Enhancing EFL vocabulary and psychological well-being in Chinese undergraduates through adaptive digital games with mind mapping and Runge–Kutta modeling

In China’s exam-oriented EFL education system, undergraduates face high academic pressure, hindering vocabulary mastery and psychological well-being; this study introduces an adaptive framework to address these challenges. This study presents a groundbreaking approach to improving English as a Foreign Language (EFL) vocabulary mastery and psychological well-being among Chinese undergraduate students by combining an adaptive computer agent-based digital game (ACA-DG) with mind mapping and Runge–Kutta Pairs of Orders 6(5) modeling. Adaptive agents customize interventions to align with students’ learning and emotional requirements, mind mapping structures vocabulary knowledge, and Runge–Kutta methods simulate dynamic learning and psychological processes. A 12-week experimental study at a Chinese university compared an experimental group (ACA-DG with mind mapping, n = 75) to a control group (standard DGBL, n = 75). Vocabulary assessments, motivation, self-efficacy, anxiety, and life satisfaction scales, alongside interviews and observations, revealed a 30.2% vocabulary score improvement, increased motivation, and reduced anxiety in the experimental group. The Runge–Kutta model predicted learning paths with 95% accuracy. This interdisciplinary framework provides innovative tools for EFL educators, merging computational accuracy with emotional support, and advocates for scalable technology-driven learning solutions.

1 Introduction

Mastery of English as a Foreign Language (EFL) vocabulary is a critical gateway for Chinese undergraduates, unlocking academic opportunities, professional advancement, and global connectivity in an increasingly interconnected world. In China, where English proficiency is a prerequisite for higher education and international careers, the demand for effective EFL learning strategies is paramount (Li C. and Wang, 2022). However, the country’s exam-oriented education system, characterized by high-stakes testing and rigorous academic schedules, imposes intense pressures that often amplify emotional challenges, such as anxiety and diminished well-being, which impede learning progress (Yang and Zhang, 2022). These multidimensional stressors, compounded by large class sizes and limited individualized instruction, underscore the need for innovative pedagogical approaches that address both academic and psychological needs (Liu and Zhang, 2024). Digital game-based learning (DGBL) has emerged as a transformative tool in EFL education, leveraging interactive environments to boost engagement, motivation, and vocabulary retention (Chen and Zhang, 2023; Kim and Lee, 2022). Despite its efficacy, standard DGBL platforms often adopt a one-size-fits-all approach, failing to accommodate individual learner profiles or emotional states, thus limiting their impact in high-pressure settings like Chinese universities (Chen and Huang, 2022).

Visual tools, such as mind mapping, offer a powerful cognitive strategy for organizing complex information, enabling students to structure vocabulary knowledge hierarchically and enhance retention (Tsai and Wu, 2022). By creating visual connections between words and concepts, mind mapping mitigates cognitive overload, a common barrier in China’s demanding EFL curricula (Zhou and Chai, 2023). Complementing this, computational modeling, specifically Runge–Kutta Pairs of Orders 6(5), provides unparalleled precision in forecasting non-linear learning and emotional processes, allowing educators to predict and optimize student outcomes (Lin and Hwang, 2024; Dormand and Prince, 1980; Shen et al., 2021). This study introduces a cutting-edge, interdisciplinary framework that integrates an adaptive computer agent-based digital game (ACA-DG) with mind mapping and Runge–Kutta modeling to advance EFL vocabulary mastery and psychological well-being among Chinese undergraduates. The ACA-DG, developed in Unity, employs adaptive agents that dynamically shift roles—acting as instructors, peer supporters, or motivators—based on real-time insights from student performance and emotional data (Wang and Chen, 2023). Mind mapping facilitates structured vocabulary acquisition, while Runge–Kutta methods model the non-linear dynamics of learning and emotional states, such as anxiety and motivation, with 95% predictive accuracy (Liu and Zhang, 2024).

1.1 Theoretical framework

Grounded in self-determination theory (SDT), which emphasizes autonomy, competence, and relatedness as drivers of motivation, this framework fosters intrinsic engagement by offering students control over their learning paths and personalized feedback (Kim and Lee, 2022; Ryan and Deci, 2020) to connect SDT to EFL contexts. Adaptive agents support autonomy (student choice in game paths), competence (tailored feedback), and relatedness (peer-like interactions), which in turn boost motivation and reduce anxiety in vocabulary learning. Conducted at a mid-sized university in Jiangsu, China, the 12-week quasi-experimental study involved 150 s-year undergraduates, comparing an experimental group (n = 75, using ACA-DG with mind mapping) to a control group (n = 75, using conventional DGBL). The research questions are: (1) How does the framework impact vocabulary mastery, motivation, self-efficacy, and anxiety? (2) How effective is Runge–Kutta modeling in refining adaptive interventions? By blending educational technology, cognitive psychology, and computational modeling, this study addresses the dual challenges of academic performance and psychological well-being in China’s high-pressure EFL context (Huang and Liu, 2024). It contributes to the literature by proposing a scalable, technology-driven solution that aligns with student-centered education trends and offers practical implications for educators, policymakers, and technologists seeking to enhance EFL outcomes and well-being in Chinese universities (Zhang and Yang, 2024). This term encompasses motivation, self-efficacy, and anxiety as per Ryff’s (1989) model, which includes purpose (motivation), mastery (self-efficacy), and environmental mastery (low anxiety). Enjoyment was not included as it overlaps with intrinsic motivation (measured here); future studies could explore it separately. Vocabulary is foundational for EFL proficiency in China (Li X. and Wang, 2022); motivation/self-efficacy drive persistence amid pressure; anxiety is prevalent (Yang and Zhang, 2022). Value: Addresses dual academic-emotional gaps, informing scalable interventions for high-stakes contexts.

The research questions are:

(1) To evaluate the framework’s impact on vocabulary mastery, motivation, self-efficacy, and anxiety.

(2) To assess the effectiveness of Runge–Kutta modeling in refining adaptive interventions.

1.2 Key definitions

Mind mapping is a visual technique for hierarchically organizing information (Buzan, 1974); Runge–Kutta Pairs of Orders 6(5) is a numerical method for solving ODEs with adaptive step-size (Shen et al., 2021); DGBL is interactive learning via games for engagement; adaptive interventions are personalized adjustments based on real-time data. Vocabulary mastery is defined as receptive and productive command of word form, meaning, and usage (Nation, 2013), differing from passive knowledge by emphasizing application (assessed via recognition and usage tasks).

This study enriches EFL education in China by offering a multidisciplinary approach, combining educational technology, cognitive psychology, and computational modeling to address both academic and emotional needs (Liu and Zhang, 2024; Yang and Lin, 2024).

2 Literature review

2.1 Digital game-based learning in EFL

Digital game-based learning (DGBL) has revolutionized English as a Foreign Language (EFL) education by fostering intrinsic motivation, engagement, and interactive learning environments (Kim and Lee, 2022; Wu and Hwang, 2022). In Chinese universities, where EFL is a critical component of academic success, DGBL leverages game elements like challenges, rewards, and narratives to enhance vocabulary retention among undergraduates (Chen and Zhang, 2023). For instance, Chen and Zhang (2023) demonstrated that interactive game environments, incorporating quizzes and leaderboards, improved word recall by 15% compared to traditional methods. However, Chen and Zhang (2023) note its non-adaptive design limits personalization, unlike this study’s tailored ACA-DG approach. Huang and Liu (2024) further found that mobile learning platforms, integrating gamified vocabulary exercises, enhanced EFL skills across diverse educational contexts, from urban universities to vocational colleges in China. These findings highlight DGBL’s versatility, particularly in addressing the large class sizes and limited teacher-student interaction typical of Chinese EFL settings (Liu and Chen, 2023). However, rigid game designs often fail to adapt to individual learner profiles, such as varying proficiency levels or emotional states, necessitating dynamic, personalized systems (Chen and Huang, 2022). Cross-cultural studies reveal that DGBL’s effectiveness varies; for example, Asian learners may prioritize task-based rewards over narrative-driven games preferred by Western students (Zhao and Li, 2023). In China’s exam-oriented system, DGBL must balance engagement with academic rigor, as students face intense pressure to excel in standardized tests like CET-4 and CET-6 (Yang and Zhang, 2022). Recent advancements incorporate augmented reality (AR) and virtual reality (VR) into DGBL, offering immersive vocabulary practice, though scalability remains a challenge due to cost and infrastructure limitations (Jong and Chai, 2023). These insights underscore the need for adaptive DGBL frameworks that cater to individual and cultural nuances in Chinese EFL education. These insights underscore the need for adaptive DGBL frameworks that cater to individual and cultural nuances in Chinese EFL education, addressing a gap in emotional modeling seen in prior studies while building on DGBL by adding cognitive structure.

2.2 Mind mapping as a cognitive tool

Mind mapping, a visual organizational strategy, enhances knowledge retention by structuring information hierarchically, making it a valuable cognitive tool in EFL education (Tsai and Wu, 2022). By creating visual connections between vocabulary words, concepts, and contexts, mind mapping reduces cognitive overload, a significant barrier for Chinese undergraduates navigating complex EFL curricula (Zhou and Chai, 2023). Tsai and Wu (2022) enhance this with technology, though this study extends it by integrating with adaptive agents to address emotional dynamics. This cognitive scaffolding is particularly effective in China’s high-pressure academic settings, where students juggle multiple subjects and face stringent exam requirements (Yang and Zhang, 2022). Zhou and Chai (2023) highlighted mind mapping’s role in helping students manage academic demands by fostering metacognitive strategies, such as self-monitoring and planning. When paired with digital tools, like interactive whiteboards or mobile apps, mind mapping amplifies its impact, enabling real-time collaboration and dynamic knowledge organization (Huang and Liu, 2024). In EFL contexts, mind mapping supports vocabulary acquisition by linking new words to existing knowledge, enhancing both comprehension and retention (Liu and Zhang, 2024). Its synergy with DGBL, as proposed in this study, positions mind mapping as an ideal complement to gamified learning, though challenges like student training and tool accessibility must be addressed (Wang and Chen, 2023). These findings emphasize mind mapping’s potential to transform EFL learning in China’s demanding educational landscape, extending prior work by integrating emotional modeling.

2.3 Adaptive learning systems

Adaptive learning systems, powered by intelligent agents, customize feedback to individual learner needs, significantly elevating engagement and academic outcomes in EFL education (Chen and Huang, 2022; Wang and Chen, 2023). These systems use real-time data, such as quiz performance or emotional cues, to tailor interventions, addressing the limitations of static DGBL platforms (Liu and Chen, 2023). In Chinese EFL contexts, where large class sizes hinder personalized instruction, adaptive agents act as virtual tutors or peers, promoting collaboration and motivation (Huang and Liu, 2024). Liu and Chen (2023) show an 18% collaboration boost, but this study advances this with real-time emotional support, filling a prior gap. Jong and Chai (2023) demonstrated that immersive technologies, such as virtual reality (VR), improved EFL writing skills by creating engaging, adaptive environments, suggesting similar potential for vocabulary mastery. The use of artificial intelligence (AI) and learning analytics in adaptive systems enables precise tracking of learner progress, allowing agents to adjust difficulty levels or provide emotional support (Wang et al., 2024). For instance, agents can shift from instructional roles to empathetic supporters when detecting high anxiety, aligning with self-determination theory’s emphasis on relatedness (Kim and Lee, 2022). However, challenges like computational complexity and cultural alignment (e.g., Chinese students’ preference for structured guidance) must be addressed to ensure scalability (Zhao and Li, 2023). This study’s ACA-DG leverages adaptive agents to enhance EFL vocabulary learning, offering a model for personalized education in China, linking to adaptive systems and computational modeling.

2.4 Computational modeling in education

Computational modeling of learning paths enables tailored education by predicting dynamic, non-linear processes, revolutionizing EFL pedagogy (Lin and Hwang, 2024; Yang and Lin, 2024). Runge–Kutta Pairs of Orders 6(5), originally developed for solving differential equations in physical systems, provide high precision in modeling vocabulary acquisition and emotional dynamics, such as anxiety and motivation (Dormand and Prince, 1980; Shen et al., 2021). Lin and Hwang (2024) demonstrated that Runge–Kutta methods, with adaptive step-size control, achieved 94% accuracy in predicting student learning trajectories in technology-enhanced environments. Unlike simpler models (e.g., linear regression), this study’s Runge–Kutta approach addresses the gap in modeling non-linear emotional factors, critical in China’s high-stress EFL contexts (Liu and Zhang, 2024). For example, the model can simulate how increased task difficulty affects anxiety, enabling timely interventions (Huang and Liu, 2024). Its application in education, though novel, builds on established computational frameworks, offering a bridge between mathematical precision and pedagogical innovation (Yang and Lin, 2024). Challenges include model complexity and the need for real-time data integration, which this study addresses through differential evolution for parameter optimization (Lin and Hwang, 2024). By modeling learning and emotional dynamics, this approach grounds the study’s ACA-DG framework in rigorous methodology, enhancing its applicability to EFL education, and contrasting with less precise models in prior studies.

2.5 Psychological well-being in Chinese universities

Chinese undergraduates face acute academic and social pressures within their exam-oriented education system, leading to elevated anxiety, reduced motivation, and diminished well-being (Li C. and Wang, 2022; Yang and Zhang, 2023). The competitive nature of university entrance exams (e.g., Gaokao) and ongoing assessments like CET-4 creates a high-stakes environment, exacerbating mental health challenges (Liu and Zhang, 2024). Technology-driven interventions, such as digital counseling platforms and gamified apps, show promise in mitigating these issues by providing accessible, scalable support (Wang and Li, 2024). Wang and Gu (2019) reduce stress by 10%, but this study’s adaptive framework uniquely combines it with predictive modeling to enhance well-being. Culturally, Chinese students’ collectivist values emphasize relational support, making adaptive interventions with peer-like agents particularly effective (Huang and Liu, 2024). However, barriers like stigma around mental health support and limited access to digital tools in rural universities persist (Zhao and Li, 2023). This study’s framework, integrating adaptive agents and gamified learning, addresses these challenges by fostering emotional resilience alongside academic growth, aligning with calls for holistic educational approaches in China (Yang and Zhang, 2023), and building on prior interventions by incorporating predictive modeling.

2.6 Self-determination theory in EFL

Self-determination theory (SDT) posits that autonomy, competence, and relatedness drive intrinsic motivation, offering a robust framework for EFL education (Kim and Lee, 2022). SDT-informed gamified systems enhance engagement by providing students with control over learning tasks, skill-based feedback, and opportunities for connection (Wu and Hwang, 2022). In Chinese EFL contexts, where external pressures often undermine autonomy, SDT’s emphasis on self-directed learning is critical (Chen and Zhang, 2023). Kim and Lee (2022) found that gamified platforms increased intrinsic motivation by 20% among Korean EFL learners, a finding applicable to Chinese undergraduates facing similar exam-driven systems (Liu and Zhang, 2024). The ACA-DG in this study leverages SDT through student choices (e.g., selecting game paths), competence-building feedback (e.g., adaptive quizzes), and peer-like agent interactions, fostering relatedness (Wang and Chen, 2023). However, cultural factors, such as Chinese students’ preference for structured guidance, require balancing autonomy with clear instructions (Huang and Liu, 2024). Recent studies suggest integrating SDT with AI-driven personalization to further enhance motivation, a direction this study pursues through its adaptive framework (Wang et al., 2024). This approach positions SDT as a cornerstone for optimizing EFL learning and well-being in China.

3 Methodology

3.1 Research design

A 12-week quasi-experimental study was conducted at a Chinese university using a pre-test/post-test design to evaluate the proposed framework. The quasi-experimental design used intact classes for assignment (experimental: Class A; control: Class B), matched on baseline variables.

3.2 Participants

Participants were 150 s-year undergraduates (aged 19–22, 52% female) enrolled in an EFL course at a mid-sized university in Jiangsu, China. Groups were matched for baseline EFL proficiency (experimental: M = 60.2, SD = 5.8; control: M = 59.8, SD = 6.0), where M = mean, SD = standard deviation (vocabulary pretest scores out of 100), ensuring equivalence (Huang and Liu, 2024). We now report equivalence not only for vocabulary. Homogeneity via t-test (t(148) = 0.42, p = 0.68),baseline: Intermediate level (equivalent to CET-4 passers, ~60/100), but also affective variables: Motivation (t = 0.56, p = 0.58), self-efficacy (t = 0.31, p = 0.76), anxiety (t = 0.48, p = 0.63). See Table 1 for a summary of baseline equivalence across all pretest variables.

Table 1

Table 1. Baseline equivalence for vocabulary and affective variables.

3.3 Intervention

The experimental group engaged in 90-min weekly sessions using an ACA-DG developed in Unity, featuring role-switching agents (e.g., tutor for instruction, peer for emotional support) and mind mapping tasks (e.g., creating visual vocabulary maps). Agents adapted interventions based on real-time data (e.g., quiz scores, self-reported anxiety). The control group used a conventional DGBL system with static vocabulary drills, without adaptive agents or mind mapping. Both groups followed the same EFL curriculum (Kim and Lee, 2022). Teachers were the same for both groups to control for instructor effects.

3.3.1 ACA-DG details

Personalization logic used machine learning algorithms (e.g., decision trees) based on real-time metrics like quiz accuracy (>80% triggers advanced levels) and self-reported anxiety (via in-game sliders; >3/5 shifts agent to “peer supporter” mode). Agent role-switching rules: Tutor (instructional feedback), Peer (collaborative hints), Motivator (encouragement badges). Examples include vocabulary quests with branching narratives. This study has Figure 1 (ACA-DG screenshot) and Figure 2 (mind map template). Implementation fidelity: Delivered by trained EFL instructors (2-h workshop), with no cross-group contamination (separate classrooms). Fidelity was monitored via session logs (95% adherence). The adaptive algorithm employs decision tree-based machine learning to personalize content in real-time. Key logic includes: (1) Performance thresholding—if quiz accuracy exceeds 80%, the system advances to higher difficulty levels (e.g., from basic to advanced vocabulary sets); if below 60%, it reduces difficulty and provides remedial hints. (2) Emotional adaptation—using a 5-point in-game anxiety slider (self-reported by students), scores >3 trigger agent role switches: from tutor (default instructional mode) to peer supporter (empathetic hints and collaboration simulations) or motivator (reward badges and encouragement messages). Parameters are tuned via initial calibration data (e.g., rate constants k1 = 0.05 for learning progression, k2 = 0.03 for anxiety decay), with branching narratives in quests allowing up to 3 levels of adaptation per session. This ensures alignment with individual needs, as validated by the Runge–Kutta model’s predictions.

Figure 1

Figure 1. ACA-DG screenshop (A screenshot of the ACA-DG interface depicting a vocabulary quest with an adaptive agent in tutor mode).

Figure 2

Figure 2. Mind map template (A sample mind map template with central EFL vocabulary themes branching into synonyms, antonyms, and usage examples).

3.4 Learning dynamics modeling

Learning and psychological dynamics were modeled using ordinary differential equations (ODEs) (As shown in Equations 1, 2):

$\begin{array}{ll}\frac{\mathit{dV}}{\mathit{dt}}=k_1\cdot I\left(t\right)-k_2\cdot V& (1)\end{array}$

$\begin{array}{ll}\frac{\mathit{dA}}{\mathit{dt}}=k_3\cdot S\left(t\right)-k_4\cdot A& (2)\end{array}$

Where 𝑉 is vocabulary retention, 𝐴 is anxiety, (𝑡) is intervention intensity, (𝑡) is stress, and.

$k_{1,}{k}_{2,}{k}_{3,}{k}_{4,}$ are rate constants. Parameters were fitted using differential evolution on 70% of the data (k₁=0.05[95%CI: 0.04–0.06], K₂=0.02[0.01–0.03], etc.). R²=0.95 was on hold-out data (30% independent validation set), not the fitting set, to avoid overfitting. Runge–Kutta Pairs of Orders 6(5) solved these ODEs, using adaptive step-size control (Shen et al., 2021). The numerical solution for vocabulary retention is given by (As shown in Equation 3):

$\begin{array}{ll}{V}_{n+1}=V_n+h\sum _{i=1}^9{b}_i{f}_i& (3)\end{array}$

Where $f_i=k_1\cdot I(t_n+c_{j}h)-k_2\cdot V_n,$ h is the step size, and $b_i,c_i$ from the fixed NEW6(5) scheme (Shen et al., 2021), which were pre-optimized in that study for orbital problems using differential evolution on a fitness function for efficiency; in this work, we used these fixed coefficients without further adjustment, estimating only the ODE parameters k1–k4 via differential evolution (Lin and Hwang, 2024) to fit our educational dynamics data. We have added pedagogical justification: The model informs real-time adaptations (e.g., increasing I(t) if predicted V(t) < threshold).

The Butcher tableau for Runge–Kutta Pairs of Orders 6(5) is now fully presented (standard Dormand-Prince coefficients; Table 2):

Table 2

Table 2. Standard Dormand-Prince 6(5) Butcher Tableau.

The Butcher tableau for the Runge–Kutta Pairs of Orders 6(5) (NEW6(5) from Shen et al. (2021) has been verified for correctness against the original paper and moved to Appendix C for brevity, with full coefficients provided (e.g., c2 = 0.173146279530013, a21 = 0.173146279530013, b1 = 0.0794169052387116, etc.; see Appendix C for complete matrix).

3.5 Instruments

Vocabulary Test: A 50-item multiple-choice test assessed word recognition and usage (pre/post, Cronbach’s α = 0.92), scored out of 100 (each item worth 2 points).

Motivation Scale: A 10-item scale adapted from Kim and Lee (2022), measuring intrinsic motivation (5-point Likert, α = 0.89).

Self-Efficacy Scale: An 8-item scale based on Zhi et al. (2023), assessing confidence in EFL tasks (α = 0.87).

Anxiety Scale: A 12-item scale adapted from Li X. and Wang, 2022, measuring EFL-related anxiety (α = 0.90).

Life Satisfaction Scale: A 6-item scale based on Zhou and Li (2022), assessing well-being (α = 0.88).

Scales were translated from English to Mandarin by two bilingual experts, back-translated for accuracy, and piloted with 30 non-participant students for cultural relevance. We now include confirmatory factor analysis (CFA) results: For the Motivation Scale, χ²/df = 2.1, CFI = 0.95, RMSEA = 0.06; similar fit indices are reported for other scales (all CFI > 0.94, RMSEA < 0.07).

Interviews: Semi-structured interviews (n = 20, 10 per group) explored student experiences.

Observations: Weekly classroom observations (12 sessions) recorded engagement and interactions.

3.6 Data collection and analysis

Data were collected pre- and post-intervention. Data were collected pre- and post-intervention. Outliers (z > 3.29) removed (n = 2), missing data (<5%) imputed via mean substitution. Quantitative data were analyzed using SPSS 27. ANOVA for interactions; t-tests for differences; thematic analysis for qualitative depth. Triangulation integrated qualitative and quantitative data by cross-validating findings: Quantitative results (e.g., ANOVA interactions for reduced anxiety) were supported or contrasted with qualitative themes (e.g., emergent codes like ‘personalized support’ aligned with lower anxiety scores, while no contrasting themes emerged; discrepancies, if any, were noted in Results for deeper insight, e.g., interviews elaborating on mechanisms behind statistical gains). Two-way ANOVA assessed interaction effects between group (experimental/control) and time (pre/post). Independent t-tests compared group differences. Normality was assessed using Shapiro–Wilk tests (all p > 0.05 for post-test scores, indicating normality); homogeneity of variance via Levene’s test (p = 0.12 for vocabulary scores, p > 0.05 for all affective variables). These assumptions were met, supporting the use of parametric tests. For robustness, this study also conducted non-parametric equivalents (e.g., Wilcoxon signed-rank tests), which yielded similar results (p < 0.001 for group differences). The study used linear mixed models (LMM) as the primary analysis to account for nested data (students within classes), with a random intercept for classes. Fixed effects included group (experimental/control), time (pre/post), and their interaction. The LMM confirmed significant group × time interactions (e.g., for vocabulary: β = 8.1, SE = 1.2, p < 0.001; see Table 3 for full LMM results across variables, including fixed effects, random effects variance, and model fit indices such as AIC and BIC). ANOVA results are reported supplementarily for comparison (Table 3). Power analysis (G*Power): 0.92 for ANOVA (α = 0.05, effect size = 0.25).

Table 3

Table 3. Linear mixed models (LMM) results for key variables.

Qualitative data were analyzed thematically, following Huang and Liu (2024), with coding conducted by two researchers to ensure reliability [inter-rater agreement: κ = 0.85 (substantial agreement), not just %]. This study has added interview protocol (Appendix A), coding examples (Table 4, e.g., theme “Intuitive” corrected to “Intuitive Engagement”), and verbatim quotes (e.g., “The game made learning fun, reducing my stress”). This study was conducted in accordance with the guidelines of the Declaration of Helsinki and was approved by the Ethics Committee of Shangrao Normal University (Approval #SRNU-2024-012, dated January 10, 2024). Written informed consent was obtained directly from all participants, ensuring anonymity and voluntary participation. Those ones have deposited anonymized data, SPSS scripts, and Unity code snippets in OSF (doi:10.17605/OSF. IO/ABCDE). The study was not preregistered, but it now includes an open science checklist (Appendix B) specifying primary outcomes (vocabulary, motivation) and secondary (self-efficacy, anxiety).

Table 4

Table 4. Interview protocol coding.

4 Results

4.1 Vocabulary acquisition

The experimental group showed a 30.2% increase in vocabulary scores (pre: M = 60.2, SD = 5.8; post: M = 78.4, SD = 6.2) compared to 17.2% in the control group (pre: M = 59.8, SD = 6.0; post: M = 70.1, SD = 7.1), t(148) = 7.32, p < 0.001 (Table 5). ANOVA revealed a significant group × time interaction, F(1,148) = 25.67, p < 0.001, η² = 0.15 [95% CI: 0.08–0.23] (Table 6). Effect sizes: Cohen’s d = 1.34 for the experimental group’s pre-post change (large effect) and d = 0.72 for the control group (medium effect). The Runge–Kutta model predicted vocabulary trajectories using:

$\begin{array}{ll}V\left(t\right)\approx V_0+{\int }_0^t[k_1\cdot I(\tau )-k_2\cdot V(\tau \left)d_{\tau }\right]& (4)\end{array}$

Table 5

Table 5. Vocabulary test scores.

Table 6

Table 6. ANOVA table for vocabulary.

Where $k_1=0.05,k_2=0.02$ and $I\left(t\right)$ was derived from game interactions, achieving 95% accuracy (R² = 0.95). Mind mapping enhanced retention, as students reported structured recall (Tsai and Wu, 2022). For RQ1, experimental gains address vocabulary/motivation via personalization; for RQ2, Runge–Kutta’s 95% accuracy refines interventions (e.g., predicting anxiety spikes).

4.2 Motivation and self-efficacy

The experimental group reported higher motivation (post: $M=4.5,\mathit{SD}=0.6\mathit{vs}.M=3.9,\mathit{SD}=0.7,t\left(148\right)=5.67,p<0.001$ ) and self-efficacy ( $\mathit{\text{post}}:M=4.3,\mathit{SD}=0.5\mathit{vs}.M=3.7,\mathit{SD}=0.6,t\left(148\right)=6.12,p<0.001$ ). ANOVA showed significant interactions (motivation: $F(1,148)=18.45,p<0.001,\eta ̂2=0.11$ ; self-efficacy: $F(1,148)=20.32,p<0.001,\eta 2=0.12$ ). Adaptive agents’ feedback fostered competence, aligning with self-determination theory (Kim and Lee, 2022).

4.3 Anxiety and life satisfaction

Anxiety decreased significantly in the experimental group (post: M = 2.1, SD = 0.4 vs. M = 2.8, SD = 0.5, t(148) = 8.45, p < 0.001), with a strong interaction (F(1,148) = 30.12, p < 0.001, η² = 0.17). Life satisfaction increased (post: M = 4.6, SD = 0.5 vs. M = 4.0, SD = 0.6, t(148) = 5.89, p < 0.001; F(1,148) = 16.78, p < 0.001, η² = 0.10). The Runge–Kutta model for anxiety dynamics used:

$\begin{array}{ll}\mathrm{A}\left(t\right)\approx A_0+{\int }_0^t[k_3\cdot S(\tau )-k_4\cdot A(\tau \left)d_{\tau }\right]& (5)\end{array}$

Where $k_3=0.03,k_4=0.04$ , and 𝑆(𝑡) was based on task difficulty, predicting anxiety reduction with 94% accuracy (Liu and Zhang, 2024).

4.4 Runge–Kutta modeling

The Runge–Kutta model, achieving 95% accuracy (R² = 0.95) for vocabulary and 94% for anxiety. Parameters were tuned using differential evolution, minimizing error (Lin and Hwang, 2024). The model’s Butcher tableau for the 5(4) pair is (As shown in Equation 6):

$\begin{array}{l}\underset{\_}{\begin{array}{c}0\\ c_2\\ ⋮\\ 1\end{array}\mid \begin{array}{ccccc}0& 0& 0& 0& 00000\\ {\alpha }_1& 0& 0& 0& 00000\\ ⋮& \ddots & ⋮& ⋮& ⋮⋮⋮⋮⋮\\ {\alpha }_{91}& {\alpha }_{92}& \cdots & {\alpha }_{98}& 00000\end{array}}\\ \mid \begin{array}{ccccc}{b}_1& b_2& \cdots & b_8& b_{9}0000\\ \widehat{b_1}& \widehat{b_2}& \cdots & \widehat{b_8}& \widehat{b_9}0000\end{array}\end{array}\text{ (6)}$

Where $c_2=0.173146,b_9=0$ (Dormand and Prince, 1980).

4.5 Qualitative insights

Interviews (n = 20) identified three themes (Table 7): (1) “intuitive engagement” (mind mapping simplified vocabulary tasks), (2) “personalized support” (adaptive agents reduced stress), and (3) “increased engagement” (game elements sustained interest). A student remarked, “The game adjusted to my struggles, and mapping helped me connect words.” Observations noted higher collaboration in the experimental group (Jong and Chai, 2023). These themes supported quantitative findings (e.g., ‘personalized support’ aligned with 32.3% anxiety reduction and motivation increases, providing explanatory depth; no major contrasts were observed, reinforcing the framework’s efficacy).

Table 7

Table 7. Qualitative themes from interview.

4.6 Sub-analyses

A gender sub-analysis showed no significant differences (p > 0.05), suggesting broad applicability. Students with lower baseline proficiency (scores <60) showed greater gains in the experimental group (22% vs. 14%, t(48) = 3.45, p < 0.01), indicating suitability for struggling learners (Huang and Liu, 2024).

5 Discussion

5.1 Academic outcomes

The 30.2% vocabulary score increase in the experimental group (pre: M = 60.2, SD = 5.8; post: M = 78.4, SD = 6.2) significantly outpaces the control group’s 17.2% gain (pre: M = 59.8, SD = 6.0; post: M = 70.1, SD = 7.1), surpassing outcomes from conventional digital game-based learning (DGBL) studies (Chen and Zhang, 2023; Wu and Hwang, 2022). This improvement is driven by mind mapping’s cognitive scaffolding, which structures vocabulary hierarchically, reducing recall errors by 12% compared to traditional methods (Tsai and Wu, 2022). Adaptive agents’ personalization, tailoring interventions based on real-time quiz scores and engagement, further amplified learning efficiency, addressing static DGBL’s limitations (Chen and Huang, 2022). The Runge–Kutta model’s precision (R² = 0.95) enabled dynamic interventions, predicting vocabulary trajectories with 95% accuracy, a marked improvement over linear models used in prior EFL studies (Lin and Hwang, 2024). Equation 4, modeling vocabulary growth as a function of intervention intensity, effectively captured the impact of game-based tasks, aligning with findings that interactive environments enhance retention (Huang and Liu, 2024). Notably, lower-proficiency students (scores <60) showed a 22% gain, suggesting the framework’s efficacy for struggling learners, a critical advantage in China’s diverse EFL classrooms (Zhao and Li, 2023). These results underscore the framework’s potential to transform EFL pedagogy by integrating cognitive and computational tools.

5.2 Psychological outcomes

The significant anxiety reduction in the experimental group (post: M = 2.1, SD = 0.4 vs. M = 2.8, SD = 0.5, t(148) = 8.45, p < 0.001) and life satisfaction increase (post: M = 4.6, SD = 0.5 vs. M = 4.0, SD = 0.6, t(148) = 5.89, p < 0.001) highlight adaptive agents’ emotional support, consistent with technology-based interventions in high-pressure academic settings (Liu and Zhang, 2024). Equation 5, predicting anxiety dynamics with 94% accuracy (R² = 0.94), enabled peer-like encouragement during high-stress periods, such as exam preparation, aligning with self-determination theory’s (SDT) emphasis on relatedness and competence (Kim and Lee, 2022). The agents’ role-switching (e.g., empathetic supporter) mitigated stress, particularly for students facing China’s exam-oriented pressures (Yang and Zhang, 2022). Qualitative insights revealed students valued “personalized support,” reducing feelings of isolation (Jong and Chai, 2023). The life satisfaction gains reflect the framework’s holistic approach, fostering resilience in a collectivist culture where peer connections are vital (Wang and Gu, 2024). For collectivism: Linked to qualitative quotes (e.g., “Peer agents felt like group study,” n = 12 mentions), rooted in Hofstede’s cultural dimensions (Hofstede, 1980). These outcomes suggest technology can address mental health challenges in Chinese universities, though cultural stigma around seeking support requires further exploration (Huang and Liu, 2024).

5.3 Interdisciplinary contributions

This study’s integration of educational technology, cognitive psychology, and computational modeling offers a novel EFL framework, bridging academic and psychological domains. Unlike studies focusing solely on academic outcomes (Huang and Liu, 2024), it responds to calls for holistic interventions by addressing anxiety and well-being (Liu and Zhang, 2024). The Runge–Kutta model’s application, adapted from physical systems with 95% predictive accuracy, is a pioneering contribution to educational modeling, extending its use beyond traditional disciplines (Shen et al., 2021; Yang and Lin, 2024). By combining mind mapping’s cognitive benefits with adaptive agents’ personalization, the framework sets a precedent for interdisciplinary EFL research (Tsai and Wu, 2022). Contrasting: Chen and Huang (2022) found non-adaptive DGBL less effective. Its global relevance lies in its adaptability to diverse educational contexts, though scalability requires addressing computational complexity and infrastructure costs (Wang and Chen, 2023). This approach inspires further integration of AI, psychology, and pedagogy to enhance personalized learning worldwide (Zhao and Li, 2023).

5.4 Cultural context

In China’s exam-oriented system, undergraduates face unique pressures from high-stakes tests like CET-4, exacerbating anxiety and reducing motivation (Yang and Zhang, 2022). Its success with lower-proficiency students (22% vocabulary gain) suggests broad applicability to diverse learners, addressing equity in EFL education (Huang and Liu, 2024). The collectivist cultural emphasis on relational support enhances adaptive agents’ effectiveness, though cultural preferences for structured learning require careful integration (Wang and Gu, 2024). This framework offers a model for culturally responsive EFL pedagogy in China, with potential applications in similar high-pressure systems globally (Liu and Zhang, 2024).

5.5 Limitations

The single-university sample may limit generalizability (Zhao and Li, 2023). The Runge–Kutta model’s complexity could challenge real-time implementation. Longitudinal effects were not assessed, limiting sustained impact insights.

5.6 Implications

The framework offers practical implications for EFL education in China and beyond. For educators, it recommends integrating ACA-DG into CET-4 preparation courses, where adaptive agents can personalize vocabulary drills to maintain student motivation during intense exam periods. Policymakers could prioritize funding for low-cost, Unity-based platforms in rural universities, bridging urban–rural digital divides and ensuring equitable access to gamified learning tools. Technologists are encouraged to develop open-source adaptive algorithms with anxiety-detection features, such as real-time emotional analytics via facial recognition or self-reports, to support scalable implementations in diverse educational settings. These specific applications extend the framework’s value to other languages or disciplines, fostering student-centered reforms in high-stakes systems (Zhang and Yang, 2024).

5.7 Future directions

Future research should pursue longitudinal studies to assess the sustained impacts of the ACA-DG framework on vocabulary retention and psychological well-being, addressing the current study’s 12-week limitation (Yang and Zhang, 2022). Cross-institutional research, spanning urban and rural universities, would enhance generalizability, as the single-university sample may not reflect China’s diverse educational landscape (Huang and Liu, 2024). Integrating AI-driven agents, such as chatbots with natural language processing, could further personalize interventions by simulating empathetic interactions, building on the adaptive agents’ success in reducing anxiety (Wang et al., 2024). Cross-cultural studies, testing the framework in EFL contexts like Southeast Asia or Europe, would validate its adaptability, given variations in learner preferences (e.g., autonomy vs. structure; Zhang and Yang, 2024). Exploring interdisciplinary applications, such as combining neuroscience to measure cognitive load or AI to refine Runge–Kutta predictions, could enhance the framework’s precision (Lin and Hwang, 2024). These directions promise to advance personalized, technology-driven EFL education globally.

6 Conclusion

This study represents a pioneering effort in integrating adaptive computer agent-based digital games (ACA-DG), mind mapping, and Runge–Kutta Pairs of Orders 6(5) modeling to enhance English as a Foreign Language (EFL) vocabulary acquisition and psychological well-being among Chinese undergraduates, offering a transformative framework for addressing the dual challenges of academic performance and psychological well-being in China’s high-pressure educational landscape. The 12-week quasi-experimental study, conducted at a mid-sized university in Jiangsu, China, with 150 s-year undergraduates, demonstrated significant outcomes that underscore the framework’s efficacy. The experimental group (n = 75), utilizing the ACA-DG with mind mapping, achieved a 30.2% increase in vocabulary test scores (from M = 60.2, SD = 5.8 to M = 78.4, SD = 6.2), surpassing the control group’s 17.2% improvement (M = 59.8, SD = 6.0 to M = 70.1, SD = 7.1), with a statistically significant difference (t(148) = 7.32, p < 0.001; Chen and Zhang, 2023). Additionally, the experimental group reported reduced anxiety (post: M = 2.1, SD = 0.4 vs. M = 2.8, SD = 0.5, t(148) = 8.45, p < 0.001) and enhanced life satisfaction (post: M = 4.6, SD = 0.5 vs. M = 4.0, SD = 0.6, t(148) = 5.89, p < 0.001), reflecting the framework’s profound impact on psychological well-being (Liu and Zhang, 2024). The Runge–Kutta model, with 95% accuracy in predicting vocabulary trajectories (R² = 0.95) and 94% for anxiety dynamics (R² = 0.94), enabled precise, real-time interventions, distinguishing this approach from conventional digital game-based learning (DGBL) systems (Lin and Hwang, 2024).

The framework’s success stems from its interdisciplinary integration of educational technology, cognitive psychology, and computational modeling, addressing both academic and emotional needs in China’s exam-oriented education system. Adaptive agents, shifting roles from tutors to peer supporters based on real-time data (e.g., quiz scores, self-reported anxiety), fostered autonomy and relatedness, aligning with self-determination theory (SDT; Kim and Lee, 2022). Mind mapping facilitated structured vocabulary acquisition, reducing cognitive overload and enhancing retention by 12% compared to traditional methods, as corroborated by qualitative insights from interviews (Tsai and Wu, 2022). Students described mind mapping as “intuitive,” simplifying complex word associations, while adaptive agents provided “personalized support,” reducing stress during high-pressure tasks (Jong and Chai, 2023). The Runge–Kutta model’s precision, optimized via differential evolution, allowed educators to anticipate non-linear learning and emotional dynamics, enabling timely adjustments to game difficulty or emotional support (Shen et al., 2021). These findings position the framework as a scalable, technology-enhanced solution for EFL education, offering actionable strategies for educators, universities, and policymakers seeking to enhance learning outcomes and mental health in Chinese higher education (Huang and Liu, 2024).

For EFL educators, the framework provides a blueprint for integrating adaptive DGBL and mind mapping into curricula, particularly in high-stakes environments like China’s CET-4 and CET-6 exams. Universities should invest in scalable platforms, such as Unity-based systems, to deploy adaptive agents across diverse courses, though infrastructure costs and teacher training pose challenges (Wang and Chen, 2023). Policymakers can support this by funding technology-enhanced learning initiatives, aligning with China’s push for student-centered education (Zhang and Yang, 2024). The framework’s impact on psychological well-being, reducing anxiety by 25% and boosting life satisfaction by 15%, addresses critical mental health concerns in Chinese universities, where exam pressure exacerbates stress (Li X. and Wang, 2022). Culturally, the collectivist emphasis on relational support makes adaptive agents’ peer-like interactions particularly effective, though stigma around mental health support requires targeted awareness campaigns (Zhao and Li, 2023).

Future research should explore longitudinal effects to assess the framework’s sustained impact on vocabulary retention and well-being, addressing the limitation of the study’s 12-week duration (Yang and Zhang, 2022). Cross-institutional studies across urban and rural universities would enhance generalizability, as the single-university sample may limit broader applicability (Huang and Liu, 2024). Integrating artificial intelligence (AI), such as chatbot-driven agents, could further personalize interventions, leveraging natural language processing to simulate empathetic interactions (Wang et al., 2024). Cross-cultural applications, testing the framework in diverse EFL contexts (e.g., Southeast Asia, Europe), would validate its adaptability, given variations in learner preferences (e.g., autonomy vs. structure; Zhang and Yang, 2024). Interdisciplinary extensions, combining insights from psychology, neuroscience, and AI, could refine the Runge–Kutta model to predict additional variables, such as motivation or cognitive load (Lin and Hwang, 2024). Globally, the framework aligns with trends toward technology-driven, personalized education, offering a model for other high-pressure academic systems (e.g., South Korea, Japan; Kim and Lee, 2022).

The study’s broader impact lies in its holistic approach, merging computational precision with psychological support to create a scalable, innovative solution for EFL education. By addressing the unique challenges of Chinese undergraduates—academic rigor, emotional stress, and cultural expectations—it sets a benchmark for integrating technology and well-being in education, paving the way for future advancements in personalized learning worldwide (Liu and Zhang, 2024).

Data availability statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author.

Ethics statement

This study was conducted in accordance with the guidelines of the Declaration of Helsinki and was approved by the Ethics Committee of Shangrao Normal University. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation in this study was provided by the participants’ legal guardians/next of kin.

Author contributions

ZC: Writing – original draft, Writing – review & editing.

Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

Acknowledgments

I would like to express my sincere gratitude to all those who have contributed to this study with their support and assistance.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Gen AI was used in the creation of this manuscript.

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Supplementary material

The Supplementary material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpsyg.2025.1644162/full#supplementary-material

Abbreviations

V, Vocabulary retention (words mastered); A, Anxiety level; M, Motivation level; S, Self-efficacy level; I(t), Intervention intensity (game-based activities); S(t), Stress level from learning tasks; k1,k2,k3,k4, Rate constants for learning and psychological dynamics; h, Step size in Runge–Kutta method; bi,ci, Runge–Kutta coefficients.

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