Thiết kế và sử dụng các tình huống dạy học nhằm hỗ trợ phát triển trí tưởng tượng không gian cho học sinh trong dạy học hình học ở trường trung học phổ thông TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY

DAU ANH TUAN

DESIGN AND USE TEACHING SITUATIONS TO
SUPPORT THE DEVELOPMENT OF SPATIAL
IMAGINATION FOR STUDENTS IN TEACHING
GEOMETRY AT HIGH SCHOOL

Major: Theory and methods of teaching Mathematics
Code: 9140111

SUMMARY OF PHD THESIS ON EDUCATIONAL SCIENCE

NGHE AN - 2021


The work has been completed at Vinh University

Scientific instructors:
1. Prof. Dr. DAO TAM
2. Assoc.Prof. Dr. NGUYEN CHIEN
THANG
Referee 1:

Referee 2:

Referee 3:

The thesis will be defended in Thesis Evaluation

Committee at University level held at Vinh University
Time…..., day: .... month ..... year 2021

Thesis can be found at:
- Vinh University Library
- Viet Nam National Library


1
Chapter 1
RESEARCH ORIENTATION
1.1. Research problem
In the dissertation, we give the research problems including:
a, Introduce the concept of spatial imagination (SI) in teaching geometry in high schools.
b, Elucidate the role of the SI for students' geometric awareness activities in
teaching spatial geometry in high schools and the impact of the SI on developing the
ability to solve problems in teaching geometry.
c, The role of SI in the study and explanation of phenomena in reality.
d, Exploring and exploiting activities that need to be practiced to develop SI for
students in the process of teaching geometry in the direction of designing and using
teaching situations.
The research problem of the above thesis comes from the following scientific bases:
- First of all, the problem posed by the thesis's research comes from considering
the concepts of SI of many domestic and international authors.
There are many different views on the concept of SI through the introduction of a
number of essential properties. However, we have not found a clear definition of the
concept of SI. Therefore, the research problem posed first is to shed more light on SI in a
way that can initially visualize the levels of SI.
- The practice of teaching geometry in high schools according to the current
program as well as the current program of innovation in mathematics education, the

level of geometrical algebra is quite high, due to the introduction of vector methods,
coordinate methods, transformation. When the geometry program emphasizes algebra, it
will slightly reduce the development of the SI. The main cause of this decrease is the
lack of importance in teaching geometry about the balanced relationship between the
content of general geometry and algorithms using vector tools and coordinate methods in
a formal way. Since then, there has arisen the phenomenon that many students do math
on vector and coordinate expressions but they do not understand the geometrical nature
of the problem solved by vector and coordinate tools. More details on this issue will be
presented in the practical basis outlined in Chapter 3.
- The researches on the development of SI for the students in high school as well as at
other school levels have not clarified which key activities to develop the SI for students. There
have not been theoretical and practical studies to clarify the elemental activities to develop the
SI of students in the process of teaching geometry at high schools.
- There have not been studies in the country as well as abroad on the design and
use of situations to organize teaching geometry in order to develop SI for students. The
design of the above situations contains many difficulties, the outstanding difficulty is to
clarify what minimum requirements a situation designed for use in teaching geometry
must satisfy. The process and steps of designing a teaching situation to use how to
develop SI for students are specific in teaching concepts, theorems, geometrical laws,
and solving geometry exercises.
1.2. The need of research to develop students' spatial imagination in teaching
geometry
a. Stemming from the requirements of the current Math program and the 2018
General Education Math program about the high school goals in the Geometry and
Measurement section that emphasizes to develop SI for students.


2
b. The need to be problem-oriented, creatively solve and develop problems.
c. Requirements of mathematics education in the direction of connection with

practice.
d. The need for active teaching. The geometric knowledge acquired is the product
of active activities of students through interaction with situations, through
communication and cooperation between students and students, between students and
teachers. This poses a requirement for the study, design and use of cases that contain
activities towards the development of the SI for students, and poses a need to consider
and provide procedures for the design and use of situations in teaching geometry in the
direction of developing SI for students.
1.3. Research purpose of the thesis
Provide a theoretical and practical approach to clarify the concept of SI and
related concepts related to the SI, the component activities to develop the SI, the process
of designing the teaching methods and the process of using the teaching methods have
been designed to organize teaching geometry to develop the SI of high school students.
1.4. Scientific hypothesis
If the elements of the SI and the activities compatible with those elements can be
identified, then it is possible to find opportunities to organize for students to practice the above
activities in order to contribute to the development of SI in teaching geometry in high schools
1.5. Research questions
a, Based on what theoretical and practical basis to give the concept of SI?
b, How is SI expressed in teaching geometry in high schools? How to detect those
symptoms?
c, What are the main activities that need to be practiced to develop SI for students?
d, Based on what basis to develop the design process and the process of using
teaching situations towards developing the SI for students?
e, What are the levels of SI development of high school students in teaching
geometry?
1.6. Research Methods
1.6.1. Theoretical research
- Research on psychological perspectives on imagination of domestic and foreign
authors.

- Researching the perspectives on spatial imagination of mathematics educators in
the country and around the world.
1.6.2. Practical research
- Research and design questionnaires to survey students and multiple-choice
questions to survey teachers, attend geometry classes in high schools. This research
activity aims to reveal the manifestations of activities compatible with the characteristics
of SI for students.
- Research the teacher's experiential activities to design teaching situations and use
them to organize teaching of typical situations in teaching geometry in high schools.
- Studying the activities of teachers in the process of design and application of
stages, including: Process building activities, discussion organization through seminars
of teachers, experiment activities on students to find the feedback for editing to choose
the right process for the implementation of teaching situations towards the development
of the SI.


3
1.6.3. Pedagogical experiment
- Conduct experimental teaching activities according to the process of situations
designed to assess the development level of students' SI.
1.7. New contributions of the thesis
- Systematize and clarify the theoretical basis of the SI, the relationship between
the SI with intuition, thinking and knowledge; the typical elements of SI; activities
aimed at developing the SI;
- Give a conception of the SI by 11 possibilities. Propose two levels on the
development of SI of high school students in teaching Geometry;
- Propose 13 main activities to practice for students in order to support the
development of SI;
- Develop a 6-step design process for teaching situations and a 5-step process to
apply the designed situations in teaching with the development orientation of SI.

1.8. Arguments to defend
+ In teaching geometry in high schools, it is necessary to develop spatial
imagination for students;
+ The concept of spatial imagination of high school students and its characteristics
is reasonable and can be developed through the support of the proposed main activities;
+ The design process, application process and teaching situations of spatial
geometry in the direction of developing spatial imagination for high school students
designed in the thesis are appropriate and feasible.
1.9. Thesis structure
The thesis is structured in 5 chapters:
Chapter 1. Research orientation
Chapter 2. Theoretical basis
Chapter 3. Practical survey
Chapter 4. Design and use the spatial geometry teaching situations in the direction
of the development of spatial imagination for high school students
Chapter 5. Pedagogical Experiment
Conclusion of the thesis and recommendations.
References.
Chapter 2
THEORETICAL BASIS
2.1. Research overview of math educators on topics related to the research topic
- Types of situations in teaching Mathematics related to topics and situations that
contribute to fostering SI in teaching geometry in high schools.
- Researches on the concept of SI in teaching geometry - geometrical intuition.
- The research sheds light on the expression of SI, the need for the development of
SI in teaching geometry.
- Analyzing and synthesizing researches on the components of activities in
developing the student's SI.
Through the research, analysis and synthesis of mathematics educators related to
the research topic, we initially obtained the following results:

1. Clarify the relationship between geometric imagination, SI and geometric intuition.
2. Be aware of the relationship between SI, intuitive thinking with logical thinking
and formal proofs in mathematics: The SI suggests logical thinking, expressions and


4
proofs; On the contrary, if there is good logical thinking, the hypotheses proposed by the
SI have a scientific basis.
3. Previous studies have not provided a clear definition of the concept of high
school students' SI. The components constituting SI are shown through studies in
different aspects. The most typical elements can be mentioned, they are the elements in
the composition of the SI:
+ The ability to visualize the results of shapes, relationships, numbers in geometry
learned in high school.
+ The ability to visualize spatial shapes, the relationships of spatial shapes through
representations.
+ Ability to orientate space that helps to study geometry and apply it in practice:
Problems related to vectors, coordinates, rotation direction, position needed in reality, ...
We find that there are still some contents related to the SI that need to be
developed for students but have not been studied fully and deeply. For example the
following problems:
- Estimate the length, magnitude, size of the geometric figures and estimate in reality.
- The problem of the relationship between shapes, partitioning a shape into
familiar shapes, spreading the spatial figure on a plain...
4. Clarifying a number of roles of the SI in teaching geometry and in practice.
- It helps students see the meaning of mathematical knowledge, the meaning of
math problems before implementing into proof arguments, explaining problems,
reasoning to solve problems.
- Through the development of SI, it helps students have practical knowledge to
help visualize the structure of objects through drawings and designs.

- It helps students approach to judge math problems, make hypotheses through
spatial imagination.
- It helps to solve problems creatively through visualization of new events, new
problems.
5. Visualize a number of component activities of the activities of formation and
development of SI, including:
- Perceiving practical models, geometric models to form correct symbols of
shapes, interdependencies and quantitative relationships in that figure to form correct
spatial symbols . From there, there is a profound SI.
- Activities of determining direction, direction, determining position from one
point to another, from one shape to another.
- Visualization of shapes, relationships and relationships in pictures through
representations; operation that determines the representation of an image. For example,
we ask students to determine the projection plane and projection so that the
representation of a nearly regular tetrahedron is a rectangle with two diagonals added.
- Visualizing the cross-section of a space figure created by a certain plane.
- The activity of visualizing the results of solving the problem without using
pictures only through imagination.
Through the review, it is found that the authors have not mentioned the following
activities that are meaningful to the formation and development of the following:
- The operation of spreading a spatial figure onto a flat figure.
- The activity of creating a spatial shape according to the given flat parts.


5
- Space rendering activities.
- Estimating the length, area, volume associated with the figures in practice.
2.2. Approaching the pedagogical perspective of spatial imagination
2.2.1. Symbols
Symbols are forms of perception which are higher than sensations, they give us

images of things that remain in our minds after the impact of things on our senses has
ceased. In psychology, it is understood: "The symbols of memory are images of objects,
processes and phenomena that are not currently perceived but have been previously
perceived".
2.2.2. Concept of Space
The concept of "Space" mentioned in the dissertation is a 2-dimensional, 3dimensional Euclide space in the high school program (Based on real space symbols that
people can perceive - Physical space). In the symbols that the SI operates reflect the
properties (or signs) of spatial characteristics.
On that basis, we think that space is understood as a structure including the
following sets:
- Geometric shapes, objects;
- Qualitative properties: shapes of shapes, relative positions between shapes,
objects; direction;
- Relationships before - after; right - left;
- Quantitative factors: Distance, circumference, area; volume of shapes, blocks, etc.
On the basis of understanding the symbol of memory, we conceive the symbol of
space as a symbol of memory about the properties and relationships of spatial objects.
2.2.3. Imagination
Intelligence is the ability to perceive, remember, think, judge, ... of human.
Imagination is creation in the mind an image of something that is not present or has
never existed. We can understand that imagination is the cognitive ability of people to
create images of things that have been perceived but are not present or reflect things that
have never been in personal experience by build new icons on the basis of existing
images and icons.
2.2.4. Spatial imagination
As the point of view of space and imagination above, we can understand that the
object of the SI is space, that is, the symbols in the process of imagining are spatial
symbols.
Thus, we can understand that spatial imagination is the cognitive ability of people
to create images of spatial objects that have been perceived but do not have before or

reflect spatial objects that unprecedented in personal experience by building new space
images and symbols on the basis of existing space symbols.
2.3. Features of spatial imagination
To give the characteristics of SI, the important bases are:
- Derived from the concept both in the country and abroad as well as the concept
of SI above;
- Based on the research results on the essential characteristics of teaching
geometry in high schools, especially the research of Academician A.D. Alexandrov
about three characteristic elements of teaching geometry are: Reality, logic, imagination;
- Be aware of students' mistakes due to not understanding the relationships and


6
relationships between geometric objects in space. A common mistake that arises in the
process of learning spatial geometry is that they are only interested in manipulating
formal operations without visualizing the relationships between geometric objects,
especially when studying spatial geometry. Geometric studies using vector tools and
coordinate methods.
From the research results mentioned above, in this thesis, we conceive that the SI
belongs to the category of geometric intuition characterized by the following
capabilities:
- The ability to visualize spatial shapes through representations;
- The ability to determine the relative position between geometric figures;
- Ability to establish dependency relationships between geometric shapes;
- The ability to visualize cross-sections, intersect spatial shapes;
- The ability to estimate the size of spatial figures;
- Ability to transform relationships, relationships into known geometric models
convenient for problem solving;
- The ability to convert from one geometry language to another to visualize the
research model;

- The ability to develop shapes convenient for calculations;
- Ability to map, coordinate to determine position, size, distance between shapes;
- Ability to model real-world phenomena using geometrical language and symbols;
- Ability to define new spatial objects on the basis of existing spatial objects.
With the understanding of the above characteristics, for high school students' SI,
there are two levels:
Level 1: Deeply understand geometric objects, relationships and relationships
between geometric objects, geometrical meanings of formal expressions expressed in
algebraic language (Vector language, coordinates). It's essentially understandable the
geometrical content through its formal expressions.
Level 2: Help to create new geometric objects on the basis of transforming
existing objects and relationships.
Example 2.1. Students can construct a rectangular box by using a nearly regular
tetrahedron ABCD through the following proposition: “Three pairs of parallel planes
pass through pairs of opposite sides of a nearly regular tetrahedron that intersect to form
a rectangular parallelepiped".
Through this example, we see that: Spatial imagination is not only the activity of
building new symbols on the basis of existing symbols, but it is also on the basis of
existing knowledge. In the above example, the existing knowledge is: There exists only
one pair of parallel planes passing through two diagonal lines.
2.4. The relationship between intuition, spatial imagination and mathematical
thinking in teaching geometry in high schools
2.4.1. Intuition
In Soviet teaching theory, intuition is explained: “As a requirement of teaching so
that students form symbols and concepts on the basis of vivid perception, objects,
phenomena studied of the objective world or its representations”.
2.4.2. Logical thinking
Because the objects of the SI are relationships and relationships, the mathematical
laws need to be verified for correctness and falsehood. Therefore, SI needs to be

7
associated with logical thinking. According to M.Iu. Koliagin: “Logical thinking is
characterized by the ability to derive consequences from given premises, the ability to
thoroughly separate individual cases, the ability to predict specific results by theoretical
means, to sum up generalize the results obtained”.
Like other types of thinking, logical thinking also has its fulcrum from vivid intuitions.
2.4.3. The relationship between spatial imagination and intuitive thinking
According to M.Iu. Koliagin: “Intuitive thinking is characterized by the absence of
clearly defined steps. It tends to perceptively reduce the whole problem at once. One can
get a “right” or “wrong” answer.
Nowadays, the development of intuitive thinking has attracted many progressive
math educators. When they talk about the role of intuitive thinking in teaching
mathematics, Academician A.N. Konmogorov of the Russian Federation wrote:
Everywhere It was possible that mathematicians tried to make the problems to be studied
by geometrically intuitive everywhere, ... Geometric imagination or it is said.
“Geometric intuition plays an enormous role in the study of almost all areas of
mathematics, even abstract problems.”
From the above considerations, we believe that the SI is a field of geometrical
imagination and since then it belongs to the category of intuitive thinking.
2.4.4. The relationship between intuition and spatial imagination
2.4.4.1. The First mode
From the above analysis, we can see that the transition from visual to SI is done
through the symbols of spatial memory.
The symbol of spatial memory is the product of direct perception of objects,
phenomena, processes or their actual visual perception. In this case, one of the primary
tasks of using visual materials is to form specific symbols in the student's memory.
Through consciously repeatedly perceiving to learn the properties of objects, the
student acquires symbols of memory. From the symbols of this memory, through
speculative and logical activities to build new spatial symbols, that is the SI.

2.4.4.2. The second mode
Intuition is also used as a fulcrum for different thinking operations to map out the
essential properties of objects, regular relationships between objects, processes and
phenomena, thereby forming knowledge in general and knowledge of spatial
relationships in particular.
When they have a fluent knowledge of spatial relationships, students can construct
new spatial symbols through indirect perception of the material.
2.4.5. The relationship between logical thinking and spatial imagination
Because the results of the SI in teaching mathematics are hypothetical statements
about the relationships and relationships of spatial objects that need to be tested. To
confirm that the SI is correct, we need to prove it by logical reasoning, using logical
thinking.
On the contrary, from logical thinking to help the subject dominate mathematical
knowledge, this knowledge is the basis for good SI.
From the analysis of the above studies, we come up with a diagram of the
relationship between intuition, logical thinking, Spatial imagination and knowledge as
follows (Figure 2.1):


8
Intuition

Logical
thinking

Spatial
imagination

Knowledge
Figure 2.1

The above diagram is made from the analysis of the relationships between the
elements: Intuition, SI, thinking, especially logical thinking. The diagram highlights the
following relationships:
- Intuition is the fulcrum for thinking in general, and logical thinking in particular.
On the contrary, from abstract thinking, in particular, high-level logical thinking is
illuminated by richer intuition.
- Through intuitive construction of space symbols, thereby building new symbols SI. On the contrary, if the person has a good SI, the visual perception will be richer and
more profound.
- Because products of SI can be right or wrong, they need to be verified by logical
thinking. On the contrary, if there is good logical thinking, the hypotheses proposed by
the SI have a scientific basis.
- Thanks to the knowledge of rich spatial relationships that allow students to
introduce new spatial symbols indirectly - that is the SI. On the contrary, thanks to the
propositions that are products of the SI, they will become new knowledge through
logical verification.
- Komensky asserted that: "There will be nothing in the mind about things that
have not been perceived before intuitively". This statement means that acquired
knowledge begins with visual objects, begins with perception. On the contrary, if there is
abstract knowledge of spatial relationships, it allows a richer visual perception.
From the above analysis, it can be seen that the focus of teaching geometry in high
schools is to well solve the dialectical relationships between the elements mentioned in
the diagram above.
2.5. Activities towards the formation and development of spatial imagination
in teaching geometry in high schools
Based on the results of the overview research, especially the conclusions from the
overview research related to the activities oriented to the formation and development of
the SI; on the basis of the constitutive features of the SI drawn from the definition of SI
mentioned in section 2.4 and based on the analysis of the dialectical relationships
between intuition, symbols, mental retardation, logical thinking and knowledge, we give

Outlining the following component activities to form and develop SI:
Activity 1: Observing and perceiving geometric models taken in practice,
representations of spatial figures with the aim for students to analyze, compare, synthesize,
generalize, and abstract intuition, thereby obtaining correct representations of geometric
figures - their constituent elements and their relationships. Since then, SI has been
formed for students.


9
Activity 2: Activity to represent a spatial figure on a plane from simple to
complex.
Activity 3: Visualizing spatial shapes through representations.
Activity 4: Visualizing steps to solve spatial problems through imagining, not
observing drawings.
Activity 5: Using the coordinate method to describe the location of places in the
city or on the sea.
Activity 6: Determining the size and size of the shapes.
Activity 7: Estimating geometrical quantities.
Activity 8: Exploring and discovering relationships between spatial shapes.
Activity 9: The activity of cutting flat shapes and forming space shapes.
Activity 10: The operation of separating the flat parts of the space figure.
Activity 11: Spreading the picture.
Activity 12: The activity of transferring problem solving from one geometric
model to another is simpler.
Activity 13: Modeling.
2.6. The role of fostering spatial imagination for students in teaching geometry in
high schools
2.6.1. Educate students to grasp the meaning of the problem before doing
geometry problem solving
The development of SI for students in teaching geometry helps students to orient

the right way to solve problems. The role of problem identification is to help students
associate problems to be solved with prior knowledge and experience. Problem
identification will help students mobilize relevant knowledge to solve problems,
especially help them imagine the process of solving that problem.
Example 2.2. Given a tetrahedron OABC, where OA, OB, and OC are
perpendicular to each other. Let H be the perpendicular projection of O onto the plane
(ABC). Please prove
1
1
1
1
=
+
+
.
2
2
2
OH
OA OB
OC 2

The identification method 1: The square tetrahedron and the right triangle are
three-dimensional and two-dimensional units, so the solution is similar to the properties
of the altitude in a right triangle.
From there, the solution to the problem is to separate the planar parts that are right
triangles AOM with altitude OH to return to the problem of flatness and use right
triangle OBC with height OM.
The identification method 2: Use the formula to calculate the volume of a
tetrahedron.

2.6.2. Contribute to the education of creative thinking
2.6.3. Help students orient to make judgments and hypotheses about a new
object, relationship, and new geometric rule
2.6.4. Helps to approach constructivist teaching and problem-solving perspectives in
geometry teaching
Constructivist theory is basically a theory based on observation and scientific
research to answer the question: How do people learn? It considered that humans
construct understandings and knowledge of the world through experience and reflection.


10
The fundamental idea of constructivism is to put the role of the perceiver at the forefront
of the cognitive process. In which perception is the process of adapting the subject to the
environment through the use of two activities: Assimilation and accommodation.
In which, the adaptive activity is directed at restructuring the existing knowledge
to be compatible with the new situation.
2.6.5. Help to detect mistakes because students do not pay attention to the
content (semantics) but only pay attention to the syntactic (formal) side of the math
operations.
Example 2.12. In the orthogonal coordinate system (Oxyz) given two lines d:
x y −1 z
x −1 y −1 z
and d':
=
=
=
= and the point M 0 . Find the equation of the line that
1
2
−1

−1
−1 1

passes through and intersects the two lines d; d'.
It is possible to check two lines on diagonals. When solving this problem, students
follow the formal (syntax) solution with the following process:
- Establish the equation of the plane passing through M 0 and d; that's the equation
x + z = 0.

- Establish the equation of the plane passing through M 0 and d', that is the equation
2 x + y − z − 3 = 0.

x+z =0
2 x + y − z = 0


- Conclusion the straight line to find is  : 

Because students do not understand the semantics, they cannot show that this line
does not intersect the line d'. Since then there is no straight line  .
2.6.6. Help students approach to solving practical problems
2.6.6.1. Orient and determine the geographical location of a certain destination,
…
2.6.6.2. Foster the ability to estimate quantities in real models
2.6.6.3. Read the drawings, the design, determine dimensions in construction,
production, in irrigation, calculate volume
2.6.6.4. Solve the extreme problems in practice related to the quality of images
2.6.7. Potentiality of developing spatial imagination for high school students of
the strand of Geometry and Measurement
First of all, through this knowledge strand, the students can form spatial symbols

enough to recognize and analyze their expressions in life as well as use them as fulcrums
to build new spatial symbols in a logical way with scientific basis.
Second, students learn about parallel and perpendicular projection, thereby
helping them to represent spatial geometry accurately, and providing a tool to solve
spatial geometry problems effectively, creating opportunities to practice the relationship
between SI and logical thinking.
Third, students are equipped with different methods to study geometry in high
school, which are synthetic method, vector method and analytical method. These three
basic methods help students build spatial representations in different forms, help solve
problems of geometry accurately, and give them the tools to make judgments. guess,
spatial intuition.
Fourth, one of the important goals of the program is to solve some simple practical
problems associated with Geometry and Metrology. Practice is the source of mathematics in


11
general, geometry in particular. Every concept, every geometric property, no matter how
abstract, finds its image and application in practice. This is the basis for students to apply
knowledge of Geometry and Measurement to life, thereby it helps to develop spatial
imagination and logical thinking.
2.7. Teaching situation in the direction of supporting the development of
spatial imagination
2.7.1. Teaching situation
As a fulcrum for this thesis, we choose the following definition of teaching theory
by Phan Trong Ngo: “Teaching situation is a situation in which there is a delegation
from the teacher. This is the process by which the teacher puts the content to be
conveyed in the events of the situation and structures the events so that it is suitable for
pedagogical logic, when the learners solve it, they will achieve the teaching goal".
2.7.2. Teaching situation in the direction of supporting the development of
spatial imagination in teaching geometry at high schools

In order to build a definition of the concept of teaching situation to support the
development of SI, we mainly rely on the issues explored in the following thesis:
a, The key activities for the formation and development of SI in teaching geometry
in high schools have been considered in Section 2.5.
b, The role of SI in mathematics education in section 2.6.
c, Definition of teaching situation mentioned in section 2.7.1.
2.7.2.1. Definition:
The geometry teaching situation supporting the development of the SI is a
teaching situation that contains activities that promote the development of the SI,
students need to interact, discover and practice in order to visualize hypotheses about
new knowledge. and logical steps to solve problems, orienting formal logical arguments
in order to acquire knowledge according to the goal of mathematics education.
2.7.2.2. Analyze the role of definition
The above-mentioned situational teaching helps to foster students:
- To develop students' SI through discovering and solving a math problem.
- To enhance the ability to imagine to visualize judgments, hypotheses and
problem-solving arguments, suggestions for problem-solving presentation logic.
- To fostering the way of finding knowledge, attaching importance to the process
of experiencing knowledge discovery.
- To help students understand the meaning of knowledge.
Conclusion of chapter 2
- The characteristic components constituting the SI is clarified through the
overview study of the works of domestic and foreign authors related to the dissertation.
Researching the definitions of the SI that can be used in mathematics education and
exploiting the inheritance of the definition of the SI. Because these definitions are
emphasized on psychological characteristics, we have been interested in exploring the key
elements, which are characteristic of the concept of SI. Typical of these elements are shown
in the works of cited authors, such as: A.N. Konmogorov, M.Iu. Koliagin, Tran Thuc Trinh,
Pham Van Hoan, Nguyen Gia Coc, Dao Tam, Le Thi Hoai Chau, Vu Thi Thai, ...
- To elucidate the role of TTKG in mathematics education, we have approached

the research: The nature of teaching geometry by A.D. Alexandrov. In addition, to see
more roles of SI, we study to elucidate the dialectical relationship between intuition, SI,


12
logical thinking and geometrical knowledge. Determining this relationship, in addition to
the above purposes, it also sheds light on the way of fostering mental health through
fostering related elements.
- From giving a definition of SI characterized by 11 component possibilities, we
consider the role of SI in mathematics education. The above research issues are the basis
for giving typical elemental activities to form and develop the SI.
- In the theoretical approach, we have presented teaching situation. This issue and
related contents, especially the typical elemental activities, are the basis for giving
teaching situation in the direction of supporting the spatial imagination development in
teaching geometry in high schools. The implementation of teaching situation to support
the development of SI is carried out through the use of teaching methods of discovery
and problem solving, and constructivist teaching, and self-study teaching methods.
Chapter 3
PRACTICAL SURVEY
3.1. Purpose of the survey
a, To find out the teacher's perception of the following issues: Concept of space,
spatial imagination, elements of SI; Their understanding of the approach to the SI;
activities towards the development of SI; the role of the development of SI in teaching
geometry and in practice for high school students; The symptoms of SI, ...
b, Student's SI ability: The ability to grasp the meaning of geometric problems
before starting to solve problems; the ability to intuitively recognize the results of spatial
problems; the ability to understand the relationships and the relationships between
spatial shapes; the ability to perceive geometrical ideas contained in practical situations.
3.2. Survey content
Exploiting ideas on education of SI in the content of the current textbook

program and the innovative program of teaching geometry at high schools in the
future. Using exercises and problems in textbook that require to solve by using the
SI; Exploiting situations in internal geometry as well as in practice plays a role in
developing students' SI.
3.3. Survey tool
- Use multiple choice questions for teachers. The basis for choosing these
questions is the theory and practice of teaching in high schools in the direction of
approaching the development of SI for students. These facilities were presented in
Chapter 2.
- Attend lessons on geometry by experienced teachers to learn the role of teachers
in the formation and development of SI for students.
- Interview experienced teachers about teaching methods in the direction of
developing SI.
- The questions are raised to achieve the survey objectives.
- Provide questionnaires; problems in order to observe students' activities in the
direction of developing mental retardation for students. These problems are either taken
from the textbook program or designed to match the students' perceptions.
3.4. Survey organization
- Have surveys on 30 teachers in Nghe An province at 6 high schools including: 2
schools in mountainous areas, 02 schools in the plains, 02 schools in the city.
- Attend 3 schools in Vinh City to find out the activities of students and the


13
fostering ability of teachers of 5 high schools in Nghe An province.
- Interview 5 experienced high school teachers about the concepts and expressions
of students and about the role of the development of the SI.
3.5. Survey of students
- Survey tool: Provide 05 problems and questionnaires, instructions to find out the
ability of students' SI according to the current textbook program.

- Organization and implementation: We conducted the students activities to solve
math problems according to the questionnaire of 5 groups of students of grade 11 and
grade 12 of high schools in Nghe An province including: Huynh Thuc Khang High
School; Ha Huy Tap High School; Do Luong 1 High School; Anh Son 1 High School;
Quy Hop 3 High School; Ky Son High School.
3.6. Evaluation about the results of the survey on the teachers and the students
This assessment aims to draw conclusions about teachers' perceptions, their
activities on fostering spatial awareness for students, and at the same time, the
advantages and disadvantages of teachers in terms of spatial imagination in teaching
geometry in high school. Through the activity of solving mathematical problems related
to the SI in order to assess the students' ability of the SI, it is the basis for designing and
using teaching situations to support the development of the SI of the students in teaching
geometry at high schools.
3.6.1. Sub-conclusion of the survey on the teachers
We conducted a survey to find out teachers' perception of the SI in order to
supplement the theoretical research and propose the types of activities towards the
formation and development of the SI as presented in Chapter 2.
The logic of this inquiry in the order that it proceeds is as follows:
To find out the teacher's perception of space symbols, the SI, the path of formation
and development of the SI; the expression of SI in teaching geometry at high schools; to
find out the teacher's perception of activities that can be exploited to practice for students
in order to form and develop SI for students;
The role of SI in effective teaching of geometry at high schools.
Through the results of the survey, qualitative and quantitative analysis, it is
necessary to practice for students activities to form and develop SI. This requires not
only theoretical but also practical aspects of teaching geometry today. This necessity
does not only arises for the role of the SI in teaching geometry, but it is also for its role
in integrated teaching, the ideology that the new program now places the leading
position in teaching. mathematics education in high schools.
3.6.7. Sub-conclusion of the survey on the student

- The students have not been not equipped with knowledge to explore and explore
different definitions of geometric shapes.
- The students have not been able to regularly apply various concepts and
properties of spatial shapes and in different chapters of geometry.
- The students have not focused on knowledge of mathematical modeling to
describe the properties of the relative positions of shapes as well as about the
geometrical quantities present in practice. For example, the problems of estimating
geometrical quantities in space as well as explaining phenomena in practice related to
the student's SI.
Conclusion of chapter 3


14
The survey results showed:
a, It is necessary to inculcate for teachers an understanding of the characteristics of
SI. These characteristics have been mentioned in Chapter 2. The SI has two levels: Level
1: On the basis of existing knowledge, deeply practiced, the learners can visualize the
relationships and positions between shapes, shapes, quantities and relationships between
quantities. At level 2: On the basis of being equipped with space symbols, the learners
can create new space symbols through their imagination. At this level, it requires the SI
to move from intuition by observing geometric objects through the use of visual models:
representations, real representations, dynamic visualizations, ... From there, the learners
continue get memory symbols through the use of mental manipulations applied in
geometry.
b, When the teachers have the specific characteristics of the SI, they need to know
the specific manifestations of the SI, the types of activities to form and develop the SI.
Especially, the teachers need to know the knowledge and methods to practice for
students the activities to form and develop the SI.
c, Through experiential activities, the teachers need to come up with situations
containing activities to help students interact to discover new knowledge and thereby

develop the SI. It's particular to emphasis here is on the practical situations that are
meaningful for the development of the SI.
Chapter 4
DESIGN AND USE THE SPATIAL GEOMETRY TEACHING SITUATIONS IN
THE DIRECTION OF THE DEVELOPMENT OF SPATIAL IMAGINATION
FOR HIGH SCHOOL STUDENTS
4.1. Prepare knowledge and skills for teachers on designing a teaching
situation in the direction of supporting the development of spatial imagination
4.1.1. In terms of knowledge
The teachers need to understand the role of the development of the SI in detecting
problems and solving geometrical problems. It means that the situations are designed to be
set up so that they contain the knowledge for the elucidation of that knowledge by using
teaching methods to discover and solve the problems. SI also helps to orientate
hypotheses, then it conducts testing of that hypothesis whose steps are visualized by
learners' SI. This shows the situations that can be installed in the process of implementing
constructivist theory when teaching geometry. In addition to that, the teachers also need to
be aware of the activities that need to be practiced for students to develop their SI and they
can be installed in self-study situations for students. From there, the teachers need to know
the knowledge related to teaching methods of self-study.
Also in the field of knowledge, the teachers need to understand the SI. The basic
characteristics of SI; the activities that need to be practiced for students to develop SI.
Understanding these activities is the basis for installing them in the teaching situation of
geometry: concept teaching situation, rules teaching situation, theorems and solving
geometry exercises teaching.
The above-mentioned theoretical issues have been presented in Chapter 2. In
addition, in terms of knowledge, the teachers also need to have a deep understanding of
the relationship between intuition, imagination and logic in teaching. geometry. This


15

proves that the potential for development of SI in teaching geometry requires teachers to
exploit.
The following is an equally important issue that should be emphasized when we
design the teaching situations for the students to interact with in order to form and
develop the SI: The spatial symbols are not only in the category of visual objects.
science, relationships between geometric objects, relative positions between geometric
figures, magnitude, size between shapes, ... but the symbol of space is also present in
life, we call it are samples. The interest in the above-mentioned specimens is useful for
exploring practical situations for students to imagine, generalize and abstract in the
process of using modeling to discover knowledge. The process of activities mentioned
above contributes to the development of SI for students.
4.1.2. In terms of skills
- The ability to determine the goal of a lesson.
- The ability to explore and detect situations taken internally in mathematics or in
practice contains activities that need to be practiced for students in the direction of
supporting the spatial imagination developmen.
- Lesson research skills in the direction of exploring situations, experimenting, and
interacting with other teachers to choose the optimal situation.
- Carefully prepare teaching skills according to selected situations.
4.2. The process of designing and using teaching situations to support the
development of spatial imagination in teaching geometry in high schools
4.2.1. Scientific and practical basis for sequential design steps
- Clarify the teaching objectives of the lesson content, including knowledge, skills,
and components of competence as required by the geometry textbook program in high
schools.
- Clarify the activities that need to be practiced in order to develop the SI through
each lesson content. Typical activities in teaching geometry such as: Observing the
representations of objects and spatial relationships; analysis, synthesis, abstraction and
generalization to form correct symbols of geometric objects and relationships; the
activity of forming judgments and hypotheses thanks to the SI; the activity of visualizing

relationships, relationships, and logic to suggest deductive steps in problem solving;
Associative activities to transform images, transform associations, transform information
to receive new knowledge.
- On the basis of clarifying the goals of the above activities, the teachers need to
experience and explore situations in order to integrate activities into situations so that
students can interact to discover problems and solve problems.
- When there are situations, the teachers need to conduct lesson research activities
to choose the optimal options. Such activities, for example: Organize discussions in
groups of teachers, exchange with experienced teachers.
- Experimenting with selected situations through implementing activities to
approach problem detection and problem solving for groups of students with
questionnaires and instructions from the teachers. To conduct this activity, the teachers
need to perform observation activities, guide discussions, record audio, and take pictures
to understand the behavior of students in the process of detecting and solving problems.
- The teacher's handling of feedback from students to correct and overcome the
disadvantages of the situation for knowledge acquisition activities, activities towards the


16
development of the SI. Thereby confirming the design process to apply in teaching
geometry.
4.2.2. The process of designing teaching situations in the direction of supporting
the development of spatial imagination in teaching geometry at high schools
Step 1: Research the objectives and content of the lesson and exploit the activities
towards the formation and development of the SI to promote cognitive activities in order
to acquire knowledge and skills according to the objectives and content of the lesson.
Step 2: The teacher explores geometry teaching situations that support the
development of SI and promotes cognitive activities through specific lessons.
Step 3: Choose a teaching situation that is suitable to the chosen goal and method.
Step 4: Discuss and adjust the teaching situation in the direction of studying the

lesson.
Step 5: Experiment with teaching situations.
Step 6: Confirm the teaching situation.
4.2.3. The process of using the designed situations in teaching geometry in high
schools in the direction of supporting the development of spatial imagination
4.2.3.1. General thought
The idea of this application process is to define the steps to organize teaching
according to the designed situations mentioned in Section 4.2.2. The above thought is
concretized according to the sequence of steps of the organization of teaching, including
the transfer of cognitive tasks to students, the teachers' controlled activities, and the
students' activities in the direction of practicing components activities towards the
formation and development of SI.
In addition to the above issues, the process of applying the designed situations
needs to master the teaching methods mentioned in section 2.9 of the thesis, these
teaching methods help guide the control activities of the teachers that aim to promote the
active learning process of students. In the process of applying teaching situations, it is
necessary to clarify the assessment activities, institutionalize the knowledge found by
students, especially attach importance to the assessment of the level of component
activities towards the development of the student's the SI.
4.2.3.2. Steps of the process
Step 1: Identify an active teaching method compatible with the specific objectives
and content of the lesson in teaching geometry in high schools.
Step 2: Transfer cognitive tasks - The situation has been designed to create
cognitive needs for students.
Step 3: The teacher's controlled activities are aimed at directing students into
cognitive activities, practicing component activities in the direction of supporting the SI
development.
Step 4: Students' activities aim to interact with situations, experience, make judgments,
detect problems, testing hypotheses, and solve problems to confirm knowledge.
Step 5: Assessment activities, institutionalize knowledge of teachers, assign selfstudy tasks to students.

4.3. Apply the design processes and use situations in teaching typical
situations in teaching geometry at high schools in the direction of developing spatial
imagination
Based on the 6-step process of the design and the 5-step process using the above


17
teaching situations, we have designed and used a number of teaching situations to
support the development of SI in teaching pictures in high schools through teaching
activities of concepts, theorems, rules, solving math problems.
Especially, while we are applying the steps of the design and the application
process above, we do not mention the steps of the process but we emphasize the
following main issues:
- We are interested in exploiting the goal of developing SI through each lesson on
teaching concepts, theorems, and rules, we also concretize through the component
activities of activities aimed at developing SI that have been introduced in Section 2.6,
Chapter 2. These activities are installed in situations designed to transfer cognitive tasks
to the students, they are created for the students to operate, experience, and discover
knowledge. This is an important step of the design process that is indispensable for
teachers' teaching activities.
- We consider activities to form and develop SI in teaching concepts; theorem
teaching; rules and teaching math problem solving.
- For example, in teaching the concepts, we are interested in a number of
component activities in the activities towards the development of the SI as follows:
Activity 1; 2; 3; 4; 8; 11; 13. In teaching theorems, we pay attention and emphasize
some of the following activities: Observation activities, through surveying individual
cases taken internally in mathematics, are more likely to develop SI for students to
experience, to analyze, to compare to generalize, to discover the rules, the discover
theorems and rules. Since then, we focus on problem solving stages by using
imagination to suggest a formal presentation to solve the problem. In the problemsolving stage, we attach great importance to the activity of separating the flat parts

related to the problem from the spatial figure to convert to the plane problem. This
activity has the meaning of fostering the SI; because the students need to visualize the
flat parts constituting the spatial shape. When they move flat parts out for study, its
representations need to be changed to accommodate planar geometry.
- When we approach the situation of solving math exercises, we emphasize the
following component activities of activities aimed at supporting the development of the SI:
+ Activities of exploiting shapes with similar properties to convert space problems
to planar problems.
+ The operation of separating flat parts from the spatial figure to convert the
spatial problem into a combination of planar problems. For example, the problem of
finding the center and radius of a sphere can be reduced to the problem of finding the
center and radius of the great circle by considering a plane passing through the center of
the sphere.
+ The activity of transforming and associating one picture to another.
+ Activities to transform spatial problems into plane problems that are familiar to
the students, they have become acquainted with in middle school and the first part of
grade 10.
- In the control activities of teachers, we are interested in the questions and the
orientations to support the development of the SI such as activities:
+ Visualize spatial shapes.
+ Associative activity, to convert properties from one shape to another in order to
solve simpler problems. This activity contributes to supporting the development of SI for


18
the students.
+ Similarization between plane and space shapes.
For the following, we will concretize the above activities through implementing
the design process and applying it to teaching typical situations that are: Teaching
concepts, theorems, solving geometry exercises.

Applying to teaching and solving geometry exercises
The design of teaching situations to support the development of SI for the students in
teaching geometry problem solving in high schools has different characteristics compared
with teaching situations of concepts and theorems because the teaching of concepts, theorems
are often organized for the students in an inductive way, but teaching and solving math
problems have the main goal of consolidating, applying, and deepening knowledge.
The characteristics of the design of situations in teaching and solving math
problems are mainly shown as follows:
- Select problems in order to solve them has the opportunity to practice the
component activities of fostering activities of SI for the students. Such problems aim to
deepen the elements of spatial shape, position relationships between shapes and quantity
relationships in space (Length, angle, perpendicular, area, volume, ... ). Problems that
require the perception of space in terms of an aggregate model, a vector model, or a
spatial model described by coordinates.
- Select the practical situations to ask the students to explain and describe by using
mathematical models, using language and geometric representations to develop SI for
the students.
Example 4.1. Considering the practical situation: Please observe Figure 4.1, can
you describe the lifting and lowering equipment in construction work? Use your
mathematical knowledge why when two equal steel bars are attached together in the
shape of an X by an axis passing through the midpoints of the two bars, the ends move
on two horizontal supports that can be raised. Lower objects up and down?
To explain this, you should use language and mathematical notation to describe
the above lifting device.
B

A

I

D

C

Figure 4.1
Figure 4.2
- The teacher asks the question: Which object in mathematics
do two equal steel bars attached to the axis in the middle relate to?.
To answer this question, it requires the students to visualize, imagine, think about


19
the two diagonals of the rectangle ABCD (Figure 4.2).
- From the lifting device, please tell me in the rectangle representing it, which
elements are fixed and which ones change?
To answer this question, it requires the students to consider constant quantities,
variable quantities: Length of diagonals AC = BD = d (Const), and dimensions AD = x;
AB = y variation.
Then, using the Pythagorean theorem, we have the relationship between the
diagonals and the sides: d 2 = x 2 + y 2 (1).
- From the equation (1), can you clarify when does the lifting device lift the object
to a high floor?
To answer this question, it requires the students to visualize and imagine space:
When we raise, the endpoints A, B move closer together, then from the equation (1)
inferred, when x is smaller the larger y is since the sum of the squares x 2 + y 2 that does
not change. It can be said that it is the mathematical principle used through the
equipment applied in practice.
Conclusion of Chapter 4
In this chapter, we have given a 6-step design process for teaching situation in the
direction of developing SI. The process is designed based on the following pedagogical

scientific ideas, these scientific bases have been presented in Chapters 2 and 3:
- The first emphasized thought is that: Knowledge, skills, thinking as well as
correct symbols of space are formed and developed through interaction of the subject
with situations and environments. The content of the interaction includes observing,
comparing, analyzing, synthesizing, generalizing, abstracting, hypothesizing, testing
them, ...
- The teaching situations are in the direction of the development of the SI that are
the place to install the component activities of the development of the SI for students.
- The design of situations in use should be compatible with the cognitive
characteristics of students, they are accepted situations that must have fulcrums from
surveying practical activities, interacting with situations of students, focusing on the
component activities of the activities towards the development of the SI. The survey of
practical activities through the design of questionnaires and instructions of teachers.
- The situations are discussed through studying the lessons of teachers by subject
groups, with the participation of teachers who have experience in teaching geometry in
high schools.
In this chapter, we give a 5-step process to apply to teaching geometry. These
steps represent selected active teaching methods to orient the control activities of
teachers and promote active and self-disciplined learning activities of students. Students'
learning activities are shown through interaction with situations in order to form and
deepen symbols of space. The fostering activities of the SI are shown through two levels:
the level of deepening the existing symbols and the level of forming new symbols
through imagination and generalization.


20
Chapter 5
PEDAGOGICAL EXPERIMENT
5.1. Objectives of the experiment
- To assess the ability of math teachers in implementing the steps of the process.

To evaluate the ability to study the lesson of the selected group of teachers; To be
mainly interested in debate activities, critical thinking when choosing situations to use.
- To assess the advantages and disadvantages of students in performing activities
towards the development of SI through the interaction with the designed situations
mentioned in Chapter 4.
5.2. Experimental content
- For teachers. We assign tasks to 02 groups of teachers from 02 high schools in
Nghe An province: Le Loi High School, Tan Ky district; Dien Chau 5 high school, Dien
Chau district; Each group consists of 10 teachers. Specific tasks are as follows:
Mission 1
1. Give the process of designing the situation towards the development of the SI for the students
when you teach the theorem about the intersection of three planes, Geometry textbook 11, page 57. This
process consists of 06 steps; The subject of the experimental teacher has approached first.
2. Clarify the process of applying the design case mentioned in part 1: State the steps, clarify the
meaning of each step.
Mission 2
1. Clarify the student's activities that are meaningful to the practice of SI when
you explain the following practical situations
by using mathematical knowledge: Figure 4.1, is an image of the actual
representation of a device "Raising and lowering" in construction work that brings the
equipment up and carries the objects down. Please tell us what mathematical ideas are
used when lifting equipment and when lowering objects?
2. Use the design process to support the development of SI for the students, lease
give an example of designing such a process when teaching the concept: "The center of
symmetry of a shape" in lesson "Central symmetry" Geometry textbook 11, page 12.
- For the students. The experimental content for the students mainly present the situations
mentioned in Chapters 2, 3, 4 for the students to interact, experience, make judgments, detect
problems and visualize the steps to solve problems to acquire the knowledge. The students
interact with situations through the teacher's questionnaire and instructions.
Mission 1

Figure 5.1 - The representation of a football field and Figure 5.2 is a mathematical
model of a football field, rectangular in shape.
A

E

D

I

B

Figure 5.1

F

Figure 5.2

C


21
On the mathematical model of the football field ABCD, there is a rectangle with
the mean line EF. I is the intersection of EF and the diagonal AC. Please clarify the
following questions:
- Prove that I is the center of rectangle ABCD. Let M be certain point on the side
of the rectangle. Prove through center symmetry I, the image of point M is point M' also
on the side of the rectangle above?.
- The same argument creates an image of the point M' that M is also on the side of
the rectangle?.

- Prove that if point M lies inside a rectangle, then its image M' through central
symmetry I also belongs to the rectangle?. Let's argue that if image M' is inside the
rectangle, make its image through center symmetry I also inside the rectangle?.
Mission 2
Given a circle (C) with center O. Show that O is the center of symmetry of circle (C).
- From the two situations mentioned above, give the definition: Point I is the
center of symmetry of the figure H?.
- Give an example to show that if I is the center of symmetry of the figure H, then
I can belong to H or not to H?.
Mission 3
In the orthogonal coordinate system (Oxyz) given two lines d: and d': and the
point. Find the equation of the line that passes through and intersects the two lines d; d'.
Mission 4
Estimate the mass of a cylindrical steel pipe with a length of 6 m, a cross-sectional
diameter measured on the outside surface of 6 cm, and a cross-sectional diameter
measured on the inside surface of 5.4 cm. The density of steel is 7,850 kg/ .
Mission 5
Are there four double straight lines that cross each other and are perpendicular to each other?
We assign cognitive tasks to 04 groups of the students, each group of 08 students in 02
high schools in Nghe An province are: Le Loi High School; Dien Chau High School 5.
5.3. Experimental form
We conduct practical research by the way of case study, by testing one or a group
of subjects on the perception and expression of SI in teaching geometry in high schools.
During the experiment, we recorded and recorded the exchanges and discussions of the
teachers and the students.
5.4. Experimental organization
The experiment was conducted by the researcher himself with 02 math team
leaders of two high schools for teachers: Le Loi High School, Dien Chau 5 High School,
Nghe An province. When we conducted the experiments for the students, we agreed to
ask for the help of the math group teachers: Mr. Nguyen Ngoc Hoang organized the

group experiment of class 11A1 (08 students); Mr. Bui Van Duc, organized the group
experiment of class 12A3 (08 students) of Le Loi High School; Mr. Thai Doan An,
organized the group experiment of class 11A1 (08 students); Ms. Vo Thi Bich Ha,
organized the group experiment of class 12A1 (08 students) at Dien Chau 5 high school.
The teachers conducting the experiment have studied the following contents:
- The characteristics of the SI in teaching geometry in high schools.
- Activities that need practice to develop SI.
- The process of designing situations to organize teaching in order to develop SI


22
for the students; the process of applying the situations that have been designed in
teaching in order to develop the SI for the students.
5.5. Evaluation of experimental results
5.5.1. A priori analysis
The a priori analysis essentially predicts the results of answers, explanations, and
performance results of the teachers and the students through assigned tasks in the
experiment. 5.3. Experimental form
We conduct practical research by the way of case study, by testing one or a group
of subjects on the perception and expression of SI in teaching geometry in high schools.
During the experiment, we recorded and recorded the exchanges and discussions of the
teachers and the students.
5.4. Experimental organization
The experiment was conducted by the researcher himself with 02 math team
leaders of two high schools for teachers: Le Loi High School, Dien Chau 5 High School,
Nghe An province. When we conducted the experiments for the students, we agreed to
ask for the help of the math group teachers: Mr. Nguyen Ngoc Hoang organized the
group experiment of class 11A1 (08 students); Mr. Bui Van Duc, organized the group
experiment of class 12A3 (08 students) of Le Loi High School; Mr. Thai Doan An,
organized the group experiment of class 11A1 (08 students); Ms. Vo Thi Bich Ha,

organized the group experiment of class 12A1 (08 students) at Dien Chau 5 high school.
The teachers conducting the experiment have studied the following contents:
- The characteristics of the SI in teaching geometry in high schools.
- Activities that need practice to develop SI.
- The process of designing situations to organize teaching in order to develop SI
for the students; the process of applying the situations that have been designed in
teaching in order to develop the SI for the students.
5.5. Evaluation of experimental results
5.5.1. A priori analysis
The a priori analysis essentially predicts the results of answers, explanations, and
performance results of the teachers and the students through assigned tasks in the experiment.
5.5.2. Post-analysis of the results of solving tasks for the teachers and the students
5.5.2.1. Analysis of experimental results for the teachers
During the experiment, we observed the spirit and attitude of the teachers when
they answered the missions. The teachers also expressed their interest in answering
questions. However, their understanding about the SI is incomplete; especially they do
not fully grasp the activities towards the development of SI. Since then, solving the tasks
set for the teachers also reveals some weaknesses due to the lack of knowledge and
skills preparation for designing situations and applying them.
5.5.2.2. Analyze and evaluate the students' experimental results
In the experimental test, we paid the special attention to the practical situations
containing activities towards the development of the SI: The activity helps the students
imagine and locate the points that are the images of the points lying on the edge of the
football ground; The image of the points lying on the football ground through the
symmetry of the center is the tee of the ground (On the mathematical model, the center
of symmetry is the intersection of two diagonals of the rectangle).
In the tasks assigned to test experimentally, the problem solving in those tasks is


23

elucidated through the results of SI. For example, the students need to imagine to calculate the
volume of a rotating cylindrical steel pipe, as the basis for estimating its mass.
Conclusion of chapter 5
- Assessing teachers' understanding (Through case studies for two groups of
teachers, each group of 10 teachers from Le Loi High School, Dien Chau 5 High School,
Nghe An province) on the process of designing situations towards Formation and
development of SI for high school students. This process consists of 6 steps with the
requirement of teachers to explain and clarify the content of the steps, to clarify the
activities installed in the situations that are meaningful to the development of the SI in
teaching geometry. It requires to test teachers on their awareness of the steps of the
application process. In which, it specially appreciates the role of steps for the
development of SI for the students.
- The experimental test for the teachers is focused on the tasks of concretizing the
steps of the design process and the application process in teaching concepts, theorems,
solving geometry exercises in the direction of supporting development. SI for the
students in teaching geometry in high schools.
- In the experimental test for the students, we are interested in installing the
following activities into situations designed by the teachers and the situations where the
process is applied to teaching concepts, theorems, and math exercises.
- Some assigned tasks require a high level of SI. To solve those tasks requires an
imagination to associate with logical thinking, with descriptive intuition, modeling
activities taken from real situations.
- The empirical assessment is mainly through qualitative analysis, observing
students' behavior, listening to students' discussions, recording, and communicating
activities to see the expression of students' efforts, interest in doing assignments and
Refer to the results through the quality of the answers, the quality of solving specific
problems. Conclusion of chapter 5
- Assessing teachers' understanding (Through case studies for two groups of
teachers, each group of 10 teachers from Le Loi High School, Dien Chau 5 High School,
Nghe An province) on the process of designing situations towards Formation and

development of SI for high school students. This process consists of 6 steps with the
requirement of teachers to explain and clarify the content of the steps, to clarify the
activities installed in the situations that are meaningful to the development of the SI in
teaching geometry. It requires to test teachers on their awareness of the steps of the
application process. In which, it specially appreciates the role of steps for the
development of SI for the students.
- The experimental test for the teachers is focused on the tasks of concretizing the
steps of the design process and the application process in teaching concepts, theorems,
solving geometry exercises in the direction of supporting development. SI for the
students in teaching geometry in high schools.
- In the experimental test for the students, we are interested in installing the
following activities into situations designed by the teachers and the situations where the
process is applied to teaching concepts, theorems, and math exercises.
- Some assigned tasks require a high level of SI. To solve those tasks requires an
imagination to associate with logical thinking, with descriptive intuition, modeling
activities taken from real situations.