ENHANCING CLASSROOM COMMUNICATION TO DEVELOP STUDENTS'
MATHEMATICAL THINKING
Tran Vui, Hue University, Vietnam
Introduction
The reform curriculum in Vietnam tries to lessen the training of basic skills and proceduresin mathematics but increases more hands-on activities to help students grasp the mathematics
ideas and develop mathematical thinking. The mathematics teachers have learnt thegenerative teaching strategies that encourage mathematical communication to implement
more effective lessons focusing on mathematical thinking. Vietnamese mathematics teachers
believe that classroom oral communication is of outmost importance for students learning
mathematics. In particular, teachers’ questioning, listening, and responding approaches in theclassroom have been suggested to characterize their pedagogical practices, and to reflect theirbeliefs about mathematics and its teaching and learning. In this article we will discuss and try
to answer three questions related to the enhancing classroom mathematical communication todevelop students’ mathematical thinking. Firstly, how does our reform curriculum enhancemathematical communication for students? Secondly, what are our kinds of components ofcommunication for developing the mathematical thinking? And thirdly, what is ourappropriate teaching model to enhance classroom communication in mathematics?
1. Reform curriculum and students' mathematical communication
In Vietnam, because the emphasis of old curriculum was on procedural knowledge andmemorization of algorithms, students often worked independently completing exercises in the
textbooks and workbooks. Students practiced mathematics independently to learn themathematical concepts. When asking students questions, most teachers seek one “right”
answer to the math problem and will explain why the answer is correct.
School reform mathematics education aims to help students achieve four following
objectives: knowledge, skills, thinking and attitudes. Especially, with thinking objective, themathematics curriculum provides opportunities for students to develop:- The ability of observing, verifying, predicting, rational reasoning and logicalreasoning;- The ability of expressing precisely and clearly their own ideas and understanding theideas of others;- Spatial imagination;- The characteristics of thinking, especially the flexible, independent and creative
thinking.
- Thinking operations: comparison, analogy, generalization, and specialization.
The assessment of students' ability to communicate mathematics should provide evidence thatthey can express mathematical ideas by speaking, writing, demonstrating and depicting themvisually.

- Tran Vui 1 -1Figure 1. Levels of thinking and opportunities for communicating at each level
The reform curriculum supports active learning that creates opportunities for students'
mathematical communication.
Traditional Curriculum vs. Reform Curriculum
Traditional Curriculum Supports

Reform Curriculum Supports

Remembering formulas, facts.

Understanding
mathematics
deriving formulas.

concepts,

Focus on procedural knowledge of algorithms Communicate thoughts, explanations, focus
on problem solving process and investigation.
Training of basic skills and procedures in More hands-on activities to help students
mathematics.
grasp the mathematics ideas and develop
mathematical thinking
Students work independently completing Students think about, discuss, extend,
workbooks and home works. Students are not elaborate, verbalize, write, listen, read, andaccustomed to “talk” about mathematical inquire about mathematical concepts.
concepts.
Textbook-driven, teachers' clear explanations Communication is about participating,and classroom control.
interpreting, and negotiating meanings, and it
involves all classroom members alike.
Disseminate knowledge, facts, algorithms Students learn “naturally” by reading,and generally expect students to identify and writing, discussing, elaborating, thinking, andreplicatethefieldsofknowledge inquiring mathematics.
disseminatedStudents' independent practice.

Students should be able to work
cooperatively in groups and independently to- Tran Vui 2 -2make the necessary mental constructions
about a particular mathematical concept.
When asking students questions, most The teachers facilitates and organizes
teachers seek one “right” answer to the math students to explore new knowledge and solve
problem and will explain why the answer is non-routine problems.
correct
2. Main components of communication for developing the mathematical thinking
From the reform curriculum we can define five main components of mathematical processthat may promote students' communication in the classrooms at each level.

Figure 2. Components of mathematical process that may promote communication ateach level of thinking
Sometime we categorize the features of communication in the classrooms as follows. These
categories and their features will help teachers define what to do in the classroom to promotestudents' communication.
Categories

Features

Tools: WHAT

Language, mathematical language, mathematical representations:concepts, symbols, rules, formulas, real materials, virtual materials(computer-based), proofs, examples, counterexamples.

Ways: HOW

Graph, pictures, models, talking, writing solutions, ways of
explanation, reasoning, reflecting, discussing, amending, negotiating,ways of proving.

Needs: WHY

To inquire, to understand, to explore, to adjust, to solve, and to check.

Contexts: WHEN

Attitudes, get stuck, meet false propositions, failure, and need help.

- Tran Vui 3 -3For very young children, non-verbal communication, such as the use of materials todemonstrate their thinking is the earliest form of communication.
Non - Verbal Communication

Verbal Communication

Using Manipulative Materials

Using Language

Communication Skills in Mathematics
Everyday Language

Specific Language

Mathematical Language

Mathematical Representation
Language

3. Teaching model to enhance classroom communication in mathematics
When students construct their own knowledge of mathematical concepts, they need to havethe opportunity to think about, discuss, extend, elaborate, verbalize, write, listen, and read in
the mathematics classroom. Normally in a mathematics class, students are not accustomed totalk about mathematical concepts. They are usually taught the concept by the teacher.
Therefore, teachers need to use a number of approaches to probe students’ thinking inmathematics. Students are not natural talkers in the mathematics classrooms. Older students
will encounter more complex concepts in higher levels of math, discussing, talking,elaborating, writing, reading, and thinking about complex themes and concepts will help
students to obtain deeper understanding in mathematics.
3.1. Types of lessons that support the curriculum
From the curriculum and textbooks we can categorize lessons into two main types as follows.
New lessonsa. Help students pose, explore and solve problems
Teacher facilitates students to explore and pose questions, problems of the new lesson
when they engage in a problematic situation. Teacher then facilitates students to mobilize
what they have experienced and learnt to recognize and seek for the relationships between
the posed problem and their known knowledge to find the appropriate strategy to solve
problem.
b. Give students opportunities to consolidate and apply knowledge constructed after
learning new lesson such that students start to achieve the new knowledge- Tran Vui 4 -4In the mathematics textbook at primary level, after new lesson usually there are three
exercises for students to consolidate and practice what they have learnt in the lesson. The
first two exercises require students know how to apply and practice directly the newknowledge. The third usually is a problem requires an indirect application of new
knowledge.
Consolidation, practice and general practice lessonsa. Help students recognize learnt and new knowledge in various problems
b. Help students self practice on their pace of ability
c. Give students opportunities to help each other in small group with effective
interactions by using worksheets.
d. Practice students having habits of checking, evaluating their works.
e. Practice students having habits of finding various strategies and choosing mostappropriate strategy to solve problem.
3.2. Four main activities in a lesson
Four main activities in a lesson that teachers should follow to develop students’ mathematicalthinking (MOET 2006):
Activity 1. Examine and Consolidate the previous knowledge involved with new lesson;
Activity 2. Teacher facilitates students explore mathematical knowledge and construct new
knowledge by themselves.
Activity 3. Students practice the new knowledge by solving exercises and problems in the
textbooks and exercise books.
Activity 4. Teacher concludes what students have learnt from new lesson and assigns thehomework.
Each teaching and learning activity in classrooms gives opportunities for students'
communication.

Figure 3. Teaching activities and opportunities for communicating at each activity- Tran Vui 5 -5Engaging to the lesson, the students will have opportunities to show their mathematicalthinking through:•The ability of observing, predicting, rational reasoning and logical reasoning;•

Knowing how to express procedures, properties by language at specific levels ofgeneralization by words, word formulas;•

Knowing how to investigate facts, situations, relationships in the process of learningand practicing mathematics;•

Developing ability on analyzing, synthesis, generalization, specifying; and starting tothink critically and creatively.
The level of difficulty and complexity of a problem is defined by the achievement objectives
in the standard curriculum for each strand of mathematics. The exercises in the practicelessons are ranked: from easy to difficult, from simple to complicated, and from direct
practice to flexible and combined applications. The communication increasing depends on thelevel of difficulty and complexity of a problem.
3.3. Teaching strategies- Promote active, initiative and self-conscious learning of the learners;- Form and develop the ability of self-study;- Cultivate the characteristics of flexible, independent, and creative thinking;- Develop and practice the logical thinking;- Apply problem solving approaches;- Apply mathematics to real life situations.
3.4. Teaching flows

3.5. The Teaching Modes- Tran Vui 6 -6Interpretive teaching is focused on the establishment of a common language inside theclassroom, valuing the social aspect of learning. Teachers teaching in an interpretive modeask fewer testing questions and more genuine and provoking questions than if they wereteaching in an evaluative mode. The interpretive listening mode is characteristic of aninterpretive teaching mode; therefore, there are increasing opportunities for classroom
interaction and discussion. However, responding to students still tends to be evaluative, andtypical instructional approaches are mainly textbook-driven, though enriched with problems
and activities. Within a classroom environment based on interpretive teaching, students’contributions to the classroom discourse still do not have a significant impact on lesson
unfolding, since considerable concern for following lesson plans and avoiding ambiguity arestill evident.
Regarding the generative teaching mode, communication is about participating, interpreting,and negotiating meanings, and it involves all classroom members alike. Genuine andprovoking questions dominate the typical discourse of these teachers, although there is room
for testing questions as well. It is the hermeneutic listening mode that characterizes generative
teaching, and teachers respond to students by stimulating further discussion (probing, giving
feedback, redirecting questions or comments). Instruction is inquiry-driven and not clearlystructured. Generative teaching necessarily implies teachers’ questioning of their own
practices and beliefs, and revising of their own mathematical knowledge while exploring andconstructing mathematical ideas with their students.
Most likely, an evaluative teaching mode is associated with an instrumentalist perspective of
mathematics teaching and learning, an interpretive teaching mode is related to teachers’
Platonist views of mathematics, and a generative teaching mode is connected to a problemsolving perception of what it means to teach and learn mathematics. It is important to notice,though, that exact correspondences between teachers’ dominant teaching modes and their key
beliefs were not anticipated beforehand.
3.6. Investigation-centered learning model
The new paradigm of teaching is to help students construct their knowledge in an active way
while working cooperatively with classmates so that their talents and competencies aredeveloped. The assumption is that teaching requires training and skill implicitly and explicitly.
We try to provide a teaching model based on student-centered learning style that should be
relevant to the socio-culture of our region. Looking ahead into the future, the learning model
should be balanced between behaviorism and constructivism. It means that by then routine
learning shall have been lessened and the process of doing and understanding mathematics
shall have been increased.
The investigation - centered learning model has been developed to create a rich educational
activity for classroom use.
A rich educational activity should (Wheatley, 1991):
be accessible to everyone at the start;
invite students to make decisions;encourage "what if" questions;- Tran Vui 7 -7encourage students to use their own methods;
promote discussion and communication;
be replete with patterns,
lead somewhere;
have an element of surprise,
be enjoyable,
be extendable.
Stages in doing investigation
There are five steps in doing mathematics investigation that can be implemented in theclassroom.
Stage 1: Introduce the problem. The teacher is to capture the students’ interest making use ofgood motivation or having puzzles.
Stage 2: Clarify the problem. Use questions to draw out the underlying mathematicalquestions within the problem.
Stage 3: Design the investigation. The teacher led the students to choose the most satisfactory
problem solving strategies.
Stage 4: Carry out the investigation. Students make and test hypotheses; make generalizations
and the teacher must ask questions to guide the students to reach a satisfactory
solution to the problem.
Stage 5: Summarize the learning. Students need time to present their findings and explain any
theories they may have about the mathematical task given. Class questions may tie
these discoveries together and reveal the processes used during investigation.
3.7. Group investigation model
Inquiries- Confrontation with a problem;- Knowledge results from group process;- Learner - not teacher - driven.
Knowledge- Different people see problems differently- Negotiating the differences creates social and personal understanding.
The four Critical Components of Group Investigation

Figure 4. The interrelation of four critical components in group investigation- Tran Vui 8 -8In investigating classrooms these four components are interrelated and occur simultaneously.
1. Investigation refers to the organization and procedures for directing the conduct ofclassroom learning as a process of inquiry.
2. Interaction identifies the interpersonal, or social dimension of the learning process as it
unfolds in the communication among members of small groups in the classroom.
3. Interpretation occurs both at the interpersonal as well as at the individual cognitive level.
4. Intrinsic motivation refers to the nature of students’ emotional involvement in the topic
they are studying and in the pursuit of the knowledge they seek to acquire.
Stages of Implementation of Group Investigation
Stage 1: A class determines the subtopics and organizes them into research groups. Students
scan sources, propose questions, and sort them into categories. The categories become
subtopics. Students join the group studying the subtopic of their choice.
Stage 2: Groups plan their investigation. Group members plan their investigationcooperatively; they decide what they will investigate, how they will go about it andhow they will divide the work among themselves.
Stage 3: Groups carry out their investigations. Group members gather, organize, and analyze
information from several sources. They pool their findings and form conclusions.
Group members discuss their work in progress in order to exchange ideas andinformation, and to expand, clarify, and integrate them.
Stage 4: Groups plan their presentations. Group members determine the main idea of theirinvestigation. They plan how to present their findings. Group representatives meet as
steering committee to coordinate plans for final presentation to class.
Stage 5: Groups make their presentations. Presentations are made to the class in a variety offorms. The audience evaluates the clarity and appeal of each presentation.
Stage 6: Teacher and students evaluate their projects. Students share feedback received about
their investigations and about their affective experiences. Teacher and students
collaborate to evaluate individual, group, and class wide learning. Evaluation includes
assessment of higher level thinking processes.
CONCLUSION
Teaching elementary school mathematics aims to equip students with basic mathematics anddevelop their mathematical thinking to solve problems. Communication is an essential part ofthe students' learning in mathematical classrooms, it helps students to achieve basic
knowledge and develop mathematical thinking. With the reform curriculum, communications
are required in the mathematics classrooms. Students may use verbal language tocommunicate their thoughts, extend thinking, and understand mathematical concepts. They
may also use written language to explain, reason, and process their thinking of mathematicalconcepts. Communication is a tool which can help students to form questions or ideas about
concepts. Most of teachers in Vietnam really need a practical framework to develop students’mathematical communication in their actual classrooms. Of course, each economy has
slightly different approach in teaching mathematics to promote the mathematicalcommunication, but we still have a lot of thing in common to share and build up together arealistic framework that helps classroom teachers in each APEC economy.

- Tran Vui 9 -9REFERENCE
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- Tran Vui 10 -10