A “Mixed” Strategy for Collaborative Group Formation and Its Learning Outcomes

Abstract

This study uses homogeneity in personal learning styles and heterogeneity in subject knowledge for collaborative learning group decomposition indicating that groups are “mixed” in nature. Homogeneity within groups was formed using K-means clustering and greedy search, whereas heterogeneity imbibed using agenda-driven search. For checking learning effectiveness, a simple schema of collaborative learning was proposed and prototype learning system developed using Android Emulator. Multiple regression analysis was applied on their learning results to derive regression coefficients for determining learning efficiency. The derived set of regression coefficients suggests more the time taken to form groups, better the student learning quality.

Keywords

collaborative learning group formation agenda-driven search Android Emulator multiple regression analysis

Introduction

Collaborative learning is a method of learning where two or more people attempt to learn something together (Dillenbourg, 1999). In this form of learning, learners capitalize on one another’s resources and skills, thereby broadening their own knowledge bases. Interaction is considered to be the integral part of collaborative learning. Participants in a group, with different knowledge levels, interact with each other and try to overcome their own shortcomings. Students engaged in collaborative learning are exposed to other viewpoints that may significantly vary from their own. This enables them to come up with ways to defend their viewpoint(s) and ensures that they have a deeper understanding of the subject. Several examples of such collaborative learning systems exist in contemporary literature (Hwang, Wu, & Ku, 2011; Suthers & Jones, 1997).

However, an important aspect of collaborative work is its group formation. Over the years, several methods of collaborative group formation have been proposed by the researchers. These have been implemented from the learner as well as instructor point of view. The simplest way of doing this is to let the learners choose their own groups (Abrami et al., 1995). It has been found that in this case, learners tend to chose groups based on friendship rather than learning background. These groups generally contain learners with similar collaborative skill and knowledge levels. A problem that may occur in forming groups in this manner is that it may create “orphan” students. These are the students who have not been able to assign themselves to any groups. Two ways may be devised to deal with them: They may be added to the members of existing groups, or these “orphans” may be asked to form a group by themselves (Slavin, 1983). In either of the cases, these learners fail to assign themselves to the group due to lack of interest and initiative and thus may not be able to function properly when working as a group. Learners and instructors often use “proximity” methods in determining groups (Ames & Ames, 1985). As an example, students with contiguous roll numbers or those residing in a location cluster may be assigned to groups. Again the problem with this method is that groups may contain students with different backgrounds and learning interests, and hence the purpose of collaboration may be defeated. Instructors sometimes use common characteristics within learners to form groups. As an example, groups may be formed on the basis of common learner research interests or time slots when the learners are free to work. However, such grouping would require the instructor to know his learners very well (Collins & Brown, 1988).

The aim of this research is to propose an automated method of group formation for collaborative work and investigate its effectiveness. However, before proceeding with this task, two aspects must be considered. First, the specific student characteristics on which grouping are based has to be decided. This study proposes two types of learner attributes for group decomposition: personal styles and subject knowledge. The second aspect relates to the type of group: homogeneous and heterogeneous. It is argued that a similar personal style facilitates group work. On the other hand, variation in subject knowledge enhances collaborative work within the group. This is because students within the group with poor knowledge can learn the subject from students with better knowledge, whereas students with better knowledge can clear their conceptions teaching students with poor knowledge. This study thus proposes collaborative group formation based on homogeneity of personal styles and heterogeneity based on subject knowledge. In other words, group are neither totally homogeneous nor heterogeneous but are mixed in nature.

In light of the earlier discussion, this study has two main goals: The first is to develop a model for automated group formation ensuring the required degree of homogeneity and heterogeneity using data mining and artificial intelligence algorithms. For ensuring homogeneity within groups on personal styles, this study uses K-means algorithm for initial group decomposition. Agenda-driven search is then used to ensure group heterogeneity on the basis of subject knowledge. Next, the working of this model with a running example is described. The proposed model is implemented by developing an Automated Group Decomposition Program (AGDP) in Java platform. AGDP tool is used for group decomposition of a set of students doing graduation in computer science for learning the subject “Introduction to Java Programming.” Collaborative Learning System (CLS) developed by Acharya and Sinha (2016) is used by these groups subsequently for the learning purpose. The Learning Experiences using AGDP and CLS and Comparative Study of Learning Experiences sections presents a comparative study on the learning outcomes of the current study with three similar works done by researchers for group formation (Christodoulopoulos & Papanikolaou, 2007; Graf & Bekele, 2006; Redmond, 2001). For this purpose, 4 sets of 24 students were selected at random to form groups of size 4 using methods proposed in these papers (3 such) and current study. Thereafter, all these groups (6 × 4 = 24 in total) were made to learn using CLS. Their learning efficiency was then compared by applying multiple regression analysis on the resulting data set. Finally, limitations of the model are discussed indicating the areas where future work may be directed.

Related Works

Over the years, several researchers have proposed various ways of collaborative group formation. This section discusses some of the pioneering works done by the researchers in this field. The most primitive work in this area has been done by Redmond (2001) where he proposed group formation based on preferred time slots. The greedy method proposed by him works by collecting preferred time slots from students using a survey and assigns them to groups based on their preferred subjects. The heterogeneity within the group is achieved on the basis of students’ computer proficiency. Christodoulopoulos and Papanikolaou (2007) proposed a method of group formation based on the learning attitude and communication skills of the learner. Learners were first divided into homogeneous clusters using Fuzzy C Means algorithm. It was found that each cluster contains unequal number of students. These clusters were then shuffled to contain equal number of students based on Fuzzy C Means probability matrix. Heterogeneity of the samples was obtained using random sampling without replacement. Graf and Bekele (2006) proposed a method of student grouping based on learner group work aptitude and subject performance. Their method initially formed a group of two high and low score student and finding two average score student in between. The optimality of these groups was determined by Goodness of Heterogeneity, Coefficient of Variation, and Euclidian Distance between groups. The optimal values of these parameters were then determined using Ant Colony Optimization algorithm.

In a different vein, Wang, Lin, and Sun (2007) have used similarity of thinking styles as a basis of collaborative group formation. Three thinking styles were identified: legislative, executive, and judicial. Initially, students’ thinking styles scores were tabulated from a questionnaire. Groups were formed by selecting a triangle with largest area generated by Random Mutation Hill Climbing; the vertices of the triangle denote three thinking styles. The greater the size of the triangle, more is the degree of heterogeneity within groups. Ounnas, Davis, and Millard (2009) have performed a detailed survey on the algorithms proposed by various researchers for collaborative group formation. They found two problems with these methods: (a) groups were created on the basis of fixed and predefined set of parameters and (b) one-dimensional data were used for model computation like questionnaire, surveys, and so forth. Based on these, they designed a framework to assist teachers in forming groups based on their chosen set of constraints. The framework was done in two levels: modeling student features using semantic web ontology and negotiating group formation using constraint satisfaction problem. A technique by the name of “Opportunistic Group Formation” was proposed by Ikeda, Go, and Mizoguchi (1997) in which they have used educational significance of collaboration in group learning. A negotiation process model is constructed that takes charge of collaboration when a learner shifts from individual learning mode to collaborative learning mode. Various system concepts like negotiation process, communication protocols, and so forth are used to characterize learning groups for Opportunistic Model Formation.

A novel approach to collaborative group formation using Intended Points of Cooperation was described by Wessner and Pfister (2001) by integrating them in a web-based course. Intended Points of Cooperations define the point of learning where collaboration should occur along with the type and size of learning group, the collaboration type, and additional material for each activity. This concept was used in development of “Lifelong Learning as Utility,” which is a German project aimed at the development of an integrated Internet-based learning infrastructure for continued training and education. De Faria, Adán-Coello, and Yamanaka (2006) described a way of group formation for collaborative learning in “Introductory Computer Programming” courses based on students’ programming styles. These styles were identified as length of identifiers, size and number of modules and number of indentations, comments, and blank lines. These features were used to develop a tool that will automatically evaluate student programs. Experiments conducted by the authors showed that collaborative learning was very effective for improving programming styles of students working in heterogeneous groups. Tobar and De-Freitas (2007) discussed the functioning of a software tool that divides a class into groups based on student characteristics and educational objectives. Groups have been created based on a set of rules defining the characteristics teachers want for group participation. The IMS data model was used to represent student characteristics. A summary of group formation techniques thus discussed is presented in Table 1.

Table 1.

Summary of Group Formation Techniques in Collaborative Learning Environment Discussed in Related Works.

Reference	Learner characteristic used for group formation	Methodology/algorithm used	Uniqueness of the application	Implementation platform
Redmond (2001)	Free working slots	Greedy Algorithm	Easy to implement	Pascal
Christodoulopoulos & Papanikolaou (2007)	Learning aptitude	Fuzzy C Means	Groups are either homogeneous or heterogeneous	PHP and MySQL
Graf & Bekele (2006)	Personality traits and performance	Ant Colony Optimization	Applicable for large data sets	Not reported
Wang et al. (2007)	Thinking styles	Random Mutation Hill Climbing	Learner psychological factors taken into account	Not reported
Ounnas et al. (2009)	Dynamic learner profiles	Constraint satisfaction search	Assist teachers in forming groups based on their choice of constraints	Semantic web technologies and logic programming
Ikeda et al. (1997)	Characteristics and functions of the learner groups	Means End Analysis	Learner transit between individual learning and collaborative mode	Logic programming
Wessner & Pfister (2001)	Dynamic collaboration points	Ontology-based integration of IPoCs’ in web-based environment	Web-based collaboration and training	Java and PHP
De Faria et al. (2006)	Programming styles	Software quality metrics for system assessment	Collaborative learning of computer programming	“C” programming language
Tobar & de-Freitas (2007)	Characteristic and educational objective	Rule-based system	Minimize the time taken by the teacher to form groups	IMS data model

Note. IPoC = Intended Points of Cooperation.

A notable feature of all these applications is that groups formed are heterogeneous in nature. In the application developed by Christodoulopoulos and Papanikolaou (2007), however, the groups are either homogeneous or heterogeneous in nature. While designing this application, it was thought that there should be certain similarity between the learners for them to be successful while working in groups. This study thus emphasizes on the fact that certain personal traits among the learners should be common for collaborative learning while stressing on heterogeneity of subject knowledge among them.

Proposed Approach

As discussed in the Introduction section, this study uses two types of parameters for collaborative group formation: personal styles and subject knowledge. Under personal styles, the attributes Communication skills, Fluency in using computers, Group work attitude are considered. Subject knowledge is used as a performance indicator for ensuring heterogeneity. Each learner is represented using a five-dimensional vector of the form (s_i1, s_i2, s_i3, s_i4, s_i5), 1 ≤ i ≤ n, where n is the number of learners. The first component represents student ID, whereas the next four components represent the attributes stated earlier. The model is built on the assumption that the similar values within the group for s_ij (1 ≤ i ≤ n, 2 ≤ j ≤ 4) enhance collaborative work, whereas heterogeneity is required for s_i5 (1 ≤ i ≤ n). In other words, groups should be formed in such a manner that it should contain learners with high, medium as well as low performance. The values for the parameters s_i2, s_i3, s_i4 (1 ≤ i ≤ n) may be assigned by their respective instructors based on observation. A maximum rating of p is assigned for each parameter. The value of the fifth parameter may be determined by conducting a pretest on the subject. The full mark of the test is q. Because similar values of s_i2, s_i3, s_i4 (1 ≤ i ≤ n) are required within a group, initial groups are formed by ensuring homogeneity of the sum of these values. The sum total of s_i2, s_i3, s_i4 (1 ≤ i ≤ n) for each learner represent vector called A, whereas s_i5 (1 ≤ i ≤ n) represents the Vector B. These elements of these vectors are represented using the Equations 1 and 2, respectively.

a_{i} = \sum_{j = 2}^{4} s_{ij}, 1 \leq i \leq n

(1)

b_{i} = s_{i 5}, 1 \leq i \leq n

(2)

whereA = {a_{i}}, 1 \leq i \leq n

(3)

B = {b_{i}}, 1 \leq i \leq n

(4)

Grouping of learners should be done in manner so as the members of the group would have “very close” a_i values and “very scattered” b_i values. In other words, Vector A would be used for ensuring homogeneity, whereas Vector B would be used for ensuring heterogeneity. Three steps are performed sequentially to achieve this:

Homogeneity within the group is achieved by clustering students using K-means algorithm on Vector A.

Each group obtained using clustering contains variable number of learners. Reshuffle within groups is performed based on Cluster Distances (CD) so that each group contains equal number of learners.

Heterogeneity on subject knowledge is required within a group to ensure quality of the group work. This is achieved using agenda-driven search on groups derived in Step 2 applying appropriate heuristic.

The process is next explained in details.

Homogeneity Using K-Means Clustering

This section performs clustering of the learners on the basis of Vector A values ensuring their homogeneity on the basis of personal styles. K-means algorithm is used for this purpose. Two reasons may be cited for using this algorithm in this context: (a) The number of clusters is known in advance (K); (b) each group consists of at least one learner (Wu & Kumar, 2009). Once an initial set of clusters are obtained, the number of learners in these clusters are equalized as discussed in the next section.

The K-means algorithm work by arbitrarily selecting K elements as cluster centers corresponding to each cluster. Elements of each cluster are computed based on similarity. The method of similarity computation used here is Square Error criterion defined by the equation

E = \sum_{i = 1}^{k} \sum a \in C_{i} | a - m_{i} |^{2}

(5)

where E is the sum of square errors for all objects in data set, a represents the object, and m_i is the mean of cluster C_i (Han, Kamber, & Pei, 2011). Based on these values, new cluster centers are computed. The procedure as applied to this study is enumerated in Algorithm 1.

Algorithm 1: Initial Groups

Input: number of groups g, {a_i}, 1 ≤ i ≤ n

Output: n learners clustered into g groups of unequal length

Step 1: Choose arbitrary g elements as cluster centers

Step 2: Repeat until no changes are made to the clusters

Step 2.1: Assign each {a_i}, 1 < i ≤ n to the cluster based on Equation 5.

Step 2.2: Update cluster means

Step 3: Return {G_j}, 1 ≤ j ≤ g

Group Equalization Based on CD

The problem however with this approach is that each cluster contains variable number of students. A way is proposed to construct groups from these clusters such that each group contains equal number of students while maintaining homogeneity in Vector A within the group. This process in which it is achieved is next discussed.

Let the desired (fixed) number of learners in each group be r. If the number of learners in the Group G_i is less than r (say k), then r − k learners are identified in Group G_i+1, and their s_i1 (1 ≤ i ≤ n) values are stored in the Vector T such that

T = Max {CD}_{(i + 1)} {G_{i + 1}}

(6)

The learners identified by the Vector T are transferred to Group G_i.

On the other hand, if the number of learners in Group G_i is greater than r (again say k), then k − r learners are identified in Group G_i, and their s_i1 (1 ≤ i ≤ n) values are stored in the Vector T such that

T = Max {CD}_{i} {G_{i}}

(7)

The learners identified by the Vector T are transferred to Group G_i+1.

This process is summarized by the following algorithm. It is to be noted that this algorithm is greedy (Brassard & Bratley, 1995) in nature because for filling the groups with deficient number of learners, a_i (1 ≤ i ≤ n) are chosen with maximum CD at each step without thinking whether it is the best decision in the long run.

Algorithm 2: Group Reshuffle

Input: n learners clustered into g groups of unequal length

Output: s groups of fixed length r

Step 1: Sort the groups in ascending order based on cluster centers

Step 2: Repeat the following steps for each Group G_i (1 ≤ i ≤ g)

Step 2.1: Find the number of learners in each group (k)

Step 2.2: if (k < r) then

Step 1.2.1: Compute t = r − k

Step 1.2.2: Compute Vector T using Equation 6

Step 1.2.3: Move these learners to Group G_i.

Step 2.3: if (k > n) then

Step 1.3.1: Compute t = k − r

Step 1.3.2: Compute Vector T using Equation 7

Step 1.3.3: Move these learners to Group G_i+1.

Step 3: Return {G_j}, each G_j containing exactly r elements, 1 ≤ j ≤ s, n = r*s

After this step, a check is performed to verify whether group homogeneity is preserved after equalization. This is done by computing the Group Decomposition Factor (GDF). GDF is the ratio of the number of students that remain in the original groups (n1) after equalization divided by the total number of students (n). In other words,

GDF = n 1 / n

(8)

This value of GDF is computed after equalization. If the value of GDF is greater than 0.6, computation proceeds to the next step.

Heterogeneity Using Agenda-Driven Search

This section first defines a parameter for checking the degree of heterogeneity of a Group G_i. This parameter is then used as a heuristic for adjusting the group members using agenda-driven search such that each group satisfies a predefined value of the degree of heterogeneity. Let the values of the Vector B for a Group G_i be divided into a set of intervals as [I₁,I₂], [I₂ + 1,I₃], [I₃ + 1,I₄],…, [I_K+1,I_K+1]. Because the full marks of the test that was used for determining Vector B is q, these intervals may be rewritten as [0,I₂], [I₂ + 1,I₃], [I₃ + 1,I₄],…, [I_K+1,q]. Of all these intervals, suppose x intervals contains nonzero members. Then, heterogeneity of Group G_i (denoted by H(G_i)) is defined as

H (G_{i}) = x / r

(9)

As an example, suppose a test was conducted on a Group G₁ of five students on full marks of 50. This range is divided into intervals [0–10], [11–20], [21–30], [31–40], [41–50]. The number of students securing marks in this range are 1, 0, 3, 0, 1, respectively. Then, H(G1) = 3/5.

This study proposes that the value of H(G_i) corresponding to the Group G_i should be greater than 0.5. This is achieved by applying agenda-driven search (Rich & Knight, 2005) on each of the groups. Agenda-driven search contains an agenda, which is a list of tasks that a system could perform. Associated with each task is a justification and overall rating of this task. In this case, the agenda is determining a set of heterogeneous groups of fixed size r. This is justified by the fact that proper learning requires group heterogeneity determined by H(G_i). The algorithm works by computing the sum total of the degree of heterogeneity (SH) of all groups for a current state. This defines the rating of the current state. This state is obtained due to improved H(G_i) of a particular group achieved by swapping s_i1 (1 ≤ i ≤ n) based on Vector B. If this value of SH is greater than the value achieved in the previous state, then this state is accepted as the new state. The algorithm works until all the groups achieve the desired degree of heterogeneity. The detailed working of the algorithm is shown in the following.

Algorithm 3: Achieve Heterogeneity

Input: s groups of fixed length r satisfying Equation 8 with Vector B.

Output: s groups of fixed length r satisfying Equation 9.

Step 1: Repeat the following steps until H(G_i) > 0.6 for all G_i, 1 ≤ i < r.

Step 1.1: Compute C== $\sum_{i}$ H(G_i). This is current task rating. Call this step CURR.

Step 1.2: for each Group G_j, 1 ≤ j ≤ r

If H(G_j) ≤ 0.5

Step 1.2.1: Find Min {k} in [1,r], k<>j such that suitable swap between

(s_j1,s_k1) based on B vector yields H(G_j) > 0.5.

Step 1.2.2: Compute new H(G_j)

Step 1.3: Compute N== $\sum_{j}$ H(G_j). This is new task rating. Call this step NEW. If N > C

then set NEW to CURR.

Step 2: Return {G_j} with desired levels of {H(G_j)} 1 ≤ j < r.

The value of GDF corresponding to this group decomposition in again computed. Due to the swaps performed within s_i1, 1 ≤ 1 ≤ n, this value to bound to be reduced compared with the value computed in Heterogeneity Using Agenda-Driven Search section. An acceptable value of GDF computed after this step is thus proposed to be greater than 0.4.

Summing It Up

This section discusses how the modules proposed in the previous sections can be used in collaborative group formation. In a nutshell, K-means algorithm is used to obtain an initial grouping of the learners based on Vector A values. It is found that each group contains unequal number of learners. Reshuffle of groups is done based on CD to obtain group equalization. After this stage, the value of GDF corresponding to this group decomposition is computed. Once it is found that GDF is greater than 0.6, then computation proceeds to the next stage. The third stage performs agenda-driven search to achieve the required degree of heterogeneity with Vector B values. Again after computing GDF, if it is found that this value is greater than 0.4, then the groups are accepted as final. These steps are enumerated in Algorithm 4.

Algorithm 4: Collaborative Group formation

Input: n learners with their respective A and B scores

Output: s groups each containing r elements ensuring group wise homogeneity in Vector A and heterogeneity in Vector B

Step 1: Apply Initial Groups

Step 2: Apply Group Reshuffle

Step 3: Compute GDF. If GDF < 0.6

Step 3.1 Print “Group decomposition not possible”

Step 3.2 Exit

Step 4: Apply Achieve Heterogeneity. If GDF < 0.4

Step 4.1 Print “Group decomposition not possible”

Step 3.2 Exit

Step 5: Return {G_j}, 1 ≤ j ≤ s

Example

The aim of this section is to illustrate the working of the algorithms proposed in the previous section using a suitable example. Data set described in the previous section corresponding to 15 students is used for this purpose. Each group is proposed to contain three learners. The algorithm works in three steps. The first step computes the Vectors A and B corresponding to the student set along with their cluster numbers and Euclidian distance of Vector A from cluster center using K-means algorithm. The corresponding results are shown in Table 2. These clusters are taken as a basis for initial group formation. For ease of computation, Statistical Package for Social Sciences (SPSS; Argyrous, 2012) was used here. It is to be noted here that marks corresponding to each of the three parameters related to personal style was allocated out of 10, whereas marks corresponding to subject knowledge was allocated out of 30.

Table 2.

Initial Clustering of Students Into Groups Using K-Means Algorithm.

Student ID	Vector A	Vector B	Cluster number	Distance
1	17	12	1	0.1428
2	18	06	1	1.1428
3	12	28	3	0.6667
4	23	09	2	1.3333
5	13	07	3	0.3333
6	17	13	1	0.1427
7	21	6	2	0.6667
8	25	27	5	0.0000
9	16	28	1	0.8572
10	21	28	2	0.6667
11	18	25	1	1.1428
12	28	05	4	0.0000
13	17	14	1	0.1428
14	13	16	3	0.3333
15	21	11	2	0.6667

For the second step, the cluster is sorted on ascending order of cluster centers. This ordering is found as {3,1,2,5,4}. However, each of these clusters are found to contain variable number of learners as {3,6,4,1,1}. The clusters obtained after this step is shown in Table 3.

Table 3.

Clusters Obtained After Sorting on Cluster Centers.

Cluster number	Student ID	Cluster centre	Cluster size
3	3, 5, 14	12	3
1	1, 2, 6, 9, 11, 13	17	6
2	4, 7, 10, 15	22	4
5	8	25	1
4	12	28	1

Cluster equalization is next performed. As an example, Group 1 contains six learners in place of three. For equalization purpose, three learners with highest CD from Group 1 are selected and transferred to Group 2. This is shown in Table 4.

Table 4.

Equalizing Group 1.

Group number	3	1	2	5	4
Student ID	3, 5, 14	1, 6, 13	2, 9, 11, 4, 7, 10, 15	8	12

This process is continued for all the groups until all the groups are finalized. The final groups obtained after equalization are shown in Table 5.

Table 5.

Final Groups Obtained After Equalization.

Group number	3	1	2	5	4
Student ID	3, 5, 14	1, 6, 13	7, 10, 15	8, 9, 11	2, 4, 12

The value of GDF factor is computed to check the preserved degree of homogeneity. The numbers of learners that remain in their original groups after decomposition are

3(Group 3) + 3(Group 1) + 3(Group 2) + 1(Group 5) + 1(Group 4) = 11. Thus, GDF = 11/15 = 0.73, and hence group decomposition proceeds to the next step.

The next task is to compute the degree of heterogeneity H(G_i) (1 ≤ i ≤ 6) with respect to Vector B for each group and determine rating of this state. The intervals are defined as [0,9], [10,19], [20,30]. This is shown in Table 6.

Table 6.

Groups With Respective H(G_i) Values.

Group number	Student ID	Vector B	H(Gi)
1	1, 6, 13	12, 13, 14	0.33
2	7, 10, 15	6, 28, 11	1.0
3	3, 5, 14	28, 07, 16	1.0
4	2, 4, 12	06, 09, 05	0.33
5	8, 9, 11	27, 28, 25	0.33

The value of C corresponding to this state is 2.99. One set of learners of G₁ and G₄ are now swapped to improve H(G₁) and H(G₄). The resulting state is shown in Table 7. The s_i1 (i = 1,2) swapped are indicated using *.

Table 7.

State Obtained After Swapping Members of Groups 1 and 2.

Group number	Student ID	Vector B	H(G_i)
1	2*, 6, 13	06, 13, 14	0.66
2	7, 10, 15	6, 28, 11	1.0
3	3, 5, 14	28, 07, 16	1.0
4	1*, 4, 12	12, 09, 05	0.67
5	8, 9, 11	27, 28, 25	0.33

The value of N obtained corresponding to this new state is 3.66. Thus, this state is accepted as the new state, and search continues till all H(G_i) are greater than 0.5 (1 ≤ i < 5). The final groups obtained in this manner are shown in Table 8.

Table 8.

Final State Obtained With Highest Rating.

Group number	Student ID	Vector B	H (Gi)
1	8, 6, 13	27, 13, 14	0.66
2	7, 10, 15	6, 28, 11	1.0
3	3, 5, 14	28, 07, 16	1.0
4	1, 4, 12	12, 09, 05	0.67
5	2, 9, 11	06, 28, 25	0.67

Finally, a check is again performed to see if homogeneity is preserved using GDF. The numbers of learners that remain in their original groups after decomposition now are

2 (Group 1) + 3 (Group 2) + 3 (Group 3) + 1 (Group 4) + 0 (Group 5) = 10

Thus, GDF is 9/15 = 0.60. Because this value is greater than 0.4, this state preserves homogeneity along proposed degree of heterogeneity within groups.

Implementation

AGDP that will compute student groups based on input of personal styles and subject knowledge in automated fashion is used for implementation purpose. However, the method of group formation will be validated only if it provides learning effectiveness. For the purpose of learning, this study uses a CLS developed by Acharya and Sinha (2016).

Automated Group Decomposition Program

The process described in previous section was used for forming groups for studying the course “Introduction to Java Programming” at the undergraduate level. Each project group contains four students. Evaluation corresponding to the parameters s_i2, s_i3, s_i4 (1 ≤ i ≤ 24) were done by the mentor of the group out of 10, whereas s_i5 (1 ≤ i ≤ n) was derived using a test conducted on the basic concepts of the subject. This test is called the pretest. The full mark of this test is 40. Roll numbers have been used to identify the students. The corresponding data set is stored in the form of files that are used as input to the program. The output of the program shows the corresponding groups along with their A and B scores and DOH (Figure 1). AGDP has been implemented using the Java programming language, and the GUI has been implemented using the lightweight Java Swing GUI widget toolkit.

Figure 1.

Group decomposition using proposed method.

The next section will describe the manner in which these groups learn among themselves.

Collaborative Learning Using CLS

Let “n” students collaborate between themselves to learn a subject. Let them be divided into “k” groups denoted by G1, G2,…, Gk using the method described earlier. Thus, number of students in each group is n/k = m (say). The students in Gi are numbered as s_i1, s_i2,…, s_im, 1 ≤ i ≤ m. The students of each group are given to perform two tasks: First, they are given to learn the subject within the group, and second, they are to construct a prototype learning system based on the concepts they have learnt. Let the learning system developed by Group Gi be called Li (1 ≤ i ≤ k). The learning system Li so constructed will have two functionalities: It will contain modules that will aid the learning process, and at the end of learning, there will be a set of i (1 ≤ i ≤ m) comprehensive tests that checks each student’s concepts for all the modules. Students of Group Gi will learn and appear in test corresponding to learning system L(i + 1)mod n (1 ≤ i ≤ k). There are thus three phases in learning process of a Group Gi: Learning the subject by collaboration within Group Gi, designing the learning system Li, and appearing in a test in the learning system L(i + 1)mod n (1 ≤ i ≤ k). The results of this test for learner j corresponding to Group 1 will be called posttest (Tij, 1 ≤ i < k, 1 < j ≤ m). This method of learning and posttest is adapted so as to make the learning process more adaptive and rigorous.

The six groups derived in the previous section are used for learning now. The prototype learning systems are developed using Android Emulator because it is found that all learners have a working knowledge of it. The groups thus are numbered G1, G2,…, G6. They were given to learn the course “Introduction to Java Programming” and told to develop a learning system on it using Android Emulator. Each learning system consists of a set four tests. After completion of learning, learners of Group Gi revise and appear in the posttest Tij (1 ≤ i ≤ 6, 1 ≤ j < 4) using L(i + 1)mod n (1 ≤ i ≤ 6). A snapshot of the study material corresponding to G3 is shown in Figure 2.

Figure 2.

Snap of study material developed by Group G3.

Learning Experiences Using AGDP and CLS

A multiple regression analysis model (Goon, Gupta, & Dasgupta, 1975) is next developed to check the effectiveness of collaborative learning achieved using CLS based on the method of group division effected by AGDP. Multiple regression analysis is used when we want to predict the value of a variable based on the value of two or more other variables. The variable we want to predict is called the dependent variable (or outcome, target, or criterion variable). The variables we are using to predict the value of the dependent variable are called the independent variables (or predictor, explanatory, or regressor variables). Multiple regression also allows you to determine the overall fit (variance explained) of the model and the relative contribution of each of the predictors to the total variance explained. In this study, student learning inputs are used as independent variables, whereas learning results are used as dependent variable. The results of the set of 24 students used for illustration purpose in the previous section are used for developing the regression model. The simulations of the statistical procedure are done using SPSS (Argyrous, 2012).

The collaborative learning methodology proposed in the previous section is divided into three parts. First, students learn the subject in traditional manner based on which a pretest is conducted for computation of B scores. A scores were assigned by the class mentor. Second, students learn the subject collaboratively. Finally, students develop the learning system for learning and appearing in the test by their peers. This test (called posttest) is conducted at the end of the learning process to evaluate the quality of collaborative learning. These steps are named as Group_form, Colla_learn, CLS_develop for further reference. These variables, being independent, are used to predict the posttest results (named Test_result). All the three independent variables are estimated by the number of hours used in completing them. The posttest is conducted out of 50. Thus, the dependent and independent variables were measured in continuous scale. Moreover, independence of observations was checked using Durbin–Watson statistic in SPSS before deriving the model. Finally, the data set showed no significant outliers. All these properties indicate multiple regression model can be applied to this data set. The derived model summary is shown in Table 9.

Table 9.

Model Summary for Proposed Learning.

Model	R	R ²	Adjusted R²	SE of the estimate
1	.973	.954	.943	1.21

The correlation coefficient R (Hutcheson & Sofroniou, 1999), which in this case is .973, is the degree to which three predictors (Group_form, Colla_learn, CLS_develop) are related to the dependent variable Test_result. R², also known as the coefficient of determination, is used to evaluate model fit. Values close to 1 indicate that the dependent and independent variables are highly related, which is the case in Table 9. Interpretation of relationship between the variables can be obtained by the regression coefficients (Table 10).

Table 10.

Regression Coefficients for Learning.

	Regression coefficient	SE	t value	Significance
Constant	31.124	7.316	4.316	.001
Group_form	0.213	0.217	2.213	.000
Colla_learn	−0.205	0.113	−3.216	.000
CLS_develop	−0.526	0.316	−1.712	.003

Note. CLS = Collaborative Learning System.

From Table 10, it can be concluded that the general form of the regression equation is

\begin{matrix} Test_result = 31.124 + 0.213 X (Group_form) \\ - 0.205 X (Colla_learn) - 0.526 X (CLS_develop) \end{matrix}

(10)

The signs indicate the direction of the relationship between variables; the positive sign of the regression coefficients indicate direct proportionality, whereas negative sign of the regression coefficient indicate inverse proportionality. The t values and degree of significance corresponding to each regression coefficient is shown in Table 10. Because p < .05 for all parameters, it can be concluded that these coefficients are statistically significant. The F ratio in the analysis of variance table (Table 11) shows that the independent variables statistically significantly predict the dependent variable, as F(4, 17) = 53.174, p < .05 (i.e., the regression model is a good fit of the data).

Table 11.

ANOVA Results for Learning.

Model	SS	Degrees of freedom	MS	F	Significance
Regression	324.714	4	127.714	53.174	.001
Residual	17.360	17	5.608
Total	342.074	21

Note. ANOVA = analysis of variance.

Based on the regression analysis done earlier, the following conclusions can be drawn:

Posttest results in experimental data are directly proportional to the time take for group formation. This is indicated by the positive sign of Group_form coefficient. This is because good-quality students do well in pretest and hence have high values of B scores.

Posttest results are inversely proportional to the time taken to learn the subject collaboratively (indicated by the negative sign of Colla_learn coefficient). This indicates good-quality students are able to collaborate with their peers in a much better fashion.

Finally, posttest results are inversely proportional to the time taken to develop CLS (indicated by the negative sign of CLS_develop coefficient). Thus, better the quality of the student, lesser is the time take to develop learning system.

Comparative Study of Learning Experiences

The results obtained is now used to compare the learning experiences of this study with certain other pioneering works relating to group formation discussed earlier. The works of Redmond (2001), Christodoulopoulos and Papanikolaou (2007), and Graf and Bekele (2006) were taken up for comparison purpose. Two reasons may be cited for choosing these three papers over others. First, it was found that they could be easily implemented to derive the groups manually. Second, they contain a high number of citations as indicated by Google Scholar and hence are popular among a large section of researchers. These studies are referred to as A, B, and C, respectively, for easy reference in this section.

For this purpose, 72 undergraduate students were selected for studying the course “Introduction to Java Programming” collaboratively. They were divided into three subsets randomly each containing 24 students. Each subset is then divided into six groups, each containing four students. For the first subset, group formation was done as per Study A; for the second subset, group formation was done as per Study B; for the third subset, group formation was done as per Study C. After group formation, these groups were given to learn using the methodology discussed in Collaborative Learning using CLS section and posttest conducted after that. Three data sets of 24 records each was prepared exactly in the format specified in Learning Experiences using AGDP and CLS section and multiple regression analysis applied on then. The regression coefficients thus obtained are enumerated in Table 12 along with the corresponding degree of freedom and F ratio. The corresponding coefficients obtained in this study are added for ready reference.

Table 12.

Comparative Statistics of the Regression Coefficients of Studies A, B, and C With Current Study.

	Redmond (2001)	Christodoulopoulos & Papanikolaou (2007)	Graf & Bekele (2006)	Present study
Group_form (t value, sig.)	0.106 (7,124, .000)	−0.312 (−3.125, .003)	−0.112 (−4.163, .001)	0.213 (2.213, .000)
Colla_learn (t value, sig.)	−0.312 (−3.142, .000)	−0.214 (−1.112, .004)	−0.132 (−8.763, .000)	−0.205 (−3.216, .000)
CLS_develop (t value, sig.)	−0.167 (−6.142, .003)	−0.412 (−8.314, .002)	−0.345 (−3.126, .001)	−0.526 (−1.712, .003)
Degrees of freedom (Reg.,Res)	(3,12)	(7,18)	(3,16)	(4,17)
F (sig.)	63.161 (.004)	51.123 (.000)	42.312 (.003)	53.174 (.001)

Note. CLS = Collaborative Learning System.

The t values and degree of significance indicate that all the three models corresponding to Studies A, B, and C are acceptable. Again, the F ratio corresponding to all three studies indicate that they are a good fit for data. These statistics may thus be used to draw the following conclusions:

Just like the current study, it is found that posttest results depend directly on the time taken to form groups in Study A. Contrary to that posttest results are inversely proportional to group formation time for Studies B and C.

All the four studies indicate that collaborative learning time is inversely proportional to posttest results. Thus, collaboration is successful in reducing study time. This is most pronounced for Study A.

Similarly, all four studies indicate that CLS development is inversely proportional to posttest results. Again, collaboration is found to be successful in reducing learning system development time. This is most pronounced for this study.

Conclusion

The aim of this article is to propose a new method for group creation for collaborative learning. To this end, a “mixed” approach is proposed for group formation. It was argued that similar personal styles and heterogeneity in subject knowledge enhances group work. For this purpose, a set of A and B scores were defined. K-means algorithm was initially used for homogeneous group division on based A scores. Equalization of the groups was done using greedy search algorithm. Finally, heterogeneity within the groups was achieved using agenda-driven search based on B scores. There groups were then given to learn using a CLS developed by Acharya and Sinha (2016). Multiple regression analysis was applied to the results obtained by learning using this method for a set of undergraduate students for the course “Introduction to Java Programming.” Furthermore, another three student sets were selected of exactly same size and given to form groups using methods proposed in three prominent studies (Christodoulopoulos & Papanikolaou, 2007; Graf & Bekele, 2006; Redmond, 2001). Multiple regression analysis was again applied on their learning results for comparison purpose.

Multiple regression analysis turned out to be a very useful statistical tool in analyzing the learning results. It was found that longer the learner preparation time, better is the group learning result. Furthermore, it was found that better the quality of student, lesser the collaboration time and the time taken to develop the corresponding learning system. The comparative study between the learning results of the four studies enumerated in Table 12 is more conclusive. Two sets of regression coefficients indicate more the group formation time, better the learning result, whereas two others indicate the converse. Even better, all the four studies suggested that better the student quality, lesser the time taken for collaborative learning as well as learning system development. Thus, it may be concluded without a shadow of doubt that the proposed method of group formation has a positive effect in collaborative learning.

However, there are certain issues ignored in this study which may form the crux of future research. First, student choice is not taken into account while forming student groups. Certain factors like “friendship,” “location proximity,” and “trust” are often seen to be factors influencing group formation. Second, the factors that are taken into account for forming groups are fixed (student personal style and subject knowledge). The factors just listed cannot thus be taken into account if desired. Finally, group decomposition is possible only if this initial information about learners is available. For a set of students corresponding to whom initial information is not available, group decomposition would not be possible.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Author Biographies

Anal Acharya is an assistant professor at the Department of Computer Science in St Xavier’s College, Kolkata. He was the Head of the Department of Computer Science from the period of 01.01.2009 to 30.06.2013. His present research interest is in Educational Data Mining (EDM). He has about 20 years of experience in undergraduate and post graduate teaching and supervised several post graduate dissertations. He has several accepted papers in International Conferences and Journals. He has been awarded PhD (Tech) in Computer Science and Engineering by the University of Calcutta under the supervision of Prof. Devadatta Sinha.

Devadatta Sinha is currently a professor at the Computer Science and Engineering Department in the University of Calcutta, Kolkata. He was the Head of the department on several occasions. His research interests include Distributed Systems. He has over 30 years of experience and has supervised several PhD students.

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