Competencies Needed to Be Formed and Developed Through Teaching Mathematics in High School

• NL uses tools and means to learn math.

- From the OECD's perspective, the above mentioned NLs can also be divided into 3 groups:

+ Tool group , including:

• NL uses Information and Communication Technology (ICT);

• Communication skills;

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• Calculation capacity.

+ NL group masters and develops themselves , including:

• Self-study ability;

• NLGQVĐ and creativity;

• Aesthetic skills;

• Physical ability.

+ Social relations group , including:

• Communication skills;

• Collaborative skills.

However, depending on the characteristics of educational fields and subjects, there is also a system of separate competencies needed for learning the subject and developing through the process of learning the subject, including Mathematics.

1.2.2. Competencies that need to be formed and developed through teaching Mathematics in high schools

The formation and development of competencies for students is carried out through many different fields and activities at school, with the main activity being teaching and learning; in which primary education plays a very meaningful role. This is a process of close interaction between students and teachers through teaching and learning mathematics in a reasonable manner. From concepts, methods to specific teaching techniques of teachers, all aim to achieve the ultimate goal of developing competencies for students. In addition, through active, proactive and creative learning activities, students also contribute to the development of competencies for themselves. That process is carried out throughout a long learning period from accessing and acquiring knowledge, forming and developing skills and over time on that basis, the competencies of students are prescribed to be achieved at an appropriate level in the teaching goal. In particular, because TH always takes up a large amount of time in the TH education program in most countries in the world with content selected because of the role and benefits of TH for

TT will have the basis to create many opportunities to contribute to the development of general and core competencies for each person. However, mathematics teaching, in addition to the requirement to contribute to the development of general competencies as mentioned above, also has the task of developing competencies specific to the subject, understood in the sense that those competencies are necessary to learn mathematics and are developed through learning mathematics as mentioned in the above section. There are many ways to list competencies formed and developed through learning mathematics from different perspectives.

According to Tran Kieu [20], the necessary competencies developed through Mathematics include:

- Thinking capacity : Including general thinking operations such as analysis and synthesis, comparison, abstraction, generalization , ...; paying special attention to specific TH thinking factors such as logical thinking capacity in deduction and argumentation, while at the same time attaching importance to critical thinking, creativity as well as factors of prediction, exploration, TH intuition, and spatial imagination .

- NLGQVĐ : This is one of the competencies that Mathematics has many advantages to develop for learners through receiving concepts, proving TH propositions and especially through solving math problems.

- Ability to model learning : From hypothetical learning situations or real life situations to convert into learning models and then use learning methods to work with the models to find solutions. This is an ability that needs to be given more attention in general schools in our country.

- Communication skills (through speaking or writing): Related to the effective use of TH language (letters, symbols, charts, graphs, logical connections,...) combined with common language. This ability is demonstrated through understanding TH texts, asking questions, answering questions, reasoning when proving the correctness of propositions, solving math problems,...

- Ability to use tools and means of learning math : Including common means, especially means closely associated with the use of information technology.

- Ability to self-study with appropriate methods and to cooperate effectively with others in the process of learning math.

Regarding this type of competence, there may be differences in its definition in countries around the world, but studying the programs of some countries or the concepts of some organizations (NAEP, NAPLAN, OECD, etc.), the author of the thesis believes that the National Curriculum and the TH model are competencies with high consensus among countries.

1.2.3. Problem solving ability

1.2.3.1. Problem

According to Nguyen Ba Kim (2011) [21, p.185]: "A problem is called an issue if the subject does not yet have an algorithm that can be applied to solve that problem". Author Le Ngoc Son [35, p.26] explains: "A problem is a problem, a question or a requirement that requires action to solve, requiring an individual or a group to come up with a solution, an answer, actions to take, without knowing which path leads to the result".

A problem (in the field of study) is represented by a proposition and question or a system of propositions, questions (or action requirements) that satisfy the condition: Up to the present time, students do not have enough knowledge or do not have a solution to answer the question (in other words, they have not learned any algorithmic rules to answer the question or carry out the given requirements).

In the above sense, a problem is not completely synonymous with a math problem. There are math problems that are not problems if they only require students to directly apply an algorithm or a formula, or students can immediately see the solution without thinking (equation solving problems only require using the solution steps or applying learned formulas to calculate area and volume when the measurements of the related factors are fully known).

It is worth noting that the problem is relative, it may be a problem for one person but not for another. The geometric proof problem "Given a pyramid

S. ABC has

SA AC , SA AB .

Proving that SA  ( ABC )” will not be a problem.

If students have learned the sign to recognize a line perpendicular to a plane, it means it is perpendicular to two intersecting lines on that plane. But it will be a problem if students have not learned that sign or learned it without understanding the theorem.

1.2.3.2. Problem solving ability

Problem solving in the usual sense is to find appropriate solutions to solve difficulties and obstacles. For a specific problem, there may be a number of solutions, including the optimal solution. Branford JD (1984) [50, p.105], when discussing the ideal problem solver , pointed out 5 components of the problem solving process:

1) Identify the problem;

2) Thoroughly understand the difficulties;

3) Offer a solution;

4) Implement the solution;

5) Evaluate the effectiveness of implementation.

Many authors have mentioned the steps of this process and they are basically the same (in terms of steps, purposes and meanings). If we look at the above components from the perspective of competence, we can also see these as the component competencies of problem-solving competence. Therefore, the author of the thesis agrees with the process mentioned above, from which problem-solving competence is described according to the above signs and problem-solving competence of students in learning mathematics will be revealed through activities in the problem-solving process.

Over the past decade, many educators have also called for authentic assessment, based on the notion that knowledge and skills must be placed in real-world contexts. And from there, a trend has emerged to link problem-solving with real-world situations . Another trend in 21st-century teaching is to focus on training students in higher-order thinking skills, including applied thinking, to meet future challenges. Teaching problem-solving can develop application skills, develop higher-order thinking, and prepare students to face and overcome new challenges in the future effectively.

For decades now, problem-solving ability has occupied a leading position in teaching activities in many countries around the world, including most of the ASEAN countries. Because of its significance and importance, the National Council of Teachers of Mathematics [52] of the United States has emphasized that "problem-solving ability must be the focus of school learning". More recently, problem-solving ability has been emphasized once again in the statement on "Vision for School Learning" of the National Council of Teachers of Mathematics of the United States [62], according to which students will become flexible and resourceful in problem-solving ability, problem-solving ability is considered both a goal of school education and a tool for learning Mathematics. In addition, although problem-solving ability is formed and developed through many subjects, many fields and many different educational activities, it can be seen that Mathematics has an important role and many advantages in developing this ability for high school students. A typical representative of this viewpoint is G. Polya - a famous TH researcher and educator from the last century who was interested in GQVĐ and some of his research results have been used to this day in his massive works, such as the TH Creation series [32]. Although G. Polya's diagram was proposed a long time ago, its relevance remains. After all, until now

There has not been any work that has provided more complete suggestions at the general level than G. Polya's Problem Solving Scheme (and can also be considered as a Problem Solving Scheme, although G. Polya did not use the term Problem Solving Scheme). Problem Solving Scheme activities in teaching mathematics today can still rely on G. Polya's Problem Solving Scheme to organize Problem Solving Scheme teaching. G. Polya's scheme includes the following steps:

Step 1: Understand the problem

- What is the unknown? What is the data? What is the condition? Can the problem condition be satisfied? Are the conditions sufficient to determine the unknown? Or is it redundant, or is it missing? Or is there a contradiction?

- Draw pictures.

- Using appropriate symbols, can conditions and data be represented as formulas? Clearly distinguish the parts of the condition.

Step 2: Find the solution to the problem

- Have you come across a problem similar to this? Or in a slightly different form?

- Do you know a theorem or a problem related to this problem?

- Look carefully at the unknown and try to remember if there is any problem with the same unknown?

- This is a problem that you have solved before, what can you apply to it?

Method? Result? Or do additional factors need to be added to make it applicable?

- Carefully consider the concepts in the problem and, if necessary, return to the definitions.

meaning

- If you can't solve this problem, try solving an easier sub-problem.

related, a particular case, similar, more general?

- Retain a part of the hypothesis then to what extent is the unknown determined? From that you can draw out what is useful for solving the problem? With what hypothesis can you solve this problem?

- Have you used all the assumptions of the problem?

Step 3: Solve the problem

Carry out the solution you have proposed. Do you think the steps are correct? You can prove it is correct.

Step 4: Exploit the problem

- Can you think of another way to solve the problem? Is the solution shorter or more interesting?

- Have you applied that solution to any problem?

- Can you apply this problem to solve other known problems?

In recent times, one of the international school quality assessment programs, PISA, has paid great attention to students' problem-solving ability. In 2003, PISA introduced an assessment framework for problem-solving ability, mainly through Mathematics and Science [74]. In 2012, it added the "interactive problem solving" (IPS) part, which is conducted on computers for all areas of knowledge that are not related to a specific subject.

In our country, problem solving in Mathematics was first approached from G. Polya's problem solving model in the last century. Over time, from considering problem solving as a method or a teaching style, it has gradually shifted to considering it as both a goal, a learning content, a thinking method and now considered as the learner's ability. It can be said that regardless of the form - teaching content, teaching methods, learning methods, thinking skills or ability - problem solving has become the focus of Vietnamese general education and is aiming at developing problem solving ability as one of the main goals. In addition to studies on teaching to develop problem solving ability, there is a new research direction related to problem solving which is assessing ability. Therefore, from the assessment perspective, there are also new concepts about problem solving ability. For example, the concept stated in the work of Nguyen Thi Lan Phuong [30]: NLGQVĐ is the ability of an individual to effectively use cognitive, motivational and emotional processes to solve problem situations where conventional solutions cannot be immediately resolved.

Also according to [30], NLGQVĐ includes 4 elements. Each element includes a number of individual behaviors when working independently or when working in groups during the GQVĐ process. The four elements are:

+ Problem recognition and understanding : Recognize problem situations; identify and interpret information; share problem understanding with others.

+ Establishing problem space : Collecting, organizing and evaluating the reliability of information; connecting information with learned knowledge (field/subject/topic); determining methods, processes, and strategies for solving problems; agreeing on actions to establish problem space.

+ Plan and implement the solution : Establish the implementation process for the selected solution (collect data, discuss, seek opinions, resolve objectives, review the solution,... and the time to resolve each objective); allocate and determine how to use

using resources (resources, human resources, funding, means, etc.); implementing and presenting solutions to problems; organizing and maintaining group activities when implementing solutions (adjusting and monitoring to suit the problem space when there are changes).

+ Evaluate and reflect on solutions : Evaluate implemented solutions; reflect on the value of solutions; confirm knowledge and generalize to similar problems; evaluate individual role in group activities.

From there, this author also provides levels of NLGQVĐ development to outline the cognitive development path or the capacity development path that students need to achieve. Through that, it creates conditions for teachers to orient teaching so that students can achieve the levels.

1.2.4. Ability to solve practical problems

Based on the concept of problem situations , issues, and NLGQVĐ in high school Mathematics, we believe that: In Mathematics, a problem for high school students is a problem arising from a problem situation, posing a "problem situation" that needs to be answered and solved, requiring students to mobilize knowledge and skills to solve it.

According to Bui Huy Ngoc [28, pp.25-26], the process of applying TH to TT in general must be carried out in 4 steps shown in the following diagram:

TH Model

(ii)

TT situation

TT problem

(iii)

(i)

(iv)

Results of solving math problems

Diagram 1.1. The process of applying TH to TT

In which: (i) Building a practical problem: From a practical situation, build a practical problem that can be solved with a TH tool; (ii) Realizing a practical situation: Converting from a practical problem to a TH problem, identifying necessary TH information, recognizing TH concepts, providing related TH structures, representations, and characteristics to bring the constructed practical problem to a specific TH model; (iii) Solving problems: selecting and using appropriate TH methods and tools to solve a problem that has been established in the form of a TH model. The final product in this step is a TH result; (iv) Converting from the result in the TH model to the solution of a practical problem: considering the TH result in the context of the situation

the original real-world situation, adjust the results accordingly, and make the results meaningful.

The above problem can also be approached from the perspective of PISA, the concept of mathmatization is described as the basic process in which students use mathematical knowledge and skills accumulated from school along with life experience to solve mathematical problems. This mathmatization process includes 5 steps, shown in the following diagram [72]:

Real world

World of Mathematics

Practical solution

Solution TH

Practical problem

TH problem

1, 2, 3

Diagram 1.2. The process of TH according to PISA

Step 1: Start with a real-world problem;

Step 2: Identify mathematical knowledge relevant to the problem, reorganize the problem according to mathematical concepts;

Step 3: Continuously select practical elements to transform the problem into a problem that represents the situation;

Step 4: Solve the problem;