Bidirectional Teaching Reform in Theoretical Mechanics: Integrating Engineering Thinking and Personalized Assignments

Abstract

Traditional theoretical mechanics courses often emphasize the rote learning of principles over practical applications. This focus can diminish student engagement and leave graduates ill prepared for applying concepts to real engineering problems. To address these challenges, this study introduces a bidirectional teaching reform that integrates a front-end focus on cultivating engineering thinking with a back-end focus on personalized assignment design. In the front-end reform, active learning methods, including case-based and project-based learning (PBL) within a structured BOPPPS lesson framework, are used to connect theoretical content with real-world engineering scenarios, thereby strengthening problem-solving skills and engagement among students. The back-end reform introduces personalized and collaborative assignments tailored to the interests and abilities of students, such as individualized problem sets, programming-based exercises, and team projects that encourage innovation and a deeper exploration of mechanics concepts. By addressing both in-class instruction and post-class work, these two reforms complement each other, providing a cohesive learning experience from initial concept acquisition to practical application. Implemented together in a second-year undergraduate mechanics course, this integrated approach was observed to increase student motivation, improve students’ ability to apply theory in practice, and enhance overall teaching effectiveness while fostering stronger collaborative skills. This bidirectional reform provides an effective model for modernizing theoretical mechanics education and prepares students to meet contemporary engineering needs by bridging the longstanding gap between theoretical knowledge and practical application.

1. Introduction

Theoretical mechanics is a foundational course for engineering undergraduates that is typically taught in their second year of university. It serves as a gateway to the study of mechanics, forming the theoretical basis for subsequent specialized courses and bridging the divide between fundamental science and applied engineering disciplines. Rooted in classical Newtonian mechanics, the course primarily addresses macroscopic phenomena at low velocities, where relativistic effects are negligible (Zhi, 2021). The curriculum is traditionally structured around three core areas: statics, kinematics, and dynamics. Statics is concerned with the equilibrium of forces, disregarding motion; kinematics describes motion without reference to the forces that cause it; and dynamics integrates force and motion, encompassing key theorems such as the work–energy principle, momentum theorem, and angular momentum theorem, and each is derived from Newton’s second law. In addition to these classical components, the course also introduces selected topics in analytical mechanics, including D’Alembert’s principle and the principle of virtual work. Through this integrated content, the course aims to cultivate systematic reasoning, logical analysis, and interdisciplinary problem-solving abilities.

The importance of practical application in mechanics education was emphasized by Qian Xuesen, a renowned scientist and student of Theodore von Kármán. Together, they made seminal contributions to supersonic flow theory and compressible boundary layers, most notably the Kármán–Tsien approximation equation. In his educational philosophy, Qian Xuesen consistently stressed the necessity of equipping students with the ability to solve real-world engineering problems (Tang & Zhu, 2018). This emphasis is evident in the nature of his examinations, which often featured a small number of open-ended, highly challenging questions. For example, one exam included a conceptual question worth 30 points and a second problem, requiring the formulation of equations to model a round-trip solar orbit using a rocket, worth 70 points. This type of question required far more from students than the routine application of formulas; it demanded the abstraction of mechanical problems from complex, real-world scenarios, thereby pushing students toward higher-order engineering thinking.

The recent educational literature supports this emphasis on developing mechanical and engineering thinking. Y. Xu et al. (2023) propose a three-level progression—memorization and comprehension at the basic level, application and analysis at the intermediate level, and evaluation and creation at the advanced level, all of which align with Qian’s pedagogical approach. In parallel, policy initiatives such as the “New Engineering Education” strategy of China have highlighted the urgent need to cultivate innovative talent in engineering and computational science to meet the demands of the Fourth Industrial Revolution (Shen et al., 2020). This vision has led to reforms in engineering education that focus on the transformation of teaching philosophy, the innovation of pedagogical content, and the development of mechanisms to foster interdisciplinary capabilities.

Despite these aspirations, traditional instruction in theoretical mechanics remains dominated by formulaic problem-solving and passive learning, often neglecting the cultivation of creativity and the ability of students to apply knowledge in unfamiliar contexts. Consequently, students frequently struggle to see the relevance of theoretical mechanics to practical engineering, leading to disengagement and poor knowledge transfer.

To address these issues, this study proposes a dual-path reform strategy that integrates both front-end and back-end instructional innovations. The front-end reform centers on the cultivation of engineering thinking, encouraging students to apply theoretical knowledge to real-world scenarios through methods such as case-based instruction and project-based learning. This aligns with recent research demonstrating the benefits of experiential learning approaches in engineering education (J. Liu et al., 2025). Structured within the BOPPPS framework (bridge-in, objective, pre-assessment, participatory learning, post-assessment, and summary), the reformed course design seeks to foster student engagement and deepen conceptual understanding. J. Liu et al. (2025) provided systematic evidence supporting the effectiveness of the BOPPPS model in improving student outcomes.

Complementing this, the back-end reform introduces a personalized homework design that accounts for individual learning needs and interests. Drawing on advances in adaptive learning systems, this component of the reform is informed by findings by Plooy et al. (2024), who demonstrate that personalized learning strategies significantly enhance student performance and engagement. Assignments are tailored in terms of difficulty and format, ranging from conceptual reviews to programming-based simulations and collaborative projects, thereby enabling students to consolidate knowledge while stimulating innovation.

Together, these two reform paths form a cohesive strategy for modernizing theoretical mechanics instruction. By integrating real-world applications, adaptive learning technologies, and structured pedagogical models, the proposed approach aims to enhance student motivation, learning outcomes, and long-term competency in engineering practices. This study addresses three core aspects: the theoretical basis and implementation of engineering thinking cultivation, the design and impact of personalized and collaborative assignments, and the synergistic role of combining front-end and back-end strategies. This study seeks to contribute a practical and adaptable model for improving mechanics education in response to the evolving needs of engineering training in the 21st century.

2. Research Methodology

To rigorously assess the effectiveness of the proposed teaching reform, a structured research methodology was developed. This section outlines the experimental design, participant characteristics, teaching implementation, and data analysis procedures adopted in this study. By combining longitudinal tracking with cross-sectional comparisons, the research aims to provide robust empirical evidence on the impact of the bidirectional reform strategy within the context of theoretical mechanics education.

2.1. Experimental Design

This research adopts a quasi-experimental design that combines longitudinal and cross-sectional comparisons to systematically evaluate the impact of the proposed bidirectional teaching reform strategy on learning outcomes in a second-year theoretical mechanics course. The experiment spans three consecutive autumn semesters (2022–2023, 2023–2024, and 2024–2025). In each semester, both an experimental group and a control group were established. Importantly, student allocation to these groups was not randomized; students self-selected their course sections without external intervention. This lack of random assignment could introduce selection bias, although all participants shared a similar academic background and met the same course prerequisites. To maintain internal validity, the course content, instructors, and total class hours (64 h per semester—4 h per week over 16 weeks) were kept consistent across all groups and semesters.

The teaching interventions were implemented in progressive phases aligned with the three semesters. In the first semester (2022–2023), both the experimental and control groups followed a traditional, lecture-based teaching approach, serving as the baseline for subsequent comparisons. In the second semester (2023–2024), both groups adopted the front-end engineering-thinking-cultivation measures, including problem-based learning (PBL) and the standardized BOPPPS instructional model, to enhance active learning and engagement. In the third semester (2024–2025), the experimental group introduced a back-end personalized assignment design in addition to the front-end strategies, establishing a full bidirectional reform that linked in-class and after-class learning. Meanwhile, the control group in the third semester continued with only the front-end reforms. This phased design allowed us to assess both the long-term effects of the teaching reforms and the differential impact of integrating personalized assignments by the final phase.

2.2. Participants and Sample Description

Participants in this study were undergraduate students from engineering majors at Northwestern Polytechnical University. All students enrolled in the theoretical mechanics course through a self-selection process without external interference, with the expectation that the experimental and control groups would be comparable in terms of academic background and prior achievement. The course was compulsory for these students, ensuring a common baseline of prerequisite knowledge.

The number of students in each group across the three semesters was as follows:

Autumn Semester of 2022–2023: 101 students in the experimental group; 70 students in the control group.

Autumn Semester of 2023–2024: 110 students in the experimental group; 135 students in the control group.

Autumn Semester of 2024–2025: 106 students in the experimental group; 159 students in the control group.

These sample sizes reflect typical enrollment in the course each year. All participating students completed the same curriculum content under the conditions specified for their group.

2.3. Teaching Implementation

The instructional approach for each group was carried out in three phases corresponding to the academic semesters:

Phase 1 (2022–2023 Autumn): All students (in both the experimental group and the control groups) received traditional, lecture-based instruction with no reform measures implemented. This phase provided a baseline for student performance under the conventional teaching method.

Phase 2 (2023–2024 Autumn): Both the experimental and control groups incorporated front-end engineering thinking cultivation strategies into the course. This included employing PBL activities and the structured BOPPPS model during lectures to foster active learning, improve classroom participation, and develop students’ problem-solving skills. Both groups experienced the same front-end reforms in this phase, as the personalized assignment component had not yet been introduced.

Phase 3 (2024–2025 Autumn): The experimental group added a back-end personalized assignment design on top of the ongoing front-end strategies, completing the bidirectional reform. Assignments in the experimental group were diversified and tailored to students’ individual learning characteristics, interests, and abilities, with the goal of strengthening the connection between theoretical knowledge and practical engineering application. These personalized tasks included individualized problem sets, programming-based exercises, and team projects that encouraged a deeper exploration of mechanics concepts. In contrast, the control group continued with the front-end active learning strategies but did not receive any personalized after-class intervention, continuing with standard uniform assignments. This divergence in Phase 3 allowed for a direct comparison of outcomes between the full bidirectional reform (the experimental group) and the front-end-only reform (the control group).

Throughout all phases, care was taken to keep teaching materials, assessment methods, and instructor involvement consistent between the groups (aside from the intended differences in instructional strategy) to isolate the effect of the reform elements being tested.

2.4. Data Collection and Analysis

The effectiveness of the teaching reform was assessed based on students’ overall course performance, with particular attention being paid to the proportion of students achieving excellent scores (defined as final course scores between 90 and 100 out of 100). Course scores were collected for each semester and group as the primary data for evaluation. The data used for evaluation were collected systematically at the end of each semester, and consistent grading criteria were applied across groups to ensure comparability. The data analysis was primarily descriptive. For each group in each semester, we calculated the excellent rate, the percentage of students whose scores fell in the excellent range, as a key performance indicator. We then examined these excellent rates to identify trends over time (longitudinal analysis across the three semesters) and differences between the experimental and control cohorts within the same semester (cross-sectional comparison). This comparative trend analysis provided a clear view of how student performance evolved with the introduction of the teaching reforms. All calculations were performed using Microsoft Excel. Notably, no formal statistical hypothesis tests (such as chi-square or t-tests) were conducted, in line with the descriptive focus on observed performance patterns rather than inferential statistics. Instead, the analysis centered on comparing the magnitude of the excellent rates to interpret the impact of each intervention phase.

The results of the study are summarized in

Table 1. Comparison of excellent student proportions between the experimental and control groups across semesters.

Semester	Teaching Method	Group Type	Number of Students	Excellent Rate (%)
2022–2023 Autumn	Traditional Teaching	Experimental	101	19.80
		Control	70	14.29
2023–2024 Autumn	Front-End Engineering Thinking Cultivation	Experimental	110	36.36
		Control	135	39.26
2024–2025 Autumn	Front-End and Personalized Assignment	Experimental	106	61.32
	Front-End Reforms Only	Control	159	36.48

. In the first semester (baseline, traditional teaching), the proportion of excellent students was very similar between the experimental group (19.80%) and the control group (14.29%), confirming that the two groups had roughly equivalent performance at the start. In the second semester (with front-end engineering thinking reforms implemented in both groups), the excellent rates increased to 36.36% for the experimental group and 39.26% for the control group, indicating a general improvement in student performance across the board but no notable difference between the two groups under the shared front-end intervention. However, in the third semester, after the introduction of the full bidirectional reform in the experimental group, the excellent rate of that group rose markedly to 61.32%, compared to 36.48% in the control group. This outcome highlights the marked positive impact of integrating personalized assignments (the back-end reform) in conjunction with the front-end active learning strategies. The substantial increase in the experimental group’s excellent rate, relative to the control group’s more modest improvement, suggests that the combined bidirectional approach was far more effective in elevating student performance than the front-end measures alone.

3. Reform Strategy Overview

One impetus for this reform is the persistent gap between theoretical instruction and practical application in traditional theoretical mechanics courses. This disconnect often leaves students disengaged and ill prepared for applying concepts to real engineering problems. As evidenced by a recent experiment (Table 1), student performance improved markedly only after personalized assignments were introduced, underscoring the need for a more comprehensive, practice-oriented teaching approach. In response, we developed a bidirectional teaching reform strategy that links classroom learning with after-class practice to unify students’ theoretical grounding with hands-on problem-solving.

This bidirectional reform strategy consists of two interrelated components: a front-end component focused on cultivating engineering thinking during class and a back-end component focused on personalized assignment design after class. The front-end reforms reframe classroom instruction to actively connect fundamental mechanics principles with real-world engineering scenarios. Instead of relying on traditional lectures alone, students engage in case studies, demonstrations, and project-based activities that require them to apply theoretical concepts to practical problems. Such an approach immerses students in the mindset of an engineer from the outset, strengthening their ability to abstract and model complex mechanical systems even as they learn core theories.

The back-end reform extends learning beyond the classroom through personalized and adaptive homework assignments. Instead of one-size-fits-all problem sets, students tackle tasks customized to their individual interests and skill levels. These assignments vary in format and difficulty, ranging from conceptual analytical questions to programming-based simulations and collaborative projects, thereby ensuring that each student is appropriately challenged while applying mechanics concepts in contexts that resonate with them. By accommodating diverse learning needs and encouraging exploration, this personalized approach reinforces key theoretical principles through practice and fosters higher motivation and creativity. Students consolidate their knowledge as they solve tailored problems, often discovering deeper insights and innovative solutions in the process.

Importantly, the front-end and back-end measures function in concert rather than in isolation. Each component amplifies the effect of the other through a feedback mechanism linking classroom and homework experiences. During in-class sessions, students gain intuitive understanding and context for mechanical principles, which prepares them to tackle the more open-ended, individualized assignments that follow. The outcomes of those assignments, in turn, inform subsequent teaching interactions: difficulties or insights encountered during homework are addressed and built upon in the next class. This cyclical reinforcement creates a bidirectional flow of learning, wherein practical problem-solving experiences continually enrich theoretical comprehension and vice versa.

Through this integrated approach, students experience clear progression from mastering fundamental theory to confidently applying it in practice. In essence, the bidirectional strategy transforms theoretical mechanics education into a learning continuum where each stage reinforces the next; this effectively bridges academic knowledge with engineering practice at every step. The subsequent sections will examine each component of this reform in detail: first the in-class cultivation of engineering thinking is addressed; then, the development and impact of the personalized assignment design are detailed.

4. Innovative Pedagogies in Engineering Education

Building upon the theoretical and structural considerations discussed previously, this section provides a broader review of innovative teaching practices in contemporary engineering education, highlighting their relevance to the reform efforts described in this study.

Contemporary engineering education has increasingly embraced learner-centered, outcome-focused pedagogies. For example, outcome-based education (OBE) frameworks define explicit graduate competencies and align the curriculum and assessments with these outcomes (Z. Xu et al., 2024). In their study, Z. Xu et al. (2024) combined an OBE framework with a structured BOPPPS (bridge-in, objectives, participatory learning, post-assessment, and summary) teaching model and found significantly higher student performance compared to traditional methods (Z. Xu et al., 2024). Likewise, active and project-oriented methods are shown to improve engagement and higher-order thinking. Lin et al. (2025) reported that a flipped-classroom approach integrated into multidisciplinary team learning greatly enhanced students’ theoretical understanding and independent learning skills; here, the “observation” (flipped-MDT) group significantly outperformed the control group on concept mastery and problem analysis. Similarly, Etemi et al. (2024) observed that engineering students in a flipped-learning environment exhibited markedly higher technology acceptance and self-directed learning perceptions, underscoring the value of such student-driven active learning approaches. Blended learning by combining face-to-face instruction with online or hybrid components also features prominently in modern reforms. By allowing students to learn asynchronously and engage with materials in varied formats, blended models can increase flexibility and motivation. In all, these innovative methods—project-based case studies, flipped and blended classrooms, and a clear OBE alignment—are reported to bridge the longstanding gap between theory and practice in engineering curricula.

5. Application and Cultivation Strategies of Engineering Thinking in Theoretical Mechanics Education

Educational reform in higher engineering education generally encompasses three interconnected dimensions: updating curriculum content, shifting educational paradigms, and transforming instructional methodologies. Historically, attempts to bring innovation to theoretical mechanics teaching can be traced back to the 1990s, when educators began incorporating computer-assisted instructional tools (E. Wang, 1993; Q. Wang et al., 1993). More recently, technological advancements such as massive open online courses (MOOCs) and virtual classroom environments have significantly enriched instructional practices (Zhang et al., 2019). However, despite these technological advances, course content and underlying educational objectives have not always evolved concurrently. This misalignment between pedagogical content and technological tools occasionally diverts students’ attention away from the substantive concepts of mechanics, thus impairing learning outcomes.

Engineering education aims to equip students not merely with exam-oriented problem-solving skills but also with the capacity to apply comprehensive theoretical knowledge to real engineering scenarios. This goal underscores the critical importance of cultivating engineering thinking within the theoretical mechanics curriculum.

5.1. Definition and Importance of Engineering Thinking

Engineering thinking can be defined as an integrative cognitive approach to solving complex problems, which involves systematically synthesizing interdisciplinary knowledge, performing analytical reasoning, designing and optimizing solutions, and ultimately implementing these solutions effectively. The notion of engineering thinking is closely intertwined with concepts such as systems thinking, design thinking, innovative reasoning, and practical problem-solving (Crawley et al., 2014). As such, cultivating engineering thinking requires students to utilize scientific and engineering principles creatively when confronted with multifaceted problems.

Within theoretical mechanics education, fostering engineering thinking implies moving beyond the mere transmission of mechanical principles. Instead, it involves actively guiding students to apply theoretical knowledge in solving practical engineering challenges. This educational strategy significantly strengthens students’ innovative capabilities, critical thinking, and practical problem-solving skills (McKenna, 2010).

Specifically, engineering thinking enables students to identify and simplify real-world engineering issues, translate these into mechanical models, and then apply theoretical principles systematically to analyze and address these issues. Traditional instruction often presents students with highly abstracted and simplified problems, which are beneficial for foundational learning; however, they are insufficient for developing the competencies required to tackle authentic engineering problems encountered in professional practices (Prince et al., 2020). This limitation becomes apparent when students struggle to transition from observing practical problems to developing appropriate mechanical models, a critical skill frequently overlooked in conventional mechanics courses.

For example, the quick return mechanism commonly employed in mechanical shapers (Figure 1) illustrates the necessity of connecting theoretical mechanics education directly with engineering practices. The mechanism is specifically designed to allow for slower motion during the cutting stroke and faster motion during the return stroke, thus enhancing operational efficiency. Designing such a system requires students to abstract complex mechanical functions from practical needs and translate these into analyzable mechanical structures, and this is a quintessential exercise in engineering thinking.

However, typical theoretical mechanics courses often initiate instruction at the stage of kinematic analysis—step four in Figure 1—bypassing critical stages of problem identification, simplification, and initial model design. Such omissions raise fundamental pedagogical questions: Why can a complex mechanical system like a shaper’s quick return mechanism be simplified into a crank–slider model? Why are certain parts, such as cranks and connecting rods, abstracted into simplified rigid bodies despite their actual geometric complexities? By prompting students to explore these fundamental questions, educators encourage active, peer-driven inquiry, significantly enhancing students’ practical engineering reasoning.

Empirical research supports the assertion that involving students early in realistic engineering contexts and prompting model simplification and abstraction significantly improves their ability to handle complex problems encountered in actual engineering practices (Guerra & Rodriguez-Mesa, 2021). In contrast, neglecting these initial design-oriented steps can result in students mastering abstract theory yet failing to apply it effectively in practical contexts, thereby diminishing the overall effectiveness of engineering education.

Hence, effective university-level instruction in theoretical mechanics must prioritize not merely problem-solving skills but also the comprehensive cultivation of abilities of students in problem identification, analytical modeling, and practical application. This integrated pedagogical approach lays a critical foundation for developing systematic reasoning, interdisciplinary integration, and logical analysis skills that are essential for future professional success in engineering practice (Lucas & Hanson, 2016).

5.2. Problem-Based Instruction in the Teaching of Theoretical Mechanics

Problem-based instruction (PBI) has been widely acknowledged as a highly effective educational approach within engineering education. Central to this method is the active engagement of students in authentic and complex problems derived from real-world scenarios. In theoretical mechanics, the adoption of PBI not only deepens students’ conceptual understanding but also significantly enhances their capability for applying theoretical principles practically, thereby fostering advanced research competencies and refined engineering intuition.

This section outlines a structured implementation of PBI specifically within a theoretical mechanics course, focusing on the topic of collision dynamics. The instructional design leverages a genuine engineering incident, illustrating how realistic problem scenarios can act as powerful catalysts for knowledge acquisition, theoretical modeling, and engineering applications.

5.2.1. Real-World Incidents as Instructional Catalysts

The instructional sequence commences by examining the real-world accident involving Jeju Air Flight 2216 in 2024, which suffered a suspected bird strike, causing landing gear malfunction and culminating in a catastrophic belly landing with significant casualties. Rather than presenting this incident merely as a case study narrative, it is strategically used to provoke scientific inquiry, prompting students to abstract pertinent mechanical questions: How can bird strikes critically damage aircraft structures? What are the magnitudes of forces experienced during emergency landings? Utilizing this inquiry-driven approach shifts the instructional focus from passive narrative toward active, analytical reasoning deeply rooted in the fundamentals of collision mechanics.

This concrete context effectively introduces collisions as dynamic processes characterized by rapid momentum transfer and high transient forces, thereby anchoring theoretical content within relatable and meaningful engineering challenges.

5.2.2. Theoretical Grounding and Conceptual Clarification

Following the initial scenario presentation, the lesson advances into the systematic construction of theoretical frameworks. Students collaboratively explore fundamental collision characteristics, including brief interaction duration, significant impulsive forces, energy dissipation, and localized structural deformation, by examining relatable everyday experiences and practical engineering examples.

To facilitate analytical clarity, standard simplifying assumptions commonly utilized in collision mechanics are critically introduced and justified: (1) external ordinary forces are neglected during the extremely brief collision duration; (2) the displacement during collision is considered negligible; and (3) colliding bodies are idealized as quasi-rigid, allowing minimal local deformation only at interfaces. These assumptions are explicitly discussed as deliberate modeling choices with clearly defined analytical implications, thus training students in making reasoned approximations that are essential to practical engineering problem-solving.

5.2.3. Analytical Application of Collision Models

With a robust theoretical foundation established, students apply collision principles analytically to quantify forces within representative scenarios derived from the initial aviation incident.

Firstly, the bird strike scenario is quantitatively explored: students calculate the impact force exerted by a 2 kg bird striking an aircraft at approximately 235 m/s over a contact duration of 5 milliseconds, resulting in a calculated impulsive force of around 94,000 N. Secondly, the scenario of a hard landing is analyzed, where an aircraft weighing approximately 70,000 kg descending at 10 m/s and experiencing ground contact for 0.5 s generates an impact force approaching 1.4 MN. These analytical exercises reinforce students’ theoretical understanding while vividly illustrating the real-world magnitude and practical relevance of collision forces within engineering contexts.

5.2.4. Reflection and Extension: Toward Open-Ended Inquiry

In the concluding stage of instruction, the pedagogical approach transitions from structured analytical problem-solving toward exploratory, inductive reasoning, prompting students to engage deeply with open-ended engineering questions. Students are encouraged to consider questions such as the following: How can future aircraft structures be optimized to minimize damage from bird strikes? What innovative modifications to landing gear systems might significantly improve shock absorption capabilities? How should future supersonic aircraft designs address even greater collision risks?

These reflective, inquiry-driven prompts are purposefully designed not merely to elicit immediate technical solutions but to cultivate a sustained intellectual curiosity, active research orientation, and continuous engagement with unresolved engineering challenges. Such pedagogical strategies resonate closely with recent findings by Etemi et al. (2024), who noted that innovative, student-driven instructional methods significantly enhance engineering students’ engagement and their readiness for independent learning. Similarly, the structured, scenario-based approach adopted here aligns with the documented effectiveness of project-based learning (PBL) methodologies highlighted by Lavado-Anguera et al. (2024), underscoring the value of experiential, real-world problem contexts for developing higher-order engineering thinking skills.

Thus, this comprehensive PBI instructional sequence—from contextual scenario introduction to conceptual framework clarification through detailed analytical applications, thus culminating in exploratory open-ended inquiry—collectively aligns with and advances the overarching objectives of contemporary engineering education.

5.3. From Theory to Practice: Transforming Collision Experiments in Engineering Education

Building upon the previous discussion of problem-based instruction (PBI) in collision mechanics, this section addresses the broader educational transformations necessary for bridging theoretical knowledge with practical engineering skills. Specifically, it reconsiders the role and design of collision laboratory experiments, emphasizing the importance of model simplification and interdisciplinary integration within contemporary engineering curricula.

5.3.1. Limitations of Traditional Collision Experiments

Conventional collision experiments conducted in undergraduate laboratories often exhibit a significant disconnect from authentic engineering scenarios. Although such experiments generally produce controlled and reproducible outcomes, they typically lack complexity and uncertainty—essential features of realistic engineering environments. Consequently, students are rarely challenged to abstract complex, real-world situations into analyzable models—a crucial skill that is necessary for engineering practice (Dym et al., 2005). Without these meaningful connections, student motivation and deep conceptual engagement may remain limited, undermining the overarching educational objectives of laboratory instruction.

5.3.2. Bridging the Gap Through Model Simplification

Real engineering problems are inherently multifaceted, characterized by multiple interacting variables and complex boundary conditions. Educators must explicitly introduce students to the rationale and methodology behind simplification processes, clarifying that such simplifications are deliberate decisions made to facilitate analytical or numerical tractability. Simplified models fulfill several crucial purposes: (1) they reduce computational complexity, thereby rendering problems manageable; (2) they highlight dominant physical mechanisms, enhancing conceptual understanding; (3) they enable validation through controlled laboratory experiments.

For instance, instructors can encourage comparative classroom analysis, asking students to juxtapose laboratory-based experiments, such as collisions between standardized metal spheres, with more sophisticated real-world simulations of automotive crashes. Such comparisons help students critically evaluate the assumptions enabling theoretical results generalization to practical engineering applications, thereby deepening their understanding of modeling practices.

5.3.3. Enhancing Student Engagement Through Finite Element Analysis

Incorporating Finite Element Analysis (FEA) into undergraduate instruction can effectively bridge the gap between theoretical mechanics and practical engineering complexity. Though software tools like ANSYS (academic free version) and ABAQUS (student version) may initially appear advanced to sophomore students, structured introductory tasks can significantly enrich their understanding of mechanical phenomena by providing realistic simulations. Students engaging in simplified crash simulations or impact tests gain firsthand experience in manipulating parameters such as boundary conditions, material properties, and geometric configurations. This experiential learning enhances their appreciation for the practical significance and applicability of theoretical mechanics, aligning closely with contemporary engineering workflows and practices.

A representative instructional activity might involve students simulating the impact of a hammer strike on a metallic plate using the FEA software, such as ANSYS (academic free version) and ABAQUS (student version), thus allowing the visualization and analysis of stress distribution, deformation patterns, and energy dissipation characteristics.

5.3.4. Promoting Inquiry-Based, Interdisciplinary Learning

Effective education in collision mechanics demands an interdisciplinary perspective that integrates knowledge from mechanics, material science, structural engineering, and data analysis. Adopting an inquiry-based learning (IBL) approach encourages students to collaboratively investigate and interpret complex phenomena, such as analyzing actual crash test footage, interpreting experimental sensor data, or critically reviewing contemporary research articles. While the complexity of the academic literature might initially present challenges for undergraduates (Katharine et al., 2022), structured reading assignments complemented by facilitated classroom discussions and reflective writing exercises can progressively build students’ scholarly literacy and analytical abilities (Chen et al., 2021).

For example, assigning students a concise research article on crashworthiness in aerospace structures, followed by guided class discussions, can foster critical reflection on how material properties and structural configurations influence impact performance; thus, such an approach can reinforce interdisciplinary comprehension and practical engineering thinking.

In summary, the transformation of collision experiments from static demonstrations toward dynamic, integrative exploration represents a necessary evolution in engineering education. By systematically incorporating FEA, explicitly addressing model simplification rationales, and promoting inquiry-based, interdisciplinary learning practices, educators can significantly enhance the authenticity, depth, and relevance of theoretical mechanics instruction. This shift will more effectively prepare students to confront complex, real-world engineering challenges, fostering both the practical competence and the innovative problem-solving capabilities that are essential for future engineering professionals.

5.4. Instructional Design Using the BOPPPS Model for Engineering Thinking Cultivation

BOPPPS (bridge-in, objective, pre-assessment, participatory learning, post-assessment, and summary) is a student-centered instructional model developed in the late 1970s at the University of British Columbia as part of the Instructional Skills Workshop program (J. Liu et al., 2025). Grounded in constructivist learning theory and communication principles, the BOPPPS framework establishes a complete, closed-loop teaching process focused on clear learning objectives and diversified methods of instruction. It breaks a lesson into six sequential stages: they start with an engaging “bridge-in” introduction to spark interest; then, setting explicit objectives are set; prior knowledge is probed via a pre-assessment; interactive participatory learning activities are delivered; outcomes are evaluated through a post-assessment; finally, students summarize to consolidate their knowledge. This structured cycle ensures that teaching is organized around desired learning outcomes and continuous feedback, thereby promoting active learner involvement and effective achievement of the teaching goals (Li et al., 2024).

Over the past decade, the BOPPPS model has gained broad recognition in higher education and has been adopted across disciplines—including engineering—as a proven approach for enhancing student engagement and performance. It meets modern engineering education standards and has been introduced in hundreds of universities worldwide as a means to implement outcome-based, student-centered learning (Li et al., 2024). Empirical evidence from recent studies consistently shows that BOPPPS-based instruction can significantly improve learning outcomes. For instance, a 2024 systematic review of 19 studies concluded that the majority reported positive effects of BOPPPS on students’ academic performance, skills development, class participation, and other learning measures (J. Liu et al., 2025). Likewise, comparative experiments have observed notably higher student engagement, deeper understanding (e.g., better mastery of knowledge and problem-solving skills), and improved exam results under BOPPPS-designed teaching as opposed to traditional lecture methods (Z. Xu et al., 2024). These advantages make BOPPPS highly suitable for engineering education reform efforts. In particular, applying BOPPPS in a theory-heavy course like theoretical mechanics can transform passive rote learning into an active learning experience, heightening student involvement and reinforcing the integration of theory with practice. Thus, in the present work, the BOPPPS model is employed as the guiding framework for cultivating engineering thinking, as detailed in the following subsections.

The effective cultivation of engineering thinking requires careful instructional planning and the strategic application of proven teaching methodologies. Among various instructional models, the BOPPPS model is widely recognized for its clarity and structured approach, making it particularly effective for engineering education contexts. The BOPPPS framework is structured around six sequential stages, each represented by a letter: bridge-in, objective, pre-assessment, participatory learning, post-assessment, and summary (Johnson, 2006).

Table 2. The structure of BOPPPS teaching model.

BOPPPS	Objective	Key Words
Bridge-in	Effective introduction of content: for example, leveraging current news or events, engineering and real-life practical problems, videos, or audio materials.	Why
Objective	Define the objectives and tasks of teaching. (This part of the content can be shared with students depending on the actual situation.)	Memorize, understand, calculate, analyze, apply, evaluate, and create
Pre-Assessment	Assess students’ knowledge and understanding as a pretest to gain insights into their basic grasp of the subject matter.	Given knowledge
Participatory Learning	Traditional classic teaching content but with an emphasis on interactive learning and the application of innovative methods and means.	Interactive teaching and learning
Post-Assessment	Assessment of students’ understanding and mastery of knowledge after instruction. Compare with teaching objectives.	Related knowledge
Summary	Summarize the main content of the class and expand the depth and breadth of knowledge. Introduce the upcoming topics for discussion.	Conclusion, elevation, and linking previous content with upcoming content

details the instructional goals and key aspects associated with each stage of the BOPPPS model.

In practice, implementing the BOPPPS model involves aligning prepared content systematically to the defined stages. For example, the bridge-in stage serves to establish relevance and arouse student curiosity, making them appreciate the practical significance of the lesson content. Similarly, clear and achievable objectives set in the objective stage guide both teaching and learning activities toward measurable outcomes, ensuring instructional coherence.

While some educators may initially perceive BOPPPS as overly structured or conventional, evidence from engineering classrooms consistently demonstrates its effectiveness, particularly in facilitating active student participation and enhancing engagement with complex theoretical materials (J. Liu et al., 2025). Nonetheless, our practical experience highlights the importance of brevity and clarity in each stage, as extended durations or excessive details can diminish the overall instructional efficacy. Empirical studies have demonstrated that the optimal application of BOPPPS typically occurs when teaching discrete knowledge points, ideally within intervals of approximately 15 min. Extended duration in initial phases, such as overly detailed introductions or exhaustive preliminary explanations, tends to weaken the effectiveness of subsequent assessment stages, negatively impacting student recall and learning retention (Etemi et al., 2024).

The concise and targeted design of “each” BOPPPS stage is therefore essential. To clearly illustrate this instructional approach, the subsequent section applies the BOPPPS framework explicitly to the teaching of planar rigid body motion, highlighting precise steps and practical implementation details.

5.4.1. Bridge-In

Although the landing gear represents only a small fraction of an aircraft’s overall structure, its function is fundamental to flight safety, facilitating both takeoff and landing. To engage students at the outset of instruction, instructors may guide them to observe and compare landing gear configurations across different aircraft shown in media or real-world footage. This seemingly simple observation often reveals notable variations in structural design, even if students are initially unable to describe or classify them with precision. Such inquiry serves as an effective entry point for discussion, shifting students’ attention from passive observation to active mechanical reasoning. At this stage, the instructor plays a guiding role, using the observed diversity in landing gear to lead students toward recognizing how structural design correlates with specific functional requirements in aircraft engineering.

Landing gear types differ primarily based on aircraft size, weight distribution, and operational conditions. For instance, the conventional landing gear, which is typically found in small aircraft, features two main wheels at the front and a single tail wheel; however, its limited ground maneuverability and nose-up position can hinder visibility during taxiing. In contrast, the tandem landing gear, used in gliders and some military aircraft, aligns a central gear linearly beneath the fuselage, with smaller supporting wheels at the wings for balance. Floater landing gear represents an adaptive design tailored for operations on water or snow, while the tricycle configuration, which is the most common in commercial aviation, combines a nose wheel with two main gear assemblies beneath the fuselage, enhancing both stability and control. Introducing these distinct configurations provides not only a concrete context for mechanical modeling but also a natural segue into the study of planar rigid body motion, as each design can be abstracted into simplified kinematic structures for analysis.

5.4.2. Objective

Clearly defined and targeted learning objectives are essential for effective teaching, as they help align instructional strategies with desired learning outcomes. In the context of this lesson, which centers on planar rigid body motion and the four-bar linkage mechanism, the objectives are designed to foster both conceptual understanding and practical modeling skills. Rather than approaching the topic as an abstract kinematic exercise, the lesson is structured to guide students in recognizing how everyday mechanical systems, such as aircraft landing gear, can be interpreted through the lens of simplified mechanical models.

Specifically, the instructional goals are threefold: first, to enable students to conceptualize the aircraft landing gear as a four-bar linkage mechanism; second, to help them identify and articulate the defining characteristics of planar rigid body motion in mechanical systems; and third, to develop their ability to abstract complex three-dimensional configurations into tractable two-dimensional models. These objectives, which are embedded within the BOPPPS framework, provide a clear direction for both teaching and learning, ensuring that the lesson is coherent, focused, and pedagogically grounded in the development of engineering thinking.

5.4.3. Pre-Assessment

The pre-assessment phase engages students by having them observe and analyze a simplified mechanical representation of landing gear motion (Figure 2). Students are asked to identify critical motion characteristics of the depicted mechanisms, which inherently require them to differentiate essential structural and functional elements from secondary details. Through this initial exercise, students practice the critical skill of simplifying complex mechanical systems into analyzable models—a cornerstone ability in engineering thinking.

5.4.4. Participatory Learning

Participatory learning constitutes the core of the BOPPPS instructional model and typically occupies the most substantial portion of class time. In the context of teaching planar rigid body motion, this stage involves progressive exploration of theoretical concepts, analytical strategies, and practical applications. The session begins with a focused explanation of the fundamental definitions and characteristics of planar rigid body motion, ensuring students grasp the conceptual underpinnings of the topic and understand its relevance to real-world mechanical systems. Building on this foundation, students are introduced to model simplification techniques, learning how to abstract essential motion behaviors from complex structures and reduce them to analyzable forms. This process not only enhances their ability to recognize underlying mechanical principles but also reflects the essential reasoning skills expected in engineering practice. Finally, the instructional sequence moves to the formulation of key equations governing planar rigid body motion, including methods for calculating velocity and angular velocity. Rather than relying solely on lectures, the teaching approach emphasizes interaction through group discussions, peer collaboration, and guided problem-solving exercises, thus encouraging students to actively construct understanding. This design enables learners to transition from conceptual comprehension to analytical application in a coherent and engaging manner.

5.4.5. Post-Assessment

For the post-assessment phase, we selected a problem involving the solution of a four-bar linkage mechanism, as presented below (Figure 3).

Crank $\mathrm{OA}$ has a length of $20 \mathrm{cm}$ and rotates at a speed of $\mathsf{\omega }=5.2 \mathrm{rad}/\mathrm{s}$. Additionally, rod ${\mathrm{BO}}_1$ has a length of $40 \mathrm{cm}$. Students were asked to determine the angular velocities of rod ${\mathrm{BO}}_1$ and link $\mathrm{AB}$ at the illustrated position.

This question involves two key concepts: (1) Solving for the angular velocity of a rigid body undergoing fixed-axis rotation essentially entails calculating the velocity of any point on the rigid body undergoing fixed-axis rotation. (2) Solving for the angular velocity of a rigid body undergoing planar motion essentially involves applying the method of the instantaneous center and the method of the fixed point to calculate the velocity of a specific point on the rigid body undergoing planar motion. The aim is to cultivate students’ clear analytical thinking and establish the fundamental connection between solving problems related to angular velocity of rigid bodies and the calculation methods of velocity.

5.4.6. Summary

The form of summary can take various approaches. For instance, it can involve a simple review focusing on the key points covered. Alternatively, it may entail testing through questions. It can also expand on the current topic to enhance understanding. Additionally, it is important to introduce the subsequent topics to bridge the learning process.

The above is a complete outline of a BOPPPS course design, including its steps and main components. Practice has shown that this course design plays a certain role in enhancing the quality of classroom teaching, as its structure is relatively fixed, making it easy to implement. Many teaching materials can be taught using the BOPPPS approach, especially in modular teaching, where each module can be designed using the BOPPPS framework. However, excessive reliance on this course design may lead to a lack of innovation. Therefore, teaching methods need to be flexibly applied, with specific analysis for each situation, as there is no one-size-fits-all method.

6. Strategies for Implementing Personalized Homework Design and Interactive Learning in Theoretical Mechanics

In traditional education, homework serves as a foundational mechanism through which students consolidate classroom learning, reflecting the longstanding pedagogical principle of “reviewing the old to understand the new”. While, historically, homework assignments have predominantly been paper-based, rapid advances in digital technology have created opportunities for more dynamic, interactive, and personalized homework systems. Analogous to the transformative effect of digital payment systems in commerce, digital innovations in education similarly aim to transcend traditional paper assignments, fostering more responsive, individualized learning experiences. Recent research corroborates the advantages of such innovations, highlighting improved student performance and engagement through personalized electronic assignment systems (Gfrerer et al., 2023).

Electronic homework platforms offering automatic grading and instant feedback exemplify this shift towards digitalization. Such platforms significantly enhance grading efficiency and the timeliness of feedback compared to conventional manual methods (Cabi & Türkoğlu, 2025). In courses such as theoretical mechanics, where class sizes often exceed 100 students, manual grading can delay feedback by up to two weeks, thereby diminishing the pedagogical value of assignments. Electronic platforms can substantially mitigate these constraints, delivering timely and informative feedback that sustains student engagement and accelerates learning processes (Y. Liu et al., 2022). To address the existing limitations that are inherent in traditional assignment practices, we propose a comprehensive reform strategy, as depicted schematically in Figure 4, focusing on a personalized homework design to elevate both student motivation and learning outcomes.

6.1. Personalized Homework Generation and Assignment Framework

To facilitate personalized learning, the proposed system leverages an extensive, carefully structured database of assignments. This database is systematically organized into three tiers—easy, intermediate, and advanced—to match the distinct learning objectives at the different stages of students’ academic progression. Individualized assignments generated from this database are assigned randomly yet adaptively, reducing the risk of academic dishonesty and ensuring that each student’s unique learning trajectory is appropriately supported. Empirical studies suggest that individualized assignments tailored through learning analytics significantly enhance students’ mastery of targeted knowledge areas and foster deeper conceptual understanding compared to uniform assignment methods (Rodríguez-Martínez et al., 2023).

To ensure that the system remains responsive and effective, a dynamic updating mechanism incorporating both manual oversight and basic analytics reviews student performance continuously. Feedback from these analyses informs periodic adjustments to the homework database, enabling incremental refinements to maintain both the relevance and challenge of assignments. Such flexibility supports ongoing pedagogical improvement, ensuring assignments remain engaging, effective, and tailored to evolving student needs.

We structure the learning process into three stages: elementary, intermediate, and advanced, each corresponding explicitly to a specific homework difficulty level. The elementary stage emphasizes the fundamental concepts, focusing on students’ memorization and comprehension of basic principles and theorems. Although seemingly simple, these foundational tasks are essential because they ensure precision in students’ understanding of core theoretical constructs—a prerequisite for more complex analyses. Research indicates that clearly staged assignments facilitate better conceptual retention and improve student confidence by aligning task complexity with learners’ developmental levels (Herset et al., 2023).

At the intermediate stage, assignments increase in complexity, challenging students to apply theoretical knowledge through mathematical modeling and computational tools. Students learn to abstract and represent complex mechanical phenomena mathematically, translating real-world engineering challenges into analytically or numerically solvable problems. Such tasks foster deeper analytical reasoning and practical competence, preparing students for sophisticated problem-solving scenarios.

Finally, at the advanced stage, students encounter challenging assignments that demand a higher level of innovation and flexible application of knowledge. These tasks replicate authentic engineering problems, requiring students to navigate and resolve practical constraints encountered in real-world scenarios. Such high-level tasks not only consolidate students’ existing knowledge but also encourage original thinking, creativity, and adaptive problem-solving—competencies that are essential to professional engineering practice and the advancement of knowledge.

Collectively, this personalized homework framework aligns closely with current educational best practices, promoting student-centered learning through structured differentiation of task complexity, personalized feedback, and interactive engagement. Thus, the reform seeks to enhance not only immediate learning outcomes but also students’ long-term analytical and innovative capabilities.

6.2. Enhancing Interactive Learning Through Group Assignments

Group-based assignments represent an effective strategy to enrich students’ learning experiences by promoting collaboration and interactive engagement. By working in small teams, students engage actively in peer discussions, exchange insights, and collectively approach complex homework tasks, thus leveraging shared knowledge and perspectives. This collaborative learning format facilitates quicker and more diverse peer feedback, allowing students to recognize and address misconceptions more promptly than traditional, instructor-driven feedback processes. Furthermore, collaborative assessments significantly alleviate the constraints associated with traditional homework grading practices, which are typically limited to instructor availability and classroom or office settings. Thus, group-based assignments not only foster teamwork but also enhance efficiency in learning assessments, potentially improving both students’ satisfaction and their sense of accomplishment.

Effective assignment design must be innovative, intellectually stimulating, and closely aligned with the pedagogical vision advocated by Qian Xuesen. Homework tasks need not be numerous, yet they should effectively promote deep learning by encouraging students to integrate and apply theoretical concepts in realistic and practical scenarios. This integration of theoretical mechanics with authentic engineering problems reinforces the essence of engineering education, thus training students to identify fundamental issues in real-world contexts and apply systematic methods and knowledge to address them. Two illustrative cases successfully implemented in recent teaching practices exemplify such effective homework design strategies.

The following instructional example was implemented during the 2024–2025 autumn semester in the experimental class consisting of 106 undergraduate engineering students. The course applied both front-end engineering thinking cultivation and back-end personalized assignment design strategies. The exercise was designed to deepen students’ understanding of planar rigid body motion by connecting theoretical mechanics principles with real-world vehicle dynamics.

6.2.1. Homework Example: Analysis of Vehicle Turning Motion

One practical illustration of applying planar rigid body motion principles is the turning motion of vehicles (Figure 5). To encourage students’ deeper analytical thinking, this homework exercise raises several probing questions: (1) Are the steering angles of the two front wheels identical during vehicle turning? (2) What are the definitions and physical significance of the turning radius and Ackermann steering geometry? (3) Why do large trucks experience increased accident risks when negotiating turns?

Although vehicle turning might appear as a commonplace occurrence, it encapsulates rich kinematic concepts and challenges students’ intuitive understanding. Unlike conventional homework tasks that typically focus on idealized rigid body motion examples, this assignment emphasizes the analysis of authentic, everyday phenomena to enhance relevance and stimulate students’ critical thinking about practical engineering challenges. Typically, students may not readily distinguish between the roles and dynamics of front versus rear wheels during turns, nor realize the practical implications underlying these differences. To address this knowledge gap, students begin by observing everyday phenomena and proceed toward deeper, structured inquiry. They initially define the steering angle explicitly as the angle formed between a front wheel and the vehicle body axis, thus establishing a simplified geometric model to examine wheel behavior.

Experimental observation reveals an intriguing phenomenon: the two front wheels’ steering angles are unequal during a turn. However, empirical results alone are insufficient to elucidate the underlying mechanics. Hence, students must employ theoretical principles from planar rigid body kinematics to rigorously explain these observations. When a vehicle turns on a horizontal plane, its motion can be approximated as planar rigid body motion because the vertical distance from any point on the vehicle body to the ground plane remains constant. Consequently, students can abstract this motion into two-dimensional analysis, simplifying the complex physical problem into a manageable theoretical model (see Figure 5).

Utilizing the instantaneous center method from rigid body kinematics, students analyze the velocities at points connecting the wheels’ axles to the vehicle body. By drawing perpendicular lines from these velocities, students locate the instantaneous center of velocity, enabling them to interpret vehicle turning as an instantaneous rotation around this point. From this analysis, they deduce that all points on the rigid body share the same angular velocity at any instant, with linear velocity magnitudes proportionate to their distances from the instantaneous center. Hence, the wheels positioned further from this center exhibit greater linear velocities, directly implying unequal front wheel velocities and steering angles. Specifically, as illustrated in Figure 5, during a left turn, the right front wheel exhibits a smaller steering angle than the left front wheel.

Further conceptual exploration addresses the notion of turning radius, which is defined as the radius of curvature traced by the vehicle during turns. Figure 5 visually demonstrates that the inner and outer wheels follow distinct trajectories, resulting in two separate turning radii whose difference defines the Ackermann angle. A geometric derivation provides the expression linking the Ackermann angle directly to the vehicle’s wheelbase length, underscoring why large trucks, with their substantially longer wheelbases, present higher Ackermann angles. This geometric characteristic explains the increased risk associated with trucks navigating sharp turns, as their rear wheels may unintentionally encroach upon pedestrian spaces or adjacent lanes, creating serious safety concerns.

Through carefully structured questions and guided analytical tasks, this example illustrates how well-designed homework can cultivate students’ ability to synthesize theoretical knowledge and practical observation. After completing this assignment, students reported a significantly improved ability to conceptualize planar rigid body motion in practical settings. Classroom discussions revealed that many students were able to articulate the mechanics behind differential wheel steering more precisely and demonstrated enhanced confidence when applying theoretical models to real-world engineering problems. These observations suggest that integrating real-life phenomena into mechanics homework can effectively bridge the gap between theoretical learning and engineering practice.

6.2.2. Homework Example: Kinematics of a Point on a Rolling Wheel

This homework task was implemented during the 2024–2025 autumn semester as part of the experimental class involving 106 undergraduate engineering students. Situated within the bidirectional teaching reform phase, this assignment integrated theoretical derivation with computational visualization, aiming to cultivate students’ ability to bridge abstract mechanics theory with dynamic real-world motion. Another illustrative homework task commonly integrated into theoretical mechanics courses involves the kinematic analysis of a point on the edge of a rolling wheel (as depicted in Figure 6). Specifically, students are required to determine the trajectory, velocity, and acceleration characteristics of a wheel undergoing pure rolling along a horizontal surface, with a given rotation angle defined as $\mathsf{\phi }=\mathsf{\omega }\mathrm{t}$, where $\mathsf{\omega }$ is a constant. The concept of pure rolling, defined as rolling without relative slipping between the wheel and ground at the point of contact, forms the conceptual foundation of this analysis.

Unlike conventional homework exercises that emphasize purely symbolic manipulations or textbook-style derivations, this assignment requires students to synthesize analytical modeling with dynamic computational simulations, thereby deepening their intuitive grasp of complex kinematic phenomena. Initially, students conduct geometric and analytical calculations to derive the equations governing the point’s motion. These calculations yield parametric equations describing the horizontal and vertical coordinates of the point as functions of time. While deriving the equations analytically provides theoretical insights, directly visualizing these trajectories from analytical expressions alone poses challenges, particularly due to nonlinearities in temporal functions. Consequently, traditional approaches may offer limited intuition regarding the physical motion involved.

To overcome these limitations, computational methods and numerical simulations are incorporated into the assignment. By programming these equations into computational software, students can visualize complex trajectories and dynamic characteristics clearly and intuitively. Figure 7 illustrates the step-by-step formation of the cycloidal trajectory traced by a point on the wheel’s rim during pure rolling. This cycloidal shape emerges vividly from the sequential states captured via numerical simulations, significantly enhancing conceptual comprehension beyond analytical methods alone.

Further detailed numerical analysis (see Figure 8) reinforces this intuitive understanding. Figure 8a presents the precise cycloidal trajectory of the point, clearly demonstrating its characteristic periodic path. Figure 8b shows the temporal variation in the point’s position coordinates, highlighting the periodicity of vertical displacement—a behavior consistent with pure rolling motion. Notably, the horizontal displacement, though continuously increasing, does not exhibit linearity; this is a subtle but important phenomenon that is not readily apparent without computational visualization.

The velocity and acceleration profiles, often the most challenging aspects of kinematic understanding for students, become readily comprehensible through numerical plotting. Figure 8c reveals the periodic variations in velocity magnitude, explicitly indicating that the point attains its maximum speed when it reaches the highest position on the wheel’s rim. Meanwhile, the acceleration profile (Figure 8d) presents profound insights: even when the point momentarily contacts the ground with zero velocity, its acceleration remains nonzero—a counterintuitive yet fundamental kinematic insight illuminated clearly through computational simulations.

Classroom observations and informal student feedback indicated that the visualization of components significantly improved their conceptual understanding of nonlinear motion and instantaneous kinematic properties. Many students expressed greater confidence in interpreting velocity and acceleration variations; these areas are traditionally considered to be challenging in rigid body kinematics.

Given the extensive time and intellectual investment required for completing such assignments, it is crucial to reflect their significance appropriately within the course’s grading structure, thereby ensuring sustained student engagement and motivation. Recent research underscores the fact that pedagogical innovations designed to increase engagement can significantly enhance student learning outcomes. For example, Ortiz-Rojas et al. (2025) reported measurable improvements in student performance when gamification elements were introduced into university-level calculus courses. Although our instructional reforms leverage personalized assignments rather than gamification directly, the underlying principle remains analogous: fostering active participation and intrinsic motivation leads to a deeper mastery of theoretical content. Recognizing the intellectual demands of such assignments, the course design placed greater emphasis on process-oriented learning, where continuous engagement through assignments and projects accounted for a significant portion of the final grade. This approach aimed to sustain motivation and foster deeper analytical capabilities throughout the semester.

6.2.3. Homework Example: Kinematic Analysis of Arbitrary Points in a Crank–Slider Mechanism

With advancements in computational programming, numerical analysis and dynamic visualization have become effective methods for illustrating complex mechanical motions. This homework task was conducted during the 2024–2025 autumn semester with 106 undergraduate engineering students enrolled in the experimental class. Implemented under the bidirectional teaching reform framework, the assignment aimed to integrate computational modeling with traditional kinematic analysis, thereby enhancing students’ understanding of complex mechanical motion through dynamic visualization. A representative application of such techniques is the kinematic analysis of arbitrary points within a crank–slider mechanism (Figure 9). In this mechanism, the crank rotates uniformly around a fixed pivot at point A, driving the connecting rod, AB, in planar rigid-body motion. The endpoint, B, of the connecting rod is attached to a slider constrained to linear motion along a fixed guide.

Traditionally, instruction on the kinematics of linkages has largely relied on manual geometric constructions and abstract symbolic calculations. In contrast, this assignment introduced a systematic computational approach that not only reduced the reliance on tedious hand derivations but also provided students with immediate, dynamic feedback, thereby fostering more intuitive and accurate understanding. Traditional analytical methods typically require deriving instantaneous velocity and acceleration expressions manually, a process which can be both labor-intensive and prone to errors. However, by leveraging computational programming, the entire motion of the system can be systematically modeled, greatly simplifying the analysis of arbitrary points within the mechanism. For example, students might investigate the midpoint, C, on the connecting rod, AB. Through computational methods, the position of point C at each instant is precisely determined, allowing its trajectory to be accurately depicted over a complete cycle of crank rotation. The computed trajectory, illustrated in Figure 10, reveals an irregular, egg-shaped closed curve, which is distinct from standard geometric forms such as circles or ellipses, highlighting the intricate motion patterns characteristic of real-world mechanical linkages.

Furthermore, dynamically animated visualizations provide students with a clear, real-time depiction of the trajectory evolution, enabling them to intuitively grasp the relationships among the moving components within the crank–slider assembly. Such visualizations significantly reinforce students’ conceptual understanding, bridging theoretical knowledge with practical mechanical insights.

Building upon basic trajectory analysis, additional programming-based tasks can explore more sophisticated kinematic relationships within the mechanism. Students may be tasked with simultaneously visualizing trajectories of multiple points, such as the crank tip, slider, and several intermediate points located at fractional distances (one quarter, half, and three quarters) along the connecting rod. Computationally derived results (Figure 11) vividly display these concurrent trajectories, illustrating how the crank tip traces a perfect circle, the slider moves linearly, and the intermediate points form distinct elliptical paths. These elliptical trajectories exhibit a pronounced geometry, wider at one end and narrower at the other, revealing subtle nuances of the connecting rod’s planar motion.

Advanced analyses may also integrate vector representations of velocity and acceleration by superimposing them directly onto the computed trajectories, as demonstrated in Figure 12. Employing distinct visual styles and colors for clarity, velocity vectors align tangentially to each point’s trajectory, while acceleration vectors clearly illustrate dynamic characteristics influenced by inertial and constraint forces. Such vector visualizations not only enhance students’ intuitive comprehension of kinematic variables but also clarify intricate relationships between geometric motion paths and associated dynamic quantities.

This computational, visualization-oriented approach substantially improves educational effectiveness by offering students a direct, interactive perspective on mechanical phenomena. Feedback collected from classroom discussions and post-assignment reflections indicated that students demonstrated enhanced proficiency in interpreting velocity and acceleration vectors. Many students also reported that the animated visualization significantly deepened their grasp of the otherwise abstract relationships between displacement, velocity, and acceleration in planar mechanisms.

6.2.4. Homework Example: Relative Motion Analysis in Multiple Reference Frames

The analysis of relative motion constitutes a fundamental yet challenging aspect of theoretical mechanics education. Situated within the bidirectional teaching reform phase, the following series of homework tasks aimed to strengthen students’ spatial reasoning and analytical abilities by integrating computational modeling techniques into traditional relative motion analysis. Traditional instructional methods, which often rely heavily on manual geometric derivations, may hinder intuitive understanding due to their abstract nature. To address this issue, computational programming and dynamic visualization techniques have been integrated into teaching strategies. The following examples demonstrate the application of numerical modeling and visualization to elucidate complex relative motions, thereby significantly enhancing conceptual comprehension and learner engagement.

Relative Motion in Opposite Circular Paths

As illustrated in Figure 13, two points move along the same circular path with identical speeds but in opposite directions; specifically, the black point moves clockwise, while the green point travels counterclockwise. Traditionally, analyzing and visualizing the relative trajectory of these points poses significant challenges, as intuitive reasoning alone may not readily predict the resulting path. However, through numerical modeling and programming-based visualizations, the relative motion becomes distinctly comprehensible. When observed from a reference frame fixed to the black point, the trajectory traced by the green point notably simplifies into a near-linear path, as demonstrated by the visualization (Figure 13, right panel). This dynamic representation allows students to directly perceive the evolving relative positions, thereby reinforcing spatial reasoning and deepening their grasp of the underlying theoretical principles of relative motion.

Complex Relative Motion Analysis Within a Stationary Reference Frame

In another scenario presented in Figure 14, two points independently trace distinct geometric paths on stationary surfaces. The black point moves along a circular path on a stationary blue surface, whereas the green point traverses a square trajectory on a stationary red surface. The challenge arises when students attempt to intuitively predict or manually derive the relative trajectory of the green point from the perspective of the black point. Computational programming addresses this difficulty effectively by generating accurate visualizations of this relative motion. The resultant numerical visualization clearly depicts the complex relative trajectory, significantly improving students’ intuitive understanding of the interplay between different moving paths and enhancing their spatial reasoning capabilities.

Relative Motion Analysis in a Rotating Reference Frame

The scenario depicted in Figure 15 further complicates the analysis by introducing rotational reference frames. Here, the green point moves along a square path defined on a red surface that itself rotates uniformly about its central axis. Consequently, from an inertial reference perspective, the absolute trajectory of the green point exhibits considerable complexity. Employing numerical modeling and visualization tools enables students to accurately plot both the absolute and relative trajectories under these conditions. The resultant visualizations clearly illustrate how rotational motion profoundly alters perceived trajectories, helping students intuitively appreciate the intricate relationship between rotational and translational motions within non-inertial reference frames. Such insights substantially enrich students’ conceptual understanding of complex mechanical motions. Classroom feedback suggested that students gained a more accurate and intuitive understanding of the principles of relative motion, particularly appreciating the visual clarity afforded by dynamic simulations over traditional static diagrams.

Complex Motion in Multiple Reference Frames

A particularly intricate case involving multiple reference frames is illustrated in Figure 16. In this example, a green point moves along a square trajectory relative to a red surface, which simultaneously undergoes fixed-axis rotation about its center. Additionally, a black point independently moves along a circular trajectory on a stationary blue surface. Observed from an inertial reference frame, the absolute trajectory of the green point becomes highly nonlinear and intricate. However, through computational modeling and visualization, students can clearly observe the relative motion of the green point with respect to the black point, yielding deeper insights into the compound effects arising from interacting reference frames. This visualization fosters an enhanced understanding of the synthesis and decomposition of complex motions, enabling students to develop sophisticated analytical skills that are critical to advanced mechanical analyses.

The four cases discussed above illustrate the substantial pedagogical benefits derived from integrating computational programming into theoretical mechanics education. Primarily, this integration allows for the automated and precise numerical computation and visualization of complex relative motions, thereby substantially reducing dependence on labor-intensive manual derivations and drawings. Consequently, abstract theoretical concepts become more tangible and visually intuitive, significantly enhancing student comprehension and engagement. Furthermore, computational tools enable instructors to prioritize conceptual explanations and interactive classroom discussions over repetitive mechanical calculations, markedly improving instructional efficiency and effectiveness.

Moreover, dynamic visualizations catalyze active learning by transforming abstract mechanics principles into observable, manipulable, and interactive experiences. Modern computational platforms such as MATLAB (freely available academic version) and Python (version 3.10), characterized by their user-friendly interfaces and robust numerical capabilities, facilitate the widespread implementation of these visualization techniques. The pedagogical implications of this approach extend well beyond theoretical mechanics, holding significant promise for broader applications in engineering education and physics instruction and ultimately promoting deeper conceptual understanding and more effective learning outcomes.

Collectively, the progressive integration of computational programming and dynamic visualization into relative motion analysis not only bridges the gap between abstract theory and intuitive understanding but also demonstrably enhances students’ spatial reasoning skills and analytical capabilities. These improvements align directly with the objectives of the bidirectional reform strategy, affirming the educational value of computationally enriched instructional practices in advanced mechanics education.

7. Discussion

Figure 17 presents the conceptual and operational framework guiding the bidirectional reform strategy proposed in this research. It systematically maps the intended pathways through which front-end engineering thinking cultivation and back-end personalized assignment practices jointly contribute to the comprehensive development of student competencies within the context of theoretical mechanics education. Front-end reforms emphasize cultivating an engineering mindset among students by closely connecting theoretical knowledge with practical engineering challenges. Concurrently, back-end reforms focus on personalized homework assignments tailored to individual student needs, stimulating innovative thinking and deepening theoretical understanding through customized practice. Although each reform targets distinct educational aspects, their integration creates mutual reinforcement, fostering a comprehensive improvement in student competencies.

The combined strategy shares a unified goal: the holistic development of students’ capabilities, including systematic reasoning, innovative thinking, logical analysis, and practical engineering skills. Pérez and Verdín (2023) highlighted the effectiveness of such integrated teaching approaches, demonstrating that iterative practice and timely feedback significantly enhance students’ content mastery and problem-solving abilities. Similarly, Burns et al. (2023) reported notable improvements in student performance through the introduction of modular instruction coupled with frequent, personalized online assessments. These findings reinforce the effectiveness of combining structured theoretical learning with targeted practical assignments to nurture comprehensive engineering competencies. The practical implementation of this framework was evaluated through a comparative analysis of three consecutive academic cohorts. In the 2024–2025 academic year, the experimental class adopting the full bidirectional reform strategy achieved a marked increase in the proportion of excellent students (61.32%) compared to the control class (36.48%), which only implemented front-end reforms. These empirical results substantiate the key linkages depicted in Figure 17, demonstrating that the integration of targeted theoretical engagement and personalized practice pathways significantly accelerates the cultivation of systems thinking, innovative reasoning, and problem-solving abilities.

The mutually reinforcing mechanism between front-end and back-end reforms operates as a feedback loop. This feedback mechanism forms a core component of the logical structure illustrated in Figure 17, where practical problem-solving experiences derived from front-end reforms enrich theoretical understanding, which in turn enhances personalized assignment performance. This cyclical interaction sustains student engagement and progressively consolidates both theoretical knowledge and practical competencies. Practical, real-world problem-solving exercises implemented through front-end reforms enable students to deepen their theoretical understanding, subsequently enhancing their performance on personalized homework tasks. Conversely, the outcomes and challenges identified from personalized assignments feed back into the front-end learning environment, providing relevant and stimulating real-world examples that enrich classroom discussions and project-based activities. Thus, this cyclical interaction promotes sustained engagement and deeper learning.

Furthermore, the effectiveness of the proposed reforms is not constrained by traditional classroom boundaries, as these teaching strategies are compatible with blended learning environments. Blended learning, incorporating online digital resources alongside traditional classroom activities, significantly expands the temporal and spatial dimensions of education. Low et al. (2023) found that blended teaching methods, such as flipped classrooms, received positive feedback from 66.1% of students in engineering dynamics courses, underscoring student acceptance and improved learning outcomes. The systematic review by De Bruijn-Smolders and Prinsen (2024) further affirmed that blended learning significantly enhances student engagement and academic performance, validating its suitability for implementing comprehensive educational reforms.

Finally, addressing the gap in practical engineering experience among university educators is essential for achieving sustainable educational outcomes. Hora and Lee (2022) emphasize that faculty with industry experience tend to prioritize teaching critical practical skills, such as communication, teamwork, and complex problem-solving. However, they also noted that industry experience alone is insufficient; educators require additional pedagogical training to translate their practical knowledge effectively into classroom instruction. Shah and Gillen’s (2024) systematic review supports this idea, highlighting that university–industry collaborations provide faculty opportunities to continually update their disciplinary knowledge and gain insights into real-world engineering practices, thereby enhancing the practical relevance and effectiveness of their teaching. Similarly, Valiente Bermejo et al. (2021) demonstrated through a collaborative curriculum model that sustained industry involvement in curriculum design bridges the gap between academic preparation and industry requirements, enhancing the employability of graduates and enriching faculty expertise.

In summary, by effectively integrating front-end and back-end reforms, adopting blended learning strategies, and strengthening university-industry collaboration, the proposed bidirectional teaching reform strategy significantly enriches theoretical mechanics education. This comprehensive approach not only addresses existing pedagogical shortcomings but also cultivates versatile and innovative engineering graduates capable of effectively responding to contemporary industry demands.

Overall, the bidirectional reform strategy, as operationalized through the logical framework in Figure 17, has demonstrated both conceptual robustness and practical effectiveness. The observed improvements in student academic outcomes and competency development align closely with the framework’s hypothesized pathways, confirming its value as both a design tool and an analytical model for future instructional innovation in engineering education.

8. Conclusions

This research proposes and empirically validates a bidirectional reform strategy for the teaching of theoretical mechanics, integrating a front-end focus on cultivating engineering thinking with a back-end emphasis on personalized assignment design. The strategy was implemented across three consecutive academic cohorts, with empirical results demonstrating a substantial increase in student excellence rates and enhanced conceptual engagement in the experimental group applying the full bidirectional model. The front-end reform employed active, problem-oriented instructional methods, such as case studies, project-based learning, and the BOPPPS model, to bridge the gap between abstract theoretical content and real-world engineering challenges. The effectiveness of this approach was reflected in the 2023–2024 academic cohort, where both the experimental and control groups exhibited noticeable improvements in course engagement and academic performance following the adoption of front-end strategies. The back-end reform complemented this by introducing personalized and collaborative homework tasks tailored to students’ learning needs, including programming-based simulations and team-based design projects. In the 2024–2025 academic year, the experimental group, which integrated personalized assignments with front-end reforms, achieved an excellent student proportion of 61.32%. This was markedly higher than the 36.48% observed in the control group that maintained front-end strategies alone, highlighting the critical role of personalized assignments in deepening theoretical comprehension and enhancing problem-solving skills.

By integrating these two mutually reinforcing strategies, the proposed reform model demonstrably improved both the quality and outcomes of theoretical mechanics instruction. It significantly enhanced students’ systems thinking, logical reasoning, and engineering problem-solving abilities, as evidenced by higher academic achievement rates and positive student feedback collected through reflection surveys. These findings substantiate the practical value of the bidirectional teaching reform and provide a replicable model for modernizing STEM education to cultivate innovation-driven, practice-oriented learning environments.

Despite these advances, the present study and much of the literature have notable limitations. First, the intervention was conducted with a relatively small student cohort in a single course, which limits statistical generalizability. Similar pilot studies often rely on quasi-experimental designs without random assignment; this can introduce selection bias and calls for caution when inferring broader effects. Future work should deploy larger, multi-site samples and randomized controlled trials to strengthen causal claims. Second, the duration of the reform was relatively short (only a few semesters). Longitudinal studies are needed to determine whether gains in motivation and understanding persist over time. Third, while personalized and collaborative assignments can deeply engage students, scaling such individualized work across large classes poses practical challenges. Crafting tailored problems and providing timely feedback for each student demands significant instructor effort and institutional support, so large-scale implementation may be difficult without dedicated resources or learning analytics tools. Finally, persistent inequities in technology and digital resources must be addressed. Addressing these gaps by ensuring equitable access to technology and training in digital skills will be crucial for scaling blended and personalized engineering curricula.

Future research should aim to address these limitations by employing randomized controlled trials across diverse institutional contexts to strengthen the validity of causal inferences. Incorporating multidimensional assessment tools, including standardized evaluations of critical thinking, collaborative problem solving, and longitudinal learning tracking, would provide a more comprehensive view of educational impacts. Future studies could also explore the differential effects of the bidirectional reform among students with varying academic backgrounds and learning preferences. Additionally, investigating the integration of emerging technologies such as adaptive learning platforms and learning analytics into the reform strategy may further enhance its effectiveness and scalability.