Applying the Concepts of “Community” and “Social Interaction” from Vygotsky’s Sociocultural Theory of Cognitive Development in Math Teaching to Develop Learner’s Math Communication Competencies

As an integral skill in Math learning at high schools, Math communication competencies are formed and developed throughout the process of Math learning in the classroom environment through student-teacher, student-student as well as student-learning material and instrument interactions. Vygotsky’s Sociocultural Theory of Cognitive Development acts as the foundation and guideline for the teaching and learning process at school, emphasizing the role of social interaction in cognitive development. The study presents the two concepts of ‘community’ and ‘social interactions’ from the theory by Vygotsky for application in Math teaching. The article starts with clarifying the two concepts above in the context of Math classrooms, then proposes several sample activities of teaching Algebra in high schools with the application of the two concepts to develop learners’ Math communication competence.


INTRODUCTION
Lev Vygotsky is considered the father of sociocultural theories. His theory on human development highlights the influence of an individual's social and cultural environment on their personal development (de Abreu, 2000). It is believed that Vygotsky's psychology is an application of Marx's theories into learning, providing a framework whereby the sociocultural roots of thought become internalised in the individual (Lerman, 2001). Thus, he has been widely acknowledged as the founder of sociocultural theories, researching the relationships between learning and development, raising issues in general education as well as education for children in the 21 st century (Mahn, 1999). The theory is applicable throughout one's lifetime and across different cultures. Specifically, children's social world is guided by language which they use to interpret and experience their own world. Eventually, the language they hear (together with other cultural tools, sign systems and practice) become internalised and help them to control their behaviour. When children grow up, they use language to convey cultural values and language acts as a tool to educate cultural standards for children. Vygotsky believes that development and education are social (Tavassolie & Winsler, 2018). In other words, people's cognitive development can be considered a consequence of their communication with others (Alkhudiry, 2022).
There are a number of studies exploiting Lev Vygotsky's theory or applying his theory in Math teaching. Forman (2013) encouraged his students to actively participate in Math practice through group work and collaboration to identify, and arrange data in open Math questions which accept multiple possible solutions; and justify their ideas for their partners. Thus, it is important to improve students' competencies to communicate and collaborate with others (Forman, 2013). requirements for the conduct of Math communication activities are the Environment for communicating and Activities of communication (in different communicative forms). Mathematical communication environment refers to one of the Math classrooms which involves teachers, learners and the 'interactive atmosphere' that the class creates. All learners are supposed to be open and active in sharing their thoughts and solutions as well as questions, misunderstandings or concerns. At first, it would be challenging for learners when transforming from the passive learning method to the active communicative method, especially ones with incompetent Math knowledge, insufficient Math communication skills and a lack of willingness to converse with teachers and other students. In those cases, teachers need to direct the lesson step-by-step, create a positive and open interactive environment for lively discussions, engage learners in Math communicative situations and build up their activeness and initiative in raising their voices.
Diversity of Math communicative activities in the Math classroom: Reading and listening activities: Learners carry out reading and listening Math communicative activities when reading materials such as textbooks, workbooks, and reference books, dealing with any Math problems; or listening to teachers, and other students or any MKO talk about Math topics. While both traditional and communicative classrooms expose learners to these kinds of activities, the latter focuses on what learners perceive through reading and listening. Teachers collect information by requiring learners to reconstruct the Math knowledge they have perceived (in spoken, written or presented forms). Learners need to discuss, exchange and reflect by speaking or writing down the perceived information. According to Luong (2021), the diversity of Math communicative activities with the focus on interaction and creating positive and open exchanging environments is a direction for Math communication competency-based teaching. Speaking activities are conducted by presenting Math ideas and solutions; discussing Math contents, and debating for agreement or disagreement with other ideas and solutions. Writing activities involve presenting solutions for a Math problem or arguments for such an answer and ideas, etc.

Applying the concepts of 'community' and 'social interactions' in Math communication competence-based teaching: Teaching Samples.
In the following samples, the activities related to the concept of 'community' are written in bold while ones related to 'social interactions' or Math communication activities are in italics.
Sample 1: Introducing the concept 'proposition' and 'proposition with variables' With the aims of establishing and consolidating the concepts of 'proposition' and 'propositional variables', Activity 1 can exploit the teaching techniques focusing on helping learners to construct Math language knowledge and enhance their Math language proficiency. Activity 1 includes the following steps: Learners are divided into groups to complete the assigned tasks by the teacher individually and then discuss in groups; finally, they present their group results on A4 papers.
Task 1: Each group writes down random statements (either about Mathematics or not) Task 2: Write down next to each statement the letter 'Đ' if the statement is true, and 'S' if the statement is false. Task 3: Categorize the written statements into groups based on their similar features.
(In case there is no group with any proposition containing variables, teachers may do Task 1 like a student with the statement 'with every real number, The teacher monitors learners' interactive process in groups and supports if required by eliciting and giving suggestions and instructions. The teacher requires learners to be a reporter of their group work results and direct the activities as follows:

Teacher/Learners Math Speaking/Writing content
Teacher A learner is randomly selected to report his/her group work results (the results are stuck on the board)

Learner 3
Report results

VIETNAM JOURNAL OF EDUCATION
 213 

Learner 4
Why is this statement of your group true?

Learner 1 Explain
Teacher Can you categorise the statements written down by all the groups based on the criteria 'True/False'? Is there any statement belonging to either of the two groups?
Learner 5 Yes, there is a statement which is neither true nor false.

Teacher
Good job. Can someone emphasize again the results we have achieved?

Learner 6 Given the results from 3 groups, there are 3 kinds of statements: True statement, False Statement, Neither-True-Nor-False Statement
Teacher Student 7, do you agree with Student 6?

Learner 7 Respond
Teacher Introduce the concept of 'proposition' and elicit to introduce characteristics of propositions with variables Therefore, learners construct knowledge by talking about the concepts of 'proposition' and 'propositions with variables'.
In the sample activity above, 'interactive environment' -'community' is created by teachers in order that learners can construct the concepts of 'proposition' and 'propositions with variables' by themselves: Learners are organised in groups. In their group, learners actively interact and discuss to create a shared group result; meanwhile, teachers act as a guide, a facilitators to ensure the groups' smooth cooperation. In such a community-like interactive environment, learners are required to engage in Math communicative activities actively, with analysis, and critical thinking towards their friends' results and ideas within a group or as a whole class; and eventually to report their group work results with confidence.
Sample 2: Teaching reading skills when instructing students' self-study on the topic 'the concept of permutation' -Algebra 10 (currently in-use curriculum) Activity 1: Teacher: 'You will read a text mentioning the concept of permutation to identify the Mathematical information mentioned in the text.' (Teacher distributes the handouts with the text to students.) Learners: Read the text, focusing on the important Mathematical information in the text (learners underline with Mathematical terms in the text while reading.) Figure 1 represents a handout completed by a student after the Activity. The Mathematical terms and symbols in the text include: + List of the 5 players with orders + Permutations for 11 players taken 5 at a time + Set A, n elements, integer ,0 k k n  + Permutations for n elements from set A taken k at time + Permutations for set A taken k at a time The teacher presents the file showing collected Mathematical terms and symbols (or the Teacher may organise groups and have group representatives write their groups' identified terms and symbols on the board).
-Activity 2: Ask students to say any mathematical content they perceived while reading, which would allow connecting separated mathematical information from Activity 1 into Mathematical knowledge (preliminary forms of speaking, writing or presenting). The teacher asks students to give peer feedback and self-check their own results to identify the key content of the activity: 'the concept of permutations for n elements taken k at a time'.
-Activity 3: Through peer discussion, students self-check again the mathematical information and achieve the precise concept of permutations for n elements taken k at a time with the teacher's support.
This is an example of reading and comprehending the information in a text, selecting necessary information and re-construct the provided mathematical knowledge (by writing, speaking or presenting). Mathematical communicative activities are designed to support learners in constructing knowledge.

CONCLUSION
Mathematical communication in certain aspects is the manifestation of the concept of 'social interaction' in the context of a Math classroom. In the journey of self-constructing Math knowledge, learners can not depart from mathematical communicative activities. Thus, Mathematical communication is the means and also the results of the learning process. With the search for effective mathematical communicative activities, it is vital that learners enjoy a 'community' with active, open, and engaging members. Regular social interactions in an open community, it's highly likely for students to not only be Math knowledge perceivers but also proactive owner.