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Erroneous Examples in Refutational Text to Address the Phenomenal Sign Misconception in Equations and Inequalities

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Title: Erroneous Examples in Refutational Text to Address the Phenomenal Sign Misconception in Equations and Inequalities
Language: English
Authors: Konstantinos P. Christou (ORCID 0000-0002-7615-4968), Courtney Pollack (ORCID 0000-0002-0144-5971), Eleni Karagiannidou
Source: Instructional Science: An International Journal of the Learning Sciences. 2025 53(2):315-335.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 21
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Grade 9
High Schools
Junior High Schools
Middle Schools
Secondary Education
Descriptors: Foreign Countries, Grade 9, Mathematics Education, Mathematical Concepts, Symbols (Mathematics), Misconceptions, Algebra, Thinking Skills, Error Correction
Geographic Terms: Greece
DOI: 10.1007/s11251-024-09688-2
ISSN: 0020-4277
1573-1952
Abstract: The ability to solve equations and inequalities is necessary for success in algebra. However, reasoning biases and misconceptions may create barriers for students to build knowledge of algebraic symbols and their values. This study investigated whether students' errors when solving equations and inequalities could be attributed to their tendency to misinterpret the phenomenal sign of an expression (e.g., -2x interpreted as representing negative numbers only). Additionally, the study examined whether an intervention using erroneous reasoning examples in refutational texts would be more effective than correct examples in helping students address the specific misconception. The study involved 119 9th-grade Greek students who underwent Pre-, Post-, and Retention tests. The Experimental Group (N = 44) saw erroneous examples of reasoning with solving inequalities in refutational text, while the Control Group (N = 65) saw correct examples in non-refutational text. The results showed that students' misinterpretation of the phenomenal sign in algebraic expressions may influence their mistakes when solving certain kinds of equations and inequalities. Both erroneous and correct examples were effective in helping students address some of their misconceptions, although the gains were not sustained in the long term.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1487889
Database: ERIC
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  Value: <anid>AN0184555982;isl01apr.25;2025Apr21.02:37;v2.2.500</anid> <title id="AN0184555982-1">Erroneous examples in refutational text to address the phenomenal sign misconception in equations and inequalities </title> <p>The ability to solve equations and inequalities is necessary for success in algebra. However, reasoning biases and misconceptions may create barriers for students to build knowledge of algebraic symbols and their values. This study investigated whether students' errors when solving equations and inequalities could be attributed to their tendency to misinterpret the phenomenal sign of an expression (e.g., − 2x interpreted as representing negative numbers only). Additionally, the study examined whether an intervention using erroneous reasoning examples in refutational texts would be more effective than correct examples in helping students address the specific misconception. The study involved 119 9th-grade Greek students who underwent Pre-, Post-, and Retention tests. The Experimental Group (N = 44) saw erroneous examples of reasoning with solving inequalities in refutational text, while the Control Group (N = 65) saw correct examples in non-refutational text. The results showed that students' misinterpretation of the phenomenal sign in algebraic expressions may influence their mistakes when solving certain kinds of equations and inequalities. Both erroneous and correct examples were effective in helping students address some of their misconceptions, although the gains were not sustained in the long term.</p> <p>Keywords: Erroneous examples; Refutational text; Phenomenal sign misconception; Equations; Inequalities; Natural number bias</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0184555982-2">Introduction</hd> <p>The ability to solve equations and inequalities is a fundamental skill in algebra. However, students often encounter persistent difficulties in both areas (Booth et al., [<reflink idref="bib4" id="ref1">4</reflink>]). The significance of algebraic symbolism, including variables and algebraic expressions, and the ability to comprehend their operations, have been identified as crucial factors contributing to student difficulties and the occurrence of mistakes that lead to low performance (Kieran, [<reflink idref="bib27" id="ref2">27</reflink>]).</p> <p>To develop conceptual understanding of equations and inequalities, as well as their solutions, one must grasp the idea that certain numerical values can be substituted for the literal symbols to satisfy the given relationship. Hence, students' understanding of the number values associated with the literal symbols used in algebra to represent generalized numbers (i.e., indeterminate number values, one at a time) plays a crucial role in comprehending the solutions of equations and inequalities (Christou, [<reflink idref="bib7" id="ref3">7</reflink>]; Lee and Wheeler, [<reflink idref="bib31" id="ref4">31</reflink>]).</p> <p>Previous research has demonstrated that students tend to interpret literal symbols as standing primarily for natural numbers, and not as any rational or real number, positive or negative, as they are instructed when introduced to the concept of variable. As a byproduct of this tendency, students appear to interpret the phenomenal sign of an algebraic expression as the actual sign of the value it represents. This means that students may assume negative-like expressions, such as − 3x− 7, represent only negative numbers, while positive-like expressions, such as 6y + 2, represent only positive numbers. This cognitive predisposition, referred to as the <emph>phenomenal sign misconception</emph> (PSM), can significantly impact students' understanding of the solutions to equations and inequalities, and under certain conditions, it can also influence their ability to solve such tasks (Christou and Vosniadou, [<reflink idref="bib8" id="ref5">8</reflink>]; Christou et al., [<reflink idref="bib11" id="ref6">11</reflink>]).</p> <p>The present study focuses specifically on equations and inequalities in which the phenomenal sign misconception could lead students to make mistakes. Additionally, we examine the effect of a refutational text intervention that uses erroneous examples of reasoning based on the phenomenal sign misconception compared to non-refutational text of correctly worked examples. The refutational text approach may be more effective in helping students correct their mistakes. By targeting and addressing this misconception in the specific domain, we aim to enhance students' understanding and improve their performance in dealing with equations and inequalities.</p> <hd id="AN0184555982-3">Theoretical and empirical background</hd> <p>Throughout the history of mathematics education, researchers have been interested in investigating students' difficulties with equations and inequalities, given the significance of this domain in students' mathematical literacy. Extensive research has shed light on various reasons underlying misconceptions and challenges in equations and inequalities, with students' understanding of algebraic symbolism, including the equal sign and the use of literal symbols as variables, being prominent factors (Kieran, [<reflink idref="bib27" id="ref7">27</reflink>]).</p> <p>Although the power of algebraic symbolism is to serve as a concise and precise representation of mathematical ideas, detached from their specific context (Arcavi, [<reflink idref="bib2" id="ref8">2</reflink>]), this characteristic presents challenges from a semantic perspective, leading to difficulties in comprehension for students (Lee and Wheeler, [<reflink idref="bib31" id="ref9">31</reflink>]). This is because the detachment of symbols from specific meanings allows for the transformation of expressions using established techniques, such as following algorithms, without considering the individual meanings of the symbols involved. For example, Arcavi ([<reflink idref="bib2" id="ref10">2</reflink>]) describes a situation where someone could solve the equation (2<emph>x</emph> + 3)/(4<emph>x</emph> + 6) = 2, using the typical algorithm, even though she shouldn't have even tried to do so, since by <emph>reading</emph> the equation she could come up with the conclusion that this equation has no solution because the denominator is double the numerator, so the result could never equal 2. Thus, the <emph>reading</emph> of the symbolic expressions is a meaning making process that can add layers of connections and reasonableness to the results of procedures.</p> <p>When students are introduced to the notion that literal symbols represent numbers in algebra, they are typically informed explicitly that these symbols can represent numbers within a specified set. This understanding implies that the symbols can represent various types of numbers, including natural or rational, positive or negative numbers, and so on. However, early research found that students initially tend to interpret these symbols as abbreviated names of people or objects rather than as specific unknown numbers, generalized numbers, or variables (Kuchemann, [<reflink idref="bib30" id="ref11">30</reflink>]). While students develop a better understanding of literal symbols as specific unknowns over time, grasping their role as generalized numbers or variables remains challenging (Knuth et al., [<reflink idref="bib28" id="ref12">28</reflink>]; Lucariello et al., [<reflink idref="bib34" id="ref13">34</reflink>]).</p> <p>Due to the strong relationship between numbers and the concept of variables, Christou and Vosniadou ([<reflink idref="bib8" id="ref14">8</reflink>]) applied the framework theory approach to conceptual change, as used in the study of developing the number concept (Vosniadou et al., [<reflink idref="bib57" id="ref15">57</reflink>]), to explore how students interpret literal symbols representing variables in algebraic expressions. This theoretical framework suggests that students already possess a complex system of interconnected concepts and beliefs related to natural numbers before being introduced to non-natural numbers. The foundational concept of natural number may influence students' understanding of rational numbers, impacting their reasoning and problem-solving strategies across various tasks that involve reasoning with rational numbers, a phenomenon characterized as a <emph>natural (or whole) number bias</emph> (Ni and Zhou, [<reflink idref="bib38" id="ref16">38</reflink>]; Van Dooren et al., [<reflink idref="bib52" id="ref17">52</reflink>]).</p> <p>In a series of studies, Christou and colleagues (Christou and Vosniadou, [<reflink idref="bib12" id="ref18">12</reflink>], [<reflink idref="bib8" id="ref19">8</reflink>]; Christou et al., [<reflink idref="bib9" id="ref20">9</reflink>]) investigated the impact of the natural number bias on students' interpretations of algebraic expressions showing a tendency to interpret literal symbols primarily as natural numbers. Specifically, students tended to think that <emph>a</emph> stands only for natural numbers, while <emph>−b</emph> stands for negative integers, and similarly that <emph>k</emph> + <emph>3</emph> stands for positive numbers bigger than 3, <emph>4g</emph> stands for multiples of 4, and <emph>−d−4</emph> stands for negative integers smaller than 5. The students also consistently believed that, for example, 5<emph>d</emph> is always greater than 4/<emph>d</emph>, relying on testing with natural numbers only (Christou and Vosniadou, [<reflink idref="bib8" id="ref21">8</reflink>]; Christou et al., [<reflink idref="bib10" id="ref22">10</reflink>]).</p> <p>An intriguing finding from these studies, which has also been replicated in other research (Van Dooren et al., [<reflink idref="bib17" id="ref23">17</reflink>]), is students' tendency to associate the negative sign of variables and algebraic expressions with negativity (i.e., negative values; see also Vlassis, [<reflink idref="bib55" id="ref24">55</reflink>]), and the absence of the negative sign with positivity (Chiarugi et al., [<reflink idref="bib6" id="ref25">6</reflink>]). Consequently, students often perceive expressions like <emph>k</emph> + 3 as representing positive numbers and <emph>−d−4</emph> as representing negative numbers. This observed pattern suggests a systematic exclusion of positive numbers when algebraic expressions have a prominent negative sign and vice versa. Importantly, the signs attached to literal symbols do not always align with their values. This distinction between algebraic expressions and numbers in arithmetic can pose additional challenges when students develop the concept of variables (Wagner, [<reflink idref="bib58" id="ref26">58</reflink>]). Two possible explanations for this finding stem from a <emph>natural number bias</emph> perspective. First, it may be a direct effect of students interpreting literal symbols as natural numbers, leading them to categorize expressions such as <emph>−b</emph> as negative and <emph>k</emph> + 3 as positive. Alternatively, students may preconceive that the expression has a fixed sign based on the presence of the + or<emph>− </emph>symbols, retaining a feature of the basic interpretation of literal symbols as natural numbers, namely positivity (for a discussion see Christou et al., [<reflink idref="bib11" id="ref27">11</reflink>]). By extension, this <emph>phenomenal sign misconception</emph> (PSM) may influence students' reasoning about the potential solutions of equations and inequalities, leading to specific errors in certain types of tasks within these domains.</p> <p>When students associate only natural numbers with literal symbols, they may accept limited solutions, as often observed in equations like <emph>x</emph><sups><emph>2</emph></sups> = 25. Thus, the presence of a PSM may hinder the process of interpreting equations and inequalities, leading to specific mistakes as previous research has shown (Christou, [<reflink idref="bib7" id="ref28">7</reflink>]). For instance, students can incorrectly identify 4 as the sole solution for the quadratic equation <emph>x</emph><sups>2</sups> = 16 due to the PSM, overlooking the possibility of <emph>x</emph> representing the negative number <emph>−</emph>4. Similarly, the equation <emph>−y</emph> = 5 has a solution of <emph>−</emph>5, as <emph>− </emph>(<emph>−</emph>5) = 5; however, the PSM might lead students to believe that this equation has no solution since a negative number (-<emph>y</emph>) could not equal a positive one. Likewise, the PSM could influence students to conclude that the inequality <emph>−y</emph> < 5 is impossible because, in their understanding, a negative number (<emph>−y</emph>) cannot be smaller than a positive number. Consequently, students may provide incorrect responses, particularly when expressions with different phenomenal signs appear on both sides of the relation or in cases when both positive and negative number solutions are involved. This study aims to investigate these specific mistakes in equations and inequalities, as well as explore potential interventions to address these misconceptions.</p> <hd id="AN0184555982-4">Refutational texts</hd> <p>Previous research has demonstrated that misconceptions like the PSM are deeply rooted in students' existing knowledge, making them resistant to change. Additionally, students often lack awareness of these misconceptions and in order to address these challenges, cognitive conflict has been suggested as an effective approach (Christou and Vosniadou, [<reflink idref="bib8" id="ref29">8</reflink>]; Christou et al., [<reflink idref="bib11" id="ref30">11</reflink>]). Cognitive conflict situations expose students to their own contradictions, prompting them to become aware of and address their incorrect conceptions. Such situations are particularly beneficial in cases where prior knowledge revision is necessary to acquire more advanced and sophisticated knowledge, as observed in the context of conceptual change (Vosniadou et al., [<reflink idref="bib56" id="ref31">56</reflink>]).</p> <p>One way to create cognitive conflict is through refutational text. Refutation argumentation is a strategy that explicitly identifies a misconception, promptly refutes it, and then offers a persuasive explanation of the correct concept, often using examples and counterexamples (Hynd, [<reflink idref="bib22" id="ref32">22</reflink>]; Mason et al., [<reflink idref="bib36" id="ref33">36</reflink>]; Tippett, [<reflink idref="bib50" id="ref34">50</reflink>]). When presented in a textual format, refutational texts are texts that confront the incorrect conception that requires correction, explicitly stating its inaccuracy and providing a new, accurate alternative (Tippett, [<reflink idref="bib50" id="ref35">50</reflink>]). In this manner, refutational argumentation addresses pre-existing misconceptions deeply ingrained in prior knowledge (Hynd, [<reflink idref="bib22" id="ref36">22</reflink>]), thereby fostering learning through conceptual change (Sinatra and Broughton, [<reflink idref="bib46" id="ref37">46</reflink>]). Conversely, non-refutational text such as expository texts offer information about a specific concept or process, concentrating solely on the correct understanding.</p> <p>Extensive research has demonstrated that the use of refutational texts helps students overcome misconceptions leading to erroneous responses and low performance rates across various domains in science (Guzzetti et al., [<reflink idref="bib21" id="ref38">21</reflink>]; Mason et al., [<reflink idref="bib36" id="ref39">36</reflink>]; Sinatra and Broughton, [<reflink idref="bib46" id="ref40">46</reflink>]; Skopeliti and Vosniadou, [<reflink idref="bib47" id="ref41">47</reflink>]), particularly benefiting those with limited prior knowledge (Diakidoy et al., [<reflink idref="bib16" id="ref42">16</reflink>]). A recent meta-analysis encompassing research in mathematics, science, and social science domains demonstrated that refutational texts were generally effective for all learners, regardless of content domain, test type, or timing of test administration (Schroeder and Kucera, [<reflink idref="bib44" id="ref43">44</reflink>]). Considering misconceptions in mathematics, a few studies have tested the effects of refutational text, yielding promising outcomes (Christou, [<reflink idref="bib13" id="ref44">13</reflink>]; Lem et al., [<reflink idref="bib32" id="ref45">32</reflink>]).</p> <p>In a previous study (Christou, [<reflink idref="bib13" id="ref46">13</reflink>]), 10th-grade students were asked to determine whether given equations, inequalities, and functions, which included square roots and absolute values, would hold true under certain conditions. In the intervention, a small experimental group attended a refutational argumentation session in the form of a brief lecture. During this session, the issue of the phenomenal and the actual sign of the algebraic expressions was raised to create cognitive conflict. Several examples were provided for illustration. However, it's worth noting that these examples were not erroneous and did not lead to incorrect reasoning in those domains. The results showed promise to some extent, as they assisted students in addressing the implications of their misconceptions. The current study builds on the previous one by incorporating erroneous examples and focusing on reasoning about possible solutions of equations and inequalities that are commonly encountered in everyday mathematical classrooms.</p> <hd id="AN0184555982-5">Erroneous examples</hd> <p>Another potentially beneficial approach to improve students' understanding of mathematical concepts is the presentation of erroneous examples (Adams et al., [<reflink idref="bib1" id="ref47">1</reflink>]). Erroneous examples involve presentations of solutions or answers that include at least one mistake (Isotani et al., [<reflink idref="bib23" id="ref48">23</reflink>]; Zhao and Acosta-Tello, [<reflink idref="bib61" id="ref49">61</reflink>]). This approach, known as the erroneous example principle, complements the well-established worked example principle, which emphasizes the importance of studying correct examples for learning (for a review, see Van Gog et al., [<reflink idref="bib53" id="ref50">53</reflink>]). According to Adams et al., ([<reflink idref="bib1" id="ref51">1</reflink>]), erroneous examples have the potential to enhance learning by providing an appropriate level of challenge that stimulates generative processing in learners. These examples have demonstrated positive effects on understanding and skill development in various domains, such as medical diagnosis (Kopp et al., [<reflink idref="bib29" id="ref52">29</reflink>]) and the comprehension of 3D diagrams (Jaeger et al., [<reflink idref="bib24" id="ref53">24</reflink>]).</p> <p>Additional support for learning from errors comes from the productive failure theoretical approach (Kapur, [<reflink idref="bib25" id="ref54">25</reflink>]). This approach, which focuses on learning a new concept through novel problem solving, has shown that teaching students in this way leads to greater conceptual understanding and mental effort compared to introducing a new concept through direct instruction. This finding is consistent with constructivist theories that emphasize the importance of learning situations that create a state of disequilibrium when students encounter information that directly contradicts their naive theories or misconceptions (Piaget, [<reflink idref="bib40" id="ref55">40</reflink>]). Such confrontations, which can occur when students are presented with errors, may be essential for the growth and refinement of knowledge, and to foster deeper understanding and cognitive development. Previous research has also suggested that promoting negative knowledge (i.e., what not to do because it is ineffective) can enhance problem-solving skills (Oser and Spychiger, [<reflink idref="bib39" id="ref56">39</reflink>]).</p> <p>It appears that when students study and learn from other students' mistakes presented in erroneous examples, they may develop a deeper understanding of mathematics. Evidence of the benefits of using erroneous examples in explaining concepts, procedures, or correcting misconceptions in mathematics has varied. To address misconceptions, studying erroneous examples has been shown to help students become aware of and address parts of the misconception by correcting the associated mistakes (Adams et al., [<reflink idref="bib1" id="ref57">1</reflink>]; Durkin and Rittle-Johnson, [<reflink idref="bib18" id="ref58">18</reflink>]; Große and Renkl, [<reflink idref="bib20" id="ref59">20</reflink>]; Loibl and Leuders, [<reflink idref="bib33" id="ref60">33</reflink>]). Erroneous examples may improve students' understanding and mistake avoidance in domains such as rational numbers (Adams et al., [<reflink idref="bib1" id="ref61">1</reflink>]; Durkin and Rittle-Johnson, [<reflink idref="bib18" id="ref62">18</reflink>]; McLaren et al., [<reflink idref="bib37" id="ref63">37</reflink>]), equation solving (Booth et al., [<reflink idref="bib3" id="ref64">3</reflink>]; Zhao and Acosta-Tello, [<reflink idref="bib61" id="ref65">61</reflink>]) or exponents (Safadi and Hawa, [<reflink idref="bib42" id="ref66">42</reflink>]). Therefore, there are grounds to believe that the use of erroneous examples could help students recognize and address the PSM, thereby reducing its impact on reasoning about the possible solutions of equations and inequalities.</p> <p>However, the effectiveness of erroneous examples compared to worked examples may depend on various factors. Students may not necessarily benefit more from erroneous examples than from correct worked examples or problem-solving practice in areas like reasoning and critical thinking (Große and Renkl, [<reflink idref="bib20" id="ref67">20</reflink>]; Pillai et al., [<reflink idref="bib41" id="ref68">41</reflink>]; Van Peppen et al., [<reflink idref="bib54" id="ref69">54</reflink>]), calculation (Wang et al., [<reflink idref="bib59" id="ref70">59</reflink>]; Wesenberg et al., [<reflink idref="bib60" id="ref71">60</reflink>]), or decimal number understanding (Isotani et al., [<reflink idref="bib23" id="ref72">23</reflink>]). Additional pertinent factors include the opportunity for comparison with correct worked examples (Siegler, [<reflink idref="bib45" id="ref73">45</reflink>]; Zhao and Acosta-Tello, [<reflink idref="bib61" id="ref74">61</reflink>]), provision of feedback and scaffolding (Booth et al., [<reflink idref="bib3" id="ref75">3</reflink>]; Wang et al., [<reflink idref="bib59" id="ref76">59</reflink>]), and prompting students to trace and self-explain the mistakes (Loibl and Leuders, [<reflink idref="bib33" id="ref77">33</reflink>]; McLaren et al., [<reflink idref="bib37" id="ref78">37</reflink>]). Frequently, erroneous examples are presented without supplementary explicit explanations regarding the underlying causes of the erroneous reasoning that led to these mistakes. In cases where a mistake arises from a specific misconception, offering students a more detailed explanation of why the mistake occurred may enhance their awareness and result in deeper knowledge acquisition. In this context, refutational argumentation can serve as a valuable tool for facilitating learning. In other words, the combination of erroneous worked examples and refutational texts may be an effective strategy to assist students in addressing their PSM when reasoning about the solutions of equations and inequalities.</p> <hd id="AN0184555982-6">The present study</hd> <p>The primary objective of the present study was to explore students' specific mistakes when reasoning about the solutions of certain equations and inequalities. We investigated whether mistakes would be consistent with the tendency to misinterpret the phenomenal sign of expressions. The first hypothesis of the study was that students' tendency to misinterpret the phenomenal sign of the algebraic expressions as the sign of the only numbers they can represent would affect students' reasoning about the possible solutions of equations and inequalities leading to specific errors. In addition, we tested a second hypothesis that an intervention using incorrect examples presented with a refutational text would be more effective in helping students address errors related to the PSM in equations and inequalities than correctly solved examples presented in a non-refutational text format. Refutational texts often offer explicit explanations about misconceptions without directly pinpointing the specific errors resulting from these misconceptions. However, in the current study, the refutational texts provided erroneous examples of reasoning and explanations that connected the presented mistake with the phenomenal sign misconception. This general approach is also consistent with prior research, which suggests that providing students with guidance to recognize and comprehend errors can enhance their overall understanding (Adams et al., [<reflink idref="bib1" id="ref79">1</reflink>]; Große and Renkl, [<reflink idref="bib20" id="ref80">20</reflink>]). Here, refutational texts provided this guidance to help students become more aware of the underlying reasons for their mistakes and, consequently, to correct them. By amalgamating elements of erroneous example and refutational text methodologies, we aimed to leverage their respective strengths to enhance the potential impact of the intervention. Thus, we hypothesized that the intervention would be more effective in addressing students' PSM compared to correct examples provided in non-refutational text.</p> <hd id="AN0184555982-7">Method</hd> <p></p> <hd id="AN0184555982-8">Participants</hd> <p>Participants in this study comprised 119 9th-grade students attending a public high school in Greece, with an age range of 14 to 15 years old, including 57 female students. The sample was divided into two groups: the Experimental Group (N = 54) and the Control Group (N = 65), selected randomly from four classes within the same school. Participants had been introduced to rational numbers as early as 3rd grade, equations employing literal symbols to represent unknown values in 7th grade, and inequalities in 8th grade. By the 9th grade, they had gained substantial experience in both equation solving and presenting solutions, primarily focused on following algorithms.</p> <hd id="AN0184555982-9">Materials</hd> <p>The study used three tests: a pretest conducted before the intervention, a posttest immediately after the intervention, and a retention test administered one month later. Each test consisted of nine equations and nine inequalities. While the format of the tasks remained consistent across the tests, different numerical values were used in the posttest compared to the pretest. The retention test replicated the structure of the posttest.</p> <p>During the tests, participants were instructed to determine whether each equation or inequality had solutions or not. They were presented with two alternative responses and asked to justify their choice. For instance, they were given an equation or an inequality such as <emph>x</emph><sups>2</sups> + 1 = 5 and the question asked was, <emph>Can you find numbers that make the equation/inequality below hold?</emph> The two response options were: (a) <emph>Yes, the solutions are...</emph> and (b) <emph>No, there are no solutions because...</emph> The purpose of this question format was to encourage students to reason about potential solutions of the given equations and inequalities without relying solely on traditional solving algorithms.</p> <p>The tasks were designed to elicit errors from students who consistently misinterpreted the phenomenal sign of the variables and the algebraic expressions that appeared in the task. This was achieved by presenting algebraic expressions with different phenomenal signs on the two sides of each given relation, or by providing equations with both positive and negative solutions. To accomplish this, square roots and quadratic equations were utilized, as they both involve positive quantities. Additionally, the equations incorporated algebraic expressions featuring additive or multiplicative relations between numbers and literal symbols.</p> <p>The tasks required simple mental calculations to identify possible solutions and arrive at accurate responses, particularly when considering negative integers. For example, in the case of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><mn>2</mn><mi>x</mi><mspace width="0.166667em" /><mo>-</mo><mn>1</mn></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> </ephtml> ,solving the inequality using the algorithm would be more challenging than testing negative integers (<emph>−</emph>1, <emph>−</emph>2, 3, etc.), which could readily provide possible solutions.</p> <p>The initial section of each test comprised nine equations, as illustrated in Table 1. These equations fell into three categories: (a) Three of the equations had a single solution (e.g., <emph>−x</emph> = 17). It was anticipated that students exhibiting a PSM would respond by asserting that there is no solution because <emph>a positive quantity cannot equal a negative quantity;</emph> (b) Three quadratic equations with two integer solutions were included (e.g., <emph>y</emph><sups>2</sups> = 25), with the correct solutions being <emph>y</emph> = ±5. Students demonstrating PSM tendencies toward the numbers associated with the literal symbols would incorrectly provide only positive numbers as possible solutions; (c) Three quadratic equations with no solution were also incorporated (e.g., <emph>y</emph><sups>2</sups> = <emph>−</emph>16). These particular tasks were included to enable the alternative response <emph>No, there are no solutions because...</emph> to be correct in certain cases.</p> <p>Table 1 The equation tasks presented in each questionnaire</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" colspan="3"><p>Number of solutions</p></th></tr><tr><th align="left" /><th align="left"><p>Two solutions</p></th><th align="left"><p>One solution</p></th><th align="left"><p>No solution</p></th></tr></thead><tbody><tr><td align="left"><p>Pretest</p></td><td align="left"><p><italic>x</italic><sup>2</sup> + 1 = 5</p><p>y<sup>2</sup> = 25</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><msup><mrow><mo stretchy="false">(</mo><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>3</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq11.gif" /></p></td><td align="left"><p>−y− 1 = y + 1</p><p>−<italic>x</italic> = 17</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><mn>5</mn><mi>x</mi></mrow></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>5</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq12.gif" /></p></td><td align="left"><p>−2ּy<sup>2</sup> = 2</p><p>y<sup>2</sup> = −16</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>-</mo><msqrt><mi>x</mi></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>2</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq13.gif" /></p></td></tr><tr><td align="left"><p>Posttest and retention test</p></td><td align="left"><p>y<sup>2</sup> + 2 = 11</p><p>y<sup>2</sup> = 36</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><msup><mrow><mo stretchy="false">(</mo><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>2</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq14.gif" /></p></td><td align="left"><p>x + 5 = −x− 5</p><p>−y = 13</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><mi>y</mi></mrow></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>2</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq15.gif" /></p></td><td align="left"><p>−3<italic>x</italic><sup>2</sup> = 9</p><p><italic>x</italic><sup>2</sup> = −25</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>-</mo><msqrt><mi>y</mi></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>4</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq16.gif" /></p></td></tr></tbody></table> </ephtml> </p> <p>The second section of the test consisted of nine inequalities, sharing a similar structure to the equation tasks mentioned earlier (refer to Table 2). Among these inequalities, six had multiple solutions (e.g., <emph>−y</emph> > 7), while three had no solution (e.g., 3<emph>x</emph><sups>2</sups> < <emph>−</emph>3). In cases where a positive-like quantity was smaller than a negative-like one (e.g., 2 < <emph>−y− </emph>2), we expected that students influenced by the PSM would erroneously conclude that there are no solutions.</p> <p>Table 2 The inequality tasks presented in each questionnaire</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Test</p></th><th align="left" colspan="3"><p>Number of solutions</p></th></tr><tr><th align="left" /><th align="left" colspan="2"><p>Multiple solutions</p></th><th align="left"><p>No solution</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Pretest</p></td><td align="left"><p>x + 2 <−x− 2</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>1</mn><mspace width="0.166667em" /><mo><</mo><mspace width="0.166667em" /><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq17.gif" /></p></td><td align="left"><p>3x<sup>2</sup> <−3</p></td></tr><tr><td align="left"><p>−2x− 1 > 5</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>4</mn><mrow><mi>y</mi><mspace width="0.166667em" /><mo>+</mo><mspace width="0.166667em" /><mn>1</mn></mrow></mfrac><mspace width="0.166667em" /><mo><</mo><mspace width="0.166667em" /><mo>-</mo><mn>1</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq18.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mn>2</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq19.gif" /></p></td></tr><tr><td align="left"><p>−y > 7</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><mn>2</mn><mi>x</mi><mspace width="0.166667em" /><mo>-</mo><mn>1</mn></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq20.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>-</mo><msqrt><mi>y</mi></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mn>4</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq21.gif" /></p></td></tr><tr><td align="left" rowspan="3"><p>Posttest and retention test</p></td><td align="left"><p>x <−x</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>2</mn><mspace width="0.166667em" /><mo><</mo><mspace width="0.166667em" /><mo>-</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq22.gif" /></p></td><td align="left"><p>−5x<sup>2</sup> > 5</p></td></tr><tr><td align="left"><p>−y—2 > 2</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>1</mn><mrow><mi>y</mi><mspace width="0.166667em" /><mo>+</mo><mspace width="0.166667em" /><mn>1</mn></mrow></mfrac><mspace width="0.166667em" /><mo><</mo><mspace width="0.166667em" /><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq23.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mn>3</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq24.gif" /></p></td></tr><tr><td align="left"><p>−y > 4</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msqrt><mrow><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn><mspace width="0.166667em" /></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq25.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>-</mo><msqrt><mi>y</mi></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mn>1</mn></mrow></math><inline-graphic mime-subtype="GIF" href="11251_2024_9688_Article_IEq26.gif" /></p></td></tr></tbody></table> </ephtml> </p> <p>At the commencement of the test, students had written instructions to guide them in completing the tasks. These instructions specifically emphasized that students should select only one of the two provided alternatives that they believed to be the correct response. A reminder was provided that the solutions to the given equations and inequalities could encompass any type of number known to the participants. It was explicitly stated that the solutions could be any number encountered in their study of mathematics. The internal consistency of the initial test was acceptable (Cronbach's alpha = 0.75).</p> <hd id="AN0184555982-10">Intervention</hd> <p>During the intervention, the Experimental Group received worksheets containing four erroneous examples presented in refutational texts. These examples consisted of incorrect answers given by a hypothetical student as solutions (or no solutions) to inequalities. The errors in these responses stemmed from the PSM. In the subsequent text, explanations for the incorrect responses were presented, alongside examples illustrating correct reasoning for the particular instance of the presented inequality. Figure 1 provides an example of one such erroneous reasoning with the given refutational text.</p> <p>Graph: Fig. 1 An erroneous and a correctly solved example used in the intervention (Text is translated from Greek to English)</p> <p>The Control Group received analogous examples that were correctly solved and were presented in non-refutational texts. These examples utilized the same inequalities as those in the Experimental Group. In the correctly solved examples, the students were reminded that both positive and negative numbers can be associated with the literal symbols.</p> <p>The intervention materials specifically focused on inequalities because they offer a context in which the solution can encompass a range of numbers. This allows students to develop a better understanding of literal symbols as generalized numbers that can represent various arithmetical values, including positive and negative numbers, integers, and non-integers. Additionally, inequalities are generally considered more challenging for students compared to equations (Kieran, [<reflink idref="bib26" id="ref81">26</reflink>]), making them suitable for facilitating the transfer of knowledge between domains. This transfer of knowledge gained in one domain to the other was tested for both conditions: erroneous and correct examples.</p> <hd id="AN0184555982-11">Procedure</hd> <p>The test underwent a pilot study involving four 9th grade students from various public schools. This pilot study served to refine the tests, making them clearer for students to comprehend and aligning them more effectively with their abilities. Additionally, the pilot testing was instrumental in determining the required time for completing each test. In the current study, both groups of participants completed the finalized version of the tests in their respective classrooms, with their mathematics teacher and one of the researchers present. Each test had a designated time allowance of 20 min, which proved to be sufficient. Following the completion of the pretest, students attended the intervention, which lasted approximately 15 min. During this intervention, the second author individually provided transcripts containing examples similar to those presented in Fig. 1. Each student independently read the given examples, and they were allowed to ask clarification questions. The researcher addressed these questions, providing detailed explanations that expanded upon the information presented in the transcripts. Subsequently, the posttest was administered immediately after the intervention, and the retention test was conducted one month later, employing the same conditions and procedures as the previous tests.</p> <hd id="AN0184555982-12">Results</hd> <p></p> <hd id="AN0184555982-13">Students' answers to the equations</hd> <p>For each phase of the experiment, we categorized students' answers to the equation tasks into three main groups: correct, partially correct, or incorrect, as shown in Table 3. Answers were correct when a student accurately chose the correct option from the two provided alternatives and presented accurate solutions for equations with solutions. Responses were partially correct if a student correctly chose between the two given alternatives but did not provide solutions or provided only one solution for equations with two solutions (e.g., for <emph>x</emph><sups>2</sups> + 1 = 5, stating only <emph>x</emph> = 2 as the sole solution) or if they offered incomplete justifications for equations with no solution. Responses were incorrect if a student selected the incorrect option from the two given alternatives.</p> <p>Table 3 Students' answers in the equations in the three phases of the experiment</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left"><p>Incorrect</p></th><th align="left"><p>Partially correct</p></th><th align="left"><p>Correct</p></th><th align="left"><p>No answer</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Experimental group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>154 (39%)</p></td><td char="(" align="char"><p>145 (36%)</p></td><td char="(" align="char"><p>63 (16%)</p></td><td char="(" align="char"><p>34 (9%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>135 (34%)</p></td><td char="(" align="char"><p>126 (32%)</p></td><td char="(" align="char"><p>101 (25%)</p></td><td char="(" align="char"><p>34 (9%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>184 (47%)</p></td><td char="(" align="char"><p>79 (20%)</p></td><td char="(" align="char"><p>120 (30%)</p></td><td char="(" align="char"><p>13 (3%)</p></td></tr><tr><td align="left" rowspan="3"><p>Control group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>215 (37%)</p></td><td char="(" align="char"><p>258 (44%)</p></td><td char="(" align="char"><p>64 (11%)</p></td><td char="(" align="char"><p>48 (8%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>138 (24%)</p></td><td char="(" align="char"><p>310 (53%)</p></td><td char="(" align="char"><p>97(16%)</p></td><td char="(" align="char"><p>40 (7%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>269 (46%)</p></td><td char="(" align="char"><p>159 (27%)</p></td><td char="(" align="char"><p>152 (26%)</p></td><td char="(" align="char"><p>5 (1%)</p></td></tr></tbody></table> </ephtml> </p> <p>As indicated in Table 3, less than one third of the students' answers in the given equations were correct, even though the tasks were well within their abilities. Approximately one third of the students' answers were incorrect.</p> <p>To test the first hypothesis of the study, we examined student mistakes and categorized them based on the explanations students provided or did not provide. Three categories emerged. First, the <emph>phenomenal sign</emph> category includes answers that explicitly mention the phenomenal sign of the expressions appearing in the equations as part of their justifications. For example, students stated that <emph>the two terms of the equation had opposite signs and that two opposite numbers cannot be equal</emph>. Second, the <emph>incorrect calculations</emph> category encompasses answers that involve any kind of incorrect calculation. For instance, students provided calculations that led to incorrect results, such as stating <emph>x</emph><sups><emph>2</emph></sups> = <emph>−25 holds for −5, because −5</emph><sups><emph>2</emph></sups> = <emph>−25</emph>. Third, the <emph>absent or meaningless explanation</emph> category comprises answers for which it is not possible to identify a clear reason for the mistake or answers that lacked any explanation altogether. For instance, students provided responses like <emph>the equation</emph><ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><mi>y</mi></mrow></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>2</mn></mrow></math> </ephtml><emph> has no solution because </emph><ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><mi>y</mi></mrow></msqrt><mspace width="0.166667em" /><mo>+</mo><mspace width="0.166667em" /><msqrt><mi>y</mi></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mo>-</mo><mi>y</mi><mspace width="0.166667em" /><mo>+</mo><mspace width="0.166667em" /><mi>y</mi><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>0</mn></mrow></math> </ephtml><emph>.</emph> Table 4 presents the distribution of students' answers across these categories of incorrect responses during the three phases of the experiment. Additionally, Table 4 includes the two categories of partially correct answers to the equations. We classified answers in the <emph>natural number trials</emph> category when students reported only the natural number solution of equations that had two solutions, excluding the negative integer solution. We classified answers in the <emph>no explanation</emph> category when students selected the correct alternative response but did not provide any specific solution or explanation, as required.</p> <p>Table 4 Students' types of incorrect answers in the equations in the three phases of the experiment</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left" colspan="3"><p>Incorrect</p></th><th align="left" colspan="2"><p>Partially correct</p></th></tr><tr><th align="left" /><th align="left" /><th align="left"><p>Phenomenal sign</p></th><th align="left"><p><italic>Incorrect calculations</italic></p></th><th align="left"><p>Absent or meaningless explanation</p></th><th align="left"><p>Natural number trials</p></th><th align="left"><p>No explanation</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Experimental group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>65 (42%)</p></td><td char="(" align="char"><p>60 (39%)</p></td><td char="(" align="char"><p>20 (19%)</p></td><td char="(" align="char"><p>78 (54%)</p></td><td char="(" align="char"><p>67 (46%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>40 (30%)</p></td><td char="(" align="char"><p>55 (40%)</p></td><td char="(" align="char"><p>40 (30%)</p></td><td char="(" align="char"><p>67 (53%)</p></td><td char="(" align="char"><p>59 (47%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>64 (35%)</p></td><td char="(" align="char"><p>65 (35%)</p></td><td char="(" align="char"><p>55 (30%)</p></td><td char="(" align="char"><p>59 (76%)</p></td><td char="(" align="char"><p>19 (24%)</p></td></tr><tr><td align="left" rowspan="3"><p>Control group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>94 (44%)</p></td><td char="(" align="char"><p>59 (27%)</p></td><td char="(" align="char"><p>62 (29%)</p></td><td char="(" align="char"><p>89 (35%)</p></td><td char="(" align="char"><p>169 (65%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>37 (27%)</p></td><td char="(" align="char"><p>44 (32%)</p></td><td char="(" align="char"><p>57 (41%)</p></td><td char="(" align="char"><p>114 (37%)</p></td><td char="(" align="char"><p>196 (63%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>93 (35%)</p></td><td char="(" align="char"><p>102 (38%)</p></td><td char="(" align="char"><p>74 (27%)</p></td><td char="(" align="char"><p>87 (55%)</p></td><td char="(" align="char"><p>72 (45%)</p></td></tr></tbody></table> </ephtml> </p> <p>Table 4 highlights that the predominant category of incorrect answers, not attributable to incorrect calculations, was associated with the misconception related to the phenomenal sign of the expressions involved. Students provided explanations such as <emph>positive numbers cannot be equal to negatives</emph>, <emph>the two sides of the equation being opposite and therefore unable to be equal</emph>, or <emph>the presence of a negative number under the square root leads to an impossible result</emph>. For instance, in the case of equations like <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msup><mrow><mo stretchy="false">(</mo><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>3</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><mn>5</mn><mi>x</mi></mrow></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>5</mn></mrow></math> </ephtml> , students asserted that no numbers could satisfy these equations due to the presence of a negative number under the square root. Similarly, in the equation −<emph>y</emph>− 1 = <emph>y</emph> + 1, students responded that <emph>the two sides were opposite and, as a result, could not be equal</emph>. Furthermore, during the pretest, 31% of the students in the experimental group explicitly stated that the equation −<emph>x</emph> = 17 had no solution <emph>because a negative number could not be equal to a positive one</emph>. This further exemplifies the effect of the PSM in students' reasoning.</p> <p>Another prevalent category of incorrect answers involved incorrect calculations after substituting variables with specific numbers. For instance, in the equation −2ּ<emph>y</emph><sups>2</sups> = 2, students incorrectly identified -1 as a possible solution. Similarly, in the no-solution equation <emph>y</emph><sups>2</sups> = −16, students reported −4 as a potential solution. Additionally, in the equation <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><msqrt><mi>x</mi></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>2</mn></mrow></math> </ephtml> students concluded that the equation would hold for <emph>x</emph> = 4. Notably, all students' answers in this category only utilized integer number substitutions.</p> <p>Another prevalent category within this group of partially correct answers involved students making conclusions about the possible solutions or lack thereof by testing exclusively with a series of natural numbers. For instance, in equations such as −<emph>x</emph> = 17 and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><mn>5</mn><mi>x</mi></mrow></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>5</mn></mrow></math> </ephtml> , students responded that no numbers could satisfy the equations because none of the tested numbers (e.g., 1, 2, 3, etc.) worked. This tendency of students to primarily test the initial sequence of natural numbers led to partially correct answers in the quadratic equations included in the test. These answers that were categorized as <emph>natural number trials</emph> emerged as the primary category of partially correct responses. For instance, in quadratic equations such as <emph>x</emph><sups>2</sups> + 1 = 5, <emph>y</emph><sups>2</sups> = 25, and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msup><mrow><mo stretchy="false">(</mo><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></msqrt><mspace width="0.166667em" /><mo>=</mo><mspace width="0.166667em" /><mn>3</mn></mrow></math> </ephtml> , students predominantly provided the natural number solutions (e.g., <emph>x</emph> = 2, <emph>y</emph> = 5, and <emph>x</emph> = 9, respectively) as the only solutions, neglecting the negative integer solutions. Examples like the ones mentioned above were representative of students' responses in all phases of the experiment, encompassing both the experimental and control groups, with varying frequencies as detailed in Table 4.</p> <hd id="AN0184555982-14">Students' answers to the inequalities</hd> <p>Similar to the equation tasks, we categorized students' answers to the inequalities into correct, partially correct, and incorrect categories. Correct responses included both the correct answer and at least one correct solution or a range of numbers that satisfied the inequality. Partially correct responses had the correct given response but an incomplete or no justification. For instance, some students in this category did not provide any possible solution to the inequality, despite selecting the correct answer indicating that the inequality has solutions. Incorrect responses had the incorrect option of the two given alternatives. Table 5 displays the frequencies of students' responses in each category for each phase of the experiment. Similar to the equation tasks, less than one-third of students' answers to the inequalities were correct. It appears that students faced even greater difficulties with inequalities compared to equations, as nearly half of the given answers were incorrect in the pretest.</p> <p>Table 5 Students' answers in the inequalities in the three phases of the experiment</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left"><p>Incorrect</p></th><th align="left"><p>Partially correct</p></th><th align="left"><p>Correct</p></th><th align="left"><p>No answer</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Experimental group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>182 (46%)</p></td><td char="(" align="char"><p>111 (28%)</p></td><td char="(" align="char"><p>52 (13%)</p></td><td char="(" align="char"><p>51 (13%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>125 (32%)</p></td><td char="(" align="char"><p>111 (28%)</p></td><td char="(" align="char"><p>116 (29%)</p></td><td char="(" align="char"><p>44 (11%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>182 (46%)</p></td><td char="(" align="char"><p>49 (12%)</p></td><td char="(" align="char"><p>101 (26%)</p></td><td char="(" align="char"><p>64 (16%)</p></td></tr><tr><td align="left" rowspan="3"><p>Control group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>267 (45%)</p></td><td char="(" align="char"><p>219 (37%)</p></td><td char="(" align="char"><p>41 (7%)</p></td><td char="(" align="char"><p>58 (10%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>197 (34%)</p></td><td char="(" align="char"><p>227 (39%)</p></td><td char="(" align="char"><p>141 (24%)</p></td><td char="(" align="char"><p>20 (3%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>259 (44%)</p></td><td char="(" align="char"><p>175 (30%)</p></td><td char="(" align="char"><p>128 (22%)</p></td><td char="(" align="char"><p>23 (4%)</p></td></tr></tbody></table> </ephtml> </p> <p>We identified four categories of incorrect answers that suggest the reasons behind students' inability to provide a correct response (see Table 6). The first category, the phenomenal sign category, included responses that focused on the signs of the algebraic expressions in the inequalities. Students in this category argued that a negative quantity (referring to a negative-like algebraic expression) cannot be greater than a positive one (referring to a positive-like algebraic expression). For instance, in the inequality <emph>x</emph> + 2 < −<emph>x</emph>—2, students explained that <emph>the left side of the inequality is positive while the right side is negative</emph>, and according to their understanding, <emph>a negative number is always smaller than a positive number</emph>. Similarly, in the inequality −<emph>y</emph> > 7, students explicitly stated that 7 <emph>is a positive number, but </emph>−<emph>y represents a negative number</emph>, leading them to conclude that <emph>this inequality is not possible</emph>. Another example in this category was the inequality <emph>x</emph> < −<emph>x</emph>, where students responded that <emph>one side of the inequality will always positive and the other side negative</emph>. Similarly, in the inequality −<emph>y</emph> > 4, students concluded that <emph>there is no solution because positive numbers will always be greater than negative numbers</emph>.</p> <p>Table 6 Students' incorrect answers in the inequalities in the three phases of the experiment</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left"><p>Phenomenal sign</p></th><th align="left"><p>Incorrect calculations</p></th><th align="left"><p>Absent or meaningless explanation</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Experimental group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>94 (52%)</p></td><td char="(" align="char"><p>35 (19%)</p></td><td char="(" align="char"><p>53 (29%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>42 (34%)</p></td><td char="(" align="char"><p>50 (40%)</p></td><td char="(" align="char"><p>33 (26%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>74 (41%)</p></td><td char="(" align="char"><p>39 (21%)</p></td><td char="(" align="char"><p>69 (38%)</p></td></tr><tr><td align="left" rowspan="3"><p>Control group</p></td><td align="left"><p>Pretest</p></td><td char="(" align="char"><p>148 (56%)</p></td><td char="(" align="char"><p>41 (15%)</p></td><td char="(" align="char"><p>78 (29%)</p></td></tr><tr><td align="left"><p>Posttest</p></td><td char="(" align="char"><p>61 (31%)</p></td><td char="(" align="char"><p>106 (54%)</p></td><td char="(" align="char"><p>30 (15%)</p></td></tr><tr><td align="left"><p>Retention test</p></td><td char="(" align="char"><p>88 (34%)</p></td><td char="(" align="char"><p>74 (29%)</p></td><td char="(" align="char"><p>97 (37%)</p></td></tr></tbody></table> </ephtml> </p> <p>The second category, the incorrect calculations category, involved mistakes students made when testing possible number solutions. This category highlighted inadequate understanding of concepts such as squares, square roots, and inequalities. For example, in the inequality <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mspace width="0.166667em" /><mo>></mo><mspace width="0.166667em" /><mn>3</mn></mrow></math> </ephtml> , students provided 3 or 9 as possible solutions. Similar to equations, students in this category primarily substituted natural numbers or negative integers for the variables. Additionally, students often drew conclusions about the solutions or lack thereof based on substituting the first few natural numbers (i.e., 1, 2, 3, ...) into the literal symbols of the algebraic expressions. For instance, students incorrectly determined that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mspace width="0.166667em" /><mo><</mo><mspace width="0.166667em" /><mo>-</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></math> </ephtml><emph>had no solution because the numbers 1 and -1 did not make the inequality hold</emph>. Similarly, in the inequality √(−<emph>x</emph>− 1) > 1/3, students concluded that there was no solution because none of the numbers 1, 2, or 3 satisfied the inequality. The <emph>absent or meaningless explanation</emph> category comprised answers for which it was not possible to identify a clear reason for the mistake or incorrect answers that lacked any explanation altogether.</p> <p>As with the equations, the most prevalent category of incorrect responses was a predisposition toward the phenomenal sign. This inclination influenced half of the participants from both groups, leading them to provide incorrect responses in the pretest (see Table 6). Furthermore, students' challenges in dealing with inequalities involving square roots and squares are evident from their frequent calculation errors, highlighting their struggles with the requisite procedural and conceptual knowledge for the necessary mathematical operations.</p> <hd id="AN0184555982-15">The impact of the intervention</hd> <p>To test the second hypothesis of the study, we evaluated the impact of the intervention using erroneous and correct examples by calculating mean scores for the experimental and control groups separately, considering equations and inequalities, and each phase of the experiment. We assigned each response a code: incorrect responses as 0, partially correct responses as 1, and correct responses as 2. If no response was provided, it was coded as a missing value. The mean scores for each group and phase are presented in Table 7.</p> <p>Table 7 Performance of the experimental and control groups, in each phase of the experiment</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left" colspan="2"><p>Pretest</p></th><th align="left" colspan="2"><p>Posttest</p></th><th align="left" colspan="2"><p>Retention test</p></th></tr><tr><th align="left" /><th align="left" /><th align="left"><p>Mean</p></th><th align="left"><p>S.D</p></th><th align="left"><p>Mean</p></th><th align="left"><p>S.D</p></th><th align="left"><p>Mean</p></th><th align="left"><p>S.D</p></th></tr></thead><tbody><tr><td align="left" rowspan="2"><p>Equations</p></td><td align="left"><p>Exp. Group</p></td><td char="." align="char"><p>0.74</p></td><td char="." align="char"><p>0.31</p></td><td char="." align="char"><p>0.90</p></td><td char="." align="char"><p>0.33</p></td><td char="." align="char"><p>0.82</p></td><td char="." align="char"><p>0.40</p></td></tr><tr><td align="left"><p>Con. Group</p></td><td char="." align="char"><p>0.72</p></td><td char="." align="char"><p>0.26</p></td><td char="." align="char"><p>0.92</p></td><td char="." align="char"><p>0.24</p></td><td char="." align="char"><p>0.80</p></td><td char="." align="char"><p>0.36</p></td></tr><tr><td align="left" rowspan="2"><p>Inequalities</p></td><td align="left"><p>Exp. Group</p></td><td char="." align="char"><p>0.61</p></td><td char="." align="char"><p>0.28</p></td><td char="." align="char"><p>1.02</p></td><td char="." align="char"><p>0.42</p></td><td char="." align="char"><p>0.71</p></td><td char="." align="char"><p>0.44</p></td></tr><tr><td align="left"><p>Con. Group</p></td><td char="." align="char"><p>0.58</p></td><td char="." align="char"><p>0.25</p></td><td char="." align="char"><p>0.90</p></td><td char="." align="char"><p>0.32</p></td><td char="." align="char"><p>0.77</p></td><td char="." align="char"><p>0.37</p></td></tr></tbody></table> </ephtml> </p> <p>Levene's test for homogeneity of variances indicated that the performance of the two groups did not significantly differ in the pretest, both for equations [<emph>F</emph>(<reflink idref="bib1" id="ref82">1</reflink>,<reflink idref="bib107" id="ref83">107</reflink>) = 1.455, <emph>p</emph> = 0.230] and inequalities [<emph>F</emph>(<reflink idref="bib1" id="ref84">1</reflink>,<reflink idref="bib107" id="ref85">107</reflink>) = 3.014, <emph>p</emph> = 0.085]. We conducted mixed ANOVAs on the mean performance, considering the three factors: group, task, and test. The analysis showed a main effect of test. Posttest performance was significantly better than pretest performance for both equations [<emph>F</emph>(1.836, 196,439) = 15,015, <emph>p</emph> < 0.001, <emph>n</emph><subs><emph>p</emph></subs><sups>2</sups> = 0.123][<reflink idref="bib1" id="ref86">1</reflink>] and inequalities [<emph>F</emph>(<reflink idref="bib2" id="ref87">2</reflink>, 214) = 34.973, <emph>p</emph> < 0.001, <emph>n</emph><subs><emph>p</emph></subs><sups>2</sups> = 0.246] among students in both groups (see Table 7). However, there were no statistically significant differences between the intervention and control group performance on equations [<emph>F</emph>(<reflink idref="bib1" id="ref88">1</reflink>,<reflink idref="bib107" id="ref89">107</reflink>) = 0.030, <emph>p</emph> = 0.864, <emph>n</emph><subs><emph>p</emph></subs><sups><emph>2</emph></sups> = 0.000] or inequalities [<emph>F</emph>(<reflink idref="bib1" id="ref90">1</reflink>,<reflink idref="bib107" id="ref91">107</reflink>) = 0.509, <emph>p</emph> = 0.477, <emph>n</emph><subs><emph>p</emph></subs><sups><emph>2</emph></sups> = 0.005].</p> <p>We conducted additional paired sample <emph>t</emph>-tests to examine the performance of the intervention group. Results indicated that the intervention group had significantly higher scores at posttest compared to the pretest for inequalities <emph>t</emph>(<reflink idref="bib43" id="ref92">43</reflink>) = 5.560, <emph>p</emph> < 0.001 and equations <emph>t</emph>(<reflink idref="bib43" id="ref93">43</reflink>) = 3.109, <emph>p</emph> = 0.003, suggesting transfer of knowledge from one domain to another. However, there were no significant differences between their performance on the pretest and the retention test for equations <emph>t</emph>(<reflink idref="bib43" id="ref94">43</reflink>) = 1.393, <emph>p</emph> = 0.171 or inequalities <emph>t</emph>(<reflink idref="bib43" id="ref95">43</reflink>) = 1.499, <emph>p</emph> = 0.141.</p> <p>There were similar patterns for the control group. The control group showed statistically significantly higher scores on the posttest compared to the pretest for equations <emph>t</emph>(<reflink idref="bib64" id="ref96">64</reflink>) = 6.894, <emph>p</emph> < 0.001 and inequalities <emph>t</emph>(<reflink idref="bib64" id="ref97">64</reflink>) = 6.351, <emph>p</emph> < 0.001. Although there were no statistically significant differences between their performance on the pretest and retention test for equations <emph>t</emph>(<reflink idref="bib64" id="ref98">64</reflink>) = 1.658, <emph>p</emph> = 0.102, the control group had better performance on the retention test compared to the pretest <emph>t</emph>(<reflink idref="bib64" id="ref99">64</reflink>) = 3.281, <emph>p</emph> = 0.002 for inequalities.</p> <hd id="AN0184555982-16">Discussion</hd> <p>This study investigated the effects of students' tendency to misinterpret the phenomenal sign of algebraic expressions as the sign of the numerical values they can only represent when reasoning about the solutions of equations and inequalities. In addition, the study tested the influence of an erroneous examples and refutational text intervention on students' solving of equations and inequalities compared to correctly worked examples. Results strongly support the first hypothesis of the study: students' tendency to misinterpret the phenomenal sign of algebraic expressions informs their reasoning about solutions to equations and inequalities. In the pretest, students from both groups commonly responded that equations like −<emph>y</emph>− 1 = <emph>y</emph> + 1 and −<emph>x</emph> = 17 have no solutions, citing the presence of opposite signs on the two sides and the belief that a positive quantity cannot be equated with a negative one. Similarly, in the case of inequalities such as −<emph>y</emph> > 7, students rejected the possibility of a (phenomenally) negative value being greater than a positive one. These findings align with and extend prior research that identified the effects of students' tendency to misinterpret the phenomenal sign of algebraic expressions as the actual sign of the numbers they can represent (Chiarugi et al., [<reflink idref="bib6" id="ref100">6</reflink>]; Christou, [<reflink idref="bib13" id="ref101">13</reflink>]; Christou et al., [<reflink idref="bib11" id="ref102">11</reflink>]; Vlassis, [<reflink idref="bib55" id="ref103">55</reflink>]).</p> <p>These findings demonstrate that students' understanding of the referential meaning of literal symbols and algebraic expressions is hindered by their tendency to misinterpret the sign an expression appears to have as the actual sign of the numbers it can represent. Establishing a connection between algebraic expressions and their referential meaning in the world of numbers is essential for students to effectively comprehend and manipulate variables, even when they possess an abstract understanding of algebraic expressions. As Arcavi ([<reflink idref="bib2" id="ref104">2</reflink>]) argued, seeking the meaning of symbols can play a pivotal role in problem-solving or merely add insight. However, as this study shows, students' inclination to misinterpret the phenomenal sign of an expression as its actual sign, restricted to the numbers it represents, can lead to misconceptions and low performance across various mathematical domains, particularly in algebra, where literal symbols are used to represent generalized relations between numbers. This finding may provide insights into the errors and low performance rates exhibited by students in other mathematical domains when the sign of algebraic expressions holds significance, such as functions or the comprehension of the concept of absolute value.</p> <p>Additionally, the findings of this study revealed that erroneous reasoning examples provided with refutational text supported students' performance on inequality tasks, which were the focus of the intervention. We observed statistically significant differences between the average scores of the intervention group in the pre- and posttests. Closer examination of the mistakes made before and after the intervention showed that students in the intervention group successfully addressed certain aspects of their misconceptions related to the possible solutions of inequalities. These results debunk the concern that learners exposed to incorrect solutions might develop erroneous procedural knowledge without understanding the underlying errors (see also Pillai et al., [<reflink idref="bib41" id="ref105">41</reflink>]). Instead, presenting erroneous examples in a refutational argumentation was beneficial in helping the students address parts of their phenomenal sign misconception when reasoning about the possible solutions to equations and inequalities, in line with previous literature on erroneous examples (Adams et al., [<reflink idref="bib1" id="ref106">1</reflink>]; Booth et al., [<reflink idref="bib3" id="ref107">3</reflink>]; Durkin and Rittle-Johnson, [<reflink idref="bib18" id="ref108">18</reflink>]; Loibl and Leuders, [<reflink idref="bib33" id="ref109">33</reflink>]; McLaren et al., [<reflink idref="bib37" id="ref110">37</reflink>]) and refutational texts (Diakidoy et al., [<reflink idref="bib16" id="ref111">16</reflink>]; Lem et al., [<reflink idref="bib32" id="ref112">32</reflink>]; Sinatra and Broughton, [<reflink idref="bib46" id="ref113">46</reflink>]; Skopeliti and Vosniadou, [<reflink idref="bib47" id="ref114">47</reflink>]).</p> <p>Furthermore, the students in the intervention group demonstrated some transfer of the acquired knowledge from inequalities to equations, leading to a reduction in phenomenal sign mistakes in equations in the posttest, despite having been presented with erroneous examples exclusively for inequalities. However, these benefits in both domains were not as robust during the retention test. One possible explanation is that the students in this study were solely exposed to the erroneous problems without first being prompted to self-explain those examples or compare them with correct solutions, which could have further enhanced their learning, as previous research on erroneous examples has indicated (Große and Renkl, [<reflink idref="bib20" id="ref115">20</reflink>]; Siegler, [<reflink idref="bib45" id="ref116">45</reflink>]). Future studies could incorporate this factor to examine its potential impact on the specific misconception.</p> <p>We observed similar patterns in the control group, where students exhibited improvements in the posttest compared to the pretest when presented with correct examples. However, these improvements were not as sustained in the retention test for equations, only for inequalities. Overall, the rate of correct responses was better maintained for equations compared to inequalities, indicating an advantage of equations over inequalities in students' understanding, in line with previous research (Kieran, [<reflink idref="bib26" id="ref117">26</reflink>]).</p> <p>However, there were no performance differences between students who received erroneous examples in refutational texts or students who received correct examples in non-refutational text. Previous research on erroneous examples has demonstrated their effectiveness in addressing students' misconceptions in domains like equations, with no apparent advantage of erroneous examples over correct ones (Isotani et al., [<reflink idref="bib23" id="ref118">23</reflink>]; Pillai et al., [<reflink idref="bib41" id="ref119">41</reflink>]; Van Peppen et al., [<reflink idref="bib54" id="ref120">54</reflink>]). However, given prior literature showing positive effects of refutational text in addressing misconceptions in mathematics (Lem et al., [<reflink idref="bib32" id="ref121">32</reflink>]), we anticipated that the intervention group, exposed to refutational texts, would outperform the control group in the posttest. Results of the present study did not support this hypothesis. This suggests that when addressing reasoning mistakes caused by the PSM, a reminder in the form of either an erroneous or correct example could be sufficient to help students overcome their PSM, at least in the short term. It may be that a refutational text would need to contain more comprehensive information (see for example Skopeliti and Vosniadou, [<reflink idref="bib47" id="ref122">47</reflink>]), such as explicitly addressing students' inclination to associate literal symbols solely with natural numbers as the primary cause of the PSM.</p> <p>The lower mean performance on the retention test compared to the posttest shows that the brief intervention with erroneous or correct examples did not lead to substantial changes in the fundamental structure of students' prior knowledge of possible arithmetical values of algebraic expressions containing literal symbols. The decrease in students' performance on the retention test compared to the posttest was not primarily attributable to a decline in the frequency of correct responses. Instead, the decrease can be largely attributed to an increase in incorrect responses during the retention test. Analyzing the types of mistakes that increased in the retention test, it was predominantly the phenomenal sign mistakes that stood out among the answers that provided meaningful explanations shedding light on the possible reasons for these errors. Additionally, calculation mistakes remained at high levels as in the posttest and the retention test (see Table 4–6 for detailed information).</p> <p>From a conceptual change perspective, the lack of sustained effects from one brief intervention may be expected. The PSM can be traced back to a more general natural number bias, which leads students to associate only natural numbers with the literal symbols in algebraic expressions (Christou and Vosniadou, [<reflink idref="bib8" id="ref123">8</reflink>]; Christou et al., [<reflink idref="bib11" id="ref124">11</reflink>]). Consequently, addressing the PSM may necessitate significant changes to how students perceive literal symbols, a process that must happen over time. To effectively tackle this issue, targeted teaching methods that emphasize the connection between literal symbols and real numbers should be implemented, ideally starting from when students are first introduced to symbols representing numbers in arithmetic equations (Switzer, [<reflink idref="bib49" id="ref125">49</reflink>]). In this regard, adopting a functional approach to algebra, wherein letters represent varying quantities rather than unknown numbers, and algebraic expressions depict relations between quantities rather than constant values, could prove to be advantageous (Kieran, [<reflink idref="bib27" id="ref126">27</reflink>]). Such approaches could promote students' understanding of the concept of variable along with the concept of number in the long run, provided that students have the opportunity to work with a variety of non-natural measures of quantities. This is because an emerging understanding of variables as symbols to represent a range of natural as well as non-natural numbers is necessary for addressing the PSM misconception. However, promoting an understanding of natural and non-natural numbers as unified systems of (rational or real) numbers is a process that requires conceptual change in the initial concept of number, which is initially organized around the properties of natural numbers, which is a difficult and time-consuming process (Vamvakoussi et al., [<reflink idref="bib51" id="ref127">51</reflink>]). Such approaches require a long-term perspective on mathematics instruction and the purposeful design of mathematics curricula.</p> <p>Even so, a short intervention targeting the already established PSM could prove beneficial to tackle the PSM in domains such as equations and inequalities, as demonstrated in the current study. The study's results provide optimism regarding the potential benefits of systematically revisiting the same topics in different contexts as a means for students to monitor and control their thinking in these contexts. This suggests that examples directly addressing specific student misconceptions, similar to those used in this study, could be incorporated into textbooks and become part of mathematics teachers' everyday classroom practices. The frequent use of both correct and erroneous examples and the use of refutational and expository texts that draw students' attention to how to avoid a mistake based on a misconception may prove more effective in the long run. Some supporting evidence in this direction comes from studies showing that in countries with a history of high math achievement, such as Japan and China, teachers probe students to discuss errors (Schleppenbach et al., [<reflink idref="bib43" id="ref128">43</reflink>]; Stigler and Hiebert, [<reflink idref="bib48" id="ref129">48</reflink>]). To support implementation, teachers should be informed about the benefits of using erroneous examples in their classrooms and trained on how to utilize them effectively. Therefore, these suggestions should first be integrated into teacher education programs that provide mathematics teachers with the necessary content knowledge and pedagogical content knowledge regarding students' misconceptions in the specific domain of mathematics learning. This integration is vital because teachers frequently do not have sufficient opportunities to develop professional competence to identify and address students' misconceptions (Depaepe et al., [<reflink idref="bib15" id="ref130">15</reflink>]), especially in algebra (Elisha, [<reflink idref="bib19" id="ref131">19</reflink>]). Of course, empirical support, ideally from future studies, is needed to validate these suggestions.</p> <p>An additional concern that arises is that interventions targeting specific misconceptions may not yield satisfactory overall results unless they are accompanied by the necessary conceptual and procedural knowledge that helps students develop a comprehensive understanding of the related concepts or procedures. In the current study, students encountered significant difficulties in understanding concepts such as square root and following the appropriate procedures to solve inequalities and equations accurately. Since the intervention applied in the study solely focused on identifying and addressing PSM errors without providing additional instruction on correct transformations for solving inequalities and equations, the students lacked the support needed to deal with their lack of knowledge, hindering their ability to reason correctly. Thus, future interventions should strive to incorporate comprehensive instructional approaches that address both specific misconceptions and the broader conceptual and procedural aspects to foster more effective learning outcomes.</p> <p>The findings of this study in mathematics education add to an existing body of literature that shows how strategies such as refutational argumentation with the use of erroneous examples can apply to various learning contexts where initial misconceptions or less sophisticated ideas need to be revised. In science education—in areas such as physics, chemistry and biology—misconceptions are often a barrier to the acquisition of scientifically accepted ideas, making refutational texts and interventions using incorrect examples particularly valuable (Vosniadou et al., [<reflink idref="bib57" id="ref132">57</reflink>]). Extensive research in these fields has already demonstrated the effectiveness of such approaches (see Danielson et al., [<reflink idref="bib14" id="ref133">14</reflink>]). Beyond mathematics and science, similar learning challenges arise in social sciences such as history (Carretero and Perez-Manjarrez, [<reflink idref="bib5" id="ref134">5</reflink>]), economics and environmental studies (Lundholm and Davies, [<reflink idref="bib35" id="ref135">35</reflink>]), where students often need to revise their initial perspectives on more elusive or even controversial issues. In these contexts, refutational argumentation and, where appropriate, examples of errors, such as those used in the current study, could help students to revise erroneous or less sophisticated conceptions. Recent meta-analyses have already demonstrated the effectiveness of these strategies across content domains (Schroeder and Kucera, [<reflink idref="bib44" id="ref136">44</reflink>]).</p> <hd id="AN0184555982-17">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0184555982-18"> <title> References </title> <blist> <bibl id="bib1" idref="ref47" type="bt">1</bibl> <bibtext> Adams DM, McLaren BM, Durkin K, Mayer RE, Rittle-Johnson B, Isotani S, Van Velsen M. Using erroneous examples to improve mathematics learning with a web-based tutoring system. 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Christou; Courtney Pollack and Eleni Karagiannidou</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib27" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib31" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib11" firstref="ref6"></nolink> <nolink nlid="nl4" bibid="bib30" firstref="ref11"></nolink> <nolink nlid="nl5" bibid="bib28" firstref="ref12"></nolink> <nolink nlid="nl6" bibid="bib34" firstref="ref13"></nolink> <nolink nlid="nl7" bibid="bib57" firstref="ref15"></nolink> <nolink nlid="nl8" bibid="bib38" firstref="ref16"></nolink> <nolink nlid="nl9" bibid="bib52" firstref="ref17"></nolink> <nolink nlid="nl10" bibid="bib12" firstref="ref18"></nolink> <nolink nlid="nl11" bibid="bib10" firstref="ref22"></nolink> <nolink nlid="nl12" bibid="bib17" firstref="ref23"></nolink> <nolink nlid="nl13" bibid="bib55" firstref="ref24"></nolink> <nolink nlid="nl14" bibid="bib58" firstref="ref26"></nolink> <nolink nlid="nl15" bibid="bib56" firstref="ref31"></nolink> <nolink nlid="nl16" bibid="bib22" firstref="ref32"></nolink> <nolink nlid="nl17" bibid="bib36" firstref="ref33"></nolink> <nolink nlid="nl18" bibid="bib50" firstref="ref34"></nolink> <nolink nlid="nl19" bibid="bib46" firstref="ref37"></nolink> <nolink nlid="nl20" bibid="bib21" firstref="ref38"></nolink> <nolink nlid="nl21" bibid="bib47" firstref="ref41"></nolink> <nolink nlid="nl22" bibid="bib16" firstref="ref42"></nolink> <nolink nlid="nl23" bibid="bib44" firstref="ref43"></nolink> <nolink nlid="nl24" bibid="bib13" firstref="ref44"></nolink> <nolink nlid="nl25" bibid="bib32" firstref="ref45"></nolink> <nolink nlid="nl26" bibid="bib23" firstref="ref48"></nolink> <nolink nlid="nl27" bibid="bib61" firstref="ref49"></nolink> <nolink nlid="nl28" bibid="bib53" firstref="ref50"></nolink> <nolink nlid="nl29" bibid="bib29" firstref="ref52"></nolink> <nolink nlid="nl30" bibid="bib24" firstref="ref53"></nolink> <nolink nlid="nl31" bibid="bib25" firstref="ref54"></nolink> <nolink nlid="nl32" bibid="bib40" firstref="ref55"></nolink> <nolink nlid="nl33" bibid="bib39" firstref="ref56"></nolink> <nolink nlid="nl34" bibid="bib18" firstref="ref58"></nolink> <nolink nlid="nl35" bibid="bib20" firstref="ref59"></nolink> <nolink nlid="nl36" bibid="bib33" firstref="ref60"></nolink> <nolink nlid="nl37" bibid="bib37" firstref="ref63"></nolink> <nolink nlid="nl38" bibid="bib42" firstref="ref66"></nolink> <nolink nlid="nl39" bibid="bib41" firstref="ref68"></nolink> <nolink nlid="nl40" bibid="bib54" firstref="ref69"></nolink> <nolink nlid="nl41" bibid="bib59" firstref="ref70"></nolink> <nolink nlid="nl42" bibid="bib60" firstref="ref71"></nolink> <nolink nlid="nl43" bibid="bib45" firstref="ref73"></nolink> <nolink nlid="nl44" bibid="bib26" firstref="ref81"></nolink> <nolink nlid="nl45" bibid="bib107" firstref="ref83"></nolink> <nolink nlid="nl46" bibid="bib43" firstref="ref92"></nolink> <nolink nlid="nl47" bibid="bib64" firstref="ref96"></nolink> <nolink nlid="nl48" bibid="bib49" firstref="ref125"></nolink> <nolink nlid="nl49" bibid="bib51" firstref="ref127"></nolink> <nolink nlid="nl50" bibid="bib48" firstref="ref129"></nolink> <nolink nlid="nl51" bibid="bib15" firstref="ref130"></nolink> <nolink nlid="nl52" bibid="bib19" firstref="ref131"></nolink> <nolink nlid="nl53" bibid="bib14" firstref="ref133"></nolink> <nolink nlid="nl54" bibid="bib35" firstref="ref135"></nolink>
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  Data: Erroneous Examples in Refutational Text to Address the Phenomenal Sign Misconception in Equations and Inequalities
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  Data: <searchLink fieldCode="AR" term="%22Konstantinos+P%2E+Christou%22">Konstantinos P. Christou</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-7615-4968">0000-0002-7615-4968</externalLink>)<br /><searchLink fieldCode="AR" term="%22Courtney+Pollack%22">Courtney Pollack</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-0144-5971">0000-0002-0144-5971</externalLink>)<br /><searchLink fieldCode="AR" term="%22Eleni+Karagiannidou%22">Eleni Karagiannidou</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Instructional+Science%3A+An+International+Journal+of+the+Learning+Sciences%22"><i>Instructional Science: An International Journal of the Learning Sciences</i></searchLink>. 2025 53(2):315-335.
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Data: 21
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  Data: Journal Articles<br />Reports - Research
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  Data: <searchLink fieldCode="EL" term="%22Grade+9%22">Grade 9</searchLink><br /><searchLink fieldCode="EL" term="%22High+Schools%22">High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+9%22">Grade 9</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Symbols+%28Mathematics%29%22">Symbols (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Misconceptions%22">Misconceptions</searchLink><br /><searchLink fieldCode="DE" term="%22Algebra%22">Algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Error+Correction%22">Error Correction</searchLink>
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  Data: 10.1007/s11251-024-09688-2
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  Data: 0020-4277<br />1573-1952
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  Data: The ability to solve equations and inequalities is necessary for success in algebra. However, reasoning biases and misconceptions may create barriers for students to build knowledge of algebraic symbols and their values. This study investigated whether students' errors when solving equations and inequalities could be attributed to their tendency to misinterpret the phenomenal sign of an expression (e.g., -2x interpreted as representing negative numbers only). Additionally, the study examined whether an intervention using erroneous reasoning examples in refutational texts would be more effective than correct examples in helping students address the specific misconception. The study involved 119 9th-grade Greek students who underwent Pre-, Post-, and Retention tests. The Experimental Group (N = 44) saw erroneous examples of reasoning with solving inequalities in refutational text, while the Control Group (N = 65) saw correct examples in non-refutational text. The results showed that students' misinterpretation of the phenomenal sign in algebraic expressions may influence their mistakes when solving certain kinds of equations and inequalities. Both erroneous and correct examples were effective in helping students address some of their misconceptions, although the gains were not sustained in the long term.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1487889
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1487889
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s11251-024-09688-2
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 21
        StartPage: 315
    Subjects:
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: Grade 9
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematical Concepts
        Type: general
      – SubjectFull: Symbols (Mathematics)
        Type: general
      – SubjectFull: Misconceptions
        Type: general
      – SubjectFull: Algebra
        Type: general
      – SubjectFull: Thinking Skills
        Type: general
      – SubjectFull: Error Correction
        Type: general
      – SubjectFull: Greece
        Type: general
    Titles:
      – TitleFull: Erroneous Examples in Refutational Text to Address the Phenomenal Sign Misconception in Equations and Inequalities
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Konstantinos P. Christou
      – PersonEntity:
          Name:
            NameFull: Courtney Pollack
      – PersonEntity:
          Name:
            NameFull: Eleni Karagiannidou
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0020-4277
            – Type: issn-electronic
              Value: 1573-1952
          Numbering:
            – Type: volume
              Value: 53
            – Type: issue
              Value: 2
          Titles:
            – TitleFull: Instructional Science: An International Journal of the Learning Sciences
              Type: main
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