Improving Mathematical Proof Comprehension through Self-Explanation: Evidence from a Mixed Methods Study

Improving Mathematical Proof Comprehension through Self-Explanation: Evidence from a Mixed Methods Study | 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 Improving Mathematical Proof Comprehension through Self-Explanation: Evidence from a Mixed Methods Study Abdela Mohammed, Moshe Phoshoko This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9507868/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 Understanding and building mathematical proofs is a constant challenge for university mathematics students, especially in proof-intensive courses. This study explored the effect of self-explanation teaching strategies on student's conceptual understanding and proof understanding and how structured self-explanation supports learning mathematical proofs. A mixed-method sequencing explanatory research design was used to provide both statistical evidence and detailed explanatory information on the effectiveness of the intervention. The quantitative phase consisted of 54 undergraduate mathematics students enroled during the academic year 2024/2025, which were divided into experimental group and control group. The experimental group received integrated instruction with structured self-explanatory prompts, guided activities, and reflective tasks while the control group was taught using conventional teaching methods. Both groups completed pre- and post-tests to assess conceptual understanding and understanding. In addition, student attitudes towards mathematics were measured using a standardized Likert scale instrument. Quantitative data was analysed using descriptive statistics, correlation analysis, variance analysis, and multivariate variance analysis. Self-explanation strategy proof comprehension conceptual understanding metacognitive learning active learning strategies proof-based instruction mathematics pedagogy higher education learning Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction 1.1 General Overview of the Study The overview of research efforts to improve mathematics education emphasises strengthening teacher capacities, updating curriculum, and applying student-centred teaching methods. However, what has yet to be adequately addressed is the integration of real application and cultural relevance into mathematics education. The main areas of attention are historical and current educational frameworks, the possibilities of self-explanatory teaching strategies for linking abstract concepts to students' experiences, and the need for further research into the application of self-explanatory methods for improving Ethiopian mathematics education. In the context of higher education, mathematics is vital to promoting critical thinking and problem solving skills. However, a considerable number of undergraduate students in Ethiopia struggle to acquire and understand mathematical proofs, reflecting the larger national challenges of mathematical education. The ability to master evidence, an essential element of advanced mathematics courses, remains vague to many students, highlighting the need to explore other ways of teaching to improve understanding and engagement. Self-explanation is a promising strategy that encourages students to talk about reasoning during problem solving. Empirical research shows that self-explanatory thinking improves cognitive processing and learning outcomes, especially in mathematical evidence that requires abstract thinking and logical reasoning (Chi 1994). The implementation of the self-explanatory teaching strategy in the transformation geometry and computational courses of public University in Ethiopia could effectively address the persistent challenges that students face in understanding mathematical evidence. Currently, such as overcrowded classrooms, insufficient resources, and the lack of qualified teachers, there are many obstacles that require active learning methods. By encouraging students to engage in proof construction, self-explanatory skills contribute to solving abstract concepts. This approach is consistent with broader initiatives aimed at contextualising mathematics education. Unlike traditional methods that focus on memory and repeated learning, self-explanatory explanations promote a deeper engagement with mathematical concepts. It can be further improved by incorporating culturally relevant examples and practical applications, making learning evidence more relatable and easier to grasp by students. The study evaluated the effects of self-explanation using quantitative and qualitative approaches. For example, before and after the intervention, proof understanding and construction were carried out, as well as student interviews and focus group discussions. This methodology provided valuable insight into the impact of self-explanation on students' understanding, attitude, and confidence in dealing with mathematical evidence. The results of this study have made a significant contribution to mathematics education not only at public University, but also at a similar level in Ethiopia and beyond. This approach helps to enhance students' reasoning abilities through self-explanatory explanations, cultivating a better understanding and appreciation of mathematics. 2 LITERATURE REVIEW 2.1 Introduction This section reviews the relevant literature related to the effect of the self-explanation strategy on conceptual understanding and comprehension in undergraduate mathematics. The review establishes a theoretical and empirical foundation for the study by examining key constructs, including mathematical understanding, conceptual and procedural knowledge, mathematical connections, and the role of self-explanation in learning proofs. Emphasis is placed on how students develop proof comprehension and how instructional strategies, particularly self-explanation, support deeper conceptual understanding beyond rote memorisation. Understanding mathematical proofs is widely recognised as one of the most challenging aspects of undergraduate mathematics learning. Traditional lecture-based instruction often emphasises procedural replication rather than conceptual reasoning, limiting the ability to internalise proof structures and underlying logical relationships Dietrich and Evans ( 2022 ). Consequently, contemporary mathematics education research increasingly advocates for active learning strategies that promote metacognition, reasoning, and meaning-making Xie et al. ( 2024 ). Self-explanation emerges as a promising strategy in this regard, as it encourages students to articulate reasoning, identify conceptual gaps, and integrate new knowledge with prior understanding. 2.2 Mathematical Understanding in Undergraduate Mathematics Mathematical understanding is a central goal of mathematics education and has been extensively discussed in the literature (Hiebert & Carpenter, 1992 ; Cai & Ding, 2017 ). It encompasses more than the ability to perform calculations; rather, it involves knowing, interpreting, explaining, and justifying mathematical ideas and relationships. According to the National Council of Mathematics Teachers (NCTM, 2000) and the National Research Council (NRC, 2001), effective mathematics instruction should prioritise active construction of knowledge through reasoning and reflection. In undergraduate mathematics, proof tasks are commonly used to assess understanding (Mejia-Ramos & Inglis, 2009; Weber, 2012 ). However, several scholars argue that proof construction alone does not necessarily reflect genuine proof comprehension, as students may rely on memorised templates or surface-level strategies (Cowen, 1991 ; Conradie & Frith, 2000 ). Research shows that many instructors assess understanding by asking students to reproduce proofs, although such practices may not capture students’ conceptual grasp of students of the logical structure and meaning of proofs. Understanding mathematics involves recognising relationships between concepts, representations, and procedures. Yang et al. ( 2021 ) describe mathematical understanding as the ability to know, see, interpret, and make sense of mathematical ideas. This view aligns with constructivist learning theory, which emphasises that learners actively construct knowledge by connecting new information to previous experiences. 2.2.1 Conceptual Understanding and Procedural Fluency Conceptual understanding refers to comprehension of mathematical concepts, principles, and relationships, while procedural fluency involves the accurate and efficient execution of mathematical procedures (Kilpatrick et al., 2001 ). Although procedural fluency is important, it is insufficient alone to support deep mathematical understanding or proof comprehension. Conceptual understanding enables students to explain why a procedure works, not merely how to apply it. Research indicates that students who lack conceptual understanding often struggle with proof construction and interpretation, particularly in abstract areas such as geometry and calculus Weber ( 2001 ). Dewi et al. (2020) highlight that procedural fluency must be supported by conceptual reasoning to develop meaningful problem-solving skills. In proof-based courses, students often encounter difficulties because they attempt to apply procedures without understanding the underlying logic. Therefore, instructional strategies that integrate conceptual reasoning with procedural knowledge are essential to enhance proof comprehension. Self-explanation supports this integration by prompting students to justify each step of a proof and relate procedures to the underlying concepts. 2.2.2 Mathematical Connections and Proof Understanding Mathematical understanding is further strengthened through the ability to make connections within and across mathematical domains (Freudenthal, 1986 ; sterman & Brting, 2019). Mathematics is not a collection of isolated topics but a coherent network of interconnected ideas. Establishing connections between algebra, geometry, calculus, and real-world applications improves students’ ability to reason mathematically and understand proofs holistically. Studies emphasise that when students recognise relationships between concepts, they are better able to construct and comprehend proofs (Majeed & Hussain, 2021 ). In calculus, for example, linking graphical, symbolic, and verbal representations helps students understand the logical basis of theorems such as continuity and differentiability (Feynman et al., 2011 ). However, research at the university level on students’ ability to make such connections remains limited (Dawkins & Epperson, 2014 ). Self-explanation encourages students to explicitly articulate these connections, thereby strengthening their conceptual networks and supporting deeper proof comprehension. 2.2.3 The Role of Self-Explanation in Proof Comprehension Self-explanation is a cognitive and metacognitive strategy in which learners generate explanations for themselves while learning (Chi et al., 1994 ). It involves articulating reasoning, identifying causal relationships, and making inferences that are not explicitly stated. Research consistently shows that self-explanation improves conceptual understanding, problem-solving ability, and knowledge transfer (Bisra et al., 2018 ; Chi, 2018 ). In mathematics, self-explanation requires students to explain the reasoning behind each step of a solution or proof, justify the use of definitions and theorems, and reflect on the logical coherence of arguments. This process helps learners detect understanding gaps, integrate new ideas with prior knowledge, and construct meaningful mental representations of the proofs. Maarif et al. ( 2020 ) note that proof is a key indicator of mathematical maturity, yet many students struggle to articulate proofs using sound reasoning. Self-explanation addresses this challenge by making implicit reasoning explicit and promoting active engagement with proof structures. Empirical studies show that students who regularly engage in self-explanation demonstrate better proof comprehension, retention, and transfer of learning. 3. RESEARCH METHODOLOGY 3.1 Introduction This section describes the research methodology used to investigate the effect of the self-explanation strategy on conceptual understanding and proof in mathematics. Given the complex and multidimensional nature of mathematical proof learning that encompasses cognitive, metacognitive, and affective components, a mixed method approach was adopted. Integrating quantitative and qualitative methods enabled a comprehensive examination of both measurable learning outcomes and lived learning experiences. The study was conducted at public University, focusing on undergraduate mathematics students enrolled in Transformation Geometry and calculus courses. 3.2 Research Design The study employed a sequential explanatory design, as proposed by Creswell and Clark ( 2017 ), in which quantitative data collection and analysis precede qualitative inquiry. This design was chosen to allow the statistical results on proof comprehension and conceptual understanding to be explained and enriched by qualitative insights from the students. The research was implemented over one academic semester using a quasi-experimental pre-test - post-test control group design. Two groups were formed for each course: an experimental group exposed to the self-explanation strategy and a control group taught using conventional instructional methods. Group Pre-test Treatment Post-test Experimental X₁ Self-explanation X₂ Control X₃ — X₄ 3.2.1 Sequential Explanatory Design In order to better understand the research problem, mixed methods is defined as a process for gathering, evaluating, and "mixing" or integrating both quantitative and qualitative data at some point during the research process within a single study (Tashakkori and Teddlie 2003). In Fig. 3.1 the Sequential Explanatory Design is a two-phase mixed-methods research technique that is illustrated in this flowchart. In this study the first step in the process is gathering and analyzing quantitative data, which offers the first numerical understanding of the study issue. The quantitative results are then further explained, interpreted, and elaborated upon by the collection and analysis of qualitative data. An integrated interpretation is taken as the last step, in which the findings from the two stages are combined to provide thorough conclusions. When quantitative data necessitate further investigation to comprehend underlying systems or participant viewpoints, this design works very well. According to (Bowen et al., 2017 ) several essential characteristics set the explanatory sequential design apart from previous mixed methods techniques. The quantitative phase comes before the qualitative phase in this kind of mixed methods research and the latter is utilized to explain the former. Two separate stages make up the explanatory sequential design first researchers gather and examine quantitative data and then they gather and examine qualitative data to further explore the quantitative findings. This order enables researchers to more thoroughly explain and analyze quantitative results using qualitative insights. The research topic and study objectives frequently influence the choice of a mixed methods design such as the explanatory sequential approach. As they asserted that when employing the explanatory sequential design researchers can use a variety of data gathering techniques, including focus groups or interviews for qualitative data and surveys for quantitative data. Also they added the interpretation stage of the research process is when the integration of quantitative and qualitative data takes place most frequently. 3.3 Participants The participants of this study were public University mathematics second-year and third-year mathematics students. The students in targeted classes that were examined in this study who enrolled during the 2024/2025 academic year. All participants were given the full right to withdraw from the participation at any time in the middle of data collection. All participants from the department of mathematics will be considered at the university. Among those classes, students will be selected purposively depending on their previous semester’s achievement in each section(s). Two groups at each of them randomly assigned as either the experimental group or the control group. There will be equal numbers students in the experimental group and in the control group. These two groups, will be pre-tested, administered a treatment, and then post-tested. 3.4 Data Collection Instruments 3.4.1 Achievement tests Parallel pre-test and post-test instruments were developed for both Transformation Geometry and Calculus to assess conceptual understanding and proof comprehension. The tests focused on the logical structure, justification of the steps and coherence of proofs, aligning with established proof-completion frameworks. 3.4.2 Attitude towards Mathematics Questionnaire Student attitudes were measured using an adapted version of the Attitudes toward Mathematics Scale (Tapia, 1996 ). The questionnaire used a five-point Likert scale and evaluated dimensions such as confidence, enjoyment, perceived value, motivation, and participation in self-explanation. 3.4.3 Focus Group Discussions and Interviews Qualitative data was collected through focus group discussions and semi-structured interviews with selected students from both experimental and control groups. These instruments explored students’ experiences with proof learning, perceptions of self-explanation, challenges faced, and suggestions for improvement. 3.5 Procedure The study was implemented in three phases: Pre-intervention phase: Administration of pre-tests and attitude questionnaires. Intervention phase: The experimental groups received instruction using structured self-explanation instructions, while the control groups followed traditional teaching methods. Post-intervention phase: Administration of post-tests, attitude questionnaires, interviews, and focus group discussions. 3.6 Data Analysis 3.6.1 Quantitative Data Analysis Quantitative data was analysed using SPSS. Descriptive statistics (means and standard deviations) were calculated, followed by inferential analyses that included ANOVA, ANCOVA, MANOVA, and correlation analyses to examine differences between groups and relationships between variables. According to (Rana et al. 2021 ) quantitative approaches are a practical way to explore a research issue. In this phase the participants who were enrolled during 2024/2025 at public university in the department of mathematics. Due to minimum number of student in the deparment all second year mathematics students for the calculus class with total numbers 29 and all third year mathematics students for the transformation geometry class with total numbers 25 were taken for this study. The researcher must select particular methods for data collection, analysis, and interpretation based on the nature of the issue or the research question being investigated As (Watson. 2015) reveal that a variety of techniques pertaining to the methodical examination of social phenomena through the use of statistical or numerical data are included in quantitative research. As a result, quantitative research uses measurement and makes the assumption that the thing being studied is measurable. Also Schutt ( 2019 ) added that quantitative methods are a group of approaches that use numbers to depict factual reality and are predicated on positivism, which holds that observers who measure the social world can understand its properties. At its inception, sociology included quantitative methodologies that placed the field within the historical progression of science and infused it with the dominant spirit of discovery. It is reasonable to associate that advancement with the Renaissance because of its spirit of inquiry and openness to question accepted wisdom. The goals of quantitative research are measurement-based data collection, trend and relationship analysis, and measurement verification. In the literature of (Holton & Burnett 2005 ) quantitative research is descriptive, correlational, experimental, or quasi-experimental. 3.6.2 Qualitative Data Analysis Qualitative data was transcribed and analysed using ATLAS.ti-25. A thematic analysis approach was used that involved coding, categorisation, and theme development. The qualitative phase were used to explain and contextualise the quantitative results. Qualitative research explores, and provides deeper insights into real-world problems (Moser and Korstjens, 2017 ). Qualitative research helps produce hypotheses to better examine and interpret quantitative data, as opposed to gathering numerical data points or intervening or providing treatments, as is the case with quantitative research. The participants of the study were randomly chosen among the participants from both experimental and control groups to both subject sections Qualitative research collects the experiences, opinions, and actions of individuals. Rather than addressing how many or how much, it addresses the hows, and whys. It may take the form of a stand-alone study that just uses qualitative data, or it may be a component of mixed-methods research that incorporates qualitative. 3.7 Reliability and Validity The reliability of the quantitative instruments was established using the Cronbach alpha, with values exceeding the acceptable threshold of 0.70. The validity of the content and construction was ensured through expert review and alignment with established theoretical frameworks. Qualitative credibility was enhanced through triangulation, participant validation, and rich verbal quotations. 4. RESULTS, ANALYSIS, AND FINDINGS 4.1 Overview of the Results This section presents the results, analysis, and key findings of the study that examines the effect of the self-explanation instructional strategy on undergraduate student conceptual understanding and proof comprehension in transformation geometry and calculus. Both quantitative and qualitative data were analysed to determine the effectiveness of the intervention. Quantitative data was analysed using descriptive and inferential statistics, while qualitative data from interviews and focus group discussions were thematically analysed using ATLAS.ti. The results are organised to address the research objective of determining whether self-explanation improves the comprehension of student proof, conceptual understanding, and attitudes toward mathematics. 4.2 Descriptive results of conceptual understanding and proof performance Table-1 Descriptive statistics of pre and post test scores Table 1 The results show a significant increase in the performance of the experimental group compared to the control group after applying the self-explanation strategy. Pre-Test Score for Calculus N Minimum Maximum Mean Std. Deviation 29 9 25 16.79 4.246 Post-Test Score for Calculus 29 13 30 20.28 4.174 Pre-test score for Transformation geometry 25 10 27 19.52 4.753 Post-Test Score for Transformation geometry 25 14 30 22.88 4.978 Valid N (listwise) 25 Descriptive statistics revealed clear improvements in student academic performance following the implementation of the self-explanation strategy. In Calculus, the mean scores increased from 16.79 (SD = 4.246) in the pretest to 20.28 (SD = 4.174) in the posttest. This gain indicates a better conceptual understanding and an improved ability to follow and justify logical steps in calculus proofs. The relatively stable standard deviation suggests that the improvement was consistent among students rather than limited to a small group of high achievers. Similarly, in transformation geometry, mean scores increased from 19.52 (SD = 4.753) to 22.88 (SD = 4.978) after intervention. This improvement reflects the increased capacity to analyze geometric transformations, connect algebraic and visual representations, and construct coherent and logically sound proofs. These findings suggest that self-explanation enabled students to articulate reasoning processes more explicitly and verify the logical consistency of the proof steps, thus strengthening conceptual understanding. 4.3 Inferential Analysis of Proof Comprehension Gains Inferential analysis using ANOVA revealed statistically significant relationships between the pre-test and post-test scores in both transformation geometry and calculus. For transformation geometry, the combined effect was significant, F (11.33) = 3.028, p = 0.031, indicating significant differences in post-test performance based on initial understanding. The significant linearity component, F(1,13) = 29.606, p < 0.001, demonstrates a strong and systematic learning progression, while the non-significant deviation from linearity confirms that learning gains followed a consistent pattern. In calculus, similar results were observed. The combined effect of ANOVA was significant, F(15,13) = 3.518, p = 0.014, with a highly significant linear relationship between pre-test and post-test scores, F(1,13) = 34.553, p < 0.001. These findings indicate that students with stronger initial understanding benefited more consistently from the self-explanation strategy and that conceptual growth and proof comprehension occurred in a structured and progressive manner at different ability levels. 4.4 Effects of self-explanation on attitudes towards mathematics The results of the general linear model (MANOVA) in Table 2 revealed statistically significant improvements in students’ attitudes toward mathematics following the self-explanation intervention. Significant effects were observed for enjoyment of learning mathematics, confidence in mathematical ability, interest and perceived importance of mathematics, and enjoyment of self-explanation activities, all at p 0.95) across these variables indicate that self-explanation had a substantial impact on affective engagement. Increased enjoyment of learning mathematics suggests that self-explanation transformed students’ emotional experiences by reducing the anxiety associated with proofs and promoting active participation. Improvements in confidence reflect improved self-efficacy in constructing and understanding proofs, aligning with constructivist and sociocultural theories that emphasise internal scaffolding and metacognitive regulation. Students also reported an increased appreciation of the importance of mathematics, indicating that self-explanation helped them perceive proofs as meaningful reasoning processes rather than memorised procedures. Table-2 Tests Source Dependent Variable SS df MS F Sig. Partial Eta Squared Observed Power e Corrected Model Post Enjoy Learning Math 64.894 a 46 1.411 4.608 0.020 0.968 0.883 Post Confident Math. Ability 83.418 b 46 1.813 2.962 0.004 0.951 0.686 Post Interest Important 60.762 c 46 1.321 3.596 0.041 0.959 0.781 Post Enjoy Self Explanation 39.048 d 46 0.849 20.797 0.000 0.993 1.000 Intercept Post Enjoy Learning Math 13.780 1 13.780 45.014 0.000 0.865 1.000 Post-confident maths. Ability 6.238 1 6.238 10.189 0.001 0.593 0.781 Post Interest Important 11.070 1 11.070 30.135 0.001 0.811 0.996 Post Enjoy Self Explanation 11.583 1 11.583 283.781 0.000 0.976 1.000 Pre Enjoy Learning Math Post Enjoy Learning Math 9.771 4 2.443 7.980 0.022 0.820 0.914 Pre ConfidentMath Ability Post-confident maths. Ability 6.709 4 1.677 2.739 0.005 0.610 0.460 Pre Interest Important Post Interest Important 9.162 4 2.290 6.235 0.040 0.781 .830 Pre Enjoy Self Explanation Post Enjoy Self Explanation 0.733 3 0.244 5.984 0.013 0.719 .769 Error Post Enjoy Learning Math 2.143 7 0.306 Post-confident maths. Ability 4.286 7 0.286 Post Interest Important 2.571 7 0.367 Post Enjoy Self Explanation 0.286 7 0.041 Corrected Total Post Enjoy Learning Math 67.037 53 Post-confident maths. Ability 87.704 53 Post Interest Important 63.333 53 Post Enjoy Self Explanation 39.333 53 R squared = 0.968 (Adjusted R squared = 0.758), R Squared = 0.951 (Adjusted R Squared = 0.630), R Squared = 0.959 (Adjusted R Squared = 0.693), R Squared = 0.993 (Adjusted R Squared = 0.945), Computed using alpha = 0.05 4.5 Quantitative evidence of the effect of Self-Explanation The quantitative findings provide strong evidence that the self-explanation strategy significantly enhances undergraduate students’ conceptual understanding and proof comprehension in both calculus and Transformation Geometry. Analysis of pre-test and post-test results revealed that students in the experimental group consistently outperformed those in the control group after the intervention, despite both groups demonstrating comparable baseline performance. Clustered boxplot analyses illustrated a pronounced upward shift in post-test scores for the experimental group, accompanied by reduced variability, indicating both improved performance and greater consistency in proof understanding. In contrast, the control group showed only marginal gains, with substantial overlap between pre- and post-test distributions. These visual patterns reinforce the statistical results, suggesting that self-explanation facilitated deeper engagement with proof structures rather than surface-level learning. Correlation analyses also demonstrated strong and statistically significant relationships between pre-test and post-test scores in both subjects (Calculus: r = 0.725, p < 0.01; Transformation geometry: r = 0.667, p < 0.01). These findings indicate that while prior knowledge remained an important predictor of achievement, the self-explanation strategy effectively built on existing conceptual frameworks to improve proof understanding. Multivariate analysis of variance (MANOVA) provided robust confirmation of the instructional impact. The significant multivariate group effect with a large effect size (Wilks’ Lambda = 0.399, p < 0.001, Partial η² = 0.601) demonstrates that the type of instruction played a decisive role in learning outcomes. The effects showed statistically significant differences in posttest scores favouring the experimental group, while no significant differences were found in pretest scores. This confirms that observed improvements can be attributed to the self-explanation strategy rather than pre-existing ability differences. The clustered boxplot in Fig. 4.3 provides a visual representation of the pre-test and post-test scores for both the experimental group (EG) and control group (CG) in transformation geometry. This visual evidence strongly supports the statistical findings obtained from the MANOVA and Between-Subjects Effects tables which discussed in the above table value. In the experimental group we csn see an increase in median scores from pre-test to post-test. The post-test scores (red box) in the EG not only have a higher median than the pre-test scores (blue box). Also show a more compact distribution indicating less variability and more consistency among students improved performance. This aligns with the MANOVA result where the group effect is significant at level of (F = 5.9840, p = 0.013) for post-test scores and partial eta squared = 0.719 which suggesting a moderate effect size due to the self-explanation intervention. In contrast the control group shows less improvement between the pre-test and post-test. In the pre-test and post-test result the median vslues are closer the boxes this implies there overlap, which indicating minimal changes in student performance. 4.6 Qualitative evidence of enhanced proof comprehension The qualitative findings of interviews and focus group discussions as indicated in Fig. 4.1 and Fig. 4.2, respectively, show that strongly supports the quantitative results. Students exposed to the self-explanation strategy consistently reported a better understanding of proof structure, logical sequencing, and conceptual connections. By articulating each step in their own words, the students moved beyond memorisation toward meaningful justification and reasoning, a hallmark of genuine proof comprehension. Participants described increased engagement and cognitive involvement during learning tasks, noting that self-explanation required them to actively process and validate each logical step. Students in Transformation Geometry reported improved integration of visual representations with algebraic reasoning, while Calculus students emphasised greater clarity in constructing proofs from foundational principles rather than relying on memorised templates. Although some students identified challenges related to the articulation of complex ideas and time demands, these difficulties were primarily associated with insufficient scaffolding. Importantly, students overwhelmingly recognised the value of self-explanation to develop deeper conceptual understanding and transferable reasoning skills. Many recommended the strategy for broader use in proof-based mathematics courses, particularly when supported by teacher feedback and structured prompts. 5. Conclusion This study was designed to examine the effect of the self-explanation strategy on the conceptual understanding and proof comprehension of undergraduate students in calculus and transformation geometry. By integrating quantitative and qualitative evidence, the findings provide strong and consistent support for the effectiveness of self-explanation as a pedagogically sound and impactful instructional strategy in proof-based mathematics courses. Quantitative results from pretest and post-test comparisons, correlation analyses, and MANOVA revealed that students exposed to the self-explanation strategy demonstrated significantly higher posttest performance than those taught using conventional instructional approaches. Although both the experimental and control groups exhibited comparable levels of prior knowledge, the experimental group showed a markedly greater improvement in proof comprehension and conceptual understanding. The large multivariate effect sizes confirm that the observed gains were attributable to the instructional intervention rather than chance or pre-existing differences. These findings indicate that self-explanation not only improves learning outcomes but also supports more stable and consistent achievement among students. Qualitative evidence further substantiated these results by providing insight into how and why self-explanation improves proof understanding. Students reported that articulating their reasoning helped them internalise the logical structure of the proofs, recognise conceptual relationships, and justify each step with greater clarity. The strategy promoted active participation, reflective thinking, and metacognitive awareness, allowing students to move beyond rote memorisation toward meaningful reasoning. In Transformation Geometry, students highlighted improved connections between visual and symbolic representations, while Calculus students emphasised increased confidence in constructing proofs independently. Declarations Author Contribution A.M. wrote the main manuscript text M.P. 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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-9507868","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":628439062,"identity":"0757774b-3459-4a0a-b3e4-cbb18da37682","order_by":0,"name":"Abdela Mohammed","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYDACCQYDhgcgBjOIqGAAconRkgCkecBazpCkBcRhbCNCC//s5o0PEmoY5O3Z2R9/+DjvsLw5e/MBhh8V23BbcudYsUHCMQbDHmYeM8mZ2w4b7uw5lsDYc+Y2bmtu5JhJJLAxMAK1sDHzbjvMuOFGjgEzYxtuLfI3csx/JPxjsO9hZn/8mXfOYXuCWgyAtjAktjEk9jAzGEjzNhxOJKjFEOgXicQ+huSew0C/zDiWnrzhzLGEg/j8Ine7eeOHD98YbNv7jz/+8KHG2nbD8eaDD35U4PE+BPyHMZrB5AFC6pFBHSmKR8EoGAWjYIQAAE8CWXdKjJc9AAAAAElFTkSuQmCC","orcid":"","institution":"Wachemo University","correspondingAuthor":true,"prefix":"","firstName":"Abdela","middleName":"","lastName":"Mohammed","suffix":""},{"id":628439066,"identity":"2bde031e-e5cf-4830-aae3-6147e35d70a7","order_by":1,"name":"Moshe Phoshoko","email":"","orcid":"","institution":"University of South Africa","correspondingAuthor":false,"prefix":"","firstName":"Moshe","middleName":"","lastName":"Phoshoko","suffix":""}],"badges":[],"createdAt":"2026-04-23 14:23:14","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9507868/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9507868/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107861148,"identity":"1b27e6e9-471f-47ea-b801-ead7ba5f8783","added_by":"auto","created_at":"2026-04-27 05:35:08","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":239086,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 3.1 Sequential Explanatory Design Flowchart\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-9507868/v1/888b5809a267e8966753bdea.png"},{"id":107870519,"identity":"9d97e498-1384-4f2e-8d79-60aa69d2bd7a","added_by":"auto","created_at":"2026-04-27 07:39:50","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":10101,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 4.3 Clustered Boxplot\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-9507868/v1/abe20adce835cb2068fccbf3.png"},{"id":107870350,"identity":"eca6ffef-162b-4f70-a7f6-747239e43f94","added_by":"auto","created_at":"2026-04-27 07:39:26","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":93738,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 4.1: Frequencies of Key Codes from interview responses\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-9507868/v1/0fd1caddafb7f3007cb9771f.png"},{"id":107861151,"identity":"9ec480e0-fe33-44c6-bcfe-6911ce8d8c5b","added_by":"auto","created_at":"2026-04-27 05:35:08","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":102224,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 4. 2:- Focus group discussion chart\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-9507868/v1/c058dc1aa3f2c748cab6192b.png"},{"id":108421343,"identity":"af41c5c6-5251-4917-8854-00692752213c","added_by":"auto","created_at":"2026-05-04 12:40:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":738350,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9507868/v1/3e944423-90ff-4763-95d3-de429f13c8fe.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Improving Mathematical Proof Comprehension through Self-Explanation: Evidence from a Mixed Methods Study","fulltext":[{"header":"1. Introduction","content":"\u003cdiv id=\"Sec2\" class=\"Section2\"\u003e \u003ch2\u003e1.1 General Overview of the Study\u003c/h2\u003e \u003cp\u003eThe overview of research efforts to improve mathematics education emphasises strengthening teacher capacities, updating curriculum, and applying student-centred teaching methods. However, what has yet to be adequately addressed is the integration of real application and cultural relevance into mathematics education. The main areas of attention are historical and current educational frameworks, the possibilities of self-explanatory teaching strategies for linking abstract concepts to students' experiences, and the need for further research into the application of self-explanatory methods for improving Ethiopian mathematics education. In the context of higher education, mathematics is vital to promoting critical thinking and problem solving skills. However, a considerable number of undergraduate students in Ethiopia struggle to acquire and understand mathematical proofs, reflecting the larger national challenges of mathematical education. The ability to master evidence, an essential element of advanced mathematics courses, remains vague to many students, highlighting the need to explore other ways of teaching to improve understanding and engagement. Self-explanation is a promising strategy that encourages students to talk about reasoning during problem solving. Empirical research shows that self-explanatory thinking improves cognitive processing and learning outcomes, especially in mathematical evidence that requires abstract thinking and logical reasoning (Chi 1994).\u003c/p\u003e \u003cp\u003eThe implementation of the self-explanatory teaching strategy in the transformation geometry and computational courses of public University in Ethiopia could effectively address the persistent challenges that students face in understanding mathematical evidence. Currently, such as overcrowded classrooms, insufficient resources, and the lack of qualified teachers, there are many obstacles that require active learning methods. By encouraging students to engage in proof construction, self-explanatory skills contribute to solving abstract concepts. This approach is consistent with broader initiatives aimed at contextualising mathematics education. Unlike traditional methods that focus on memory and repeated learning, self-explanatory explanations promote a deeper engagement with mathematical concepts. It can be further improved by incorporating culturally relevant examples and practical applications, making learning evidence more relatable and easier to grasp by students. The study evaluated the effects of self-explanation using quantitative and qualitative approaches. For example, before and after the intervention, proof understanding and construction were carried out, as well as student interviews and focus group discussions. This methodology provided valuable insight into the impact of self-explanation on students' understanding, attitude, and confidence in dealing with mathematical evidence. The results of this study have made a significant contribution to mathematics education not only at public University, but also at a similar level in Ethiopia and beyond. This approach helps to enhance students' reasoning abilities through self-explanatory explanations, cultivating a better understanding and appreciation of mathematics.\u003c/p\u003e \u003c/div\u003e"},{"header":"2 LITERATURE REVIEW","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Introduction\u003c/h2\u003e \u003cp\u003eThis section reviews the relevant literature related to the effect of the self-explanation strategy on conceptual understanding and comprehension in undergraduate mathematics. The review establishes a theoretical and empirical foundation for the study by examining key constructs, including mathematical understanding, conceptual and procedural knowledge, mathematical connections, and the role of self-explanation in learning proofs. Emphasis is placed on how students develop proof comprehension and how instructional strategies, particularly self-explanation, support deeper conceptual understanding beyond rote memorisation.\u003c/p\u003e \u003cp\u003eUnderstanding mathematical proofs is widely recognised as one of the most challenging aspects of undergraduate mathematics learning. Traditional lecture-based instruction often emphasises procedural replication rather than conceptual reasoning, limiting the ability to internalise proof structures and underlying logical relationships Dietrich and Evans (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Consequently, contemporary mathematics education research increasingly advocates for active learning strategies that promote metacognition, reasoning, and meaning-making Xie et al. (\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Self-explanation emerges as a promising strategy in this regard, as it encourages students to articulate reasoning, identify conceptual gaps, and integrate new knowledge with prior understanding.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Mathematical Understanding in Undergraduate Mathematics\u003c/h2\u003e \u003cp\u003eMathematical understanding is a central goal of mathematics education and has been extensively discussed in the literature (Hiebert \u0026amp; Carpenter, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Cai \u0026amp; Ding, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). It encompasses more than the ability to perform calculations; rather, it involves knowing, interpreting, explaining, and justifying mathematical ideas and relationships. According to the National Council of Mathematics Teachers (NCTM, 2000) and the National Research Council (NRC, 2001), effective mathematics instruction should prioritise active construction of knowledge through reasoning and reflection.\u003c/p\u003e \u003cp\u003eIn undergraduate mathematics, proof tasks are commonly used to assess understanding (Mejia-Ramos \u0026amp; Inglis, 2009; Weber, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). However, several scholars argue that proof construction alone does not necessarily reflect genuine proof comprehension, as students may rely on memorised templates or surface-level strategies (Cowen, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1991\u003c/span\u003e; Conradie \u0026amp; Frith, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). Research shows that many instructors assess understanding by asking students to reproduce proofs, although such practices may not capture students\u0026rsquo; conceptual grasp of students of the logical structure and meaning of proofs. Understanding mathematics involves recognising relationships between concepts, representations, and procedures. Yang et al. (\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) describe mathematical understanding as the ability to know, see, interpret, and make sense of mathematical ideas. This view aligns with constructivist learning theory, which emphasises that learners actively construct knowledge by connecting new information to previous experiences.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Conceptual Understanding and Procedural Fluency\u003c/h2\u003e \u003cp\u003eConceptual understanding refers to comprehension of mathematical concepts, principles, and relationships, while procedural fluency involves the accurate and efficient execution of mathematical procedures (Kilpatrick et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Although procedural fluency is important, it is insufficient alone to support deep mathematical understanding or proof comprehension. Conceptual understanding enables students to explain \u003cem\u003ewhy\u003c/em\u003e a procedure works, not merely \u003cem\u003ehow\u003c/em\u003e to apply it.\u003c/p\u003e \u003cp\u003eResearch indicates that students who lack conceptual understanding often struggle with proof construction and interpretation, particularly in abstract areas such as geometry and calculus Weber (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Dewi et al. (2020) highlight that procedural fluency must be supported by conceptual reasoning to develop meaningful problem-solving skills. In proof-based courses, students often encounter difficulties because they attempt to apply procedures without understanding the underlying logic.\u003c/p\u003e \u003cp\u003eTherefore, instructional strategies that integrate conceptual reasoning with procedural knowledge are essential to enhance proof comprehension. Self-explanation supports this integration by prompting students to justify each step of a proof and relate procedures to the underlying concepts.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Mathematical Connections and Proof Understanding\u003c/h2\u003e \u003cp\u003eMathematical understanding is further strengthened through the ability to make connections within and across mathematical domains (Freudenthal, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; sterman \u0026amp; Brting, 2019). Mathematics is not a collection of isolated topics but a coherent network of interconnected ideas. Establishing connections between algebra, geometry, calculus, and real-world applications improves students\u0026rsquo; ability to reason mathematically and understand proofs holistically.\u003c/p\u003e \u003cp\u003eStudies emphasise that when students recognise relationships between concepts, they are better able to construct and comprehend proofs (Majeed \u0026amp; Hussain, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In calculus, for example, linking graphical, symbolic, and verbal representations helps students understand the logical basis of theorems such as continuity and differentiability (Feynman et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). However, research at the university level on students\u0026rsquo; ability to make such connections remains limited (Dawkins \u0026amp; Epperson, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Self-explanation encourages students to explicitly articulate these connections, thereby strengthening their conceptual networks and supporting deeper proof comprehension.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 The Role of Self-Explanation in Proof Comprehension\u003c/h2\u003e \u003cp\u003eSelf-explanation is a cognitive and metacognitive strategy in which learners generate explanations for themselves while learning (Chi et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e1994\u003c/span\u003e). It involves articulating reasoning, identifying causal relationships, and making inferences that are not explicitly stated. Research consistently shows that self-explanation improves conceptual understanding, problem-solving ability, and knowledge transfer (Bisra et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Chi, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). In mathematics, self-explanation requires students to explain the reasoning behind each step of a solution or proof, justify the use of definitions and theorems, and reflect on the logical coherence of arguments. This process helps learners detect understanding gaps, integrate new ideas with prior knowledge, and construct meaningful mental representations of the proofs. Maarif et al. (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) note that proof is a key indicator of mathematical maturity, yet many students struggle to articulate proofs using sound reasoning. Self-explanation addresses this challenge by making implicit reasoning explicit and promoting active engagement with proof structures. Empirical studies show that students who regularly engage in self-explanation demonstrate better proof comprehension, retention, and transfer of learning.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3. RESEARCH METHODOLOGY","content":"\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Introduction\u003c/h2\u003e \u003cp\u003eThis section describes the research methodology used to investigate the effect of the self-explanation strategy on conceptual understanding and proof in mathematics. Given the complex and multidimensional nature of mathematical proof learning that encompasses cognitive, metacognitive, and affective components, a mixed method approach was adopted. Integrating quantitative and qualitative methods enabled a comprehensive examination of both measurable learning outcomes and lived learning experiences. The study was conducted at public University, focusing on undergraduate mathematics students enrolled in Transformation Geometry and calculus courses.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Research Design\u003c/h2\u003e \u003cp\u003eThe study employed a sequential explanatory design, as proposed by Creswell and Clark (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), in which quantitative data collection and analysis precede qualitative inquiry. This design was chosen to allow the statistical results on proof comprehension and conceptual understanding to be explained and enriched by qualitative insights from the students. The research was implemented over one academic semester using a quasi-experimental pre-test - post-test control group design. Two groups were formed for each course: an experimental group exposed to the self-explanation strategy and a control group taught using conventional instructional methods.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"4\"\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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGroup\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePre-test\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTreatment\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePost-test\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExperimental\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX₁\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSelf-explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX₂\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eControl\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX₃\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026mdash;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX₄\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e3.2.1 Sequential Explanatory Design\u003c/h2\u003e \u003cp\u003eIn order to better understand the research problem, mixed methods is defined as a process for gathering, evaluating, and \"mixing\" or integrating both quantitative and qualitative data at some point during the research process within a single study (Tashakkori and Teddlie 2003).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e the Sequential Explanatory Design is a two-phase mixed-methods research technique that is illustrated in this flowchart. In this study the first step in the process is gathering and analyzing quantitative data, which offers the first numerical understanding of the study issue. The quantitative results are then further explained, interpreted, and elaborated upon by the collection and analysis of qualitative data. An integrated interpretation is taken as the last step, in which the findings from the two stages are combined to provide thorough conclusions. When quantitative data necessitate further investigation to comprehend underlying systems or participant viewpoints, this design works very well. According to (Bowen et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) several essential characteristics set the explanatory sequential design apart from previous mixed methods techniques. The quantitative phase comes before the qualitative phase in this kind of mixed methods research and the latter is utilized to explain the former. Two separate stages make up the explanatory sequential design first researchers gather and examine quantitative data and then they gather and examine qualitative data to further explore the quantitative findings. This order enables researchers to more thoroughly explain and analyze quantitative results using qualitative insights. The research topic and study objectives frequently influence the choice of a mixed methods design such as the explanatory sequential approach. As they asserted that when employing the explanatory sequential design researchers can use a variety of data gathering techniques, including focus groups or interviews for qualitative data and surveys for quantitative data. Also they added the interpretation stage of the research process is when the integration of quantitative and qualitative data takes place most frequently.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Participants\u003c/h2\u003e \u003cp\u003eThe participants of this study were public University mathematics second-year and third-year mathematics students. The students in targeted classes that were examined in this study who enrolled during the 2024/2025 academic year. All participants were given the full right to withdraw from the participation at any time in the middle of data collection. All participants from the department of mathematics will be considered at the university. Among those classes, students will be selected purposively depending on their previous semester\u0026rsquo;s achievement in each section(s). Two groups at each of them randomly assigned as either the experimental group or the control group. There will be equal numbers students in the experimental group and in the control group. These two groups, will be pre-tested, administered a treatment, and then post-tested.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Data Collection Instruments\u003c/h2\u003e \u003cdiv id=\"Sec15\" class=\"Section3\"\u003e \u003ch2\u003e3.4.1 Achievement tests\u003c/h2\u003e \u003cp\u003eParallel pre-test and post-test instruments were developed for both Transformation Geometry and Calculus to assess conceptual understanding and proof comprehension. The tests focused on the logical structure, justification of the steps and coherence of proofs, aligning with established proof-completion frameworks.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section3\"\u003e \u003ch2\u003e3.4.2 Attitude towards Mathematics Questionnaire\u003c/h2\u003e \u003cp\u003eStudent attitudes were measured using an adapted version of the Attitudes toward Mathematics Scale (Tapia, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e1996\u003c/span\u003e). The questionnaire used a five-point Likert scale and evaluated dimensions such as confidence, enjoyment, perceived value, motivation, and participation in self-explanation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section3\"\u003e \u003ch2\u003e3.4.3 Focus Group Discussions and Interviews\u003c/h2\u003e \u003cp\u003eQualitative data was collected through focus group discussions and semi-structured interviews with selected students from both experimental and control groups. These instruments explored students\u0026rsquo; experiences with proof learning, perceptions of self-explanation, challenges faced, and suggestions for improvement.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Procedure\u003c/h2\u003e \u003cp\u003eThe study was implemented in three phases:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003ePre-intervention phase: Administration of pre-tests and attitude questionnaires.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eIntervention phase: The experimental groups received instruction using structured self-explanation instructions, while the control groups followed traditional teaching methods.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003ePost-intervention phase: Administration of post-tests, attitude questionnaires, interviews, and focus group discussions.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e3.6 Data Analysis\u003c/h2\u003e \u003cdiv id=\"Sec20\" class=\"Section3\"\u003e \u003ch2\u003e3.6.1 Quantitative Data Analysis\u003c/h2\u003e \u003cp\u003eQuantitative data was analysed using SPSS. Descriptive statistics (means and standard deviations) were calculated, followed by inferential analyses that included ANOVA, ANCOVA, MANOVA, and correlation analyses to examine differences between groups and relationships between variables. According to (Rana et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) quantitative approaches are a practical way to explore a research issue. In this phase the participants who were enrolled during 2024/2025 at public university in the department of mathematics. Due to minimum number of student in the deparment all second year mathematics students for the calculus class with total numbers 29 and all third year mathematics students for the transformation geometry class with total numbers 25 were taken for this study. The researcher must select particular methods for data collection, analysis, and interpretation based on the nature of the issue or the research question being investigated As (Watson. 2015) reveal that a variety of techniques pertaining to the methodical examination of social phenomena through the use of statistical or numerical data are included in quantitative research. As a result, quantitative research uses measurement and makes the assumption that the thing being studied is measurable. Also Schutt (\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) added that quantitative methods are a group of approaches that use numbers to depict factual reality and are predicated on positivism, which holds that observers who measure the social world can understand its properties. At its inception, sociology included quantitative methodologies that placed the field within the historical progression of science and infused it with the dominant spirit of discovery. It is reasonable to associate that advancement with the Renaissance because of its spirit of inquiry and openness to question accepted wisdom. The goals of quantitative research are measurement-based data collection, trend and relationship analysis, and measurement verification. In the literature of (Holton \u0026amp; Burnett \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) quantitative research is descriptive, correlational, experimental, or quasi-experimental.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section3\"\u003e \u003ch2\u003e3.6.2 Qualitative Data Analysis\u003c/h2\u003e \u003cp\u003eQualitative data was transcribed and analysed using ATLAS.ti-25. A thematic analysis approach was used that involved coding, categorisation, and theme development. The qualitative phase were used to explain and contextualise the quantitative results. Qualitative research explores, and provides deeper insights into real-world problems (Moser and Korstjens, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Qualitative research helps produce hypotheses to better examine and interpret quantitative data, as opposed to gathering numerical data points or intervening or providing treatments, as is the case with quantitative research. The participants of the study were randomly chosen among the participants from both experimental and control groups to both subject sections Qualitative research collects the experiences, opinions, and actions of individuals. Rather than addressing how many or how much, it addresses the hows, and whys. It may take the form of a stand-alone study that just uses qualitative data, or it may be a component of mixed-methods research that incorporates qualitative.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e3.7 Reliability and Validity\u003c/h2\u003e \u003cp\u003eThe reliability of the quantitative instruments was established using the Cronbach alpha, with values exceeding the acceptable threshold of 0.70. The validity of the content and construction was ensured through expert review and alignment with established theoretical frameworks. Qualitative credibility was enhanced through triangulation, participant validation, and rich verbal quotations.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. RESULTS, ANALYSIS, AND FINDINGS","content":"\u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Overview of the Results\u003c/h2\u003e \u003cp\u003eThis section presents the results, analysis, and key findings of the study that examines the effect of the self-explanation instructional strategy on undergraduate student conceptual understanding and proof comprehension in transformation geometry and calculus. Both quantitative and qualitative data were analysed to determine the effectiveness of the intervention. Quantitative data was analysed using descriptive and inferential statistics, while qualitative data from interviews and focus group discussions were thematically analysed using ATLAS.ti. The results are organised to address the research objective of determining whether self-explanation improves the comprehension of student proof, conceptual understanding, and attitudes toward mathematics.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec25\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Descriptive results of conceptual understanding and proof performance\u003c/h2\u003e \u003cp\u003eTable-1 Descriptive statistics of pre and post test scores\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\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\u003eThe results show a significant increase in the performance of the experimental group compared to the control group after applying the self-explanation strategy.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePre-Test Score for Calculus\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eN\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMinimum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMaximum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eStd. Deviation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e29\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e16.79\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.246\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePost-Test Score for Calculus\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e20.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.174\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre-test score for Transformation geometry\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e19.52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.753\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePost-Test Score for Transformation geometry\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22.88\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.978\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eValid N (listwise)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eDescriptive statistics revealed clear improvements in student academic performance following the implementation of the self-explanation strategy. In Calculus, the mean scores increased from 16.79 (SD\u0026thinsp;=\u0026thinsp;4.246) in the pretest to 20.28 (SD\u0026thinsp;=\u0026thinsp;4.174) in the posttest. This gain indicates a better conceptual understanding and an improved ability to follow and justify logical steps in calculus proofs. The relatively stable standard deviation suggests that the improvement was consistent among students rather than limited to a small group of high achievers. Similarly, in transformation geometry, mean scores increased from 19.52 (SD\u0026thinsp;=\u0026thinsp;4.753) to 22.88 (SD\u0026thinsp;=\u0026thinsp;4.978) after intervention. This improvement reflects the increased capacity to analyze geometric transformations, connect algebraic and visual representations, and construct coherent and logically sound proofs. These findings suggest that self-explanation enabled students to articulate reasoning processes more explicitly and verify the logical consistency of the proof steps, thus strengthening conceptual understanding.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Inferential Analysis of Proof Comprehension Gains\u003c/h2\u003e \u003cp\u003eInferential analysis using ANOVA revealed statistically significant relationships between the pre-test and post-test scores in both transformation geometry and calculus. For transformation geometry, the combined effect was significant, F (11.33)\u0026thinsp;=\u0026thinsp;3.028, p\u0026thinsp;=\u0026thinsp;0.031, indicating significant differences in post-test performance based on initial understanding. The significant linearity component, F(1,13)\u0026thinsp;=\u0026thinsp;29.606, p\u0026thinsp;\u0026lt;\u0026thinsp;0.001, demonstrates a strong and systematic learning progression, while the non-significant deviation from linearity confirms that learning gains followed a consistent pattern. In calculus, similar results were observed. The combined effect of ANOVA was significant, F(15,13)\u0026thinsp;=\u0026thinsp;3.518, p\u0026thinsp;=\u0026thinsp;0.014, with a highly significant linear relationship between pre-test and post-test scores, F(1,13)\u0026thinsp;=\u0026thinsp;34.553, p\u0026thinsp;\u0026lt;\u0026thinsp;0.001. These findings indicate that students with stronger initial understanding benefited more consistently from the self-explanation strategy and that conceptual growth and proof comprehension occurred in a structured and progressive manner at different ability levels.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Effects of self-explanation on attitudes towards mathematics\u003c/h2\u003e \u003cp\u003eThe results of the general linear model (MANOVA) in Table\u0026nbsp;2 revealed statistically significant improvements in students\u0026rsquo; attitudes toward mathematics following the self-explanation intervention. Significant effects were observed for enjoyment of learning mathematics, confidence in mathematical ability, interest and perceived importance of mathematics, and enjoyment of self-explanation activities, all at p\u0026thinsp;\u0026lt;\u0026thinsp;0.05. Large effect sizes (partial η\u0026sup2; \u0026gt; 0.95) across these variables indicate that self-explanation had a substantial impact on affective engagement. Increased enjoyment of learning mathematics suggests that self-explanation transformed students\u0026rsquo; emotional experiences by reducing the anxiety associated with proofs and promoting active participation. Improvements in confidence reflect improved self-efficacy in constructing and understanding proofs, aligning with constructivist and sociocultural theories that emphasise internal scaffolding and metacognitive regulation. Students also reported an increased appreciation of the importance of mathematics, indicating that self-explanation helped them perceive proofs as meaningful reasoning processes rather than memorised procedures.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTable-2 Tests\u003c/b\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabb\" border=\"1\"\u003e \u003ccolgroup cols=\"9\"\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=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" 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=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSource\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDependent Variable\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003edf\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eSig.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003ePartial Eta Squared\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eObserved Power\u003csup\u003ee\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eCorrected Model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e64.894\u003csup\u003ea\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.411\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.608\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.968\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.883\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Confident Math. Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e83.418\u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.813\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.962\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.951\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.686\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e60.762\u003csup\u003ec\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.321\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.596\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.041\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.959\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.781\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e39.048\u003csup\u003ed\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.849\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e20.797\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eIntercept\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e13.780\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13.780\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e45.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.865\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost-confident maths. Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.238\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.238\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e10.189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.593\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.781\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e11.070\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11.070\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e30.135\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.811\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e11.583\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11.583\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e283.781\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.976\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.771\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.443\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.022\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.820\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.914\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre ConfidentMath Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost-confident maths. Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.677\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.739\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.610\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.460\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.162\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.290\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.235\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.040\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.781\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e.830\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePre Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.733\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.244\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.984\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.719\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e.769\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eError\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.143\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.306\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost-confident maths. Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.286\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.286\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.571\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.367\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.286\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.041\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eCorrected Total\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Learning Math\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e67.037\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost-confident maths. Ability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e87.704\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Interest Important\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e63.333\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePost Enjoy Self Explanation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e39.333\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"9\"\u003e\u003csup\u003eR squared = 0.968 (Adjusted R squared = 0.758), R Squared = 0.951 (Adjusted R Squared = 0.630), R Squared = 0.959 (Adjusted R Squared = 0.693), R Squared = 0.993 (Adjusted R Squared = 0.945), Computed using alpha = 0.05\u003c/sup\u003e\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec28\" class=\"Section2\"\u003e \u003ch2\u003e4.5 Quantitative evidence of the effect of Self-Explanation\u003c/h2\u003e \u003cp\u003eThe quantitative findings provide strong evidence that the self-explanation strategy significantly enhances undergraduate students\u0026rsquo; conceptual understanding and proof comprehension in both calculus and Transformation Geometry. Analysis of pre-test and post-test results revealed that students in the experimental group consistently outperformed those in the control group after the intervention, despite both groups demonstrating comparable baseline performance.\u003c/p\u003e \u003cp\u003eClustered boxplot analyses illustrated a pronounced upward shift in post-test scores for the experimental group, accompanied by reduced variability, indicating both improved performance and greater consistency in proof understanding. In contrast, the control group showed only marginal gains, with substantial overlap between pre- and post-test distributions. These visual patterns reinforce the statistical results, suggesting that self-explanation facilitated deeper engagement with proof structures rather than surface-level learning. Correlation analyses also demonstrated strong and statistically significant relationships between pre-test and post-test scores in both subjects (Calculus: r\u0026thinsp;=\u0026thinsp;0.725, p\u0026thinsp;\u0026lt;\u0026thinsp;0.01; Transformation geometry: r\u0026thinsp;=\u0026thinsp;0.667, p\u0026thinsp;\u0026lt;\u0026thinsp;0.01). These findings indicate that while prior knowledge remained an important predictor of achievement, the self-explanation strategy effectively built on existing conceptual frameworks to improve proof understanding. Multivariate analysis of variance (MANOVA) provided robust confirmation of the instructional impact. The significant multivariate group effect with a large effect size (Wilks\u0026rsquo; Lambda\u0026thinsp;=\u0026thinsp;0.399, p\u0026thinsp;\u0026lt;\u0026thinsp;0.001, Partial η\u0026sup2; = 0.601) demonstrates that the type of instruction played a decisive role in learning outcomes. The effects showed statistically significant differences in posttest scores favouring the experimental group, while no significant differences were found in pretest scores. This confirms that observed improvements can be attributed to the self-explanation strategy rather than pre-existing ability differences.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe clustered boxplot in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e4.3\u003c/span\u003e provides a visual representation of the pre-test and post-test scores for both the experimental group (EG) and control group (CG) in transformation geometry. This visual evidence strongly supports the statistical findings obtained from the MANOVA and Between-Subjects Effects tables which discussed in the above table value. In the experimental group we csn see an increase in median scores from pre-test to post-test. The post-test scores (red box) in the EG not only have a higher median than the pre-test scores (blue box). Also show a more compact distribution indicating less variability and more consistency among students improved performance. This aligns with the MANOVA result where the group effect is significant at level of (F\u0026thinsp;=\u0026thinsp;5.9840, p\u0026thinsp;=\u0026thinsp;0.013) for post-test scores and partial eta squared\u0026thinsp;=\u0026thinsp;0.719 which suggesting a moderate effect size due to the self-explanation intervention. In contrast the control group shows less improvement between the pre-test and post-test. In the pre-test and post-test result the median vslues are closer the boxes this implies there overlap, which indicating minimal changes in student performance.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec29\" class=\"Section2\"\u003e \u003ch2\u003e4.6 Qualitative evidence of enhanced proof comprehension\u003c/h2\u003e \u003cp\u003eThe qualitative findings of interviews and focus group discussions as indicated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4.1\u003c/span\u003e and Fig.\u0026nbsp;4.2, respectively, show that strongly supports the quantitative results. Students exposed to the self-explanation strategy consistently reported a better understanding of proof structure, logical sequencing, and conceptual connections. By articulating each step in their own words, the students moved beyond memorisation toward meaningful justification and reasoning, a hallmark of genuine proof comprehension.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e Participants described increased engagement and cognitive involvement during learning tasks, noting that self-explanation required them to actively process and validate each logical step. Students in Transformation Geometry reported improved integration of visual representations with algebraic reasoning, while Calculus students emphasised greater clarity in constructing proofs from foundational principles rather than relying on memorised templates.\u003c/p\u003e \u003cp\u003eAlthough some students identified challenges related to the articulation of complex ideas and time demands, these difficulties were primarily associated with insufficient scaffolding. Importantly, students overwhelmingly recognised the value of self-explanation to develop deeper conceptual understanding and transferable reasoning skills. Many recommended the strategy for broader use in proof-based mathematics courses, particularly when supported by teacher feedback and structured prompts.\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study was designed to examine the effect of the self-explanation strategy on the conceptual understanding and proof comprehension of undergraduate students in calculus and transformation geometry. By integrating quantitative and qualitative evidence, the findings provide strong and consistent support for the effectiveness of self-explanation as a pedagogically sound and impactful instructional strategy in proof-based mathematics courses. Quantitative results from pretest and post-test comparisons, correlation analyses, and MANOVA revealed that students exposed to the self-explanation strategy demonstrated significantly higher posttest performance than those taught using conventional instructional approaches. Although both the experimental and control groups exhibited comparable levels of prior knowledge, the experimental group showed a markedly greater improvement in proof comprehension and conceptual understanding. The large multivariate effect sizes confirm that the observed gains were attributable to the instructional intervention rather than chance or pre-existing differences. These findings indicate that self-explanation not only improves learning outcomes but also supports more stable and consistent achievement among students.\u003c/p\u003e \u003cp\u003eQualitative evidence further substantiated these results by providing insight into how and why self-explanation improves proof understanding. Students reported that articulating their reasoning helped them internalise the logical structure of the proofs, recognise conceptual relationships, and justify each step with greater clarity. The strategy promoted active participation, reflective thinking, and metacognitive awareness, allowing students to move beyond rote memorisation toward meaningful reasoning. In Transformation Geometry, students highlighted improved connections between visual and symbolic representations, while Calculus students emphasised increased confidence in constructing proofs independently.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eA.M. wrote the main manuscript text M.P. Reviewed and supervised the manuscript\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBisra, K., Liu, Q., Nesbit, J. C., Salimi, F., \u0026amp; Winne, P. H. (2018). 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The emergence of mathematical understanding: Connecting to the closest superordinate and convertible concepts. \u003cem\u003eFrontiers in Psychology\u003c/em\u003e, \u003cem\u003e12\u003c/em\u003e, 525493.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"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":"Self-explanation strategy, proof comprehension, conceptual understanding, metacognitive learning, active learning strategies, proof-based instruction, mathematics pedagogy, higher education learning","lastPublishedDoi":"10.21203/rs.3.rs-9507868/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9507868/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUnderstanding and building mathematical proofs is a constant challenge for university mathematics students, especially in proof-intensive courses. This study explored the effect of self-explanation teaching strategies on student's conceptual understanding and proof understanding and how structured self-explanation supports learning mathematical proofs. A mixed-method sequencing explanatory research design was used to provide both statistical evidence and detailed explanatory information on the effectiveness of the intervention. The quantitative phase consisted of 54 undergraduate mathematics students enroled during the academic year 2024/2025, which were divided into experimental group and control group. The experimental group received integrated instruction with structured self-explanatory prompts, guided activities, and reflective tasks while the control group was taught using conventional teaching methods. Both groups completed pre- and post-tests to assess conceptual understanding and understanding. In addition, student attitudes towards mathematics were measured using a standardized Likert scale instrument. Quantitative data was analysed using descriptive statistics, correlation analysis, variance analysis, and multivariate variance analysis.\u003c/p\u003e","manuscriptTitle":"Improving Mathematical Proof Comprehension through Self-Explanation: Evidence from a Mixed Methods Study","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-27 05:34:59","doi":"10.21203/rs.3.rs-9507868/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":"7f823b08-3dd7-4293-861d-a4d8da1a8978","owner":[],"postedDate":"April 27th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Rejected","date":"2026-05-04T12:25:04+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T12:39:58+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-27 05:34:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9507868","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9507868","identity":"rs-9507868","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}