Since reform of mathematical education is currently underway in most countries of the world, the problem of setting goals in the school course of mathematics has become the main and very important in the development of teaching. The ability to solve problems is the most striking characteristic of the state of education. How do students and teachers understand this goal today in the school math course?
Student learning
Almost all students of the school think that when the right solution is found, and the received answer of the problem coincides with what is proposed in the textbook, their work is finished, you can forget about the problem.
The student or teacher does not take into account the fact that the role of each task is to develop orientation skills in problem situations, increase knowledge and experience. If you do not pay attention to the actualization of acquired knowledge, the process of mathematical thinking is disrupted, which helps to reduce the development of skills.
But before you deal with this issue, you need to find out what the task is and what its role in learning is.
What is a task?
This term has several interpretations. Consider one of them applied to mathematics. Here the task is problem situation (question), which requires a solution through the use of certain skills, knowledge, and thoughts. This is a goal that is within the framework of a problem situation that needs to be achieved, as well as a condition and a requirement.
Thus, to solve a problem means transforming a given problem situation or revealing that such reconstruction is impossible under these conditions. It is important to define the process of solving the problem as a mental activity aimed at achieving the goal.
Task format
In each mathematical problem, it is customary to distinguish the components of the situation, the rules of transformation, the desired goal or conclusion. The solution itself can be set in different ways:
a) as the formation of relations between the components of the situation (for example, when it is necessary to find out which of the objects is harder);
b) as the final state of the situation (for example, collecting a puzzle);
c) how to obtain new knowledge (for example, solving an example).
The role of the task in learning
Since the task is a problem situation that needs to be solved, its role in human education is very important. So, with its help, the theoretical question is illustrated - it is studied, its content is explained. Through simple exercises that are performed according to the template that the theory gives, the assimilation of the fact being studied is achieved. The problem and its solution forms the students' ability to navigate in new situations, to collect information to perform other tasks or study new sections of science, as well as knowledge of reality.
Learning objectives through tasks
The task is a tool used in teaching, designed to interest and motivate students, to form the concept of a mathematical model in them . Properly posed, it reveals the modern teaching methodology, since its solution serves many learning objectives. For example, tasks (Grade 7) can be used to study a new topic or to control (self-control) knowledge and develop interest in mathematics. Most importantly, they serve to familiarize the student with search and creative activity, develop his thinking and logic.
Problem and Solution
The decision takes place in four stages:
- Understanding the conditions of the assignment, as well as its individual components.
- Building a solution plan.
- Implementation of the plan and all its details.
- Final check of the decision, revision for the purpose of assimilation of the material, identification of what may be useful in the future when mastering other tasks.
To get the right solution, you must clearly present the whole situation proposed in the problem. It is necessary to find out what is given, what needs to be found. It is recommended to sketch a visual drawing, this will help identify possible solutions. The mathematics of the problem puts forward those that are solved by logical thinking, the scheme allows you to visually see the right direction.
Hint system
In order to optimally activate the students βmental activity, it is recommended to use a didactic technique called theβ Prompt System β. This technique consists of secondary tasks or questions that give the right direction to the thought stream, making the search for a solution orderly. Solving tasks requires the presence of combining abilities, that is, the ability to make the right choice in the conditions of a glut of knowledge. This search and selection should be focused. The choice will be much faster and easier if we turn to a suitable analogy. For example, you can ask the question: βWhere has something similar happened before?β Using the method of analogy in solving tasks, it is recommended to change their wording. It is best to apply this technique at the initial stage of solving problems. If it is here that it is possible to compare this task with those that were solved earlier, then the similarity of the conditions and methods of solution directs students to the right path, develops the appearance of fruitful ideas in the preparation of a solution plan.
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Methods for solving mathematical problems
Since the problem is a question (situation) that needs to be solved, to find the correct answer to the mathematical problem means to identify the sequence of mathematics that are used to deduce the correct result. Today there are several methods for solving mathematical problems:
- Arithmetic. The answer is due to the implementation of mathematical operations on the numbers that are given in the task. So, the same problem can often be solved using different arithmetic methods, which differ from each other in the logic of reasoning.
- Algebraic. The answer is due to the preparation and solution of the equation. First, the quantities are distinguished and the relationship between them is established, then the variables are introduced, denoting them with letters, they make up the equation and solve it. After that, a solution is checked and the response is recorded.
- Combined. This method includes both arithmetic and algebraic methods for solving problems.
Summarizing
A mathematical problem is a problematic situation, which is solved by using mathematical techniques that require certain skills and knowledge. Tasks are divided into simple and composite, depending on the number of actions. When solving a task involves applying only one action, we are talking about a simple task. In the case of using more than two actions, we will talk about composite tasks. But both of them can be solved in several ways.
Solving one task in various ways is very useful, since in this case various mental operations begin, such as analysis, generalization, comparison, and others. This, in turn, has a positive effect on the development of mathematical thinking among students. To correctly solve the task, it is necessary to analyze and synthesize the problem situation, reformulate the problem, find an inductive method for solving it, using analogies and forecasting. You always need to remember that any task is solvable, you only need to find the right path, using the knowledge, skills that come in the learning process.