Vygotsky's Social Constructivism in Mathematics Teaching

Introduction and Personal Learning Philosophy

Growing up, I constantly remember hearing the phrase "no child left behind." As a student, it kept me motivated and gave me a sense of security. No matter what my abilities were, my teachers were there for support. As a teacher, this statement is just as important to me. Student involvement—whereby students feel important in the learning process and "as if it could not continue without their presence"—is imperative for meaningful, constructive learning to take place.

My theory of learning consists of not only what I view as effective from a teacher's perspective but also what I remember being effective for me as a student. It is true that not all students learn in the same manner or at the same rate, so it is important to vary approaches to account for this diversity. One of my main goals is to offer my students assistance and guidance in their own learning process. I want them to actively contribute and participate in the classroom; therefore, I often use games, role play, hands-on activities incorporated with group work and scaffolding. This gives students a chance to understand and conceptualize topics in a more intellectual manner through peer interactions and the sharing of ideas.

I find these methods are extremely important because I believe there is a need for improvement in mathematics learning in classrooms, and this is fundamentally related to development in teaching. Our role as teachers is not to transfer mathematical knowledge to students but to assist, guide, and facilitate in the reconstruction of particular ways of thinking. There must be a higher function of thought, which is reached by the student with the aid of the teacher. This can be done through social interaction between teacher and student, and student to student, which is fundamental for higher cognitive development.

Social Constructivism and Vygotsky's Theory

Reflecting upon my own learning theory and practice, I realize that it favors Vygotsky's social constructivist theory, which asserts that social interaction precedes development. I integrate Vygotsky's theory in my own practice through "scaffolding, small groups, cooperative learning [and] group problem solving" (Blake and Pope, 2008: 63). In this paper I will discuss Vygotsky's social constructivist theory and how it is relevant to my own learning theory. I will then discuss my practice of Vygotsky's zone of proximal development (ZPD), scaffolding, play, and collaborative work in my Year 6 mathematics classroom.

Russian psychologist Lev Semenovich Vygotsky (1896–1934) founded the sociocultural theory of development. Vygotsky's theory emphasizes the complex but critical interactions between social, cultural, and personal elements that weave together to create a learning environment that is as unique as the individual and that guides a child's cognitive development. He sees "cognitive growth as a collaborative process" (Papalia et al., 2008, p. 34). It is not a process that can be completed by one single party but rather a team or through collaboration in order to reach a higher level of thinking.

Jean Piaget was also a psychologist whose work evolved around the same time as Vygotsky. Piaget's scientific background assisted him in discovering a way to link how children acquire knowledge to their development in age. However, Piaget's view was that a child's development precedes their learning, while Vygotsky argued, "learning is a necessary and universal aspect of the process of developing culturally organized, specifically human psychological function" (1978, p. 90). In other words, social learning tends to pave the way for development. Therefore, in my classroom I find Vygotsky's theory more relevant.

Vygotsky's theory stresses the crucial role of social interaction in the development of reasoning, and that is an essential part of every lesson I conduct. During my early years of teaching, I was following a traditional style in which the classroom is teacher-centered. Today, my practices and theory of teaching have been altered completely for the advantage of my students. I found that when I incorporated more teacher-student interaction, my students were able to grow and become independent learners and thinkers.

For example, I now begin class by asking students questions, which starts a discussion and creates a safe environment for students to share their opinions and ideas. With this increased level of interactivity, students are constructing knowledge in a meaningful way. Also, I found that when I include more social interaction, my students developed more as learners but more importantly as thinkers. The next step to social interaction is knowing the students' abilities and what can be achieved. Our goals as teachers must be realistic and feasible in an allotted timeframe. Vygotsky's zone of proximal development acts as a guide for what can be taught and what a student can learn, which I will discuss further in the next section.

Zone of Proximal Development in Mathematics

Vygotsky's term zone of proximal development refers to the difference between what children can do on their own and what they could do with the assistance of others. According to Vygotsky, the zone of proximal development is "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers" (1978, p. 86).

Within the zone of proximal development, there are two levels. First, we have the actual development level, which is the set of tasks a student can perform independently. The second level is the level of potential development, which is the set of tasks a student can perform with the assistance of a more competent individual—this could be a peer or teacher. Vygotsky viewed the zone of proximal development as the area where the most sensitive instruction or guidance should occur, since it should in turn develop their higher mental functions. The teacher's role here is to provide a framework to guide the learner through the zone, ultimately reaching the level in which they are independent from assistance. The teacher is successful if the student completes the journey to the objective and is able to operate independently. The student is successful if the objective is attained.

Students learn at different paces based on, for example, varying levels of intelligence and motivation, which can be linked with each student's zone of proximal development. There might be two students at the same actual developmental level, meaning they have the same knowledge and level of understanding of a specific topic, but they both reached this level at different rates, even with assistance. Therefore, we should not only concern ourselves with what our students know but also with their ability to learn with assistance. Consequently, it can be said that the zone of proximal development is the area where actual teaching and learning takes place, where new information and knowledge is shared and assimilated. This knowledge is acquired through the use of "social interaction with more knowledgeable others to move development forward" (Blake and Pope, 2008: 62). The more knowledgeable can not only be the teacher but peers within the classroom as well.

The role of the teacher in the ZPD is to facilitate learning and provide assistance where it is needed, while decreasing the amount of assistance provided gradually as the transfer of knowledge increases. Vygotsky believed that an effective way to help struggling students would be to "direct their attention to the key features of the task and prompt them in ways that will facilitate their understanding" (Kyriacou, 2009: 30)—in other words, through scaffolding.

In my practice, I have found that the key to incorporating the zone of proximal development is knowing the student's actual developmental level. I teach in an American curriculum school where American Common Core Standards are followed. My students are English second language learners and most are below a Year 6 level. Knowing the level of my students is extremely important as a starting point for lesson planning so that new learning can take place. I have to be aware of my target as a teacher but take into consideration whether it is feasible based on my students' current level. Since I teach mathematics, I must take into consideration both their language and mathematical skills.

This does not mean I can only attempt to teach knowledge that is familiar to the students. As the zone of proximal development suggests, they might not be able to perform a given task independently as yet, but they may be able to at some later point. One of my standards in my Year 6 mathematics class was to "apply and extend previous understandings of multiplication and division to divide fractions by fractions" (2010, p. 42). For about 60 percent of my class, this objective would be out of their zone of proximal development.

In the beginning of the school year, I attempted to teach students long division, and although some students were grasping the division aspect, they were unable to master the skill. This is attributed to their lack of basic multiplication skills and also because they were being taught a concept that was out of their zone of proximal development. Then I began teaching them through thinking, talking, and doing. "Students gain meaning through discussion and debate—speaking and hearing mathematics—with each other and with teachers" (Irvin, 2008, p. 287).

I started by revising the basics of subtraction and addition. I would give them opportunities to think as I retaught the concept. Then I would allow them to discuss with each other through group work. The students who were on or above level would guide the lower-level students. Since the task at hand was below some students' zone of proximal development, I had to find a way to keep them interested. I did this by pairing them with students who were struggling and having stronger students assist struggling students. Eventually they were all solving worksheets and engaging in deeper-level thinking.

Although addition and subtraction seem like very basic concepts, if there is no deeper level of thinking, even with a topic so basic, you cannot move on to a more complex topic and expect students to grasp the concept. I would focus on mathematical vocabulary and encourage students to use these words whenever referring to a mathematical concept. This was a skill they were not engaging in before, as they did not fully understand the meaning of the words on a higher level of thinking. By the end of the class, they were all using mathematical vocabulary, whether they were explaining to a peer or asking questions.

Scaffolding as Instructional Support

Vygotsky's theory links to this particular lesson because the students were now constructing knowledge in a meaningful way due to collaboration, social interaction, and discussions. All students became more motivated and engaged because of this increased interactivity.

Scaffolding is directly related to zone of proximal development in that it is a support mechanism that helps a student successfully perform a task within their zone of proximal development. Generally, this process is completed by a more capable individual supporting the learning of a less capable individual. Scaffolding can be defined as: "a form of adult assistance that enables a child or novice to solve a problem, carry out a task or achieve a goal which would be beyond his unassisted efforts" (Wood, Bruner, and Ross cited in Capello and Moss, 2010: 182).

Although Vygotsky did not define the term scaffolding himself, it is a "concept that is associated with [him]" (Swain, Kinnear, and Steiman, 2010: 26). A lot of what is understood by the term can be directly related to his ideas of the zone of proximal development. There are a wide variety of techniques to implement scaffolding within the classroom.

"Scaffolding occurs by breaking down the skill into small units and guiding performance to a higher level" (Bergin and Bergin, 2011: 113)—in other words, it is the breaking of new work into smaller, more manageable parts. My lesson on division first began with a bag of M&Ms. Whenever I begin a new mathematical concept, I try to always relate it to everyday life and use words they can associate the concept with. As soon as I entered the class with the M&Ms, all students were intrigued and engaged immediately.

I asked them if they would like me to share. I then demonstrated and modeled how I would share the M&Ms equally and unequally. Modeling is crucial as it is the "ability to perform an action when imitating a model is an indication of a budding developmental stage" (Gordon, 2007: 177). At this point, they are still at the stage where they need guidance and assistance from a more competent individual in order to complete the next task. Although some students would already be familiar with the division concept, in this process I am linking what they know with an idea they are yet to learn.

I then gave each group a bag of M&Ms paired with a worksheet and a vocabulary list. They would have to work together using the words on the vocabulary list to complete the activity. The use of group work is important here, as the less competent student can be assisted by the more competent student. I also provided assistance to each group when needed, using some other Vygotskian methods such as providing a first step in a solution, asking leading questions, explaining further or supplying more information, asking questions, correcting, and making the students explain their own work.

Once each group completed the activity, a member would come and present their findings to the class. This gives everyone a chance to see what each group found and learn from each other. I always give them a timeframe in which the work must be completed in order to keep them on task. Throughout the activity, I give them multiple opportunities to discuss their findings and I continuously ask questions. This encourages them to think deeply and challenges their thought process. By the end of class, they have collaborated as groups and as a class, and no one is left unengaged. Even if they are not all on the same level, they have a chance to express themselves and what they understood from the lesson using social interaction.

Conclusion

I do not believe that I can choose one teaching theory and call it my own. There is no single theory that is completely flawless and can be established in all classrooms. Many factors must be taken into consideration. Although this is my belief, in my essay I discuss how Vygotsky's social constructivist theory applies in my classroom. In my particular situation, I feel that it is the most relevant and appropriate. Vygotsky established this theory many years ago; it has proved to be adaptable enough to withstand the constantly changing world.