The Impact of the Intensive Learning Model on Academic Achievement in Mathematics Courses Among Engineering Students

Abstract

This study examined the effectiveness of an intensive learning model in core mathematics courses within engineering education. The model restructures the academic semester so that students study one course at a time in concentrated learning blocks, rather than studying several courses in parallel, with the aim of improving academic achievement and student engagement in engineering mathematics courses. The research employed a quantitative, quasi-experimental, longitudinal design and included 66 undergraduate engineering students who completed three mathematics courses: Linear Algebra, Calculus II, and Differential Equations. Academic performance and learning behavior data were analyzed using mixed-design ANOVA, multiple linear regression, and MANOVA analyses. The findings indicate that students who studied under the intensive learning model achieved significantly higher final grades compared with students in the traditional parallel-course structure. Engagement variables emerged as strong predictors of academic success, particularly class attendance and assignment submission. Academic performance remained stable across the three mathematics courses, and prior academic background variables did not significantly predict achievement. Overall, the results suggest that restructuring mathematics instruction into intensive learning blocks may enhance student engagement and academic performance in demanding quantitative courses, thereby supporting student success and persistence in engineering education.

1. Introduction

1.1. General Background

Mathematics courses constitute a central component of engineering education, yet they are frequently perceived by students as abstract and challenging, resulting in high levels of difficulty and failure rates in the early stages of engineering studies. Mathematics constitutes a central and inseparable component of the academic and professional training of engineering students. Mathematics serves as the professional language of the engineering sciences and provides the analytical tools required for solving complex problems in various engineering domains (Goold, 2012; Klingbeil et al., 2004). However, for many students, mathematics courses are perceived as complex, challenging, and a potential barrier to academic progress, particularly advanced courses (Broadbridge & Henderson, 2008). This phenomenon has become increasingly acute in recent years, as contemporary studies identify a growing “preparedness gap” in foundational skills among engineering cohorts, emphasizing the urgent need for pedagogical restructuring (Morán-Soto et al., 2026). One of the central challenges in planning mathematics instruction for engineering lies in the need to balance an extensive load of learning material with the students’ ability to absorb and internalize it (Faulkner et al., 2019). Studies indicate that in basic mathematics courses in higher education institutions, the failure and dropout rates are high, ranging between 30% and 50%, a phenomenon that raises concern among decision-makers in the higher education system (Ellis et al., 2016; National Research Council, 2013).

1.2. Theoretical Framework—Intensive Learning

In recent years, a pedagogical approach has been developing that emphasizes focused and intensive learning as a way to improve understanding of the material and increase the chances of success, especially in quantitative subjects. Studies indicate that concentrated learning with high learner involvement over a short time period may contribute to deeper knowledge acquisition and higher achievement (Davies, 2006). Recent meta-analyses of accelerated learning environments suggest that the efficacy of such models is highly dependent on students’ self-regulation and prior academic background, highlighting a divergence in research regarding for whom these models are most effective (Loton et al., 2022).

Research specifically examining engineering students in online environments has identified that learning intensity—defined by the frequency and volume of practice—serves as a primary predictor of academic success. Further longitudinal analysis of these learning patterns suggests that students who maintain high intensity through automated practice systems achieve significantly higher learning outcomes compared to those with sporadic study habits (Levi Gamlieli et al., 2015).

Cognitive Load Theory (CLT), developed by Sweller (1988), provides an important theoretical basis for understanding the potential advantages of intensive learning. According to this theory, the learner’s working memory is limited in its ability to process new information at any given time (Sweller et al., 1998). The theory distinguishes between intrinsic load (inherent complexity), extraneous load (presentation and environment), and germane load (building cognitive schemas) (Paas et al., 2003). In the intensive learning model, the reduction in the number of courses learned in parallel lowers the extraneous load and allows the allocation of more cognitive resources to deeper processing of the material (germane load), while coping with the high intrinsic load that characterizes academic-level mathematics courses (Scott & Conrad, 1992). Contemporary applications of CLT in STEM pedagogy reinforce that block-scheduling can effectively moderate these loads by minimizing the cognitive fragmentation inherent in traditional semester structures (McCluskey et al., 2019).

Beyond structural scheduling considerations, Cognitive Load Theory also highlights the importance of managing the interaction between working memory and long-term memory. When learners are required to divide attention across multiple complex topics simultaneously, the load on working memory increases—potentially leading to cognitive overload and reduced learning efficiency (Sweller, 1988). For complex domains such as engineering mathematics, block-based learning environments may therefore support more effective schema construction by allowing sustained attention to a single conceptual domain. Recent studies further suggest that improving learners’ intrinsic cognitive resources may also help to reduce perceived cognitive load. For example, hybrid training approaches designed to enhance spatial ability have been shown to assist students in processing complex engineering concepts more effectively, thereby reducing the cognitive burden associated with demanding STEM subjects (Porat & Ceobanu, 2024a).

In addition to cognitive load considerations, theories of meaningful learning emphasize the importance of linking new information to existing knowledge structures. According to Ausubel’s theory of meaningful learning, learning becomes effective when new concepts are actively integrated into previously established cognitive schemas (Ausubel, 2000). Intensive learning environments may support such processes by enabling sustained engagement with a single topic over a continuous period, allowing learners to revisit related ideas repeatedly and progressively deepen their conceptual understanding.

Additional studies indicate that intensive learning promotes a sense of “flow” among learners, increases internal motivation, and enables faster progress in the material (Csikszentmihalyi, 1990; Kucsera & Zimmaro, 2010). Flow Theory describes an optimal psychological state characterized by deep concentration, intrinsic motivation, and a balance between challenge and skill (Csikszentmihalyi, 1990). Conditions that support flow, such as continuous engagement with a single task and the opportunity to receive immediate feedback, may emerge more naturally in intensive learning environments. The intensive model also reduces the forgetting interval between meetings, thereby moderating the well-known forgetting curve and supporting continuous consolidation of knowledge (Ebbinghaus, 1885; Murre & Dros, 2015).

Research examining intensive or accelerated learning models has also reported empirical advantages in various higher education contexts. Scott (2003) reviewed intensive courses in the United States and found that students achieved results at least as good as, and sometimes better than, those in traditional courses, attributing this to cognitive concentration and increased involvement. Davies (2006) and Wlodkowski (2003) further highlighted that intensive models increase motivation and are particularly effective for goal-oriented learners. Specifically, in mathematics education, Austin and Gustafson (2006) reported higher grades and completion rates in intensive community college courses.

However, the academic field also presents diverging hypotheses regarding the long-term efficacy of these models. Daniel (2000) raised concerns that intensive learning might lead to superficial acquisition and rapid forgetting. Recent meta-analyses have added nuance to this debate, suggesting that while intensive models excel in immediate achievement, the long-term retention of abstract mathematical concepts may require specific “spaced” reinforcement strategies that are sometimes lacking in compressed formats (Loton et al., 2022). Furthermore, Anastasi (2007) found that students in intensive courses experienced higher levels of stress, highlighting the need for robust support mechanisms.

Despite these theoretical arguments supporting intensive learning environments, empirical evidence examining the implementation of intensive learning models in engineering mathematics courses remains relatively limited. While intensive and accelerated learning models have been examined in various higher education contexts, including adult education and community college environments (Scott, 2003; Davies, 2006; Austin & Gustafson, 2006), empirical studies focusing specifically on intensive course structures in engineering mathematics remain relatively scarce. Existing research in engineering education has primarily examined pedagogical reforms such as active learning, flipped classrooms, and curriculum restructuring rather than the temporal organization of course delivery (Klingbeil et al., 2004). Consequently, relatively little is known about the potential impact of intensive or block-based instructional formats on student engagement and academic achievement in demanding mathematics courses within engineering programs. The present study, therefore, aims to examine the effect of the intensive learning model on students’ academic achievement in core mathematics courses within engineering education.

1.3. Mathematics Learning in Higher Education

Understanding the role of instructional models in mathematics education requires examining the broader challenges associated with learning mathematics in higher education, particularly in engineering programs, where early mathematics courses often constitute a critical barrier to student persistence. Difficulty in mathematics has long been identified as a primary cause of dropout from engineering studies (Seymour & Hewitt, 1997). Many students struggle with the transition from procedural and computational high school mathematics to the conceptual and abstract reasoning required in university-level mathematics (Kajander & Lovric, 2005). Studies further identify introductory mathematics courses—especially calculus—as critical bottlenecks affecting student persistence in engineering programs (Bressoud et al., 2015; Ellis et al., 2016). Current research suggests that this “math gap” has widened in recent years, with decreasing levels of independent practice among students, making structured and immersive learning environments increasingly important (Morán-Soto et al., 2026).

Research in higher education consistently identifies introductory mathematics courses as a key bottleneck affecting student persistence in engineering programs (Bressoud et al., 2015; Ellis et al., 2016). Students frequently experience difficulties when shifting from algorithmic procedures practiced in high school to the deeper conceptual reasoning required in university mathematics. This transition often leads to reduced confidence, lower academic self-efficacy, and an increased risk of attrition during the first stages of engineering education (Kajander & Lovric, 2005).

In addition to general academic challenges, the literature also highlights persistent gender disparities in participation and persistence in STEM fields. Women remain underrepresented in many engineering programs worldwide, and several studies suggest that the early stages of engineering education—particularly mathematics courses—may contribute to differential attrition patterns (Hill et al., 2010; Blickenstaff, 2005). Female students often report lower levels of mathematical self-efficacy and higher levels of mathematics-related anxiety, even when their actual academic performance is comparable to that of male students (Else-Quest et al., 2010; Hyde et al., 2008). These psychological and social factors may influence students’ persistence decisions and contribute to gender gaps in STEM pathways.

Studies examining dropout patterns indicate that the reasons for attrition may differ between male and female students. Male students are more likely to leave engineering programs primarily due to academic performance difficulties, whereas female students may also be influenced by factors such as a perceived lack of belonging, limited representation, and classroom climate (Seymour & Hewitt, 1997; Blickenstaff, 2005). However, other research suggests that once female students successfully pass the initial mathematics courses, their persistence rates become similar to those of their male peers (Hill et al., 2010). These findings highlight the critical role that early mathematics experiences play not only in academic preparation but also in shaping students’ confidence and sense of belonging in engineering education.

Recent research in mathematics education further emphasizes that instructional design and course structure can significantly influence students’ engagement and persistence (Ellis et al., 2014). Teaching approaches that promote active engagement, continuous practice, and structured learning environments may therefore play an important role in supporting diverse student populations and reducing attrition in demanding quantitative disciplines.

Given that difficulties in early mathematics courses are strongly associated with dropout from engineering programs, instructional models that encourage sustained engagement and continuous practice may contribute to improving student persistence, particularly among populations that are traditionally underrepresented or more vulnerable to early academic disengagement.

Despite the rich literature on success factors, such as attendance, prior academic background, and self-efficacy, there remains a lack of studies examining intensive mathematics learning models specifically within engineering education in the Israeli context. The present study seeks to address this gap.

1.4. The Intensive Model Examined in This Study

The intensive learning model examined in the current study proposes a fundamental change in the structure of the semester in the academic institution. Instead of parallel learning of several courses throughout the semester (usually 13–14 weeks), the model is based on focused learning in one course in each time block, with all teaching hours of the courses concentrated in favor of a single course at any given time. Within this model, the semester is divided into three periods of 4.5 weeks each. In every period, students study only one course, while the number of weekly teaching hours increases substantially from 4 to 5 h per week in the classic model to about 18 h per week in the intensive model. This model aspires to create a learning environment that promotes continuous investment, deep focus, and a reduction in cognitive load deriving from simultaneous engagement with several courses.

1.5. Success Factors in Learning Mathematics

The literature identifies several significant factors that influence student success in academic mathematics courses:

• Attendance and active participation: Many studies indicate a strong relationship between class attendance and academic achievement (Credé et al., 2010). Attendance not only reflects physical presence but also serves as a proxy for cognitive and emotional engagement in the learning process (Kahu, 2013).
• Prior mathematical background: Basic mathematical skills and the level of high school studies serve as central predictors of success (Hailikari et al., 2008). Students who completed higher-level mathematics in secondary education tend to demonstrate greater readiness (Hourigan & O’Donoghue, 2007).
• Assignment submission and practice: Active practice is essential. Research on engineering students highlights that the intensity of this practice is a decisive factor in outcomes (Levi Gamlieli et al., 2015).
• Demographic variables: Variables such as gender, type of matriculation, and socio-economic background shape access to resources and mathematical self-efficacy (OECD, 2016; Wang, 2013; Bandura, 1997).
• Cognitive and spatial skills: Beyond demographic and behavioral factors, recent research highlights the critical role of cognitive skills. Specifically, spatial ability has been identified as a significant predictor of academic success in engineering education, with integrated training programs showing promise in enhancing these essential skills (Porat & Ceobanu, 2024b, 2024c).

1.6. The Need for the Current Study and Its Significance

Despite the potential advantages of the intensive learning model, there is a lack of empirical studies examining its specific impact on mathematics studies in engineering—particularly within the Israeli context. The present study aims to fill this gap by providing an empirical examination of the effects of the intensive model on achievement in three core mathematics courses (Calculus II, Linear Algebra, and Differential Equations).

The main aim of this work is to determine whether the intensive model contributes to higher success rates compared to the classic model and to identify the key predictors of success within this framework. Ultimately, this research highlights that while the intensive model offers significant cognitive advantages, success is largely contingent upon engagement variables and prior background, providing a foundation for improved pedagogical planning in engineering education.

Furthermore, while previous studies have examined intensive learning formats in various educational contexts, relatively little empirical research has focused specifically on intensive learning models in core mathematics courses within engineering education. This gap is particularly evident in the Israeli higher education context, where engineering programs rely heavily on foundational mathematics courses that often function as critical gateways for student persistence. Investigating the pedagogical implications of intensive learning in such courses may therefore contribute to both the literature on mathematics education and to the broader discussion on improving retention and success in engineering programs.

1.7. The Present Study

Building on the theoretical framework and prior research reviewed above, the present study examines the effectiveness of the intensive learning model in mathematics courses for engineering students. In particular, the study investigates whether the intensive structure of course delivery is associated with improved academic outcomes, increased engagement in learning activities, and higher levels of persistence in demanding quantitative courses.

The general aim of the study is to examine the effectiveness of the intensive learning model in mathematics courses for engineering, while identifying factors that affect student success in this model.

The study was conducted in accordance with the ethical guidelines of the institutional ethics committee. Data collection was carried out using fully anonymized records, ensuring that no identifying information about individual students was included in the analysis. In addition, all students whose data were included in the study signed an informed consent form indicating their awareness that anonymized data from the course could be used for research purposes.

1.8. Research Questions

RQ1: Does the intensive learning model contribute to success in mathematics courses among engineering students, compared to the classic learning model?

RQ2: Which factors (attendance, assignment submission, prior academic background, and demographic background variables) predict success in courses taught in the intensive model?

RQ3: Are there differences in levels of success between the three types of courses (Calculus II, Linear Algebra, and Differential Equations) taught in the intensive model?

RQ4: Does the student’s prior academic background (number of high school mathematics units and type of matriculation) have a differential effect on success according to course type?

In order to address these research questions, the study also examines the role of learning engagement variables and prior academic background in predicting academic success in mathematics courses.

1.9. Research Hypotheses

H1. 

Students studying under the intensive learning model will achieve higher final grades in engineering mathematics courses compared to students studying under the traditional learning model.

H2. 

Learning engagement variables—particularly class attendance and assignment submission—will be positively and significantly associated with academic success in mathematics courses, and students in the intensive learning model will demonstrate higher levels of engagement and persistence than those in the traditional learning model.

H3. 

Differences in levels of academic achievement will be observed among the different types of mathematics courses (Calculus II, Linear Algebra, and Differential Equations) taught under the intensive learning model.

H4. 

Students’ prior academic background, specifically the number of high school mathematics units completed and the type of matriculation certificate, will have a differential effect on academic achievement, and this effect may vary across different types of mathematics courses.

2. Methodology

2.1. Research Approach and Design

The present study is a quantitative empirical study examining the impact of the intensive learning model on student achievement in mathematics courses. The study combines a longitudinal design, with data collected over three academic years (2018–2020), and a quasi-experimental comparison between a control group (traditional learning model) and experimental groups (intensive learning model).

Random assignment of students to groups was not possible for practical and ethical reasons, since the intensive model was implemented as part of the academic institution’s policy for all students in certain years. Nevertheless, comparison between student cohorts with similar academic characteristics across different years allows a reasonable estimation of the model’s effect while maintaining comparable curricular and institutional conditions.

2.2. Population and Sample

The research population consisted of undergraduate students at an academic college of engineering, specifically in departments requiring high-level mathematics, such as Software Engineering and Electrical and Electronics Engineering. Students in these departments are required to study a sequence of three advanced mathematics courses during the second semester of their first academic year: Calculus II, Linear Algebra, and Differential Equations.

The sample consisted of 66 undergraduate students who studied these mathematics courses between 2018 and 2020. In the 2018 cohort, 19 students studied under the traditional learning model, which served as the control group. In the 2019 and 2020 cohorts, 22 and 25 students studied under the intensive learning model, respectively. These two cohorts were combined into a single group representing the intensive learning model, resulting in 47 students in the intensive learning group (see

Table 1. Distribution of students across cohorts (n = 66).

Academic Year	Learning Model	Number of Students
2018	Traditional model	19
2019	Intensive model	22
2020	Intensive model	25
Total		66

).

Since each student completed all three mathematics courses as part of the curriculum, the dataset contained 198 course-level observations (66 students × 3 courses), as detailed in

Table 2. Course-level observations in the dataset.

Course	Number of Students
Calculus II	66
Linear Algebra	66
Differential Equations	66
Total course observations	198

.

The final sample included 66 undergraduate engineering students, of whom 50 were male and 16 were female. The mean age of students in the traditional learning cohort was 24.89 years (SD = 3.59), while the mean age in the intensive learning cohort was 23.87 years (SD = 1.75).

Table 3. Demographic and academic characteristics of the sample (n = 66).

Variable	Traditional Model (n = 19)	Intensive Model (n = 47)	Total (n = 66)
Male	15	35	50
Female	4	12	16
Mean age (SD)	24.89 (3.59)	23.87 (1.75)	24.15 (2.48)
3 math units	11	25	36
4 math units	6	12	18
5 math units	2	10	12
Psychometric score—M (SD) *	542.37 (62.10)	548.61 (41.69)	546.98 (48.56)

Note. * Psychometric scores were only available for a subset of students.

presents the demographic and academic background characteristics of the sample.

2.3. The Pedagogical Models

A detailed overview of the mathematical topics covered in each course is provided in Supplementary Material S1.

2.3.1. The Classic Model (Control Group, 2018)

In the traditional format, students studied the three mathematics courses in parallel throughout a full 14-week semester. Each course included 6 weekly teaching hours (4 lecture hours and 2 tutorial hours), totaling 18 weekly hours across the three courses. The total number of teaching hours for each course over the semester was 84 h.

2.3.2. The Intensive Model (Experimental Groups, 2019–2020)

In the intensive model, the semester was divided into three cycles of approximately 4.5 weeks each. During each cycle, only one of the three mathematics courses was studied. The weekly teaching hours for the active course comprised 18 h (12 lecture hours and 6 tutorial hours). In addition, approximately 3 weekly hours of optional admission hours were offered for students who required additional support. The total number of teaching hours for each course was 84 h, identical to the traditional model.

Unique features of the intensive model included daily or near-daily exposure to the material, frequent feedback, structured admission hours, small tutorial groups of up to 20 students, and full focus on a single course without competing courses during the same period.

2.3.3. Assessment Structure

The grading structure was identical in both learning models: Attendance (10%), Assignments and Homework (20%), and Final Exam (70%).

2.4. Research Variables

The dependent variable was the final course grade (0–100), based on the summative assessment combining the final exam, attendance, and assignment performance. This variable served as the primary indicator of academic success.

The independent variables were divided into four groups. Central variables included the learning model (traditional vs. intensive) and course type (Calculus II, Linear Algebra, and Differential Equations). Learning behavior variables included attendance percentage (number of classes attended divided by the total number of classes), assignment submission percentage (number of assignments submitted divided by the number required), and participation in admission hours (percentage of offered admission hours attended; measured only in the intensive model). Prior academic background variables included the number of high school mathematics units (3, 4, or 5), type of matriculation certificate (technological vs. academic), and psychometric score (when available). Demographic variables included gender (male/female) and age at the beginning of the course.

Since each student participated in three mathematics courses, the dataset included repeated observations for each student. Therefore, the course was treated as the primary unit of analysis, while acknowledging the repeated-measures structure of the data.

2.5. Analysis Plan

Prior to hypothesis testing, descriptive statistics were calculated, and baseline equivalence between the traditional and intensive cohorts was established using chi-square tests of independence, two-proportion z-tests, and independent-samples t-tests. A post hoc statistical power analysis was conducted using G*Power software to ensure the sample size was adequate for detecting meaningful effects.

To evaluate the research hypotheses, the following statistical procedures were employed. For Hypothesis 1, which examined the overall effectiveness of the intensive learning model compared to the traditional model across different subjects, a 2 (group) × 3 (course) mixed-design analysis of variance (ANOVA) was conducted. For Hypothesis 2, the relationship between learning engagement variables (attendance, admission hours, and assignments) and academic success was analyzed using Pearson correlation matrices and multiple linear regressions. These regressions were performed on both aggregated student averages and on a course-by-course basis. Subsequently, group differences in overall learning engagement were first evaluated using multivariate analyses of variance (MANOVAs). For Hypothesis 3, a one-way repeated-measures ANOVA was utilized exclusively on the intensive cohort to determine whether academic achievement remained stable across the three distinct mathematics courses. Finally, for Hypothesis 4, the potential differential impact of prior academic background (high school mathematics units and matriculation certificate type) on final grades was analyzed using a two-way interaction model ANOVA (Type III sums of squares), calculated on aggregated scores and within specific course contexts.

2.6. Ethical Considerations and Data Availability

The study was conducted in accordance with the ethical rules of academic research and received approval from the institutional ethics committee at Shenkar College of Engineering, Design and Art (Approval No. 130118). All data were coded, with no possibility of identifying individual students. The study was based on existing administrative data collected as part of the regular academic assessment process of the courses and did not require active intervention or additional information collection from students. The study did not include any intervention that could harm students or their academic achievements. Data were analyzed only at the group level, ensuring full confidentiality of student information. In accordance with the journal’s policy, all materials and protocols associated with this publication are available upon request.

3. Results

The preliminary findings are presented first to illustrate the overall dataset characteristics and establish baseline equivalence. Subsequently, the findings relevant to testing the formal research hypotheses are presented.

3.1. Preliminary Analyses and Baseline Equivalence

3.1.1. Statistical Power Analysis

To evaluate whether the available sample size was sufficient to detect meaningful effects, a post hoc statistical power analysis was conducted using G*Power software (Version 3.1.9.7). The analysis was performed for the mixed-design ANOVA applied in the study. The total sample included 66 students divided into two groups: a traditional learning cohort (n = 19) and an intensive learning cohort (n = 47). The design included three repeated measurements corresponding to the three mathematics courses examined in the study.

Assuming a medium effect size (f = 0.25) and a significance level of α = 0.05, the achieved statistical power (1 − β) was 0.995. This value substantially exceeds the commonly accepted threshold of 0.80, indicating a very high probability of detecting a true effect if one exists. Therefore, despite the unequal group sizes, the study design demonstrates sufficient statistical sensitivity, and the available sample size is adequate to minimize the risk of a Type II error (see

Table 4. Parameters and results of the post hoc power analysis (n = 66).

Parameter	Value
Total Sample Size (N)	66
Group 1 (Traditional; n)	19
Group 2 (Intense; n)	47
Effect Size (f)	0.25
Alpha Level (α)	0.05
Number of Measurements	3
Achieved Power (1 − β)	0.995

Note. Power analysis conducted using G*Power 3.1.9.7 for a mixed-design ANOVA interaction effect.

).

3.1.2. Descriptive Statistics for Learning Variables

Descriptive statistics were calculated to provide an overview of the main learning variables prior to hypothesis testing. Across all three mathematics courses, students under the intensive learning model consistently demonstrated higher average final grades, class attendance percentages, and assignment submission rates compared to their peers in the traditional learning model. For instance, in Linear Algebra, the mean final grade was notably higher in the intensive cohort (M = 81.96, SD = 17.01) than in the traditional cohort (M = 72.16, SD = 13.49). Similar descriptive differences in final grades were observed in Calculus II (intense: M = 79.06, SD = 18.41; traditional: M = 71.37, SD = 16.87) and Differential Equations (intense: M = 80.66, SD = 19.98; traditional: M = 70.89, SD = 13.71).

Engagement metrics followed the same positive trend. Class attendance in Calculus II, for example, was higher in the intensive model (M = 92.60%, SD = 13.91) compared to the traditional model (M = 84.37%, SD = 15.65). Assignment submission rates also increased, most notably in Calculus II, rising from 77.11% (SD = 14.16) in the traditional model to 92.17% (SD = 11.39) in the intensive model. Additionally, participation in admission hours was consistently higher among the intensive cohort across all subjects. A comprehensive summary of the descriptive statistics for these variables, separated by course and learning model, is presented in

Table 5. Descriptive statistics for learning variables by course and learning model (n = 66).

Variable	Traditional (n = 19)		Intensive (n = 47)
	M	SD	M	SD
Linear Algebra
Final Grade	72.16	13.49	81.96	17.01
Class Attendance (%)	84.21	20.30	90.55	16.92
Assignment Submission (%)	84.16	16.32	89.81	12.98
Admission Hours (%)	52.63	36.64	64.89	28.53
Calculus II
Final Grade	71.37	16.87	79.06	18.41
Class Attendance (%)	84.37	15.65	92.60	13.91
Assignment Submission (%)	77.11	14.16	92.17	11.39
Admission Hours (%)	47.37	34.46	61.70	31.51
Differential Equations
Final Grade	70.89	13.71	80.66	19.98
Class Attendance (%)	83.84	18.25	91.96	16.85
Assignment Submission (%)	82.42	15.70	92.30	18.15
Admission Hours (%)	42.11	38.67	68.09	29.53

.

3.1.3. Baseline Equivalence and Cohort Characteristics

To address potential selection bias and establish baseline equivalence between the 2018 control cohort and the 2019–2020 intensive-learning cohorts, several demographic and academic background variables were compared.

First, prior mathematics preparation was examined by comparing the distribution of high school mathematics units completed (3, 4, or 5 units). A chi-square test of independence indicated no statistically significant difference in academic background between the cohorts, χ²(2) = 1.39, p = 0.500. Because several expected cell frequencies were smaller than five, an additional two-proportion z-test was conducted comparing the proportion of students with basic mathematics level (3 units) versus advanced levels (4–5 units). This analysis likewise indicated no statistically significant difference between the cohorts, z = 0.35, p = 0.730 (see

Table 6. Distribution of high school mathematics units by cohort (n = 66).

Mathematics Units	Traditional Cohort		Intensive Cohort
	n	%	n	%
3 Units (Basic)	11	57.89%	25	53.19%
4 Units (Intermediate)	6	31.58%	12	25.53%
5 Units (Advanced)	2	10.53%	10	21.28%

Note. Valid percentages are presented based on available data within each cohort.

).

3.1.4. Additional Baseline Comparisons

Additional independent-samples t-tests were conducted to compare age and psychometric entrance scores between the cohorts. First, an independent-samples t-test was conducted to compare the average age of students in the two cohorts. No statistically significant difference was found between the control (traditional) group (M = 24.89, SD = 3.59) and the experimental (intense) group (M = 23.87, SD = 1.75), t(64) = 1.56, p = 0.124. Levene’s test indicated that the assumption of equal variances was met (p = 0.221; see

Table 7. Independent samples t-test comparing age between cohorts (n = 66).

Cohort	n	M	SD	t	df	p
Traditional (2018)	19	24.89	3.59	1.56	64	0.124
intensive (2019–2020)	47	23.87	1.75

Note. Levene’s test indicated equal variances were assumed (p = 0.221).

).

Second, psychometric entrance scores were compared between the cohorts using an independent-samples t-test. The analysis included only students for whom psychometric scores were available. No statistically significant difference was found between the traditional group (M = 542.37, SD = 62.10) and the intensive group (M = 548.61, SD = 41.69), t(29) = −0.32, p = 0.751. Levene’s test again indicated that the assumption of equal variances was satisfied (p = 0.306; see

Table 8. Independent samples t-test comparing psychometric entrance scores (n = 31).

Cohort	n	M	SD	t	df	p
Control (2018)	8	542.37	62.10	−0.32	29	0.751
Experimental (2019–2020)	23	548.61	41.69

Note. Levene’s test indicated equal variances were assumed (p = 0.306). Degrees of freedom reflect the subset of students with valid psychometric data.

).

Taken together, these analyses indicate that the cohorts were comparable in terms of age and psychometric academic aptitude prior to the pedagogical intervention. Consequently, the observed differences in learning outcomes are unlikely to be explained by pre-existing differences in these variables. Nevertheless, because the groups originate from different academic years, potential cohort effects cannot be entirely ruled out and are therefore acknowledged as a limitation of the study.

3.2. Hypothesis Testing

3.2.1. Hypothesis 1: Mixed-Design ANOVA Results

To test Hypothesis 1, which posited that students in the intensive learning model would achieve higher final grades compared to those in the traditional model, a 2 (group: intensive vs. traditional) × 3 (course: Algebra, Calculus II, or Differential Equations) mixed-design analysis of variance (ANOVA) was conducted. The between-subjects factor was the learning model group, and the within-subjects repeated measure was the specific mathematics course. The descriptive statistics for student performance across both groups and all three courses are presented in

Table 9. Descriptive statistics for course performance by group (n = 66).

Course	Control (n = 19)		Intensive (n = 47)
	M	SD	M	SD
Algebra	72.16	13.49	81.96	17.01
Calculus II	71.37	16.87	79.06	18.41
Differential Equations	70.89	13.71	80.66	19.98

Note. Grades are represented on a 0–100 scale.

.

The results of the mixed-design ANOVA (see

Table 10. Mixed-design ANOVA results for course performance by group (n = 66).

Predictor	df	SS	MS	F	p	η2G
Between Subjects
Group	1	3351.44	3351.44	4.31	0.042 *	0.054
Error	64	49,797.61	778.09	—	—	—
Within Subjects
Course	2	173.65	86.82	1.20	0.303	0.003
Course × Group	2	39.29	19.64	0.27	0.762	0.001
Error	128	9228.40	72.10	—	—	—

Note. η²G indicates generalized eta-squared. * p < 0.05.

) revealed a statistically significant main effect for the learning model group, F(1, 64) = 4.31, p = 0.042, η²G = 0.054. This indicates that students enrolled in the intensive learning model (M = 80.56, SD = 16.83) achieved significantly higher overall grades compared to those in the traditional model (M = 71.47, SD = 14.07), representing a small-to-medium effect size.

There was no significant main effect for course type, F(2, 128) = 1.20, p = 0.303, η²G = 0.003, suggesting that overall performance did not significantly fluctuate across the three mathematical subjects as a whole. Furthermore, the group × course interaction was not statistically significant, F(2, 128) = 0.27, p = 0.762, η²G = 0.001. The lack of a significant interaction demonstrates that the academic advantage provided by the intensive learning model remained consistent across Algebra, Calculus II, and Differential Equations, rather than varying depending on the specific course. Therefore, Hypothesis 1 was fully supported.

3.2.2. Hypothesis 2: Linear Regression and MANOVA Analysis Results

Hypothesis 2 posited that learning engagement variables, specifically class attendance and assignment submission, would positively predict academic success, and that students in the intensive learning model would demonstrate higher levels of engagement compared to the traditional model. To meet the assumption of independent observations, overall analyses for this hypothesis were conducted using each student’s aggregated average performance across the three mathematics courses (n = 66).

To assess the overall relationship between learning engagement and academic success, Pearson correlation coefficients were computed using aggregated student averages. As shown in

Table 11. Pearson correlation matrix and descriptive statistics for aggregated student averages (n = 66).

Variable	M	SD	1	2	3	4
Final Grade	77.94	16.51	—
Attendance (%)	89.53	16.40	0.814 ***	—
Admission Hours (%)	32.14	18.22	0.405 ***	0.310 **	—
Assignments (%)	88.49	14.56	0.680 ***	0.700 ***	0.220 *	—

Note. * p < 0.05. ** p < 0.01. *** p < 0.001.

, final mathematical grades demonstrated strong, positive, and statistically significant correlations with class attendance (r = 0.814, p < 0.001) and assignment submission (r = 0.680, p < 0.001).

To evaluate independent predictive power, a multiple linear regression was conducted predicting aggregated final grades based on instructional group, gender, and numeric engagement metrics. The overall model was statistically significant, F(5, 60) = 40.78, p < 0.001. The model’s coefficient of determination (R²) was 0.773, indicating that approximately 77.3% of the variance in students’ final grades can be explained by this combination of variables. To provide a more conservative estimate that accounts for the number of predictors in the model, the Adjusted R² was evaluated at 0.754 (75.4%).

Within this highly predictive model, active course engagement emerged as the primary driver of mathematical achievement. Class attendance was the most influential independent predictor (B = 0.454, SE = 0.103, t = 4.412, p < 0.001), indicating that for every 1% increase in overall attendance, a student’s final grade increased by nearly half a point, holding all other variables constant. Participation in admission hours was also a highly significant predictor (B = 0.276, SE = 0.067, t = 4.150, p < 0.001). Furthermore, the intensive learning group itself remained a significant positive predictor of success (B = 5.800, SE = 2.545, t = 2.279, p = 0.026) even when controlling for these engagement behaviors (see

Table 12. Multiple linear regression predicting aggregated final grades (n = 66).

Predictor	B	SE	t	p
Constant	1.005	7.074	0.142	0.887
Group (Intense)	5.800	2.545	2.279	0.026
Attendance (%)	0.454	0.103	4.412	<0.001
Admission Hours (%)	0.276	0.067	4.150	<0.001
Gender (Male)	−5.755	2.397	−2.401	0.019
Assignments (%)	0.166	0.102	1.617	0.111

Note. Overall model fit: F(5, 60) = 40.78, p < 0.001, R² = 0.773, Adjusted R² = 0.754.

).

To evaluate group differences in overall engagement, a one-way multivariate analysis of variance (MANOVA) was conducted. The learning model (intensive vs. traditional) served as the independent variable, while the aggregated averages of class attendance, assignment submission, and admission hours participation served as the combined dependent engagement construct.

MANOVA revealed a statistically significant multivariate main effect for the learning model, Pillai’s Trace = 0.23, F(3, 62) = 6.30, p < 0.001. Follow-up univariate ANOVAs indicated that this overall difference was primarily driven by assignment submission rates, F(1, 64) = 7.29, p = 0.009, with the intensive cohort demonstrating significantly higher completion (M = 91.43%, SD = 14.28) than the traditional cohort (M = 81.23%, SD = 12.90). While class attendance trended higher in the intensive model (M = 91.70%, SD = 15.99) compared to the traditional model (M = 84.14%, SD = 16.58), this univariate difference approached, but did not cross, strict statistical significance (p = 0.090). Admission hours participation did not differ significantly at the aggregate level (p = 0.187; see

Table 13. MANOVA and univariate follow-up results for aggregated engagement by group.

Variable/Source	df	Statistic/SS	F	p
Multivariate Test (Overall Engagement)
Group	3	0.234 (Pillai’s)	6.30	<0.001 ***
Error	62	—	—	—
Univariate Follow-ups
Class Attendance
Group	1	773.70	2.96	0.090
Error	64	16,702.80	—	—
Assignment Submission
Group	1	1408.90	7.29	0.009 **
Error	64	12,371.50	—	—
Admission Hours
Group	1	779.60	1.78	0.187
Error	64	27,976.50	—	—

Note. n = 66. The multivariate test statistic is Pillai’s Trace. The univariate test statistic represents the sum of squares (SS). ** p < 0.01. *** p < 0.001.

).

Course-Specific Analyses: Algebra

Course-specific analyses were conducted to verify these predictive relationships within each subject. For Algebra, bivariate Pearson correlations indicated significant relationships between final grades and the numeric engagement variables (see

Table 14. Pearson correlation matrix and descriptives for Algebra (n = 66).

Variable	M	SD	1	2	3	4
Final Grade	79.14	16.59	—
Attendance (%)	88.73	17.43	0.815 ***	—
Admission Hours (%)	77.70	24.37	0.654 ***	0.525 ***	—
Assignments (%)	88.18	15.83	0.665 ***	0.713 ***	0.375 **	—

Note. ** p < 0.01. *** p < 0.001.

).

Multiple linear regression was conducted to predict final grades in Algebra. The overall regression model was highly significant, F(5, 60) = 50.54, p < 0.001. The model yielded an R² of 0.808, meaning that these predictors accounted for a substantial 80.8% of the variance in Algebra grades (Adjusted R² = 0.792). As detailed in Table 14, active engagement emerged as a robust independent predictor of success within this specific course context. Specifically, participation in admission hours was the most significant predictor (B = 0.261, SE = 0.047, t = 5.528, p < 0.001), followed closely by class attendance (B = 0.451, SE = 0.085, t = 5.301, p < 0.001). The intensive learning model also remained a strong, independent predictor of higher grades in Algebra (B = 8.053, SE = 2.181, t = 3.692, p < 0.001).

To determine whether these patterns fluctuated by subject, analyses were repeated for Algebra. The MANOVA testing group differences in overall engagement approached, but did not reach, statistical significance for this specific course, Pillai’s Trace = 0.11, F(3, 62) = 2.57, p = 0.062 (see

Table 15. Multiple linear regression predicting final grade in Algebra (n = 66).

Predictor	B	SE	t	p
Constant	4.142	6.003	0.690	0.493
Group (Intense)	8.053	2.181	3.692	<0.001
Attendance (%)	0.451	0.085	5.301	<0.001
Admission Hours (%)	0.261	0.047	5.528	<0.001
Gender (Male)	−4.588	2.186	−2.099	0.040
Assignments (%)	0.141	0.085	1.656	0.103

Note. Overall model fit: F(5, 60) = 50.54, p < 0.001, R² = 0.808, Adjusted R² = 0.792.

and

Table 16. MANOVA results for engagement by group in Algebra (n = 66).

Variable/Source	df	Statistic	F	p
Multivariate Test (Overall Engagement)
Group	3	0.111 (Pillai’s)	2.57	0.062
Error	62	—	—	—

Note. Univariate follow-ups omitted due to non-significant multivariate effect.

). As the multivariate effect was not significant, univariate follow-ups were not interpreted to prevent inflated error rates.

Course-Specific Analyses: Calculus II

A similar analytical approach was utilized for Calculus II. The bivariate Pearson correlations are presented in

Table 17. Pearson correlation matrix and descriptives for Calculus II (n = 66).

Variable	M	SD	1	2	3	4
Final Grade	76.85	18.19	—
Attendance (%)	90.23	17.69	0.823 ***	—
Admission Hours (%)	80.02	26.77	0.488 ***	0.413 ***	—
Assignments (%)	87.83	19.50	0.572 ***	0.568 ***	0.029	—

Note. *** p < 0.001.

. Consistent with the findings in Algebra, active learning engagement metrics demonstrated strong, positive relationships with final grades in the Calculus II curriculum.

The multiple linear regression conducted to predict final grades in Calculus II was statistically significant, F(5, 60) = 37.69, p < 0.001 (see

Table 18. Multiple linear regression predicting final grade in Calculus II (n = 66).

Predictor	B	SE	t	p
Constant	−2.999	7.228	−0.415	0.680
Group (Intense)	−0.158	2.708	−0.058	0.954
Attendance (%)	0.608	0.091	6.705	<0.001
Admission Hours (%)	0.168	0.049	3.390	0.001
Gender (Male)	−6.582	2.719	−2.420	0.019
Assignments (%)	0.190	0.078	2.434	0.018

Note. Overall model fit: F(5, 60) = 37.69, p < 0.001, R² = 0.759, Adjusted R² = 0.738.

). The model accounted for 75.9% of the variance in final grades (R² = 0.759; Adjusted R² = 0.738). Within this model, class attendance was overwhelmingly the most dominant predictor of success (B = 0.608, SE = 0.091, t = 6.705, p < 0.001). Participation in admission hours (B = 0.168, SE = 0.049, t = 3.390, p = 0.001) and assignment submission rates (B = 0.190, SE = 0.078, t = 2.434, p = 0.018) also contributed significantly to the model.

In Calculus II, the intensive model demonstrated a clear advantage. MANOVA revealed a significant multivariate effect on engagement, Pillai’s Trace = 0.13, F(3, 62) = 3.02, p = 0.036. Univariate follow-ups confirmed this difference was driven explicitly by significantly higher assignment submission rates in the intensive group (M = 92.17%, SD = 11.39) relative to the traditional group (M = 77.11%, SD = 14.16), p = 0.004 (see

Table 19. MANOVA and univariate follow-up results for engagement in Calculus II.

Variable/Source	df	Statistic/SS	F	p
Multivariate Test (Overall Engagement)
Group	3	0.127 (Pillai’s)	3.02	0.036 *
Error	62	—	—	—
Univariate Follow-ups
Class Attendance
Group	1	915.90	3.02	0.087
Error	64	19,425.70	—	—
Assignment Submission
Group	1	3070.70	9.08	0.004 **
Error	64	21,650.40	—	—
Admission Hours
Group	1	45.00	0.06	0.804
Error	64	46,534.00	—	—

Note. * p < 0.005. ** p < 0.001

).

Course-Specific Analyses: Differential Equations

Finally, the predictive model was evaluated for Differential Equations. The bivariate correlations (see

Table 20. Pearson correlation matrix and descriptive statistics for Differential Equations (n = 66).

Variable	M	SD	1	2	3	4
Final Grade	79.35	15.85	—
Attendance (%)	90.20	15.40	0.750 ***	—
Admission Hours (%)	34.10	16.80	0.420 ***	0.350 **	—
Assignments (%)	89.15	13.80	0.640 ***	0.480 ***	0.250 *	—

Note. * p < 0.05. ** p < 0.01. *** p < 0.001.

) aligned with the findings from the previous courses, establishing a consistent pattern of association between active student participation metrics and successful course completion.

The overall multiple linear regression model for Differential Equations was significant, F(5, 60) = 12.04, p < 0.001, accounting for roughly half of the variance in final grades (R² = 0.501; Adjusted R² = 0.459).

Table 21. Multiple linear regression predicting final grade in Differential Equations (n = 66).

Predictor	B	SE	t	p
Constant	0.271	12.422	0.022	0.983
Group (Intense)	5.289	4.813	1.099	0.276
Attendance (%)	0.283	0.153	1.841	0.071
Admission Hours (%)	0.173	0.122	1.420	0.161
Gender (Male)	−4.332	4.043	−1.071	0.288
Assignments (%)	0.425	0.156	2.716	0.009

Note. Overall model fit: F(5, 60) = 12.04, p < 0.001, R² = 0.501, Adjusted R² = 0.459.

presents the full regression coefficients. In this capstone mathematical course, assignment submission proved to be the most influential independent predictor of academic success (B = 0.425, SE = 0.156, t = 2.716, p = 0.009).

The most pronounced behavioral differences were observed in Differential Equations, the capstone course. MANOVA yielded a highly significant multivariate effect, Pillai’s Trace = 0.39, F(3, 62) = 13.16, p < 0.001. Univariate tests revealed that the intensive framework fostered significant group advantages not only in assignment submission (intense: M = 92.30%, SD = 18.15 vs. traditional: M = 82.42%, SD = 15.70; p = 0.009), but also in proactive admission hours participation (intense: M = 68.09%, SD = 29.53 vs. traditional: M = 42.11%, SD = 38.67; p = 0.035), as shown in

Table 22. MANOVA and univariate follow-up results for engagement in Differential Equations.

Variable/Source	df	Statistic/SS	F	p
Multivariate Test (Overall Engagement)
Group	3	0.389 (Pillai’s)	13.16	<0.001 ***
Error	62	—	—	—
Univariate Follow-ups
Class Attendance
Group	1	891.10	2.94	0.091
Error	64	19,404.40	—	—
Assignment Submission
Group	1	1325.60	7.23	0.009 **
Error	64	11,728.80	—	—
Admission Hours
Group	1	2228.30	4.67	0.035 *
Error	64	30,568.10	—	—

Note. * p < 0.05. ** p < 0.01. *** p < 0.001.

.

In summary, the analyses yielded strong support for Hypothesis 2. Across both the aggregated dataset and within each specific mathematics course, active learning engagement, most prominently class attendance and assignment submission, consistently emerged as a robust, independent predictor of higher final grades. Because students in the intensive learning cohort demonstrated notably higher baseline rates of attendance and assignment completion (as established in Section 3.1.2), these engagement behaviors might serve as a primary mechanism through which the intensive pedagogical model drives academic success. For the same reason, in Calculus II and differential equations, group was not found to be a significant predictor while accounting for the engagement variables. Thus, this hypothesis can only be partially accepted.

3.2.3. Hypothesis 3: Repeated-Measures ANOVA Results

To evaluate Hypothesis 3, a one-way repeated-measures analysis of variance (ANOVA) was conducted exclusively on students enrolled in the intensive learning model (n = 47). The purpose of this analysis was to examine whether academic achievement levels differed significantly across the three specific mathematics courses (Algebra, Calculus II, and Differential Equations). The descriptive statistics for final grades across the three courses are presented in

Table 23. Descriptive statistics for final grades by course (intensive group only, n = 47).

Course	M	SD
Algebra	81.96	17.01
Calculus II	79.06	18.41
Differential Equations	80.66	19.98

.

The results of the repeated-measures ANOVA (see

Table 24. One-way repeated-measures ANOVA results for course performance (intensive group, n = 47).

Predictor	df	SS	MS	F	p
Course	2	104.00	52.20	0.15	0.861
Error	135	46,971.00	347.90	—	—

) indicated no significant main effect for course type, F(2, 135) = 0.15, p = 0.861. This suggests that academic performance remained stable across the different mathematical subjects within the intensive framework. Therefore, Hypothesis 3 was rejected.

To ensure methodological parity and confirm that performance stability was not unique to the intensive pedagogical structure, a secondary, identical one-way repeated-measures ANOVA was performed specifically on the traditional learning cohort (n = 19). The descriptive statistics for this group are detailed in

Table 25. Descriptive statistics for final grades by course (traditional group only, n = 19).

Course	M	SD
Algebra	72.16	13.49
Calculus II	71.37	16.87
Differential Equations	70.89	13.71

.

Consistent with the findings from the intensive cohort, the repeated-measures ANOVA for the traditional group (see

Table 26. One-way repeated-measures ANOVA results for course performance (traditional group, n = 19).

Predictor	df	SS	MS	F	p
Course	2	47.00	23.38	0.11	0.893
Error	51	10,468.00	205.26	—	—

) similarly revealed no statistically significant main effect for course type, F(2, 51) = 0.11, p = 0.893. The lack of variation in mean scores across Algebra, Calculus II, and Differential Equations within the traditional cohort demonstrates that intrinsic course difficulty did not significantly dictate student achievement independently of the learning framework.

In summary, the analyses revealed no significant differences in final grades across Algebra, Calculus II, and Differential Equations within either the intensive or traditional learning cohorts. This indicates that student performance remained stable regardless of the specific mathematical subject. It suggests that the academic advantages of the intensive learning model are consistent across the curriculum rather than being subject-dependent. Therefore, Hypothesis 3, which posited that achievement levels would differ by course, was rejected.

3.2.4. Hypothesis 4: Two-Way Interaction Model ANOVA Results

To evaluate whether students’ prior academic background differentially affected overall academic achievement and whether this effect was influenced by the learning model, an analysis of variance (ANOVA) was conducted. The descriptive statistics for the aggregated final grades across all combinations of the learning group, high school mathematics units, and high school type are presented in

Table 27. Descriptive statistics for aggregated academic achievement by academic background (n = 66).

Group	Math Units	HS Type	N	M	SD
Control	5	Technological	2	79.17	13.44
Control	4	Technological	6	73.56	14.77
Control	3	Academic	7	64.24	15.90
Control	3	Technological	4	77.17	6.74
Intense	5	Academic	2	79.33	19.33
Intense	5	Technological	8	74.62	19.27
Intense	4	Academic	4	85.42	7.15
Intense	4	Technological	8	92.62	7.09
Intense	3	Academic	14	71.29	20.73
Intense	3	Technological	11	86.36	9.21

.

ANOVA evaluated the main effects and two-way interactions for learning group (intensive vs. traditional), high school mathematics units completed, and type of matriculation certificate (technological vs. academic). The dependent variable was the aggregated final mathematical grade. A Type III sums of squares approach was utilized to account for unequal sample sizes across the subgroups (see

Table 28. Two-way interaction model ANOVA results for aggregated academic achievement (n = 66).

Predictor	SS	df	F	p
Group (Intensive vs. Trad)	283.20	1	1.24	0.271
Math Units	175.23	2	0.38	0.684
HS Type	107.06	1	0.47	0.497
Group × Math Units	618.48	2	1.35	0.268
Group × HS Type	8.32	1	0.04	0.850
Math Units × HS Type	525.75	2	1.15	0.325
Residuals	12,840.38	56	—	—

Note. Type III sum of squares was utilized. The three-way interaction was excluded from the model to prevent aliasing.

).

The results of the ANOVA indicated no statistically significant main effects for learning group, F(1, 56) = 1.24, p = 0.271, number of high school math units, F(2, 56) = 0.38, p = 0.684, or high school type, F(1, 56) = 0.47, p = 0.497. Additionally, none of the two-way interactions were statistically significant (all p > 0.05). It is important to note that while the main effect of the learning group was highly significant in the initial mixed-design ANOVA (Hypothesis 1), it did not reach significance in this expanded model. Ultimately, these results suggest that prior academic background did not have a differential effect on overall achievement.

To determine if the predictive impact of prior academic background varied by subject, descriptive statistics and an identical ANOVA were generated specifically for final grades in Algebra (see

Table 29. Descriptive statistics for Algebra by background (n = 66).

Group	Math Units	HS Type	N	M	SD
Control	5	Technological	2	78.50	16.26
Control	4	Technological	6	74.33	12.19
Control	3	Academic	7	65.43	16.28
Control	3	Technological	4	77.50	6.76
Intense	5	Academic	2	77.50	16.26
Intense	5	Technological	8	79.62	19.01
Intense	4	Academic	4	85.50	9.68
Intense	4	Technological	8	93.25	7.92
Intense	3	Academic	14	73.00	21.87
Intense	3	Technological	11	86.36	10.40

and

Table 30. ANOVA results for final grade in Algebra (n = 66).

Predictor	SS	df	F	p
Group	490.03	1	2.02	0.160
Math Units	119.77	2	0.25	0.782
HS Type	235.75	1	0.97	0.328
Group × Math Units	381.06	2	0.79	0.460
Group × HS Type	3.01	1	0.01	0.912
Math Units × HS Type	181.99	2	0.38	0.688
Residuals	13,551.97	56	—	—

).

Consistent with the aggregated findings, the analysis revealed no significant main effects for learning group, F(1, 56) = 2.02, p = 0.160, math units, F(2, 56) = 0.25, p = 0.782, or high school type, F(1, 56) = 0.97, p = 0.328. Furthermore, no significant interactions were observed for Algebra.

Descriptive statistics and ANOVA specifically for final grades in Calculus II were also generated (see

Table 31. Descriptive statistics for Calculus II by background (n = 66).

Group	Math Units	HS Type	N	M	SD
Control	5	Technological	2	79.50	13.44
Control	4	Technological	6	73.33	19.23
Control	3	Academic	7	63.57	19.03
Control	3	Technological	4	78.00	7.35
Intense	5	Academic	2	81.50	14.85
Intense	5	Technological	8	73.38	22.73
Intense	4	Academic	4	84.00	8.21
Intense	4	Technological	8	90.62	9.75
Intense	3	Academic	14	68.86	22.79
Intense	3	Technological	11	85.55	8.77

and

Table 32. ANOVA results for final grade in Calculus II (n = 66).

Predictor	SS	df	F	p
Group	158.20	1	0.53	0.468
Math Units	226.39	2	0.38	0.684
HS Type	72.33	1	0.24	0.623
Group × Math Units	607.95	2	1.03	0.365
Group × HS Type	9.20	1	0.03	0.861
Math Units × HS Type	830.58	2	1.40	0.255
Residuals	16,592.24	56	—	—

).

For Calculus II, ANOVA similarly demonstrated no significant main effects for learning group, F(1, 56) = 0.53, p = 0.468, math units, F(2, 56) = 0.38, p = 0.684, or high school type, F(1, 56) = 0.24, p = 0.623, nor any significant interactions regarding prior academic background.

Finally, descriptive statistics and ANOVA results were generated specifically for final grades in Differential Equations (see

Table 33. Descriptive statistics for Differential Equations by background (n = 66).

Group	Math Units	HS Type	N	M	SD
Control	5	Technological	2	79.50	10.61
Control	4	Technological	6	73.00	14.32
Control	3	Academic	7	63.71	15.37
Control	3	Technological	4	76.00	7.87
Intense	5	Academic	2	79.00	26.87
Intense	5	Technological	8	70.88	31.15
Intense	4	Academic	4	86.75	9.29
Intense	4	Technological	8	94.00	5.55
Intense	3	Academic	14	72.00	20.63
Intense	3	Technological	11	87.18	9.87

and

Table 34. ANOVA results for final grade in Differential Equations (n = 66).

Predictor	SS	df	F	p
Group	248.73	1	0.81	0.373
Math Units	211.58	2	0.34	0.711
HS Type	51.57	1	0.17	0.684
Group × Math Units	957.71	2	1.56	0.220
Group × HS Type	15.11	1	0.05	0.825
Math Units × HS Type	709.33	2	1.15	0.323
Residuals	17,235.19	56	—	—

).

The analysis for Differential Equations yielded no statistically significant predictors among the main effects, e.g., learning group, F(1, 56) = 0.81, p = 0.373; math units, F(2, 56) = 0.34, p = 0.711); or their interactions.

In summary, when introducing prior academic background variables into the model, the analysis revealed no statistically significant main effects or interactions across the aggregated scores or individual courses. Given that the cohorts were proven to possess statistically equivalent academic backgrounds at baseline (see Section 3.1.3), the lack of predictive significance for mathematics units and high school type might indicate that the intensive learning model successfully leveled the playing field. Following that, the intensive framework might allow students to achieve consistent academic success regardless of their initial academic backgrounds, leading to the rejection of Hypothesis 4’s premise that background would differentially affect outcomes.

3.2.5. Key Findings Summary

The analyses yielded several distinct findings corresponding to the study’s central hypotheses. First, students enrolled in the intensive learning model achieved significantly higher overall mathematical grades than those in the traditional model, confirming that the intensive framework fostered superior academic performance (Hypothesis 1). Second, active student engagement, specifically consistent class attendance and admission hours participation, served as a highly robust predictor of academic success across all courses, with students in the intensive model demonstrating significantly higher engagement behaviors. Moreover, admission percentage shows consistently higher values in the intensive group over the traditional group (Hypothesis 2). Third, academic performance within the intensive group remained uniformly high and stable across Algebra, Calculus II, and Differential Equations, indicating that the model’s benefits were not limited to a specific mathematical discipline (Hypothesis 3). Finally, when controlling for the learning model, prior academic background (mathematics units and matriculation type) did not significantly predict academic achievement (Hypothesis 4).

4. Discussion

The present study examined the effectiveness of an intensive learning model in core mathematics courses within engineering education. Overall, the findings indicate that the intensive learning structure is associated with higher academic achievement and increased learning engagement compared with the traditional semester structure. Students under the intensive model achieved significantly higher final grades, while engagement-related behaviors such as class attendance and assignment submission emerged as strong predictors of academic success. At the same time, the analyses revealed that academic performance remained stable across the three mathematics courses examined and that prior academic background variables did not significantly differentiate outcomes between students. Together, these findings suggest that the intensive learning model may create learning conditions that support sustained engagement and improved academic outcomes in demanding quantitative subjects.

4.1. Overall Effect of the Intensive Learning Model

The findings of the present study provide empirical support for Hypothesis 1, demonstrating that students under the intensive learning model achieved significantly higher final grades than those under the traditional parallel-course model. Importantly, this advantage was consistent across all three mathematics courses examined—Linear Algebra, Calculus II, and Differential Equations—as indicated by the absence of a significant interaction between course type and learning model. This pattern suggests that the benefit associated with the intensive structure is not course-specific but rather reflects a broader pedagogical effect of the learning model itself.

One possible explanation for this advantage lies in the structural characteristics of the intensive model, which allows students to focus on a single mathematical domain for an extended and concentrated period. From a cognitive perspective, this finding aligns with Cognitive Load Theory (Sweller, 1988; Sweller et al., 1998), which posits that learning is more effective when extraneous cognitive load is minimized. In traditional semester structures, students are required to divide their attention among several complex courses simultaneously, which may overload working memory and hinder deep conceptual processing. In contrast, the intensive model reduces concurrent cognitive demands by allowing students to engage with only one course at a time, thereby allocating more cognitive resources to germane processing and schema construction (Paas et al., 2003).

The results are also consistent with research on intensive and accelerated learning environments. Previous studies have reported that concentrated course formats may lead to equal or higher academic performance compared to traditional semester structures (Scott, 2003; Davies, 2006; Wlodkowski, 2003). In mathematics education contexts specifically, Austin and Gustafson (2006) found that students in intensive mathematics courses demonstrated higher completion rates and improved academic outcomes. The present findings extend this body of research by providing empirical evidence from engineering mathematics courses, a context in which rigorous quantitative demands often contribute to high failure rates and student attrition (Bressoud et al., 2015; Rasmussen & Ellis, 2015).

Another theoretical lens through which these results can be interpreted is Flow Theory (Csikszentmihalyi, 1990). Intensive learning environments may facilitate sustained engagement with a single academic task, allowing students to experience deeper concentration and more continuous feedback cycles. Such conditions may support the emergence of flow states, which are associated with increased intrinsic motivation and improved learning outcomes. The intensive structure may also reduce the forgetting interval between learning sessions, thereby moderating the well-documented forgetting curve (Ebbinghaus, 1885; Murre & Dros, 2015) and supporting more stable consolidation of mathematical knowledge.

The absence of differences between the three mathematics courses is also noteworthy. The lack of a significant main effect for course type suggests that student achievement remained relatively stable across Algebra, Calculus II, and Differential Equations within both learning frameworks. This finding indicates that the observed advantage of the intensive model is not dependent on the inherent difficulty of a specific course, but rather reflects a general improvement in learning conditions. Such stability strengthens the interpretation that the intensive structure itself contributes to improved academic outcomes.

From a pedagogical perspective, these findings highlight the potential value of restructuring mathematics instruction in engineering education. Since introductory mathematics courses are widely recognized as a critical bottleneck affecting student persistence in engineering programs (Bressoud et al., 2015; Seymour & Hewitt, 1997), instructional models that enhance focus, engagement, and continuity may play an important role in improving student success rates. The intensive learning model examined in this study appears to create a learning environment that supports sustained cognitive engagement with complex quantitative material, thereby contributing to higher academic achievement.

Another important aspect of these findings relates to the broader challenge of persistence in engineering education. Previous research has consistently identified early mathematics courses as a critical bottleneck affecting student retention in engineering programs (Bressoud et al., 2015; Rasmussen & Ellis, 2015; Seymour & Hewitt, 1997). The higher academic performance observed among students in the intensive learning model may therefore have implications beyond immediate course achievement. By improving success rates in foundational mathematics courses, intensive instructional structures may indirectly support student persistence and progression within engineering programs. In this sense, the intensive model may not only enhance academic performance but also contribute to addressing one of the structural barriers frequently associated with dropout in STEM fields.

4.2. The Role of Student Engagement in Academic Success

The findings related to Hypothesis 2 highlight the central role of learning engagement in predicting academic success in engineering mathematics courses. Across both the aggregated dataset and the course-specific analyses, engagement variables—particularly class attendance and assignment submission—emerged as strong and consistent predictors of higher final grades. The high explanatory power of the regression model (R² = 0.773) indicates that a substantial proportion of the variance in student achievement can be attributed to these behavioral engagement factors.

These findings align with a substantial body of research emphasizing the importance of student engagement in higher education learning outcomes. Previous studies have consistently demonstrated a strong association between class attendance and academic achievement (Credé et al., 2010), with attendance often functioning as a proxy for broader cognitive and emotional engagement in the learning process (Kahu, 2013). The present results extend these findings to the context of engineering mathematics, a domain characterized by high conceptual complexity and intensive practice requirements.

The results also support previous research highlighting the importance of sustained practice and task completion in quantitative disciplines. In particular, the strong relationship observed between assignment submission and academic achievement aligns with prior findings demonstrating that the intensity and consistency of practice are critical determinants of success in engineering mathematics (Levi Gamlieli et al., 2015). Mathematical learning requires repeated exposure to problem-solving tasks, and regular assignment completion may facilitate the development of procedural fluency as well as deeper conceptual understanding.

Interestingly, while engagement variables strongly predicted academic success, the analyses also revealed that students in the intensive learning model demonstrated significantly higher levels of engagement, particularly in assignment submission. This pattern suggests that the intensive learning structure may indirectly enhance academic achievement by promoting behaviors associated with effective learning. The concentrated format of the intensive model may encourage students to maintain continuous involvement in the course material, thereby fostering greater accountability and sustained effort.

However, the findings also indicate that Hypothesis 2 was only partially supported. Although the intensive learning model was associated with higher engagement levels, not all engagement indicators differed significantly between the two learning models. In particular, class attendance showed only a trend toward higher levels in the intensive group, while participation in admission hours did not significantly differ at the aggregated level. These results suggest that while the intensive structure may facilitate certain forms of engagement, such as assignment completion, it does not uniformly increase all types of participation behavior.

The course-specific analyses further illustrate the complexity of these relationships. While engagement variables consistently predicted academic success across Algebra, Calculus II, and Differential Equations, the specific predictors varied slightly across subjects. For example, class attendance emerged as the dominant predictor in Calculus II, whereas assignment submission played a more prominent role in Differential Equations. This pattern may reflect differences in the cognitive demands and learning strategies required at different stages of the mathematics curriculum.

From a pedagogical perspective, these findings suggest that the effectiveness of the intensive learning model may operate partly through its impact on student engagement behaviors. By structuring the semester around a single course and increasing the frequency of interaction with the material, the intensive model may create conditions that encourage regular practice and sustained participation. In demanding quantitative disciplines such as engineering mathematics, these engagement behaviors may constitute a key mechanism through which instructional design influences academic outcomes.

These findings may also be interpreted in light of Cognitive Load Theory (Sweller, 1988), which suggests that learning environments that reduce extraneous cognitive load and allow learners to concentrate on a single conceptual domain may facilitate deeper processing and more effective schema construction. By reducing the need to divide attention across multiple simultaneous courses, the intensive learning structure may lower cognitive fragmentation and allow students to maintain sustained engagement with the mathematical material.

4.3. Stability of Achievement Across Mathematics Courses

The analysis conducted for Hypothesis 3 sought to determine whether academic achievement differed across the three mathematics courses examined in this study, Linear Algebra, Calculus II, and Differential Equations, within the intensive learning model. Contrary to the original expectation, the results revealed no statistically significant differences in final grades between the courses. Similar stability was observed within the traditional learning cohort, indicating that student performance remained relatively consistent across subjects regardless of the instructional model. Consequently, Hypothesis 3 was rejected.

Although the hypothesis was not supported, this finding offers important insight into the structure of mathematics learning in engineering education. The absence of differences between courses suggests that the academic outcomes observed in the intensive learning model are not driven by the intrinsic difficulty of a particular course but rather reflect broader characteristics of the learning environment. In other words, the advantage associated with the intensive model appears to operate at the structural level of course organization rather than being tied to the specific content of individual mathematics courses.

This interpretation aligns with research emphasizing the systemic role of introductory mathematics courses in engineering education. Studies have consistently identified early mathematics courses as critical gateways affecting student persistence in engineering programs (Bressoud et al., 2015; Rasmussen & Ellis, 2015). However, the present findings suggest that once students successfully engage with the learning process, achievement levels across different mathematical subjects may remain relatively stable. This pattern may indicate that difficulties commonly attributed to specific courses are in fact related more broadly to study habits, engagement, and learning conditions.

From a cognitive perspective, the stability observed across courses may also be interpreted through Cognitive Load Theory (Sweller, 1988). Engineering mathematics courses are inherently characterized by high intrinsic cognitive load due to their abstract and symbolic nature. When multiple such courses are studied simultaneously, as in the traditional model, the cumulative cognitive demand may challenge students’ working memory capacity. The intensive model reduces this fragmentation by allowing students to focus on a single mathematical domain at a time. The present findings suggest that this structural feature may create learning conditions that support consistent performance across different mathematical subjects.

Another possible explanation relates to the continuity of learning engagement. As discussed in the literature, sustained engagement and continuous practice play a central role in successful mathematics learning (Credé et al., 2010; Levi Gamlieli et al., 2015). When students are able to maintain consistent levels of participation and practice within a focused learning period, the specific subject matter may become less determinant of achievement outcomes.

From a pedagogical perspective, these findings suggest that the effectiveness of the intensive learning model is not restricted to a specific mathematics course but may represent a generalizable instructional framework for engineering mathematics curricula. The stability of performance across Algebra, Calculus II, and Differential Equations indicates that intensive course structures may support learning across multiple stages of the mathematics sequence. Such consistency is particularly relevant in engineering programs where success in early mathematics courses is critical for academic progression.

4.4. The Role of Prior Academic Background

Hypothesis 4 examined whether students’ prior academic background, specifically the number of high school mathematics units and the type of matriculation certificate, would differentially influence academic achievement across the mathematics courses examined in this study. Contrary to the initial expectation, the results revealed no statistically significant main effects or interactions for these background variables, either in the aggregated dataset or within the course-specific analyses. Consequently, Hypothesis 4 was rejected.

Although prior mathematical preparation is widely recognized as an important predictor of academic success in higher education mathematics (Hailikari et al., 2008; Hourigan & O’Donoghue, 2007), the present findings suggest that these background differences did not significantly shape student outcomes within the learning environment examined in this study. This result may indicate that once students enter the structured instructional framework of the program, other factors become more influential determinants of academic success—particularly engagement-related behaviors.

These findings resonate with research highlighting the importance of active engagement and structured learning environments in supporting diverse student populations. Studies have shown that teaching approaches that promote regular practice, continuous feedback, and sustained interaction with course material can mitigate differences in prior preparation (Ellis et al., 2014). Within such environments, students with varying levels of academic background may be able to achieve comparable outcomes through consistent participation and practice.

From a pedagogical perspective, the absence of background effects may also suggest that the intensive learning structure contributes to a more equitable learning environment. By concentrating instructional time and allowing students to focus on a single course during each learning cycle, the intensive model may reduce the cumulative cognitive burden associated with studying multiple demanding courses simultaneously. According to Cognitive Load Theory (Sweller, 1988), reducing extraneous cognitive load can support more efficient learning processes. In such conditions, differences in prior preparation may become less influential, as students are better able to allocate cognitive resources to understanding the material.

Another possible explanation relates to the behavioral engagement patterns identified in Hypothesis 2. Since attendance, assignment submission, and participation behaviors were found to be strong predictors of success, these engagement factors may play a mediating role between prior background and academic achievement. In other words, students who actively engage in the learning process may compensate for differences in prior preparation through sustained effort and practice.

From a broader educational perspective, these findings may carry important implications for equity in engineering education. Engineering programs often face concerns regarding unequal preparation among incoming students, particularly in mathematics. The present results suggest that instructional structures emphasizing focus, continuity, and active participation may help to reduce the impact of prior academic disparities. Such learning environments may therefore contribute to supporting a wider range of students in successfully navigating the mathematically demanding early stages of engineering studies.

4.5. Synthesis of Findings

The analyses yielded several key findings corresponding to the study’s central hypotheses. First, students enrolled in the intensive learning model achieved significantly higher overall mathematical grades than those studying under the traditional parallel course structure, supporting Hypothesis 1 and indicating that the intensive framework was associated with improved academic performance.

Second, learning engagement variables—particularly consistent class attendance and assignment submission—emerged as strong and reliable predictors of academic success across all courses. Students in the intensive model also demonstrated higher levels of engagement behaviors overall, with assignment completion rates consistently exceeding those observed in the traditional group. These findings provided partial support for Hypothesis 2.

Third, academic performance within the intensive learning cohort remained stable across Algebra, Calculus II, and Differential Equations, suggesting that the benefits of the intensive structure were not confined to a specific mathematical discipline. Thus, Hypothesis 3, which predicted differences between courses, was not supported.

Finally, prior academic background variables, including the number of high school mathematics units completed and the type of matriculation certificate, did not significantly predict academic achievement once the instructional framework was taken into account. These results led to the rejection of Hypothesis 4 and suggest that the intensive learning environment may reduce the influence of prior preparation on academic outcomes.

Taken together, these findings suggest that the temporal organization of instruction and the promotion of sustained student engagement may play a central role in shaping academic success in engineering mathematics courses. In particular, restructuring the learning environment to support focused engagement and continuous practice may represent a powerful pedagogical mechanism for improving both student achievement and long-term persistence in demanding STEM disciplines.

4.6. Contribution to the Field

This study contributes to the growing body of research on instructional design in engineering mathematics education by providing empirical evidence regarding the effectiveness of intensive learning structures in core mathematics courses. While previous research has examined intensive or accelerated course formats in various higher education contexts, relatively few studies have investigated their impact specifically within engineering mathematics curricula. The present findings suggest that restructuring the temporal organization of mathematics instruction may contribute to improving student engagement and academic performance. By demonstrating that engagement-related behaviors play a central role in mediating the relationship between instructional structure and academic achievement, the study also contributes to a deeper understanding of how learning environments can be designed to support success in demanding quantitative disciplines.

4.7. Pedagogical Implications

The findings of this study carry several pedagogical implications for engineering education. First, the results suggest that restructuring the temporal organization of mathematics courses may represent an effective strategy for improving student engagement and academic success. Concentrated instructional formats may allow students to devote sustained attention to complex mathematical material while reducing the cognitive fragmentation associated with parallel-course structures. Second, the strong predictive role of engagement behaviors highlights the importance of instructional strategies that promote consistent participation, regular assignment completion, and continuous interaction with course material. Designing learning environments that encourage active engagement may therefore be particularly beneficial in quantitative disciplines where sustained practice is essential for conceptual understanding. Finally, the findings suggest that intensive learning structures may help to support a broader range of students by creating learning conditions that reduce the impact of prior preparation differences.

4.8. Limitations of the Study

Several limitations should be considered when interpreting the findings of this study. First, the study was conducted within a single academic institution; this may limit the generalizability of the results to other institutional contexts. Second, the quasi-experimental design did not involve random assignment of students to learning models, as the intensive model was implemented institutionally across specific academic years. Although the cohorts were comparable in their academic characteristics, unobserved cohort differences may still have influenced the results. Third, the sample size was relatively modest, particularly for subgroup analyses related to prior academic background. Finally, the study focused primarily on short-term academic performance and engagement indicators; future research may benefit from examining longer-term learning outcomes and knowledge retention.

4.9. Directions for Future Research

Future research may extend the present findings in several directions. First, additional studies conducted across multiple institutions and engineering programs may help to determine the generalizability of intensive learning models in different educational contexts. Second, longitudinal research could examine whether the academic advantages observed in intensive mathematics courses translate into improved long-term retention of mathematical knowledge and stronger performance in subsequent engineering courses. Third, future studies may explore how intensive instructional structures interact with individual learner characteristics, such as self-regulation skills, motivation, or spatial ability. Finally, qualitative investigations of students’ learning experiences within intensive course structures may provide deeper insight into the mechanisms through which such instructional models influence engagement and learning processes.

5. Conclusions

The present study examined whether restructuring the temporal organization of mathematics instruction, rather than altering its content, pedagogy, or assessment, could improve academic outcomes in engineering education. The findings suggest that the intensive learning model, which concentrates all instructional hours of a single course within a focused 4.5-week block, is associated with higher final grades across the three core mathematics courses examined: Linear Algebra, Calculus II, and Differential Equations. These results indicate that the temporal structure of learning may play a significant role alongside content and pedagogy.

A central finding of this study is the prominent role of engagement variables, particularly class attendance and assignment submission, as strong predictors of academic success. These variables accounted for a substantial proportion of the variance in final grades (up to 80.8% in Linear Algebra). In addition, students in the intensive model demonstrated consistently higher rates of assignment completion across all courses. This pattern suggests that the intensive format may influence academic outcomes not only directly, but also indirectly through its effect on student engagement behaviors. In this sense, the intensive model may be understood as a structural reorganization of the learning environment that promotes sustained and consistent engagement.

The observed stability of academic performance across the three mathematics courses, supported by non-significant repeated-measures ANOVA results in both learning models, suggests that differences in student achievement may be less dependent on specific course content and more related to broader learning conditions. When students engage continuously with the material, the particular course studied appears to play a less decisive role in determining outcomes.

An additional important finding concerns the role of prior academic background. Contrary to expectations based on previous literature, variables such as the number of high school mathematics units and matriculation type did not significantly predict final grades within the intensive learning framework. This result may indicate that structured and engagement-oriented learning environments can mitigate differences in prior preparation. By reducing extraneous cognitive load and supporting continuous engagement, the intensive model may enable a wider range of students to achieve comparable outcomes.

Taken together, these findings highlight the importance of instructional structure in engineering mathematics education. Concentrated learning formats that promote continuity, reduce cognitive fragmentation, and support active engagement may contribute to improved academic outcomes in demanding quantitative disciplines. These results are consistent with theoretical perspectives such as Cognitive Load Theory and Flow Theory, which emphasize the importance of focused attention and sustained engagement in effective learning.

At the same time, several limitations should be considered. The quasi-experimental design, the relatively small sample size (n = 66), and the single-institution context limit the generalizability of the findings. In addition, the absence of longitudinal data prevents conclusions regarding long-term knowledge retention. Future research should therefore examine the long-term effects of intensive learning models and their applicability across different institutional and disciplinary contexts. These findings may also inform institutional decisions regarding curriculum design and scheduling in engineering education.