Journal of Computer Assisted Learning

In this paper, a new approach for the formation of four‐member collaborative learning groups is presented. Group formation is presented by the mathematical optimization problem. Based on the proposed approach and the variable neighbourhood search (VNS) algorithm, the application that solves the presented problem and provides the appropriate division into groups is created. The proposed approach considers the scores of a pretest, interpersonal relationships, and prosocial behaviour/openness skill of students. In order to validate our approach, an experiment was designed with 108 first‐year university students of Belgrade Business School—Higher Educational Institution for Applied Studies. Experimental and control groups were divided into four‐member groups. The experimental group was divided by using the proposed method and the control group by student selection and random selection. Multilevel analysis is used to determine whether there is a significant difference in learning outcomes between the two groups. The experimental results showed that students from the experimental group achieved significantly higher success than the students from the control group. In addition, computational results obtained with the proposed VNS algorithms are compared and verified with the results obtained by random (Monte Carlo) method.

Lay Description

What is currently known about the subject matter of this paper:

Group formation have great influence on the success of collaborative learning.
Consideration of multiple variables for group formation can be very demanding.
Previous computer‐based methods for group formation do not use VNS.

What this paper adds:

In this paper, new metaheuristic approach for group formation is proposed.
The proposed approach considers the scores of a pretest.
The proposed approach considers interpersonal relationships.
The proposed approach considers prosocial behaviour/openness skill of students.

The implications of study findings for practitioners:

The proposed approach have positive impact on students' academic performance.
This approach can divide a large number of students into smaller groups based on multiple variables.

1 INTRODUCTION

There are many examples of positive outcomes of collaboration and collaborative knowledge sharing (Johansson, 2004). Group formation is the first stage of the collaborative learning group development life cycle (Tuckman & Jensen, 1977). Researchers in the area of collaborative learning claim that many of the unsuccessful outcomes of group work originate from the group formation process (Johnson & Johnson, 1985; Slavin, 1987). Group formation plays a critical role in terms of enhancing the success of collaborative learning (Johnson & Johnson, 1990; Slavin, 1983), therefore increasing the learning progress of students (Graf & Bekele, 2006).

Teachers who are willing to consider multiple variables to create collaborative learning groups must deal with major computational requirements (Hwang, Yin, Hwang, & Tsai, 2008). Dividing students into groups is a very difficult task due to the large number of possibilities, and it is often impossible to be carried out without the use of computers. Computational tools that help teachers in the formation of groups according to specific criteria are very important (Notari, Baumgartner, & Herzog, 2014). Recently, several studies have explored the computer‐based methods assisting in group formation procedure (Graf & Bekele, 2006; Hubscher, 2010; Hwang et al., 2008; Lin, Huang, & Cheng, 2010; Moreno, Ovalle, & Vicari, 2012; Sadeghi & Kardan, 2015; Tanimoto, 2007; Wang, Lin, & Sun, 2007).

In this paper, the new approach for group formation is proposed. It considers the scores of a pretest, interpersonal relationships, and prosocial behaviour/openness skill of students. Characteristics used in our approach were selected not only based on their effect on the process of collaborative learning but also based on the ability to choose a clear strategy for the division of students, which will cover all possibilities and that will enable the efficient implementation of the proposed approach. Based on the proposed approach, the application is developed, which generates collaborative learning groups from the input data given in the form of list of students and their characteristics.

The model creates division of students into heterogeneous groups of approximately equal quality regarding the scores of a pretest, in a way that these groups do not contain students who have negative feelings towards one another, nor are they close friends, and the groups are homogenous regarding the prosocial behaviour/openness skill of the students. Such an optimization problem can be solved by using the exhaustive method, which examines all possible solutions, but this approach is not always feasible, depending on the number of students, students' characteristics, and groups (Moreno et al., 2012). In cases where exhaustive method is not feasible because of the complexity of the problem, the metaheuristic search methods may be a good alternative (Glover & Kochenberger, 2003). Metaheuristics are problem‐specific higher level procedures designed to efficiently find a sufficiently good solution to a mathematical optimization problem (Blum & Roli, 2003). Metaheuristic methods cannot guarantee that the obtained solution is the optimal solution of the mathematical optimization problem as well. However, practice has shown that metaheuristics have given very good solutions to various problems. Metaheuristics very often find optimal solution with significantly lower computational effort when compared to exhaustive method applied to the same problem (Marić, 2010; Stanimirović, Marić, Božović, & Stanojević, 2012). Some well‐known metaheuristics are simulated annealing, taboo search, genetic algorithm, particle swarm optimization, variable neighbourhood search (VNS; Glover & Kochenberger, 2003; Kennedy & Eberhart, 1995; Mladenovic & Hansen, 1997; Reeves, 1993; Resende & Pinho De Sousa, 2004), and so forth. VNS metaheuristic is based on systematic change of neighbourhoods, and because the proposed problem is suitable for the definition of the neighbourhood structure, VNS for solving the proposed problem is implemented.

Results obtained by presented metaheuristic method are compared with the results obtained by the random method in order to evaluate the performance of the proposed approach. In addition, the research is conducted in order to prove whether the proposed approach affected the collaborative learning process and consequently the students' achievements.

2 THEORETICAL BACKGROUND

2.1 Methods for collaborative learning groups formation

Groups are generally formed by using student selection, random selection, or teacher selection (Wang et al., 2007). Groups formed by student selection are very often based on friendship or common interests in a topic, which may lead to the formation of homogenous groups (Abrami et al., 1995). Friendship between group members can facilitate the process of collaborative learning but can lead to weaker learning outcomes due to a lack of multiple perspectives. Good friends are not necessarily good learning partners (Sadeghi & Kardan, 2015). Also, there is a possibility that members of homogeneous groups exclude students with less developed social skills (Wang et al., 2007).

By using the random method for group formation, the possibility that the individual student feels rejected or singled out is reduced. On the other hand, groups formed by random methods may consist of low‐ability students. Furthermore, there is the possibility that members of random selected groups cannot work together, so the groups need to be rearranged (Moreno et al., 2012). Teacher selection implies that teachers use specific characteristics, usually ability or prior achievement, to form groups (Wang et al., 2007). Teachers who do not know their students very well cannot take full advantage of this strategy because they do not have enough reliable information to form groups.

2.2 Group formation based on specific characteristics

The use of collaborative learning as a teaching method requires two important decisions. The teacher first needs to choose the students' characteristics that can affect the outcome of collaborative learning. Characteristics suggested in previous research include race, gender, ability, self‐efficacy, and learning style (Bandura, 1997; E. G. Cohen & Lotan, 1997; Cordero, Di Tomaso, & Farris, 1996; Savicki, Kelley, & Lingenfelter, 1996; Sternberg, 1998). In addition, the students' social skills may significantly affect the outcomes of collaborative learning, influencing the interaction among group members (Notari et al., 2014).

Secondly, the group type, regarding all specific characteristics that teachers use for selection, must be considered—either heterogeneous or homogeneous. Group type, according to general criteria, can also be combined, for instance, by preferring homogeneity for some characteristics and heterogeneity for others (Gogoulou, Gouli, Boas, Liakou, & Grigoriadou, 2007). Some group composition studies show that homogeneous group compositions pose an advantage over heterogeneous group compositions provided that the variable in question is positively associated with the learning or collaborative processes, for example, high cohesiveness or high sociability (S. G. Cohen & Bailey, 1997; Gully, Devine, & Whitney, 1995; Holen, 2000; Hsu, Chou, Hwang, & Chou, 2008).

On the other hand, there are some research papers considering the group formation procedure (Graf & Bekele, 2006; Hubscher, 2010; Johnson, Johnson, & Holubec, 1998; Lin et al., 2010; Moreno et al., 2012; Wang et al., 2007) in which the heterogeneous groups (in regard to academic achievement, gender, ethnicity, task orientation, and abilities) are preferred over homogeneous grouping with the similar instructional level of students.

2.2.1 Students' ability

Heterogeneous groups based on ability have a positive effect on students regardless of their ability (Sadeghi & Kardan, 2015). However, there is a possibility that high‐ability students may be dissatisfied, as assisting low‐ability students requires additional effort. Moreover, low‐ability students can feel discomfort because they need help from other group members (Abrami et al., 1995). Although homogeneous groups based on these characteristics have a positive influence on high‐ability students, such groups can leave low‐ability students without sufficient opportunities for improvement (Wang et al., 2007). Students in all‐high or all‐low ability homogeneous groups more often ask for correct answers or surface information, whereas students in all‐low ability groups are more hesitant to ask for help (Webb, 1989). Collaborative learning process works best when groups contain a combination of high‐ and low‐ability members, but great differences among group members can decrease the effects of cooperation (Webb, 1989).

2.2.2 Interpersonal relationships between students

Current research indicates that most computer‐based methods for group formation do not consider interpersonal relationships between students. Poor interpersonal relationships among students can lead to the dissolution of groups based on purely academic principles. In the research of Moreno et al. (2012), two of the nine groups formed regardless of the interpersonal relationships were unable to function because of personal problems among students. There are several methods that take into account interpersonal relations, but only in order to avoid the negative relationships among group members. In these methods (Hubscher, 2010; Sadeghi & Kardan, 2015; Tanimoto, 2007), students can choose the members of their group, provided that this division does not interfere with the parameters set by the teacher.

Positive interpersonal relationships and a sense of community are a prerequisite for enhanced task accomplishment that may be achieved, especially in a computer‐supported collaborative learning environments where it is more difficult to achieve such psychosocial processes than in face‐to‐face learning groups (Kreijns, Kirschner, & Jochems, 2002). The lower levels of social interaction can have negative effect on efficacy of collaborative learning (Kreijns, Kirschner, & Jochems, 2003). However, groups formed by student selection are very often based on friendship, which can lead to weaker learning outcomes due to a lack of multiple perspectives (Abrami et al., 1995). There are studies that indicate that good friends are not always the best choice for collaborative group members because some preferred groupmates were only good social friends (Sadeghi & Kardan, 2015; Takači, Stankov, & Milanović, 2015). From the above stated reasons, we can conclude that higher levels of positive interpersonal relationships between students should positively affect the degree of collaboration but under the condition that casual conversations do not became dominant compared to task‐oriented discussions. Therefore, it is desirable that the collaborative learning group does not involve students who are close friends or who cannot cooperate with each other (Takači et al., 2015) in order to create a working and professional atmosphere.

2.2.3 Prosocial behaviour/openness skill

Prosocial behaviour refers to “a broad category of acts that are defined by some significant segment of society and/or one's social group as generally beneficial to other people” (Penner, Dovidio, Piliavin, & Schroeder, 2005, p. 2). Openness positively influences person's willingness to accept novel ideas and readiness to experience both positive and negative emotions (Costa & McCrae, 1992). Prosocial behaviour/openness skill is a construct based on prosocial behaviour and openness. Notari et al. (2014, p. 5) described prosocial behaviour/openess as an “openness to other people's opinions, being able to take someone else's perspective, being ready to help someone.”

Heterogeneity regarding prosocial behaviour/openness can be the main cause for dissatisfaction with performance, the perceived lack of efficiency of collaboration, and the division of responsibilities (Notari et al., 2014). Increased heterogeneity can cause the situation in which tasks are more often divided among group members (Karau & Williams, 1993). Personal social skill levels of group members has less impact on the effectiveness of the collaborative learning process than the social skill configuration within the learning group (Notari et al., 2014). Homogeneous prosocial behaviour/openness skill groups should be more efficient than heterogeneous groups regardless of the average level of this skill (Notari et al., 2014).

2.3 Computer‐based methods assisting in group formation procedure

Recently, several studies have explored the computer‐based methods assisting in group formation procedure. Graf and Bekele (2006) presented a method based on an ant colony optimization approach, using general student performance and an estimation of students personality traits as criteria for group formation. The method presented in Hwang et al. (2008) uses an enhanced genetic algorithm approach, considering the scores of a pretest and the understanding level of a certain course domain. Lin et al. (2010) also focus on two characteristics. Their method is based on a particle swarm optimization approach, considering student interests and understanding levels as grouping criteria. The method presented in Wang et al. (2007) also uses a genetic algorithm approach, but it considers three characteristics related to particular thinking styles.

Bekele (2005) used a vector space model. Gender, group work attitude, interest in mathematics, achievement motivation, self‐confidence, shyness, English, and mathematics performance were considered as grouping criteria. Moreno et al. (2012) proposed a genetic algorithm approach using multiple student characteristics. In their study, an example using three characteristics (an estimate of student knowledge levels, an estimate of student communicative skills, and an estimate of student leadership skills) is presented.

None of the above‐mentioned methods considers interpersonal relationships among students as a parameter for group formation in collaborative learning. In several methods (Hubscher, 2010; Sadeghi & Kardan, 2015; Tanimoto, 2007), compatibility is used, which is based on a rating provided by the students, reflecting how much they would like to work with a fellow student. These methods are designed to satisfy the wishes of students when it comes to group formation, which represents a combination of teacher and students selection approach for the creation of collaborative groups.

3 THE PROPOSED GROUP FORMATION METHOD

In this paper, we propose group formation method considering the scores of a pretest, interpersonal relationships, and prosocial behaviour/openness skill of students. The group formation process including several parameters is presented as a mathematical programming problem. Formulation of the problem is proposed and the approach for solving the problem is presented.

3.1 The group formation procedure

3.1.1 Description of the validation population

The collaborative learning approach is applied to the one‐semester course of Statistics attended by the first‐year students of Belgrade Business School—Higher Educational Institution for Applied Studies. It is proposed to create four‐member groups, which are heterogeneous regarding students' previous knowledge of the mentioned subject, which contain the students with neutral mutual relations (who do not have negative opinion about other group members nor are they close friends) and which are homogeneous regarding prosocial behaviour/openness skill.

In order to establish the students' previous knowledge level of the subject, we used the results of the colloquium, which will be called pretest in further text. Pretest consisted of four mathematical problems (student‐produced response questions), which the students were expected to solve in 3 hr. Besides providing the exact solution, students had to explain in detail in what way they came up with the solution. The internal consistency of the pretest was decided according to Cronbach's α, a reliability of 0.78 was obtained. The number of points that a student obtained on the colloquium should be scaled to the value of [0,100], so the students' previous knowledge level of the subject is represented with one number from the interval [0,100]. The students should be divided into reasonably heterogeneous groups of approximately equal performance regarding the scores of a pretest. A reasonably heterogeneous group is a group composed of students with low, average, and high student scores (Graf & Bekele, 2006). This is based on the recommendation of Slavin (1987) who proposes that collaborative learning groups should be mixed‐ability groups, which include one high achiever, two average achievers, and one low achiever.

In order to determine the interpersonal relationships between students, students were given questionnaire in which they could indicate colleagues they wished to absolutely avoid, those who are not the best choice to work with, and those who they preferred to work with. Less preferable relationships should be represented with a higher number and more preferable relationship with a smaller number. There are no limits to the number assigned to specific relationships, and teachers have freedom to choose ratio between numbers representing different relationships. Therefore, the attitude of one student towards another student is represented with one number greater or equal to zero.

In order to determine the level of prosocial behaviour/openness skill, students were given a questionnaire containing self‐referential statements that students rated on a 4‐point scale—do not agree at all (1)–totally agree (4)—(Notari et al., 2014). The average rate of some students' answers was used to represent the level of prosocial behaviour/openness skill with one number between 1 and 4. Based on the results of the questionnaire, the students were ranked according to the level of prosocial behaviour/openness skill.

3.1.2 Method formulation

In order to form reasonably heterogeneous groups regarding previous knowledge, the ranking list of all students (from least successful to the best one) is created, based on the results of a pretest. Graf and Bekele (2006) defined the measure of goodness of heterogeneity A_i by the formula:

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where S₁ is the maximum student score, S₄ is the minimum student score, and S₂ and S₃ are the student scores of the remaining students from the ith group. Based on this measure of heterogeneity, the division into groups should be carried out in the following way, according to this criterion. The first four‐member group G₁ consists of the student with the maximum score and the student with the minimum score, and two students whose scores are the closest to the arithmetic mean of the best and worst students' results from the four‐member group. The selected students are removed from the ranking list, and the process is repeated in order to form the next four‐member group. However, due to the existence of other criteria in this method, the final division will be different from the division obtained in this procedure.

Drawback of the previous measure of heterogeneity of the group is in the fact that the small differences in scores S_j can cause great variations in measure of heterogeneity A_i, which can lead to disproportion with the measures of other characteristics used in this paper. Therefore, we use the modified version of the previous formula:

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where Max is the maximum score from the pretest. When the value of A_iis smaller, the heterogeneity of group G_i is better. Due to proportion with the measures of other characteristics, it is recommendable to scale the results of the pretest to 100.

Let N be a total number of students and F_xy be the matrix which represents interpersonal relations, where 1 ≤ x, y ≤ N (Hubscher, 2010). F_xy is not necessarily equal to F_yx. In Hubscher (2010), only one possibility is allowed in case when student X does not want to be in the same group as student Y. The proposed method introduces two possibilities for avoiding foes, so that in the case when all students' wishes cannot be fulfilled, there is still a difference between more and less wanted foes. Therefore, we use the following formula:

$urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0003$

When the values of I_i are higher, placing students x and y in the same group is less desirable. In this research, the values I₀ = 10,000, I₁ = 1,000, I₂ = 10 are used.

The measure of interpersonal relations in the group is given by the formula:

$urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0004$

When B_i value is lower, the interpersonal relations in group G_i are more desirable.

Regarding prosocial behaviour/openness skill, the groups are proposed to be homogeneous. The ranking list of all students based on the level of prosocial behaviour/openness skill should be created. If two students share the same level of prosocial behaviour/openness skill, they should have equal rank. The differences in group members' ranks should be the smallest possible, in order to make the group more homogeneous regarding this characteristic. The measure of homogeneity of the four‐member group G_i is given by the formula:

$urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0005$

where R_i represents the student's rank regarding prosocial behaviour/openness skill. When the value of C_iis lower, group G_i is more homogeneous.

In order to group the students in the optimal way, based on previous characteristics, the mathematical optimization model is proposed. The aim of the proposed model is to minimize the maximal number of unmet demands of the four‐member groups:

$urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0006$

The priorities of characteristics are represented by constants w₀, w₁, w₂ (weight). In this research, the values used are w₀ = 10, w₁ = 1, w₂ = 1.

3.2 Variable neighbourhood search

VNS is a metaheuristic proposed by Mladenovic and Hansen (1997) for solving mathematical optimization problems. Based on the systematic change of the neighbourhood, it was implemented on many classes of problems, some of which are location theory, cluster analysis, scheduling, vehicle routing, network design, lot sizing, artificial intelligence, engineering, pooling problems, biology, phylogeny, reliability, geometry, telecommunication design (Hansen, Mladenovic, Brimberg, & Prez, 2010), and so forth.

VNS consists of two main phases:

The shake phase—phase of leaving the local optimum.
The local search phase—phase of finding a local optimum.

Let variable x represent a solution to a proposed problem and variable N_k, k = 1, …, k_max the sequence of a preselected neighbourhood structures, then N_k(x) denotes the set of solutions in the kth neighbourhood of x.

During the shake phase, a random solution is considered in the kth neighbourhood of current solution x.

In the phase of local search, the whole neighbourhood of the current solution x is searched (mainly N₁(x) and N₂(x)), and if there is a better solution than x, the new solution is accepted and the local search continues from the new improved solution. Otherwise, if there is no better solution in the neighbourhood than the current one, the local optimum is hereby reached and the phase of the local search is completed.

VNS algorithm iteratively fulfils a shake phase and a local search phase (Algorithm .).

3.2.1 VNS for group formation

The solution x of the proposed problem represents a division of students into groups. A move (G_i, S_a)↔(G_j, S_b) is defined by swapping student S_a from group G_i with student S_b from group G_j, resulting in S_a being in group G_j and S_b in group G_i, respectively. Solution x^′ is in the kth neighbourhood of solution x, if the division x^′could be obtained from the division x by k swaps.

The objective function of one solution is defined by the formula max(w₀A_i + w₁B_i + w₂C_i) presented in Section 3.1.2.

3.3 Application

For the study purposes, an application that solves the proposed mathematical optimization problem is created. The application is developed on the.NET Framework, Version 3.5, C# language. To solve the optimization problem, the program uses VNS metaheuristic described in Section 3.2.1. As input data, the application requires an excel file with information about students. For each student, the excel document contains the following information:

ID
Student identifier
Score numbers in the pretest
Prosocial behaviour/openness skill
List of students to be completely avoided
List of students who are not the best choice to work with
List of students preferred to work with

ID represents the unique ID of the student. Student identifier represents a textual description of the student, showing his name, surname, and index number. The number of scores in the pretest represents the number of points that a student obtained on the colloquium and should be scaled to the value of [0,100]. Prosocial behaviour/openness skill provides the position of a student on the ranking list regarding that characteristic. If two or more students have the same level of prosocial behaviour/openness skill, then they are assigned the same number (rank) on the list. Therefore, the next or several following numbers will be skipped. The list of students, consisting of the ones to be absolutely avoided, those who are not the best choice to work with and those who are preferred to work with, is represented by the corresponding IDs that are separated by commas.

After the program loads the data from excel file, it runs an algorithm that divides students into groups. In the program, there is a parameter related to the number of iterations of the VNS algorithm. The higher parameter gives a higher chance for the algorithm to obtain a better solution, but on the other hand, it increases the working time of the program.

When the program finishes its work, as a result, it gives the user a new excel file containing the obtained group distribution. Each of the obtained four‐member groups consists of selected students, placed in four rows, together with the corresponding calculated characteristics of each group A_i, B_i, C_i.

This application is publicly available at the address: http://geogebra.matf.bg.ac.rs/software/forming_groups/app.zip.

4 RESEARCH METHOD

4.1 Research objectives

In this paper, a new method for the formation of groups for collaborative learning is presented. The main questions of this research are whether the use of this approach to divide students into groups can improve the outcome of collaborative learning of students and whether the presented application effectively leads to the satisfactory solution of the problem.

The following hypotheses are formulated:

H1.The presented application can obtain the division of students into groups, which to a greater extent fulfils the set of conditions than the division obtained by the random method, within a short computation time.

H2.The students from groups formed by using the proposed approach achieve a better result in the posttest than the students learning in groups composed by random or student selection.

4.2 Experimental design

4.2.1 Algorithm performance evaluation

In order to test the efficiency and performance of the proposed application based on VNS algorithm, the application is tested on various problem dimensions and the obtained results are compared with those obtained by the random (Monte Carlo) method.

4.2.2 Experimental design regarding the influence of the proposed method on students' academic performance

The research was conducted with two groups of students, the experimental and the control groups. The collaborative learning is applied to lectures and exercises as part of a one‐semester course of Statistics, for the first‐year students of the class of 2014 at Belgrade Business School. The specific course subject was Regression and Correlation Analysis. Students from the experimental group were divided into four‐member groups by using the proposed approach, and students from the control group were divided by student selection and random selection. Students from the experimental and control groups were physically separated during the study in order to prevent contamination of conditions. The students in the experimental group are expected to perform significantly better in the posttest regarding their learning domain.

4.3 Participants

The study included 108 first‐year university students from Belgrade Business School—Higher Educational Institution for Applied Studies. All students were domain novices, and they had never before been engaged in collaborative learning activity.

Students were assigned to one of two groups by using a matching procedure, taking into account that in both groups, students of different gender are equally represented and that both groups are equal regarding the results of the pretest. In the experimental group, there were 56 students (43 females, 13 males; average age: 20 years old) divided into four‐member groups using VNS algorithm explained above, whereas in the control group, there were 52 students (40 females, 12 males; average age: 20 years old) divided into four‐member groups by student selection (nine groups) and random selection (four groups). Random selection was used for the division of students who failed to divide into groups by themselves. The results of the pretest for experimental and control groups are presented in Table 1.

Table 1. Pretest results

	Number	Means	Stand dev.
Experimental group	56	22.96	8.70
Control group	52	23.00	9.60

Because students had never been engaged in collaborative learning before the pretest, the t test (α = 0.05) was used. The obtained value of p = 0.984 showed that there is no significant difference between the results of experimental and control groups in the pretest.

The average level of prosocial behaviour/openness skill for experimental group was 3.20 and for the control group was 3.22. Regarding interpersonal relations, students from the experimental group listed on average, 1.73 names of students to be completely avoided, 0.86 names of students who are not the best choice to work with, and 3.43 names of students preferred to work with. Students from the control group listed on average, 1.69 names of students to be completely avoided, 0.88 names of students who are not the best choice to work with, and 3.36 names of students preferred to work with. These data indicate that there are no important differences between the experimental and control groups on any of the above criteria.

The average measure of heterogeneity of the four‐member groups based on the results of the pretest in the experimental group was 75.57 (min. 66, max. 83) and in the control group was 86 (min. 47, max. 100), which means that heterogeneity of the four‐member groups in the experimental group was better. The average measure of homogeneity of the four‐member groups regarding the prosocial behaviour/openness skill in the experimental group was 67.36 (min. 0, max. 152) and in the control group was 88.7 (min. 27, max. 151), which indicates that the experimental group was better divided into homogeneous groups. The experimental group contains 10, and the control group three 4‐member groups in which the students are neither friends nor foes. In the experimental, group there are four 4‐member groups containing students who asked to be together in the group.

4.4 Procedure

4.4.1 Procedure of algorithm performance evaluation

The used test instances include 16, 24, 32, 48, 64, 96, and 100 students who should be assigned to groups. We used the real data of first‐year students of Belgrade Business School. The program was tested on a computer with an AMD FD‐7500 (2.10 GHz) processor and 8GB of RAM, running the 64‐bit version of Windows 8.1.

To verify the obtained results, a random division of students into groups is generated for each instance and its performance is calculated. In order to provide a fair comparison, in the case of random division, the Monte Carlo method is used, that is, a random algorithm is performed in 2,000n iterations (where n presents the total number of students) and the best obtained division is selected. It is chosen not to use the constant number of iterations in random method, because when the number of students increases, the number of possible combinations of group divisions also increases. Therefore, it would be less probable for random method to achieve good division. The stopping condition of VNS algorithm used in the experiment is fulfilled when the performance of obtained divisions in 10 consecutive iterations of the algorithm is the same. An iteration of VNS algorithm includes a shake procedure call and a local search procedure call.

4.4.2 Experimental procedure regarding the influence of the proposed method on students' academic performance

The study took place over a period of 1 month. During the first week, the students were given the assignment to learn the material from the field of Regressive and Correlation Analysis. The next week, the students studied a new lesson, with the help of professors, assistants and colleagues in collaborative groups, and through an interactive learning process, adopting new knowledge and solving various problems. At the end of the week, each four‐member group in the experimental and control groups was assigned to do a project. The project consisted of a set of problem‐based questions. Each member of the group had to solve problems individually after which all group members had to discuss and define the group solutions. All groups were required to submit all individual solutions together with the group solution. In the third week, students were assigned to solve problems together in collaborative groups during the lectures and exercises, working for 3 hr a day. At the end of the month, the students were rewarded for successfully finishing the project with maximal 10 points (out of 100, which is the maximum for the excellent result achieved in the whole exam).

After this period of collaborative learning, the students were tested. The posttest was conducted in the same way as the pretest. Each student did the posttest individually, as they did the pretest. The internal consistency of the posttest was decided according to Cronbach's α, a reliability of 0.81 was obtained. Let us remark that the maximum number of points on the pretest and posttest was the same, 30 points.

5 RESULTS

5.1 Results of algorithm performance evaluation

The test results are given in Table 2, where the comparison of the Monte Carlo algorithm and VNS algorithms is presented. The values of objective function from Table 2 are calculated according to the corresponding formula presented in Section 3.1.2.

Table 2. Algorithm performance

Number of students	Random		Variable neighbourhood search application
Number of students	Objective function	Time (s)	Objective function	Time (s)
16	1,006	0.41	1,006	0.28
24	957.3	0.84	920	0.47
32	959	1.45	909	0.81
48	1,000	3.28	848	3.36
64	1,070	6.23	848	5.07
96	1,207.3	14.33	1,010	8.91
100	1,221.7	15.57	955.4	11.33

5.2 Experimental results regarding the influence of the proposed method on students' academic performance

Multilevel analysis was employed to investigate the effect of group formation method on the results of the posttest because students worked in four member groups during the study, which violates the assumption of nonindependence of observations of individuals (Kirschner, Paas, Kirschner, & Janssen, 2011). The data were analysed by using a random intercept multilevel model that includes the group formation method (dummy coded with 0 = control group and 1 = experimental group) as a predictor variable. The dependent variable was the academic performance of students represented by the results of the posttest. The random intercept regression equation is given by

$urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0007$

where γ₀₀ is the intercept, γ₀₁ is the regression coefficient of the group level variable “group formation method,” U_0j is the group level variance, and R_ij is the individual level variance. This model was estimated with MLwiN version 2.36.

Deviance of the presented multilevel model is 846.58, whereas the deviance of the multilevel model without predictor (empty model) is 850.08. The decrease in deviance is significant, χ²(1) = 4.5, p < 0.05, which indicates that the estimated model better fits the data than the empty model. Estimated value of intercept γ₀₀ is 19.08 (standard error = 1.82). Estimated value of γ₀₁ is 4.89 (standard error = 1.865) p < 0.01. The multilevel analysis shows a significant effect of group formation method on the results of the posttest. The positive sign of γ₀₁ shows that the students from the experimental group performed better on the posttest than the students from the control group.

6 DISCUSSION

In Table 2, it is shown that the Monte Carlo and VNS methods gave the same results only in the case of the smallest instance when the number of students is 16. In this case, a VNS algorithm found the solution in less time because it requires a smaller number of iterations. In solving all other instances, the VNS algorithm gave much better results than the Monte Carlo algorithm in less time, which confirms H1. The Monte Carlo algorithm could achieve better speed if it uses smaller number of iterations. However, in that case, obtaining good solution with Monte Carlo algorithm would be much less probable.

If the presented problem is solved by the exhaustive method, where n students are divided into m = n/4 groups, the possible number of different divisions will be $urn:x-wiley:02664909:media:jcal12299:jcal12299-math-0008$ . For example, if 24 students are divided into six groups, the number of possible combinations is approximately 2⁴². For each combination, it is necessary to go through the list of students in one iteration, in order to calculate the objective function of the obtained solution. Taking into account the time needed to calculate the objective function for all combinations on the processor where an experiment is performed, it would require more than 913 days to solve this problem.

In addition to the good results, it can be seen that the algorithm is very time efficient compared to the dimension of the problem to be solved. The average time for solving instances is 4.32 s, whereas solving the highest instances of 100 students divided into 25 groups needs 11.33 s, which is a very short time, taking into account the number of possible combinations.

Regarding the effect of the proposed method of group formation on student achievement, the data from Section 5.2 shows that the students from the experimental group were more successful in learning contents from the course of Statistics, which confirms H2.

These results are consistent with previous research and confirm that group formation method has an impact on the outcomes of the process of collaborative learning (Graf & Bekele, 2006; Hwang et al., 2008; Johnson & Johnson, 1990; Lin et al., 2010; Moreno et al., 2012; Slavin, 1983; Wang et al., 2007).

Higher success of students in the experimental group can be explained as the benefit of the applied method on the experimental group. In contrast to the control group, in the experimental group, it was taken into consideration that the small groups for collaborative learning should be reasonably heterogeneous in terms of the students' abilities (Graf & Bekele, 2006; Slavin, 1987). Heterogeneous groups based on ability enhance learning process for students of all levels of ability (Abrami et al., 1995). Previous studies have also shown that the heterogeneous groups based on ability perform better than random selected groups (Moreno et al., 2012; Wang et al., 2007).

The collaborative learning groups should include students who have neither a positive nor a negative attitude towards each other (Sadeghi & Kardan, 2015; Takači et al., 2015). In the experimental group, ten 4‐member groups fulfil this condition, whereas four groups include preferred students. The algorithm should avoid the preferred students in the division, but the weight of that demand for this experiment was relatively small compared with other criteria, because previous studies indicate that heterogeneity of groups based on ability (Abrami et al., 1995; Moreno et al., 2012; Wang et al., 2007) as well as avoiding the negative relationships among group members (Hubscher, 2010; Kreijns et al., 2002, 2003; Tanimoto, 2007) should have a priority over this demand. On the other hand, in the control group, there are three 4‐member groups that include students who have neither a positive nor a negative attitude towards each other, and 10 groups formed by student selection. Even though the experimental group to a greater extent fulfils the demand that four‐member groups do not include the preferred students, we believe that an additional study is needed in order to determine whether this demand makes a significant influence on the success of collaborative learning.

In order to ensure that this division has the best possible effect on the process of collaborative learning, the configuration of the group regarding the prosocial behaviour/openness skill of students was taken into consideration. Forming homogeneous groups regarding prosocial behaviour/openness skill of students reduced the possibility for dissatisfaction with performance, the perceived lack of efficiency of collaboration, and the division of tasks (Notari et al., 2014). All the above could have an impact on the better results achieved by the experimental group.

7 RECOMMENDATIONS FOR FUTURE RESEARCH

This study demonstrated that the combination of three observed characteristics has a significant positive impact on the effects of collaborative learning in heterogeneous groups based on ability. Future research should explore what effect the division based on ability has on students with different ability levels. Regarding interpersonal relations, the experimental group fulfilled to a greater extent the demand that four‐member groups do not include the preferred students but due to small number of groups and overall sample used in this research, an additional research is needed in order to determine whether the groups that include students who have neither a positive nor a negative attitude towards each other achieve different effects of collaborative learning when compared to the groups formed by student selection.

In line with the research of Maqtary, Mohsen, and Bechkoum (2017), there is a need for development of comprehensive paradigm that expresses all the details of group formation process in different situations. In this paradigm, the classroom learning and the computer‐supported collaborative learning should be considered as different contexts. Further research should also focus on comparing the effects of single students' characteristics, as well as the effects of different combinations of students' characteristics on the outcomes of collaborative learning. Also, there is a need for the development of more general solution (application), which could consider more than three characteristics and which could divide students into groups of arbitrary size.

8 CONCLUSIONS

In this paper, a novel metaheuristic group formation approach based on VNS algorithm was proposed to address the multiple criteria grouping problem. The application based on the proposed approach can model the composition of a set of collaborative learning groups from a large number of participating students according to requirements specified by instructors.

To evaluate the proposed method, an experiment was conducted to compare the objective function values and computation time of the proposed method with those of random method. The performance analysis results showed that the proposed method can obtain better results than random method within a short computation time. In the case of large instances, the obtained solutions have a small number of unmet demands, which was the aim of the calculations. Time‐saving factor is significant when dealing with large‐scale problems.

From a pedagogical perspective, this study uses the scores of a pretest, interpersonal relationships, and prosocial behaviour/openness skill of students to form collaborative learning groups. Regarding the effect of the proposed method of group formation on student achievement, this research shows that the collaborative learning in groups formed by using the VNS method is more successful than learning in groups composed by random or student selection.

The use of the proposed approach for forming four‐member collaborative learning groups can be very helpful for students and teachers. The positive impact of the proposed approach to students is reflected in better learning outcomes. This approach can be very useful for the teachers in situations where a large number of students are to be divided into smaller groups based on multiple characteristics of the students.

ACKNOWLEDGEMENTS

This research was partially supported by the Serbian Ministry of Education and Science under Grant 174010. The authors are grateful to all of the anonymous reviewers for their useful comments leading to the improvement of this paper.

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Volume34, Issue6

December 2018

Pages 907-916

Journal of Computer Assisted Learning

A novel metaheuristic approach for collaborative learning group formation

Abstract

Lay Description

1 INTRODUCTION

2 THEORETICAL BACKGROUND

2.1 Methods for collaborative learning groups formation