A Visual Prompt-Based Mobile Learning System for Improved Algebraic Understanding in Students With Learning Disabilities

This study aims to address the significant learning challenges faced by students with learning disabilities, particularly in mastering algebraic concepts. Traditional rule-based instruction often exacerbates these challenges, highlighting the need for more effective teaching methodologies. While existing literature confirms the effectiveness of both prompt-based learning and computer-assisted instruction in math education, very few studies have looked at combining mobile technology with prompt-based learning approaches for algebra instruction. In response to this gap, we propose a Visual Prompt-Based Mobile Learning Strategy (VPML). This innovative method is specifically designed to enhance the understanding of basic algebra, focusing on linear equations. Utilizing a quasi-experimental design, we conducted a comprehensive 12-week study to assess the impact of VPML on learning outcomes among students with learning disabilities. The results indicate significant improvements in the students’ abilities to comprehend and solve linear equations when taught via the VPML strategy. Therefore, this study serves not only as a validation of the VPML approach but also as a meaningful contribution to the ongoing discourse on optimizing educational interventions for students with learning disabilities in algebra.


I. INTRODUCTION
Mathematical proficiency is intrinsically linked to both academic success and career opportunities, making it an essential skill for all students [1].Mastery of mathematics not only equips students with problem-solving abilities that are applicable to more advanced mathematical contexts but also to everyday scenarios, such as budgeting for groceries [2].Algebra stands out as a particularly crucial area within the mathematics discipline, often serving as the foundation for higher-level educational and career pathways [3], [4].However, students with learning disabilities encounter unique The associate editor coordinating the review of this manuscript and approving it for publication was Chin-Feng Lai .challenges, including deficits in working and long-term memory [5], which create substantial barriers to their success in algebra [6].Consequently, there is an urgent need to develop effective strategies aimed at improving the algebraic understanding of students with learning disabilities, thereby fostering their academic and long-term success.Conventional approaches to mathematics instruction have been shown to impact not just academic outcomes but also the emotional well-being of students.Traditional teaching methods could contribute to emotional disturbances such as depression, anxiety, or even learned helplessness among students [7].Cortiella and Horowitz [8] outline common difficulties that students with learning disabilities in mathematics face, including issues with calculations, measurement, estimation of number quantities, and problem-solving strategies.Given these challenges, there is a pressing need for more effective, emotionally-sensitive instructional methods.One such strategy is 'prompting,' designed to assist students in employing a particular skill or behavior when they find it challenging to respond to conventional instructional cues.Implementing this and other supportive strategies can help ameliorate both the academic and emotional obstacles students face in the mathematics classroom, particularly those with learning disabilities.Prompt-based learning has emerged as a promising approach, incorporating specifically designed prompts within learning tasks to enhance comprehension and engagement.Numerous studies underscore its efficacy for different learner profiles.Hayes [9], for instance, advocates for prompt-based strategies as effective in bolstering language and communication skills among children with Autism Spectrum Disorder (ASD).Beyond special education, prompt-based learning has proven useful in general educational settings as well.Studies by Chabani and Hommel [10] as well as Sung et al. [11] have demonstrated the multifaceted benefits of this approach.Specifically, Chabani and Hommel focus on its efficacy in enhancing visuospatial processing skills, while Sung et al. underscore its role in fostering learner self-efficacy and overall achievement.Particularly relevant to this study, prompt-based learning has shown significant potential in improving mathematical abilities in students [12].While prompting strategies hold promise, they are not a standalone solution for effective learning.Advancements in technology have paved the way for computer-assisted learning, which has been especially beneficial for students with learning disabilities [13].This computer-aided approach allows for more visually rich and adaptive learning experiences [14], complementing traditional prompt-based strategies by offering more flexibility and context-specific prompts.Given the foundational importance of algebraic understanding in both academic and practical contexts, there is a significant need for effective instructional methods that are tailored to students with learning disabilities.Traditional rule-based instruction often creates barriers for these students, hindering their comprehension of algebra-related concepts.
This study introduces the Visual Prompt-Based Mobile Learning (VPML) strategy, an innovative approach that combines the strengths of both prompt-based and computerassisted learning.By leveraging mobile technology, VPML aims to offer a flexible, adaptive, and effective way for students with learning disabilities to grasp basic algebraic concepts, with a focus on solving linear equations.

A. OUTLINE AND SUMMARY OF CONTRIBUTIONS
The contributions of this work can be summarized as follows: Certainly, here are the three major contributions in a more compact form: • Approach Integration: This study combines their strengths to create the Visual Prompt-Based Mobile Learning (VPML) strategy.By doing so, the work provides a new framework that can be both adaptive and contextually sensitive, thereby making the learning process more effective and inclusive for students with learning disabilities.
• Algebraic Focus for Learning Disabilities: This study aims to fill a gap in the literature by focusing on the challenges and needs of students with learning disabilities in the context of algebra.The VPML strategy is tailored to improve their understanding of algebraic concepts, particularly in solving linear equations, which is crucial for both academic and practical life skills.
• Empirical Validation: By utilizing quasi-experimental methodologies, this study provides compelling evidence that the VPML approach is a viable and impactful educational intervention.This quantifiable proof strengthens the study's argument for the practical implementation of VPML in educational settings to improve algebraic understanding among students with learning disabilities.

II. RELATED WORKS
The landscape of educational strategies for improving mathematical proficiency, particularly in algebra, has been shaped by two significant lines of inquiry: Prompt-Based Learning and Computer-Assisted Learning.These two threads converge in a broader discourse about alternative methods for mathematics instruction, especially for students with learning disabilities.

A. PROMPT-BASED LEARNING
The instructional paradigm of prompt-based learning has attracted escalating scholarly interest, especially for its robust capacity to augment both student engagement and the depth of subject matter comprehension [15], [16], [17], [18], [19], [20].This approach offers a significant departure from more traditional pedagogical techniques, which often adopt a onesize-fits-all methodology and may not be as dynamic or adaptive to each student's unique learning needs.Promptbased learning stands out for its active inclusion of learners in the academic journey.It utilizes strategically designed cues or questions that serve as scaffolding mechanisms to guide students' cognitive processes, enrich their problem-solving skills, and promote critical thinking.
A substantial body of research in special education substantiates the potential merits of employing prompt-based learning methodologies [16], [20], [21], [22].In environments catering to learners with cognitive or developmental impairments, this approach has been found particularly invaluable.For example, Delisio and Isenhower effectively utilized video prompting to foster independent behaviors among students with Autism Spectrum Disorder [20].Köse and Sönmez-Kartal ventured into assessing the efficacy of animated video prompting in imparting car washing skills to individuals with mild intellectual disabilities [23].These studies demonstrate how prompt-based learning can transcend the acquisition of academic skills and can be tailored to aid learners in mastering a wide range of life skills, all within structured yet adaptive educational settings.
Beyond the confines of specialized educational environments, the tentacles of prompt-based learning have also reached into mainstream education [15], [17], [18], [19].Research reveals that the incorporation of targeted prompts into the standard curriculum fosters better study habits, critical thinking, and a more profound understanding of varied subjects.For example, Estapa and Amador found that the utilization of specific prompts was intrinsically linked with higher engagement levels and improved mathematical performance [19].Marthaliakirana et al. showcased that students exposed to learning environments enriched with metacognitive prompts exhibited elevated levels of argumentation and critical thinking skills [17].

B. COMPUTER-ASSISTED LEARNING
Computer-assisted learning has emerged as a transformative force in contemporary education, fundamentally reshaping traditional pedagogical paradigms [24], [25], [26], [27], [28].By synthesizing foundational educational practices with the capabilities of digital technology, computer-assisted learning has overhauled the instructional approaches to various subjects, including mathematics.This digital revolution offers an array of interactive platforms characterized by visually rich experiences, thus accommodating diverse learning styles and preferences to expand its efficacy and applicability.
In the realm of customized and adaptive educational experiences, computer-assisted learning stands out as particularly effective.Traditional pedagogical methods often subscribe to a one-size-fits-all approach, which is increasingly recognized as insufficient for diverse learning needs [25], [26], [29].Computer-assisted platforms can be tailored to individual learning trajectories, an essential feature for accommodating students with learning disabilities.Tejedor-García et al, for example, designed a minimal-pair-based Computer-Assisted Pronunciation Training tool focusing on exposureperception-production cycles.This tool provides automatic feedback and specifically targets adult native Spanish speakers for mastering difficult English phonemes [25].Similarly, Rho et al. developed a virtual reality-based sign language learning system that employs validation-based feedback, thereby implementing an experiential learning model [26].These platforms typically offer real-time feedback and adjustments based on the learner's immediate performance, making educational interventions more targeted and timely.
Finally, the impact of computer-assisted instruction on students' performance in mathematics is noteworthy.Studies have reported significant improvements in students' mathematical achievements when utilizing computerassisted instruction, further cementing the importance of computer-assisted instruction in this subject [30].

C. PROMPTING IN COMPUTER-ASSISTED LEARNING
While computer-assisted learning and prompt-based instructional strategies have independently shown promise, a burgeoning and multidisciplinary avenue for research is examining the synergistic effects at their intersection.This promising domain combines the adaptability, engagement, and individualized focus inherent to both computer-based learning and prompt-based educational techniques [31], [32], [33].
Emerging scholarship illuminates the transformative potential of harmonizing these approaches.Chen et al. ventured into the realm of language acquisition, examining how computer-assisted platforms integrated with targeted prompting could optimize the experience for English as a Foreign Language (EFL) learners [31].Their study revealed not just incremental but significant gains in participants' speaking proficiency and self-efficacy when compared to peers relying on traditional language learning methodologies.
In a different but equally challenging educational context, Castronovo et al. probed the impact of digital platforms that employ adaptive metacognitive prompts in game-based settings, specifically designed for engineering students grappling with construction-related problems [32].Their findings substantiate the efficacy of this blended approach, evidencing notable improvements in students' problem-solving capacities.These advances were especially remarkable in complex scenarios requiring higher-order cognitive skills.
Expanding the applicability of this integration, Bouck and Park focused on enhancing the educational experiences of students with disabilities in the subject area of mathematics [33].They engineered an app-based learning platform that utilized real-time analytics to issue personalized prompts designed to scaffold learning.The prompts were customized based on a blend of ongoing performance metrics and individual learning style assessments.Notably, their work illustrated that the fusion of computer-assisted and prompt-based learning is not just versatile but indispensable for meeting the diverse needs of specialized educational pathways.
This confluence of computer-assisted learning and promptbased instructional designs, as the cited studies illustrate, transcends subject boundaries and pedagogical contexts.It offers both the specialized attention that prompt-based techniques can provide and the adaptive, large-scale reach that technology facilitates, culminating in a more versatile and impactful educational paradigm.

III. MATERIALS AND METHODS
In this study, we developed a VPML system aimed at enhancing algebraic understanding among students with learning disabilities.Informed consent was obtained from all participants or their legal guardians before initiating the study.The VPML system integrates two primary components: a Visual Prompt-based Learning framework and a mobile application.Within the context of algebra education, fundamental axioms and methods are crucial not only for practical everyday applications but also for the acquisition of more complex algebraic concepts in future studies.

A. MATERIALS
The VPML system is specifically designed to utilize colorcoded prompts, aiding students in associating variables with their respective roles within equations.Acknowledging the variance in individual student capabilities, the system employs tiered levels of visual prompts.These stratified prompts are intended to guide students incrementally toward a deeper understanding of fundamental algebraic principles.

1) VPML SYSTEM
The VPML architecture is segmented into three core systems: a Prompting System, a Learning System, and an Evaluation System.These systems are buttressed by four databases: the Student Profile Database, the Prompt Database, the Learning Material Database, and the Evaluation Database, as depicted in Fig. 1.Within the Prompting System are three specialized coloring units: the Equation-Coloring Unit, the Variable-Coloring Unit, and the Coefficient-Coloring Unit.These units operationalize the visual prompt strategy by assigning distinct colors to specific elements within mathematical equations.Elements sharing similar characteristics are colorcoded identically, thereby aiding students in systematically approaching equation-solving tasks.The details of VMPL's user interface are elaborated upon in Fig. 2.
The Learning System is subdivided into three key areas: Topic-Browsing, Ability Assessment, and Practice.These areas facilitate multifaceted engagement with the learning process.In the Topic-Browsing Area, students can choose specific subjects they wish to delve into.The Ability Assessment Area serves as a platform for gauging students' mathematical proficiency.Meanwhile, the Practice Area offers students targeted problems based on their selected topic, allowing them to refine their equation-solving skills through repetition.
The Evaluation System comprises an Examination Unit, a Level-Assessment Unit, and a Feedback Unit, functioning together to establish a comprehensive assessment framework.The Examination Unit dispenses topic-specific test questions, which students then complete.Once the examination phase concludes, the Level-Assessment Unit calculates a standardized score that encapsulates the student's performance in the chosen subject area.Lastly, the Feedback Unit enables instructors to offer real-time, personalized feedback to students based on their examination outcomes.

2) APP DESIGN AND CONCEPT
As depicted in Fig. 3, the VPML app consists of two primary modules: (i) the Learning System and (ii) the Evaluation System.The Learning System is designed to offer adaptive practice on selected algebraic topics, while the Evaluation System focuses on assessing a student's proficiency post-practice, along with providing detailed feedback and discussion opportunities.
The user journey commences with Topic Selection within the Learning System, allowing students to select specific algebraic topics they wish to improve upon.Once a topic is chosen, an adaptive Pre-test Level Assessment is administered to evaluate the student's baseline abilities.Immediate feedback on the correctness is provided for each question, and the difficulty level of subsequent questions is dynamically calibrated based on student performance.Following the initial assessment, the app transitions to a Practice Problems module.During this phase, an interactive Prompting Unit is engaged, supplying students with guided problem-solving tasks.The extent to which this unit is utilized serves as an indicator of the student's mastery of the chosen topic.If the student's performance falls below a preset benchmark accuracy rate even with the aid of prompts, the app reverts to the Pre-test Level Assessment to fine-tune the learning level.Conversely, if student's performance surpasses the benchmark accuracy rate while using prompts, the student advances to the formal examination within the (ii) Evaluation System.This benchmark rate is adjustable to accommodate specific instructional needs; for the purposes of this study, it has been set at 70%.
Upon completing the formal examination, students undergo a comprehensive Post-test Assessment, featuring a range of questions across multiple algebraic topics.This assessment serves dual purposes: it measures performance gains in the initially chosen subject and recommends additional areas that could benefit from further practice.After this, students have the option to either continue their practice within the (i) Learning System or transition to the Discussion and Feedback Section in the (ii) Evaluation System.The Discussion and Feedback Section facilitates real-time communication between teachers and students, as well as among peers.Instructors can also leverage this section to monitor real-time performance metrics and to provide instantaneous, actionable feedback, which is directly accessible to students via the app.

3) ASSESSMENT
To evaluate students' existing mathematical proficiency, a Preliminary Level Assessment is conducted, which categorizes algebraic questions into three tiers of difficulty: Level 1 (Easy), Level 2 (Intermediate), and Level 3 (Challenging).Initially, students are administered a set of five intermediate-level questions (Level 2) following their topic selection.The ensuing difficulty of questions is dynamically calibrated based on the students' accuracy rate on these initial five questions.The instructor configures the scoring thresholds for these levels; in this specific study, the thresholds were set as follows: Students achieving an 80% accuracy rate or above are escalated to Level 3, targeting the development of advanced problem-solving capabilities.Those scoring between 50% and 79% continue at Level 2 to solidify their current understanding.Students falling below the 50% mark are guided to Level 1 questions to strengthen their foundational skills.

B. METHODOLOGY
In order to investigate the effects of differing instructional approaches on student achievement, this study adopted a quasi-experimental design, illustrated in Fig. 4. The design facilitated a systematic comparison between two distinct pedagogical strategies: VPML and traditional rule-based instruction.
Participants were carefully allocated into one of two groups.The experimental group underwent an educational experience framed by the VPML strategy.Leveraging the capabilities of mobile technologies such as smartphones and tablets, VPML offers an immersive, multifaceted learning environment.Within this framework, students have the opportunity to capture images and make digital annotations, effectively bridging the gap between visual and conceptual understanding.This interactive approach aims to augment academic performance by fostering active engagement and promoting cognitive scaffolding.On the other hand, the control group was subjected to traditional rule-based instruction, which relies on well-established but less interactive teaching techniques.Instruction for this cohort was conducted using traditional classroom tools such as chalkboards and penand-paper methods.This comparison allowed for a nuanced evaluation of the relative merits and shortcomings of both the VPML strategy and conventional teaching methods.
By employing a quasi-experimental design, the study controlled for extraneous variables to the extent possible, aiming to attribute observed differences in learning outcomes specifically to the instructional methods under investigation.The aim was to ascertain whether the VPML approach offers statistically significant improvements in student learning when compared to traditional methodologies, thereby offering empirical evidence to inform educational strategies for enhanced learning outcomes.

1) INSTRUMENTS
The research methodology was structured around two key phases: an initial pretest and a concluding posttest.Each of these assessments consisted of 20 questions specifically tailored to evaluate understanding of linear equations, adhering to the standards of the junior high school algebra curriculum.The questions were designed not just to test mathematical skills, but also to measure problem-solving abilities in the context of algebra.The independent variable under scrutiny was the type of instructional methodology-either the Visual Prompt-Based Mobile Learning (VPML) strategy for the experimental group or traditional rule-based teaching for the control group.These instructional sets were methodically developed to provide nuanced approaches to algebraic problem-solving, offering divergent pathways for the acquisition of mathematical competence.
To establish a foundation for the study, pretests were initially administered to participants in both the experimental and control groups, as shown in Fig. 4.This helped in capturing the baseline academic abilities and mathematical competencies of all participants, providing a starting point for subsequent instructional intervention.Following the pretest, students from both groups participated in a comprehensive 12-week training program centered around standard classroom instruction, tailored in accordance with their designated instructional methodology.At the end of the 12week instructional period, posttests were conducted for both groups.These post-assessments were designed to measure the extent to which each instructional approach impacted students' algebraic understanding and problem-solving skills.The comparative analysis of pretest and posttest scores allowed for an in-depth evaluation of the efficacy of each instructional method.Through this phased approach, the study sought to offer empirical evidence regarding the influence of the chosen instructional methodologies on educational outcomes in algebra.

A. PARTICIPANTS
The study engaged a specific cohort of 80 seventh-grade students, all of whom have been formally diagnosed with math learning disabilities and are enrolled in specialized math education programs.Informed consent was obtained from all participants or their legal guardians prior to the commencement of the study.The age range of the participants was tightly clustered around an average age of 13 years.Students were assigned to either the experimental group or the control group through a randomized allocation process, with each group containing 40 students to ensure statistical robustness.All participants were educated by the same certified special education teacher, ensuring a consistent teaching approach and mitigating potential biases introduced by different teaching styles or methods.This homogeneity in instruction also facilitates a more straightforward interpretation of the study's results, as any performance differences can readily be attributed to the intervention itself rather than teacher variability.
Eligibility for inclusion in the study was rigorously defined to maintain a focused and relevant participant pool.Drawing on established criteria outlined by [34], students were required to meet three specific conditions: (a) a formal, medically or educationally diagnosed learning disability specifically in the domain of mathematics; (b) a clearly outlined math goal focused on linear equations as part of their most recent Individualized Education Program (IEP); and (c) sufficient technological literacy to operate a tablet without assistance, ensuring that the study's outcomes would not be skewed by disparities in basic tech-savviness.

B. DATA COLLECTION
Data collection was executed using a two-pronged assessment strategy that comprised both pretest and posttest evaluations.The objective was to capture a snapshot of the students' baseline algebraic understanding prior to the introduction of the VPML approach, as well as to document any changes in their understanding subsequent to the implementation of this pedagogical method.Each of these assessments featured a set of 20 multiple-choice questions, formulated with pedagogical rigor to scrutinize different dimensions of algebraic knowledge.The questions were methodically constructed to evaluate a range of skills, from foundational understanding to more advanced problem-solving capabilities in algebra.Importantly, these questions were not uniform in their level of difficulty but were categorized into three separate difficulty tiers-basic, intermediate, and advanced.This stratification was designed to furnish a multi-layered evaluation, allowing for more nuanced inferences about the students' abilities in algebra.

C. ANALYSIS
Statistical analyses were carried out using R software, version 3.5.1.The data were analyzed employing both descriptive (e.g., means, standard deviations) and inferential statistics to contrast the learning outcomes between the experimental and control groups.To ensure robustness and internal validity, participants were assigned to either group using stratified random assignment based on their initial pretest scores.
In this quasi-experimental study, we implemented a two-pronged statistical approach to address sample size limitations and enhance the robustness of our findings.Recognizing the constraints imposed by the original sample size of 80 seventh-grade students with math learning disabilities, we employed bootstrapping techniques alongside noise injection to simulate a larger and more diverse dataset.Bootstrapping was employed to generate 1000 number of resampled datasets from the original 80-student sample.This technique allowed us to repeatedly resample with replacement, thereby creating an ensemble of datasets that closely approximate the characteristics of a larger population.By applying bootstrapping, we enhanced the generalizability of our findings, giving us greater confidence in the statistical estimates derived from our analyses.
Complementing the bootstrapping method, we incorporated noise injection as an additional layer of rigor to our methodology.Random Gaussian noise was added to the features of the dataset, subtly altering the data points without significantly affecting the overall distribution.The purpose of this noise injection was to simulate the inherent variability and ''noise'' that would be present in a larger, more diverse sample.This simulated variability added robustness to our model, allowing it to generalize better to unseen or future data.The use of bootstrapping and noise injection was particularly beneficial given the quasi-experimental design of our study, which lacked the random assignment characteristic of true experiments.These statistical techniques acted as safeguards against potential biases and confounding  variables, thereby increasing the credibility and reliability of our findings.Through this methodological enhancement, we were able to deliver more nuanced insights into the efficacy of the VPML strategy in comparison to traditional rule-based instruction, despite the initial limitations in sample size.
To rigorously assess the assumptions underlying the employed statistical tests and to validate our research findings, multiple graphical and analytical techniques were utilized on the enhanced sample set generated through bootstrapping and noise injection.In the first step of our validation process, box plots were constructed to undertake a comparative analysis of the initial distribution of WISC-IV scores and pretest scores among both the experimental and control groups, as presented in Fig. 5.The outcomes of this preliminary scrutiny substantiated the similarity in the baseline attributes of both groups.This validation bolstered our confidence in attributing observed differential outcomes to the unique instructional methodologies employed in each group.Secondly, we took steps to verify the assumption of normal distribution, a requisite condition for executing various parametric tests.This was accomplished by constructing a Quantile-Quantile (QQ) plot (see Fig. 6, first subplot) and conducting a Shapiro-Wilk test.The close alignment of data points to the 45-degree line in the QQ plot, coupled with the results of the Shapiro-Wilk test, corroborated the hypothesis that the data follows a normal distribution.Thirdly, the assumption of homogeneity of variances was put to the test through an examination of a scale-location plot (see Fig. 6, second subplot) supplemented by Levene's test.The appearance of a nearly horizontal line in the scale-location plot, along with the absence of any significant deviations in Levene's test, validated that the assumption of homoscedasticity-or equal variances-was satisfied.Fourthly, the residual vs. fitted plot (see Fig. 6, thrid subplot) was carefully analyzed to confirm linearity and constant variance of the error terms.The pattern of residuals 'bouncing' randomly around the 'residual = 0' line was indicative of a linear relationship.Furthermore, the formation of a 'horizontal band' around this line attested to the equal variances of the error terms.
Upon the successful verification of these foundational assumptions, we proceeded to the main phase of our statistical analysis.Utilizing a paired-sample T-test, we set the significance threshold at α = 0.05, as is standard in the social sciences.In terms of baseline capabilities, the control group displayed a mean pre-test score of 50.86 with a standard deviation (S.D.) of ±23.54.Comparatively, the experimental group started with a mean pre-test score of 44.07 and a higher standard deviation of ±32.16.The paired-sample T-test yielded instructive results, as illustrated in Table 1.Firstly, it revealed that both groups made statistically significant strides in their algebraic problemsolving capabilities, as evidenced by a p-value of less than 0.05.This initial finding was crucial as it established that both the traditional and VPML teaching methodologies were  effective to some extent in enhancing the mathematical proficiencies of the participating students.However, what was even more illuminating was the degree of improvement exhibited by each group.The experimental group, which was exposed to the VPML system, showed not just an improvement, but a statistically significant leap in their post-test scores.The t-value for the post-test comparison between the experimental and control groups was t = 11.87,accompanied by a p-value of less than 0.01.In addition, we also use a box plot to show that there is potentially better improvement in the experimental group compared to the control group.
To bolster the robustness and credibility of our research findings, we employed a one-way Analysis of Covariance (ANCOVA) as an additional layer of analytical scrutiny.The purpose of this statistical technique was to dissect and ascertain the impact of the different instructional strategies on student outcomes, while simultaneously controlling for pre-existing differences in student abilities.In our ANCOVA model, pretest scores functioned as the covariate, posttest outcomes served as the dependent variable, and the type of instructional strategy (VPML or traditional) constituted the independent variable.Prior to performing ANCOVA, it was crucial to confirm the assumption that no significant interaction exists between the covariate and the independent variable.The results pertinent to this assumption are presented in Table 2. Notably, the p-value associated with the interaction term was 0.928, suggesting that the interaction between the covariate and the independent variable was not statistically significant.This finding confirms that the effect of the covariate on the dependent variable does not differ across levels of the independent variable, thereby satisfying the necessary conditions for proceeding with an ANCOVA.
To deepen our understanding of the relationships among the pretest scores, posttest scores, and the two student groups, an ANCOVA analysis was conducted.The interaction term was omitted from the analysis because it was found to be statistically insignificant.As indicated in Table 3, the analysis yielded compelling results.Specifically, the pretest was found to have a statistically significant impact on the posttest, as evidenced by a p-value of 2 × 10 −16 , which is well below the 0.05 significance level.Similarly, the posttest scores were significantly different between the experimental and control groups, as indicated by a p-value of 4.93 × 10 −6 , also below the 0.05 threshold.This ANCOVA analysis confirms that the pretest serves as a meaningful covariate affecting the posttest outcomes.Moreover, there exists a statistically significant difference in posttest scores between students in the experimental group, who were taught using the VPML strategy, and those in the control group, who received traditional instruction.
Following the ANCOVA results, which accounted for initial student abilities, a regression coefficient test was employed to evaluate the true effectiveness of the two instructional strategies.This test aimed to measure post-test differences between the experimental and control groups after neutralizing the influence of initial ability differences.The regression coefficient test results, as depicted in Fig. 7, revealed that the VPML teaching method was markedly more effective than traditional blackboard instruction in enhancing students' proficiency in solving linear equations.In our analysis, we found that both the VPML and traditional teaching methods demonstrated parallel lines when plotted, with the covariate (pre-test scores) on the x-axis and the dependent variable (post-test scores) on the y-axis.Intriguingly, while the slopes were identical-indicating that the covariate had a consistent impact across both instructional methods-the VPML line was consistently higher by approximately 10 units.This higher intercept for the VPML group strongly suggests that, on average, students exposed to the VPML instructional strategy scored 10 units higher in their post-test scores compared to those in the traditional teaching group.This difference is not only numerically significant but could also bear educational importance, particularly when considering the challenge of enhancing learning outcomes, especially for students with learning disabilities.Given that the slopes were similar, the VPML's effectiveness appears to be universally beneficial across diverse levels of initial abilities.
As a complement to the quantitative realm of our study, we also harvested valuable qualitative insights through postintervention surveys.Students from the experimental group reported a significantly more favorable academic experience under the VPML pedagogy.These subjective reports extend beyond mere academic metrics, revealing heightened levels of student engagement, confidence, and self-efficacy in learning algebra.The concordance between these qualitative findings and our quantitative results deepens the evidential base supporting the efficacy of the VPML approach, suggesting that its benefits are manifold, influencing both the cognitive and affective dimensions of student learning.

A. ACADEMIC AND COGNITIVE IMPACT
One of the most salient and impactful outcomes from our study revolves around the remarkable acceleration in algebraic skill development within the group utilizing the VPML strategy.But what sets these findings apart is not just the statistical significance-a point underscored by both the paired-sample T-test and the ANCOVA results.Beyond the p-values and confidence intervals, the real-world implications of these results are of particular interest.The magnitude of the difference between pre-and post-test scores points to more than just academic curiosity; it establishes a compelling case VPML's practical utility in pedagogical settings.

1) VERSATILITY IN PROBLEM-SOLVING
Moreover, the nuanced capabilities that students acquired were not confined to rudimentary algebraic manipulations.
The VPML group demonstrated an increased versatility in problem-solving-able to maneuver through more complex algebraic problems that had previously posed considerable challenges.They did not just improve; they expanded their repertoire of algebraic problem-solving strategies, a skill that is likely to translate into broader academic and real-world benefits.These results carry implications that reach beyond the classroom walls.Mathematics, particularly algebra, often acts as a stumbling block in the educational trajectory for many students, creating a bottleneck effect that may impede progress in other STEM-related disciplines [35], [36], [37].VPML shows promise not just as another tool in the toolkit but as a pivotal means to bridge these traditional educational gaps.By transforming the way algebra is taught and understood, VPML has the potential to break down the barriers that have made this subject so formidable for many students, including those with learning disabilities [38], [39].

2) A PARADIGM SHIFT IN TEACHING ALGEBRA
The capability of students to tackle complex problems also suggests a potential paradigm shift in how algebra can be taught more broadly [40], [41], [42].Traditional methods often focused on procedural understanding and memorization, may be effectively supplemented by VPML's conceptually robust and interactive approach.This opens the door to making advanced algebraic concepts more accessible earlier in a student's educational journey, thereby enriching their entire academic landscape [43], [44].Given the observable benefits of both basic and complex problemsolving skills, educational institutions might consider the integration of VPML as a component in algebra courses.Its demonstrated efficacy suggests that it could serve as a cornerstone for modern mathematical pedagogy, radically transforming the way algebra is approached in educational settings and setting students up for long-term success in mathematics and related fields.

B. PEDAGOGICAL AND TECHNOLOGICAL SYNERGIES 1) EDUCATOR INSIGHTS
The VPML system does more than just serve as an educational adjunct; it transforms into a pedagogical compass that guides educators through the maze of individual learning preferences, abilities, and roadblocks.The wealth of granular analytics furnished by VPML elevates the teaching experience from a generic, one-size-fits-all model to a highly tailored, student-centric approach.

a: NAVIGATING THE COMPLEXITY OF LEARNING DISABILITIES
VPML's capability for micro-assessments is especially groundbreaking.Traditional teaching methods often rely on periodic assessments to gauge student performance.In contrast, VPML provides real-time analytics that allows teachers to dynamically adjust the instructional tempo and material for each student [45], [46].Whether it's deciding to spend more time on a particularly challenging concept or choosing to incorporate more hands-on activities for kinesthetic learners, VPML's analytics enable an unprecedented level of customization.For students with learning disabilities, this feature is nothing short of revolutionary.In addition, individual learning needs can differ significantly, especially when disabilities are factored in [47], [48], and [49].The VPML system enables the identification of these nuanced learning requirements by generating data not just on what students are struggling with but also on how they are engaging with the material.This deep level of understanding opens the door for differentiated instruction that can meet students where they are, instead of forcing them to adapt to a standardized curriculum.The system serves as a form of navigational support, offering real-time observations that can be leveraged to better understand and cater to each student's learning style and pace.This does not only make the educational journey smoother, but it makes it more effective, targeted, and fulfilling for both students and teachers alike.

2) TECHNOLOGICAL INTEGRATION
The VPML system stands as a testament to the transformative power of integrating technology with education, epitomizing how contemporary teaching and learning paradigms can be elevated to new heights.This integration isn't an ancillary feature but a central component that aligns with the educational demands and lifestyle habits of the 21st-century learner [50].

a: MEETING STUDENTS IN THEIR DIGITAL HABITAT
Modern students, often referred to as ''digital natives,'' grow up in an ecosystem populated by smartphones, tablets, and laptops.VPML capitalizes on this ubiquity of technology to meet students in their natural digital habitats.Far from being a gimmick or simply jumping on the tech bandwagon, this approach reflects a deeper understanding that educational methods must not only adapt but also thrive within the current technological milieu to achieve maximum efficacy.One of the most profound implications of this technological integration is the ability to extend the learning environment beyond the physical classroom [51], [52], [53].Whether it's through asynchronous learning modules that students can engage with at their own pace or through real-time, cloud-based collaboration, the digital features of VPML expand the opportunities for educational interaction.This fluidity between physical and virtual spaces effectively elongates the classroom's four walls to encompass anywhere a student has a mobile device and internet connectivity [54], [55], [56].

b: REAL-TIME ADAPTATION AND PERSONALIZATION
Notably, the VPML system's digital aspects also contribute to making education more accessible and inclusive.For students who may face logistical constraints-such as health issues or geographical distances-the virtual extensions of the classroom offer a lifeline to maintain consistent educational engagement [45], [57].This is especially pertinent in the era of remote learning and hybrid education models, where the symbiotic relationship between technology and pedagogy transitions from being merely beneficial to being fundamentally necessary for equitable access to educational opportunities.Technological integration enables another layer of real-time adaptation and personalization.Unlike traditional educational settings where immediate feedback is often impractical, this system continuously monitors students' engagement and performance.This dynamic tracking enables instantaneous feedback mechanisms, allowing the system to adapt instructional materials on the fly.Depending on the student's current performance metrics, the system can tailor problem sets, insert remedial or advanced lessons as appropriate, and even notify educators of any significant shifts in a student's learning trajectory.

C. AFFECTIVE AND PSYCHOLOGICAL DIMENSIONS
The influence of the VPML system is multidimensional, impacting not just educational metrics but also the psychological factors crucial to learning.The true effectiveness of this instructional resource is not confined to test results or quantitative metrics; it also extends to its beneficial impact on student engagement and self-efficacy.

1) THE FEEDBACK LOOP OF ENGAGEMENT AND SELF-EFFICACY
One of the standout features of the VPML system is its interactive design, which actively engages students in their learning journey.The system does not simply deliver information; it demands active participation, encouraging cognitive investment and critical thinking [58], [59], [60].This type of engagement forms a positive feedback loop with academic performance.Engaged students are more likely to participate, practice, and perform, and these performance milestones further heighten their engagement-a cycle that is particularly beneficial for students who have learning disabilities and often struggle with academic selfconcept [61].Additionally, VPML's impact on self-efficacythe belief in one's ability to achieve specific outcomes-is perhaps one of its most transformative aspects.Self-efficacy has been widely recognized as a strong predictor of academic success and is especially significant for students with learning disabilities, who may frequently encounter academic and psychological challenges [62], [63].VPML's adaptive, personalized approach enables small wins, incrementally building a student's confidence and reinforcing their belief that they are capable of learning and succeeding.This is not just theory; numerous psychological studies corroborate the strong link between self-efficacy and educational outcomes.

2) THE ROLE OF GAMIFICATION AND FEEDBACK
Another contributing factor to both engagement and self-efficacy is the VPML system's use of gamification elements and immediate feedback [64].By incorporating game-like features and real-time feedback, the VPML system not only makes learning more enjoyable but also offers immediate validation or constructive feedback.This immediacy is instrumental in helping students make real-time adjustments to their learning strategies, thereby enhancing their meta-cognitive awareness and self-efficacy [65], [66].One of the most striking features of VPML is its ability to balance autonomy and structure, providing students with the freedom to explore at their own pace while also offering structured learning pathways.This fosters an educational environment where students feel in control of their own learning, thereby further enhancing their engagement and boosting their sense of self-efficacy.

VI. CONCLUSION
Our study provides compelling evidence that the VPML strategy serves as a comprehensive educational framework, offering advantages that extend across multiple dimensions.This is especially noteworthy for students with learning disabilities, a group that often faces unique educational challenges and for whom VPML appears to have particular resonance.VPML's efficacy is not constrained to a single metric or outcome.Its advantages manifest themselves in a variety of ways, whether viewed through the lenses of academic performance, pedagogical innovation, technological integration, or psychological well-being.This makes it a uniquely versatile educational tool that does more than just keep pace with traditional rule-based methods-it substantially outperforms them.One of the key strengths of VPML is its adaptability and focus on personalized learning pathways.By tailoring the educational experience to individual students, VPML not only respects but embraces the variability in learning styles and paces.This makes it a particularly potent tool for accommodating the needs of a diverse student population, including those with learning disabilities.VPML's integration of technology serves to extend the reach and accessibility of educational content, aligning with the tech-savvy tendencies of modern learners.This is not a mere nod to contemporary digital lifestyles but represents a deeper pedagogical strategy, leveraging technology to expand educational horizons and make learning more engaging and versatile.
Beyond cognitive gains and academic performance, VPML also has a profound impact on students' psychological wellbeing.By engaging learners in a more interactive and personally resonant way, it boosts both their self-efficacy and their engagement with the educational material.These factors are not just side benefits but are integral to the overall success and retention of educational outcomes.The evidence garnered from this study suggests that VPML is not merely an alternative to traditional methods but a superior, nextgeneration solution.Its manifold advantages position it as a catalyst for broader change in educational practices, offering a blueprint for a more inclusive and effective educational landscape.

FIGURE 1 .
FIGURE 1. Diagrammatic Representation of VPML Architecture: The framework features three core systems-Prompting, Learning, and Evaluation-each anchored by a corresponding database: Student Profile, Prompt, Learning Material, and Evaluation.

FIGURE 2 .
FIGURE 2. The VMPL User Interface -Elements with similar characteristics are uniformly color-coded to facilitate a systematic approach to problem-solving.

FIGURE 3 .
FIGURE 3. Workflow of the VPML App: The diagram outlines the Learning and Evaluation Systems.The Learning System begins with Topic Selection and a Pre-test Level Assessment, leading to tailored Practice Problems.The Evaluation System includes a formal exam, a Post-test Assessment, and a Discussion and Feedback Section for real-time interactions.

FIGURE 4 .
FIGURE 4. Consort Flow Diagram of Quasi-Experimental Design Comparing VPML and Traditional Rule-Based Instruction: This diagram outlines the systematic allocation of participants into experimental and control groups, the interventions they received, and the metrics used for evaluation.The study aims to provide an assessment of VPML's potential for improving learning outcomes in comparison to traditional teaching methods.

FIGURE 5 .
FIGURE 5. Comparative Box Plots for Experimental and Control Groups -The first two plots display the initial distribution of WISC-IV and pretest scores, affirming the baseline similarity between both groups.The final plot highlights a potentially superior improvement in the experimental group compared to the control group.

FIGURE 6 .
FIGURE 6. Diagnostic Plots for Model Validation -The QQ-Plot confirms data normality with points along a 45-degree line.The Scale-Location Plot shows a horizontal line, indicating variance homogeneity.The Residual vs. Fitted Value Plot reveals an even scatter around zero, validating the linear model's appropriateness.

FIGURE 7 .
FIGURE 7. Impact of VPML versus Traditional Teaching on Post-Test Scores: Both regression lines share a slope of 0.47, indicating a uniform effect of the covariate (pre-test scores) on post-test performance across the two teaching strategies.VPML consistently scores 10-unit higher post-test score compared to traditional teaching, signifying the superior efficacy of the VPML approach.

TABLE 1 .
Descriptive statistics on pretest and posttest.

TABLE 2 .
The relation between the dependent variable and the interaction term.

TABLE 3 .
The relation between variables after removing the interaction term.