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MINISTRY OF EDUCATION AND TRAINING
UNIVERSITY OF PEDAGOGY HO CHI MINH CITY

HOA ANH TUONG
Specialization: Theory and Methods of Teaching
and Learning Mathematics
Scientific Code: 62.14.01.11
SUMMARY OF DOCTORAL THESIS ON EDUCATIONAL SCIENCE
HO CHI MINH CITY– 2014
THE THESIS COMPLETED IN:
UNIVERSITY OF PEDAGOGY HO CHI MINH CITY
Supervisor: Assoc. Prof. Dr. Tran Vui
Reviewer 1: Prof. Dr. Dao Tam
Vinh University
Reviewer 2: Assoc. Prof. Dr. Nguyen Phu Loc
Can Tho University
Reviewer 3: Dr. Le Thai Bao Thien Trung
University of pedagogy Ho Chi Minh city

The Thesis Evaluation University Committee:
UNIVERSITY OF PEDAGOGY HO CHI MINH CITY
Thesis can be found at:
- General Science Library of Ho Chi Minh City
- Library of University of Pedagogy Ho Chi Minh City
THE PUBLISHED WORKS OF AUTHOR
RELATED TO CONTENT OF THESIS
1. Hoa Anh Tuong (2009), Lesson study-a view in researching mathematical
education, Journal of Science and Education, Hue University’s College of
Education, ISSN 1859-1612, No. 04/2009, pp 105-112.
2. Hoa Anh Tuong (2009), Research to make opportunity for students to
communicate mathematics, Journal of Education, Ministry of education and

training, ISSN 0866-7476, No. 222 (period 2-9/2009), pp 50-52.
3. Hoa Anh Tuong (2010), Mathematical creativity in teaching exercises, Journal
of Saigon University, ISSN 1859-3208, No. 04 (9/2010), pp 54-60.
4. Hoa Anh Tuong (2010), Using lesson study in the lesson “Area of a Polygon”,
Journal of Science, Ho Chi Minh city University of Education, ISSN 1859-
3100, No. 24 (12/2010), pp 133-140.
5. Hoa Anh Tuong (2011), Thalès theorem-A research to improve the quality of
teaching and learning, Journal of Science, Ho Chi Minh city University of
Education, ISSN 1859-3100, No. 27 (4/2011), pp 54-61.
6. Hoa Anh Tuong (2011), Approach a problem by solving different ways,
Teaching and learning today, Journal of Vietnam central learning promotion
association, ISSN 1859-2694, No. 9 (2011), pp 59-60.
7. Hoa Anh Tuong (2011), To look at the problem in different ways, Journal of
Science, Saigon University, ISSN 1859-3208, No. 07 (9/2011), pp 105-111.
8. Hoa Anh Tuong (2011), Using the external visual representations in
mathematics for teaching students grade 6, Journal of Science, Vinh
University, ISSN 1859-2228, Vol 40, No.1A (2011), pp 56-65.
9. Hoa Anh Tuong (2011), Using “Open–ended problem” stimulus student to
communicate mathematics, Journal of Science, Ho Chi Minh city University of
Education, ISSN 1859-3100, No. 31 (10/2011), pp 121-124.
10. Hoa Anh Tuong (2012), Approaching “Open–ended problem” helps students
study geometry actively, Journal of Science, Vinh University, ISSN 1859-
2228, Vol 41, No. 1A (2012), pp 85-91.
11. Hoa Anh Tuong (2013), Focus on innovative teaching method-a view of
practical researcher, Journal of Science, Saigon University, ISSN 1859-3208,
No. 14 (6/2013), pp 81-87.
12. Hoa Anh Tuong (2010), Theoretical basis of constructivism theory in
mathematics teaching, Proceedings of the scientific conference of master
students and PhD students in 2010, Ho Chi Minh city University of Education,
pp 92-102.

13. Hoa Anh Tuong (2010), Lesson study- Theoretical basis và applying in
mathematics teaching, Proceedings of the scientific conference of master
students and PhD students in 2010, Ho Chi Minh city University of Education,
pp 103-116.
14. Hoa Anh Tuong (2012), The Use Of Visual Representation In Reasoning And
Expanding Mathematics Problem: Lesson Study On The Area Polygon,
Proccedings of the 5th International Conference on Educational Research
(ICER) 2012, Challenging Education for Future Change, September 8-9, 2012,
Khon Kaen University, Thailand, pp. 417-424.
15. Hoa Anh Tuong (2013), Applying "open - ended task" to help secondary
students to communicate mathematics, Proccedings of the 6th International
Conference on Educational Research (ICER) 2013, ASEAN Education in the
21
st
century, February 23-24, 2013, Mahasarakham University, Cambodia, pp.
394-405.
16. Hoa Anh Tuong (2013), Solution to decrease distance between training
teachers of education mathematics and teaching mathematics of new teachers
in vietnamese secondary school, International Conference on Mathematical
Research, Education and Application, December 21
st
-23
rd
, 2013, UEL, VNU-
HCMC 2013, pp.105. (abstract)
17. Hoa Anh Tuong (2014), Apply model of lesson study in teaching mathematics,
Proceedings of the scientific conference on the teaching of natural sciences in
2014, An Giang province, pp. 127-134.
1
INTRODUCTION

1. Definition of terms
Mathematical communication is a way of sharing ideas and clarifying
understanding. Through communication, ideas become the objects of
reflection, refinement, discussion, and amendment. The communication
process also helps to build meaning, permanence ideas and makes them
public (Lim, 2008).
Mathematical communication competence: including the disclosure is
our own political opinions about the mathematics problems, understand
people's ideas when they present the matter, express their own ideas crisply
and clearly, use mathematical language, conventions and symbols (Pham
Gia Đuc và Pham Đuc Quang, 2002; Mónica Miyagui, 2007).
Open-ended problems are often thought of as tasks for which more than
a single correct solution is possible (Erkki, 1997). Foong (2002) describes
the open-ended problems as “ill-structured” because they comprise missing
datas or assumptions with no fixed procedures that guarantee a correct
solution.
Lesson study is a professional development form in which research on
teaching and learning in classroom is carried out systematically and
collaboratively by a group of teachers in order to improve their teaching
practices (James W.Stigler & nnk, 2009; Nguyen Thi Duyen, 2013). A
group of teachers collaboratively designs the lesson plan, implements and
observes the lesson in the classroom, discusses and reflects on the lesson
which is taught, revises the lesson plan, and teaches the new version of the
edited lesson plan.
A study lesson is a lesson that the lesson study group chooses to explore
in the lesson study process.
2. Introduction
Mathematical communication and lesson study have been much
interested in many countries:
• “Communication process helps students understand mathematics more

deeply” (NCTM, 2007).
• “Communication has been identified as one of the core competencies
for students to develop” (Luis Radford, 2004).
2
• Chang (2008) stated “The first goal of mathematical communication is
to understand the mathematical language”. Emori (2008) stated “All the
mathematical experiences are done through communication. Mathematical
communication is needed to develop mathematical thinking because
thinking development is explained by the manner's language and ways of
communication”.
• Lesson study helps teachers continuously innovate teaching and
improve learning for students. In lesson study, teachers play a central role
in deciding what is new in teaching and learning and directly implement
innovation in the real classroom. Through lesson study, teachers do
accumulate real experience, and improve lesson study.
In this study, we tried to design lesson plan discussed colleagues by
the process of lesson study in order to provide the opportunities for students
to show, debate, deduce, and present the proof. Since then, they need to
communicate and evoke mathematical ideas in the process of constructing
new knowledge.
We choose the research topic: "Using lesson study to develop
mathematical communication competence for secondary school students."
3. Purpose of the study
• How to organize classroom to promote and develop students’
mathematical communication competencies.
• To research and design a number of lesson contents in mathematics
grade 8 to promote students to communicate mathematics.
• To look at the scale levels of mathematical communication
competence are used in evaluating students through some of study lessons
been studied experimentally.

4. Research Questions
The first research question: How to use basic way of communicating
mathematics effectively (mathematical representation, interpretation,
argumentation, and presenting the proof) in mathematics classroom?
The second research question: How to organize mathematics classroom
that can promote and develop students’ mathematical communication
competencies?
The third research question: Which lesson contents in mathematics
grade 8 and how to design lesson plans create opportunities for students to
promote mathematical communication process?
3
The fourth research question: How to evaluate the development of
communicative competence of students through studied lessons?
5. Research tasks
• To find out the basic way of communicating mathematics is suitable
for secondary school students.
• To find out the conditions or situations in the classroom can occur the
basic way of communicating mathematics.
• To choose some of study lessons implemented by lesson study process
can create conditions for students to show the basic way of communicating
mathematics.
• To give the scale levels of mathematical communication competence.
6. The significance of the study
The thesis will be meaningful for education by:
• Surveying the basic way of communicating mathematics which is
expressed by Vietnamese students in the classroom.
• Proposing forms of teaching methods to develop mathematical
communication competence of students according to their mathematical
ability; thereby, forming confidence for Vietnamese students in sharing,
discussing with peers and teachers.

• Designing some lesson plans in mathematics grade 8 has many
opportunities to promote students to communicate.
• Proposing the scale levels of mathematical communication competence.
7. The layout of the thesis
The thesis included 6 chapters except for the introduction and conclusion
remark. Chapter 1. Mathematical communication in classrooms. Chapter
2. Lesson study and open-ended problems. Chapter 3. Methods. Chapter
4. Developing mathematical communication competence through lesson
study. Chapter 6. Conclusion and recommendation. Chapter 5. The
results of the research questions.
8. Summary of introduction
Chapter 1. Mathematical communication in classrooms
1.1. Origin of mathematical communication
“Mathematical communication is a kind of communication. Greek origin of
the word communication is related with community… Mathematical
communication is the communication in mathematics” (Isoda, 2008).
4
1.2. Communication in mathematics classrooms
Communication in mathematics classrooms is the interaction between
students-teacher-students, through verbal communication and using
everyday language.
1.3. Other studies in mathematical communication
We present some mathematical communication practices in some countries.
In thesis, we choose the meaning of communicate mathematical is the way
students express their mathematical perspectives (Brenner, 1994).
Mathematical communication has three distinct aspects: Communication
about mathematics, communication in mathematics, communication with
mathematics.
1.4. The role of mathematical communication in classrooms
Emori (2008), “Mathematical communication is a key idea which is

important not only for the improvement to learn mathematics but also for
the development of necessary skills in the development of sustainable
society knowledge”.
1.5. The scale levels of mathematical communication competence
1.5.1. The six proficiency levels in mathematics
In six proficiency levels in mathematics, from the third level, it has
proficiencies: Representation, interpretation, argumentation and reasoning.
1.5.2. The basic way of communicating mathematics
In this research, I choose the basic ways of communicating mathematics:
Representation, interpretation, argumentation, presenting the proof because
these basic ways are related to communicating mathematics.
1.5.3. Standard about communicating mathematics
1.5.3.1. Four forms of communication in Mathematics classroom
Oral communication; listerning; speaking communication; and writing.
1.5.3.2. Standard about mathematical communication
Organize and consolidate their mathematical thinking through
communication. Communicate their mathematical thinking coherently and
clearly to peers, teachers, and others. Analyze and evaluate the
mathematical thinking and strategies of others. Use the language of
mathematics to express mathematical ideas precisely (NCTM, 2007).
1.5.4. The scale levels of mathematical communication competence
1.5.4.1. The scale levels of mathematical communication competence
5
Level zero. No communication
Level 1. Expressing initial idea
- Students describe and present methods or algorithms to solve the given
problem (not to mention the method is right or wrong).
- Students know how to use the mathematical concepts, terminologies,
symbols and conventions formally.
Level 2. Explaining

- The students explain the method acceptably and present reasons why they
choose this method.
- Students use the mathematical concepts, terminologies, symbols and
conventions to support their ideas logically and efficiently.
Level 3. Argumentating
- Students argue the validity of a method or algorithm. Students can use
examples or counter-examples to test the validity of the method or
algorithm.
- Students can argue which appropriate mathematical concepts,
terminologies, symbols and conventions they should use.
Level 4. Proving
- Students use mathematical concepts, mathematical logic to prove the
given results.
- Students use mathematical language to present mathematical results.
1.5.4.2. Example of mathematical communication
We illustrated a lesson on October 3, 2010 in class 6A3 (51 students) of
Saigon Practical High school.
Friday, August 26th 2011 was Vi’s birthday.
a) After 7 days was of her mother’s birthday. What should be the day and
date of her mother's birthday? Why?
b) What should be the day after 52 days from the birthday of Vi? Why?
c) November 20th 2011 was Vi father’s birthday. What should be the day?
Why?
• Student showed the basic way of communicating mathematics as
follows:
Representation: Students could use calendar to find the date in a week
from 26/8 to 2/9; Monday schedule of every month from 17/10 to 21/11 to
6
find a solution. They knew 7 days respectively 1week, 30 days or 31 days
respectively 1 month.

Interpretation: Students tried to find a solution. Depending on their
abilities, they had different ideas. The easiest way was writtern a specific
schedule in a week. If you change the assumptions of the problem, such as
sentences b) changes 52 to 520 and change the sentence c) November 20,
2011 to December 27, 2014; this way will be not suitable. Student should
understand which solutions could still be used when the requirements have
changed. It means that students are aware of the rationality of expression.
Argumentation: Students know how to use the law of 7 day cycle, the
day will repeat. From there, students know how to find remainder in the
division by 7 to find the day. In addition, students remembered how many
days have in August and September, then to perform operations unless
suitable to find how many days have in the month. Students recognized the
problems may have link to each other.
Presenting the proof: Students themselves understood how to solve the
problem by listening to peers who demonstrated the problem.
• Evaluate the scale levels of mathematical communication competence:
Expressing initial idea: Students described a way to solve the problem
by writing the calendar in the week from 26/8 to 2/9, Monday schedule in
the month from 17/10 to 21/11. They applied the algorithm based on the
remainder in the division by 7. They used reasonable mathematical
operations: addition, subtraction, division.
Explaining: Students recognized the validity of each solution. Students
realized algorithm to find the remainder in the division by 7 more
reasonable than other solution.
Argumenting: Students expressed logical reasoning, solution of each
problem clearly.
Proving: Students used algorithms to find the remainder in division 7,
the language of mathematics, logical reasoning in the presenting the proof.
1.6. Summary of chapter 1
Chapter 2. LESSON STUDY AND OPEN-ENDED PROBLEMS

2.1. Lesson study
2.1.1. Origin of lesson study
7
“Lesson Study” (jugyou-kenkyu) in Japanese thus came to be known
around the world as a unique Japanese method of lesson improvement
designed to facilitate the development of high quality lessons.
2.1.2. Other researchs in Lesson Study
• “Japanese Lesson Study in Mathematics: its impact, diversity and
potential for educational improvement” (Isoda, 2005).
• Thailand implements lesson study: To investigate changes in teachers’
pedagogy and their professional development when they are using the open-
approach teaching method. To clarify how teachers recognize their learning
experiences in the classroom where open-approach teaching method has
been implemented.
• Fernandez also investigated how teachers took advantage of learning
opportunities that were created by lesson study.
• In Vietnam: Tran Vui (2006a, 2006b, 2007), wrote a number of articles
about the effectiveness of applying the model lesson study in practice
teaching mathematics in elementary school and secondary school. Nguyen
Duan and Vu Thi Son (2010) wrote a paper on approaching lesson study to
develop professional capacity of teachers. Nguyen Thi Duyen (2013) has a
number of articles on applied lesson study in the practice of teaching
mathematics in high schools.
2.1.3. Process of lesson study
There are many different variations of lesson study process, however a
lesson study process generally involves a group of teachers collaboratively
designing the lesson plan, implementing and observing the lesson in the
classroom, discussing and reflecting on the lesson which is taught, revising
the lesson plan, and teaching the new version of the edited lesson plan
(James W.Stigler & nnk, 2009).

2.1.4. The factor of implementing process of lesson study
To be successful implementation of lesson study process has many factors
such as teachers, students, schools, programs, textbooks.
2.1.5. Example of implementing process of lesson study
From the orientation of textbook to prove theorem sum of 4 angles of a
quadrilateral and the formula to find sum of angles of the polygon n vertices
(textbook mathematics grade 8, volume 1, page 65 and page 115). Through
discussions with teachers about students' difficulties in proving theorem and
8
solving the problem, we adjust and improve lessons plan to help students
find other ways to prove theorem “sum of 4 angles of a quadrilateral” and
set formula to find “sum of angles of the polygon with n edges” by n.
2.2. Open-ended problems
2.2.1. Origin of open-ended problems
The Open-Ended Problem Solving is based on the research conducted by
Shimada S., which is called “The Open-Ended Approach”. The Open-
Ended Approach provides students with “experience in finding new
something in the process” (Shimada 1997)
2.2.2. The role of open-ended problems
2.2.3. Example of study lesson using open-ended problems
Applying the “open-ended problems” in teaching is a question which is
typical teaching situations but the purpose of the question is open-ended
such as: there are many different solutions, there are many results, orientate
involved issues.
Example 2.1. To find the kind of quadrilateral.
Give a triangle ABC (AB < AC) with M, N and P respectively are
midpoints of segments AB, AC và BC; AH is altitude. Prove that
quadrilateral MNPH is an isosceles trapezoid?
• The requirements of the problem “To prove the quadrilateral MNPH is
an isosceles trapezoid”, is a closed-ended problem, we adjust the problem

as follows: Give a triangle ABC (AB < AC) with M, N and P respectively
are midpoints of segments AB, AC và BC; AH is the altitude. What is the
kind of quadrilateral MNPH? Why?
• The requirements of the problem “What is the kind of quadrilateral
MNPH? Why?” is an open-ended problem because students actively find
out many different results according to the ability to apply knowledge.
In particularly, students argue, explain: why quadrilateral MNPH is a
trapezoid or an isosceles trapezoid. Students should have figure reading
skills, then thinking and applying the hypothesis of the problem to find out
the ways to solve the problem. So the teacher evaluates student’s ability to
apply.
• In addition, teachers create opportunities for students to convert
problems to similar contents through open-ended problem, such as: Find the
pair of equal segments in quadrilateral MNPH? Find the pair of equal
9
angles in quadrilateral MNPH? Find the pair of equal segments and angles
in two triangles MNH và MNP? Explain? Then, students try to find as many
solutions as possible. This stimulates student to learn actively and apply
assumptions to solve given problem.
• In addition, teachers give students another open-ended problem “What
properties are there in quadrilateral MNPH?”. Students have skills in
reading figure and create a number of conclusions:
The edge: opposite sides are parallel, adjacent sides are equal. The
diagonal: diagonals are equal. Angles: 2 angles adjacent based side are
equal, 2 angles adjacent side are complement and 2 opposite angles are
complement. Symmetry: There is one axis of symmetry which is the line
passed two midpoints of the two based sides and center of symmetry
doesn’t have.
When the students listed the
characteristics above, they have mastered

how to prove the quadrilateral become a
trapezoid or an isosceles trapezoid and the
properties of an isosceles trapezoid, and
know whether the axis of symmetry and
center of symmetry of the isosceles
trapezoid exist? Students practice to prove
two equal triangles.
Figure 2.4 Quadrilateral MNPH
2.3. Summary chapter 2
Chapter 3. METHODS
3.1. Research Design
Research process was conducted according to the following steps:
- Survey the learning environment through the investigation process.
- Look the available research results in using open-ended problems,
mathematical representations.
- Research on the integration to the basic way of communicating
mathematics for students.
- Make the lesson plans through experiment to determine the strengths of
the lesson plan designed to develop mathematical communication
competence for students.
10
3.2. Research Subjects
The research object of the thesis include: How does the basic way of
communicating mathematics apply? How can classroom be organized to
create demands and opportunities to express mathematical communication
and skills design lesson plans in mathematics grade 8 that can make
students express mathematical communication? Or how can students’
mathematical communication competence be evaluated?
3.3. Research scopes
- Research the lesson plans using open-ended problems to develop

mathematical communication competence for students.
- Students participate in the experimental lessons: 166 students.
- References: mainly in the references listed in the references section.
- Survey contents: student’s thinking about learning mathematics, ways
to learn mathematics, what happens regularly in the mathematics
classrooms.
3.4. Methods of data collection
- Gather information from the research topic presented in textbooks as
well as the perfection of teachers teaching on that subject.
- Gather information from surveying students.
- Collect data from observing students and assess showing basic ways of
communicating mathematics in the experimental lessons.
3.5. Methods of data analysis
From the collected data mentioned in section 3.4, we:
- Analyze and propose adjusting through the lesson plans.
- Conduct statistical data to assess the students' perspection. Since then,
design study lessons.
- Evaluate the effectiveness of lesson plans and adjust to promote
students’ mathematical communication competence.
3.6. Research tool by the process of lesson study
3.7. The study of mathematics contents
3.7.1. Objectives and requirements of teaching mathematics in secondary
school
3.7.2. Research topic
• We chose the theme "The area of the polygon" to experiment which is
consistent to research topics:
11
- To use flexible representations: represented by language, visual
images and symbols.
- To train the ability to use language for students.

- To establish and develop the wisdom qualities.
- This topic can integrate real life situations in lesson plans and is
not heavy proving. Mathematical knowledge is not too difficult for
students to understand.
• Solving problems by using equations is a difficult form (Algebra 8) for
lower secondary school students. Through this theme, we want:
- To give students some analytical skills, write a solution, express,
choose unkowns to solve problem simply and briefly.
- To help students find the representing the correlations between
quantities by method of establishing tables which has many benefits.
- To guide the students detect problems. Students have the
opportunity to debate, explore, and give comments.
3.7.3. An overview of the study lesson
3.8. Summary chapter 3
Chapter 4. DEVELOPING MATHEMATICAL COMMUNICATION
COMPETENCE THROUGH LESSON STUDY
In teamwork, the leader prepares ideas and sends email to other members
before proceeding to discuss:
• Set the example question: To teach this lesson for all students to be
interested positively, how will teachers design a lesson plan?
• Provide ideas to design lesson plans to promote mathematical
communication competence for students. After that, team members
exchange ideas to reach agreement.
• After the experimental teaching, team leader exchanged data collection to
colleagues, colleagues comment the effect of lesson plan and adjust to
further promote mathematical communication competence for students.
• Focus on discussion: Whether the activities are suitable for students or
not? How do these need to adjust reasonably? What kind of questions or
teachers’ suggestions posed to students are suitable? How do teachers’
activities promote students to communicate effectively?

4.1. Study lesson 1. The area of trapezoid
h
a
A
C
D
B
12
a) Designing the lesson plan
Mr Hoa: Students are good at mathematics, they themselves can find a way
to prove by. It is difficult for other students to implement. We should have
a clearer suggestion. Mr. Tuan: For example, “We can divide the trapezoid
into two triangular areas that can be found”. Mr. Long: Students learned
about the formula for the area of a triangle, square, and rectangle in
elementary school. Mr. Tuong: If teacher suggests: Connect the diagonal
AC to divide the trapezoid ABCD into two triangles. Based on the formula
for calculating the area of a triangle, set the formula for calculating the area
of a trapezoid. This is forcing for students! Ms. Phan: If the teacher doesn’t
suggest, can the students themselves establish a formula for calculating?
Mr.Thong, Ms. Trinh: When students have experience in setting up
formulas for calculating the area of a trapezoid, they can set the formula for
calculating the area of a parallelogram. Mr. Si: Based on the property “If
the area of a shape is divided into H
1
, H
2
,…,H
n
without common points, the
area S of H will be calculated S = S

1
+ S
2
+ + S
n
. Teachers guide students
to divide a trapezoid into triangles, squares, or rectangles that students have
already known how to calculate the area (maybe they have many ways).
Activity 1: What are the areas of figures having in figure 4.1. Why? (unit is
one small square). Note: students are not allowed to use the formula to
compute the area of trapezoid and parallelogram.
Figure 4.2 Trapezoid ABCD
Figure 4.3 Parallelogram ABCD
Activity 2: Give trapezoid ABCD with two base sides are a, b and altitude h
as figure 4.2. Find different ways to set up the formula to calculate the area
the trapezoid ABCD?
Activity 3: Give parallelogram ABCD with base side a and altitude h as
figure 4.3. Find different ways to set up the formula to calculate the area the
parallelogram ABCD?
13
Activity 4: In figure 4.4, give trapezoid ABCD with mid-segment EF and
rectangle GHIK. Find different ways to compare the area of trapezoid
ABCD with rectangle GHIK.
Figure 4.4 Trapezoid and
rectangle have the same area
Figure 4.5 Land of three family
Activity 5: A piece of rectangular land ABCD belong to three families: An
(trapezoid ABHG), Ba (trapezoid HGFE), and Ca (trapezoid FECD) as
shown in figure 4.5. One day, three families discussed how the lines GH
and FE can be changed to divide the rectangle ABCD into 3 pieces of

rectangular land that have the same area as the area of 3 trapeziums at first.
Find way to help them.
b) Implementing and observing the lesson in the classroom
When students are working, teachers monitor, observe and record the
activities of students.
c) Discussing and reflecting on the lesson
Activity 1: Visual figures support students in the exploitation to find
different reasonable solutions.
Figure 4.6 Divided the figure into
triangles and rectangles
Figure 4.7 Rearrange the figure into the
polygon has to know the area
Activity 2:
Students
demonstrate the capability and know
how to use activity 1 in the general case.
Figure 4.8 Divide the trapezoid
into two triangles
Figure 4.9 Divide the trapezoid into two triangles
and a rectangle
E
K
F
G
C
I
D
H
B
A

14
Activity 3: Students find different ways and express their deduction in
every proof. Students use mathematical language, mathematical
conventions and mathematical symbols in describing the proof. In addition,
students know the parallelogram is a special trapezoid with two equal bases
and transform the parallelogram into rectangles that had the same area.
Activity 4: The figure is available, students demonstrate more rapidly.
Activity 5: Students applied activity 4 into activity 5. In addition, from
mistakes of other students, through interaction between teachers and
students, they know how to solve the problem.
Teacher: “If we draw HM is perpendicular to AD at M. How will we chance
HM to KL that is parallel to HM (figure 4.12) so rectangle BKLA and
trapezoid BHGA have equal areas?”. Student: “The area of BHGA minuses
the area of triangle IKH and adds the area of triangle IKH then equal the
area of BKLA”. Then, student deduces if the area of BKLA and the area of
BHGA have equal areas, KL pass
through midpoint I of GH
segment because triangles IKH
and IGL are equal and have
equal areas.
Figure 4.12 Land of three family An, Ba, Ca reorganization
d) Edited lesson plan
• Find different ways to calculate the area of the figures in figure 4.1.
• According to you, the following representations may help you have
more ways to find the area of the figures in figure 4.1? Why?
Figure 4.17 Orientation finding the area of the figure
• “The following representation can
help you transform the trapezoid into
a triangle had the same area? Why?”
F i

g
ur e
4.18 Trapezoidal and triangular
have the same area
15
4.2. Study lesson 2. Practice 1. Area of a polygon
4.3. Study lesson 3. Practice 2. Area of a polygon
4.4. Study lesson 4. Solving problems by using equations
The lesson plan
Argue about four available solutions in problem 1.
Problem 1: The distance between An’s house and school is 1200 m, The distance
between Binh’s house and school is 1650 m. Velocity of An is equal to Binh. Time
for Binh go to school is more than An 5 minutes. Calculate the velocity of An.
In your opinion, which solution is right or wrong? If solution is wrong
which step is wrong? Why?
In your opinion, which solution you should choose? Why?
To solve this problem well, what is your experience?
Solution 1:
Denote velocity of An is x.
Because velocity of An is equal to
Binh so velocity of Binh is x.
Time for An go to school is
1200
x

Time for Binh go to school is
1650
x
Time for Binh go to school is more
than An 5 minutes so we have

equation:
1650 1200
5
x x
− =
1650 1200 450
5 5 90x
x x
−
⇔ = ⇔ = ⇔ =
In conclusion, velocity of An is 90.
Solution 2:
Denote velocity of An is x (km/h)
(x>0).
Because velocity of An is equal to
Binh so velocity of Binh is x
(km/h).
Time for An go to school is
1200
x

(hour).
Time for Binh go to school is
1650
x

(hour).
Time for Binh go to school is more
than An 5 minutes so we have
equation:

1650 1200
5
x x
− =
1650 1200 450
5 5 90x
x x
−
⇔ = ⇔ = ⇔ =
In conclusion, velocity of An is 90
km/h.
S olution 3:
1200m= 1,2km; 1650m= 1,65km;
5 minutes =
1
12
hour.
Denote velocity of An is là x
Solution 4:
Denote velocity of An is x (m/min)
(x> 0).
Because velocity of An is equal to
Binh so velocity of Binh is x
16
(km/h) (x > 0).
Because velocity of An is equal to
Binh so velocity of Binh is x
(km/h).
Time for An go to school is
1,2

x

(hour).
Time for Binh go to school is
1,65
x

(hour).
Time for Binh go to school is more
than An 5 minutes so we have
equation:
1,65 1,2 1
12x x
− =
1,65 1,2 1 0,45 1
12 12x x
−
⇔ = ⇔ =
5,4x⇔ =
(condition satisfied).
In conclusion, velocity of An is 5,4
km/h.
(m/min).
Time for An go to school is
1200
x

(minute).
Time for Binh go to school is
1650

x

(minute).
Time for Binh go to school is more
than An 5 minutes so we have
equation:
1650 1200
5
x x
− =
1650 1200 450
5 5 90x
x x
−
⇔ = ⇔ = ⇔ =
(condition satisfied).
In conclusion, velocity of An is 90
m/min.
Problem 2: A train goes from A to B in 10h40'. If the train decreases the speed
10km/h, it will come 2h8’ later than to B. Calculate the distance AB and the speed
of the train.
Firstly, please select an unknown represented one quantity. Secondly,
tabulate to represent quantities and establish equations. Finally, students
write detail of the solution.
4.5. Summary chapter 4
Chapter 5. THE RESULTS OF THE RESEARCH QUESTIONS
5.1. The results of the first research question
a) Teachers make the mathematical representation suitably to help student
solve open-ended problems and facilitate opportunities for students to
communicate mathematically. Students used visual representations to

communicate with peers when they study by working group to form and
consolidate new mathematical knowledge.
b) When students are asked to explain the understanding of mathematics as
well as the results of their work with others, they can have self-regulation
and develop mathematical knowledge certainly. When students are asked to
17
explain their proof in the class, they argue which results need to be used to
solve problem.
c) Students convince others by giving exact or creative solutions.
5.2. The results of the second research question
5.2.1. Students’ communicational abilities in mathematical classroom
Students are not good at mathematics, they don’t have any idea. Students
are good at mathematics, they actively build and present their ideas in class.
5.2.2. Learning environment survey
- Students will learn more effectively when they discuss in groups or
need the teachers’ help in time. In addition, lesson content has visual
representations and to be suitable for students promote them study
positively.
- When students study mathematics, students look for connectional
problems, apply old knowledge to solve new problems, and find different
ways to solve.
- Learning integrated to solve the real-life situations would actually help
students loving mathematics.
- Students learn mathematics positively depending on making friendly
learning environment and the lesson content.
5.2.3. Organize class to promote mathematical communication
a) The situations containing the conflict between old and new knowledge
were really impacted on students’ cognition, urged them to perceive the
benefits of learning mathematics from mathematical communication.
b) Collaborative learning environment help students more confidently

present and post their opinions.
c) The open-ended problems encourage students communicate
mathematically because they have different ways to solve problem.
5.3. The results of the third research question
5.3.1. The role of lesson study
Using lesson study to organize classroom communicate mathematically is
expressed through the practicality of lesson study.
5.3.2. Design some lesson plans
The lesson plans put the strongest into the mathematical thinking and
multiple representaions. Teacher advantages the opportunity for students to
18
consolidate, inculcate contents, and base on old content to have new
knowledge.
5.3.3. A number of lesson contents in mathematics grade 8 have many
opportunities to promote students to communicate mathematics
Lesson plan integrates open-ended problems and mathematical
representation to real-life situations with the aim of:
- Students mobilize and apply the old knowledge to solve problem;
- Through specific case, students can predict the outcome in the general
case;
- Develop the student’s capabilities such as inference, reasoning,
explaining the nature of the problem.
5.4. The results of the fourth research question
5.4.1. Evaluate the basic way of communicating mathematics
Representation
- Students know how to use algebraic notation reasonably for unknown
represented one quantity to present the proof simply and briefly.
- Students use mathematical conventions to give the condition and the
unit for unknowns which are illustrated through solving problem by using
equation.

- Students apply mathematical concepts appropriately.
- Students understand the meaning of algebraic notation, and express in
speech by different ways.
- Students exploit visual figures that are reasonable and effective.
Interpretation
- Students know how to explain the reasoning, realize their mistake
when peers present the proof.
- Students find many different solutions and will realize the affect of
each solution. Students know how to explain the validity of each solution.
- Students know how to make argument step by step of each solution.
Argumentation
- Students argue which knowledge should be used to solve problems.
- Students argue the validity of each solution.
Presenting the proof
Students present the proof by writing on the paper clearly and accurately.
When students say, there are some errors.
19
5.4.2. Evaluate the scale levels of mathematical communication
competence
Study lesson. The area of trapezoid.
Level 1 and level 2: Students know how to demonstrate and give different
ways to solve the problem. Students have a long or shorter or brief proof.
Students use algebraic notation to support their mathematical ideas.
Students present their solution by speaking fluently, clearly, rather
accurately.
Level 3: Students communicate reflectively and explain why they choose
solution in each activity. Students express a logical inference when they do
activities. To solve new problems, students transform solved problems to
familiar problem.
Level 4: Students use mathematical language, mathematical conventions,

mathematical symbols and logical reasoning in presenting the proof.
Study lesson: Solving problems by using equations.
Problem 1:
- Through the reading of the solution is available, students understand
the content and express their opinion which solution is right or wrong and
analyze the error of wrong solutions (level 1 and 2).
- Student comments should choose the best solution to apply solving the
actual problem (level 2).
- Students themselves draw experiences when they solve this problem:
depending on asking of the problem, we selected an unknown represented
one quantity and units of this quantity appropriately (level 2 and 3).
Problem 2:
- Students themselves selected an unknown represented one quantity and
units of this quantity appropriately (level 1).
- Students can communicate with peer through setting up tables
represented the quantities (11 groups setting up right table and 1 group
setting up wrong table).
- Since the withdrawal of experience in problem 1, students carefully
change time unit, create right condition for unknown represented one
quantity. Students actively learn, develop thinking depending on their
cognitive capacity. Students transfer a realistic situation to set up table
represent and give simple or complex equations.
20
- From reading the problem carefully, students understand problems,
connect the quantities to unkown represented one quantity and set to tables
represent quantities (level 2).
- Through dialogue between teachers and students: Students confidently
speak and write detailed answers. At the same time, students have the
opportunity to regulate the written and expressed ability (level 1, 2, 3).
-

5.5. Summary chapter 5
Chapter 6. CONCLUSION AND RECOMMENDATION
6.1. Conclusion
6.1.1. The conclusion to the first research question
- Students used multiple representations flexibly.
- The basic way of communicating mathematics: interpretation,
argumentation increasingly promoted when students are asked to solve
open-ended problems, the intellectually conflicted situations stimulate
awareness of students.
6.1.2. The conclusion to the second research question
- Need to build mathematical learning environment cooperation in pairs
or small group, situations that integrate to open-ended problems, realistic
situations, visual representation.
- Teachers organize language activities for students.
21
- Teachers create an enviroment for students not only to have solution,
argue with teachers, but also discuss, debate with classmates to find out the
answer and solve the posed problem by their opinion.
6.1.3. The conclusion to the third research question
To promote students to communicate mathematically, study lesson should
be utilized:
- Attractive, interesting content, actual visual figure about the daily life
rather than pure mathematics. Students that read content can deduce
immediately, do not take much time for them to think. Contents are not too
complex, combine the use of different representation inferences: symbols,
tables, diagrams.
- Contents that are not only computing but also focusing on students’
thinking to find different ways to solve the problem and have unique,
creative solutions.
6.1.4. The conclusion to the fourth research question

Evaluate mathematical communication process of student through:
- Students can express how to solve problems and refer reasoning about
solution or basis that causes them to think how to solve it.
- Students select and use appropriately mathematical representations.
- Students express reasonable inferences in finding results. Students
explain the rationale for each solution.
- Students use mathematical concepts, conventions, mathematical
language in presenting the proof.
6.1.5. The conclusion to the study lesson
Study lessons of the research topic are different from lessons according
to the current teaching methods in Vietnam as follows:
In each lesson, lesson plans focus on students-centered:
• Promoting the students’ ability to look figure carefully as well as
understand the language, and use and link mathematical representations.
• Each action and each hint of teachers have a non-imposition and
suggestive nature, provoke the learning ability of students.
• Students communicate reflectively, teacher’s oral foster students to
express their thoughts and solutions.