Mathematics Mastery through Innovative Pedagogical Approaches

Mathematics Mastery through Innovative Pedagogical Approaches

*Grester P. Galope., Razil M. Gumanoy

Union National Highschool, North Eastern Mindanao State University

DOI: https://doi.org/10.51584/IJRIAS.2025.10020040

Received: 10 January 2025; Accepted: 14 February 2025; Published: 13 March 2025

ABSTRACT

The persistent low performance in mathematics among Grade 7 students at Madrid National High School underscores the need for innovative teaching strategies that foster deeper understanding, engagement, and problem-solving skills. This study investigates the effectiveness of five pedagogical approaches—Concrete-Pictorial-Abstract (CPA), problem-solving, differentiated instruction, modeling and simulation, and manipulative-based teaching—in enhancing students’ mathematical proficiency. Grounded in constructivist and cognitive learning theories, these methods aim to bridge the gap between abstract mathematical concepts and students’ comprehension by providing interactive, student-centered learning experiences. Utilizing a mixed-methods, explanatory research design incorporating quasi-experimental methods, this study examines the impact of these approaches through pre- and post-test assessments, teacher adeptness evaluations, and student perception and engagement surveys. The findings reveal a significant improvement in students’ mathematical performance following the implementation of these strategies. Additionally, the study explores the correlation between learning outcomes and students’ engagement levels, as well as the relationship between teacher proficiency in these methods and student achievement. Despite limitations such as sample size and study duration, the results highlight the potential of these innovative teaching strategies to transform mathematics education. The study provides practical recommendations for educators, curriculum developers, and policymakers to integrate these methods effectively, fostering a more inclusive and dynamic learning environment. Furthermore, the research contributes to the broader field of mathematics education by offering evidence-based solutions to enhance student comprehension and achievement.

INTRODUCTION

Innovative pedagogical approaches are modern teaching methods designed to address the diverse needs of learners, promoting active engagement, deeper understanding, and meaningful connections with the subject matter. The Concrete-Pictorial-Abstract (CPA) method is a step-by-step approach that begins with hands-on use of manipulatives (concrete), progresses to visual representations (pictorial), and culminates in abstract reasoning using symbols and equations, ensuring a smooth transition to conceptual understanding. Differentiated instruction personalizes learning by tailoring content, processes, and outputs to accommodate varying levels of readiness, interests, and learning styles, creating an inclusive and supportive environment for all students. The problem-solving approach engages learners in exploring real-world challenges, fostering critical thinking, creativity, and collaboration, while also encouraging reflection on strategies and solutions. Manipulative-based teaching employs physical tools, such as base-ten blocks or algebra tiles, to help students visualize abstract concepts, making learning interactive and accessible. Similarly, modeling and simulation use physical or digital representations to explore complex ideas, enabling students to manipulate variables, observe outcomes, and develop analytical skills in a safe and controlled environment. Together, these innovative strategies empower educators to create engaging, inclusive, and effective learning experiences that build confidence, foster mastery, and prepare students for real-world challenges.

Recent studies underscore the effectiveness of innovative teaching strategies in mathematics education. A 2022 study by Johnson & Williams reaffirmed the efficacy of the Concrete-Pictorial-Abstract (CPA) approach in helping students transition from tangible experiences to abstract reasoning, showing its positive impact on problem-solving skills and mathematical understanding, particularly for middle school students. Additionally, a study by Greenwood and Powell (2021) emphasized the role of differentiated instruction, which tailors lessons to diverse learning styles and abilities, significantly improving conceptual understanding and achievement, especially for students with learning difficulties. A meta-analysis by Hattie et al. (2023) further highlighted the positive impact of problem-solving approaches, finding that when teachers modeled problem-solving techniques and engaged students in authentic tasks, student achievement in mathematical reasoning increased. In a similar vein, modeling and simulation techniques were found to significantly enhance students’ understanding by allowing them to visualize complex mathematical processes and engage with real-world applications. Moreover, Clements and Sarama (2020) emphasized the continued relevance of manipulatives, showing that physical objects like cubes and geometric shapes helped students internalize abstract mathematical concepts, especially among younger learners. These studies collectively emphasize the importance of incorporating problem-solving, differentiated instruction, CPA, modeling and simulation, and manipulatives to enrich students’ mathematical learning experiences.

The consistently low performance in mathematics among Grade 7 students at Madrid National High School highlights an urgent need for change. Recent assessments show that only 40% of students reached proficiency, far below the expected standards. This reflects a widespread struggle to understand key mathematical concepts and develop essential problem-solving skills, despite the use of strategies like peer-assisted and cooperative learning. These traditional methods, while helpful in some ways, appear insufficient to meet the unique needs of these students. Research has largely focused on broad teaching approaches or interventions for high school students, leaving a gap in understanding how innovative methods can support middle school learners. For instance, Green and Spector (2020) found that traditional approaches often produce uneven results, with many students disengaged or struggling to comprehend abstract ideas. This lack of targeted research for middle school mathematics underscores the need to explore and evaluate new teaching strategies tailored to this level. This study aims to address this gap by examining innovative pedagogical approaches to improve the mathematical performance of Grade 7 students. By identifying effective methods, this research seeks to provide practical, evidence-based solutions for educators to better support their students and help them succeed in mathematics.

The significance of this study lies in its potential to enhance mathematics education for Grade 7 students at Madrid National High School by providing an evidence-based strategy to improve their performance through innovative pedagogical approaches. By tailoring these innovative teaching methods to the specific socio-economic and cultural context of the schools, this research aims to meet the unique needs of these students, fostering greater engagement, confidence, and interpersonal skills. The study will offer practical implementation guidelines for educators and contribute to educational research by addressing existing gaps, combining quantitative and qualitative outcomes, and exploring long-term impacts. Additionally, the findings could inform educational policy and decision-making, advocating for broader adoption of creative and interactive teaching strategies to support students’ academic and personal development.

The statistical treatment applied in this study provided a robust framework to analyze and interpret data collected from students’ mathematical abilities and the effectiveness of various teaching approaches. By employing a combination of Weighted Means, ANOVA, Reliability Analysis, and Correlation Analysis, the study aimed to draw comprehensive insights into how different teaching strategies influenced student performance and engagement. In addition to descriptive statistics, ANOVA was used to compare pretest and posttest scores within each group. This analysis determined whether the observed differences in students’ performance were statistically significant, helping to evaluate the effectiveness of the innovative pedagogical approaches. The results from this analysis provided insight into the impact of the intervention and how different groups of students responded to the methods. Furthermore, correlation analysis, using either Pearson or Spearman correlation coefficients, was employed to explore the relationships between learning outcomes and students’ perceptions, engagement levels, and teachers’ implementation proficiency. This helped identify whether higher student engagement and positive perceptions correlated with improved learning outcomes, and whether teachers’ skill in implementing the innovative approaches was linked to better student performance. Finally, qualitative data gathered from surveys, interviews, and focus group discussions underwent thematic analysis. This analysis identified common themes in student and teacher experiences, providing valuable insights into the effectiveness of the teaching methods. Based on these findings, suggestions for further innovative activities were made to enhance students’ mathematical learning.

Theoretical/Conceptual Framework

Students’ comprehension and mastery of mathematics are greatly improved by cutting-edge pedagogical strategies including the Concrete-Pictorial-Abstract (CPA) method, manipulative-based courses, problem-solving strategies, modeling and simulation, and customized instruction. Based on educational theories that support active learning and accommodate different cognitive types, these methods make mathematics more approachable, interesting, and useful for students of different skill levels. Empirical evidence substantiates this assertion, emphasizing the significance of shifting from tangible encounters to abstract reasoning, cultivating conceptual comprehension, and delivering customized education to fulfill individual student requirements.

Piaget’s Constructivist Theory (1952). According to constructivist theory, students build their own knowledge and understanding of the world by having experiences and thinking back on them. The phases of cognitive development proposed by Jean Piaget highlight the fact that children learn best when they are actively engaged in the educational process. This theory backs the use of manipulative-based teaching and the Concrete-Pictorial-Abstract (CPA) approach, which involve students in practical tasks that foster active learning and help them develop a strong grasp of mathematical ideas.

This study focuses on integrating five innovative pedagogical approaches—Concrete-Pictorial-Abstract (CPA), problem-solving, differentiated instruction, modeling and simulation, and manipulative-based teaching—to enhance students’ mastery of mathematics. The CPA approach facilitates a gradual transition from concrete experiences to abstract understanding, enabling students to build a strong foundation for mathematical concepts. Manipulative-based teaching complements CPA by using physical tools like blocks and counters to represent abstract ideas, fostering hands-on learning. Problem-solving strategies encourage students to apply mathematical concepts to real-world situations, promoting critical thinking and collaboration. Modeling and simulation provide dynamic, interactive experiences that simplify complex ideas and help students visualize abstract concepts. Differentiated instruction ensures that teaching methods and materials are tailored to meet the diverse needs and abilities of individual learners, making mathematics more accessible and engaging for all students.

The schematic diagram illustrates the relationship between innovative pedagogical approaches and students’ academic achievement in mathematics, as measured by pre- and post-tests. The independent variable encompasses five key teaching strategies: The Concrete-Pictorial-Abstract (CPA) approach, manipulative-based teaching, the problem-solving approach, modeling and simulation, and differentiated instruction. The pre-test establishes students’ baseline performance, while the post-test measures the effectiveness of these strategies in enhancing academic achievement. The framework highlights a direct relationship, where the innovative approaches are expected to significantly improve students’ mathematical proficiency, as reflected in the comparison of pre- and post-test scores.

RESEARCH METHODOLOGY

This study carefully explored how different teaching strategies impact students’ mathematical performance and engagement. By combining numbers with real student experiences, it aimed to capture both measurable progress and the deeper learning process. Approaches like the Concrete-Pictorial-Abstract (CPA) method, manipulative – based teaching, problem-solving approach, modeling and simulations, and differentiated instruction were examined to see how they help students understand math better. The chosen methodology provided a well-rounded view, ensuring that both academic results and personal learning experiences were taken into account.

This study used a mixed-method, explanatory, and quasi-experimental design to gain a deep and well-rounded understanding of how different teaching strategies impact students’ mathematical performance and engagement. Each aspect of this design played a key role in ensuring that the research was thorough and meaningful. A mixed-method approach was employed to integrate both quantitative and qualitative data, as Creswell and Plano Clark (2018) emphasize that combining numerical data with personal experiences leads to a more comprehensive understanding of research problems. The pretest and posttest assessments provided measurable data on students’ progress, while observations, interviews, and surveys captured their thoughts, challenges, and engagement. This combination offered a fuller picture of how these teaching strategies influenced learning. The explanatory aspect of the study was crucial because it went beyond simply recording changes. As Maxwell (2013) highlights, explanatory research focuses on understanding why and how specific factors influence outcomes. In this case, the study sought to determine why strategies such as the CPA approach and hands-on activities helped students grasp mathematical concepts more effectively. This deeper analysis provided insights that could help teachers refine their instructional methods to better support student learning.

The data collection process in this study unfolded in five stages, each designed to assess and improve the mathematics understanding of Grade 7 students at Madrid National High School using innovative teaching strategies. In Step 1, the pre-test was administered to a selected group of Grade 7 students. The students took a math pre-test to assess their baseline skills. The test covered key topics from the second quarter of their curriculum, such as irrational numbers, unit conversions, Venn diagrams, and the volumes of geometric shapes. This pre-test allowed the researchers to measure the students’ initial understanding of the material before the intervention. Step 2 focused on implementing the innovative teaching strategies. The students were divided into groups, with each group assigned to a different approach to learning. The strategies included the Concrete-Pictorial-Abstract (CPA) method, manipulative-based teaching, problem-solving strategies, modeling and simulation, and differentiated instruction. These methods were carefully applied in a structured manner by different teachers, with the aim of giving students the best opportunity to understand and engage with mathematical concepts. The proponent did not conduct the intervention but instead facilitated the process, allowing the teachers to lead the implementation of the strategies in the classroom. The Concrete-Pictorial-Abstract (CPA) method, in particular, played a significant role in helping students grasp abstract mathematical concepts in a more accessible way. The CPA method is a teaching approach that starts with concrete representations, such as manipulatives (e.g., blocks, shapes, or objects), to make the abstract ideas more tangible. Next, students progress to pictorial representations, such as diagrams or pictures, to help them visualize mathematical concepts. Finally, students move to the abstract stage, where they work with symbols and equations. By gradually moving from hands-on experiences to abstract reasoning, the CPA method helps students build a deeper understanding of mathematical concepts. Teachers employed the CPA method to help students better understand topics such as fractions, geometric shapes, and algebraic expressions.  During Step 3, after the intervention, all students took a post-test. This post-test was designed to measure the progress made in their mathematical understanding as a result of the innovative teaching strategies. The post-test covered the same concepts as the pre-test, allowing for a direct comparison of students’ performance before and after the intervention.

The next stage involved the survey and interview. Teachers and students filled out surveys to share their thoughts on how the new teaching methods impacted their understanding and engagement with math. The surveys included both Likert-scale questions and open-ended questions to capture a range of feedback. In addition, a smaller group of students participated in semi-structured interviews, where they reflected on their personal experiences with the teaching strategies. These interviews provided deeper insights into how the different methods, including CPA, influenced their learning journey and motivation. The interviews with the teachers were a crucial part of the study. Since the teaching approaches were implemented by different teachers, the researchers interviewed each teacher to gather detailed feedback on the strengths and challenges of the strategies they used. Teachers were asked to discuss how well the methods worked in their classrooms, how the students responded to the CPA approach and other methods, and what challenges they faced in applying the strategies. This feedback helped to further understand the practical application of these approaches and how they can be improved for better student outcomes. Finally, in Step 5, the data were analyzed. To evaluate the effectiveness of the teaching strategies, the researcher compared the pre-test and post-test results using both descriptive and inferential statistics. This allowed the researcher to identify patterns and assess whether the innovative strategies led to significant improvements in students’ mathematical performance. Additionally, a thematic qualitative analysis was conducted on the interview transcripts and survey responses. This helped to uncover recurring themes related to how the teaching strategies impacted both students and teachers, providing a deeper understanding of their experiences. The combination of quantitative and qualitative data offered a well-rounded perspective on the outcomes of the intervention.

RESULTS AND DISCUSSIONS

To ensure the effectiveness of the lesson plans and pretest and posttest tools used in this study, validity and reliability tests were conducted. Evaluators assessed the lesson plans based on key criteria, such as measurability, realism, and alignment with curriculum standards. The reliability of the pre-test and post-test tool was also analyzed to determine its consistency in measuring student learning outcomes.

Table 1: The results of the validity and reliability tests for the pretest and posttest tool and the lesson plan.

The results from the validity and reliability tests emphasize the robustness of the lesson plans and assessment tools used in this study. The unanimous “Strongly Agree” ratings indicate that the lesson plans effectively support student learning by ensuring measurable, realistic, and attainable objectives. This aligns with research by Wiliam (2018) and Marzano (2017), which highlights the role of well-structured lesson plans in fostering student achievement. Furthermore, the high reliability coefficient (0.889) of the pre-test and post-test tool suggests that it is a consistent and dependable instrument for measuring student progress. This supports previous findings by Fraenkel & Wallen (2019), who stress the importance of reliable assessment tools in educational research.

These findings imply that implementing structured and curriculum-aligned lesson plans enhances the effectiveness of innovative teaching approaches. Teachers can use these insights to refine instructional strategies, ensuring that they meet student learning needs while maintaining alignment with educational standards. Additionally, the high reliability of the assessment tool provides educators with confidence in measuring and evaluating student learning outcomes accurately.

To evaluate the effectiveness of different teaching approaches, pre-test and post-test assessments were conducted. The results provide insights into how each instructional method influenced student learning outcomes. The table below presents a comparative analysis of student performance before and after the interventions.

Table 2: Pre-Test and Post Test result of the innovative pedagogical approaches.

The pre-test and post-test results indicate varying levels of effectiveness among the five pedagogical approaches. The Concrete–Pictorial–Abstract (CPA) Approach group had the lowest pre-test mean (14.74) but showed the highest post-test mean (21.77), reflecting the greatest improvement (+7.03). This suggests that CPA, which progresses from hands-on experiences to abstract reasoning, was highly effective in enhancing students’ conceptual understanding and retention based on the study of Hattie, 2017; Kirschner, Sweller, & Clark, 2018. The Differentiated Instruction Approach demonstrated notable improvement (+4.63), despite having the highest variability in pre-test scores (SD = 3.71). This result aligns with Tomlinson & Moon (2017), who posit that differentiated instruction accommodates diverse learning styles, thereby benefiting a wide range of students. However, the variation in student performance suggests that some students may require additional support or alternative differentiation strategies to maximize their learning outcomes.  The Problem-Solving Approach had the highest pre-test mean (17.58) and maintained consistent results with a post-test mean of 19.56. This indicates that problem-solving strategies foster steady cognitive engagement, particularly among students with stronger foundational skills (Jonassen, 2017; Schoenfeld, 2019). The Modeling and Simulation Approach exhibited the lowest post-test mean (18.28) and the smallest improvement (+3.17), suggesting limited effectiveness. Research by Clark & Mayer (2017) emphasizes that while modeling and simulation can enhance higher-order thinking, its effectiveness depends on students’ prior exposure to such methods and their ability to engage in self-directed learning. Additional scaffolding may be needed to improve student comprehension in this approach. These results highlight the importance of choosing the right teaching strategies based on student needs. The strong performance of CPA and Differentiated Instruction suggests that hands-on learning and personalized teaching approaches significantly enhance understanding.

Table 3: Significant difference on the pretest and posttest scores of the students.

The significant difference between the pretest and posttest scores, as indicated by the F-value of 6.56 and a p-value of 0.000, aligns with recent educational research highlighting the effectiveness of innovative teaching approaches in improving student learning outcomes. Studies by Hattie (2020) emphasize that active learning strategies, such as the CPA (Concrete-Pictorial-Abstract) approach and problem-solving methods, lead to significant gains in student performance when compared to traditional instruction. Similarly, Tomlinson and Imbeau (2019) stress that differentiated instruction enhances learning by addressing individual student needs, leading to measurable improvements in academic achievement. The role of technology in education, particularly through modeling and simulation, has also been shown to positively impact student comprehension and retention (Mouza et al., 2020). Furthermore, research on assessment methods highlights that varied instructional strategies, when aligned with formative and summative assessments, result in significant learning gains (Popham, 2020). These studies reinforce the findings of the current study, demonstrating that the innovative teaching approaches employed had a statistically significant effect on student performance, as evidenced by the marked improvement from pretest to posttest scores.

Table 4: The level of the perception and engagement towards the different approaches.

The data presented in Table 5 indicates that the teaching approaches implemented in the classroom were generally effective in engaging students and enhancing their learning experiences, with an average weighted mean of 3.93, corresponding to “Agree.” The highest-rated indicator, “The teaching approaches used in class were interesting and engaging” (4.27), suggests that students found the methods stimulating and motivating. The supportive learning environment (4.03) and the effectiveness of teaching methods in helping students understand lessons (3.94) further emphasize the positive impact of the approaches. However, the slightly lower score for collaboration (3.73) points to an area for improvement in promoting peer interaction and teamwork. Research supports these findings, highlighting the importance of engaging, interactive, and cooperative learning strategies in fostering motivation and understanding (Dörnyei & Ushioda, 2021; Ryan & Deci, 2000). The implication of the study is that while the teaching approaches were largely successful in increasing student engagement, there is a need to further enhance opportunities for collaboration to improve teamwork and peer learning experiences. Integrating more cooperative learning activities could strengthen these aspects, creating a more comprehensive and interactive learning environment.

Table 5: The level of the teacher’s adeptness on the use of the different pedagogical approaches.

The results indicate that the teacher demonstrated a high level of proficiency in using different pedagogical approaches, with an average weighted mean of 4.70, signifying “Strongly Agree.” The highest-rated indicators, such as “Identify the appropriate approach to use based on the learning objectives” (5.00) and “Design engaging and meaningful learning activities aligned with the chosen approaches” (5.00), emphasize the teacher’s skill in selecting and aligning methods with learning goals. These findings align with the research by Shulman (2022), who underscores the importance of teachers’ knowledge in selecting appropriate strategies to achieve educational objectives.

Student engagement and perception play a crucial role in shaping learning outcomes, as they influence motivation, participation, and overall academic performance. When students perceive learning experiences as meaningful, interactive, and aligned with their needs, they are more likely to engage actively in the learning process, leading to improved comprehension and retention of knowledge. Research suggests that teaching strategies that prioritize student-centered approaches, such as CPA, problem-solving, manipulative – based teaching, differentiated instruction and modeling and simulation, contribute to higher levels of engagement and better learning outcomes. To examine this relationship, the study analyzed the correlation between students’ learning outcomes and their perception and engagement with the implemented teaching approaches. The results of this analysis are presented in the table below.

Table 6: Significant relationship between learning outcomes and students’ perception and engagement.

The significant relationship between learning outcomes and students’ perception and engagement, as indicated by a p-value of 0.043, aligns with recent research emphasizing the impact of student engagement on academic achievement. According to Fredricks, Blumenfeld, and Paris (2022), students who are more engaged—behaviorally, emotionally, and cognitively—tend to perform better academically because engagement fosters deeper learning and motivation. Zepke and Leach (2020) further highlight that students’ perception of their learning environment, including instructional strategies and classroom interactions, significantly affects their motivation and persistence in learning. Similarly, Ryan and Deci’s (2021) Self-Determination Theory suggests that students who perceive their learning experience as meaningful and supportive of their autonomy, competence, and relatedness show higher levels of engagement and improved learning outcomes. Research by Bond et al. (2023) also confirms that active and student-centered teaching methods, such as CPA, problem-solving, and modeling and simulation, contribute to higher engagement levels, which in turn enhance comprehension and retention. These studies support the conclusion that instructional strategies that promote positive student perception and engagement play a crucial role in improving academic performance.

The findings indicate that student engagement and perception are key factors influencing learning outcomes. This suggests that educators may prioritize strategies that create an engaging and supportive learning environment, such as interactive and student-centered teaching methods. Schools may focus on fostering a positive classroom culture where students feel motivated and valued, as this can lead to greater academic success. Furthermore, professional development programs may equip teachers with effective engagement techniques, such as differentiated instruction and technology-enhanced learning, to ensure active student participation. Educational institutions may also consider implementing feedback mechanisms to assess student perceptions of their learning experiences, allowing for instructional adjustments that enhance engagement. Future research could explore the long-term impact of engagement-focused strategies on different learning domains and student populations, further strengthening the link between engagement, perception, and academic achievement.

Table 7: Significant relationship between learning outcomes and teacher’s adeptness.

Recent research confirms the strong relationship between teachers’ adaptability and student learning outcomes. According to Hattie (2023), teacher effectiveness—particularly the ability to modify instructional strategies based on student needs—has one of the highest effect sizes on student achievement. Darling-Hammond et al. (2022) emphasize that teachers who continuously refine their instructional approaches based on classroom dynamics and student feedback create a more inclusive and responsive learning environment, leading to improved academic performance. Similarly, Tomlinson (2021) highlights that differentiated instruction, which relies on teacher adaptability, significantly enhances student engagement and comprehension. The role of technology in modern education also requires teachers to be adaptable, as Mouza et al. (2023) argue that integrating digital tools effectively into lessons requires teachers to continuously adjust their strategies to maximize student interaction and learning. Furthermore, Gore et al. (2022) stress the importance of ongoing professional development to help teachers cultivate adaptive teaching practices, ensuring they remain responsive to the evolving needs of students. These studies reinforce the findings of the current study, highlighting that teachers’ ability to modify their teaching approaches significantly impacts student learning outcomes.

The findings emphasize the importance of teacher adaptability in fostering effective learning environments. Since a significant relationship exists between student learning outcomes and teacher adaptiveness, educators may be encouraged to continuously refine their teaching methods to meet students’ evolving needs. This highlights the necessity for ongoing professional development programs that focus on instructional flexibility, differentiated teaching, and the integration of innovative pedagogical strategies. Schools and policymakers may invest in training initiatives that enhance teachers’ ability to assess student progress in real-time and adjust their teaching accordingly. Additionally, fostering a culture of reflective teaching—where educators actively evaluate and modify their approaches—can lead to sustained improvements in student achievement. Future research may explore specific adaptive teaching strategies that have the greatest impact on different subject areas, further strengthening the link between instructional flexibility and student success.

CONCLUSION

The validity and reliability tests confirmed that the lesson plans and assessment tools used in this study were effective. The high ratings for the lesson plans indicate that they were well-structured, realistic, and aligned with curriculum standards, providing a solid foundation for student learning. The reliability of the pre-test and post-test tools, with a score of 0.889, confirms that they consistently measured student learning outcomes, giving confidence in the assessment methods used throughout the study. The pre-test results revealed varying levels of prior knowledge among students, with the Problem-Solving Approach group showing higher initial understanding. The Concrete-Pictorial-Abstract (CPA) Approach group had the lowest pre-test scores, highlighting the need for structured instructional methods to address gaps in student knowledge. These findings emphasize the importance of pre-assessment in identifying student needs and tailoring instruction accordingly.

After the innovative teaching approaches were implemented, all groups showed improvement, with the CPA Approach achieving the greatest increase in scores. This indicates that the CPA approach was particularly effective in enhancing students’ conceptual understanding of mathematics. The Differentiated Instruction approach also led to significant improvements, suggesting that tailoring lessons to individual needs is a key factor in student success. However, the Modeling and Simulation Approach had less impact, suggesting that this method may require additional support for better student engagement and comprehension. The statistical analysis confirmed a significant difference between pre-test and post-test scores, indicating that the improvements in student performance were not due to chance but were the result of the innovative teaching approaches used. This reinforces the effectiveness of student-centered teaching methods in improving learning outcomes and suggests that these approaches can lead to meaningful and measurable academic progress. Students responded positively to the teaching methods, with the highest-rated aspect being the interest and engagement generated by the approaches. However, the lower score on collaboration indicates that students may benefit from more opportunities for peer interaction and teamwork. This highlights the need for educators to further enhance opportunities for collaboration within these innovative teaching strategies, improving both social interaction and learning outcomes.

The students agreed that the innovative approaches significantly improved their learning outcomes. The highest-rated aspect, related to applying learning to real-life situations, emphasizes the relevance of the lessons. However, the slightly lower rating for explaining topics confidently suggests that students may require more opportunities to practice verbalizing and presenting their understanding. This points to the need for activities that encourage students to communicate their learning clearly and confidently. Teachers demonstrated a high level of proficiency in using various teaching strategies, particularly in selecting appropriate approaches based on learning objectives and adapting methods to meet student needs. The findings indicate that teacher adaptability is a key strength in implementing effective teaching practices. The study suggests that continued professional development focused on adapting teaching methods and responding to diverse student needs will further enhance instructional quality and student success. The study found a significant relationship between student engagement and learning outcomes, confirming that when students perceive the teaching methods as engaging and meaningful, they perform better academically. This highlights the importance of designing lessons that actively involve students in the learning process, fostering motivation and a deeper understanding of the material. A strong correlation was found between teacher adaptability and student success, suggesting that teachers who adjust their strategies to meet individual student needs have a significant positive impact on learning outcomes. This underscores the importance of flexible teaching and the need for ongoing professional development that supports teachers in adapting their methods to ensure all students succeed. The study identified several innovative activities, such as interactive workshops, student-led projects, gamified learning challenges, real-world application tasks, and collaborative problem-solving activities, which can further enhance student engagement and learning. These activities support the findings that active, collaborative, and real-world connected learning environments are key to improving student understanding and retention of mathematical concepts.

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