Abstract
The rapid shift to online teaching during the COVID-19 pandemic introduced both challenges and opportunities for effective classroom management. This review explores 20 empirical studies published between 2021 and 2025, examining how educators have adapted to the virtual learning environment. The findings highlight key strategies that contribute to successful classroom management, including maintaining student engagement, fostering clear and timely communication, and leveraging technological tools effectively. A blend of traditional pedagogical approaches and innovative digital practices appears essential in creating structured yet flexible online learning spaces. Strategies such as interactive elements, personalized feedback, and well-defined expectations help sustain student attention and participation. Despite these advancements, gaps remain in understanding the long-term effectiveness and adaptability of these methods across diverse educational settings. By synthesizing recent research, this study offers valuable insights into best practices for managing online classrooms in the post-pandemic era, providing educators with practical strategies to enhance engagement and learning outcomes.
Plain Language Summary
The switch to online teaching during the COVID-19 pandemic created new challenges for teachers. This study looks at how teachers have adapted to online classes, focusing on what has worked well. We reviewed 20 studies published between 2021 and 2025 that explored different strategies for managing online classrooms. The research shows that teachers who successfully managed their online classes did so by keeping students engaged, communicating clearly and often, and using technology effectively. These strategies helped create online learning environments where students could stay focused and motivated. However, we found that there’s still a lot to learn about how these strategies work over time and in different types of schools. This study helps educators and schools understand what makes online teaching successful and what areas need more attention in the future.
Keywords
Introduction
Classroom management is widely recognized as a cornerstone of effective teaching, as it fosters a structured and engaging learning environment where every student has the opportunity to succeed (Marzano et al., 2003; Pianta et al., 2020). Beyond minimizing disruptions and maximizing instructional time, effective classroom management has been strongly linked to improved student engagement, academic performance, and positive teacher-student relationships (Bond & Bedenlier, 2019; Martin & Bolliger, 2018).In recent years, the philosophy of classroom management has evolved to include trauma-informed approaches, equity-driven discipline models, and the integration of social-emotional learning (SEL), all of which aim to promote not only behavioral order but also student connectedness and psychological well-being (Darling-Hammond, 2023). These contemporary approaches emphasize the importance of clear rules and expectations, consistent consequences, and nurturing a positive classroom climate (Emmer & Sabornie, 2015).
Building on these evolving perspectives, classroom management also plays a pivotal role in supporting students’ social-emotional development. A well-managed classroom creates a safe and supportive environment where students feel respected and valued, fostering positive interactions and a sense of belonging (Bear, 2010). Integrating social-emotional learning (SEL) practices can further enhance students’ self-regulation, empathy, and cooperation, contributing to a more inclusive and harmonious learning atmosphere (Jones & Bouffard, 2012). Beyond promoting emotional well-being, effective classroom management also improves student focus, strengthens social-emotional skills, and supports overall academic success (Osher et al., 2016).
In traditional, face-to-face classrooms, teachers rely on a combination of physical presence, nonverbal cues, and real-time feedback to maintain engagement and order. Strategies such as proximity, eye contact, and immediate intervention help manage disruptions and sustain student attention (Emmer & Sabornie, 2015). These approaches not only support discipline but also enhance student-teacher relationships by providing instant, personalized interactions. However, the increasing shift to online learning has introduced unique challenges, as digital classrooms lack the immediacy of physical and social dynamics. Ensuring participation, addressing disruptions, and maintaining engagement in virtual environments often requires teachers to adapt their strategies, leveraging digital tools and asynchronous communication methods (Borup et al., 2014; Martin & Bolliger, 2018).
The shift to online learning has fundamentally transformed classroom management dynamics, requiring educators to adopt more flexible, student-centered strategies (Johnson et al., 2021). Without physical presence and direct teacher-student interactions, maintaining engagement, fostering community, and ensuring student accountability presents significant challenges (Hodges et al., 2020; Herdi et al., 2025). As a result, teachers must develop alternative strategies, such as structured discussion forums, interactive multimedia content, and real-time virtual check-ins, to replicate the supportive and engaging environment of a well-managed physical classroom.
One key adaptation in online classroom management is the way teachers establish and enforce rules and expectations. Without the ability to use physical presence, direct eye contact, or immediate interventions, teachers must rely more on clear communication and structured routines (Borup et al., 2014). For instance, setting explicit guidelines for participation, camera use, and communication etiquette becomes essential in an online environment, where students might feel less accountable due to the lack of physical presence.
Beyond establishing rules and expectations, maintaining student engagement presents another challenge where online classroom management diverges from traditional practices. In a physical classroom, teachers can assess engagement through visual and auditory signals like eye contact and body language. Online, these cues are often missing, making it harder to gauge if students are actively participating or merely going through the motions (Martin & Bolliger, 2018). To address this, educators may need to adopt more interactive and student-centered teaching methods, such as live polls, breakout rooms, and collaborative projects, to maintain student involvement and engagement.
In addition to engagement challenges, online teaching can also amplify issues related to equity and access, which may be less pronounced in face-to-face settings. Technical difficulties, such as poor internet connections and lack of appropriate devices, can create significant barriers for some students (Muilenburg & Berge, 2005). These barriers can lead to disengagement and absenteeism, making classroom management in virtual settings even more complex. Moreover, the digital divide may exacerbate existing inequalities, as students from lower socioeconomic backgrounds face greater hurdles in accessing and fully participating in online learning environments.
Beyond addressing engagement and accessibility challenges, the online environment also necessitates a rethinking of how to build classroom community and foster positive student-teacher relationships. In a traditional classroom, these relationships develop organically through daily interactions and informal conversations. However, in an online setting, these opportunities are more limited, requiring teachers to be intentional about creating spaces for connection and collaboration (Keengwe & Kidd, 2010). Strategies such as discussion forums, virtual office hours, and group projects can help enhance students’ sense of belonging and strengthen relationships with both peers and instructors.
In summary, while the core principles of classroom management are essential in both online and traditional settings, the virtual environment presents unique challenges that require adapted strategies. The shift to online learning has necessitated new approaches to maintaining order and fostering engagement, with a focus on clear communication, structured routines, and innovative technologies. Educators also need to be mindful of equity and access issues, which can significantly affect classroom management effectiveness in a digital context. As online education continues to evolve, understanding these differences and developing effective management practices will be critical for creating successful and inclusive learning environments.
Research on online classroom management has expanded significantly since the onset of the COVID-19 pandemic, highlighting the challenges and adaptations required for effective virtual instruction (e.g., Bond & Bedenlier, 2019; Johnson et al., 2021; Martin & Bolliger, 2018). Studies have explored engagement strategies, communication practices, and technology integration, shedding light on best practices for digital learning environments (e.g., Hodges et al., 2020; Herdi et al., 2025). Additionally, research on educators’ experiences has underscored the need for professional development and institutional support in navigating online teaching (e.g., Cutri et al., 2020). However, despite these contributions, a comprehensive review synthesizing empirical studies on classroom management strategies in online settings remains limited. This review aims to address that gap by critically analyzing recent research, identifying effective strategies for online classroom management, and outlining key areas for further investigation.
This review seeks to fill existing gaps by critically examining recent empirical studies on online classroom management published between 2021 and 2025. Grounded in Social Constructivism (Vygotsky, 1978) and Engagement Theory (Fredricks et al., 2004), it explores how social interaction, collaboration, and multidimensional engagement influence effective management practices in virtual classrooms. By synthesizing empirical evidence, the review identifies key strategies for overcoming engagement barriers, addressing equity concerns, and fostering inclusive digital learning environments. Although various aspects of digital teaching have been explored in prior research, a comprehensive synthesis of online classroom management strategies remains limited. This work contributes to the evolving discourse on digital pedagogy by offering research-based insights to support educators in enhancing student engagement and instructional effectiveness in online settings.
Theoretical Framework
This study is grounded in Social Constructivism (Vygotsky, 1978) and Engagement Theory (Fredricks et al., 2004), which together provide a robust foundation for examining classroom management in online settings. These frameworks address both the relational and motivational dimensions of teaching and learning, offering complementary perspectives on how educators can enhance engagement and order in virtual classrooms.
Engagement Theory (Fredricks et al., 2004; Redmond et al., 2018) conceptualizes engagement as a multidimensional construct comprising behavioral, emotional, and cognitive components. Behavioral engagement involves students’ active participation in learning tasks, such as attending class and completing assignments. Emotional engagement reflects students’ affective reactions to teachers, peers, and learning, including interest, belonging, and enjoyment. Cognitive engagement refers to the mental effort and learning strategies students employ to master content. In the context of online education, maintaining all three dimensions is critical for sustaining attention, promoting motivation, and managing classroom dynamics effectively.
Social Constructivism and Collaboration
Social Constructivism posits that learning is constructed through interaction with others in a shared cultural and social context (Vygotsky, 1978). Traditionally, this is enacted through group discussions, cooperative learning, and guided feedback from teachers (Laurillard, 2013). In online settings, however, the lack of physical co-presence can weaken these interactions and pose challenges for classroom management, particularly in fostering active participation and collaborative norms (Hrastinski, 2009).
To mitigate this, educators use digital tools like breakout rooms, shared documents (e.g., Google Docs), Padlet, and Jamboard to recreate collaborative dynamics virtually (Dabbagh & Kitsantas, 2012). These platforms enable synchronous and asynchronous interaction, promoting the co-construction of knowledge even in distributed learning environments. From a classroom management perspective, structuring these tools with clear expectations and roles helps establish behavioral norms and accountability, aligning with constructivist principles while fostering order and engagement (Sun & Rueda, 2012).
In this review, Social Constructivism provides a lens for evaluating how collaborative tools and peer-based strategies contribute not only to knowledge construction but also to maintaining classroom structure, fostering active participation, and cultivating a sense of shared responsibility among learners in online environments. By promoting interaction, dialogue, and community-building, constructivist-informed approaches help mitigate the isolation often associated with virtual learning and support the development of a cohesive and well-managed digital classroom culture.
Engagement Theory and Motivation
Engagement Theory offers a framework for understanding how students interact with learning activities, emphasizing the need for meaningful participation, motivation, and connection in educational settings (Fredricks et al., 2004). At its core, the theory suggests that students are more likely to be engaged when learning experiences are active, collaborative, and purposeful. This is particularly important in online learning environments, where maintaining student motivation and participation can be challenging.
The theory identifies three interconnected dimensions of engagement:
Behavioral Engagement: This refers to students’ active participation in academic tasks, such as attending classes, completing assignments, and contributing to discussions. However, in online settings, behavioral engagement can be hindered by technological barriers, home distractions, and a lack of direct accountability (Moore, 1989; Muilenburg & Berge, 2005).
Emotional Engagement: This dimension relates to students’ sense of belonging, enthusiasm, and motivation toward learning. In virtual environments, the absence of spontaneous social interactions can lead to feelings of isolation, reducing emotional engagement (Martin & Bolliger, 2018; Redmond et al., 2018).
Cognitive Engagement: This involves deep investment in learning, critical thinking, and problem-solving. In asynchronous settings, where immediate feedback is often lacking, students may struggle to maintain the same level of cognitive engagement as in traditional face-to-face environments (Herdi et al., 2025).
By recognizing these three dimensions, Engagement Theory highlights the need for intentional strategies to foster participation, motivation, and deeper learning in online education. Creating structured, interactive, and socially connected learning environments is essential to overcoming the unique challenges posed by digital classrooms (Borup et al., 2020; Kahu, 2013).
In this review, Engagement Theory serves not only as a conceptual lens but also as a guiding framework for evaluating how different classroom management strategies promote student motivation and participation in online settings. The literature is analyzed through the lens of behavioral, emotional, and cognitive engagement, allowing for a nuanced understanding of which strategies effectively address these dimensions. This framework supports the categorization of empirical findings and helps identify practices that sustain student engagement in digital learning environment.
Integration of Theories
By integrating Social Constructivism and Engagement Theory, this study adopts a holistic approach to understanding classroom management in online settings. Social Constructivism highlights the importance of collaboration, while Engagement Theory emphasizes the need for sustained motivation and active participation. Together, these frameworks provide a comprehensive lens for analyzing and mitigating the challenges of managing online classrooms.
Understanding how these theories apply to digital learning environments is essential for developing effective classroom management strategies that foster engagement, inclusivity, and student success. This theoretical foundation will guide the analysis of empirical studies on online classroom management and inform recommendations for educators navigating virtual learning environments.
Together, these theories provide a lens for identifying and interpreting online classroom management strategies that foster order, engagement, and emotional well-being. Specifically, Social Constructivism informs how collaborative tools and peer interaction can replace physical proximity, while Engagement Theory helps assess how different approaches support participation, motivation, and deep learning. In this review, empirical studies were selected and categorized based on how their strategies align with one or both frameworks, offering a structured way to evaluate their effectiveness in managing online classrooms.
Methodology
This study employs a narrative review methodology to synthesize empirical research on online classroom management strategies in the post-pandemic era. While this approach is not systematic, it allows for a comprehensive yet flexible exploration of key themes by selecting studies that provide meaningful insights into engagement, discipline, and communication strategies in virtual settings. Narrative reviews are particularly useful for capturing emerging themes and synthesizing diverse perspectives, making them suitable for this study’s objective (Snyder, 2019). The focus is on identifying practical implications for educators by drawing from methodologically rigorous and contextually relevant sources.
Search Strategy
A systematic search was conducted across multiple academic databases, including ERIC, Google Scholar, SpringerLink, Wiley Online Library, and SAGE Journals. Search terms included a combination of keywords such as:
“classroom management” AND “online teaching”
“virtual learning” AND “student engagement in remote learning”
“effective online instruction” AND “behavior management in digital classrooms”
Additional synonyms and related terms were used to ensure a comprehensive search across different terminology variations. Boolean operators and advanced search filters (e.g., limiting results to peer-reviewed journals and English-language publications) were applied to refine the results. Searches were conducted between January 2021 and April 2025 to ensure the inclusion of studies reflecting the shift to online education during and after the COVID-19 pandemic.
Inclusion and Exclusion Criteria
To ensure a rigorous and relevant selection of studies, this review applied specific inclusion and exclusion criteria. Studies were included if they met the following conditions:
Publication timeline: The study was published between January 2021 and April 2025, ensuring relevance to post-pandemic educational contexts.
Study type: The research reported primary empirical data on classroom management strategies, engagement techniques, or behavioral interventions in online or hybrid learning environments.
Publication source: The study must have been published in a peer-reviewed academic journal or conference proceedings.
Thematic relevance: The study addressed at least one key aspect of online classroom management, such as student engagement, discipline, communication strategies, or instructional techniques.
Studies were excluded if they focused solely on theoretical or conceptual discussions without empirical data, examined classroom management only in traditional (face-to-face) settings without a virtual or hybrid component and lacked sufficient methodological detail or did not align with the research objectives.
Screening and Selection Process
The initial database search yielded 246 studies. After removing duplicate entries, 189 unique studies remained. A two-phase screening process was applied to refine the selection:
Title and abstract screening: One hundred thirty-six studies were excluded for not meeting the inclusion criteria, such as those focusing on traditional classrooms or presenting only conceptual frameworks without empirical findings.
Full-text review: The remaining 53 studies were thoroughly assessed based on their methodology, research scope, and relevance. Of these, 33 were excluded due to factors such as insufficient methodological clarity or a lack of direct connection to online classroom management strategies. Ultimately, 20 studies met all inclusion criteria and were retained for the final analysis (see Table 1 for details).
Post-Pandemic Empirical Studies on Online Classroom Management.
Study quality appraisal: Although no formal critical appraisal tool (e.g., MMAT or CASP) was employed, the methodological quality of the selected studies was systematically evaluated during full-text screening. This appraisal focused on key indicators such as clarity of research aims, appropriateness of study design, robustness of data collection and analysis procedures, ethical transparency, and credibility of findings. Studies were prioritized if they provided detailed methodological descriptions and clear alignment between research questions, analytical strategies, and conclusions. Studies lacking methodological transparency or raising concerns about internal validity were excluded. This approach ensured the inclusion of methodologically sound and contextually relevant evidence in line with the aims of this review.
To emphasize quality over quantity, the final sample of 20 studies was selected not merely based on thematic relevance but also on methodological rigor and contextual applicability. Given the complexity of online classroom management, this review foregrounds high-quality empirical research to ensure that the synthesized findings are both actionable and meaningful for educators and policymakers seeking evidence-based strategies in virtual learning environments.
Data Synthesis and Analysis
Following the selection of studies, a thematic synthesis approach was employed to analyze and interpret the findings. This method allowed for the identification of key patterns, challenges, and effective strategies in online classroom management. The analysis was conducted in the following stages:
Categorization of Studies: Each study was classified based on its primary focus, such as student engagement, discipline management, instructional communication, and technological integration in virtual learning environments. This categorization facilitated a structured comparison of findings across different aspects of online classroom management.
Comparative Analysis of Findings: Commonalities and differences among the studies were examined to determine the effectiveness of various classroom management strategies. Particular attention was given to studies that provided empirical evidence on engagement techniques, behavioral interventions, and instructional adaptations in online settings.
Alignment with Theoretical Frameworks: The findings were interpreted through the lens of Social Constructivism (Vygotsky, 1978) and Engagement Theory (Fredricks et al., 2004). This analysis helped contextualize the results, illustrating how collaborative learning, motivation, and student-centered approaches contribute to effective classroom management in online environments.
Identification of Research Gaps and Limitations: The studies were also assessed for limitations, inconsistencies, and unexplored areas. Gaps in research were identified, particularly regarding the long-term impact of online classroom management strategies, variations across educational levels, and the role of institutional support in sustaining engagement. These insights highlight opportunities for future research.
This systematic synthesis provided a comprehensive understanding of best practices while also recognizing the evolving nature of online education. By integrating findings from multiple empirical studies, this review offers educators and policymakers practical, evidence-based strategies to enhance virtual classroom management in post-pandemic learning environments.
Ethical Considerations
As this study is a narrative review synthesizing previously published research, it did not involve the direct participation of human subjects. Therefore, no ethical approval or informed consent procedures were necessary. All included studies were assumed to have received the appropriate ethical clearance from their respective institutions, where applicable.
Results
This section synthesizes findings from 20 empirical studies to identify six key themes in online classroom management: technical challenges and accessibility, engagement and motivation, discipline and classroom order, synchronous versus asynchronous instruction, assessment and feedback, and pedagogical adaptation. These themes reflect the most prominent concerns educators face in virtual learning environments and highlight diverse strategies developed to address them.
Each theme is examined through the dual lenses of Social Constructivism and Engagement Theory, offering a theoretical foundation for interpreting how specific practices influence online teaching. Grounding the findings in both empirical data and theoretical perspectives enables a coherent and critical framework for understanding and enhancing classroom management in digital contexts.
Research emphasizes that well-structured online management strategies—such as clear expectations and structured interaction protocols—significantly enhance student participation and engagement (Johnson et al., 2021). These strategies reduce ambiguity, increase predictability, and foster more effective learning environments, echoing earlier studies that stress the importance of communication, pre-established norms, and active engagement (Borup et al., 2014; Martin & Bolliger, 2018).
Technical Challenges and Accessibility
One of the most pervasive challenges in online classroom management is unequal access to technology and persistent technical difficulties, both of which can significantly disrupt teaching and learning processes. Several studies underscore how issues such as unstable internet connections, limited device availability, and lack of institutional support hinder effective classroom management.
Adsız and Dinçer (2025), drawing on interviews with 97 teachers, identified low student participation, lack of accountability, and frequent digital disruptions as the most critical barriers to managing online classrooms. Teachers emphasized that without adequate institutional training and technological infrastructure, sustaining student engagement and maintaining order became increasingly difficult.
Supporting these findings, Fabriz et al. (2021) examined the effectiveness of synchronous versus asynchronous strategies and reported that students with unreliable internet or limited digital tools were significantly less likely to participate consistently. While asynchronous methods offered greater flexibility, they failed to fully address the digital divide, as resource-constrained students continued to struggle with course requirements.
Similarly, Yaseen et al. (2025) explored the role of adaptive learning technologies in supporting students with varied access levels and learning paces. Their findings suggest that while personalized digital tools accommodated students with differing technological proficiency, those lacking stable internet or modern devices remained at a disadvantage. The authors emphasize that institutional investment is critical to ensuring equitable access and supporting inclusive learning environments.
From a Social Constructivist perspective, the lack of reliable access to digital tools inhibits collaborative learning, as students are unable to participate fully in peer discussions, group activities, or interactive learning tasks—essential components of co-constructing knowledge (Hrastinski, 2009; Vygotsky, 1978). Similarly, Engagement Theory (Fredricks et al., 2004; Redmond et al., 2018) highlights how ongoing technical issues disrupt behavioral engagement, leading to student frustration, withdrawal, and reduced participation in learning activities.
Improving Access and Addressing Technical Challenges
To mitigate technical barriers and ensure equitable participation, institutional support is critical in equipping both students and instructors with the resources necessary for navigating online learning environments. Zalat et al. (2021) found that institutions offering structured training programs, reliable IT support, and access to learning platforms significantly reduced technical disruptions. Their study underscores that institutional investment in infrastructure and professional development not only enhances teaching effectiveness but also contributes to more manageable online classrooms.
In addition to institutional backing, instructional flexibility is vital for addressing disparities in student access. Conner (2024) emphasizes that permitting alternative submission methods—such as completing assignments offline or submitting via email—allowed students to work around connectivity issues. His findings reveal that this flexibility alleviated student anxiety and sustained engagement, thereby preserving instructional continuity despite technological obstacles.
Addressing access-related challenges thus requires a multi-pronged approach: (1) institutional investment in digital infrastructure and faculty training, (2) use of low-bandwidth teaching resources, and (3) implementation of alternative engagement strategies tailored to students with limited access. Across the reviewed studies, proactive practices such as providing downloadable materials, asynchronous support mechanisms, and financial aid for digital tools consistently emerged as effective strategies for maintaining equity and engagement in online learning settings.
Engagement and Motivation
Maintaining student engagement and motivation in online learning environments presents a persistent challenge in classroom management. The absence of physical presence and spontaneous interaction makes it difficult for instructors to assess student interest, encourage active participation, and cultivate a sense of community. Several studies have shown that such barriers contribute to passive learning, emotional detachment, and reduced academic motivation in virtual classrooms.
Limbu and McKinley’s (2025) systematic review of 50 studies identified instructor presence, peer interaction, and students’ self-efficacy as key drivers of online engagement. Their findings underscore the need for structured support systems and a consistently active teaching presence to sustain motivation and involvement in virtual settings. These components directly relate to emotional and cognitive engagement, which are crucial for promoting persistence and performance.
Building on this, Lee and Kim (2022) found that prompt feedback and clear communication were instrumental in increasing both academic achievement and student participation. Similarly, Hu and Wang (2023) reported that instructor immediacy—characterized by responsiveness, approachability, and frequent communication—significantly enhanced students’ sense of connection and willingness to engage. Conversely, a lack of instructor presence led to feelings of isolation and disengagement.
Interactive teaching methods further contribute to engagement by creating opportunities for students to take an active role in their learning. Zhao (2025) examined the effects of real-time polling, breakout rooms, and collaborative projects, concluding that these strategies increased attentiveness and reduced passivity. His findings support the shift toward student-centered learning, particularly in asynchronous or minimally interactive courses where disengagement is more likely.
Viewed through the lens of Engagement Theory, these findings highlight the importance of supporting behavioral, emotional, and cognitive engagement (Fredricks et al., 2004; Redmond et al., 2018). Without meaningful interaction, timely feedback, and a responsive learning environment, students are more likely to disengage—both emotionally and behaviorally—resulting in weaker learning outcomes and lower retention in online courses.
Enhancing Student Engagement and Motivation
To address the challenges of disengagement and passive learning in online environments, research consistently underscores the value of interactive, student-centered teaching strategies that foster a sense of connection and participation. These strategies not only enhance motivation but also contribute to building a more cohesive online learning community.
Hu and Wang (2023) highlight the importance of instructor immediacy, emphasizing practices such as frequent communication, virtual office hours, and personalized feedback. Their findings reveal that students who received timely and individualized responses from instructors reported stronger feelings of connection to the course and were more likely to remain engaged throughout the semester.
Complementing this, Zhao (2025) advocate for active learning approaches, including real-time polling, structured discussions, and gamified classroom activities. Their study demonstrates that these interactive techniques increased students’ sense of accountability and sustained focus, thereby reducing the passivity often observed in online settings.
Building further, Lampropoulos and Sidiropoulos (2024) explored the role of gamification in promoting student motivation. Their research found that elements such as progress tracking, interactive challenges, and reward-based tasks enhanced both participation and academic performance. However, the authors caution that gamified strategies must be thoughtfully designed to avoid distracting students or encouraging superficial competition. When implemented with care, gamification can serve as a powerful tool for reinforcing meaningful engagement.
Taken together, these studies suggest that enhancing student engagement in online classrooms requires a multi-layered approach. Effective practices include responsive instructor communication, active and collaborative learning opportunities, and well-integrated gamification. When aligned with the principles of Engagement Theory—particularly behavioral and emotional engagement—such strategies can transform online classrooms into dynamic, participatory environments that support motivation, connection, and sustained learning.
Discipline and Classroom Order
While student engagement is essential for sustained learning, maintaining discipline and classroom order is equally critical in virtual settings. However, this remains a distinct challenge due to the limited real-time supervision, reduced visibility, and absence of physical behavioral cues. The relative anonymity afforded by online platforms often undermines student accountability, making it easier for learners to disengage or behave disruptively—issues less prevalent in face-to-face environments.
From a Social Constructivist perspective, such disruptions stem from the lack of a shared physical space, which ordinarily supports the co-construction of classroom norms through continuous social interaction and visible teacher presence (Vygotsky, 1978). Engagement Theory further highlights that behavioral engagement—students’ willingness to follow rules, participate responsibly, and stay task-focused—relies heavily on the clarity of expectations and consistent teacher feedback (Fredricks et al., 2004).
In a study exploring the role of structure, Croce and Salter (2022) found that students were more likely to adhere to behavioral norms when participation guidelines, communication protocols, and consequences for misconduct were explicitly communicated at the beginning of a course. These clear expectations contributed to a more predictable and respectful learning environment, reinforcing both behavioral and emotional engagement. Conversely, when rules were ambiguous or inconsistently applied, students often perceived the classroom as informal or unmonitored, leading to distraction and detachment.
Building on this, Gosak et al. (2022) examined the effectiveness of digital tools in reinforcing classroom discipline. Their findings indicate that real-time monitoring, structured discussion forums, and participation tracking not only helped instructors uphold behavioral standards but also encouraged active peer collaboration. When implemented alongside clearly defined norms, these digital features enhanced accountability and fostered a shared sense of responsibility among learners.
Taken together, these studies suggest that maintaining order in online classrooms requires more than reactive discipline—it involves the proactive establishment of transparent behavioral expectations, supported by digital tools that promote visibility and collaboration. When such structures are in place, students are more likely to internalize classroom norms, engage constructively with peers, and remain focused, thereby sustaining both discipline and learning continuity in virtual environments.
Maintaining Discipline and Classroom Order
Maintaining discipline and classroom order to effectively manage discipline in an online environment, research highlights the importance of clear guidelines, structured reinforcement, and the use of digital tools to promote classroom order. Croce and Salter (2022) emphasize that setting behavioral expectations early in a course plays a crucial role in preventing disruptive behavior. Their study suggests that instructors should clearly communicate participation rules, discussion guidelines, and consequences for violations to ensure students understand what is expected in an online learning space.
Gosak et al. (2022) advocate for the integration of digital tools to help maintain discipline and focus. Their findings suggest that features such as automated attendance tracking, structured participation grading, and moderated discussion forums provide students with a sense of accountability, making them more likely to stay engaged and follow classroom norms. Similarly, Kirişçi-Sarıkaya and Parlar (2024) found that online teachers who implemented structured routines, clearly communicated rules, and interactive digital tools were more successful in sustaining student attention and minimizing behavioral issues.
Additionally, Chen et al. (2022) examined the impact of positive reinforcement strategies on student behavior in online settings. Their study found that recognizing and rewarding positive behaviors, such as active participation, respectful communication, and constructive engagement in discussions, helped foster a culture of respect and discipline in virtual classrooms. Their findings suggest that reinforcing desirable behaviors rather than focusing solely on punitive measures leads to a more structured and orderly online learning environment. Taken together, these findings indicate that maintaining discipline in online classrooms requires a multi-faceted approach that combines clear expectations, structured digital monitoring, and positive reinforcement techniques. By implementing consistent behavioral guidelines and leveraging engagement-focused tools, instructors can create a more disciplined and well managed online learning space.
Synchronous and Asynchronous Learning
Effectively balancing synchronous and asynchronous learning remains a central challenge in online classroom management. While asynchronous instruction offers greater flexibility and autonomy, it often results in disjointed communication, delayed feedback, and reduced student engagement. In contrast, synchronous sessions enable real-time interaction and immediate clarification but may create accessibility issues for students navigating time zone differences, scheduling conflicts, or unreliable internet connections.
Fabriz et al. (2021) compared classroom management strategies across both modes and found that students with unstable internet and limited access to digital tools were less likely to participate consistently—particularly in synchronous settings. These technical barriers not only disrupted lesson flow but also led to disengagement. Although asynchronous learning helped mitigate some accessibility issues, it did not fully address the challenges faced by students with persistent technological limitations.
Similarly, Fabriz et al. (2021) investigated peer interaction in asynchronous learning environments. Their findings revealed that students in fully asynchronous courses often experienced isolation and a lack of connection with classmates. The absence of real-time collaboration contributed to lower motivation and course completion rates, highlighting the importance of integrating synchronous elements to sustain a sense of community.
From the perspective of Engagement Theory, effective learning environments must support three interrelated dimensions: behavioral engagement (active participation), emotional engagement (sense of belonging), and cognitive engagement (intellectual investment; Fredricks et al., 2004; Redmond et al., 2018). An overreliance on asynchronous delivery risks diminishing the structure, accountability, and real-time feedback essential for engagement. Conversely, an overemphasis on synchronous instruction can overwhelm learners with competing responsibilities or connectivity challenges.
Overall, the findings suggest that online instructors should seek a thoughtful balance between synchronous and asynchronous elements—leveraging the flexibility of the former with the immediacy and community-building potential of the latter. Such a blended approach not only supports accessibility but also promotes consistent participation and deeper learning engagement.
Balancing Synchronous and Asynchronous Learning
To address the challenges posed by both synchronous and asynchronous formats, research increasingly supports a blended learning approach—one that strategically combines the strengths of each while minimizing their limitations. A hybrid model allows for flexibility in content delivery alongside opportunities for real-time interaction and feedback, promoting both accessibility and engagement.
Fabriz et al. (2021) recommend using asynchronous tools for delivering core content—such as recorded lectures and discussion boards—while supplementing them with regular synchronous sessions to enhance student interaction and understanding. Their study found that students who engaged in this structured combination reported greater satisfaction and demonstrated a deeper grasp of the material compared to those in purely asynchronous courses.
In a complementary finding, Fabriz et al., 2021 emphasize the value of recording synchronous sessions to accommodate students with scheduling conflicts. Allowing students to view recordings and participate in asynchronous follow-up activities enhanced inclusivity and ensured more equitable access, particularly for learners managing diverse time zones, jobs, or caregiving responsibilities.
Chen et al. (2022) further explored the role of virtual breakout rooms in synchronous sessions. Their results showed that students participating in small-group, real-time discussions felt more socially connected and were more likely to contribute meaningfully to classroom activities. These structured interactions also supported more effective classroom management by reinforcing accountability and peer collaboration.
Collectively, these findings point to the importance of intentional instructional design that blends asynchronous flexibility with synchronous presence. When thoughtfully integrated, such an approach supports a broader range of learners and helps maintain both classroom order and student motivation. From the lens of Engagement Theory, this balance enhances behavioral, emotional, and cognitive engagement—key components of successful online learning environments.
Assessment and Feedback
Assessment and feedback are central components of effective online classroom management, as they foster accountability, maintain motivation, and guide learning. However, digital learning environments introduce distinctive challenges, such as delayed responses, increased instructor workload, and reduced opportunities for personalized interaction. The literature reviewed underscores the importance of implementing structured formative assessments and leveraging technology-enhanced feedback systems to mitigate these issues and support student engagement.
From a Social Constructivist perspective (Vygotsky, 1978), feedback serves as a critical scaffold that enables learners to refine their understanding through guided support. In online settings—particularly asynchronous ones—students often lack immediate access to instructors, making formative assessments an essential tool for providing structure and direction. Clegg et al. (2021) found that low-stakes, formative assessments not only facilitated student progression but also offered immediate insights into learning gaps. Their study highlights the value of frequent, targeted feedback in helping instructors adjust instruction and support student autonomy.
Engagement Theory (Fredricks et al., 2004) similarly emphasizes the motivational role of feedback, especially in sustaining cognitive engagement—students’ investment in and effort toward deep learning. Timely, constructive feedback helps students stay focused and take ownership of their learning process. Yet in practice, the demands of large online class sizes often limit instructors’ ability to deliver individualized feedback consistently.
To address this, Clegg et al. (2021) explored the use of automated assessment tools, such as AI-driven feedback platforms and digital grading systems. Their findings showed that students who received instant, system-generated feedback were more likely to remain engaged and persist in their studies compared to those awaiting instructor responses. Automation thus offers a practical solution for maintaining momentum and providing consistency in feedback delivery.
However, over-reliance on automation poses its own risks. Clegg et al. (2021) caution that purely automated grading may overlook students’ individual learning needs and limit opportunities for deeper reflection. They advocate for a hybrid feedback model—using automated tools for objective, low-complexity tasks (e.g., quizzes), while reserving instructor feedback for open-ended assignments that benefit from nuance and dialogue. This approach aligns with constructivist principles by ensuring that feedback remains dialogic and adaptive rather than merely transactional.
Taken together, these findings suggest that a strategic combination of formative assessment, timely feedback, and appropriate technological support can significantly improve classroom management and learning outcomes in online environments. Designing feedback systems that are both scalable and personalized helps ensure that students remain cognitively engaged and supported throughout their learning journey.
Refining Assessment and Feedback
Optimizing assessment and feedback in online classrooms requires a multifaceted approach that supports both efficiency and meaningful learning. Research emphasizes that frequent formative assessments play a critical role in tracking student progress and enabling early intervention for those experiencing difficulties (Wang & Wu, 2021). These low-stakes assessments not only provide timely insights into learners’ understanding but also foster cognitive engagement by offering structured opportunities for reflection, self-monitoring, and continuous improvement.
To manage the demands of large online classes, instructors increasingly turn to automated feedback tools, such as AI-based grading systems and adaptive learning technologies. These tools reduce instructor workload while delivering instant, consistent feedback that supports student persistence and engagement (Wang & Wu, 2021). However, while automation improves scalability, Clegg et al. (2021) caution that over-reliance on such tools can diminish personalization. They advocate for a hybrid feedback model, where automated systems handle objective tasks—such as quizzes or multiple-choice items—while instructors provide personalized, developmental feedback on open-ended assignments. This combination ensures that students benefit from both timely support and deeper, individualized guidance.
In addition to textual feedback, incorporating interactive modalities—such as video comments or audio recordings—can significantly enhance the perceived presence of the instructor. These approaches foster emotional engagement, a core component of Engagement Theory, by helping students feel more connected to their instructors and less isolated in the digital environment. When learners perceive that their instructors are accessible and invested in their progress, motivation and participation tend to increase.
In sum, effective assessment and feedback in online classrooms hinge on a deliberate balance between automation and human interaction. By integrating formative assessments, scalable feedback tools, and interactive communication methods, educators can create a more responsive, engaging, and inclusive learning environment that supports sustained achievement.
Adapting Pedagogical Approaches
Adapting traditional pedagogical strategies for online environments poses a significant challenge, especially in sustaining instructional effectiveness, student engagement, and classroom management. Many hands-on, discussion-based, and interactive techniques used in face-to-face settings do not translate seamlessly to virtual platforms, requiring instructors to reconceptualize their teaching methods without compromising pedagogical integrity. These challenges can be effectively interpreted through the lenses of Social Constructivism, which emphasizes the role of interaction and scaffolding in learning (Vygotsky, 1978), and Engagement Theory, which identifies behavioral, emotional, and cognitive engagement as essential to meaningful learning (Fredricks et al., 2004).
Chen et al. (2022) explored how instructors integrated virtual breakout rooms and collaborative learning tasks to replicate the interactivity of traditional classrooms. While the use of breakout rooms supported peer interaction and mirrored constructivist principles, the study found that many instructors struggled to structure these sessions effectively. Without clear guidance or monitoring, participation was often uneven, and the learning potential diminished. These findings reinforce Engagement Theory’s assertion that interactivity alone is insufficient—it must be purposefully designed and supported to maintain consistent student engagement.
Similarly, Oruikor et al. (2023) investigated how online classroom layout and instructional design influenced teaching effectiveness. Their research revealed that disorganized digital platforms and poorly structured course materials increased students’ cognitive load, leading to confusion and disengagement. This highlights a core tenet of both theories: that effective learning environments must be intentionally designed to facilitate navigation, interaction, and knowledge construction.
From a Social Constructivist standpoint, pedagogical adaptation in online learning should prioritize collaboration, dialogue, and shared meaning-making. This requires structured opportunities for interaction and scaffolding that support students’ active participation. At the same time, Engagement Theory emphasizes that pedagogical strategies must foster all three dimensions of engagement: behavioral (active involvement), emotional (sense of connection), and cognitive (deep processing). When instructional design fails to support these dimensions, students may become disengaged, undermining both classroom management and learning outcomes.
Innovative Pedagogical Adaptations for Online Learning
Enhancing teaching effectiveness and managing classrooms in online environments requires instructors to thoughtfully adapt traditional methods while leveraging the affordances of digital tools. Research consistently highlights the importance of integrating multimedia, fostering interactive learning, and designing inclusive and well-structured digital environments to sustain student engagement and maintain pedagogical integrity.
Chen et al. (2022) found that the structured use of breakout rooms, collaborative projects, and discussion forums can foster both behavioral and cognitive engagement by promoting active participation and deeper content processing. These findings align with Engagement Theory, which underscores the role of social interaction in strengthening emotional and behavioral investment. From a Social Constructivist perspective, such collaborative activities also support the co-construction of knowledge through peer interaction and shared responsibility. However, the researchers caution that these benefits are contingent on the presence of clearly defined participation expectations and structured scaffolding, which are essential for ensuring equitable engagement across diverse learners.
Similarly, Oruikor et al. (2023) emphasize the role of online classroom layout and instructional design in supporting effective pedagogy. Their study demonstrated that intuitive navigation, visually organized content, and coherent learning modules reduced cognitive overload and enhanced student comprehension. These outcomes reflect the cognitive engagement dimension of Engagement Theory and further support the constructivist principle that thoughtfully scaffolded learning environments enable students to meaningfully engage with content and construct knowledge in collaborative digital spaces.
In addition to structural design, relational aspects of online teaching are equally important. Rahman et al. (2021) explored the impact of communication strategies on classroom management and found that fostering a supportive and inclusive virtual environment—through practices such as regular check-ins and attention to social-emotional learning (SEL)—significantly enhanced participation and adherence to classroom norms. These findings align closely with the emotional engagement component of Engagement Theory, which emphasizes students’ need to feel valued, connected, and supported. From a Social Constructivist viewpoint, such community-building efforts are foundational to co-creating shared norms and cultivating a positive learning culture.
Gamification has also emerged as an effective pedagogical adaptation in online learning. Lampropoulos and Sidiropoulos (2024), in their longitudinal study, found that gamified environments incorporating challenge-based tasks, progress tracking, and interactive components significantly improved both student motivation and academic outcomes. These results affirm Engagement Theory’s emphasis on the importance of feedback, challenge, and interactivity in sustaining behavioral and cognitive engagement. When carefully designed, gamified strategies also align with constructivist principles by fostering goal-oriented participation, peer collaboration, and active knowledge construction.
Taken together, these studies demonstrate that successful pedagogical adaptation in online education involves a strategic combination of interactive teaching methods, intuitive course design, responsive communication, and inclusive community-building. By thoughtfully integrating digital tools, encouraging student collaboration, and scaffolding learning experiences, instructors can address classroom management challenges while maintaining high levels of engagement and learning. This approach is grounded in both Social Constructivist and Engagement Theory frameworks, which together offer a robust foundation for creating dynamic, inclusive, and effective virtual classrooms.
Discussion
The findings of this study offer valuable insights into the multifaceted nature of classroom management in online education, particularly as it relates to student engagement, collaborative learning, discipline, digital equity, and pedagogical adaptation. Framed within Social Constructivism and Engagement Theory, the results illuminate both the persistent challenges of virtual instruction and the strategies that can support more inclusive, effective, and well-managed online learning environments.
Social Constructivism and Collaborative Learning
Consistent with Social Constructivist theory, the study underscores the centrality of interaction and collaboration in fostering meaningful learning. Digital tools such as breakout rooms, collaborative projects, and discussion forums were found to enhance peer-to-peer engagement and promote the co-construction of knowledge. These tools facilitated student-centered learning by creating spaces for shared responsibility and dialogue. However, the effectiveness of such strategies was often limited by unequal access to digital infrastructure. The digital divide—manifesting in unreliable internet access or limited access to devices—remained a substantial barrier for some students, preventing full participation in synchronous or group-based activities. These findings echo Hrastinski’s (2009) model, which emphasizes that active participation is vital for engagement in online contexts. Accordingly, institutional support—including provision of devices, internet subsidies, and mobile-friendly platforms—is essential to ensure equitable opportunities for collaborative learning.
Engagement Theory and Student Motivation
Engagement Theory (Fredricks et al., 2004) provides a useful lens for analyzing how different dimensions of engagement—behavioral, emotional, and cognitive—are influenced by online learning design. The findings indicate that behavioral engagement was effectively promoted through interactive technologies such as polling tools, real-time chats, and gamified activities, which prompted consistent participation. Emotional engagement was fostered through relational practices like personalized feedback, virtual office hours, and peer discussion forums that built a sense of belonging in digital classrooms. However, cognitive engagement remained a greater challenge, particularly in asynchronous environments where students reported difficulties with focus, time management, and self-motivation. Although adaptive learning technologies and gamification elements show potential to improve cognitive investment, further longitudinal research is needed to determine their long-term efficacy in sustaining student motivation, learning outcomes, and course completion rates.
Discipline and Order in Virtual Classrooms
Maintaining discipline and structure in online classrooms requires intentional management strategies that are tailored to the affordances and limitations of digital platforms. The study found that clearly communicating participation expectations, providing structured tasks, and using positive reinforcement mechanisms contributed to orderly and productive learning environments. These strategies resonate with the behavioral dimension of Engagement Theory, highlighting that effective classroom management is less about compliance and more about designing environments that encourage responsible, self-regulated participation. Tools such as attendance trackers, engagement analytics, and moderated discussion boards enabled instructors to monitor and support student behavior proactively. Future studies might explore how students’ self-regulation capacities relate to disciplinary issues in online learning and how teachers can cultivate greater student accountability and autonomy.
Equity and Access as a Prerequisite for Engagement
Perhaps the most pressing issue highlighted by this review is the critical role of digital equity in supporting sustained student engagement. Students who lacked reliable internet access or appropriate devices were consistently at a disadvantage—unable to attend synchronous sessions regularly, submit assignments on time, or fully participate in group work. These challenges disproportionately affected students from marginalized or low-income backgrounds, reinforcing pre-existing educational inequalities. The findings align with broader research on digital inclusion (Redmond et al., 2018), which stresses that equitable access to technology is a prerequisite for meaningful participation and learning. Institutional and governmental efforts—such as device loan programs, subsidized internet services, and the provision of offline learning alternatives—are therefore essential to reduce these disparities and ensure that all students can benefit from online learning opportunities.
Practical Implications for Educators and Institutions
The findings underscore the urgent need for sustained professional development to support educators in transitioning from traditional pedagogies to effective digital practices. Online classroom management demands not only technological fluency but also the ability to design student-centered, interactive, and engagement-driven learning experiences. Teachers should be trained in the strategic use of collaboration tools, formative assessment techniques, digital feedback systems, and inclusive communication strategies. Equally important is institutional investment in instructional design support, technological infrastructure, and community-building initiatives. These systemic supports can empower educators to create more inclusive, responsive, and well-managed virtual classrooms.
Conclusion and Future Directions
Taken together, the findings of this study contribute to a deeper understanding of how online classroom management must evolve in response to shifting educational demands. As online and hybrid learning modalities continue to expand, educators and institutions must prioritize equity, engagement, and pedagogical innovation. Future research should focus on longitudinal analyses of digital classroom management practices, the impact of AI-driven learning tools on engagement, and culturally responsive strategies for online environments. By grounding practice in theory and evidence, and by ensuring access for all learners, educational stakeholders can shape online learning into a more inclusive, effective, and sustainable model for the future.
Practical Implications for Educators and Institutions
In light of these findings, several key implications emerge for educators and institutions aiming to improve classroom management in online learning environments. First, there is a pressing need for sustained professional development to help educators transition from traditional pedagogies to effective digital practices. Managing online classrooms requires not only technological proficiency but also the capacity to design student-centered, interactive, and engagement-focused learning experiences. Educators should receive targeted training in the strategic use of collaboration tools, formative assessment techniques, digital feedback systems, and inclusive communication strategies. Equally important is institutional investment in instructional design support, robust technological infrastructure, and initiatives that foster community and connection. These systemic supports are essential to empower educators to build more inclusive, responsive, and well-managed virtual classrooms.
Limitations
While this review provides valuable insights into online classroom management strategies, certain limitations should be acknowledged. First, the sample of twenty studies, though selected based on rigorous inclusion criteria, may not fully capture the diversity of online teaching practices across all educational settings. Expanding future research to incorporate a broader dataset would enhance the generalizability of the findings.
Second, as a narrative review, this study synthesizes existing literature qualitatively rather than quantitatively. While this approach allows for a nuanced discussion of emerging themes, it does not measure the statistical impact of specific strategies. Future research should consider systematic or scoping reviews that apply meta-analytic techniques to assess effectiveness across various studies.
Third, potential publication bias must be considered. This review primarily includes peer-reviewed journal articles, which tend to emphasize successful interventions, potentially underrepresenting studies with null or negative findings. Future research could incorporate unpublished dissertations, institutional reports, or gray literature to provide a more comprehensive perspective.
Finally, the rapid evolution of online learning presents challenges in maintaining up-to-date findings. Given the continuous advancements in educational technology and digital pedagogy, classroom management strategies will likely evolve further. Longitudinal studies and ongoing reviews will be essential for tracking these developments and assessing the long-term effectiveness of emerging strategies.
Conclusion and Future Directions
This review underscores the complexities of online classroom management and the critical strategies required for fostering inclusive, structured, and engaging virtual learning environments. The findings emphasize that successful classroom management in online settings requires a multifaceted approach that balances technological adaptation, pedagogical strategies, and student-centered engagement practices.
A key challenge in online education is digital equity, as disparities in access to technology continue to impact student participation. Addressing this issue requires institutional investments in technology infrastructure, digital accessibility initiatives, and alternative learning solutions for students with limited resources. Beyond access, student engagement and motivation remain central to effective online instruction. Research highlights the value of interactive tools, gamification, structured discussions, and personalized feedback in sustaining engagement and reducing passive participation.
Another critical consideration is classroom discipline and order in virtual settings. The findings suggest that proactive strategies, such as establishing clear participation expectations, structured engagement techniques, and positive reinforcement, contribute to a more orderly and productive online learning environment. Furthermore, balancing synchronous and asynchronous learning has emerged as a vital strategy for maintaining instructional coherence and ensuring flexibility for diverse learners. A blended approach that combines real-time discussions with self-paced learning opportunities optimizes both engagement and accessibility.
Assessment and feedback mechanisms also require adaptation in online education. Traditional assessment models often fall short in digital environments, necessitating the integration of formative assessments, automated grading tools, and real-time feedback systems. These strategies not only help instructors manage workload efficiently but also provide students with timely insights into their learning progress.
Looking ahead, further research and innovation are essential to refine online classroom management practices. Future studies should explore emerging technologies such as artificial intelligence (AI), virtual reality (VR), and adaptive learning systems, which have the potential to personalize learning experiences and improve engagement. Additionally, longitudinal research is needed to assess the long-term impact of various management strategies on student retention, learning outcomes, and instructor workload. Addressing digital inequities through policy-driven initiatives and institutional reforms will also be crucial in ensuring an inclusive and accessible online learning landscape.
By implementing evidence-based strategies, educators, researchers, and policymakers can work toward more resilient and engaging digital learning environments. As online education continues to evolve, continuous research and cross-disciplinary collaboration will be key to shaping pedagogical best practices and ensuring that virtual classrooms remain effective, inclusive, and student-centered.
Footnotes
Acknowledgements
The author would like to thank the peer reviewers and the editorial team for their valuable suggestions and guidance.
Ethical Considerations
This manuscript is a narrative literature review that does not involve human participants, identifiable personal data, or intervention-based procedures. Therefore, ethical approval was not required. The study was conducted in accordance with the APA Ethical Principles of Psychologists and Code of Conduct (Section 8.05) and complies with the SAGE Publishing Ethics and Research Integrity Policy.
Consent to Participate
Not applicable. No human participants were involved in this study, and no informed consent was necessary.
Funding
The author received no financial support for the research, authorship, and/or publication of this article.
Declaration of Conflicting Interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Data Availability Statement
No primary data were generated or analyzed for this study. The data supporting the findings are derived from publicly available literature sources cited in the manuscript.