Content of Geometry Teaching in High School

- Documents written about practicing TDBC for students are very rare and hard to find. Moreover, if there are any, they do not provide specific implementation methods suitable for high school math teaching.

- Teachers are afraid that the amount of knowledge in a lesson is too much, training students in basic vocabulary takes a lot of time.

- In general, students' level is still weak, they do not think much but only focus on listening, taking notes and remembering. This is the consequence of the current situation of teaching methods not being renewed.

1.5. Conclusion of chapter 1

Maybe you are interested!

In chapter 1, we present the theoretical basis of TD, Mathematical TD, types of TD, especially BC TD.

The thesis presents the basic characteristics of TDBC. Compares TD types (logical TD for TDBC, functional TD for TDBC, ...).

Section 1.2 presents the student's activity to practice and develop basic knowledge for students through teaching Chemistry in high school, demonstrated through teaching typical situations ( concepts, theorems, exercises ).

In this chapter, the topic raises the necessity of training and development for students in teaching Mathematics in general and teaching Geometry in high school in particular.

The thesis presents arguments through exercises affirming that the subject of Chemistry in high school has many advantages in training and developing basic knowledge for students.

Chapter 2

SOME MEASURES TO CONTRIBUTE TO TRAINING AND DEVELOPING DIALECTICAL THINKING FOR STUDENTS THROUGH TEACHING GEOMETRY IN HIGH SCHOOL

2.1. Geometry in high school

2.1.1. Objectives of teaching Geometry in high school

- Mathematics in general, and Mathematics in particular, play an important role in achieving the general goal of general education: helping students acquire basic and practical knowledge, skills, and methods of mathematics; contributing significantly to the development of intellectual capacity, forming the ability to reason necessary for life. At the same time, it contributes to the formation and development of qualities, scientific working styles, knowing how to cooperate in work, having the will and habit of regular self-study, creating a foundation for students to be able to study at a higher level according to the orientation of each department ( natural sciences, social sciences ). Specifically, Mathematics helps students acquire systematic and basic knowledge of plane and geometric figures (combining synthetic methods, vector methods and coordinate methods), developing mathematical imagination, the ability to apply learned knowledge to solve simple exercises and some practical exercises, as well as the ability to reason logically and reasonably in specific situations, the ability to receive and express problems accurately. In addition, it also cultivates the virtues of curiosity, love of science, especially mathematics, seriousness in work, dynamism and creativity, diligence and overcoming difficulties...

- Mathematics aims to meet the subject's objectives: thoroughly grasp the spirit of comprehensive technical education (pay attention to related subjects: Physics, Biology, ...), focus on basic knowledge, skills and thinking methods specific to HH in accordance with the orientation of each department, present traditional Geometry knowledge in the light of modern Mathematical ideas. Enhance practicality and pedagogy, reduce academicity (but the content is still not reduced) which are the requirements for the rigor of theory, focus on practice and practice, to help students improve their spatial imagination and form aesthetic emotions, the ability to express ideas through learning HH.

2.1.2. Content of Geometry teaching in high school

Content of knowledge of geometry in primary school : Basic, simple knowledge of geometry in planes ( point, line segment, angle, broken line, ray, line, circle, triangle, quadrilateral - rectangle, square, trapezoid, cylinder, rhombus ) and geometry in KG ( cube, rectangular prism, cylinder, sphere ).

Contents of the subject of geometry in secondary school : Line segments, angles ( Grade 6 ); Perpendicular lines, parallel lines, triangles, relationships between elements of a triangle, concurrent lines in a triangle ( Grade 7 ); Quadrilaterals, polygons, areas of polygons, similar triangles, upright prisms, regular pyramids ( Grade 8 ); Proportional relations in right triangles, circles, angles with circles, cylinders, cones, spheres ( Grade 9 ). All are taught in 239 periods (29-70-70-70).

Content of the subject of HH in high school : Vectors, scalar product of two vectors, coordinate methods in geometry (grade 10); Transformations and similarity in geometry, straight lines and planes in geometry, parallel relations, vectors in geometry, perpendicular relations in geometry (grade 11); Transformations and similarity in geometry, polyhedron, sphere, cylinder, cone, coordinate methods in geometry (grade 12). All are taught in 150 periods for Natural Sciences (50 – 50 – 50), 134 periods for Social Sciences and Humanities (43 – 45 – 46),

2.1.3. Teaching methods of Geometry in high school

- Teaching methods must promote students' positivity and initiative in learning, especially their self-study ability. Take advantage of the advantages of each teaching method, focusing on using teaching methods to discover and solve problems. Pay attention to both providing knowledge, training skills and applying knowledge into practice. To achieve this requirement, we use KG models and encourage the use of teaching software. In addition, we must use assessment ( essay, multiple choice ). Tests and assessments must be appropriate to the level of program requirements and pay attention to students' creativity.

- The general viewpoint is " organizing for students to learn in activities and through activities that are self-conscious, active, proactive and creative" [5, p. 2], [74, p. 13]. Therefore, the directions for innovation in teaching methods are: it is necessary to change the habit of writing teaching objectives for teachers by writing learning objectives for students, teachers must plan learning activities (motivational situations)

of students in each lesson, as well as how to organize students' learning activities, so it is necessary to prepare questions ( can also use study sheets ). Teachers need to exploit positive factors in traditional teaching methods ( presentation, visualization, practice, ... ) and also need to update new teaching methods ( teaching to discover and solve problems, teaching cooperation in groups, ... ). With the current program and textbooks, the teacher's testing and evaluation methods need to be calculated right from the time of determining learning objectives and designing lessons, must be evaluated comprehensively ( knowledge, skills, methods, ... ) can also let students evaluate each other, teachers can apply multiple choice tests in evaluation.

For example : When teaching the cosine theorem [126, p. 40] with the presentation as in the textbook and teacher's book, it meets the current orientation of innovation in teaching methods.

2.1.4. Characteristics of Geometry textbooks in high schools

a. General characteristics of Math textbooks

- Eliminate knowledge that is not really basic.

- Reduce academic and scholarly elements, even if it means "sacrificing"

somewhat accurate KH.

- Promote pedagogical factors to unify symbols and terminology used.

For example : Previously, the terms sin  , cos  ... were used for both  measured in degrees or radians, now there is a distinction:

The trigonometric function of an angle or an arc is replaced by "Trigonometric value of an angle or an arc" . Therefore, in HH grade 10, it is called the cosine theorem, sine theorem in triangle and not called "Cosine function theorem, sine" like in the old book. The term trigonometric function y = sinx, ... is only used when x is a real variable.

- Textbooks meet the requirements of innovation in teaching methods, testing and evaluation (enhancing objective multiple choice assessment).

- Compared with the old program, the current program has a lot of new knowledge, many recognized properties and theorems.

b. Characteristics of Geometry textbooks in high schools

The HH textbook is more beautiful in form, larger in size, and has more visual images. The book has a question section [ ? ] to help students remember certain knowledge, or to suggest, or to

orienting thoughts... questions do not present answers (already in the textbook). The book provides activities that require students to work and calculate to reach a certain result (for proofs or calculations that are not too difficult, a few steps of student activities can replace the teacher's lecture). Thus, questions and activities aim to help students not be passive when listening to lectures, but to brainstorm and do activities at different levels to be able to answer questions or to carry out the requirements set forth by the activities.

Textbooks are a document used for both teachers and students . Textbooks reduce the theoretical part (reduce the proofs of properties or theorems) because if presented, it would be too difficult, for example: the properties of multiplying a number by a vector, but here the textbook has presented some specific cases to describe.

The textbook has been linked to reality a lot. The book has biographies of mathematicians related to that content right at the beginning of each chapter. The sections "Did you know? " and " Maybe you don't know?" are very interesting.

The textbook is equipped with a system of objective multiple choice questions, which meets the need for innovation in testing and evaluating students' learning outcomes.

The textbook provides knowledge about solving math problems using a calculator, which is an effective tool to help students calculate more easily.

The current textbook has a difference in content compared to the 2000 textbook in that: it brings the transformations in planes (central symmetry, axial symmetry, translation, rotation and similarity) to grade 11. Moves the part on geometry methods in planes (vectors, lines, circles and conic sections) of grade 12 to the end of the current grade 10 textbook.

The number of exercises is not too much, the exercises are arranged from easy to difficult, first are exercises that directly apply theory, then are exercises that relate old knowledge and difficult exercises. For example: Geometry textbook 10, Natural Science has: 49 vector exercises; 52 exercises on trigonometric relations in triangles and in circles; 70 exercises on coordinate methods in the plane; 10 year-end review lessons. Some theoretical problems because there is no more time to present are also included in the exercises.

* Some recommendations on current Math textbooks in general and HH textbooks in particular

- The current textbook policy is to present less academic, not perfectionist, simplify the program while still ensuring the amount of knowledge, and acknowledge some properties, theorems, ... to facilitate teachers in different regions, the author group should demonstrate in detail, meticulously and tightly in the textbook.

- The exercises should be more multiple choice.

- The system of symbols and the way of using words in textbooks should be unified, for example, the symbol M (x; y) should be unified without using M = (x; y) even though in some cases it is not clear, but only one symbol should be used. Therefore, if possible, after the table of contents of the 10th grade textbook, a table of symbols used in the entire grade should be printed.

- The content and program of the optional topics need to be presented more clearly and in detail. For example: when presenting the coordinates of a point, the textbook has completely omitted the formula for dividing a line segment by a given ratio k (including exercises), only stating the midpoint coordinates and the centroid coordinates, so this part should be included in the optional topic, or the introduction of coordinates is to introduce students to a method of solving math problems with many advantages. But when presenting the conic section with the focal point on the vertical axis, it should also be presented in detail in the topics because this time the textbook did not mention it, or the tangent problem... It is also necessary to clearly indicate to teachers that if they study the optional topic, what will the test and evaluation be like?

- To increase the visual appeal, if possible, the Education Publishing House should print in many colors drawings, pictures, ... with captions about those pictures.

2.2. Characteristics of Geometry program in high school

2.2.1. Geometry and space Geometry in high school

- HH space in high school textbooks

+ Space : Is a non-empty set that has been equipped with certain structures.

For example: Topological KG has topological structure, affine KG has affine structure...

+ Geometric KG : Is KG with geometric structures.

+ Direction of KG HH : A direction of KG is a set of coordinate systems such that the conversion of two coordinate systems in that set has a positive determinant. In the equation, KG HH is

choose direction according to physical principles (corkscrew rule; specifically: Clockwise is counterclockwise, counterclockwise is clockwise).

A KG is called oriented if it has chosen a fixed direction . In a KG M that has chosen a direction, then M is said to have a direction. When it is not possible to choose a direction on M, then M is said to be non-oriented.

In high school, when we say KG Oxyz, it means the KG is oriented; when we say KG for straight lines, tetrahedrons... it means the KG is not oriented.

- Geometry

+ Group acting on KG : Group G is called a group acting on KG M if there is a mapping  : G  M  M satisfying:

i). g 1 .g 2 (m) = g 1 (g 2 (m)) : g 1 , g 2  G; m  M. ii). e(m) = m : m  M; e is the identity element of G.

+ Invariant : A subset H has property  and g(H) also has property  then 

is called invariant with respect to G, for all g  G. For example:

* Affine invariant :

i). Through the group @ n (group of affine transformations)

ii). Important affine invariants: Proportionality, parallelism, straightness, concurrency, tangent, quadratic...

* Euclidean invariants : Angles, distances, ...

+ Geometry : The geometry of a group G is a subject that studies the invariants over an action group G.

Thus, Affine Mathematics is a subject that studies affine invariants over the group of affine transformations @ n ; Euclid Mathematics is also a subject that studies transformations D n .

+ HH in PT : In PT, invariants are expressed in relation to the shape and size of

The objects and the nature of the geometry in the Euclidean geometry are Euclidean geometry (2 dimensions: plane geometry or 3 dimensions: KG geometry), mainly studying important invariants such as: parallelism, collinearity, planarity, proportionality, angles, distances... These are the invariants through groups of displacements.

- Geometric Figure

+ Polyhedron : Is a set of non-empty points in the geometrical form M. For example: A point, a triangle, a tetrahedron, ... are polyhedrons.

+ Convex shape : A shape H is called convex if and only if, with two points A, B belonging to H, line segment AB also lies completely inside H.

In the high school textbook, the concept of convex polygons has been mentioned and defined as follows: a polygon is called a convex polygon when it lies on one side of a straight line passing through any edge, but this definition is very limited and cannot be applied to shapes with at least one edge that is not a straight line (eg: circle, ellipse ...) or shapes that have no limits in a plane (eg: an angle, half a plane ...), although these concepts have been mentioned in elementary school.

2.2.2. Characteristics related to the training of dialectical thinking

a. Change the survey object

For example : From the familiar angle HH measured in degrees 0 o    180 o , it leads to the trigonometric angle (Ox, Oy) which has countless angles with the same symbol, measured in radians with the observed object being a real number. It is quite confusing and "strange" for students to write: sinx = sin (x + k.2  ), k  Z.

b. Take a deeper look at the relationships between objects

Example: Problem type " consider the relative position of line (  ) and sphere (S) with center I, radius r in space ". When solving, we see 3 cases:

When d(I, (  ) ) > r  (  )  (S) =  ;

When d(I, (  ) ) = r (  )  (S) = M;

When d(I, (  ) ) < r (  )  (S) = {M, N}.

For the third case IH = d(I, (  ) ) < r (  )  (S) = {M, N}, we see that there will be 3 possibilities

potential in right triangle IHM, specifically:

+ Knowing MN (or HM) and r (or IM), find IH.

+ Knowing MN (or HM) and IH, find r (or IM).

+ Knowing r (or IM) and IH, find MN (or HM).

Figure 2.1

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