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Research article

Promoting peer learning in education: Exploring continuous action iterated dilemma and team leader rotation mechanism in peer-led instruction


  • Received: 30 July 2023 Revised: 24 September 2023 Accepted: 07 October 2023 Published: 11 October 2023
  • This paper promotes teacher-guided peer learning in education with continuous action iterated dilemma (CAID) based on team leader rotation mechanism. In previous teaching activity models, the learning communication relationship between peers was described as static, but the static relationship will hinder the learning efficiency, which does not match the real world. In addition, in view of the independence of individual students, it is necessary to establish a dynamic model which considers the complex behavior of every student. In this paper, we first propose a team leader rotation mechanism that makes sure each student has the opportunity to become a team leader, which enhances students' sense of participation and improves classroom efficiency. Next, we establish a multi-layer nonlinear student dynamic model based on continuous action iteration dilemma and involved complex and unknown nonlinear environmental factors to fit different environmental influences on different students. Also, in order to demonstrate the convergence of the proposed model, we devise the Lyapunov function as a means of mathematical proof. Through this analysis, we establish the stability of the proposed model and verify its independence from parameters, thereby enhancing its applicability in practical contexts. By incorporating the team leader rotation mechanism proposed in this paper, teachers will be able to ensure diverse student engagement to achieve information consistency, thereby ensuring the effectiveness of the classroom.

    Citation: Xiangwen Yin. Promoting peer learning in education: Exploring continuous action iterated dilemma and team leader rotation mechanism in peer-led instruction[J]. Electronic Research Archive, 2023, 31(11): 6552-6563. doi: 10.3934/era.2023331

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  • This paper promotes teacher-guided peer learning in education with continuous action iterated dilemma (CAID) based on team leader rotation mechanism. In previous teaching activity models, the learning communication relationship between peers was described as static, but the static relationship will hinder the learning efficiency, which does not match the real world. In addition, in view of the independence of individual students, it is necessary to establish a dynamic model which considers the complex behavior of every student. In this paper, we first propose a team leader rotation mechanism that makes sure each student has the opportunity to become a team leader, which enhances students' sense of participation and improves classroom efficiency. Next, we establish a multi-layer nonlinear student dynamic model based on continuous action iteration dilemma and involved complex and unknown nonlinear environmental factors to fit different environmental influences on different students. Also, in order to demonstrate the convergence of the proposed model, we devise the Lyapunov function as a means of mathematical proof. Through this analysis, we establish the stability of the proposed model and verify its independence from parameters, thereby enhancing its applicability in practical contexts. By incorporating the team leader rotation mechanism proposed in this paper, teachers will be able to ensure diverse student engagement to achieve information consistency, thereby ensuring the effectiveness of the classroom.



    Due to the effective improvement of teaching quality in peer learning, it has been widely researched [1,2,3]. It emphasizes interaction and sharing between students [4]. Students can learn knowledge from their peers, pass on their own knowledge to them, and learn from each other to bring about common progress. It can be seen that the way students communicate with others significantly affects the quality of teaching [5,6,7] because the lack of communication within the group leads to the inability to form a consensus. In addition, it should be noted that there are large individual differences among students and we need to use a more accurate mathematical model to describe [8,9,10]. In fact, there are network structures similar to teaching activities in the field of multi-agents, such as [11,12]. In order to provide better theoretical guidance for teaching activities, we designed a continuous action iteration dilemma peer learning with a team leader rotation mechanism.

    Many researchers have conducted research on peer learning [13,14,15,16]. Peer learning is an activity different from adversarial differential games. For more about adversarial differential games, one can refer to [17]. So far, existing results only analyze the static connections between students and do not fully take into account the characteristics of different students. When students are sitting in the classroom, the continuous change of communication is more conducive to the improvement of knowledge absorption. Therefore, this paper designs a method of rotating team leaders, in which all students can be called team leaders, thereby enhancing each student's sense of participation and improving the quality of teaching.

    For the study of peer learning under the guidance of teachers, the key is to abstract student behavior into a mathematical model [18,19,20]. In [21], the authors present a method that utilizes survey data analysis to enhance student performance. [22] studies the behavior of students in the presence of noise and gives its mathematical description. The author uses a filtering idea to reduce the impact on student behavior. The authors of [23] used automatic speech recognition technology to obtain data from actual classrooms to constitute imitation of teacher-student behavior. Clearly, current student models lack individual uniqueness in the mathematical modeling process, despite the fact that every student is different in reality. In addition, environmental factors such as teachers can also have an impact on education and teaching, such as random interference in field control theory [24] and actuator and sensor failures [25]. Therefore, it becomes imperative to consider how to incorporate mathematical methods to fit environmental influences. To this end, this paper proposes a CAID-based model to describe external influences.

    The main contributions can be summarized into three points:

    1) We propose a team leader rotation mechanism and mathematically model it. On the basis of information gathering among students, each student has the opportunity to become a team leader, which enhances students' sense of participation and improves classroom efficiency.

    2) Considering that the model may be subject to nonlinear external disturbances, we design the CAID model, which can cope with the external disturbances received by students in the classroom.

    3) We propose a model that combines the team leader rotation mechanism and student dynamic model. By employing the Lyapunov function, we demonstrate the convergence and stability of the proposed model, showcasing its resilience to parameter variations. This analysis enhances the model's applicability in various contexts.

    The rest of this paper is organized as follows. In Section 2, we present an overview of the model CAID. This section also introduces some key concepts. In Section 3, we model the CAID with external disturbance and team leader rotation mechanism. In Section 4, the stability of the rotation mechanism is taken into account. Section 5 provides a simulation. The obtained results demonstrate the positive outcomes of the model. Finally, in Section 6, the conclusion is presented, summarizing the key findings and implications of the study.

    In this section, we provide an overview of the fundamental concepts related to the dynamic model of CAID. This knowledge serves as a foundation for understanding the subsequent discussions and analysis in the paper.

    First, the definition of peer learning is introduced in this paper as Definition 1.

    Definition 1: Peer learning, also known as peer education or peer teaching, is a collaborative learning approach in which individuals of similar age or status engage with each other to share knowledge, skills, information and experiences to facilitate mutual learning and personal development.

    The interconnection between students can be denoted by an undirected graph G=(V,E), where V=v1,v2,,vm denotes the set of students, while ERm×m represents connections between vertices. Note that the existence of a connecting relationship between student vi and student vj can be determined by the values of eij. Specifically, if a connection exists, we have eij=1, otherwise, eij=0. We begin by conceptualizing the exchange of information among classmates as a game model. In order to describe the behavior more accurately, we introduce a model called CAID, which serves as a dynamic representation of student behavior in the information exchange process. Let xi[0,1], i{1,2,,m}, where xi represent the strategy students i adopting. Obviously, the value xi=0 means it is full detection. xi=1 implies full cooperation.

    The evolutionary dynamics are significantly influenced by payoff matrix, which serves as the foundation for students to make choices. The payoff matrix is a crucial determinant in the decision-making process. We select the payoff matrix P as follows:

    [λ0λ1λ2λ3], (1)

    where λ0,λ1,λ2,λ3 represent the payoffs associated with students utilizing a binary strategy. Therefore, the the strategy fitness H(xi) can be defined as

    H(xi)=mj=1eij[(λ0λ1λ2+λ3)xixj+(λ1λ3)xi+(λ2λ3)xj+λ3]. (2)

    Obviously, it holds that

    ΔHji=H(xj)H(xi). (3)

    According to imitation dynamics, we have

    xi(n+1)=1deg(vi)mj=1[(1sijeij)xi(n)+sijeijxj(n)], (4)

    where n is the number of evolutionary progress and sij=sig(ξ|ΔHji|) is defined as the sigmoid function of ξ multiplied by the absolute difference in the fitness values ΔHji. The sigmoid function is denoted as sig(ξ|ΔHji|)=1/(1+exp(ξ|ΔHji|)), where ξ is a positive parameter. To put it simply, Eq (4) indicates that during iteration n, a student has a probability of sij to switch to a neighboring student. Additionally, the term deg(vi)=mj=1eij represents the degree of connectivity for student i, which is the sum of the adjacency matrix elements eij for that student.

    Derivation of xi can be obtained as

    ˙xi(t)=1deg(vi)[mj=1sijeij(xj(t)xi(t))]. (5)

    Let ψij=sijeijdeg(vi). Thus, the model (5) can be reformulated as:

    ˙xi(t)=mj=1ψij(xj(t)xi(t))=mj=1sijeijdeg(vi)(xj(t)xi(t)). (6)
    Figure 1.  The dynamic model of CAID with teacher-guided peer leaning and team leader rotation mechanism.

    In this section, we design a CAID model based on the team leadership mechanism considering external disturbance.

    The behavior of students under external interference can be expressed as

    ˙xi(t)=mj=1eij(sijdeg(vi)+ui)(xj(t)xi(t)), (7)

    where ui represents the influence of complex environment on student i.

    Note that the China Unicom network only determines the interactive relationship between students. In addition to the influence of teachers on students' behavior and thinking, the team leader also played an important role. Serving as a group leader can cultivate students' organizational and management skills very well and build a bridge of communication between students and teachers. Therefore, being a group leader of a student group is what many students hope for. But at the same time, there are only a few group leaders in a student group. Always letting a fixed student serve as the group leader will also bring some problems, such as reducing the enthusiasm and participation of the rest of the students. Therefore, it is necessary to establish a team leader rotation mechanism. Based on this, this paper first introduces the dynamic model of the group leader as

    ˙xI=mj=1sIjeIjdeg(vI)(xj(t)xI(t)). (8)

    where I represents the team leader and pIj=min(p1j,,pmj). Other students are updated based on (6). The main consideration of setting the dynamics of the group leader to (8) is to highlight the leadership and decision-making role of the group leader. Therefore, when the student served as the group leader, he tried his best to stick to his point of view and reduce the interference of other students' strategies on him.

    Then, the important question is how to choose the team leader. We analyze from the characteristics of the team leader. The most important responsibility of the team leader is to coordinate the relationship between the team members. Based on this principle, we set the student with the highest cooperation rate at the current moment as the team leader, but because the status of the students exists uncertain influencing factors, the concept of cooperation is constantly changing. Therefore, we rotate team leaders based on the real-time changes in the cooperation rate of students. Every student has the opportunity to become a team leader to promote the sense of participation and cooperation among students.

    In this part, in order to analyze the convergence of the models which have been introduced, Lyapunov method is employed.

    Consider the following system (6)

    ˙x(t)=Lkx(t), (9)

    where Lk=L(Gk) is the Laplacian of graph Gk the belongs to a set Γ and calculated by (10).

    lk,ij={nj=1,jiψk,ij,j=i,ψk,ij,ji,nj=1,jIψk,Ij,j=i,ψk,Ij,jI=i. (10)

    Let α be the equilibrium point of the dynamic model. The distance between x and α, which is denoted as h, represents the error

    x(t)=α+h. (11)

    Obviously, it follows that ˙α=0. Consequently, we can derive the derivative of h based on Eq (11).

    ˙h=˙x(t). (12)

    Thus,

    ˙hi=˙xi(t)=mj=1ψlij(hjhi), (13)

    where ψlij=slijeij/deg(vi). Let V be

    V=Ni=112h2i, (14)

    where ˙h is derived by (13). Subsequently, the derivative of V can be obtained

    ˙V=(i,j)Gk(hjhi)ψlmax(hjhi)]0. (15)

    Thus, the proof is concluded, establishing the convergence of the dynamic model for CAID with a team leader rotation mechanism.

    However, in the case of the dynamic model presented in Eq (7), it is evident that the definition of error as stated in Eq (12) is not satisfied, rendering the calculation of Eq (11) infeasible. In order to ensure that the mentioned model effectively enhances the level of cooperation while maintaining system convergence, we redefine the error for student i as

    hi=xi(t)xi, (16)

    where, xi represents the strategy value which have maximum fitness. Hence, we get

    ˙hi=˙xi(t). (17)

    In particular, if the relationship of all students is completely connected, then it can be deduced that x1=x2==xN. This implies that

    hihj=xi(t)xixj(t)+xj=xi(t)xj(t). (18)

    Theorem 1. Let the weight matrix W is bounded and the communication digraph be strongly connected. Select the optimal guidances of teacher δi(t) as

    δi(t)=argminξ[sξijsij+deg(v)ˉf]. (19)

    Denote tuning law as

    ˙ˉWi=ϕ(x)Nj=1hiwij(hjhi). (20)

    Then, the behavior of system (7) exhibits convergence.

    Proof: Let the Lyapunov function be the form of

    V=mi=112h2i+12ˆWTiˆWi, (21)

    where ˆWi denotes the deviation in estimation for student i.

    Thus, one can derive the derivative

    ˙V=mi=1hi˙hi+mi=1ˆWTi˙ˆWi. (22)

    Substitute ˙hi=˙xi=mj=1eij(sξijdeg(vi)+ui)(hjhi) into (22), it leads to

    ˙V=mi=1mj=1hieij(sξijdeg(vi)+ui)(hjhi)+mi=1ˆWTi˙ˆWi. (23)

    When (19) is satisfied, it can be obtained that

    sξijsijdeg(vi)¯fi. (24)

    Therefore,

    ˙V=mi=1mj=1hiwij(sijdeg(vi)+ζ)(hjhi)+mi=1mj=1hieijˆWTiϕ(x)(hjhi)+mi=1ˆWTi˙ˆWi. (25)

    Note that if the tuning law described in Eq (20) is fulfilled, it can be inferred that the derivative ˆWi is given by

    ˆWi=ϕ(x)Nj=1hiaij(hjhi). (26)

    Define ψξij=aijpij+deg(vi)εdeg(vi), we can get

    ˙VNi=1ψmax(hjhi)20. (27)

    Consequently, the proof is completed, demonstrating the convergence of the proposed dynamic model (7) for CAID.

    In this section, a series of experiments are conducted to validate the effect of the mentioned model. If there are disagreements or misunderstandings among students, it can lead to confusion and communication barriers, thus hindering the learning process. Peer learning is typically aimed at achieving common learning objectives. If students have different interpretations of these objectives, it can result in an unclear direction of the learning process or even deviation from the intended learning goals. Therefore, we theoretically validated the proposed peer learning model. In this section, we will validate it further based on experimental data after ensuring that student states have achieved consistency.

    We first generate a fully connected graph G with N=20 students. Next, the performance of the CAID model on G and G are respectively revealed. The parameters of CAID are set to p=5,q=1,ξ=1. From Figure 2(a), it can be seen that the dynamic model without the guidance of the teacher, with the change of team leader and the learning among team members, the final convergence strategy is betrayal. After the proposed teacher-guided peer learning, the result of high cooperation rate as shown in Figure 2(b) is obtained, which proves that teacher-guided peer learning can make students with a team leader rotation mechanism more efficient which is good for teamwork. From Figure 3(a), it can be seen that students cannot reach a consensus with each other as the team leader changes and the learning among team members without the guidance of the teacher. After the proposed teacher-guided peer learning, the result of high cooperation rate as shown in Figure 3(b) is obtained, which also proves that teacher-guided peer learning can make students with a team leader rotation mechanism trend towards better teamwork. At the same time, in order to prove the effectiveness of the proposed method with a large number of students, the number of students is increased to 50. From the comparison of Figure 3(c), (d), it can be found that peer learning with teacher guidance can make students with a team leader rotation mechanism better reach consensus.

    Figure 2.  The convergence for CAIPD.
    Figure 3.  The convergence for CAISD.

    This paper introduces an innovative approach to teacher-guided peer learning by using a continuous action iterated dilemma and a team leader rotation mechanism. In the previous results, the interactive relationship between students is required to be fixed, but it is difficult to achieve in the actual classroom. On the other hand, each student has his own unique personality. Therefore, this paper adopts some new ideas. First, a team leader rotation mechanism is proposed to make sure each student has the opportunity to become a team leader, which enhances students' sense of participation and improves classroom efficiency. Second, based on CAID, we design a model to describe students' behavior in complex environments. The advantage of the CAID model is that it uses multi-layer nonlinearity, which has higher accuracy in describing the behavior of students. In terms of application, we use the Lyapunov method to prove the stability to show that the model is reliable. It is worth noting that changes in the parameters do not affect the Lyapunov function, which reveals that the range of application of the model is wide. For example, teachers can implement a team leader rotation mechanism to encourage student participation and foster leadership skills. Additionally, the CAID model can be used to more accurately describe student behavior in complex learning environments, contributing to personalized education. However, the model does not account for the additional cost of teacher guidance. Under the rotation leadership mechanism, management and supervision may become more complex. Ensuring that everyone fulfills their leadership responsibilities may require extra effort, which is a further consideration for the ongoing development of this model.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author declares there is no conflicts of interest.



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