interviews on 44 eighth-grade students in West Java, Indonesia. In-depth interviews were conducted with representative students whose answers were representative of the other students. All students' answers were analyzed using the thematic analysis software ATLAS.ti. According to the characteristic indicators of reversible thinking. This study found that junior high school students' ability in reversible thinking is not optimal. Some students who successfully solve problems are limited to using backward thinking processes rather than invertible ones. The other students still have difficulties constructing answers due to the limited context when students first learn the concept (problems with forward-thinking). Thus, this ability needs to be understood by students as one factor that supports the success of the problem-solving process.


Introduction
The capacity to solve problems is essential to the learning process of mathematics (Flanders, 2014).This is based on the statement contained in the decree of the Head of the Education Standards, Curriculum, and Assessment Agency (Ministry of Education, Culture, Research, and Technology, 2022) that one of the goals of mathematics education is to develop students' ability to solve problems, which includes understanding the problem, creating mathematical models, working with these models, or describing the results produced.
Additionally, mathematical problem-solving correlates with other mathematical abilities such as reasoning, decision-making, critical thinking, and creative thinking.According to Krutetskii and Hackenberg (Krutetskii, Teller, Kilpatrick, & Wirszup, 1976;Hackenberg, 2005), one of the abilities associated with problem-solving is the capacity to think or work in reverse.This cognitive process, referred to as reversible thinking, was originally introduced by Piaget.
The reversible thinking process is the mathematical ability to reverse the sequence of occurrence or arrange the direction of logical thinking from the end to return to the starting point (Saparwadi, 2022;Steffe & Olive, 2009).In other words, reversible thinking is a process of cognitive activity in finding a solution to a problem after determining the final result and being instructed to find the initial condition.Reversible thinking includes cognitive tasks where individuals must think logically in two interchangeable directions, establishing bidirectional relationships between concepts, principles, and procedures to strengthen their understanding (Flanders, 2014).It means the students' ability to think logically in two connected ways is required.Deep concept understanding combined with inventive logical reasoning assists students in navigating both forward and reversible thinking.To reduce mistakes in decisionmaking, students need to consider perspectives from both sides (Maf'ulah, Fitriyani, Yudianto, 2019).Additionally, Bruner mentioned that reversible thinking shapes students' cognitive understanding or knowledge (Simon, Kara, Placa, & Sandir, 2016).
Piaget expanded the concept of reversible thinking into two factors: negation and reciprocity.He said that every direct operation could be canceled, as shown by the negation indicator.In other words, every direct operation has its opposite (Oakley, 2004).For example, the multiplication operation is canceled by the division operation.Furthermore, the reciprocity indicator displays equal treatment of an equation or inequality.For instance, an equation 4+1 is equal to 2 + 2 +1.This formula can be perceived as the set of numbers on the right side and the compound of the numbers on the opposite side.Tzur (2004) defines reversibility within the realm of fractions.He conceptualized   as a fraction of a specific whole unit.For instance, the size of half a triangle cut from a wood piece is equivalent to that of half a rectangle cut from that same wood.Often, this perspective leads students to mistakenly believe that half of the triangle is bigger than half of the rectangle.
Reversible reasoning is required to understand that the whole combines each part and, conversely, each part is integrated to form a whole.This part-whole scheme can be examined in the fraction domain.For example, a rectangular is 2 3 of a shape unknown; what is the shape as a whole?Furthermore, Hackenberg (2005) defines the concept of reversibility as a part having a relationship to the whole whose solution can be described as a schema of quantitative equations.
She constructs questions and solutions that are based on quantitative situations.For example, twenty-eight ounces of juice is four times the amount you drank; how much did you drink?The process of finding one of these quantities can be written in the form of a linear equation ax = b to multiplicatively connect the unknown and known quantities.In this case, students must reason " inversely" to solve the splitting problem.According to Steffe (2001), the problem can be constructed more easily in the splitting operation because splitting involves separating a posited part from the given whole.
Mathematical subjects that necessitate understanding reversible thinking include the theme of comparison.In mathematical comparison, problems are often structured around the part-whole concept.Ramful and Olive's research (2008) highlighted that reversible thinking is frequently tied to mathematical operations, fractions, comparisons, algebra, and other mathematical aspects.Learning the comparison topic is crucial as it is a primary subject for school students.Moreover, comparison problems are also present in a PISA examination.For instance, in a PISA test, students are presented with data on two electric companies, each having an average percentage of faulty players per day but with different total players.Students are tasked with identifying which company has fewer faulty players.
Studies conducted in various countries varied in terms of levels, from elementary school students to higher education, have underscored the significance of fostering reversible thinking abilities to bridge gaps in this cognitive process (Maf'ulah, Juniati, & Siswono, 2016;Maf'ulah & Juniati, 2019;Ramful, 2014;Sangwin & Jones, 2017).A study by Maf'ulah et al. (2016) shows that students struggled with establishing reversible connections between functions and graphs.Similarly, they encountered difficulties when addressing problems involving reversible thinking concepts in arithmetic.The errors made by students include number manipulation problems, calculation inaccuracies, and problems in designing solution algorithms.Furthermore, students struggle with understanding the concept of reversible multiplication, especially in the context of fractions, which results in higher cognitive effort required to solve such problems.
Research on reversible thinking processes has been conducted previously to identify difficulty or error on functions and integer operations.Other studies related to reversible thinking are still rare.By using the keyword "reversible thinking," only 14 Scopus-indexed documents were obtained.However, this research will focus on the characteristics of students' reversible thinking on comparison.This study investigates the characteristics of reversible thinking among junior high school students.This research is expected to shed light on students' reversible thinking characteristics, which can serve as a basis for effective mathematics teaching strategies.In relation to this objective, the research questions include: (1) what is the dominant thought process for junior high school students?(2) how do junior high school students navigate the reversible thinking process?

Method
This research used qualitative methods with a case study approach.The stages in the research used were (Bungin, 2003): (1) collecting data from students' responses to reversible problems and conducting interviews; (2) filtering data to highlight what is most relevant and appropriate to the research.The data obtained will be assessed using reversible thinking indicators; (3) presenting data, revealing the results of data analysis, enabling the drawing of conclusions; (4) making conclusions, interpreting the data that has been collected and researched.
In this study, the indicators for reversible thinking were adopted from the indicator framework suggested by Maf'ulah and Hackenberg.Table 1 below shows the indicators of reversible thinking employed in this research (Hackenberg, 2010;Maf'ulah, Juniati, & Siswono, 2017) Data were collected from 44 students from grade VII of a junior high school in West Java, Indonesia.Participants who took the test were students who had received comparison topics.The test consists of four questions related to reversible thinking processes.The details of the questions are depicted in Figure 1.Data obtained from student written test results and transcripts of interview results were analyzed using ATLAS.tiapplication.This application has been applied in various qualitative research in education (Lukman, Istiyono, Kartowagiran, Retnawati, Kistoro, & Putranta, 2021).
The following steps are carried out in processing data by using ATLAS.ti: (

Results and Discussion
A total of 44 students who completed the test constructed answers through backward thinking strategies in each part of the problem.This statement is shown in Table 2, which analyzes the results of students' written answer responses based on reversible thinking indicators using ATLAS.tisoftware.
The codes in Table 2 state the definition related to students' reversible thinking ability.
For example, code 1-B2 indicates Problem 1 on the backward thinking indicator with the second code, namely "Determining multiples of a value".Labeling this code aims to facilitate a more detailed explanation of the work by students.The results of the ATLAS.tisoftware analysis display the intensity of the appearance of the code in students' written test answers included at the end of the code definition.Thus, the code labelled 1-B1 with the definition of "Constructing the process of finding the unit value" in Problem 1 appeared 30 times in students' responses.

Students' Thinking Strategies in Constructing Answer Number 1
In problem 1, students were asked to find the amount of rice that must be added for all disaster victims to get donations with the same amount of rice.Students were expected to find the relationship between the number of disaster refugees and the amount of rice donated.
Through this relationship, students could construct a comparison equivalence through both backward thinking and invertible thinking strategies.
Based on the analysis of students' answers to Problem 1, shown in Table 2, all students preferred to work with the backward thinking strategy.The strategy was carried out by students who determined the rice one refugee family obtained.Next, found the amount of rice needed by the number of refugees.For example, Figure 2 shows the students' answers to question 1.
Student's writing Translate Known: 300kg of rice consumed by 150 families Question: how many kg of rice should be added?Answer: 300: 15 = 2 30 x 2 = 60 So the rice that must be added so that all the remaining families get the same amount is 60 kg.In-depth interviews related to the students' answers in Figure 2 were also conducted to explore the backward thinking strategy.

Researcher : What is the known information in number 1? S1
: There are 30 families who don'

t get rice Researcher : Then what's next S1
: There are 300 kg for 150 families, so 1 family head gets 2 kg.Because 30 families have not received rice, 30 is multiplied by 2 kg.

Students' Thinking Strategies in Constructing Answer for Problem 2
In Problem 2, students were asked to find the chickens needed for a certain number of crocodiles.Students were expected to find a comparison relationship between the number of chickens as crocodile feed and whether each crocodile got the same amount of feed.Based on the analysis of students' answers in Table 2, students' thinking strategies in solving Problem 2 were more diverse.In the first strategy, the same as working on number 1, students found the unit value of the amount of feed needed, then looked for multiples to answer the question problem.
Furthermore, some students also work with an invertible thinking process, defining a statement to find something that is asked directly.However, the students' thinking process to find the answer needed to be reversed.For example, if 10 crocodiles ate 30 kilos of chicken in a day, how many additional crocodiles were needed to eat 42 kilos of chicken?The answer was constructed by the student in Figure 3.
Student's writing Translate Known: -10 crocodiles consume the same 30kg of meat -x crocodiles consumed 42 kg of meat Question: how many crocodiles (x) were recently delivered?Answer: number of crocodiles amount of food 10 30 Increased increased x 42 Figure 3. Answer for Problem 2 requiring students' invertible thinking processes From this answer, students added a note that showed a comparison: when there was an increase in meat to feed the crocodiles, the number of crocodiles should also increase.In this process, students' invertible thinking ability had gone well.Students could capture the initial information (10 crocodiles consume 30 kilos of meat) and then translate it into their minds.
Afterward, they made an inverted statement: x crocodiles must spend 42 kilos of meat.

Students' Thinking Strategies in Constructing Answer Number 3
In Problem 3, students were asked to determine the days of rest used by Mr Budi as a plait maker.When it was known that the number of plaits produced in the first week and the second week were different (with the statement that the number of plaits produced each day was the same).Just like the previous problem, students were expected to be able to make a linear relationship between the number of plaits and the time needed.In constructing the solution process, students also applied two strategies, as shown in Table 2. Some students worked using backward thinking strategies, and some used inverse thinking strategies.
The backward thinking process was shown when students found the number of plaits that could be produced in a day.Then, the value was used to determine the number of days used to make a certain number of plaits.Furthermore, the invertible thinking process was applied by making a linear relationship using a table.The students could interpret the sentence with the right variables, model the statement into comparison form, and perform the comparison operation well.

Students' Thinking Strategies in Constructing Answer Number 4
In Problem 4, students were expected to understand the function of variables to make the right equation model.Furthermore, students could find the relationship between the number of books and the amount of money that might be provided.
The concept of variables had been learned before comparison in grade 7, hence the temporary assumption that students had no difficulty defining these variables.However, in Problem 4, some successfully solved the problem with the backward thinking strategy, and other students did not succeed in the process of finding the comparison value because they were wrong in defining the variable to be sought.For example, the student in Figure 4 immediately adds up the amount of money that is thought to be Wati and Andi's money.

Students' writing
Translate Andi 20.000 Watii 30.000 20 + 30 = 50.000So, the answer is 50.000Furthermore, a minority of students who had successfully used the backward thinking strategy managed to interpret the sentence with the right variables, were able to find the unit price of a book, determine multiples of a value, and find the difference between two values.

Students Who Failed to Construct Answers
Based on the analysis of student answers and interviews presented in Table 2, there were students who had difficulties in each problem (Problems 1 to 4).The main cause of students who experience obstacles in constructing answers is students' lack of understanding of the comparison concept.Figure 5 shows the answers of students who had difficulty solving problems requiring backward thinking processes.

Student's writing
Translate Known: 10 crocodiles can consume 30 kilograms of chicken meat in a day.Assuming each crocodile gets the same amount of meat.Question: the crocodile keeper buys 42 kilograms of chicken meat because there are more crocodiles.How many crocodiles are newly sent?Answer: 30 : 10 = 3 42 : 3 = 14  (Tzur, 2004) because they can find a solution without memorizing the formula.
This strategy helps the students understand the problem so they can find solutions through various directions or points of view.Furthermore, invertible thinking ability is developing a solution strategy by directly changing the statement into its opposite.This thinking strategy is more difficult to apply because students must be able to interpret the sentence with the right variables when there is an addition or subtraction to the entity in question.
Nevertheless, not all students have the ability to think.Some students were unable to answer the questions correctly.This is due to several factors, including students' misunderstanding of the comparison topic.As stated by Rismayantini, Kadarisma, and Rohaeti (2021), high school students have limitations in the concept of comparison.Moreover, the presentation of different contexts leads to students' inability to understand the problem properly.
The different presentation referred to in this context is the problem that requires students to reverse their thinking process.Generally, mathematical problems are often presented as a forward-thinking process.Students who successfully solve comparison problems in the forwardthinking process are not necessarily successful when the thinking process is reversed (Pebrianti, Prabawanto, & Nurlaelah, 2023).This is in accordance with Suryadi (2019), who stated that students may have limited context when working on different problems.The existence of context limitations will generate perceptions in students that the problem being solved is a new problem that has never been studied before.This can indicate that students cannot understand the concept well and comprehensively (Wahyuningrum, Suryadi, & Turmudi, 2019).Therefore, the characteristics of reversible thinking of junior school students in reversible thinking are not optimal to use.As for the few students who can do it, the majority use the backward flow.
On the other hand, reversible thinking stands as one of the essential mathematical competencies that students must possess to enhance their problem-solving capabilities (Saparwadi, Sa'dijah, As'ari, & Chandrad, 2020;Simon et al., 2016).Students must grasp the underlying concept comprehensively through engagement in reversible thinking, enabling them to work inversely or backward from the worked-out procedure.Furthermore, students must establish connections between various concepts, allowing them to think in two directions and generate alternative solutions.This ability is important in tackling non-routine problems demanding high-level problem-solving ability.Students with reversible thinking ability can approach problems from multiple points of view, not limiting themselves to a single perspective.
The limitation in reversible thinking arises from students' struggle to create meaningful connections between mathematical concepts and their inability to establish bidirectional relationships.According to Maf'ulah et al. (2019), high school students encounter challenges in forming meaningful two-way associations between functions and their corresponding graphs.
Similarly, Ramful (2014) has proposed that students face difficulties conceptualizing multiplicative relationships in reverse, leading them to opt for more basic strategies.Moreover, there are scenarios where problems become simpler when approached with division, which is the reverse of multiplication.This observation is corroborated by other studies indicating that students across various educational levels, including elementary, secondary school, and prospective mathematics teachers, encounter difficulties when confronted with problems demanding reversible thinking ability (Maf'ulah et al., 2016;Maf'ulah & Juniati, 2020;Sutiarso, 2020;Sangwin & Jones, 2017).

Conclusion
This study has revealed the junior high school students' thinking strategies when given problems that require reversible thinking.In solving the problems, students dominantly use backward thinking (going back several steps to simplify calculations).This process is easier because students can analyze problems without formulas.However, not all students have this ability.Some students experience difficulties in making quantitative equations and limited context when they learn the concept of comparison for the first time.
This research has limitations.The students' thinking process investigated is only on the topic of comparison.Thus, it cannot be generalized to other topics.Future research within this area should be carried out on other mathematics topics, and the research results can be used as a basis for making learning designs.

Figure 1 .
Figure 1.Problems are solved by reversible thinking To obtain a deeper understanding of the student's responses, interviews were conducted with the eleven participants regarding their answers.The eleven students became representatives the sentence with the right variables, but the steps are correct(11)

Figure 2 .
Figure 2. Answer for Problem 1 requiring students' backward thinking processes

Figure 4 .
Figure 4. Student's answer for Problem 3 without using the right variables

Figure 5 .
Figure 5. Answer for Problem 2 without using the correct concept Based on the interview process, students also explained steps that did not follow the proper backward-thinking process.Researcher : Try to explain how to work No. 2! S2 : 30 divided by 10 Researcher : Why is 30 divided by 10? S2 : Initially, I wanted 10 divided by 30, but it's the same method as no.1.Because I doubt the answer number 1 is wrong, so I just flip the

Table 1 .
: Indicators of reversible thinking ability ReciprocityStudents can put equivalent relationships in equations.
1) Generate Code.Coding was done on each student's response on the writing test relevant to the research question.The code represents the abstract concept of each statement.
(2) Grouping Codes into Categories.Codes with the same basic concept are placed in one category.This aims to focus on ideas that have similar underlying ideas.There were 12 categories determined.(3)Grouping Categories into Themes.The next step is to group the categories into themes containing eight themes.

Table 2 .
Results of themes, categories and codes on student responses