[Sep. 13, 2007]

In this paper I have explored the idea of group learning and its benefits. There are many methods and strategies of teaching and learning mathematics in groups. Some of them are: collaborative learning, cooperative learning, discovery based learning, engaged learning, and problem based learning. I have tried to reflect on my past memories and nodal moments in learning mathematics. I found informal cooperative learning in a small group to be a very effective method from school to university level. Although I was not aware of the word cooperative learning at the time, I was involved in group work; cooperative learning took place within classes less or so and outside more or so, consciously or unconsciously. A community of learners that I was engaged in all the moment from school to university led a strong foundation of group cohesion and basis of learning mathematics (and other subjects) from each other.

Inquiry from Self

I have followed an autoethnographic method of inquiry in this paper. In autoethnographic method of inquiry, the author of an evocative narrative writes in the first person, making themselves the object of research and thus breaching the conventional separation of researcher and subjects the story often focuses on a single case and thus breaches the traditional concerns of research from generalization across cases to generalization within a case (Ellis & Bochner, 2000, as cited in Newton, 2004).
Autoethnography is “…research, writing, and method that connect the autobiographical and personal to the cultural and social context. This form usually features concrete action, emotion, embodiment, self-consciousness, and introspection…and claims the conventions of literary writing” (Ellis, 2004, p. xix, as stated by Jones, 2005).
Further, Spry (2001) states that autoethnography is a self-narrative that critiques the situatedness of self with others in social context (p.710). Jones (2005) states that autoethnography involves setting a scene, telling a story, weaving intricate connections among life and art, experience and theory, evocation and explanation … and then letting go, hoping for readers who will bring the same careful attention to our words in the context of their own lives (p.765).
Autoethnographic inquiry subscribes to the nomolithic worldview (Denzin & Lincoln, 2005) what reacts radically against the realist agenda of ethnography. Autoethnographic writing can be depicted as the metaphor of a camera (Ellis & Bochner, 2000 as cited in Luitel, 2003), which focuses on the rarely heard stories (Van Manen, 1988). In my understanding this is not the only data source; research narratives, but particular genres of story and a resource for data outside of the idiosyncrasy of my own biography.
I have tried to look me from myself not from others' self because that makes me more aware of my pedagogical practices and research methodology from postmodern perspectives. It also makes me more responsible in the process of narrating my experiences from the past to the present in order to interpret my own consciousness in the sociocultural contexts.

Dealing with Rigor Criteria

I have taken into account Van Manen’s notion of writing as lived experience for creating dialogic relationship between the author and the readers. Further, I have put an emphasis on creating pedagogically thoughtful text by considering the four criteria - orientation, strength, depth, and richness-as discussed by Van Manen (1990) and Geelan and Taylor (2001). These criteria establish the rigor of the research which emphasize more on process than input and outcome of the research.
I have written some of my narratives in present tense in order to give dramatic impression that helps readers to reflect at their present with my narrative accounts through dialogues and stories.

My Journey of Cooperative Learning

It was a day of 1979; I was admitted in a primary school nearby my village in Dang-Nepal. From the very beginning of my school life I started learning mathematics in-groups. Though I could count from one to hundred, I was poor at writing numbers. One of my friends, Gir Bahadur, was good at writing these numbers. I requested him to help me and he agreed. We used to sit together on the same bench. On my first day of school, I learnt how to write “EK (One) and Dui (Two) ” with the help of my friend, Gir Bahadur. Slowly I became familiar with other boys and girls in the class. We formed a team to sit together. When there was a spare period in the school we used to practise mathematics together in group. We learnt from each other in a group more than what we could learn from the teacher.
Our mathematics teacher often came in last period and told us to count from one to a hundred in turn by turn standing at the front of the class. Then he gave us writing work. I had a small slate board in my bag and wrote on it with a clay chalk. My friend, Gir Bahadur, helped me a lot to be able to write from one to a hundred. He also taught me multiplication table from two to ten.
There was no initiation of teachers to form learning teams in groups. If that had happened, there would be more learning of mathematics and classroom learning of mathematics would be cooperative learning. But I can say that at least there was cooperative learning in my informal group of learning of mathematics. Not only in grade one, it continued in grade two, grade three and others too. My cooperative learning was the main dynamics of my learning of mathematics. It started on the first day of my school and lasted until the end of my graduation. So, cooperative learning was the main dynamics of my learning of mathematics in the past and it is equally applicable to the present as well.
What is cooperative learning? How does cooperative learning help students to learn mathematics? What are the basic elements of cooperative learning? What are the different types of cooperative learning? To what extent can cooperative learning help students build social relationships, learn from each other, share responsibility, and solve their problems in a group?
These are some of the key questions that I have tried to explore in this paper. Although the school system had not incorporated a cooperative learning situation in the classroom, how students themselves created learning in cooperative groups and shared their ideas, opinions and knowledge in a responsible manner has been explored from my experiences through autoethnographic genre of writing for research.

Episodes of Cooperative Learning
The following two episodes give an idea about how I practised informal coop-erative learning of mathematics from school to university level.
Episode – One
It is an evening in 1985. Some boys gather at a room in Shashi's home. They are going to do their mathematics homework. They open their bags and every one is ready with a book, a notebook and a pen.
Kul Raj (opening up the maths book): Let's start our practice from the beginning.
Madhav: No, lets practice the difficult ones.
Bharat: Let's start from simplification.
(Shyam and Madhav remain silent for a while)
Smoky cloud from the west is moving to the east as if it has to reach its destination within a few hours and it has no time to wait and look downward. The sound of a thunderstorm in the western horizon makes us cautious about the weather conditions. We need to finish the day’s homework and prepare for examination. It must be at 8.30 p.m. on a summer day in 1983; I am among my friends Bharat, Kul Raj, Madhav and Shyam in my study room. The first terminal examination is near. Where to start from is a problem for the evening. We discuss how to start our practice.
I (seeking their consent): If you agree, let’s do what Madhav says.
Kul Raj (opposing): How can you say which is difficult for me and which is difficult for Madhav?
I (closing the discussion): Okay, let’s start from the beginning. It will cover all.
All of us agreed on my point.
We started practice of mathematics from the beginning of the mathematics exercises from the textbook of grade six.
One of the problems was like this:
, Bharat did as follows: =

Kul Raj (frowning at Bharat): It’s wrong.
Madhav (pointing at Kul Raj): It’s correct.
Shyam (turning the copy to me): Shashidhar, you tell us whether it is right or wrong.
I (after looking at the answer): Let me do it myself and then I will tell you.
Kul Raj (turning his copy up and holding pen with right hand): I will also do with Shashidhar.
Madhav: I see the next problem.
I (After few minutes): Friends, this is the solution:

Kul Raj (turning to last pages of the book): Yes, it’s correct.
Shyam and Madhav also agree that it is correct. But Bharat is still in the opposite. I try to convince him about why it is necessary to multiply by 5/5 and 3/3. He comes to a compromise saying that he will do the next one and see the answer in the same way.
The five boys from the same class stay till twelve of midnight. There is discussion and sharing of ideas among them. Nobody can go ahead without making everyone satisfied. They sometimes divide their responsibility to prepare different chapters and discuss together. The five boys have good unity in the school and it is a unique way to learn mathematics and other subjects in the school by them. They are open to teach and learn each other.
As the thunderstorm and lightening become more vigorous, the boys put the lantern out and go to their bed. At the same moment a loud thunderstorm rumbles in the sky makes them afraid and so they hide in their bed.T
Episode – Two
It could be a rainy day of 1998. Four men gather in one room in the Office of Teachers’ Records with a view to do combined study on “Operations Research”. Ajay is a new member in the group. Ujjwal starts to discuss on the topic Linear Programming. He at first asks his friends what they know about Linear Programming. Sujit says that he does not have any idea about it. Krishna says the professor did not solve any problem for them except one or two. “The professor, Mr. Write, came up in the class with a photocopied note and handed it over to us for copying.” Krishna explains how their class in the university campus is taught.
Ujjwal starts from the graph of linear algebraic system and inequality. Then he draws a graph for inequalities such x + y <5, 2x - y < 1, x > 0 and y > 0. He discusses the common solution set for all the inequalities in the graph. Krishna, Sujit and Ajay all draw a graph for the inequalities. Then all of them discuss about the vertices of the feasible region. They determine the value of the objective function max.(z) = 3x + 6y in those vertices and discuss about the maximum of the function. In this way they reach a conclusion that they can solve any Linear Programming problem by graphical method. Ujjwal says that he will start simplex method of solving Linear Programming next day.
(The office helper brings four cups of black tea and serves them)
Sujit comes up with his preparation in game theory. He starts with questions: What is a game? What are the types of games? What is two person zero sum game? And so on. He also discusses about some definitions such as strategy, matrix, players, and value of game, win, lose and so on. He, then, starts two persons zero sum game. He writes a matrix and starts discussion about how to solve the game. At first, he discusses about the Saddle Point method. It is so clear to all others that they like the technique of joint study. Sujit also stops his discussion for the day after the Saddle Point method with promise to continue next day with other methods.
Now it is time for Krishna. He says that he is not well prepared for the discussion but he will be ready next day. Ajay also chooses a topic, inventory management, for the discussion next day. Krishna tells Sujit that he can not get his points about Saddle Point method of solving a game. Ujjwal comes forward and helps him to understand with next example. During the discussion Sujit supplements the points of Ujjwal.
Ujjwal tells his grief of being a tourist student among his friends. He tells his stories about how his friends do not cooperate him when he asks for a note. One of his friends teaching in the same private school does not let him read his note. He is upset with the attitude of the classmates. One of his classmates gives him the rough copy in which he has practiced some problems of Real Analysis. Krishna consoles him that the regulars are also the same as tourists. The professors do not teach well. Those who teach well leave the campus and the oldies do not take their responsibilities of teaching in the classroom.
The four men come out of the office and enter the canteen to have some snacks. They all express their view of the day’s discussion and exclaim with sorrow that professors cannot do as well on the campus; otherwise they will not bother to be together.
The four men continued their cooperative learning till the end of their graduation. They found it very effective, interesting and an efficient way of learning. Though university classes were not in the process of conducting cooperative learning, the four men continued this approach as an informal cooperative group to learn mathematics.
My working in a group to learn mathematics not only encouraged me but it made my colleagues realize the power of group work even in study. Due to our continuous effort in group groups, I learnt the concept building and understanding the problem rather than rote learning. Later on this group performed better than other average students in the class.

Reflection on the Two Episodes

In my understanding cooperative learning has a strong philosophical foundation in constructivism. The constructivist theory, to me, takes the view that learning is construction of ideas by the active participation of learners and through the experiences they gain during participation in activities. According to constructivism, new knowledge is built using what students already know. That is, their prior knowledge influences what they construct. It seems to me that in the constructivist model, learning is active, not passive as in a traditional model. In the constructivist model, students confront their understanding in light of what they encounter during the process of learning.
In my point of view there are a number of implications for teachers in a constructivist approach with respect to cooperative learning. In this approach, first of all, the teacher is a guide, not the lead, as students construct their own knowledge. In addition, due to various backgrounds, not all students will understand everything in the same way. Hence the teacher, through cooperative learning, can have them engaged in activities which will allow them to understand their own thought processes and those of their peers, just by giving them a chance to voice them in group settings.
According to Vygotsky (1978), the zone of proximal development is the difference between what a student can do alone and what he/she can do with supportive collaboration or cooperative activities in group. It seems to me that according to Vygotsky, all learning must take place in cooperative settings as, cognitively, connections cannot be made without this collaboration.
In my understanding, adopting a structured, cooperative approach offers faculty members both the philosophical approach and the specific tools to transform their teaching. So, the philosophy to me is a constructivist theory of learning that places the responsibility for students' learning on the students themselves. Students receive support from their teachers and from their peers. Cooperative structures include a wide variety of activities suitable for different objectives. In this way, I think, cooperative learning offers a systematic, student-centered approach to instruction without putting anyone into a pedagogical strait jacket. It seems that lecturing and other approaches thus complement the cooperative principles.
I was highly committed to informal cooperative learning which built up on a set of pedagogical values that I took up (Johnson & Johnson 1991). When a classroom is characterized by ethnic, class, or religious diversity, a classroom climate shaped by cooperative learning is of particular value. Because cooperative learning methods enable all students to speak in class, share ideas, and challenge each other, the unique contributions of students are brought to the fore. Though it was our own effort to form such learners team would it have been more beneficial to all the students in the classroom if it was formally applied by our teacher.
I think, I should appreciate the use of cooperative learning in the progressive classroom where communication is a catalyst for learning. It seems to me that cooperative learning taught me to work together as a group, contribute responsibly and other important social skills. It is believed that students can learn from each other and teach each other. Students assist each other in working toward a common goal that we did in our informal cooperative learning team.
So far as I have understood, students and teachers of a progressive classroom work together to set attainable goals that they want to reach through the cooperative learning experience. I believe the best way to evaluate students then, is if they have reached the goal, or how close they come, and how they went about reaching the goal. I think the evaluation process is then individualized for each student. Progressivist style evaluation avoids standardized tests and focuses more on skill achievement and understanding of material. I believe in using authentic assessments that make the process relevant and meaningful to the student that was lacking when I was a student from primary school to the university degree.
In the progressive classroom and from the progressive teacher, students learn creativity, critical thinking and problem solving skills, and social skills. Students accomplish all of this through cooperative learning, experimentation, hands-on activities and guidance from a teacher who understands their individual needs, interests, and ability levels. But with my sadness, I have to say that, no teacher from my early school days to the university conducted such activity to promote group work and active participation in the learning process through sharing, interaction and dialogue in the classroom. Whatever we did in the informal group was our own culture of sharing, learning from each other and forming our collective view about any mathematical idea. That not only made us confident but sometimes saved us from the punishment from the teacher when we all made mistakes of same order or degree.
In relation to my style of learning mathematics in group, it was a constructivist way of learning mathematics through socio-cultural relations and positive interdependence among the members in the group. The dialogic narrations in episode one and two depict my role and the role of my group members in the mutual sharing of ideas, efforts and resources. Though we were not guided by teachers in the group, we were guided informally by principles of cooperative learning through self and group consciences.
The first episode portrays my practices of informal cooperative learning with my friends at home. I now realize that we did not have the formal set up of cooperative learning in the school classroom. We had informal group discussion and problem solving in the classroom in absence of the teacher and it was in good practice at home too. There was more cooperation with little competition in the group. We used to share our ideas while solving a mathematics problem at leisure time in school or at home or elsewhere. The collective effort benefited all the members in the group to master the contents of mathematics from the textbooks.
Episode two is about how I was able to learn mathematics in my Master’ Degree although I did not attend the class regularly. Final examination was nearly coming and I was not well prepared. I had a problem of how to master the contents. I could not get much idea when I tried to see some contents of real analysis and operations research.
I was successful to form a cooperative team of five friends from the same level and practiced mathematics everyday for two months. That practice in group enabled me to do well in the final examination. The group work became good example for others too. The learning of mathematics in group/collaborative work was really beneficial to me as I could learn every concept clearly and meaningfully.
In the same way, cooperative learning became a way of knowing mathematics for me. It was the way of solving mathematical problems. The cooperative group was formed around me from the beginning of my school days till the last day of my Masters degree. So it has become a major source of inspiration in the learning of mathematics, teaching mathematics and now research in mathematics education. It helped me to write numerals at the beginning of schooling, solve simple mathematical operations, discuss difficult problems in group, build close tie with colleagues, create a learning group, develop social and cultural understanding, boost my morale up, think positively about mathematics and be a student of mathematics for ever.
In my understanding every mathematical concepts and ideas can be learnt and well understood through cooperative learning. Arithmetic, geometry, algebra, sets, probability, trigonometry, ….in school mathematics can be well grasped by cooperative approach that can be formal or informal setup gourps. Similarly, in higher mathematics: analysis, algebra and higher geometry are the prime area in which cooperative learning becomes fruitful to understand the abstract nature of mathematics. In my experience statistics, probability, calculus, operation research, history and philosophy of mathematics could be the more relevant areas for group work and discussion. But in fact to me all mathematical concepts from school to university level can be well understood through cooperative learning provided that the group members are adaptive and cooperative to each other.
Now I would like to discuss some theoretical referents of cooperative learning.

Basic Elements of Cooperative Learning

Cooperative learning has some basic elements that form group learning as a cooperative learning. The basic elements of cooperative learning are: positive inter dependence, promotive interaction, individual and group accountability, interpersonal and small group skills, healthy social and psychological development, and group processing.
Positive Interdependence
For me and my group members and me the first and most important element in a structuring cooperative learning was positive interdependence. Positive interdependence was successfully structured when the group members perceived that they were linked with each other in a way that one cannot succeed unless everyone succeeds. Group goals and tasks, therefore, were designed and communicated to members in such ways that made them believe they would sink or swim together (Johnson & Johnson, 1989). In the words of Johnson and Johnson (1989), when positive interdependence was solidly structured, it highlighted that (a) each group member's efforts was required and indispensable for group success and (b) each group member had a unique contribution to make to the joint effort because of his or her resources and/or role and task responsibilities. I realized that in doing so we created a commitment to the success of group members as well as one's own and it was the heart of our cooperative learning. If there were no positive interdependence, there would be no cooperation.

Promotive Interaction
The second basic element of cooperative learning, in my experience, was promotive interaction, preferably face-to-face. Johnson and Johnson (1989) state that students need to do real work together in which they promote each other's success by sharing resources and helping, supporting, encouraging, and applauding each other's efforts to achieve. They further claim that there are important cognitive activities and interpersonal dynamics that can only occur when students promote each other's learning. I think that working together to achieve a common goal produces higher achievement and greater productivity than does working alone. This is confirmed by much research that it stands as one of the strongest principles of social and organizational psychology. Reputedly, Cooperative Learning, furthermore, resulted in more higher-level reasoning, more frequent generation of new ideas and solutions (i.e., process gain), and greater transfer of what is learned within one situation to another (i.e., group to individual transfer) than did competitive or individualistic learning.
In my understanding, promotive interaction includes orally explaining how to solve problems, teaching one's knowledge to others, checking for understanding, discussing concepts being learned, and connecting present with past learning. I think that each of these activities can be structured into group task directions and procedures in our cooperative learning circle (see Episodes One through Five). We did so in our group shared by helping ensure that the cooperative learning group was both an academic support system and a personal support. It was through promoting each other's learning face-to-face that my colleagues and I became personally committed to each other as well as to our mutual goals throughout Episodes one to two.

Individual and Group Accountability
The third basic element of our cooperative learning was individual and group accountability. I found two levels of accountability structured into our cooperative lessons. The group had to be accountable for achieving its goals and each member had to be accountable for contributing his or her share of the work. Our individual accountability existed but there was no mechanism to assess it formally. So the performance of each individual was assessed informally based on how much other members benefited from him. We tried to share our ideas openly in order to ascertain who needed more assistance, support, and encouragement in learning. The purpose of cooperative learning groups was to make each member a stronger individual in his or her own right (Johnson and Johnson, 1989). We learned together so that we subsequently could gain greater individual competency and succeed in the final examination.
Once the relationship is established, the next question becomes "why?" I agree that the social judgments individuals make about each other can increase or decrease the liking they feel towards each other. Such social judgments are the result of either a process of acceptance or a process of rejection (Johnson & Johnson, 1989). The process of acceptance is based on the individuals promoting mutual goal accomplishment as a result of their perceived positive interdependence. The promotive interaction for me tends to result in frequent, accurate, and open communication; accurate understanding of each other's perspective; inducibility; differentiated, dynamic, and realistic views of each other; high self-esteem; success and productivity; and expectations for positive and productive future interaction.

Interpersonal and Small Group Skills
The fourth basic element of our cooperative learning was the required interpersonal and small group skills. Cooperative learning is inherently more complex than competitive or individualistic learning because students have to engage simultaneously in task-work (learning academic subject matter) and teamwork (Johnson & Johnson, 1989).
I think Johnson and Johnson (1989) are right to say, “Social skills for effective cooperative work do not magically appear when cooperative lessons are employed. Instead, social skills must be taught to students just as purposefully and precisely as academic skills.” But in my days of schooling, teaching of mathematics never incorporated these social skills. Learning of mathematics in the classroom was more individualistic and nobody cared about others except in my cooperative learning circle. So, there was always a higher rate of failure in mathematics in comparison to other subjects.
In my opinion, leadership, decision-making, trust-building, communication, and conflict-management skills empower students to manage both teamwork and task-work successfully. Since cooperation and conflict are inherently related (Johnson & Johnson, 1995), the procedures and skills for managing conflicts constructively are especially important for the long-term success of learning groups (Johnson, 1991; 1993, Johnson & Johnson, 1994). In my understanding these elements were rare in my mathematics classroom although the class teacher or headmaster assigned some non-academic tasks in groups, such as preparation for national celebrations, sport and some co-curricular activities.

Healthy Social and Psychological Development
For me, when individuals work together to complete assignments, they interact (mastering social skills and competencies), they promote each other's success (gaining self-worth), and they form personal as well as professional relationships thereby creating the basis for healthy social development. I think that individuals' psychological adjustment and health tend to increase when schools are dominated by cooperative efforts. I think that the more individuals work cooperatively with others, the more they see themselves as worthwhile and as having value, the greater their productivity, the greater their acceptance and support of others, and the more autonomous and independent they tend to be. It seems to me that a positive self-identity is developed basically within supportive, caring, cooperative relationships while a negative self-identity is developed within competitive, rejecting, or uncaring relationships. So I think that children who are isolated usually develop the most self-rejecting identities.
It is generally accepted that cooperative experiences are not a luxury. They are a necessity for the healthy social and psychological development of individuals who can function independently. I think that healthy social and psychological development of individuals is the fifth important element of cooperative learning.
It might be that our school system was not aware of cooperative learning as a powerful method of learning in groups. The same was the case in university study. There was no cooperative environment in the day to day classroom discussion. Teachers used to come, give lectures without enough explanation and discussion and vanish from the classroom. It did not make much difference to me as I was a tourist student at the time but I did realize that the students were so paralyzed that they were not ready to share ideas, resources and knowledge. I think that my colleagues in university were so selfish due to the impact of the individualistic and competitive approach of learning. So, they did not accept my proposal to form a cooperative learning circle.
I could hardly find four/five people who agreed to join for cooperative learning. Anyway, I was successful in my mission of learning mathematics through cooperative learning though it was informal group learning, but we were fully committed to sharing ideas, opinions, resources and knowledge for our own benefit which was to understand mathematics and finally to pass the examinations.

Group Processing
The sixth basic element of our cooperative learning was group processing. Group processing exists when group members discuss how well they are achieving their goals and maintaining effective working relationships (Johnson & Johnson, 1989). “Groups need to describe what member actions are helpful and unhelpful and make decisions about what behaviors to continue or change. Continuous improvement of the processes of learning results from the careful analysis of how members are working together and determining how group effectiveness can be enhanced” (Johnson & Johnson, 1989).
I think, at first, caring and committed friendships come from a sense of mutual accomplishment, mutual pride in joint work, and the bonding that results from joint efforts. The more students care about each other, on the one hand, the harder they will work to achieve mutual learning goals. Second, joint efforts to achieve mutual goals can promote higher self-esteem, self-efficacy, personal control, and confidence in their competencies. So far as I have understood, the healthier psychologically individuals are, on the other hand, the better able they are to work with others to achieve mutual goals. Third, psychological health is built on the internalization of the caring and respect received from loved-ones. Friendships are developmental advantages that promote self-esteem, self-efficacy, and general psychological adjustment. The healthier people are psychologically, on the other hand, the more caring and committed to their relations. To me, each outcome can induce the others; they are likely to be found together. So, I think, they are a package with each outcome a door into all three. And together they induce positive interdependence and promotive interaction that was abundant in our informal cooperative learning circle all the moment from school to the university.

Epilogue

In this way the first and most important element in structuring our cooperative learning was positive interdependence. The second basic element of cooperative learning was promotive interaction, preferably face-to-face. The third basic element of in our cooperative learning was individual and group accountability. The fourth basic element of our cooperative learning was the required interpersonal and small group skills. I think healthy social and psychological development of individuals was the fifth important element of cooperative learning. The sixth basic element of our cooperative learning was group processing. We were weaker in these aspects at the beginning at school level but these characteristics were dominant in the university level. I did not find any such formal effort from schools and university campuses to encourage cooperative learning but I feel that I could practice cooperative learning of mathematics forming informal cooperative groups from school level to the university level.
Such cooperative learning in group not only helped me to understand mathematics, but it helped me to adjust myself in the group, develop a positive attitude towards mathematics learning, learning as concept building and social interaction in the group and, lastly, it helped me to develop my career in the field of mathematics education. So, cooperative learning was in true sense a dynamics of learning mathematics since I joined my primary school to till I completed my master’s degree in mathematics education. The phenomenon of cooperation made me confident all the time towards learning mathematics irrespective of classroom teaching and learning methods and approaches at that time.

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