Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6, 3230-3247 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: rita_lefrida@untad.ac.id Students’ commognitive conflicts in solving absolute value inequalities Rita Lefrida1*, Sutji Rochaminah2, Bakri M3, Alfisyahra4 1,2,3,4Department of Mathematics Education, Universitas Tadulako, Palu, Indonesia; rita_lefrida@untad.ac.id (R.L.) sucipalu@gmail.com (S.R.) bakri88oke@gmail.com (B.M.) alfisyahra27@gmail.com (A.). Abstract: The material of absolute value inequalities is considered more difficult to study. There are still many undergraduate students who still make mistakes in understanding and solving problems related to this topic. Moreover, some of them do not yet know how to read the symbols. Therefore, these issues need to be studied more thoroughly. This research was conducted to explore the undergraduate students’ discourse on absolute value inequality tasks through commognitive analysis. A qualitative approach was applied, involving 78 participants; 11 of them were further interviewed. The analysis of the absolute value inequality tasks and the interview results revealed that there were commognitive conflicts that needed to be addressed. The findings of this research showed that commognitive conflicts arose when the undergraduate students applied the definition of absolute value and determined the solution set for the absolute value inequality problems. Keywords: Absolute value, Commognitive, Discourse, Meta-level learning. 1. Introduction Language will always be present wherever we are, even when we are silent (Sfard, 2021). Furthermore, according to Sfard (2021), attention to language in mathematics education is relatively new, as researchers have been keen on investigating mathematical activities without much consideration of the language used in those activities. Baroody (1993) stated that: (1) “Mathematics as language’; mathematics is not only a tool for thinking, finding patterns, or solving problems, but also a tool for conveying ideas precisely and clearly, and (2) “mathematics learning as social activity”; as a social activity, mathematics learning involves interactions between students and students. Additionally, communication between teachers and students is a crucial part of uncovering students’ mathematical potential. Mathematics is a unique language consisting of words, tables, and illustrations, such as graphs and symbols (Tong et al., 2021). Mathematics can be viewed as an activity of training a certain form of communication called ‘discourse.’ Thus, mathematics is a discourse that cannot be equated with the often-heard and criticized statement that mathematics is a language (Sfard, 2012; 2021). This is because there are several elements of mathematics that make the ‘discourse’ different, namely keywords, visual mediators, narratives, and routines. Based on the previous teaching experiences done by the researcher, some undergraduate students read the same symbols differently, and some of them even did not understand the symbols at all. There was also confusion in understanding mathematical terms, for instance, the students often equated the definition of derivative and differential terms. Lefrida et al. ’s (2023) earlier research found that students still made errors by saying that a function having a derivative meant it also had a limit. The students also stated that a function being said to be a ‘continuous’ function had a limit. From these findings, the responses done by the students are incorrect. According to the theory, a function that has a derivative is the function that has equal derivatives in the left and right. The gap in the use of words by people has been a phenomenon worth studying until now, especially in mathematics classrooms (Sfard, 2021). This is also in line with Akçakoca et al. (2024), who stated that communication in the classroom will always be equivalent to the thinking process. https://orcid.org/0000-0002-6915-3668 https://orcid.org/0009-0009-3943-0423 https://orcid.org/0000-0003-4266-537X https://orcid.org/0009-0005-2940-7335 3231 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Several previous studies have emphasized the relevance of the context in which learning occurs, based on Sfard’s cognitive approach (Emre-Akdoğan, Güçler, & Argün, 2018; Heyd-Metzuyanim & Graven, 2019; Nachlieli & Tabach, 2019; Lefrida et al., 2023; Akçakoca et al., 2024). Commognitive theory assumes that learning is not a process where someone changes a certain cognitive structure in their mind, but rather a process of changing participation routines within a particular community (Heyd-Metzuyanim, Smith, Bill & Resnick, 2019). Consequently, discourse analysis in the context of classroom learning provides insights into how learning occurs. This is linked to Nachlieli and Heyd- Metzuyanim (2022)’s research which explored commognitive conflict in mathematics education research by focusing on students' mathematical discourse. In this research, a commognitive perspective (Sfard, 2020) was applied to conceptualize mathematics learning as a change in a person’s mathematical discourse. Incommensurable discourses usually arise when inconsistent words, visual mediators, or routines are used in communication. When communicating, incommensurable discourse can lead to commognitive conflicts between one or more interlocutors. Hence, it can be said that commognitive conflict is a situation where participants use the same words in different ways (Sfard, 2007; Ben Zvi & Sfard, 2007). 2 Literature Review 2.1. Mathematics is a Discourse Given that communication is inherently collective, the term individual or personal discourse may seem like an oxymoron. Thought has been defined as self-communication. Indeed, personal discourse, mostly internal and silent, is largely beyond direct investigation. In fact, not all participants in a mathematical conversation group are capable of doing self-communication mathematically. Mathematics can be described as discourse about mathematical objects, such as numbers, functions, sets, and geometric shapes. A specific form of communication must be used to ensure that the mathematical narratives produced are reliable and unambiguous. Thus, mathematical activity can be seen as a practice of a certain form of communication, namely discourse. Specifically, the elements of discourse include: 1) Keywords in mathematical discourse; they primarily express quantity and form (numbers, geometric objects), as well as relationships between them (equality, inequality, similarity, equivalence, etc.). A symbolic system consists of vocabulary, a set of basic signs called words, and syntax, a set of rules for combining vocabulary elements into meaningful (or valid) utterances. 2) Visual mediators; they are defined as visible means that support communication, such as written words, symbols, and graphs. Moreover, literacy in discourse is defined as visual media, primarily through symbolic artifacts. In addition to algebraic symbols, symbolic artifacts also include icons, such as diagrams, graphs, and other images. Visual mediation involves the entire series of written symbols, such as numbers, tables, algebraic formulas, and even lines (imaginary). 3) Written or spoken texts; the written or spoken texts "framed as descriptions of objects, relationships between objects, or processes with or by objects" are considered narratives; for example, axioms, definitions, and proofs are examples of narratives in mathematical discourse. Endorsed narratives, as potentially useful narratives, are produced with the help of language, mediators, and certain routines. The elements that shape discourse are sets of routines with recurring forms of communicative actions. A repeated pattern of communication is called a routines. Keywords and visual mediators used regularly in narratives are also called routines. Routine mathematical discourse includes, for example, computation, problem solving and verification. It also includes new narratives that are proved (Sfard, 2008, 2020). Activities involving mathematical objects and mathematical narratives are discourse routines (Sfard, 2020). In the mathematics classroom, students' task is to individualize a discourse that historically known as mathematical discourse. The term individualizing mathematics means being able to participate competently in this discourse, not only when talking with others but also when communicating with oneself. Commognitive theory assumes that learning is not a process where 3232 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate someone changes a particular cognitive structure in their mind, but rather a process of changing participation routines within a particular community (Heyd-Metzuyanim et al., 2019). 2.2. Commognitive Conflict as a Stimulus for Meta-Level Learning If learning mathematics is a change in discourse, then we can distinguish between two types of learning: object-level learning and meta-level learning. First, object-level learning, which manifests itself in the expansion of existing discourse, can be achieved through vocabulary expansion, building new routines, and generating endorsed narratives. Second, meta-level learning, which involves changes in the metarules of discourse, is usually associated with exogenous changes in discourse. These changes mean that some familiar tasks, such as defining words or identifying geometric shapes, may now be performed differently in unfamiliar ways. One of the examples of these changes can be seen when certain familiar words may change in how they are read or used. Meta-level learning is most likely to occur from direct encounters between students and new discourse. Since this new discourse is governed by different metarules from the discourse that students are previously familiar with, it requires a commognitive conflict, a situation where different discourses act according to different metarules. The idea of commognitive conflict differs from the acquisitionist notion of cognitive conflict. There are three substantial differences, as shown in Table 1 below. Table 1. Concepts: cognitive conflict and commognitive conflict. Cognitive conflict Commognitive conflict Conflict happens between: Interlocutors and the world Incommensurable discourses Role in learning An alternative to eliminate misunderstanding Practically essential for meta-level learning How is it resolved? By students’ rational efforts By students’ acceptance and individualization of the expert interlocutor’s discursive way Source: Sfard (2008). Commognitive conflicts also exist in mathematical discovery. The conflicts may occur within a person between two overlapping discourses that are already embedded within him/her. Sometimes, during the transition from familiar to new discourse, a mathematician may endorse conflicting narratives. One of famous cases of such inner conflict is that of George Cantor, the inventor of set theory, who, in letters to mathematician Richard Dedekind, lamented his inability to resolve the contradiction between the familiar "truth" that a part was smaller than the whole and the conclusion he reached from his new theory, which stated that a subset of an infinite set could be "as big as" the whole set (Sfard, 2008). Another example, Ioannou (2016), reported commognitive conflicts experienced by students during Group Theory exercises. As a summary, thinking, according to the participatory framework, is conceptualized as communicating with oneself, either synchronously or asynchronously. Communicating with others or with oneself is largely verbal, or aided by other symbolic systems (Sfard, 2008). According to the psychology of communication about human thought, language is no longer viewed as a window into thinking but as its determinant element. Therefore, as long as thoughts exist in language, both thought and speech cannot be separated. 3. Research Method 3.1. Type of Research This research is an exploratory research which applies a qualitative approach. Qualitative research has four characteristics: (1) it focuses on processes, understanding, and meaning-making, (2) the researcher is the main instrument in data collection and analysis, (3) the research process occurs 3233 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate inductively, and (4) data presentation is in descriptive form, consisting of words and images (Merriam & Tisdell, 2016). In this research, the subjects were selected from Mathematics Education undergraduate students for the 2024/2025 academic year. In Figure 1, the Absolute Value Inequality Task was given to 78 undergraduate students: 50 second-year students and 28 third-year students. In the data collection process, supporting instruments used to gather research data were: (a) Absolute Value Inequality Task, (b) interviews, and (c) recording devices. The data analyzed in this study included interview transcripts, documents, and field notes. Data analysis was carried out using the interactive model analysis technique proposed by Miles et al. (2014), which included data condensation, data display, and conclusion drawing and verification. The quality of qualitative research was ensured through prolonged engagement with the subjects and member-checking. Figure 1. Absolute value inequality task. After completing the task, subjects were asked to determine the criteria according to themselves. The criteria are presented in Table 2 below. Of the 70 students who completed the task, 14 said the task was difficult. The rest described it as moderate or easy. Table 2. Question criteria according to the subjects. Students Difficult Moderate Easy Total Fourth semester 8 35 7 50 Sixth semester 6 11 11 28 14 36 18 78 4. Results and Discussion The task in Figure 1 was assigned to second-year and third-year students. They were asked to comprehend the task. Each student was required to propose an initial idea for solving the absolute value inequality problem. Each idea they mentioned was also to be written down. The ideas were further grouped, as shown in Tables 2 and Tables 3. 3234 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Table 2. Grouping of initial ideas (Second year students). No. Students Understanding and initial idea Notation Code 1 CHE,KAR, SAL, NUR, TIK, SUL, SIT, KHA, YUL, YUR, FIQ, OLI The property of absolute value is always positive SM41 2 ANN, SAKI Utilizing the definition of absolute value SM42 3 NIL, FIT, VIA, SRI, FAD Starting with |𝑥| < 𝑎 SM43 4 CIN, FAD, NUR, NI, ILU SRI, ZUL, FEL, AZIZ, WAH There are four possibilities, because 𝑥 − 3 and 𝑥 − 2 are within absolute value symbols SM44 5 FIT Removing the absolute value SM45 6 PAT, WID, YOS Removing the absolute value symbol SM46 7 RAH Consisting of two cases SM47 8 EKA, KAD, LUK, NAZI, TTY, WAN, NUR Determining the value of x SM48 9 EET, MIF, ISMI, ZIAN, GUS Replacing the < symbol with = symbol SM49 10 RIN, AZIM, MAL Shifting the absolute value SM10 3235 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Table 3. Grouping of initial ideas (Third year students). No. Students Understanding and initial idea Notation Code Subject Using the properties of inequalities 1 RO, ALI, SAN, ANGG, RAF, MAR, ZAIN, INDRI To operate on absolute value inequalities, the first step is to remove the symbol of absolute value SM61 2 SAL Expanding the inequality into four forms, namely SM62 3 RIF To simplify the inequality of absolute value functions, first multiply 1 and -1 into the absolute value functions of |𝑥 − 3| + |𝑥 + 2| < 11 SM63 4 SYA, FEB, DWI Based on the properties of absolute value SM64 5 SI Replacing the inequality symbol with = symbol SM65 6 RAT Consisting of four cases SM66 7 NUR, RUL, DIAN, HERI, MAG, ANI Using the definition of absolute value SM67 8 EVA Determining the value of x There are 3 solution regions SM68 9 ARW Squaring both sides SM69 There are ten initial ideas in Table 2 and nine initial ideas in Table 3. A difference of narratives among a group of students was observed. It indicates the presence of conflict. Furthermore, the researcher revisited the data to determine if any of the ideas mentioned and written could be combined. Seven discourses were identified from the combination of the narrative of second-year and third-year students.To further explore this commognitive conflict, 11 students were selected as subjects. Interviews were conducted with the subjects, and for each identified discourse, 1 or 2 subjects were 3236 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate interviewed. The subjects who were interviewed are highlighted in black and bold. The following presents the findings from the differing subject discourses. 4.1. The Property of Absolute Value is Always Positive (SM41, SM64) CHE is a second-year student at the university. The subject described the task as an easy category and mentioned the initial idea for solving the absolute value inequality problem as “The property of absolute value is always positive”. The subject responses are shown in Figure 2. Figure 2. Answer by subject CHE. To gain further insight into the subject’s ideas and work process, an interview was conducted, as shown in Table 4. Table 4. Discussion transcripts subject CHE. Interview Discussion Researcher Do you understand the problem? CHE Yes, ma’am. The absolute value symbol, it’s always positive. Researcher Do you remember the definition of absolute value? (Asking 1SM416) CHE The absolute value of |𝑥 − 3| becomes x -3, and |𝑥 + 2| becomes 𝑥 + 2. Once the absolute signs are removed, the value of x can be determined. Peneliti Any other thoughts? CHE No, ma’am. Subject CHE focused solely on substituting the value of 𝑥 into the function, stating that the result would always be positive. For example, |𝑥 − 3| becomes 𝑥 − 3, and |𝑥 + 2| becomes 𝑥 + 2. According to the subject, since the absolute value is always positive, they simply removed the absolute value signs. It aligns with the findings of Gagatsis & Panaoura (2014), which suggest that students often assume that |𝑥 + 3| = 𝑥 + 3 if 𝑥 > 0. The partial understanding of the absolute value definition may be due to a tendency of the students to memorize the definitions (Papadouris et al., 2024). 1. 4.2. Using |𝑥| < 𝑎 (SM43, SM61) 3237 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Subject RO’s approach to solving the absolute value inequality task is by using the property |𝑥| < 𝑎. Below is the process conducted by subject RO. Figure 3. Answer by subject RO. To gather information about the subject’s idea in solving the task, an interview was conducted, as shown in Table 5. Table 5. Discussion Transcripts Subject RO. Interview Discussion RO To solve this problem, I used the property of absolute value inequalities |𝑥| < 𝑎 Researcher How did you do it? RO By removing the absolute value sign, it becomes −𝑎 < 𝑥 < 𝑎 (SM4361a) Researcher Why did it become [SM4361b]? RO |𝑥 − 3| and |𝑥 − 2| are positive. Then I performed algebraic operations by grouping like terms. Next, I solved for x, which is the solution to the inequality of the value that fulfills the inequality. After that, I wrote down the range of the solution set and the inequality in its form without absolute values. Finally, I drew the graph of the solution. Subject RO used the property of absolute value inequalities |𝑥| < 𝑎. However, in code (SM4361b), the absolute value signs were removed without showing the process. Based on the interview in Table 5, 3238 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate RO explained that absolute values are always positive. RO and CHE share the same narrative in this regard, relying on the definition of absolute value. However, neither of them paid attention to the addition sign between the two absolute values. Additionally, there is a difference in how they applied the inequality property of −𝑎 < 𝑥 < 𝑎. CHE believed it was sufficient to solve only the right side of the inequality. On the other hand, RO worked on both sides. Nevertheless, both agreed that the absolute value is always positive. 4.3. Four Possible Solutions The discourse from subjects who suggested the solution based on four possibilities is coded as (SM42, SM44, SM45, SM46, SM47, SM62, SM66), while two possibilities are coded as (SM47). Two students, FEL and RAT, were selected for interview. Their responses are shown in Figure 4 and Figure 5. Figure 4. Answer by subject FEL. Table 6. Discussion transcripts subject FEL. Interview Discussion Researcher Do you understand the problem? FEL Yes, ma’am. Researcher Which four possibilities are you referring to? FEL This one, ma’am (points to the marked SM4266c) Researcher Do you mean −(𝑥 − 3) + 𝑥 + 2 < 11, 𝑥 − 3 < 0, 𝑥 + 2 ≥ 0 (referring to code SM4266c) FEL Yes, by removing the absolute value of x−3 , and for 𝑥 − 3 < 0 SM4266d SM4266e SM4266c 3239 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate can be written as – (𝑥 − 3). The same applies to the other terms, ma’am. Researcher Why did you write the inequalities 𝑥 < 6, 5 < 11, − 5 < 11 , 𝑥 > −5? FEL To find the value of x 5 < 11 is true for all x). Similarly, −5 < 11 is true for all x, so 𝑥𝜖𝑅 Researcher Look at the graph you made. Where is the region < 6 ? (SM4266e) FEL It’s in the region 𝑥 ∈ [3, ∞) Figure 5. Answer by subject RAT. Table 7. Discussion transcripts subject RAT. Interview Discussion Researcher Why did you approach the inequality with four possibilities? (SM4266f) RAT Based on the properties of absolute value, ma’am. Researcher What is that property? RAT |𝑥| = { 𝑥, 𝑥 ≥ 0 −𝑥, 𝑥 < 0 Researcher Can you explain this graph? (referring to SM4266g) RAT The graph divides the regions into three: 𝑥 < 2, 2 < 𝑥 < 3 𝑎𝑛𝑑 𝑥 ≥ 4 Peneliti Explain your reasoning. 3240 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate RAT Points 2 and 3 are determined by the limits of x from the four possibilities. 𝑥 < 2 is from the fourth possibility. 2 < 𝑥 < 3 because of the boundary for x I’m not sure about 𝑥 ≥ 4, but I think it’s because 𝑥 ≥ 3. Researcher Why did you test 𝑥 < 2 and 𝑥 < 3? (referring to SM4266h and SM4266i) RAT Based on points 2 and 3. Researcher You turned it into an equation, can you explain why?. RAT To find the value of x, ma’am. According to FEL, there are four boundary intervals, which imply four possible solution sets for the inequality. In contrast, subject RAT also wrote four intervals but divided the number line into three regions to determine the solution set for the absolute value inequality. There is a noticeable difference in the narratives regarding the domain of absolute values. Both students struggled to fully understand the use of boundaries, despite having written them down. It is evident when they write the interval 𝑥 < 2 and change the inequality to −𝑥 + 3 + (−𝑥 + 2) = 11. However, RAT was unable to complete the entire process of solving the task. 4.4. Multiplying 1 and -1 Into the Absolute Value Function Subject RIF stated to solve absolute function inequalities, we first multiply the mediator in the form of numbers, namely 1 and -1 into the absolute value function |𝑥 − 3| + |𝑥 + 2| < 11 (SM63, SM67). Figure 6. Answer by subject RIF and subject NUR. Table 8. Discussion transcripts subject RIF and subject NUR. Interview Discussion Researcher What process did you need to do? RIF First, I multiplied the inequalities by 1 and -1 Researcher What is the reason? RIF To remove the absolute sign, ma’am. NUR I used 𝑥 > 0 and 𝑥 < 0 Researcher How did you get 𝑥 > 0 and 𝑥 < 0 NUR By using the definition of absolute value, ma’am. 3241 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Researcher Next, what did you do? RIF I removed the absolute value symbol after the inequality is multiplied by 1 and -1. Researcher Are you sure? RIF Yes, ma’am. Researcher What are 𝑥 < 6 and 𝑥 > −5 for? RIF They are the final result, ma’am. NUR Solution set of inequalities, ma’am. Researcher How did you obtain it? RIF Solution set of inequalities is the combination of two inequalities. The narratives presented by RIF and NUR differ when they removed the absolute value from the inequality. However, they claimed to have used the definition of absolute value in inequalities. Subject RIF stated that to remove the absolute sign in inequalities, the left side of the inequalities was multiplied by -1 and 1. The narrative said by NUR was almost similar to RIF’s. However, NUR added the interval conditions, which were x>0 and <0. The subjects obtained the same solution set. However, there were differences between each subject in using absolute value. Based on commognitive perspective, definitions are endorsed through narratives. This is in accordance with the research by Gagagtisis & Panaoura (2014), which found that students assume |x + 3| = x + 3 if x > 0, and |x + 3| = - x + 3 if x < 0. According to Papadouris et al. (2024), difficulties were found in understanding the definition of absolute value as a piecewise function. This definition is directly related to the challenges students face when understanding basic relationships and ultimately leads to difficulties in solving simple equations. 4.5. Determining x (SM48, SM68) Figure 7. 3242 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate Answer by subject EKA. Table 9. Discussion transcripts subject EKA. Interview Discussion Peneliti You wrote the equation 𝑥 − 3 = 0 and 𝑥 − 2 = 0, what were they for? EKA I would determine the value of x, ma’am. Then, I divided it into three regions 3, namely 𝑥 < −2, −2 < 𝑥 < 3 𝑑𝑎𝑛 𝑥 ≥ 4 Peneliti How did you decide these regions? EKA I refer to the examples provided Peneliti You wrote −𝑥 + 3 + (𝑥 − 2) < 11. How did you come up with that? EKA It was based on equality 𝑥 < −2. As were the other regions, ma’am. Peneliti How about 5 < 11? EKA That is a true statement, ma’am, in the interval in between. Peneliti Hp={−5 < 𝑥 < 6}. What does it mean? EKA Himpunan penyelesaian (Solution set) Researcher Can you explain the process? EKA I took the points -5 and 6, ma’am. Figure 8. Answer by subject EVA. Table 10. Discussion transcripts subject EVA. Interview Discussion Peneliti You wrote 𝑥 − 3 = 0 and 𝑥 − 2 = 0, what were they for? EVA To determine x, to create a number line, ma’am. Peneliti Why did you create a number line? 3243 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate EVA To determine the solution region. For example, in the region 𝑥 < 2, it is determined that −𝑥 + 3 + (−𝑥 − 2) < 11 Peneliti. Alright. Next, how did you get the solution set? EVA From the solutions 1), 2) and 3) Peneliti What did you do? EVA It’s the combination of 𝑥 > −5 and 𝑥 < 6 Peneliti Okay Subject EKA and EVA created three intervals to determine the solution set of the absolute value inequality. Both subjects wrote the equations x-3=0 and x-2=0 and used them to create points on the intervals. EKA mentioned that the interval in the middle produced a true statement because it showed 5<11. Subject EVA stated that five minus three is a true statement. In the process of obtaining the solution set, EKA said it can be seen from points -5 and 6. Meanwhile, subject EVA stated that it is the union of x>-5 and x<6. EKA and EVA differed in their interpretations. 4.6. Replacing the Less Than Sign to an Equal Sign (SM49, SM65), Moving Absolute Value (SM410) Figure 9. Answer by subject EET. Table 11. Discussion transcripts subject EET. Interview Discussion Researcher Why did you change < symbol with = symbol? EET To create an equation, ma’am, to make it easier to determine x Researcher How is the process? EET Subtract |𝑥 − 3| from both sides of the equation. Researcher Is there still an absolute value? EET To eliminate the absolute value, I used the property |𝑥| = ±𝑥 The algebraic process gives 𝑥 = 6 and 𝑥 = −5 Researcher How did you obtain the interval (-5,6)? 3244 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate EET The value 𝑥 = 6 and 𝑥 = −5 were used to determine the interval of the inequalities that fulfill the condition. Researcher Why did it change back to an inequality? EET It is based on the question, ma’am. Figure 10. Answer by subject MAL. Table 12. Discussion transcripts subject MAL Interview Discussion Researcher Why did you change < symbol to = symbol? MAL To make it easier to determine x. Then, I used the property |𝑥| = ±𝑥. Researcher There is still an absolute value on the right side (SM4266) MAL I will work on it step by step, ma’am. Researcher This result is in an equation. Why did it become an interval? MAL Try it out. Subject EET stated that to simplify his work, he changed the less than sign to an equal sign. Next, EET used the definition |𝑥| = ±𝑥 to eliminate the absolute value. The two subjects did not simultaneously eliminate the absolute value, as seen in codes (SM4965e) and (SM4965f). There were commognitive conflicts of the subjects in the endorsed narratives when applying the definition|𝑥| = ±𝑥 . This aligns with the findings of Gagatsis & Panaoura (2014), where students assumed that ∣ |𝑥| = ±𝑥 𝑖mplies that for every 𝑥, |x| has two values. 3245 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate 4.7. Are Squared (SM69) Figure 11. Answer by subject ARW. Table 13. Discussion transcripts subject ARW. Interview Discussion Researcher What is your idea? ARW Absolute value inequalities can be solved by squaring both sides. Researcher Why must it be squared? ARW It uses the property |𝑥|2 = 𝑥2 Researcher Explain this (SM4266) ARW Separate the plus sign. I’m not able to do it yet, ma’am. Subject ARW used endorsed narratives that also represented the property of absolute value, namely |𝑥|2 = 𝑥2. However, based on interview results, they only knew the property without deeper understanding. What they did was merely replace the absolute value sign with a square sign, followed by the process of squaring the function. Based on findings from 78 students working on absolute value inequality assignments, there were 10 different ideas identified as commognitive conflicts. To explore further, in-depth interviews were conducted with 11 students selected as research subjects. This study found commognitive conflicts among subjects regarding the use of visual mediators and endorsed narratives. Thus, understanding the concept of absolute value is considered fundamental. In secondary and higher education, the concept of absolute value is regarded as one of the most fundamental concepts because it encompasses many mathematical concepts (Baştürk, 2023). 3246 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 3230-3247, 2024 DOI: 10.55214/25768484.v8i6.2687 © 2024 by the authors; licensee Learning Gate When subjects used the property of absolute value inequalities |x|