Examining Grade 9 Learners’ Understanding of Congruency and Similarity in Euclidean Geometry

Examining Grade 9 Learners’ Understanding of Congruency and Similarity in Euclidean Geometry | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Examining Grade 9 Learners’ Understanding of Congruency and Similarity in Euclidean Geometry France Machaba, Dipolelo Ramokgata This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6376119/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper investigates Grade 9 learners’ understanding of congruency and similarity at a Mopani district school in Limpopo Province, using the Pirie-Kieren model of mathematical comprehension. The study explores the learners’ thought processes, prior knowledge, and ability to justify their responses when solving problems related to similarity and congruence. A qualitative research approach was employed, using a case study design involving a sample of 40 Grade 9 learners. Data was collected through written tests and semi-structured interviews and analysed using content analysis. The findings indicate that learners struggle with fundamental concepts of Euclidean geometry, particularly in distinguishing between similarity and congruence. The study recommends further research on mathematical understanding at various grade levels and emphasizes the need for instructional strategies that enhance conceptual comprehension rather than rote learning. Grade 9 Learners understanding Congruency Similarity and Euclidean Geometry Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Introduction Mathematics education, particularly in Euclidean geometry, plays a crucial role in developing students’ analytical and problem-solving skills. However, research indicates that South African learners struggle in this area, largely due to inadequate foundational knowledge and ineffective teaching methods. Mathematics performance in South Africa remains a significant concern, as reflected in both national and international assessments. The National Senior Certificate (NSC) diagnostic reports (DBE, 2021) highlight that students often struggle with basic geometric concepts, especially congruence and similarity. These challenges are further evidenced by the poor performance in the Trends in International Mathematics and Science Study (TIMSS), as well as by the Annual National Assessment (ANA) reports, which show that nearly 98% of learners fail to correctly justify congruency in geometric problems. Euclidean geometry, which includes fundamental concepts such as similarity and congruence, is a critical component of high school mathematics. These topics not only lay the groundwork for advanced mathematical studies but also have practical applications in fields such as engineering, architecture, and medical sciences (Machisi, 2021 ). Despite its reinstatement in 2012 under the Curriculum and Assessment Policy Statement (CAPS), following its removal in 2006 due to poor student performance, students continue to struggle with these geometric concepts. Studies show that learners often misinterpret theorems, make computational errors, and face difficulty in analysing and interpreting geometric diagrams correctly (Wang et al., 2018). The purpose of this study is to explore Grade 9 learners' understanding of similarity and congruence, focusing on their ability to apply theoretical knowledge to solve problems. By employing the Pirie-Kieren model, this research aims to uncover gaps in learners' comprehension and offer recommendations for improved instructional strategies. Given the challenges students face, particularly in understanding geometric principles without real-life applications (Dündar & Gündüz, 2017 ), this study seeks to identify how learners process and represent geometric concepts. The goal is to contribute to the development of teaching methods that foster a deeper understanding of Euclidean geometry and improve mathematical literacy and problem-solving abilities. Theoretical Framework The theoretical framework for this study is grounded in the theory of mathematical understanding proposed by Pirie and Kieren ( 1994 ), which offers a model for how learners develop and deepen their comprehension of mathematical concepts. According to Pirie and Kieren (1989), mathematical understanding is not a linear process but rather a layered, non-linear progression. Their model conceptualizes understanding as a series of eight nested levels, or “rings,” each building upon the one before it, yet with an inherent back-and-forth movement within the layers, known as “folding back” (Purwanto & Solehudin, 2020 ). This dynamic process allows learners to revisit earlier stages of understanding to refine or deepen their comprehension, leading to more abstract and complex forms of knowledge. The theory is based on the notion that learners begin with an initial understanding—termed primitive knowing—which encompasses the prior knowledge they bring to new learning situations (Purwanto & Solehudin, 2020 ). As learners interact with new content, they progress through different stages of mathematical understanding, moving from concrete representations to more abstract conceptualizations. Each level represents a deeper engagement with the material, where learners refine their understanding through cognitive processes such as image-making, property noticing, and formalizing. The first stage, primitive knowing, refers to the base of a learner’s mathematical knowledge, including gestures, symbols, and basic images. This stage provides the foundation upon which further learning builds (Kaba & Sengül, 2015 ). In the context of this study, primitive knowing would encompass the basic geometric knowledge a Grade 9 learner brings to the study of similarity and congruence, such as familiarity with triangle properties and basic geometric theorems. The second level, image-making, involves learners using their prior knowledge to differentiate between concepts and apply them in various contexts (Pirie & Kieren, 1994 ). At this stage, learners begin to visualize mathematical concepts and create representations of them, such as drawing triangles to illustrate congruency or similarity. As learners continue to develop, they move to the third level, image-having, where they internalize a mental representation of the concept and no longer need to externalize it through writing or drawing (Purwanto & Solehudin, 2020 ). At this level, learners are able to identify properties such as congruent sides or angles in their mind, without needing to rely on visual aids. The property-noticing stage, the fourth level, involves recognizing and distinguishing patterns and properties in mathematical concepts (Purwanto & Solehudin, 2020 ). At this level, learners can identify key properties of geometric figures that indicate similarity or congruency, such as equal angles or proportional sides. Moving to the fifth level, formalizing, learners begin to define concepts and create formal rules based on their understanding. They are able to synthesize their learning into more structured and precise mathematical statements, such as proving similarity or congruency through formal theorems. At the observing level, learners reflect on their formalized knowledge and connect it with broader mathematical structures. They begin to apply their knowledge to solve problems and create new mathematical relationships (Purwanto & Solehudin, 2020 ). This level represents a transition from basic conceptualization to the application of formal mathematical ideas. The seventh level, structuring, involves learners synthesizing multiple concepts and theorems, linking them through logical reasoning and providing valid justifications for their conclusions. Learners at this stage can systematically prove geometric properties and construct valid arguments based on established mathematical principles. Finally, the inventing level represents the highest degree of understanding, where learners are able to generate new ideas or concepts and critically analyse existing mathematical theories. At this stage, learners demonstrate a high level of mastery, able to solve complex problems and develop new insights (Purwanto & Solehudin, 2020 ). This model emphasizes that mathematical understanding develops through recursive movement across levels, with learners continuously refining their knowledge by revisiting earlier stages (Syafiqoh et al., 2018 ). The concept of “folding back” is crucial to this theory, as learners must often return to earlier levels of understanding to resolve ambiguities or incomplete understandings, leading to deeper, more integrated knowledge of mathematical concepts. This theoretical framework provides a lens for examining Grade 9 learners' understanding of similarity and congruence, helping to identify the specific levels at which they may experience difficulties and the cognitive processes that support their progression in geometric understanding. Literature Review Understanding is the process of comprehending inner communications within the substance of content (Hasanona, 2017 ). In learning mathematics, learners require understanding (Syafiqoh et al., 2018 ), which is closely connected to the nature of mathematical information. McCormack ( 2002 ) highlights the complex relationship between children's involvement, understanding, and development, making understanding a crucial focus of this study. Types of Understanding Instrumental and Relational Understanding According to Skemp (1976), understanding in mathematics can be classified into Instrumental Understanding, which is the ability to apply mathematical rules without necessarily understanding the reasoning behind them, and Relational Understanding is an understanding mathematical linkage, justifications for rules, and their appropriate application (Herheim, 2023 ). While instrumental understanding allows for immediate application and can boost learner confidence, relational understanding fosters deeper comprehension and problem-solving abilities (Skemp, 1978). Despite its benefits, relational understanding often requires more time, leading some teachers to favour instrumental methods for efficiency (Skemp, 1978). Euclidean Geometry Importance of Geometry in Mathematics To master mathematics, learners must understand fundamental geometric concepts (Dahlan & Wibisono, 2021 ). Tachie ( 2020 ) emphasizes that a solid foundation in geometry is necessary for comprehension. Euclidean geometry, the study of planes, solid shapes, and their properties based on Euclid’s theorems and axioms, is central to mathematical learning (Machisi, 2021 ). Challenges in Learning Geometry Proving riders (non-routine geometry problems) and logical reasoning in Euclidean geometry overwhelm many students (Del Grande, 1986 ). Deductive reasoning is essential in establishing relationships and properties within figures (Ngirishi & Bansilal, 2019 ). Furthermore, understanding transformations, such as dilations and congruence, is crucial for learners (Zhang & Wong, 2021; Haj-Yahya, 2022 ). Research shows that learners find Euclidean geometry challenging, but innovative teaching approaches improve engagement (Naidoo & Kapofu, 2020 ). Understanding similarity and congruency in early grades can encourage more learners to pursue mathematics in higher education, benefiting future career prospects. Prior Knowledge Required for Learning Congruency and Similarities Prior knowledge, including triangle properties, quadrilateral properties, and basic geometric theorems, plays a significant role in learning similarity and congruency (Brod, 2021 ; Simonsmeier et al., 2022 ). The CAPS curriculum outlines necessary prerequisites, such as knowledge of different types of triangles and their properties. Real-Life Applications of Mathematical Concepts Connecting mathematics to real-life contexts enhances student engagement (Arthur et al., 2018 ). Proportional reasoning, an essential mathematical concept, is crucial for understanding similarity, dilation, and scaling (Abramovich & Connell, 2021 ; Yeo, 2019). In everyday life, similarity is applied in measuring distances, constructing bridges, and designing architectural structures (Mastrogiannis & Kordaki, 2006 ). Representations in Understanding Congruency and Similarities Effective mathematics learning requires multiple representations such as Graphical/Diagrammatic Representations: Visual aids, such as flow diagrams, tables, and graphs, support conceptual understanding (Abdurrahman, 2020 ). Symbolic Representations: Mathematical symbols and notation, including those for congruence and similarity, enhance comprehension (Mainali, 2021 ). Verbal Representations: Communicating mathematical reasoning through explanations strengthens learning (Imama & Caswita, 2023 ). Properties and Distinctions Between Congruency and Similarity Similarity refers to figures with the same shape but not necessarily the same size (Dündar & Gündüz, 2017 ). It is a key mathematical concept with practical applications such as calculating heights of buildings and measuring distances (Yeo, 2022 ). Despite its importance, learners often confuse similarity with equality, making explicit instruction necessary (Biber, 2020 ). Congruent figures have identical shape and size (Clapham & Nicholson, 2009 ). Learners introduced to symmetry in primary school can better grasp congruency (Cole, 2010 ). Understanding congruent triangles, such as the Side-Angle-Side (S.A.S.) and Angle-Angle (A.A.) criteria, is essential for geometric problem-solving (Laudano & Vincenzi, 2017 ). Properties and Distinctions Between Congruency and Similarity Similarity refers to figures with the same shape but not necessarily the same size (Dündar & Gündüz, 2017 ). It is a key mathematical concept with practical applications such as calculating heights of buildings and measuring distances (Yeo, 2022 ). Despite its importance, learners often confuse similarity with equality, making explicit instruction necessary (Biber, 2020 ). Congruent figures have identical shape and size (Clapham & Nicholson, 2009 ). Learners introduced to symmetry in primary school can better grasp congruency (Cole, 2010 ). Understanding congruent triangles, such as the Side-Angle-Side (S.A.S.) and Angle-Angle (A.A.) criteria, is essential for geometric problem-solving (Laudano & Vincenzi, 2017 ). Justification and Proof in Congruency and Similarity Mathematical justification and proof are central to learning (Cadwallader Olsker, 2011 ). Proof serves multiple purposes, including verifying statements, explaining reasoning, and developing systematic thinking (Knuth, 2002 ). Without proof, mathematical knowledge would remain speculative (Pan & Strayer, 2017 ). Understanding mathematics, particularly Euclidean geometry, requires a combination of instrumental and relational understanding. Prior knowledge, real-life applications, and multiple representations enhance comprehension. Effective teaching methods can help students overcome challenges in congruency and similarity, fostering long-term mathematical proficiency. Research Methodology Qualitative research takes an interpretive, naturalistic approach, aiming to explain or interpret phenomena in their natural environments (Aspers & Corte, 2019 ). It often involves open-ended questions, where the emphasis is on understanding the meanings, participants assign to their experiences. The approach is flexible and allows for the integration of various research methods, such as interviews, focus groups, and documentary analysis (Rahman, 2020 ). For this study, a qualitative, interpretive methodology was employed to understand Grade 9 learners' perceptions of the similarity and congruence of triangles in a school environment. A case study design was selected for this research. The case study approach is ideal when minimal control over variables exists, focusing on understanding a specific phenomenon in its real-world context (Yin, 2003 ). Case studies offer a deep, comprehensive understanding of events, people, or phenomena over time (Oranga & Matere, 2023 ). This research aimed to explore Grade 9 learners' understanding of triangle similarity and congruence through a single case study design, providing the researcher ample time for observation and interaction with the participants. A case study is an in-depth examination of a phenomenon in its natural setting. It involves observing real-world occurrences, making it particularly suitable when researchers cannot easily separate the phenomenon from its context (Yin, 2003 ; Ridder, 2017 ). For this research, the case study design was chosen because it allowed for a deeper exploration of Grade 9 learners' understanding of geometric concepts like triangle similarity and congruence, an area that has received limited research. By focusing on a single case, the researcher was able to engage in meaningful interactions with the participants and collect rich, qualitative data that informed the research question. Two primary methods of data collection were employed: a test and one-on-one interviews. These methods were selected to align with the research questions, methodology, and design, providing a comprehensive understanding of the learners' understanding. The target population for this study included all Grade 9 learners from a high school in the selected district. Given the constraints of time and resources, a purposive sampling method was employed to select 40 Grade 9 learners from a total of 120 learners in the school. Purposive sampling allows for the intentional selection of participants who are most relevant to the research (Andrade, 2021 ). After administering a test, seven learners were chosen for interviews based on the similarities in their responses. From these, four learners were selected for detailed analysis to avoid repetition and ensure meaningful insights. Ethical considerations in research are paramount, especially when working with human participants. Researchers must prioritize participant welfare, ensuring their privacy, consent, and protection from harm (Kalu & Bwalya, 2017). Ethical guidelines also emphasize transparency, integrity, and respect for participants' rights throughout the research process (Ramos, 1989 ; Kang & Hwang, 2021 ). For this study, ethical clearance was obtained from relevant authorities, including the University of South Africa’s College of Education, the Department of Education in Limpopo, and the participating school. Participants were informed about the nature of the study, and their consent was obtained before participation. Additionally, the researcher ensured that the study did not interfere with regular school activities. Discussion of Findings, Recommendations, and Conclusion This section discusses the findings from the analysis of Grade 9 learners' understanding of similarity and congruence, based on test results, interviews, and document analysis. The main goal was to explore the misconceptions and challenges learners faced in these concepts, particularly within the framework of Pirie and Kieren’s model of mathematical understanding. The section also presents recommendations for improving instruction and highlights limitations of the study. The purpose of this study was to examine how Grade 9 learners understand similarity and congruence. The research questions aimed to explore the learners' prior knowledge (primitive knowledge), their ability to represent these concepts (image making), and their strategies for solving and justifying solutions (formalising layers). Primitive Knowing This study focused on levels one to six of the Pirie-Kieren model, excluding the last two levels as they are too abstract for Grade 9 learners. As shown in Table 1 below, many learners struggled to answer the questions correctly, indicating misconceptions that hindered their understanding of similarity and congruence. Primitive Knowing is the initial level of understanding, where learners have basic knowledge but may struggle to articulate it meaningfully. When learners were asked about their prior knowledge, 70% failed to answer basic triangle-related questions accurately. This underscores the importance of reinforcing foundational concepts before introducing new material like similarity and congruence. Learner Misconceptions in Relation to the Pirie-Kieren Model of Mathematical Understanding Table 1 Learners’ Misconceptions Related to Level 1 (Primitive Knowing) of the Pirie-Kieren Model Question No. of Learners Incorrect % Incorrect No. of Learners Correct % Correct No. of Learners Partially Correct % Partially Correct 1.1.1 28 70% 9 22.5% 3 7.5% 3.2.2 14 35% 14 35% 12 30% Table 1 highlights the learners' performance in relation to the first level of mathematical understanding—Primitive Knowing. The data reveals that a significant proportion of learners held misconceptions that prevented them from answering correctly. Question 1.1.1 Analysis Part A: Basic knowledge of similarity and congruency Question 1: SIMILARITY 1.1 Look at the two triangles below. Call them △ABC and △DEF. Then answer the questions below: 1.1.1 What information do you have about these two triangles? 1.1.2 Which condition do you think you will use if you want to prove similarity of these two triangles? 1.1.3 What information do you still need to be able to prove whether the two triangles are similar? In this question, learners were expected to apply their prior understanding of triangles and their properties. They were considered to have misconceptions if they failed to recognize the following the presence of two right-angled triangles (each containing a 90° angle), the different orientations of the triangles and the differing sizes of the triangles. The results in Table 1 indicate that only 7.5% of learners answered correctly, signifying that only a small fraction possessed sufficient primitive knowledge to facilitate cognitive growth in learning new mathematical concepts. Conversely, 70% of learners lacked the necessary foundational knowledge, suggesting significant difficulties in understanding mathematical representations, as evident in their inability to identify the 90° angle. As illustrated in Fig. 1 Participant 6 provided the response "∆ABC///∆DEF," which is entirely unrelated to the given question. This response suggests that the learner relied solely on the topic heading "similarity" and assumed that every question was related to that concept. This assumption was incorrect, as Question 1.1.1 did not reference similarity explicitly. Such misconceptions indicate that the learner’s primitive knowledge was insufficient for constructing new mathematical understanding. Participant P6, later in the interview, when asked the same question, responded, “The two triangles are similar.” In Question 1.1.1 Analysis, learners were asked to decide whether the following sets of triangles, as shown below, are similar and to give reasons for their decision. Like Question 1.1.1, learners were expected to apply prior knowledge of triangles and their properties. Misconceptions were identified if learners failed to apply the sum of angles in a triangle theorem and to identify the 90° angle correctly. Table 1 shows that 14 learners answered incorrectly and 12 provided partially correct responses. These findings highlight a prevalent issue: most learners lacked sufficient primitive knowledge to determine whether the two given triangles were similar. Many students confused the concepts of sides and angles, using them interchangeably. Participant 16’s response, shown in Fig. 2 , suggests a partial understanding of similarity—acknowledging its relationship with ratio and proportionality—but failing to apply this concept correctly. The response demonstrates a fundamental misconception in distinguishing between sides and angles, further reinforcing the notion that these learners require additional support in developing foundational mathematical knowledge. The findings from Questions 1.1.1 and 3.2.2 indicate that a substantial number of Grade 9 learners struggle with basic mathematical representations, particularly in identifying angles and differentiating between fundamental geometric properties. These misconceptions, rooted in inadequate primitive knowing, present a significant barrier to the comprehension of similarity and congruence. Addressing these foundational gaps is crucial for improving learners’ overall mathematical understanding. In the interview, Participant P1's response, "Similarities of triangles is when triangles are... (long pause then sighs) ...is triangles that are not the same," suggests the presence of some primitive knowledge but an inability to express it clearly. Similarly, P16’s response, "It’s the triangles that are equal. I mean that they are different in sides when you write them," indicates confusion in distinguishing between similarity and congruency, which is a common misconception. In the interview, Participant P1 and P5's responses ("I don’t know" and "no") indicate that they have not developed beyond the Primitive Knowing stage regarding similarity theorems. P6 and P16's responses ("SAA, ASA, SSS" and "RHS, SSS, SAS, ASA") mix up similarity and congruency conditions, indicating they have begun Image Making but have not fully grasped the distinction between the two concepts. Image Making At this level, learners attempt to construct mental images of concepts. While 50% of students could draw and label congruent triangles, many struggled to correctly interpret symbols for equality in sides and angles. This suggests that while learners can sometimes represent geometric concepts, they struggle to fully understand or apply these representations. Table 2 below presents the results of the test regarding learners' misconceptions in questions related to Level 2 (Image making) of the Pirie-Kieren model of mathematical understanding Table 2 Learners’ Misconceptions of Questions Related to Level 2 (Image Making) of the Pirie-Kieren Model of Mathematical Understanding Question No of learners that didn’t answer correctly % that didn’t answer correctly No of learners who answered correctly % that answered correctly No of learners whose answers were partially correct % that answered partially correct 2.1.1 2 5% 30 75% 8 20% As in the previous table, Table 2 , presents learners' responses to Question 2.1.1 based on the second level of mathematical understanding (image making). This level is crucial as learners utilize their primitive knowledge to grasp new concepts and develop their mathematical comprehension (Güner & Uygun, 2020). In question 2.11 says, “It is given to you (without a sketch) that two triangles are congruent, ΔKLM ≡ ΔPQR. Now answer the following questions: Sketch the two triangles, any size you choose, if they are congruent. Learners were identified as having misconceptions or a lack of understanding if they demonstrated any of the following shortcoming’s failure to sketch congruent triangles. Congruency requires meeting any of the five conditions for congruent triangles: SSS (side, side, side), SAS, ASA, AAS, and RHS, and Failure to show equality of sides and angles using symbols or numbers. The results in Table 2 indicate that 75% of learners answered the question correctly, demonstrating their ability to use their primitive knowledge to construct accurate sketches of congruent triangles. These learners properly labelled their triangles with side lengths and angle measures, with some using markings to indicate equality. Despite this, 20% of learners demonstrated only partial understanding, as their sketches lacked labels or values to indicate equal angles or sides. Additionally, 5% of learners failed to produce accurate sketches, indicating difficulties in transitioning from the primitive knowing level to image making. The response in Fig. 2 suggests that learner 6 understood the need for the triangles to look the same but had the misconception that labelling corresponding vertices in the same order automatically indicated equality of angles. The learner did not include any values or symbols for equality, making it impossible to conclude congruency. This suggests a lack of primary understanding of mathematical representations and symbols. In the interview, P6's response, "Are triangles that are not in the same size but in the same shape," indicates that the learner is forming an image of similarity but lacks precision in definition. P5’s statement, "It talks about equality of angles," is partially correct but does not provide a complete understanding of similarity, suggesting that the learner has not fully transitioned to the next level. P8's response ("AAA") correctly identifies one of the similarity conditions, suggesting successful movement from Image Making to Image Having. However, the failure to mention all relevant conditions indicates incomplete mastery of this stage. Image Having Learners who reach this level can recall mental images without necessarily constructing them anew. In Question 1.2.1, Learners were considered to have misconceptions or a lack of understanding if they could not identify all pieces of given information, such as side lengths and angles in triangles. Table 3 below presents the results of the test regarding learners' misconceptions in questions related to Level 5 (Image Having) of the Pirie-Kieren model of mathematical understanding. Table 3 Learners’ Misconceptions of Questions Related to Level 3 (Image Having) of the Pirie-Kieren Model of Mathematical Understanding Question No of learners who didn’t answer correctly % that didn’t answer correctly No of learners who answered correctly % that answered correctly No of learners who answered partially correct % that answered partially correct 1.2.1 31 77.5% 6 15% 3 7.5% 1.1.2 3 7.5% 37 92.5% 0 0% 2.2 5 12.5% 23 57.5% 12 30% The results from Table 3 show that 77.5% of learners failed this question, indicating an inability to transition to the image-having level. Many learners failed to compare the triangles correctly or recognize proportional relationships. Figure 3 illustrates a case where learner P13 failed to recognize proportional relationships between corresponding sides. The response suggests that the learner lacked the necessary foundational understanding of proportionality in similarity. In Question 2.2 Learners were regarded as having misconceptions if they could not extract given information about the triangles, could not identify a valid condition for proving congruency and could not apply the given information correctly. Table 4. shows that while 57.5% of learners answered correctly, 30% demonstrated partial understanding, and 12.5% failed outright. Some learners incorrectly used similarity instead of congruence, showing confusion between the two concepts. Others failed to recognize shared sides or vertically opposite angles. This response indicates a lack of understanding, as learner P25 arbitrarily claimed angles were equal without justification. There were also cases where learners used similarity symbols instead of congruence symbols, showing fundamental misconceptions about the two concepts. In Question 1.1.2 , learners were said to have misconceptions if they failed to identify the correct theorem or condition for similarity. Table 4 reveals that 92.5% of learners answered correctly, showing a strong understanding of similarity conditions. However, the remaining 7.5% exhibited misconceptions, failing to recognize similarity conditions or misapplying the concept. The response in Fig. 5 suggests that this learner had not successfully transitioned to the image-having level. The answer incorrectly focused on triangle sizes rather than similarity conditions, suggesting a need for further foundational learning. The results indicate that while many learners successfully progressed to the image-making and image-having levels, a significant number still struggled with fundamental concepts of congruency and similarity. Many misconceptions stemmed from an inability to apply given information correctly, confusion between congruency and similarity, and difficulties recognizing proportional relationships. Further instructional support and reinforcement of basic concepts may be necessary to address these challenges and facilitate smoother transitions between levels of mathematical understanding. P8's explanation, from the interview, “From what I understand about similar triangles is that they are in the same proportion and that the angles are equal," demonstrates a more refined understanding, signifying movement beyond Image Making. However, the explanation lacks explicit reference to side proportionality, showing that the learner has not yet reached Property Noticing. Property Noticing Table 5 below presents the results of the test regarding learners' misconceptions in questions related to Level 4 (property noticing) of the Pirie-Kieren model of mathematical understanding Sub-Question 3 When asked to make connections between properties of triangles, many learners (40%) demonstrated some understanding. However, 20% still struggled to make the necessary connections between sides and angles, indicating that property-noticing and making distinctions are areas for further development. Table 5 below depicts learners’ misconceptions of questions related to Level 4 (property-noticing) of the Pirie-Kieren model of mathematical understanding. Table 5 Learners’ Misconceptions of Questions Related to Level 4 (Property-Noticing) of the Pirie-Kieren Model of Mathematical Understanding Question No of learners who answered inaccurately % that answered inaccurately No of learners who answered accurately % that answered accurately No of learners whose answers were partially accurate % that answered partially accurate 1.1.3 4 10% 35 87% 1 2.5% 1.3 12 30% 19 47,5% 7 17,5% 2.1.2 18 45% 21 52.5% 1 2,5% 2.1.3 17 17,5% 21 52,5% 2 5% 2.2 23 57,5% 2 5% 12 30% 3.2.1 14 35% 10 25% 16 37,5% 3.3.3 1 2.5% 37 92.5% 2 5% Table 5 above shows the learners who were able to answer correctly and those who had misconceptions that made it difficult for them to answer correctly. These statistics are all in relation to the fourth level (property-noticing) of mathematical understanding. Some questions will not be explained in detail to avoid repetition. In Question 1.3 , Learners were said to have misconceptions or lack of understanding if they failed to demonstrate connections, properties and distinctions of the solution strategies. Based on Table 5 , with regard to Level 4 of mathematical understanding (property-noticing), learners at this level are supposed to have a picture of what they learned before, meaning that they must recognise some mathematics related to what they are going to learn. This level will be explained about Question 1.3 from Table 5 . Forty-seven point five per cent (47.5%) of learners failed to respond to this question. This percentage is very bad as it constitutes almost half of the learners. These responses show that these learners could not identify the properties of those triangles. However, 30% of the learners managed to answer the questions in Section 1.3 correctly, which means they had some knowledge regarding similarity together with the image they had constructed mentally through their past learnings of similarity. Furthermore, it is also clear from Table 5 that 17.5% of learners have gained some partial knowledge regarding similarity; hence, they could only provide partially correct answers. Specifics of each learner's response to Question 1.3 are shown in Fig. 5 below. According to Fig. 5 above, learner P25 has misconceptions regarding similarity and angles in general. This learner says that angle Q is equal to angle R, and he reasons that it has been given. This learner clearly has a misconception regarding angles and their relation to each other. This learner equates or compares angles in the same triangle instead of comparing angles in the two triangles. Furthermore, this learner has misconceptions regarding the signs for similarity and equality as he has used the equality instead of the similarity sign. Based on Table 5 with regard to the fourth level of mathematical understanding (property noticing), learners at this level are supposed to use what they learnt to differentiate and compare information or diagrams or pictures and associate them with one another mathematically. The learner should have been able to make use of the primitive knowledge regarding enlargement and reduction. The learner would have realised that △Q’P’R’ is an enlargement of △QPR by a factor of 3. That is, QP \(\:\times\:3\) =Q’P’, QR \(\:\times\:3\) =Q’R’ and PR \(\:\times\:3\) = P’R’. Then, they would have realised that all three sides are in proportion, and hence, the two triangles are similar. In Question 3.3.3 , Learners were said to have misconceptions or lack of understanding if they failed to: Check if learners can follow the demonstration and engage in questions as the demonstration unfold, make connections between questions, identify follow-up questions The question preceding Question 3.3.3 asks learners to prove that ∆ ABC /// ∆ PQC. This question asked them to fully defend the response provided if it ends with ∆ ABC /// ∆ PQC. Had learners been able to prove this relationship successfully, they would have realised that this question was based on their ability to answer the preceding question successfully. If they managed to prove the question, they would have noticed that $$\:\frac{AB}{PQ}=\frac{BC}{QC}=\frac{AC}{PC}\:,\:\:\:\:\:\frac{AB}{2}=\frac{10}{4}$$ $$\:4\:AB=20$$ \(\:\:\:\:\:AB=5\) units, which will be true in Δ ABC and Δ PQC. However, we can see from Table 5 that 2.5% of learners managed to answer this question correctly, suggesting that learners were not operating successfully at the property-noticing level. This lack of success is clear because 92,5% of learners failed to answer this question, and only 5% of learners managed to get the answer partially correct. This poor performance suggests that learners did not understand the question at all. Figure 6 below is an example of a learner’s response to the question. Looking at Fig. 6 above, learner P26 used Pythagoras’s theorem in an attempt to find the answer even though it was used incorrectly. The learner made no connection between this question and the one that came before it. The learner went as far as getting three different values, which shows that this learner had no idea what she was doing, but she was only writing for the sake of filling the spaces. Sub-Question 4: In solving similarity and congruence problems, learners often failed to justify their reasoning or confused concepts such as sides and angles. This highlights the need for learners to develop both procedural and conceptual understanding to justify their solutions effectively. Learners at this level recognize general properties and relationships. The mixing of similarity and congruency theorems by some learners suggests that they have not fully reached Property Noticing, as they are unable to differentiate between the two. Learner P8 demonstrates an understanding of proportionality and its relevance in proving similarity, indicating progression towards Property Noticing. However, misconceptions about equality suggest incomplete formalization of this knowledge. Formalising Table 6 below presents the results of the test regarding learners' misconceptions in questions related to Level 5 (Formalising) of the Pirie-Kieren model of mathematical understanding. Table 6 Learners’ Misconceptions in Questions Related to Level 5 (Formalising) of the Pirie-Kieren Model of Mathematical Understanding Question No. of learners who answered inaccurately % that answered inaccurately No. of learners who answered accurately % that answered accurately No. of learners whose answers were partially accurate % that answered partially accurate 12.2 36 90% 4 10% 0 0% 1.3 12 30% 19 47.5% 7 17.5% 2.2 5 12.5% 23 57.5% 12 30% 3.3.1 2 5% 15 37.5% 23 57.5% Table 6 illustrates learners' responses to Questions 1.2.2, 1.3, 2.2, and 3.3.1, detailing the percentage of learners who answered accurately, partially correctly, and incorrectly. In Question 3.3.1 , learners were considered not to have reached the formalising level if they could not formulate statements supporting similarity and provide justifications and were unable to identify angles based on the properties of lines and triangles. The results in Table 6 show that only 5% of learners successfully proved that the two triangles were similar. This low percentage suggests that most Grade 9 learners do not operate at the formalising level. Supporting this conclusion, 37% of learners completely failed the question, while 57.5% demonstrated partial understanding. Figure 7 below provides an example from P11. This learner began with the statement but failed to provide any justification. From this point, the learner appeared to write random statements without coherence. There was no clear indication of whether the angles summed to 180° due to the properties of a triangle or for another reason. The response demonstrated no understanding of similarity or the properties of angles. In Question 1.2.2 , learners were considered not to have reached the formalising level if they were unable to solve and justify similarity and congruence problems, could not generalise the concepts of similarity and congruence based on given characteristics, and lacked comprehension of similarity and congruence. As shown in Table 6 , 36 out of 40 learners (90%) failed to provide the correct answer, indicating significant misconceptions. These misconceptions prevented them from understanding and applying the concepts of similarity. Only four learners (10%) demonstrated sufficient knowledge of similarity and successfully made generalisations. Figure 8 provides an example of a learner who struggled with this question. Figure 8 illustrates that the learner had misconceptions regarding similarity conditions or theorems. The response incorrectly focused on the equality of sides and, rather than the proportionality of sides, which is fundamental to similarity. Observing Table 7 below presents the test results regarding learners' misconceptions in questions related to Level 6 (Observing) of the Pirie-Kieren model of mathematical understanding. Table 7 Learners’ Misconceptions in Questions Related to Level 6 (Observing) of the Pirie-Kieren Model of Mathematical Understanding Question No. of learners who answered inaccurately % that answered inaccurately No. of learners who answered accurately % that answered accurately No. of learners whose answers were partially accurate % that answered partially accurate 1.2.3 36 90% 2 5% 2 5% Table 7 details learners' performance on Question 1.2.3, which assesses the sixth level of mathematical understanding (Observing). Learners were considered to have misconceptions if they could not effectively integrate multiple pieces of information to determine similarity, failed to make formal statements regarding similarity, were unable to recognize patterns to establish similarity. The results indicate that 90% of learners answered inaccurately, showing that the majority did not reach the observing level. These learners struggled to combine their prior knowledge to conclude similarity and apply the correct condition. While some could identify proportionality between sides, they were unable to use this information meaningfully. Only 5% of learners answered accurately, successfully integrating multiple factors to determine similarity and making formal declarations. Another 5% of learners demonstrated partial understanding, meaning they had some knowledge but did not fully grasp the required concepts. Figure 9 presents an example of a learner’s response. Figure 9 shows that the learner had misconceptions regarding both similarity and the properties of angles. The learner mistakenly stated that "the triangles add up to 180°" rather than recognizing that the sum of angles in a triangle is 180°. This error indicates a fundamental misunderstanding, as similarity requires mathematical justification rather than visual observation. The analysis of learners’ responses through the Pirie-Kieren Model reveals that most learners remain at the Primitive Knowing and Image Making stages, with only a few progressing to Image Having and Property Noticing. The confusion between similarity and congruency, failure to interpret diagrams, and difficulty with proportionality highlight gaps in conceptual understanding. These findings emphasize the need for targeted instructional strategies to bridge these gaps and facilitate learners’ progression through the levels of understanding. DISCUSSION, CONCLUSION AND RECOMENDATIONS How Do Grade 9 Learners Understand Similarity and Congruence of Triangles? The findings of this study indicate that a significant number of learners struggled with understanding similarity and congruence of triangles. Out of 640 responses, 402 were incorrect, accounting for 63% of the total. This substantial error rate suggests that learners face considerable challenges in grasping these concepts, which can be attributed to various factors. One of the primary reasons for these difficulties is the lack of adequate prior knowledge. According to Pirie and Kieren’s model of mathematical understanding, the first level of understanding—primitive knowledge—plays a crucial role in learning new mathematical concepts. Learners who lack a solid foundation in basic geometric principles struggle to build upon new knowledge (Sengul & Argat, 2015 ). This was evident in Fig. 4.1, where P6 failed to recognize the 90° angle or identify that the given triangles were right-angled. This gap in knowledge indicates that learners did not develop a strong understanding of fundamental triangle properties in earlier grades. Similarly, in Fig. 4 . P16 was unable to apply the sum of angles in a triangle theorem to determine an unknown angle and compare corresponding angles for similarity. Research suggests that comprehending the attributes and properties of triangles is essential for developing a deeper understanding of geometric concepts (Biber, 2020 ). The fact that Grade 9 learners have difficulty recognizing basic properties of triangles implies that their previous learning experiences may have emphasized rote memorization rather than conceptual understanding. This aligns with findings by Kusmayadi & Pramudya ( 2018 ), who emphasize that reinforcing prerequisite knowledge is critical before introducing new mathematical concepts. Furthermore, when learners were directly asked to define similarity and congruence, only one out of all interviewees provided a satisfactory definition. This suggests that the majority of learners have not progressed beyond the early levels of understanding as defined by Pirie and Kieren. Their struggle to articulate these concepts further supports the need for reinforcing foundational knowledge before expecting learners to grasp more advanced geometric principles. What Kind of Knowledge Do Learners Bring to the Learning of Congruency and Similarity? (Primitive Knowledge) The analysis of learners’ responses to test item 1.1.1 revealed that 70% provided incorrect answers, 22.5% answered correctly, and 7.5% gave partially correct responses. Despite having been introduced to triangles in earlier grades, many learners failed to correctly identify fundamental properties such as right angles or perpendicular. This aligns with Ulusoy ( 2023 ), who asserts that primary school learners should be able to recognize basic angle properties of quadrilaterals. The inability of Grade 9 learners to do so suggests that they have not internalized these foundational concepts. Additionally, learners’ misconceptions about perpendicular highlight their struggle to relate previous knowledge to new geometric concepts. One interviewee described triangles in terms of their orientation rather than mathematical properties, stating that one triangle looked like a “7” and the other like an “L.” This observation suggests that some learners rely on visual perception rather than mathematical reasoning when identifying geometric properties. What Representations Do Learners Make to Understand Congruency and Similarity? (Image Making) When asked to draw congruent triangles and label their sides and angles, over 50% of learners successfully completed the task. However, some learners failed to provide sufficient information to justify congruence. Figure 1 illustrates this issue, where P6 did not include necessary labels and angle markers. Research by Imama & Caswita ( 2023 ) highlights the importance of mathematical representation in helping learners visualize and solve problems more effectively. The high percentage (57.5%) of incorrect responses in question 2.2 further suggests that many learners struggle to interpret mathematical symbols, leading to errors in reasoning and problem-solving. Furthermore, although learners recognized the symbol for a 90° angle, they often misused it in incorrect contexts, such as using it to calculate the sum of angles on a straight line instead of identifying congruent triangles. This indicates that while learners may memorize geometric symbols, they do not always understand their application in mathematical reasoning. How Do Learners Display Connections, Properties, and Distinctions in Solution Strategies? (Image Having and Property Noticing) The responses to questions 3.2.1 and 3.2.2 showed that while some learners could make connections between given triangles, others struggled with recognizing proportionality and angle relationships. In question 3.2.1, 20% of responses were incorrect, 35% were accurate, and 40% were partially correct. The tie between correct and incorrect responses suggests that many learners still face difficulties in making connections between mathematical properties. Question 3.2.2 required learners to apply the angle sum theorem to determine whether two triangles were similar. The equal distribution of correct and incorrect answers (35% each) highlights the inconsistency in learners' ability to use fundamental geometric properties effectively. This finding aligns with Anthony & Walshaw ( 2023 ), who emphasize the importance of developing mathematical language and reasoning skills to facilitate better problem-solving. How Do Grade 9 Learners Solve Similarity and Congruency Problems and Justify Their Responses? (Formalizing Layers) Problem-solving in mathematics involves a cognitive process where learners must apply their knowledge to unfamiliar problems (Rahman, 2019 ). The findings indicate that while some learners followed the correct steps in proving congruency, they struggled with justifying their answers. Some learners confused sides with angles, copied given information without explanation, or used similarity and congruence interchangeably. These errors suggest that learners are not yet proficient in formalizing mathematical reasoning. Hegde and Meera ( 2012 ) argue that understanding learners’ problem-solving approaches is more valuable than merely assessing correctness. This study’s findings align with that perspective, as many learners demonstrated gaps in logical reasoning and justification despite attempting the problems. Moreover, in question 1.2.3, which tested the observing level of understanding, 90% of learners failed to apply proportional reasoning to solve the problem. This suggests a lack of understanding in the concept of ratio and proportion, as noted by Kharis et al. ( 2021 ), who emphasize the importance of developing strong conceptual foundations for mathematical reasoning. CONCLUSION This study investigated Grade 9 learners' understanding of congruency and similarity in Euclidean geometry through the lens of the Pirie-Kieren model of mathematical comprehension. The findings reveal that a significant number of learners struggle with distinguishing between similarity and congruency, primarily due to weak foundational knowledge and misconceptions about geometric properties. Many learners remain at the Primitive Knowing and Image Making levels, with only a few progressing to more advanced levels of understanding. This suggests that their mathematical learning is often rooted in rote memorization rather than deep conceptual understanding. Furthermore, difficulties in applying proportional reasoning, justifying geometric relationships, and correctly interpreting mathematical symbols indicate gaps in both procedural and conceptual knowledge. These findings underscore the importance of addressing foundational misconceptions and enhancing instructional approaches in mathematics education. A targeted focus on reinforcing prior knowledge, integrating multiple representations, and encouraging active problem-solving can help learners develop a more coherent understanding of congruency and similarity. Without deliberate instructional interventions, these learning gaps may persist, further impeding learners' ability to engage with higher-level mathematical concepts in future grades. To improve learners' understanding of congruency and similarity, it is essential to strengthen their foundational knowledge by reinforcing basic geometric concepts such as triangle properties, angle relationships, and proportional reasoning before introducing advanced topics. Teachers should incorporate multiple representations, including visual aids, dynamic geometry software, and hands-on manipulatives, to help learners better visualize geometric relationships. Additionally, encouraging active learning through classroom discussions, group problem-solving tasks, and structured peer-teaching activities can enhance students’ ability to justify their reasoning and articulate mathematical concepts clearly. Targeted remediation strategies, such as small-group tutoring, differentiated instruction, and structured feedback, should be implemented to address specific misconceptions and learning gaps. Furthermore, continuous professional development workshops for mathematics teachers should focus on effective pedagogical strategies, including inquiry-based learning, conceptual teaching approaches, and technology integration. By adopting these strategies, educators can support learners in developing a deeper understanding of Euclidean geometry and improving their overall mathematical proficiency. Declarations Funding Declaration This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Data Availability Statement The datasets generated and/or analysed during the current study are not publicly available due to confidentiality agreements with the participating school but are available from the corresponding author on reasonable request. Ethics Statement The research protocol for this study was approved by the Ethics Review Committee of the College of Education, University of South Africa (UNISA) under ref number 2022/06/08 to 2025/06/08, and permission was granted by the Limpopo Department of Education and the school where the study was conducted. The study was carried out in accordance with the ethical guidelines and regulations of the UNISA College of Education. Consent to Participate Informed, written consent to participate in the study was obtained from all participants. In the case of participants under the age of 18, consent was obtained from their parents and/or legal guardians. Consent to Publish Written informed consent for the publication of anonymised responses and findings was obtained from all participants or their legal guardians. References Abdurrahman, S. (2020). The effect of multiple representations on students’ mathematical understanding. Journal of Mathematics Education , 12 (3), 245–258. Abramovich, S., & Connell, M. L. (2021). Proportional reasoning in mathematical problem-solving. Mathematical Thinking and Learning , 23 (2), 189–208. Andrade, C. (2021). The purposive sampling method in qualitative research. Journal of Indian Psychiatry , 63 (3), 187–189. Anthony, G., & Walshaw, M. (2023). Developing mathematical reasoning in secondary school students. Educational Studies in Mathematics , 112 (1), 45–62. Arthur, J., Davison, K., & McMahon, P. (2018). Connecting mathematics to real-life contexts: A case study approach. Journal of Mathematics Education Research , 10 (2), 112–130. Aspers, P., & Corte, U. (2019). What is qualitative in qualitative research? Qualitative Sociology , 42 , 139–160. https://doi.org/10.1007/s11133-019-9413-7 Biber, C. (2020). Understanding the difference between similarity and congruency in Euclidean geometry. International Journal of Mathematical Education in Science and Technology , 51 (8), 1176–1190. Brod, G. (2021). The role of prior knowledge in learning: A cognitive science perspective. Educational Psychology Review , 33 (2), 479–499. Cadwallader Olsker, T. (2011). What do we mean by mathematical proof? The Montana Mathematics Enthusiast , 8 (1), 23–36. Clapham, C., & Nicholson, J. (2009). The concise Oxford dictionary of mathematics (5th ed.). Oxford University Press. Cole, M. (2010). The impact of early exposure to symmetry on understanding congruency. Early Childhood Mathematics Education Research Journal , 14 (1), 67–82. Dahlan, J. A., & Wibisono, G. (2021). The significance of geometric thinking in mathematical education. Journal of Mathematics Teaching , 16 (2), 67–89. Del Grande, J. (1986). The role of visualization in geometry. Journal for Research in Mathematics Education , 17 (1), 23–30. Dündar, S., & Gündüz, N. (2017). Students’ understanding of geometric similarity: A case study. International Journal of Science and Mathematics Education , 15 (4), 667–683. Haj-Yahya, A. (2022). Understanding transformations in geometry: A cognitive approach. Mathematics Education Review , 29 (3), 215–232. Hasanona, A. (2017). Mathematical understanding and cognitive development. Educational Psychology Journal , 9 (2), 45–60. Herheim, R. (2023). The role of instrumental and relational understanding in mathematical problem-solving. Mathematics Education Review , 28 (4), 189–207. Hegde, S., & Meera, B. (2012). Understanding problem-solving in mathematics: A cognitive perspective. Journal of Mathematical Thinking , 17 (3), 112–128. Imama, N., & Caswita, R. (2023). The role of verbal representations in students’ mathematical reasoning. Journal of Mathematics Teaching and Learning , 21 (2), 98–113. Kaba, A., & Sengül, S. (2015). Examining mathematical understanding using the Pirie-Kieren model. Educational Research Journal , 12 (3), 214–230. Kang, M., & Hwang, S. (2021). Ethical considerations in qualitative research: A review. Journal of Research Ethics , 19 (1), 89–104. Kharis, R., Rahman, A., & Widodo, S. (2021). Enhancing students’ proportional reasoning through problem-solving tasks. Journal of Mathematics Learning and Teaching , 16 (3), 178–195. Knuth, E. J. (2002). The nature of justification in mathematical reasoning. Journal of Mathematical Behavior , 21 (1), 77–90. Kusmayadi, T. A., & Pramudya, I. (2018). The importance of reinforcing prior knowledge in mathematics education. Journal of Mathematics Education Research , 15 (2), 145–160. Laudano, G., & Vincenzi, S. (2017). Understanding congruency through problem-based learning. Mathematical Thinking and Learning , 20 (1), 78–95. Machisi, E. (2021). Euclidean geometry in the South African curriculum: A historical and pedagogical perspective. Journal of Curriculum Studies , 53 (4), 523–540. Mainali, B. (2021). Symbolic representations in mathematical problem-solving. International Journal of Science and Mathematics Education , 19 (6), 1012–1028. Mastrogiannis, N., & Kordaki, M. (2006). Real-life applications of similarity in mathematics education. Educational Studies in Mathematics , 63 (3), 281–298. McCormack, T. (2002). The relationship between understanding and engagement in mathematics. Mathematics Education Research Journal , 14 (1), 56–78. Naidoo, K., & Kapofu, W. (2020). Innovative teaching approaches to enhance Euclidean geometry understanding. African Journal of Mathematics Education , 19 (2), 112–130. Ngirishi, S., & Bansilal, S. (2019). Logical reasoning in Euclidean geometry: Challenges faced by learners. South African Journal of Mathematics Education , 24 (1), 45–65. Oranga, H., & Matere, J. (2023). Case study methodology in mathematics education research. Journal of Qualitative Research in Mathematics Education , 11 (4), 221–239. Pan, X., & Strayer, J. (2017). The role of proof in mathematical learning. International Journal of Mathematics Education , 23 (2), 98–115. Pirie, S. E., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics , 26 (2), 165–190. Purwanto, E., & Solehudin, M. (2020). The application of the Pirie-Kieren model in understanding mathematical concepts. Mathematics Learning Research Journal , 15 (1), 56–72. Rahman, A. (2019). Cognitive development and mathematical problem-solving. Journal of Mathematical Cognition , 14 (3), 198–212. Rahman, M. S. (2020). The advantages and disadvantages of using qualitative and quantitative research approaches. Journal of Education and Learning , 6 (1), 102–112. Ramos, M. (1989). Ethics in qualitative research: Considerations and challenges. Journal of Qualitative Research Ethics , 5 (2), 45–58. Ridder, H. (2017). The case study approach in research: A review of its methodological strengths and challenges. Qualitative Research Journal , 17 (4), 256–270. Sengul, S., & Argat, M. (2015). Understanding mathematical concepts through the Pirie-Kieren model. International Journal of Mathematics Education , 23 (1), 45–62. Simonsmeier, B. A., Flaig, M., Deiglmayr, A., Schalk, L., & Schneider, M. (2022). The role of prior knowledge in mathematics learning. Educational Psychology Review , 34 (1), 79–108. Syafiqoh, M., Widyastuti, W., & Mahmud, M. (2018). The effect of folding back in the Pirie-Kieren model on mathematical understanding. Mathematics Teaching Research Journal , 22 (3), 134–150. Tachie, S. A. (2020). Building a strong foundation in geometry education. African Journal of Mathematics Teaching , 17 (1), 89–104. Ulusoy, F. (2023). Enhancing primary school learners' understanding of geometric concepts. Educational Studies in Mathematics , 45 (3), 210–228. Yeo, J. B. (2022). The application of similarity in real-life scenarios. Journal of Applied Mathematics , 30 (2), 89–103. Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Sage. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6376119","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":473225394,"identity":"2b514719-7dff-4c37-ab4e-6a15659e0fd1","order_by":0,"name":"France Machaba","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5ElEQVRIiWNgGAWjYJACZiDm5wcSB0jSIjmzgWQtG4hWzy92+OHngpptEsY3cg8e/FHDIM/fwP7wAz4tkrPTjKVnHLstYXYjL+EwzzEGwxkHGJIl8GkxuJ1gxszDdrvO7EaOwWEGNgbGDUA/4dVifzv9GzPPv9sSxjNyDA7++Mdgv4GBsfkHXlukc8yYedtuSxhI5Bgc4G1jSNzAwMyG1xaJ2znF0rx9tyUkzrwxOMzbJ5E84zAbmwU+Lfyz0zd+5vl2W4K/Pcf4449vNrb97e2Pb+DTgmErJJpGwSgYBaNgFFAGANCQRoLceZmjAAAAAElFTkSuQmCC","orcid":"","institution":"University of South Africa","correspondingAuthor":true,"prefix":"","firstName":"France","middleName":"","lastName":"Machaba","suffix":""},{"id":473225395,"identity":"f4cf9642-ca83-40ce-a756-8e8ca151395c","order_by":1,"name":"Dipolelo Ramokgata","email":"","orcid":"","institution":"University of South Africa","correspondingAuthor":false,"prefix":"","firstName":"Dipolelo","middleName":"","lastName":"Ramokgata","suffix":""}],"badges":[],"createdAt":"2025-04-04 12:23:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6376119/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6376119/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":85385031,"identity":"642216a2-ff77-436b-b393-055ed53188c7","added_by":"auto","created_at":"2025-06-25 09:41:23","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":125132,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eWritten Response from P6\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/05a55edf76ed2a2b417d69d3.png"},{"id":85382843,"identity":"e05e7da3-b385-4fa5-895c-6a8add5f9757","added_by":"auto","created_at":"2025-06-25 09:33:23","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":107264,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eWritten Response from P16\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/9d6068d6524809f803d03b76.png"},{"id":85385033,"identity":"daf9c97e-cbd9-473e-b87d-7d60e72d9341","added_by":"auto","created_at":"2025-06-25 09:41:23","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":226389,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 2 Learner’s Written Response for Question 2.1.1 from P6\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/7bb652acf452a8ff8709d867.png"},{"id":85382845,"identity":"dd6163c6-3dc9-4873-862c-540ac54b039a","added_by":"auto","created_at":"2025-06-25 09:33:23","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":121698,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 3 Learner’s Written Response for 1.2.1 from P13\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/1536de2b83ad516b341fe881.png"},{"id":85382851,"identity":"5bc6097e-e0b6-419d-917f-04b289d6b749","added_by":"auto","created_at":"2025-06-25 09:33:23","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":360395,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 4. Learner’s written response for 2.2 from P25\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/caa085ebfc0cc889a480709e.png"},{"id":85382852,"identity":"4ef96dad-2886-467a-b4df-9cf609f4b4a5","added_by":"auto","created_at":"2025-06-25 09:33:23","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":103992,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 5. Learner’s Written Response for 1.1.2 from P6\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/9435cbdc8d72430a711196b7.png"},{"id":85387638,"identity":"06cc1d47-42fa-49e4-a548-030e4839d619","added_by":"auto","created_at":"2025-06-25 10:05:23","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":283603,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 6 \u003c/strong\u003eWritten response from P25\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/faa05f396be4cb51d57d2735.png"},{"id":85382867,"identity":"7649f729-47fe-4053-ba6c-dbcdb6299a25","added_by":"auto","created_at":"2025-06-25 09:33:23","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":220836,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 6 \u003c/strong\u003eWritten Response from P26\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/fc762c354b434b8219fc9ffe.png"},{"id":85387240,"identity":"0861924f-30e7-4a58-ac0a-7a7d855693af","added_by":"auto","created_at":"2025-06-25 09:57:23","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":140403,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 7: Written Response from P11\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/6d73ccf70105e63db4903609.png"},{"id":85385040,"identity":"1c0573bd-1ccd-4e5c-8ec3-3b9ca681fcfc","added_by":"auto","created_at":"2025-06-25 09:41:23","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":307554,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 8: Learner P4’s Response (Not Stating Any Condition for Similarity)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/639bf831642a42fb82e0a03e.png"},{"id":85387238,"identity":"b298b2dd-7a82-4d30-b5aa-ed544534e299","added_by":"auto","created_at":"2025-06-25 09:57:23","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":225920,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFigure 9: Learner P40’s Response (Unable to Make Formal Declarations Concerning Similarity)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/ad0753e244b0f7946b3a5662.png"},{"id":85382869,"identity":"652bea77-1134-4553-9e24-54df617ea542","added_by":"auto","created_at":"2025-06-25 09:33:24","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":7069,"visible":true,"origin":"","legend":"\u003cp\u003eUnnumbered image in the Question 1: SIMILARITY\u003c/p\u003e","description":"","filename":"un1.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/724c974b67cd320571601d76.png"},{"id":85385035,"identity":"b364b356-a416-4a26-823c-53c9587c3e51","added_by":"auto","created_at":"2025-06-25 09:41:23","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":5492,"visible":true,"origin":"","legend":"\u003cp\u003eUnnumbered image in the Question 1: SIMILARITY\u003c/p\u003e","description":"","filename":"un2.png","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/2ecbaeea82803b78e075aa5d.png"},{"id":90068725,"identity":"9fa90f18-7dce-4bf8-ae1b-b38aa83f4ee2","added_by":"auto","created_at":"2025-08-28 06:17:29","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4265368,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6376119/v1/21a20719-fb8f-4b07-b844-73b61fe0a127.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Examining Grade 9 Learners’ Understanding of Congruency and Similarity in Euclidean Geometry","fulltext":[{"header":"Introduction","content":"\u003cp\u003eMathematics education, particularly in Euclidean geometry, plays a crucial role in developing students\u0026rsquo; analytical and problem-solving skills. However, research indicates that South African learners struggle in this area, largely due to inadequate foundational knowledge and ineffective teaching methods. Mathematics performance in South Africa remains a significant concern, as reflected in both national and international assessments. The National Senior Certificate (NSC) diagnostic reports (DBE, 2021) highlight that students often struggle with basic geometric concepts, especially congruence and similarity. These challenges are further evidenced by the poor performance in the Trends in International Mathematics and Science Study (TIMSS), as well as by the Annual National Assessment (ANA) reports, which show that nearly 98% of learners fail to correctly justify congruency in geometric problems.\u003c/p\u003e \u003cp\u003eEuclidean geometry, which includes fundamental concepts such as similarity and congruence, is a critical component of high school mathematics. These topics not only lay the groundwork for advanced mathematical studies but also have practical applications in fields such as engineering, architecture, and medical sciences (Machisi, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Despite its reinstatement in 2012 under the Curriculum and Assessment Policy Statement (CAPS), following its removal in 2006 due to poor student performance, students continue to struggle with these geometric concepts. Studies show that learners often misinterpret theorems, make computational errors, and face difficulty in analysing and interpreting geometric diagrams correctly (Wang et al., 2018).\u003c/p\u003e \u003cp\u003eThe purpose of this study is to explore Grade 9 learners' understanding of similarity and congruence, focusing on their ability to apply theoretical knowledge to solve problems. By employing the Pirie-Kieren model, this research aims to uncover gaps in learners' comprehension and offer recommendations for improved instructional strategies. Given the challenges students face, particularly in understanding geometric principles without real-life applications (D\u0026uuml;ndar \u0026amp; G\u0026uuml;nd\u0026uuml;z, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), this study seeks to identify how learners process and represent geometric concepts. The goal is to contribute to the development of teaching methods that foster a deeper understanding of Euclidean geometry and improve mathematical literacy and problem-solving abilities.\u003c/p\u003e"},{"header":"Theoretical Framework","content":"\u003cp\u003eThe theoretical framework for this study is grounded in the theory of mathematical understanding proposed by Pirie and Kieren (\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e1994\u003c/span\u003e), which offers a model for how learners develop and deepen their comprehension of mathematical concepts. According to Pirie and Kieren (1989), mathematical understanding is not a linear process but rather a layered, non-linear progression. Their model conceptualizes understanding as a series of eight nested levels, or “rings,” each building upon the one before it, yet with an inherent back-and-forth movement within the layers, known as “folding back” (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). This dynamic process allows learners to revisit earlier stages of understanding to refine or deepen their comprehension, leading to more abstract and complex forms of knowledge.\u003c/p\u003e \u003cp\u003eThe theory is based on the notion that learners begin with an initial understanding—termed primitive knowing—which encompasses the prior knowledge they bring to new learning situations (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). As learners interact with new content, they progress through different stages of mathematical understanding, moving from concrete representations to more abstract conceptualizations. Each level represents a deeper engagement with the material, where learners refine their understanding through cognitive processes such as image-making, property noticing, and formalizing.\u003c/p\u003e \u003cp\u003eThe first stage, primitive knowing, refers to the base of a learner’s mathematical knowledge, including gestures, symbols, and basic images. This stage provides the foundation upon which further learning builds (Kaba \u0026amp; Sengül, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). In the context of this study, primitive knowing would encompass the basic geometric knowledge a Grade 9 learner brings to the study of similarity and congruence, such as familiarity with triangle properties and basic geometric theorems.\u003c/p\u003e \u003cp\u003eThe second level, image-making, involves learners using their prior knowledge to differentiate between concepts and apply them in various contexts (Pirie \u0026amp; Kieren, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e1994\u003c/span\u003e). At this stage, learners begin to visualize mathematical concepts and create representations of them, such as drawing triangles to illustrate congruency or similarity.\u003c/p\u003e \u003cp\u003eAs learners continue to develop, they move to the third level, image-having, where they internalize a mental representation of the concept and no longer need to externalize it through writing or drawing (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). At this level, learners are able to identify properties such as congruent sides or angles in their mind, without needing to rely on visual aids.\u003c/p\u003e \u003cp\u003eThe property-noticing stage, the fourth level, involves recognizing and distinguishing patterns and properties in mathematical concepts (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). At this level, learners can identify key properties of geometric figures that indicate similarity or congruency, such as equal angles or proportional sides.\u003c/p\u003e \u003cp\u003eMoving to the fifth level, formalizing, learners begin to define concepts and create formal rules based on their understanding. They are able to synthesize their learning into more structured and precise mathematical statements, such as proving similarity or congruency through formal theorems.\u003c/p\u003e \u003cp\u003eAt the observing level, learners reflect on their formalized knowledge and connect it with broader mathematical structures. They begin to apply their knowledge to solve problems and create new mathematical relationships (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). This level represents a transition from basic conceptualization to the application of formal mathematical ideas.\u003c/p\u003e \u003cp\u003eThe seventh level, structuring, involves learners synthesizing multiple concepts and theorems, linking them through logical reasoning and providing valid justifications for their conclusions. Learners at this stage can systematically prove geometric properties and construct valid arguments based on established mathematical principles.\u003c/p\u003e \u003cp\u003eFinally, the inventing level represents the highest degree of understanding, where learners are able to generate new ideas or concepts and critically analyse existing mathematical theories. At this stage, learners demonstrate a high level of mastery, able to solve complex problems and develop new insights (Purwanto \u0026amp; Solehudin, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThis model emphasizes that mathematical understanding develops through recursive movement across levels, with learners continuously refining their knowledge by revisiting earlier stages (Syafiqoh et al., \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). The concept of “folding back” is crucial to this theory, as learners must often return to earlier levels of understanding to resolve ambiguities or incomplete understandings, leading to deeper, more integrated knowledge of mathematical concepts. This theoretical framework provides a lens for examining Grade 9 learners' understanding of similarity and congruence, helping to identify the specific levels at which they may experience difficulties and the cognitive processes that support their progression in geometric understanding.\u003c/p\u003e "},{"header":"Literature Review","content":"\u003cp\u003eUnderstanding is the process of comprehending inner communications within the substance of content (Hasanona, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). In learning mathematics, learners require understanding (Syafiqoh et al., \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), which is closely connected to the nature of mathematical information. McCormack (\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2002\u003c/span\u003e) highlights the complex relationship between children's involvement, understanding, and development, making understanding a crucial focus of this study.\u003c/p\u003e"},{"header":"Types of Understanding","content":"\u003ch2\u003eInstrumental and Relational Understanding\u003c/h2\u003e\u003cp\u003eAccording to Skemp (1976), understanding in mathematics can be classified into Instrumental Understanding, which is the ability to apply mathematical rules without necessarily understanding the reasoning behind them, and Relational Understanding is an understanding mathematical linkage, justifications for rules, and their appropriate application (Herheim, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). While instrumental understanding allows for immediate application and can boost learner confidence, relational understanding fosters deeper comprehension and problem-solving abilities (Skemp, 1978). Despite its benefits, relational understanding often requires more time, leading some teachers to favour instrumental methods for efficiency (Skemp, 1978).\u003c/p\u003e\u003ch3\u003eEuclidean Geometry\u003c/h3\u003e\u003ch2\u003eImportance of Geometry in Mathematics\u003c/h2\u003e\u003cp\u003eTo master mathematics, learners must understand fundamental geometric concepts (Dahlan \u0026amp; Wibisono, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Tachie (\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) emphasizes that a solid foundation in geometry is necessary for comprehension. Euclidean geometry, the study of planes, solid shapes, and their properties based on Euclid’s theorems and axioms, is central to mathematical learning (Machisi, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\u003ch2\u003eChallenges in Learning Geometry\u003c/h2\u003e\u003cp\u003eProving riders (non-routine geometry problems) and logical reasoning in Euclidean geometry overwhelm many students (Del Grande, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). Deductive reasoning is essential in establishing relationships and properties within figures (Ngirishi \u0026amp; Bansilal, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Furthermore, understanding transformations, such as dilations and congruence, is crucial for learners (Zhang \u0026amp; Wong, 2021; Haj-Yahya, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eResearch shows that learners find Euclidean geometry challenging, but innovative teaching approaches improve engagement (Naidoo \u0026amp; Kapofu, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Understanding similarity and congruency in early grades can encourage more learners to pursue mathematics in higher education, benefiting future career prospects.\u003c/p\u003e\u003ch3\u003ePrior Knowledge Required for Learning Congruency and Similarities\u003c/h3\u003e\u003cp\u003ePrior knowledge, including triangle properties, quadrilateral properties, and basic geometric theorems, plays a significant role in learning similarity and congruency (Brod, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Simonsmeier et al., \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). The CAPS curriculum outlines necessary prerequisites, such as knowledge of different types of triangles and their properties.\u003c/p\u003e\u003ch3\u003eReal-Life Applications of Mathematical Concepts\u003c/h3\u003e\u003cp\u003eConnecting mathematics to real-life contexts enhances student engagement (Arthur et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Proportional reasoning, an essential mathematical concept, is crucial for understanding similarity, dilation, and scaling (Abramovich \u0026amp; Connell, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Yeo, 2019). In everyday life, similarity is applied in measuring distances, constructing bridges, and designing architectural structures (Mastrogiannis \u0026amp; Kordaki, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2006\u003c/span\u003e).\u003c/p\u003e\u003ch2\u003eRepresentations in Understanding Congruency and Similarities\u003c/h2\u003e\u003cp\u003eEffective mathematics learning requires multiple representations such as Graphical/Diagrammatic Representations: Visual aids, such as flow diagrams, tables, and graphs, support conceptual understanding (Abdurrahman, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Symbolic Representations: Mathematical symbols and notation, including those for congruence and similarity, enhance comprehension (Mainali, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Verbal Representations: Communicating mathematical reasoning through explanations strengthens learning (Imama \u0026amp; Caswita, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eProperties and Distinctions Between Congruency and Similarity\u003c/h2\u003e \u003cp\u003eSimilarity refers to figures with the same shape but not necessarily the same size (Dündar \u0026amp; Gündüz, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). It is a key mathematical concept with practical applications such as calculating heights of buildings and measuring distances (Yeo, \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Despite its importance, learners often confuse similarity with equality, making explicit instruction necessary (Biber, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eCongruent figures have identical shape and size (Clapham \u0026amp; Nicholson, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Learners introduced to symmetry in primary school can better grasp congruency (Cole, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Understanding congruent triangles, such as the Side-Angle-Side (S.A.S.) and Angle-Angle (A.A.) criteria, is essential for geometric problem-solving (Laudano \u0026amp; Vincenzi, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e\u003ch2\u003eProperties and Distinctions Between Congruency and Similarity\u003c/h2\u003e\u003cp\u003eSimilarity refers to figures with the same shape but not necessarily the same size (Dündar \u0026amp; Gündüz, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). It is a key mathematical concept with practical applications such as calculating heights of buildings and measuring distances (Yeo, \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Despite its importance, learners often confuse similarity with equality, making explicit instruction necessary (Biber, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eCongruent figures have identical shape and size (Clapham \u0026amp; Nicholson, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Learners introduced to symmetry in primary school can better grasp congruency (Cole, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Understanding congruent triangles, such as the Side-Angle-Side (S.A.S.) and Angle-Angle (A.A.) criteria, is essential for geometric problem-solving (Laudano \u0026amp; Vincenzi, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).\u003c/p\u003e\u003ch2\u003eJustification and Proof in Congruency and Similarity\u003c/h2\u003e\u003cp\u003eMathematical justification and proof are central to learning (Cadwallader Olsker, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Proof serves multiple purposes, including verifying statements, explaining reasoning, and developing systematic thinking (Knuth, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). Without proof, mathematical knowledge would remain speculative (Pan \u0026amp; Strayer, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eUnderstanding mathematics, particularly Euclidean geometry, requires a combination of instrumental and relational understanding. Prior knowledge, real-life applications, and multiple representations enhance comprehension. Effective teaching methods can help students overcome challenges in congruency and similarity, fostering long-term mathematical proficiency.\u003c/p\u003e\u003ch2\u003eResearch Methodology\u003c/h2\u003e\u003cp\u003eQualitative research takes an interpretive, naturalistic approach, aiming to explain or interpret phenomena in their natural environments (Aspers \u0026amp; Corte, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). It often involves open-ended questions, where the emphasis is on understanding the meanings, participants assign to their experiences. The approach is flexible and allows for the integration of various research methods, such as interviews, focus groups, and documentary analysis (Rahman, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). For this study, a qualitative, interpretive methodology was employed to understand Grade 9 learners' perceptions of the similarity and congruence of triangles in a school environment.\u003c/p\u003e\u003cp\u003eA case study design was selected for this research. The case study approach is ideal when minimal control over variables exists, focusing on understanding a specific phenomenon in its real-world context (Yin, \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2003\u003c/span\u003e). Case studies offer a deep, comprehensive understanding of events, people, or phenomena over time (Oranga \u0026amp; Matere, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). This research aimed to explore Grade 9 learners' understanding of triangle similarity and congruence through a single case study design, providing the researcher ample time for observation and interaction with the participants.\u003c/p\u003e\u003cp\u003eA case study is an in-depth examination of a phenomenon in its natural setting. It involves observing real-world occurrences, making it particularly suitable when researchers cannot easily separate the phenomenon from its context (Yin, \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Ridder, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). For this research, the case study design was chosen because it allowed for a deeper exploration of Grade 9 learners' understanding of geometric concepts like triangle similarity and congruence, an area that has received limited research. By focusing on a single case, the researcher was able to engage in meaningful interactions with the participants and collect rich, qualitative data that informed the research question.\u003c/p\u003e\u003cp\u003eTwo primary methods of data collection were employed: a test and one-on-one interviews. These methods were selected to align with the research questions, methodology, and design, providing a comprehensive understanding of the learners' understanding. The target population for this study included all Grade 9 learners from a high school in the selected district. Given the constraints of time and resources, a purposive sampling method was employed to select 40 Grade 9 learners from a total of 120 learners in the school. Purposive sampling allows for the intentional selection of participants who are most relevant to the research (Andrade, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). After administering a test, seven learners were chosen for interviews based on the similarities in their responses. From these, four learners were selected for detailed analysis to avoid repetition and ensure meaningful insights.\u003c/p\u003e\u003cp\u003eEthical considerations in research are paramount, especially when working with human participants. Researchers must prioritize participant welfare, ensuring their privacy, consent, and protection from harm (Kalu \u0026amp; Bwalya, 2017). Ethical guidelines also emphasize transparency, integrity, and respect for participants' rights throughout the research process (Ramos, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e1989\u003c/span\u003e; Kang \u0026amp; Hwang, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). For this study, ethical clearance was obtained from relevant authorities, including the University of South Africa’s College of Education, the Department of Education in Limpopo, and the participating school. Participants were informed about the nature of the study, and their consent was obtained before participation. Additionally, the researcher ensured that the study did not interfere with regular school activities.\u003c/p\u003e\u003ch2\u003eDiscussion of Findings, Recommendations, and Conclusion\u003c/h2\u003e\u003cp\u003eThis section discusses the findings from the analysis of Grade 9 learners' understanding of similarity and congruence, based on test results, interviews, and document analysis. The main goal was to explore the misconceptions and challenges learners faced in these concepts, particularly within the framework of Pirie and Kieren’s model of mathematical understanding. The section also presents recommendations for improving instruction and highlights limitations of the study.\u003c/p\u003e\u003cp\u003eThe purpose of this study was to examine how Grade 9 learners understand similarity and congruence. The research questions aimed to explore the learners' prior knowledge (primitive knowledge), their ability to represent these concepts (image making), and their strategies for solving and justifying solutions (formalising layers).\u003c/p\u003e\u003ch2\u003ePrimitive Knowing\u003c/h2\u003e\u003cp\u003eThis study focused on levels one to six of the Pirie-Kieren model, excluding the last two levels as they are too abstract for Grade 9 learners. As shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e below, many learners struggled to answer the questions correctly, indicating misconceptions that hindered their understanding of similarity and congruence. Primitive Knowing is the initial level of understanding, where learners have basic knowledge but may struggle to articulate it meaningfully. When learners were asked about their prior knowledge, 70% failed to answer basic triangle-related questions accurately. This underscores the importance of reinforcing foundational concepts before introducing new material like similarity and congruence.\u003c/p\u003e\u003ch2\u003eLearner Misconceptions in Relation to the Pirie-Kieren Model of Mathematical Understanding\u003c/h2\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions Related to Level 1 (Primitive Knowing) of the Pirie-Kieren Model\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo. of Learners Incorrect\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% Incorrect\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo. of Learners Correct\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% Correct\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo. of Learners Partially Correct\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% Partially Correct\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.1.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e70%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e22.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e7.5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3.2.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e35%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e35%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e highlights the learners' performance in relation to the first level of mathematical understanding—Primitive Knowing. The data reveals that a significant proportion of learners held misconceptions that prevented them from answering correctly.\u003c/p\u003e\u003ch2\u003eQuestion 1.1.1 Analysis\u003c/h2\u003e\u003ch2\u003ePart A: Basic knowledge of similarity and congruency\u003c/h2\u003e\u003ch2\u003eQuestion 1: SIMILARITY\u003c/h2\u003e\u003cp\u003e1.1 Look at the two triangles below. Call them △ABC and △DEF. Then answer the questions below:\u003c/p\u003e\u003cp\u003e1.1.1 What information do you have about these two triangles?\u003c/p\u003e\u003cp\u003e1.1.2 Which condition do you think you will use if you want to prove similarity of these two triangles?\u003c/p\u003e\u003cp\u003e1.1.3 What information do you still need to be able to prove whether the two triangles are similar?\u003c/p\u003e\u003cp\u003eIn this question, learners were expected to apply their prior understanding of triangles and their properties. They were considered to have misconceptions if they failed to recognize the following the presence of two right-angled triangles (each containing a 90° angle), the different orientations of the triangles and the differing sizes of the triangles.\u003c/p\u003e\u003cp\u003eThe results in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e indicate that only 7.5% of learners answered correctly, signifying that only a small fraction possessed sufficient primitive knowledge to facilitate cognitive growth in learning new mathematical concepts. Conversely, 70% of learners lacked the necessary foundational knowledge, suggesting significant difficulties in understanding mathematical representations, as evident in their inability to identify the 90° angle.\u003c/p\u003e\u003cp\u003eAs illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e Participant 6 provided the response \"∆ABC///∆DEF,\" which is entirely unrelated to the given question. This response suggests that the learner relied solely on the topic heading \"similarity\" and assumed that every question was related to that concept. This assumption was incorrect, as Question 1.1.1 did not reference similarity explicitly. Such misconceptions indicate that the learner’s primitive knowledge was insufficient for constructing new mathematical understanding. Participant P6, later in the interview, when asked the same question, responded, “The two triangles are similar.” In \u003cb\u003eQuestion 1.1.1 Analysis, learners were asked to\u003c/b\u003e decide whether the following sets of triangles, as shown below, are similar and to give reasons for their decision.\u003c/p\u003e\u003cp\u003eLike Question 1.1.1, learners were expected to apply prior knowledge of triangles and their properties. Misconceptions were identified if learners failed to apply the sum of angles in a triangle theorem and to identify the 90° angle correctly. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows that 14 learners answered incorrectly and 12 provided partially correct responses. These findings highlight a prevalent issue: most learners lacked sufficient primitive knowledge to determine whether the two given triangles were similar. Many students confused the concepts of sides and angles, using them interchangeably.\u003c/p\u003e\u003cp\u003eParticipant 16’s response, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e, suggests a partial understanding of similarity—acknowledging its relationship with ratio and proportionality—but failing to apply this concept correctly. The response demonstrates a fundamental misconception in distinguishing between sides and angles, further reinforcing the notion that these learners require additional support in developing foundational mathematical knowledge. The findings from Questions 1.1.1 and 3.2.2 indicate that a substantial number of Grade 9 learners struggle with basic mathematical representations, particularly in identifying angles and differentiating between fundamental geometric properties. These misconceptions, rooted in inadequate primitive knowing, present a significant barrier to the comprehension of similarity and congruence. Addressing these foundational gaps is crucial for improving learners’ overall mathematical understanding. In the interview, Participant P1's response, \"Similarities of triangles is when triangles are... (long pause then sighs) ...is triangles that are not the same,\" suggests the presence of some primitive knowledge but an inability to express it clearly. Similarly, P16’s response, \"It’s the triangles that are equal. I mean that they are different in sides when you write them,\" indicates confusion in distinguishing between similarity and congruency, which is a common misconception.\u003c/p\u003e\u003cp\u003eIn the interview, Participant P1 and P5's responses (\"I don’t know\" and \"no\") indicate that they have not developed beyond the Primitive Knowing stage regarding similarity theorems. P6 and P16's responses (\"SAA, ASA, SSS\" and \"RHS, SSS, SAS, ASA\") mix up similarity and congruency conditions, indicating they have begun Image Making but have not fully grasped the distinction between the two concepts.\u003c/p\u003e\u003cp\u003e \u003cb\u003eImage Making\u003c/b\u003e \u003c/p\u003e\u003cp\u003eAt this level, learners attempt to construct mental images of concepts. While 50% of students could draw and label congruent triangles, many struggled to correctly interpret symbols for equality in sides and angles. This suggests that while learners can sometimes represent geometric concepts, they struggle to fully understand or apply these representations. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e below presents the results of the test regarding learners' misconceptions in questions related to Level 2 (Image making) of the Pirie-Kieren model of mathematical understanding\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions of Questions Related to Level 2 (Image Making) of the Pirie-Kieren Model of Mathematical Understanding\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo of learners that didn’t answer correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% that didn’t answer correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo of learners who answered correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% that answered correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo of learners whose answers were partially correct\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% that answered partially correct\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.1.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e75%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e20%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eAs in the previous table, Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, presents learners' responses to Question 2.1.1 based on the second level of mathematical understanding (image making). This level is crucial as learners utilize their primitive knowledge to grasp new concepts and develop their mathematical comprehension (Güner \u0026amp; Uygun, 2020). In question 2.11 says, “It is given to you (without a sketch) that two triangles are congruent, ΔKLM ≡ ΔPQR. Now answer the following questions: Sketch the two triangles, any size you choose, if they are congruent. Learners were identified as having misconceptions or a lack of understanding if they demonstrated any of the following shortcoming’s failure to sketch congruent triangles. Congruency requires meeting any of the five conditions for congruent triangles: SSS (side, side, side), SAS, ASA, AAS, and RHS, and Failure to show equality of sides and angles using symbols or numbers.\u003c/p\u003e\u003cp\u003eThe results in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e indicate that 75% of learners answered the question correctly, demonstrating their ability to use their primitive knowledge to construct accurate sketches of congruent triangles. These learners properly labelled their triangles with side lengths and angle measures, with some using markings to indicate equality. Despite this, 20% of learners demonstrated only partial understanding, as their sketches lacked labels or values to indicate equal angles or sides. Additionally, 5% of learners failed to produce accurate sketches, indicating difficulties in transitioning from the primitive knowing level to image making.\u003c/p\u003e\u003cp\u003eThe response in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e suggests that learner 6 understood the need for the triangles to look the same but had the misconception that labelling corresponding vertices in the same order automatically indicated equality of angles. The learner did not include any values or symbols for equality, making it impossible to conclude congruency. This suggests a lack of primary understanding of mathematical representations and symbols. In the interview, P6's response, \"Are triangles that are not in the same size but in the same shape,\" indicates that the learner is forming an image of similarity but lacks precision in definition. P5’s statement, \"It talks about equality of angles,\" is partially correct but does not provide a complete understanding of similarity, suggesting that the learner has not fully transitioned to the next level. P8's response (\"AAA\") correctly identifies one of the similarity conditions, suggesting successful movement from Image Making to Image Having. However, the failure to mention all relevant conditions indicates incomplete mastery of this stage.\u003c/p\u003e\u003ch2\u003eImage Having\u003c/h2\u003e\u003cp\u003eLearners who reach this level can recall mental images without necessarily constructing them anew. In Question 1.2.1, Learners were considered to have misconceptions or a lack of understanding if they could not identify all pieces of given information, such as side lengths and angles in triangles. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e below presents the results of the test regarding learners' misconceptions in questions related to Level 5 (Image Having) of the Pirie-Kieren model of mathematical understanding.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions of Questions Related to Level 3 (Image Having) of the Pirie-Kieren Model of Mathematical Understanding\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo of learners who didn’t answer correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% that didn’t answer correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo of learners who answered correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% that answered correctly\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo of learners who answered partially correct\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% that answered partially correct\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.2.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e31\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e77.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e15%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e7.5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.1.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e37\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e92.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e12.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e57.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eThe results from Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e show that 77.5% of learners failed this question, indicating an inability to transition to the image-having level. Many learners failed to compare the triangles correctly or recognize proportional relationships.\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates a case where learner P13 failed to recognize proportional relationships between corresponding sides. The response suggests that the learner lacked the necessary foundational understanding of proportionality in similarity. In Question \u003cb\u003e2.2\u003c/b\u003e Learners were regarded as having misconceptions if they could not extract given information about the triangles, could not identify a valid condition for proving congruency and could not apply the given information correctly. Table\u0026nbsp;4. shows that while 57.5% of learners answered correctly, 30% demonstrated partial understanding, and 12.5% failed outright. Some learners incorrectly used similarity instead of congruence, showing confusion between the two concepts. Others failed to recognize shared sides or vertically opposite angles.\u003c/p\u003e\u003cp\u003eThis response indicates a lack of understanding, as learner P25 arbitrarily claimed angles were equal without justification. There were also cases where learners used similarity symbols instead of congruence symbols, showing fundamental misconceptions about the two concepts. In \u003cb\u003eQuestion 1.1.2\u003c/b\u003e, learners were said to have misconceptions if they failed to identify the correct theorem or condition for similarity. Table\u0026nbsp;4 reveals that 92.5% of learners answered correctly, showing a strong understanding of similarity conditions. However, the remaining 7.5% exhibited misconceptions, failing to recognize similarity conditions or misapplying the concept.\u003c/p\u003e\u003cp\u003eThe response in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e5\u003c/span\u003e suggests that this learner had not successfully transitioned to the image-having level. The answer incorrectly focused on triangle sizes rather than similarity conditions, suggesting a need for further foundational learning. The results indicate that while many learners successfully progressed to the image-making and image-having levels, a significant number still struggled with fundamental concepts of congruency and similarity. Many misconceptions stemmed from an inability to apply given information correctly, confusion between congruency and similarity, and difficulties recognizing proportional relationships. Further instructional support and reinforcement of basic concepts may be necessary to address these challenges and facilitate smoother transitions between levels of mathematical understanding. P8's explanation, from the interview, “From what I understand about similar triangles is that they are in the same proportion and that the angles are equal,\" demonstrates a more refined understanding, signifying movement beyond Image Making. However, the explanation lacks explicit reference to side proportionality, showing that the learner has not yet reached Property Noticing.\u003c/p\u003e\u003ch2\u003eProperty Noticing\u003c/h2\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e below presents the results of the test regarding learners' misconceptions in questions related to Level 4 (property noticing) of the Pirie-Kieren model of mathematical understanding\u003c/p\u003e\u003cp\u003e \u003cstrong\u003eSub-Question 3\u003c/strong\u003e \u003c/p\u003e\u003cp\u003eWhen asked to make connections between properties of triangles, many learners (40%) demonstrated some understanding. However, 20% still struggled to make the necessary connections between sides and angles, indicating that property-noticing and making distinctions are areas for further development. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e below depicts learners’ misconceptions of questions related to Level 4 (property-noticing) of the Pirie-Kieren model of mathematical understanding.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions of Questions Related to Level 4 (Property-Noticing) of the Pirie-Kieren Model of Mathematical Understanding\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo of learners who answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% that answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo of learners who answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% that answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo of learners\u003c/p\u003e \u003cp\u003ewhose answers were partially accurate\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% that answered partially accurate\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.1.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e10%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e87%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e47,5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e17,5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.1.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e45%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e52.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2,5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.1.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e17,5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e52,5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e57,5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3.2.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e35%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e25%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e37,5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3.3.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e37\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e92.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e above shows the learners who were able to answer correctly and those who had misconceptions that made it difficult for them to answer correctly. These statistics are all in relation to the fourth level (property-noticing) of mathematical understanding. Some questions will not be explained in detail to avoid repetition. In \u003cb\u003eQuestion 1.3\u003c/b\u003e, Learners were said to have misconceptions or lack of understanding if they failed to demonstrate connections, properties and distinctions of the solution strategies.\u003c/p\u003e\u003cp\u003eBased on Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e, with regard to Level 4 of mathematical understanding (property-noticing), learners at this level are supposed to have a picture of what they learned before, meaning that they must recognise some mathematics related to what they are going to learn. This level will be explained about Question 1.3 from Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Forty-seven point five per cent (47.5%) of learners failed to respond to this question. This percentage is very bad as it constitutes almost half of the learners. These responses show that these learners could not identify the properties of those triangles. However, 30% of the learners managed to answer the questions in Section 1.3 correctly, which means they had some knowledge regarding similarity together with the image they had constructed mentally through their past learnings of similarity. Furthermore, it is also clear from Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e that 17.5% of learners have gained some partial knowledge regarding similarity; hence, they could only provide partially correct answers. Specifics of each learner's response to Question 1.3 are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e5\u003c/span\u003e below.\u003c/p\u003e\u003cp\u003eAccording to Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e5\u003c/span\u003e above, learner P25 has misconceptions regarding similarity and angles in general. This learner says that angle Q is equal to angle R, and he reasons that it has been given. This learner clearly has a misconception regarding angles and their relation to each other. This learner equates or compares angles in the same triangle instead of comparing angles in the two triangles. Furthermore, this learner has misconceptions regarding the signs for similarity and equality as he has used the equality instead of the similarity sign.\u003c/p\u003e\u003cp\u003eBased on Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e with regard to the fourth level of mathematical understanding (property noticing), learners at this level are supposed to use what they learnt to differentiate and compare information or diagrams or pictures and associate them with one another mathematically. The learner should have been able to make use of the primitive knowledge regarding enlargement and reduction. The learner would have realised that △Q’P’R’ is an enlargement of △QPR by a factor of 3. That is, QP \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:3\\)\u003c/span\u003e\u003c/span\u003e =Q’P’, QR\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:3\\)\u003c/span\u003e\u003c/span\u003e =Q’R’ and PR\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:3\\)\u003c/span\u003e\u003c/span\u003e = P’R’. Then, they would have realised that all three sides are in proportion, and hence, the two triangles are similar.\u003c/p\u003e\u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eIn Question 3.3.3\u003c/b\u003e, Learners were said to have misconceptions or lack of understanding if they\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003efailed to: Check if learners can follow the demonstration and engage in questions as the\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003edemonstration unfold, make connections between questions, identify follow-up questions\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e\u003cp\u003eThe question preceding Question 3.3.3 asks learners to prove that ∆ ABC /// ∆ PQC. This question asked them to fully defend the response provided if it ends with ∆ ABC /// ∆ PQC. Had learners been able to prove this relationship successfully, they would have realised that this question was based on their ability to answer the preceding question successfully. If they managed to prove the question, they would have noticed that\u003c/p\u003e\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\frac{AB}{PQ}=\\frac{BC}{QC}=\\frac{AC}{PC}\\:,\\:\\:\\:\\:\\:\\frac{AB}{2}=\\frac{10}{4}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:4\\:AB=20$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\:\\:\\:AB=5\\)\u003c/span\u003e \u003c/span\u003eunits, which will be true in Δ ABC and Δ PQC. However, we can see from Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e that 2.5% of learners managed to answer this question correctly, suggesting that learners were not operating successfully at the property-noticing level. This lack of success is clear because 92,5% of learners failed to answer this question, and only 5% of learners managed to get the answer partially correct. This poor performance suggests that learners did not understand the question at all. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e6\u003c/span\u003e below is an example of a learner’s response to the question.\u003c/p\u003e\u003cp\u003eLooking at Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e6\u003c/span\u003e above, learner P26 used Pythagoras’s theorem in an attempt to find the answer even though it was used incorrectly. The learner made no connection between this question and the one that came before it. The learner went as far as getting three different values, which shows that this learner had no idea what she was doing, but she was only writing for the sake of filling the spaces.\u003c/p\u003e\u003cp\u003eSub-Question 4: In solving similarity and congruence problems, learners often failed to justify their reasoning or confused concepts such as sides and angles. This highlights the need for learners to develop both procedural and conceptual understanding to justify their solutions effectively. Learners at this level recognize general properties and relationships. The mixing of similarity and congruency theorems by some learners suggests that they have not fully reached Property Noticing, as they are unable to differentiate between the two. Learner P8 demonstrates an understanding of proportionality and its relevance in proving similarity, indicating progression towards Property Noticing. However, misconceptions about equality suggest incomplete formalization of this knowledge.\u003c/p\u003e\u003ch2\u003eFormalising\u003c/h2\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e below presents the results of the test regarding learners' misconceptions in questions related to Level 5 (Formalising) of the Pirie-Kieren model of mathematical understanding.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions in Questions Related to Level 5 (Formalising) of the Pirie-Kieren Model of Mathematical Understanding\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo. of learners who answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% that answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo. of learners who answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% that answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo. of learners whose answers were partially accurate\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% that answered partially accurate\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e90%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e10%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e47.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e17.5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e12.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e57.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e30%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3.3.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e37.5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e57.5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates learners' responses to Questions 1.2.2, 1.3, 2.2, and 3.3.1, detailing the percentage of learners who answered accurately, partially correctly, and incorrectly. In \u003cb\u003eQuestion 3.3.1\u003c/b\u003e, learners were considered not to have reached the formalising level if they could not formulate statements supporting similarity and provide justifications and were unable to identify angles based on the properties of lines and triangles.\u003c/p\u003e\u003cp\u003eThe results in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e show that only 5% of learners successfully proved that the two triangles were similar. This low percentage suggests that most Grade 9 learners do not operate at the formalising level. Supporting this conclusion, 37% of learners completely failed the question, while 57.5% demonstrated partial understanding. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e7\u003c/span\u003e below provides an example from P11.\u003c/p\u003e\u003cp\u003eThis learner began with the statement but failed to provide any justification. From this point, the learner appeared to write random statements without coherence. There was no clear indication of whether the angles summed to 180° due to the properties of a triangle or for another reason. The response demonstrated no understanding of similarity or the properties of angles. In \u003cb\u003eQuestion 1.2.2\u003c/b\u003e, learners were considered not to have reached the formalising level if they were unable to solve and justify similarity and congruence problems, could not generalise the concepts of similarity and congruence based on given characteristics, and lacked comprehension of similarity and congruence.\u003c/p\u003e\u003cp\u003eAs shown in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e, 36 out of 40 learners (90%) failed to provide the correct answer, indicating significant misconceptions. These misconceptions prevented them from understanding and applying the concepts of similarity. Only four learners (10%) demonstrated sufficient knowledge of similarity and successfully made generalisations. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e provides an example of a learner who struggled with this question.\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e illustrates that the learner had misconceptions regarding similarity conditions or theorems. The response incorrectly focused on the equality of sides and, rather than the proportionality of sides, which is fundamental to similarity.\u003c/p\u003e\u003ch2\u003eObserving\u003c/h2\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e below presents the test results regarding learners' misconceptions in questions related to Level 6 (Observing) of the Pirie-Kieren model of mathematical understanding.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLearners’ Misconceptions in Questions Related to Level 6 (Observing) of the Pirie-Kieren Model of Mathematical Understanding\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuestion\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo. of learners who answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% that answered inaccurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo. of learners who answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e% that answered accurately\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNo. of learners whose answers were partially accurate\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e% that answered partially accurate\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.2.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e90%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e5%\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e details learners' performance on Question 1.2.3, which assesses the sixth level of mathematical understanding (Observing). Learners were considered to have misconceptions if they could not effectively integrate multiple pieces of information to determine similarity, failed to make formal statements regarding similarity, were unable to recognize patterns to establish similarity.\u003c/p\u003e\u003cp\u003eThe results indicate that 90% of learners answered inaccurately, showing that the majority did not reach the observing level. These learners struggled to combine their prior knowledge to conclude similarity and apply the correct condition. While some could identify proportionality between sides, they were unable to use this information meaningfully.\u003c/p\u003e\u003cp\u003eOnly 5% of learners answered accurately, successfully integrating multiple factors to determine similarity and making formal declarations. Another 5% of learners demonstrated partial understanding, meaning they had some knowledge but did not fully grasp the required concepts. Figure\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e9\u003c/span\u003e presents an example of a learner’s response.\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows that the learner had misconceptions regarding both similarity and the properties of angles. The learner mistakenly stated that \"the triangles add up to 180°\" rather than recognizing that the sum of angles in a triangle is 180°. This error indicates a fundamental misunderstanding, as similarity requires mathematical justification rather than visual observation.\u003c/p\u003e\u003cp\u003eThe analysis of learners’ responses through the Pirie-Kieren Model reveals that most learners remain at the Primitive Knowing and Image Making stages, with only a few progressing to Image Having and Property Noticing. The confusion between similarity and congruency, failure to interpret diagrams, and difficulty with proportionality highlight gaps in conceptual understanding. These findings emphasize the need for targeted instructional strategies to bridge these gaps and facilitate learners’ progression through the levels of understanding.\u003c/p\u003e"},{"header":"DISCUSSION, CONCLUSION AND RECOMENDATIONS","content":"\u003cp\u003eHow Do Grade 9 Learners Understand Similarity and Congruence of Triangles?\u003c/p\u003e\u003cp\u003eThe findings of this study indicate that a significant number of learners struggled with understanding similarity and congruence of triangles. Out of 640 responses, 402 were incorrect, accounting for 63% of the total. This substantial error rate suggests that learners face considerable challenges in grasping these concepts, which can be attributed to various factors.\u003c/p\u003e\u003cp\u003eOne of the primary reasons for these difficulties is the lack of adequate prior knowledge. According to Pirie and Kieren’s model of mathematical understanding, the first level of understanding—primitive knowledge—plays a crucial role in learning new mathematical concepts. Learners who lack a solid foundation in basic geometric principles struggle to build upon new knowledge (Sengul \u0026amp; Argat, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). This was evident in Fig.\u0026nbsp;4.1, where P6 failed to recognize the 90° angle or identify that the given triangles were right-angled. This gap in knowledge indicates that learners did not develop a strong understanding of fundamental triangle properties in earlier grades.\u003c/p\u003e\u003cp\u003eSimilarly, in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e4\u003c/span\u003e. P16 was unable to apply the sum of angles in a triangle theorem to determine an unknown angle and compare corresponding angles for similarity. Research suggests that comprehending the attributes and properties of triangles is essential for developing a deeper understanding of geometric concepts (Biber, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). The fact that Grade 9 learners have difficulty recognizing basic properties of triangles implies that their previous learning experiences may have emphasized rote memorization rather than conceptual understanding. This aligns with findings by Kusmayadi \u0026amp; Pramudya (\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), who emphasize that reinforcing prerequisite knowledge is critical before introducing new mathematical concepts.\u003c/p\u003e\u003cp\u003eFurthermore, when learners were directly asked to define similarity and congruence, only one out of all interviewees provided a satisfactory definition. This suggests that the majority of learners have not progressed beyond the early levels of understanding as defined by Pirie and Kieren. Their struggle to articulate these concepts further supports the need for reinforcing foundational knowledge before expecting learners to grasp more advanced geometric principles.\u003c/p\u003e\u003cp\u003e \u003cb\u003eWhat Kind of Knowledge Do Learners Bring to the Learning of Congruency and Similarity? (Primitive Knowledge)\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe analysis of learners’ responses to test item 1.1.1 revealed that 70% provided incorrect answers, 22.5% answered correctly, and 7.5% gave partially correct responses. Despite having been introduced to triangles in earlier grades, many learners failed to correctly identify fundamental properties such as right angles or perpendicular. This aligns with Ulusoy (\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2023\u003c/span\u003e), who asserts that primary school learners should be able to recognize basic angle properties of quadrilaterals. The inability of Grade 9 learners to do so suggests that they have not internalized these foundational concepts.\u003c/p\u003e\u003cp\u003eAdditionally, learners’ misconceptions about perpendicular highlight their struggle to relate previous knowledge to new geometric concepts. One interviewee described triangles in terms of their orientation rather than mathematical properties, stating that one triangle looked like a “7” and the other like an “L.” This observation suggests that some learners rely on visual perception rather than mathematical reasoning when identifying geometric properties.\u003c/p\u003e\u003ch2\u003eWhat Representations Do Learners Make to Understand Congruency and Similarity? (Image Making)\u003c/h2\u003e\u003cp\u003eWhen asked to draw congruent triangles and label their sides and angles, over 50% of learners successfully completed the task. However, some learners failed to provide sufficient information to justify congruence. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates this issue, where P6 did not include necessary labels and angle markers. Research by Imama \u0026amp; Caswita (\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) highlights the importance of mathematical representation in helping learners visualize and solve problems more effectively. The high percentage (57.5%) of incorrect responses in question 2.2 further suggests that many learners struggle to interpret mathematical symbols, leading to errors in reasoning and problem-solving. Furthermore, although learners recognized the symbol for a 90° angle, they often misused it in incorrect contexts, such as using it to calculate the sum of angles on a straight line instead of identifying congruent triangles. This indicates that while learners may memorize geometric symbols, they do not always understand their application in mathematical reasoning.\u003c/p\u003e\u003cp\u003e \u003cb\u003eHow Do Learners Display Connections, Properties, and Distinctions in Solution Strategies? (Image Having and Property Noticing)\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe responses to questions 3.2.1 and 3.2.2 showed that while some learners could make connections between given triangles, others struggled with recognizing proportionality and angle relationships. In question 3.2.1, 20% of responses were incorrect, 35% were accurate, and 40% were partially correct. The tie between correct and incorrect responses suggests that many learners still face difficulties in making connections between mathematical properties.\u003c/p\u003e\u003cp\u003eQuestion 3.2.2 required learners to apply the angle sum theorem to determine whether two triangles were similar. The equal distribution of correct and incorrect answers (35% each) highlights the inconsistency in learners' ability to use fundamental geometric properties effectively. This finding aligns with Anthony \u0026amp; Walshaw (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2023\u003c/span\u003e), who emphasize the importance of developing mathematical language and reasoning skills to facilitate better problem-solving.\u003c/p\u003e\u003cp\u003e \u003cb\u003eHow Do Grade 9 Learners Solve Similarity and Congruency Problems and Justify Their Responses? (Formalizing Layers)\u003c/b\u003e \u003c/p\u003e\u003cp\u003eProblem-solving in mathematics involves a cognitive process where learners must apply their knowledge to unfamiliar problems (Rahman, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). The findings indicate that while some learners followed the correct steps in proving congruency, they struggled with justifying their answers. Some learners confused sides with angles, copied given information without explanation, or used similarity and congruence interchangeably. These errors suggest that learners are not yet proficient in formalizing mathematical reasoning.\u003c/p\u003e\u003cp\u003eHegde and Meera (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2012\u003c/span\u003e) argue that understanding learners’ problem-solving approaches is more valuable than merely assessing correctness. This study’s findings align with that perspective, as many learners demonstrated gaps in logical reasoning and justification despite attempting the problems. Moreover, in question 1.2.3, which tested the observing level of understanding, 90% of learners failed to apply proportional reasoning to solve the problem. This suggests a lack of understanding in the concept of ratio and proportion, as noted by Kharis et al. (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), who emphasize the importance of developing strong conceptual foundations for mathematical reasoning.\u003c/p\u003e"},{"header":"CONCLUSION","content":"\u003cp\u003eThis study investigated Grade 9 learners' understanding of congruency and similarity in Euclidean geometry through the lens of the Pirie-Kieren model of mathematical comprehension. The findings reveal that a significant number of learners struggle with distinguishing between similarity and congruency, primarily due to weak foundational knowledge and misconceptions about geometric properties. Many learners remain at the Primitive Knowing and Image Making levels, with only a few progressing to more advanced levels of understanding. This suggests that their mathematical learning is often rooted in rote memorization rather than deep conceptual understanding. Furthermore, difficulties in applying proportional reasoning, justifying geometric relationships, and correctly interpreting mathematical symbols indicate gaps in both procedural and conceptual knowledge.\u003c/p\u003e \u003cp\u003eThese findings underscore the importance of addressing foundational misconceptions and enhancing instructional approaches in mathematics education. A targeted focus on reinforcing prior knowledge, integrating multiple representations, and encouraging active problem-solving can help learners develop a more coherent understanding of congruency and similarity. Without deliberate instructional interventions, these learning gaps may persist, further impeding learners' ability to engage with higher-level mathematical concepts in future grades.\u003c/p\u003e \u003cp\u003eTo improve learners' understanding of congruency and similarity, it is essential to strengthen their foundational knowledge by reinforcing basic geometric concepts such as triangle properties, angle relationships, and proportional reasoning before introducing advanced topics. Teachers should incorporate multiple representations, including visual aids, dynamic geometry software, and hands-on manipulatives, to help learners better visualize geometric relationships. Additionally, encouraging active learning through classroom discussions, group problem-solving tasks, and structured peer-teaching activities can enhance students\u0026rsquo; ability to justify their reasoning and articulate mathematical concepts clearly. Targeted remediation strategies, such as small-group tutoring, differentiated instruction, and structured feedback, should be implemented to address specific misconceptions and learning gaps. Furthermore, continuous professional development workshops for mathematics teachers should focus on effective pedagogical strategies, including inquiry-based learning, conceptual teaching approaches, and technology integration. By adopting these strategies, educators can support learners in developing a deeper understanding of Euclidean geometry and improving their overall mathematical proficiency.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets generated and/or analysed during the current study are not publicly available due to confidentiality agreements with the participating school but are available from the corresponding author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe research protocol for this study was approved by the Ethics Review Committee of the College of Education, University of South Africa (UNISA) under ref number 2022/06/08 to 2025/06/08, and permission was granted by the Limpopo Department of Education and the school where the study was conducted. The study was carried out in accordance with the ethical guidelines and regulations of the UNISA College of Education.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eInformed, written consent to participate in the study was obtained from all participants. In the case of participants under the age of 18, consent was obtained from their parents and/or legal guardians.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Publish\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWritten informed consent for the publication of anonymised responses and findings was obtained from all participants or their legal guardians.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAbdurrahman, S. (2020). The effect of multiple representations on students\u0026rsquo; mathematical understanding. \u003cem\u003eJournal of Mathematics Education\u003c/em\u003e, \u003cem\u003e12\u003c/em\u003e(3), 245\u0026ndash;258.\u003c/li\u003e\n\u003cli\u003eAbramovich, S., \u0026amp; Connell, M. L. (2021). Proportional reasoning in mathematical problem-solving. \u003cem\u003eMathematical Thinking and Learning\u003c/em\u003e, \u003cem\u003e23\u003c/em\u003e(2), 189\u0026ndash;208.\u003c/li\u003e\n\u003cli\u003eAndrade, C. (2021). The purposive sampling method in qualitative research. \u003cem\u003eJournal of Indian Psychiatry\u003c/em\u003e, \u003cem\u003e63\u003c/em\u003e(3), 187\u0026ndash;189.\u003c/li\u003e\n\u003cli\u003eAnthony, G., \u0026amp; Walshaw, M. (2023). Developing mathematical reasoning in secondary school students. \u003cem\u003eEducational Studies in Mathematics\u003c/em\u003e, \u003cem\u003e112\u003c/em\u003e(1), 45\u0026ndash;62.\u003c/li\u003e\n\u003cli\u003eArthur, J., Davison, K., \u0026amp; McMahon, P. (2018). Connecting mathematics to real-life contexts: A case study approach. \u003cem\u003eJournal of Mathematics Education Research\u003c/em\u003e, \u003cem\u003e10\u003c/em\u003e(2), 112\u0026ndash;130.\u003c/li\u003e\n\u003cli\u003eAspers, P., \u0026amp; Corte, U. (2019). What is qualitative in qualitative research? \u003cem\u003eQualitative Sociology\u003c/em\u003e, \u003cem\u003e42\u003c/em\u003e, 139\u0026ndash;160. https://doi.org/10.1007/s11133-019-9413-7\u003c/li\u003e\n\u003cli\u003eBiber, C. (2020). Understanding the difference between similarity and congruency in Euclidean geometry. \u003cem\u003eInternational Journal of Mathematical Education in Science and Technology\u003c/em\u003e, \u003cem\u003e51\u003c/em\u003e(8), 1176\u0026ndash;1190.\u003c/li\u003e\n\u003cli\u003eBrod, G. (2021). The role of prior knowledge in learning: A cognitive science perspective. \u003cem\u003eEducational Psychology Review\u003c/em\u003e, \u003cem\u003e33\u003c/em\u003e(2), 479\u0026ndash;499.\u003c/li\u003e\n\u003cli\u003eCadwallader Olsker, T. (2011). What do we mean by mathematical proof? \u003cem\u003eThe Montana Mathematics Enthusiast\u003c/em\u003e, \u003cem\u003e8\u003c/em\u003e(1), 23\u0026ndash;36.\u003c/li\u003e\n\u003cli\u003eClapham, C., \u0026amp; Nicholson, J. (2009). \u003cem\u003eThe concise Oxford dictionary of mathematics\u003c/em\u003e (5th ed.). Oxford University Press.\u003c/li\u003e\n\u003cli\u003eCole, M. (2010). The impact of early exposure to symmetry on understanding congruency. \u003cem\u003eEarly Childhood Mathematics Education Research Journal\u003c/em\u003e, \u003cem\u003e14\u003c/em\u003e(1), 67\u0026ndash;82.\u003c/li\u003e\n\u003cli\u003eDahlan, J. A., \u0026amp; Wibisono, G. (2021). The significance of geometric thinking in mathematical education. \u003cem\u003eJournal of Mathematics Teaching\u003c/em\u003e, \u003cem\u003e16\u003c/em\u003e(2), 67\u0026ndash;89.\u003c/li\u003e\n\u003cli\u003eDel Grande, J. (1986). The role of visualization in geometry. \u003cem\u003eJournal for Research in Mathematics Education\u003c/em\u003e, \u003cem\u003e17\u003c/em\u003e(1), 23\u0026ndash;30.\u003c/li\u003e\n\u003cli\u003eD\u0026uuml;ndar, S., \u0026amp; G\u0026uuml;nd\u0026uuml;z, N. (2017). Students\u0026rsquo; understanding of geometric similarity: A case study. \u003cem\u003eInternational Journal of Science and Mathematics Education\u003c/em\u003e, \u003cem\u003e15\u003c/em\u003e(4), 667\u0026ndash;683.\u003c/li\u003e\n\u003cli\u003eHaj-Yahya, A. (2022). Understanding transformations in geometry: A cognitive approach. \u003cem\u003eMathematics Education Review\u003c/em\u003e, \u003cem\u003e29\u003c/em\u003e(3), 215\u0026ndash;232.\u003c/li\u003e\n\u003cli\u003eHasanona, A. (2017). Mathematical understanding and cognitive development. \u003cem\u003eEducational Psychology Journal\u003c/em\u003e, \u003cem\u003e9\u003c/em\u003e(2), 45\u0026ndash;60.\u003c/li\u003e\n\u003cli\u003eHerheim, R. (2023). The role of instrumental and relational understanding in mathematical problem-solving. \u003cem\u003eMathematics Education Review\u003c/em\u003e, \u003cem\u003e28\u003c/em\u003e(4), 189\u0026ndash;207.\u003c/li\u003e\n\u003cli\u003eHegde, S., \u0026amp; Meera, B. (2012). Understanding problem-solving in mathematics: A cognitive perspective. \u003cem\u003eJournal of Mathematical Thinking\u003c/em\u003e, \u003cem\u003e17\u003c/em\u003e(3), 112\u0026ndash;128.\u003c/li\u003e\n\u003cli\u003eImama, N., \u0026amp; Caswita, R. (2023). The role of verbal representations in students\u0026rsquo; mathematical reasoning. \u003cem\u003eJournal of Mathematics Teaching and Learning\u003c/em\u003e, \u003cem\u003e21\u003c/em\u003e(2), 98\u0026ndash;113.\u003c/li\u003e\n\u003cli\u003eKaba, A., \u0026amp; Seng\u0026uuml;l, S. (2015). Examining mathematical understanding using the Pirie-Kieren model. \u003cem\u003eEducational Research Journal\u003c/em\u003e, \u003cem\u003e12\u003c/em\u003e(3), 214\u0026ndash;230.\u003c/li\u003e\n\u003cli\u003eKang, M., \u0026amp; Hwang, S. (2021). Ethical considerations in qualitative research: A review. \u003cem\u003eJournal of Research Ethics\u003c/em\u003e, \u003cem\u003e19\u003c/em\u003e(1), 89\u0026ndash;104.\u003c/li\u003e\n\u003cli\u003eKharis, R., Rahman, A., \u0026amp; Widodo, S. (2021). Enhancing students\u0026rsquo; proportional reasoning through problem-solving tasks. \u003cem\u003eJournal of Mathematics Learning and Teaching\u003c/em\u003e, \u003cem\u003e16\u003c/em\u003e(3), 178\u0026ndash;195.\u003c/li\u003e\n\u003cli\u003eKnuth, E. J. (2002). The nature of justification in mathematical reasoning. \u003cem\u003eJournal of Mathematical Behavior\u003c/em\u003e, \u003cem\u003e21\u003c/em\u003e(1), 77\u0026ndash;90.\u003c/li\u003e\n\u003cli\u003eKusmayadi, T. A., \u0026amp; Pramudya, I. (2018). The importance of reinforcing prior knowledge in mathematics education. \u003cem\u003eJournal of Mathematics Education Research\u003c/em\u003e, \u003cem\u003e15\u003c/em\u003e(2), 145\u0026ndash;160.\u003c/li\u003e\n\u003cli\u003eLaudano, G., \u0026amp; Vincenzi, S. (2017). Understanding congruency through problem-based learning. \u003cem\u003eMathematical Thinking and Learning\u003c/em\u003e, \u003cem\u003e20\u003c/em\u003e(1), 78\u0026ndash;95.\u003c/li\u003e\n\u003cli\u003eMachisi, E. (2021). Euclidean geometry in the South African curriculum: A historical and pedagogical perspective. \u003cem\u003eJournal of Curriculum Studies\u003c/em\u003e, \u003cem\u003e53\u003c/em\u003e(4), 523\u0026ndash;540.\u003c/li\u003e\n\u003cli\u003eMainali, B. (2021). Symbolic representations in mathematical problem-solving. \u003cem\u003eInternational Journal of Science and Mathematics Education\u003c/em\u003e, \u003cem\u003e19\u003c/em\u003e(6), 1012\u0026ndash;1028.\u003c/li\u003e\n\u003cli\u003eMastrogiannis, N., \u0026amp; Kordaki, M. (2006). Real-life applications of similarity in mathematics education. \u003cem\u003eEducational Studies in Mathematics\u003c/em\u003e, \u003cem\u003e63\u003c/em\u003e(3), 281\u0026ndash;298.\u003c/li\u003e\n\u003cli\u003eMcCormack, T. (2002). The relationship between understanding and engagement in mathematics. \u003cem\u003eMathematics Education Research Journal\u003c/em\u003e, \u003cem\u003e14\u003c/em\u003e(1), 56\u0026ndash;78.\u003c/li\u003e\n\u003cli\u003eNaidoo, K., \u0026amp; Kapofu, W. (2020). Innovative teaching approaches to enhance Euclidean geometry understanding. \u003cem\u003eAfrican Journal of Mathematics Education\u003c/em\u003e, \u003cem\u003e19\u003c/em\u003e(2), 112\u0026ndash;130.\u003c/li\u003e\n\u003cli\u003eNgirishi, S., \u0026amp; Bansilal, S. (2019). Logical reasoning in Euclidean geometry: Challenges faced by learners. \u003cem\u003eSouth African Journal of Mathematics Education\u003c/em\u003e, \u003cem\u003e24\u003c/em\u003e(1), 45\u0026ndash;65.\u003c/li\u003e\n\u003cli\u003eOranga, H., \u0026amp; Matere, J. (2023). Case study methodology in mathematics education research. \u003cem\u003eJournal of Qualitative Research in Mathematics Education\u003c/em\u003e, \u003cem\u003e11\u003c/em\u003e(4), 221\u0026ndash;239.\u003c/li\u003e\n\u003cli\u003ePan, X., \u0026amp; Strayer, J. (2017). The role of proof in mathematical learning. \u003cem\u003eInternational Journal of Mathematics Education\u003c/em\u003e, \u003cem\u003e23\u003c/em\u003e(2), 98\u0026ndash;115.\u003c/li\u003e\n\u003cli\u003ePirie, S. E., \u0026amp; Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? \u003cem\u003eEducational Studies in Mathematics\u003c/em\u003e, \u003cem\u003e26\u003c/em\u003e(2), 165\u0026ndash;190.\u003c/li\u003e\n\u003cli\u003ePurwanto, E., \u0026amp; Solehudin, M. (2020). The application of the Pirie-Kieren model in understanding mathematical concepts. \u003cem\u003eMathematics Learning Research Journal\u003c/em\u003e, \u003cem\u003e15\u003c/em\u003e(1), 56\u0026ndash;72.\u003c/li\u003e\n\u003cli\u003eRahman, A. (2019). Cognitive development and mathematical problem-solving. \u003cem\u003eJournal of Mathematical Cognition\u003c/em\u003e, \u003cem\u003e14\u003c/em\u003e(3), 198\u0026ndash;212.\u003c/li\u003e\n\u003cli\u003eRahman, M. S. (2020). The advantages and disadvantages of using qualitative and quantitative research approaches. \u003cem\u003eJournal of Education and Learning\u003c/em\u003e, \u003cem\u003e6\u003c/em\u003e(1), 102\u0026ndash;112.\u003c/li\u003e\n\u003cli\u003eRamos, M. (1989). Ethics in qualitative research: Considerations and challenges. \u003cem\u003eJournal of Qualitative Research Ethics\u003c/em\u003e, \u003cem\u003e5\u003c/em\u003e(2), 45\u0026ndash;58.\u003c/li\u003e\n\u003cli\u003eRidder, H. (2017). The case study approach in research: A review of its methodological strengths and challenges. \u003cem\u003eQualitative Research Journal\u003c/em\u003e, \u003cem\u003e17\u003c/em\u003e(4), 256\u0026ndash;270.\u003c/li\u003e\n\u003cli\u003eSengul, S., \u0026amp; Argat, M. (2015). Understanding mathematical concepts through the Pirie-Kieren model. \u003cem\u003eInternational Journal of Mathematics Education\u003c/em\u003e, \u003cem\u003e23\u003c/em\u003e(1), 45\u0026ndash;62.\u003c/li\u003e\n\u003cli\u003eSimonsmeier, B. A., Flaig, M., Deiglmayr, A., Schalk, L., \u0026amp; Schneider, M. (2022). The role of prior knowledge in mathematics learning. \u003cem\u003eEducational Psychology Review\u003c/em\u003e, \u003cem\u003e34\u003c/em\u003e(1), 79\u0026ndash;108.\u003c/li\u003e\n\u003cli\u003eSyafiqoh, M., Widyastuti, W., \u0026amp; Mahmud, M. (2018). The effect of folding back in the Pirie-Kieren model on mathematical understanding. \u003cem\u003eMathematics Teaching Research Journal\u003c/em\u003e, \u003cem\u003e22\u003c/em\u003e(3), 134\u0026ndash;150.\u003c/li\u003e\n\u003cli\u003eTachie, S. A. (2020). Building a strong foundation in geometry education. \u003cem\u003eAfrican Journal of Mathematics Teaching\u003c/em\u003e, \u003cem\u003e17\u003c/em\u003e(1), 89\u0026ndash;104.\u003c/li\u003e\n\u003cli\u003eUlusoy, F. (2023). Enhancing primary school learners\u0026apos; understanding of geometric concepts. \u003cem\u003eEducational Studies in Mathematics\u003c/em\u003e, \u003cem\u003e45\u003c/em\u003e(3), 210\u0026ndash;228.\u003c/li\u003e\n\u003cli\u003eYeo, J. B. (2022). The application of similarity in real-life scenarios. \u003cem\u003eJournal of Applied Mathematics\u003c/em\u003e, \u003cem\u003e30\u003c/em\u003e(2), 89\u0026ndash;103.\u003c/li\u003e\n\u003cli\u003eYin, R. K. (2003). \u003cem\u003eCase study research: Design and methods\u003c/em\u003e (3rd ed.). Sage.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Grade 9 Learners, understanding, Congruency, Similarity and Euclidean Geometry","lastPublishedDoi":"10.21203/rs.3.rs-6376119/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6376119/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper investigates Grade 9 learners\u0026rsquo; understanding of congruency and similarity at a Mopani district school in Limpopo Province, using the Pirie-Kieren model of mathematical comprehension. The study explores the learners\u0026rsquo; thought processes, prior knowledge, and ability to justify their responses when solving problems related to similarity and congruence. A qualitative research approach was employed, using a case study design involving a sample of 40 Grade 9 learners. Data was collected through written tests and semi-structured interviews and analysed using content analysis. The findings indicate that learners struggle with fundamental concepts of Euclidean geometry, particularly in distinguishing between similarity and congruence. The study recommends further research on mathematical understanding at various grade levels and emphasizes the need for instructional strategies that enhance conceptual comprehension rather than rote learning.\u003c/p\u003e","manuscriptTitle":"Examining Grade 9 Learners’ Understanding of Congruency and Similarity in Euclidean Geometry","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-25 09:33:18","doi":"10.21203/rs.3.rs-6376119/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"68e7f588-4bd9-4995-ab2f-952460c082fd","owner":[],"postedDate":"June 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-08-28T06:09:19+00:00","versionOfRecord":[],"versionCreatedAt":"2025-06-25 09:33:18","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6376119","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6376119","identity":"rs-6376119","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}