Activating Learners When Teaching Typical Situations

s' .

time . point t 0

t 0 + . ∆ t s

OS 0

M 0 ∆ s M

At time t 0 + t, material point M has abscissa s 0 + s = f(t 0 + t)

So in the time interval t, the material point M has traveled a distance of:

s = f(t 0 + t) f(t 0 ). If the material point M moves with uniform velocity v (v is a constant)

number) then ratio

 s v . If the motion of M is not uniform, then the ratio  s

 t  t

is called velocity

average v m of motion over time interval t from time t 0 to time t

t 0 + t: v m

 s

 t

f  t 0 t  f  t 0

 t

If The smaller t is, the more accurately the average velocity describes the fast or slow nature of the movement during that time period.

From this, of course, comes the problem of finding the limit of average velocity.

when t 0. If that limit exists, then that limit is called the instantaneous velocity of the motion at time t 0 , denoted by v(t 0 ).

Therefore:

v  t

 lim  s lim

f  t 0 t  f  t 0

0  t  0  t  t  0  t

+ Derivative at a point: Given the function y = f(x) defined in a neighborhood of the point x 0 . When the variable receives an increment x at point x 0 then the function has the corresponding increment:

y = f(x 0 + x)  f(x 0 ).

If there is a limit

lim  y , we say the function has a derivative at x 0 and that limit is

 x  0  x

derivative of a function at x 0 , denoted by f ' (x 0 ) or

y ' .

So, we have the definition of derivative:

f '  x

 lim  y lim

f  x 0 x  f  x 0

0  x  0  x  x  0  x

* Comment : + The concept of derivative is one of the concepts born from practice, originating from the practical exercise of finding the instantaneous velocity of a point moving in a straight line, that is, originating from " objective " reality to form for students the " objective " nature of derivatives.

+ From discovering a form of movement in "reality", Mathematics has generalized and "abstracted", thereby giving a purely mathematical definition: The concept of derivative. From this definition, the concept of derivative is effectively applied in practice such as: Tangent to a flat curve, instantaneous intensity of electric current at time t. This helps students perceive the rule: " From vivid intuition to abstract thought, from abstract thinking to practice " as the rule of BC logic [85, p. 294] .

Example 3 : Consider the objective phenomenon "A straight line perpendicular to a plane" comprehensively . From the phenomenon of a flagpole buried vertically above the ground, Mathematics has generalized and abstracted it into the concept of a straight line perpendicular to a plane: " A straight line a is considered perpendicular to a plane (P) if it is perpendicular to every straight line in the plane (P)" and consider all forms of this "phenomenon" , giving signs to recognize it.

Mathematics constructs these signs using mathematical tools, through theorems about parallel and perpendicular relationships.

Consider the object "Line perpendicular to plane" in relation to:

1) Perpendicular line:

d a

  d  b



 a  b   0 

  a, b  

a 

2) Parallel lines:

a 



a // b

b 

3) Parallel MP:

a 



  // 

a 

4) Perpendicular Mp:

   



 a   



 a    a  

a 





  





   

Through examples, students can form a "comprehensive" view of TDBC.

Through the above example, we can see that: Learning math knowledge, especially solving exercises, has contributed greatly to the practice of basic math skills, that is, students gradually absorb the rules of basic math skills in a "not really self-conscious" way, but this is the foundation for forming basic math skills for students. This not only helps students learn math and other subjects while still in high school, but also creates conditions for them to practice in a basic way to apply in their working life, whether manual or mental, later on.

However, training and developing logical reasoning through teaching mathematics is not about directly and explicitly presenting the properties (rules) and activities of logical reasoning. This is the task of the subject of materialistic philosophy of logic. Within the scope of teaching mathematics, it is necessary to meet 2 requirements:

- Through teaching mathematics, forming theoretical knowledge contributes to building a foundation for students to understand the laws and activities of theoretical materialism and thereby improve the quality of teaching mathematics.

- To some extent, when students understand these rules (implicitly), they will apply them when learning math and when solving real-life situations.

Example 4 : The " contradiction and unity " of TDBC is shown:

- When considering the formation and development of number sets in Mathematical Physics .

The development of number sets is not due to the subjective reasoning of Mathematicians but due to practical needs in life or practical needs within Mathematics itself.

+ Set of natural numbers (Math 6): N =  0; 1; 2; 3; ...  Set N of natural numbers has a contradiction:

* In real life: Not yet reflecting the real phenomena of the objective world such as: profit and loss , going forward and going backward , hot and cold temperatures ...

* In Mathematics: Subtraction is not always possible: 5 – 3 = 2; 3  5 = ?

+ Set Z of integers ( Algebra 7 ): Z =  ...;  3;  2;  1; 0; 1; 2; 3; ... 

The extension from N  Z or the set Z of integers was born to solve the contradictions of the set N of natural numbers.

However, in the set Z of integers new contradictions appear (one contradiction disappears, another one forms):

* In real life: It does not reflect the real phenomena of the objective world such as: due to floods, we have to redistribute fields, land or divide the number of fish caught, divide the number of prey hunted, divide gifts for children... there are divisions that are not integers.

* Within Mathematics:

Division is not always possible: 8 : (  4) =  2; (  7) : 3 = ?

+ Set Q of rational numbers (Algebra 7): Q =  m/n: m, n  Z, n > 0 

The extension from Z  Q or the set Q of rational numbers was created to resolve the contradictions of the set Z of integers.

However, in the set Q of rational numbers, new contradictions appear:

* In real life: Not meeting the needs of measurement or calculation, there exist line segments whose length is not a rational number. For example, measuring the length of the diagonal of a square with side equal to 1 or measuring the circumference C of a circle, d and C are not rational numbers.

* Within Mathematics: The square root of a non-negative number is not always valid.

shown: 4 / 9 = 2/3  Q but

+ Set of real numbers (Algebra 9).

 Q.

The extension from Q to R or the set R of real numbers was born to resolve the contradictions of the set Q of rational numbers.

1

However, the set R of real numbers presents a new contradiction:

* The square root operation is not always possible:

= 2;

 R but = ?

Comment : Through the example, students can form some viewpoints of BC logic:

* Students can perceive the "historical" law of TDBC: When examining things, one must perceive things in their development, in their self-movement.

* Students can feel the "negation of negation" rule of materialistic mathematics. New sets of numbers that replace old sets of numbers are the "result" of "contradictions" within the old sets of numbers being "resolved" . Therefore, new sets of numbers that are born are "objective" , an inevitable factor of development.

New sets of numbers are born on the basis of "inheriting" old sets of numbers, showing the property

"inheritance" of BC negation.

* Students can feel: "Contradiction is the origin and internal driving force of development. When one contradiction disappears, another contradiction forms" is a point of view of BC logic.

- Consider: two objects Triangle and right triangle

With: "Pythagorean Theorem" then triangle and right triangle are contradictory (theorem does not hold when triangle is any triangle).

With: "Cosine theorem" then triangle and right triangle are unified (theorem is always true for any triangle).

Example 5 : The " comprehensive " nature of TDBC is also clearly demonstrated when encouraging students to find many different solutions for an exercise, by looking at the exercise from different aspects in relation to each other.

Specifically, given the BT: In the coordinate plane Oxy, triangle ABC has vertices A(  4 ; 1); B(2 ; 4); C(2 ;  2). Prove that: the centroid G, the orthocenter H and the center I of the circle circumscribing triangle ABC are collinear [126, p. 52] .

To prove that G, H, I are collinear, if the teacher guides and encourages students to see the problem as containing many different aspects, then this problem will have many different solutions:

+ Method 1. Using vectors: To prove that G, H, I are collinear, we will prove: GH kgI

+ Method 2. Using coordinate method,

Calculate the coordinates of points G, H, I then check the

GH and GI vectors are in the same direction.

+ Method 3. Prove two

Equal angles at opposite positions:

IGM = HGB

(-4 ; 1)A

-4

y 4

IG 1 HO

B(2 ; 4)

2 x

by proving:  IGM  HGB

+ Method 4. Using Thales theorem…

1.2. Thinking activities in teaching Math

1.2.1. Concept of operation

-2 C(2 ; -2)

Figure 1.17

Activity as a philosophical concept has existed for a long time. But it has only become a psychological concept since the beginning of the 20th century [42, p. 39] .

There are many different ways to define Activity, we will present two commonly used ways: Way 1: Activity is the expenditure of human nervous and muscular energy to impact objective reality, in order to satisfy one's needs .

Method 2: Activity is the way of human existence in the world. Activity is the interactive relationship between human and the world (object) to create products for both the world and human (subject) [144, p. 44] .

- An activity has the following characteristics [144, p. 44] : An activity is always an activity with an object; an activity always has a subject; an activity always has a purpose; an activity operates according to the indirect principle.

- Types of contracts [144, p. 46] :

First classification: From an individual perspective, there are 4 types of activities (play, study , work, social).

Second classification: In terms of products, there are 2 major types of activities ( practical, theoretical ). Another classification: There are 4 types (transformation, cognition , value orientation, exchange).

Structure of the contract:

Subject

Object

Specific contract

Engine

Act

Purpose

Operation

Product

Vehicle

Diagram 1.4 : Activity Flow

According to Leonchiev, the general structure of the HĐ line can be summarized as above [cited in 144, p. 48] .

- Human psychology is a product of activity and communication [144, p. 51] :

The formation and development of human psychology is summarized in Diagram 1.5 :

Communicate

Human (Psychology - personality)

Active Subject - Communication

Communication object


Active Object

Maybe you are interested!

Work

Diagram 1.5 : The formation and development of human psychology

1.2.2. Operational perspective in teaching Mathematics

In teaching Mathematics, based on Diagrams 1.4 and 1.5, we see that teaching cannot be separated from activities, or in other words, to carry out the process of teaching Mathematics, teachers and students must actively participate in activities.

Nguyen Ba Kim believes that: Each teaching content is closely related to certain activities. These are activities that have been carried out in the process of forming and applying that content. Discovering potential activities in a content is to outline a path to convey that content and carry out other teaching purposes, and at the same time

specify the purpose of teaching that content and indicate how to check the implementation of these purposes. The basic thing of teaching method is to exploit the potential activities in the content to achieve teaching purposes [74, p. 65] .

- According to Pham Gia Duc: Learning activity is an organized activity ... Learning activity is the process of working to create educational products... Teaching activity is the process of organizing student activities [42, p. 39] .

- Math activities: In high school, teaching math to students means teaching math activities , which are basically solving math problems [42, p. 40] .

- Geometry activity: Each Geometry activity is a situation that stimulates learning motivation.

A Geometry activity usually consists of many component activities with separate purposes. Once the component activities are completed, the overall purpose of the entire activity is also achieved [42, p. 40] .

1.2.3. Innovation in teaching methods of Math

According to Nguyen Ba Kim: The current orientation for innovation in teaching methods is to organize learners to learn in activities and through activities that are self-motivated, active and creative. This orientation can be called learning in activities and through activities [74, p. 40] . In innovation in teaching methods of Geometry, we pay more attention to activities that approach new knowledge [42, p. 40] .

1.2.4. Activating learners when teaching typical situations

According to Vu Duong Thuy, in the process of teaching Mathematics in high school, teaching concepts and definitions, theorems and proofs, and solving mathematical exercises are repeated many times, we call these typical situations in teaching Mathematics [40, p. 134] .

a. Activating learners when teaching Geometry concepts

- Teaching activities for HH concepts : Including 5 basic activities ( introducing concepts - helping students approach concepts; forming concepts - helping students acquire concepts; reinforcing concepts - through 2 component activities: identifying and expressing concepts; initially applying concepts in simple exercises; applying concepts in comprehensive exercises ).

Recognizing a concept is to detect whether a given object satisfies that definition [74, p. 20] .

To express a concept is to create an object that satisfies that definition.

- Train and develop students' basic knowledge through teaching activities on HH concepts.