MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF EDUCATION
DEVELOPMENT OF GEOMETRIC THINKING FOR UPPER KINDERGARTEN AND PRIMARY SCHOOL
CHILDREN THROUGH GEOMETRIC ACTIVITIES.
Major: Theory and methods of teaching Mathematics
Code: 62. 14. 01. 11
SUMMARY OF PEDAGOGICS DOCTORAL THESIS
HA NOI – 2013
Science instructor:
1.ASS.Prof, Dr. Nguyen Anh Tuan
2.ASS.Prof, Dr. Vu Quoc Chung
Judge 1: ASS.Prof, Dr. Tran Kieu
Judge 2: ASS.Prof, Dr. Vu Duong Thuy
Judge 3: Dr. Nguyen Van Thuan
Thesis will be defended in front of Council at University
Hanoi University of Education, date:…………………….
Thesis can be seen at National Library and Library of Hanoi National University of Education.
LIST OF THE AUTHOR’S RESẺACH PROJECT RELATED TO THESIS
A.Articles have been published
1. Nguyen Manh Tuan (2009), Van hiele’s viewpoint on forming geometric symbols in kindergatern children and pupils of early
primary school, Journal of Education No. 224, pp.35-38.
2. Nguyen Manh Tuan (2010), Spatial imagination and development of spatial imagination for students in early years of primary
level with educational software. Journal of Education No. 248, pp.5-7.
3. Nguyen Manh Tuan (2011), Developing Spatial Imagination in Children Aged 5-6 years by Formatting Shape
Representations. Proceedings 13, Global COE Program "Science of Human Development for Restructuring the Gap - Widening
Society", Ochanomizu University, pp.153-156.
4. Nguyen Manh Tuan (2011), Training geometric thinking for children aged 5-6 and early primary (grade 1, 2, 3) through
teaching the relative position of things, Journal of Science National University of Education, No. 6, pp.96-103.
5. Nguyen Manh Tuan (2011), Studying the impact of spatial sense and spatial imagination in upper kindergatern children,
Journal of Education No. 266, pp.9-11.
6. Nguyen Anh Tuan - Nguyen Manh Tuan (2011), Drilling geometry thinking for kindergatern and primary school children

in the future in higher levels. In which, the method of using the geometric activities plays an important role in teaching geometric
elements and geometric quantities and develops thinking for learners. Through geometric activities, the children can learn in the
activities and by the activities, can self-doing, self-deciding, can experience practical life leading to contribute to promote
positiveness, independence, creativeness, increase interest in the Math class that keep children innocent and happy in early school
age .
However, organizing geometric activities at school has some limitations. Although the content of kindergarten and primary
education program now focus on more geometric activities, but teachers do not really evaluate the level of thinking and
awareness of children in geometric activities. The geometric activities also occur scatteredly and disorderly during the process of
geometric elements and geometric quantities teaching, the teachers have not really applied effectively in the Review class or
extracurricular activities, especially the 2 sessions - class / one day in elementary school, the teachers usually recommend or
deduct from reference books, primary school children mainly take time to do homework in class, the teachers focus on teaching
knowledge more than training and developing thinking for children, organize less activities for children. The reason is that
teachers do not pay proper attention to the role of geometric activities for the development of thinking, the establishment and
organization of geometric activities which required time-consuming, the teachers are difficult and confused in evaluating
children, do not control children.
Before the children enter elementary school, children have certain awareness and understanding through activities which
help chidren be familiar with Math in kindergarten, five-year-old universal project of the Ministry of Education and Training has
been implemented with the goal of completing the whole country in 2015 and also raised new questions about the "connection"
the content of program of making familiar with Math for children at upper kindergarten and curriculum in mathematics for
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elementary children school. So researching on teaching Math in general, geometric elements and geometric quantities teaching in
particular from children at kindergarten (5 years) to children at primary school is necessary, the teacher must not teach before the
program , "primarized education" children at kindergarten, at the same time avoid children’s psychology to be"boring" when
entering the first grade.
For these reasons, the chosen topic is: " Development of geometric thinking for upper kindergatern and primary school children through
geometric activities”
2. The research purposes
Development of geometric thinking for upper kindergarten children and at primary school children through the selection,
construction, orgnization some geometric activities.
3. The objects of research

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- Methods of statistic: Using statistical methods to handle statistics to measure the level of geometric thinking
capacity in children to confirm the validity of scientific theories and the feasibility of the method which is raised in the
topic.
8. Issues defended in thesis
- The concept of geometric thinking in upper kindergarten children and at primary school children, the performance of
geometric thinking capacity in children through some geometric activities.
- Assess the geometric thinking capacity in upper kindergarten children and at primary school children in some geometric
activities.
- Develop geometric thinking through the estalishment and organization some geometric activities in upper kindergarten
children and at primary school children
9. The structure of the thesis
Besides the Introduction, Conclusion, References and Appendix, the thesis consists of three chapters:
Chapter 1: Fundamentals of theory and practice
Chapter 2: A number of methods to develop geometric thinking in upper kindergarten children and at primary school children
Chapter 3: Pedagogical experiment
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CHAPER 1 : FUNDAMENTAL OF THEORY AND PRACTICE
1.1. History of research.
1.1.1 The research in Viet Nam
In general, these researches focus on methods of establishing and organizing thinking activities to develop thinking
through geometric elements and geometric quantities teaching in upper kindergarten children and at primary school children,
however the geometric activities is built mainly based on the teaching content in each class,the research of geometric activity
throughout the classes from kindergarten and at primary school has not really been interested and focused.
1.1.2 The research in the world
Dutch mathematics educators Van Hiele gave level of geometric thinking from low to high, from that he made the
evaluation tests for children and teaching methods which is applicable to each level are widely used in design of teaching
program content (in kindergarten children, primary school children and high school) in many countries in the world
Geometric thinking requires a combination of spatial concepts and symbols, between logical thinking and spatial
perception. Clements said that even when the children know a triangle has three sides and three angles, they called the "▲" is a

thinking or not.
From conception above, The characterics of geometric thinking can be described :
1) The intellectual activities of geometric thinking is based on the perception of space
In work of thinking by eyes ,Arnheim argues that "the most important intellectual activities derived
directly from our perception of the world, and the view is considered premium sensory system which
covered and formed our cognitive processes. "Spatial perception itself is not thinking but is necessary to
carry out the geometric thinking, it will be difficult to solve geometric Mathematics without drawing
observation or at least thinking about drawing. Spacial perception is included in the perception of spatial
objects and spatial relations and movement combination between the senses (mostly the combination
between hand and eye ). In teaching the first age at school , the early basis visual is more significant because the child's thinking
is mainly visual thinking.
2) Geometric thinking has a close relationship with space imagination
- Geometric thinking is the condittion for children to carry out space imagination
For example : “How many cubes are there in the picture ?” This exercise requires children to use space imagination in
solving problems.However, that imagination is only formed in the process of observation, puzzle, analyze, compare cubes in
practice.
For example: The exercise for grade 1 in China: "Can you guess that how many door are
there in the picture, how many flowerpots are there in the picture (look for the most reasonable
answer)?"(Picture 1.7)
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Picture 1.7
Picture 1.6
This problem requires the children to have space imagination.To do this, the children must have training activities of
observation and analysis - synthesis objects in space there is the analysis of what they "see" and what they "know but do not see
"(geometric thinking activities).
With a problem that can not be solved by geometric thinking, spatial imagination can help predict results, as “solving”
method for the situation.Moreover it creates a destination for thingking towards, or in other words, it orients for geometric
thinking.Thus the difficulties of geometric thinking is a premise for spatial imagination, and spatial imagination becomes the
destination for geometric thinking to develop.
3) Language is a part of geometric thinking.

- The result of activity : How is the level of completeness of activities?
2) The capacity of the language:
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- How do the children use the term and common language in describing objects and spatial relationships?
- How do the children espress the way to solve problems with other people ?
3) Capacity of applying: What can the children do in practice? Can them answer and explain the specific objects and spatial
relationships in the environment around them?
In addition, each activity has its own characteristics that can be considered more performances of intellectual activities
such as : the activity to classify shapes, which criteria can the children use to classify? Do children use one or many criterias for
classification?
1.4.The content of geometric elements and geometric quantities teaching in big kindergarten and elementarychildren
1.4.1 In Vietnam
1.4.2.Compare the content of geometric elements and geometric quantities teaching in big kindergarten and elementarychildren in
Vietnam and some developed countries
1.5.Investigation under the direction of sorting the level of geometric thinking capacity in big kindergarten and
elementary school children in geometric activities.
1.5.1. Purposes of investigation
1.5.2. Methods of investigation
1.5.3. Objects and time of investigation
1.5.4. Contents and results of investigation
1.5.4.1. Assessment of capacity to implement the child's intellectual activities
1.5.4.2. Assessment of children of capacity to apply in practice
1.6. Conclusions of chapter 1
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The research of basis theory shows an general overview of concept of geometric thinking and concepts related to geometry,
the characteristics of thinking of upper kindergarten and elementary children. The author has proposed the concept of geometric
thinking in children, depend on these bases to analyze the characteristics of geometric thinking, the performances of geometric
thinking capacity through some geometric activities.
The research of pratical base of the topic indicates that children who are in each class, and even in the same age, a class
with the guidance of teachers will have different perceptions, the different level of geometric thingking , in the process of

contains the previous development and contains the necessary elements to achieve the next level of thinking (Picture
2.2).Children have not fully completed a level of geometric thinking capacity before they grow to the next level.This plays an
important role in designing geometric activities of teachers: While organizing activities to help children practice thinking levels
1 and 2, children should be prepared to the requirements at level 3 and higher level.
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Picture 2.1 Picture 2.2
Current teaching programs designed as logic of content development.With the hierarchy above, each content of teaching of
geometric objects or geometric relationships in the program can be designed activities for children to go through the levels of
geometric thinking as the hierarchy. For example:
+ Unit 4, page 48, Mathematic 3: Give 8 equal triangles: Please put into the shape of a fish in the sample.Teacher organizes
geometric activities including 8 triangles as the sample; fish-shaped as the sample. Ask them to pave
Then we can graduated the level of geometric thinking capacity.
Level 1: Children "trial and error" in the whole process of paving
Level 2: Require children flat paving and tell the way to move of shape.
Level 3: Visualize before moving triangle: “How many triangles do you need to creat the fishtail? How many triangles do
you need to creat the body of fish?” “How do you creat fishtail?”
- The hierarchy of level of geometric thinking capacity does not mean that the children have to achieve every level in all
the activities ,depending on The content of Mathematics teaching which can be achieved sooner or later but the general situation
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children can overcome the steps of thinking as above. For example: Introduction rectangular parallelepiped in the program of
Grade 5 when teacher requires children to draw one given rectangular parallelepiped , children can draw 1 rectangle
parallelepiped (level 1), then the child can draw two “ consecutive” rectangular parallelepipeds (level 2); or draw a
performance shape from their position; imagine to draw (perception) rectangular parallelepiped from different observation
positions (level 3).
Suggested implementing of methods
- Step 1: Prepare for geometric activities
- Step 2: Make observation paper , analyze and assess children (detailed in Appendix 3).
- Step 3: Carry out activities, observe and record the performance of children’s geometric thinking capacity (preferably
with the help of the media such as camera, photocamera),
- Step 4: Assess the level of geometric thinking capacity in children based on the results in Step 3, for comparison with

Mathematics equipped with a number of rules to represent cube which are widely used and the children need to learn the
rules to explain and re-create them. Children must learn to know that:
- They can show what can be seen from a specific position and ignore something that we know they are there but can not
seen (we do not draw all four wheels if we can only see three wheels).
- They can "distort" the shape and size to make the drawings look more realistic, for example, drawing wheel as ellip and
display the further objects are, the smaller they are as the perspective drawings.
- They can draw differently from the reality to create a emphasized point on some characteristics, as the oblique and edge
drawings
The way to solve the problem 4: We can move the object around in space by moving and rotating the image. The action
does not change the size or shape.
Contents of the way to solve the problem:
- Looking at the entire figure which is need to be paved; constituent parts;
- Analysis the figure that can not be paved; the figure can be paved based on shape features;
- The children use the rotate action, movement action to lock into position to pave;
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The way to solve the problem 5: The object can have the same characteristics and different characteristics, when classification,
we arrange the objects in the same group which they have the same characteristics .
Classification is the common activity in mathematics, such as classification of even and odd number in series of natural
numbers, or classification, arrangement the data is a common activity in content of Descriptive statistics.The content of the way
to solve this problem in the teaching of geometric elements and geometric quantities is when classifying we help children orient
the criterias for classification, children have to analyze, synthesize, compare the characteristics of the geometric objects.
Conclusion: The formation of the way to solve problems helps teachers with an general overview of the organization of
geometric activities, with a clear goal in the establíhment and organization of geometric activities, each geometric activity
not only improves and understands learnt symbols and concepts but also contributes to build the way of solving problem
for a class of Mathematics, a class of activities. We also need to understand that in order to formulate ways to solve the
problem, the children also has step-by-step implementation of the intellectual activities such as comparison and analysis -
synthesis, generalization. Thus the oriention of the main ways to solve the problem is to develop the intellectual activities,
contributing to the development of geometric thinking in children. The statement’s way to solve the problem in the form of
clause also helps teachers be easy to remember, to carry out and to make deeplier, more detail mathematical knowledge of
pre-school and primary school teachers.