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

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

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.

• 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

- 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.

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).

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

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.

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

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

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

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

(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

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.

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).
-

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.