Since the reform of mathematical education is currently taking place in most countries of the world, the problem of setting tasks in the school mathematics course 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 today understand this goal in a school mathematics course?
Teaching students
Practically all schoolchildren think that when the correct solution is found, and the received answer of the problem matches that proposed in the textbook, their work is over, they can forget about the problem.
A student or a teacher does not take into account the fact that the role of each task is to develop skills of orientation in problem situations, to increase knowledge and experience. If you do not pay attention to updating the acquired knowledge, the process of mathematical thinking is disrupted, which contributes to a decrease in the development of skills.
But before dealing with this issue, it is necessary to find out what is the task and what is its role in learning.
What istask
This term has several interpretations. Consider one of them applied to mathematics. Here, the task is a problem situation (question) that requires a solution through the use of certain skills, knowledge, and reflections. This is a goal that is within the problem situation, what needs to be achieved, as well as a condition and a requirement.
Thus, to solve a problem means to transform a given problem situation or to reveal that such a reconstruction is impossible under these conditions. Here it is important to define the process of solving a problem as a mental activity aimed at achieving a goal.
Problem format
In each mathematical problem, it is customary to highlight the components of the situation, the rules of transformation, the required goal or conclusion. The solution itself can be specified in different ways:
a) as the formation of relationships between the components of the situation (for example, when it is necessary to find out which of the objects is heavier);
b) as the end state of the situation (eg solving a puzzle);
c) as acquiring new knowledge (for example, solving an example).
The role of the task in learning
Since a task is a problematic situation that needs to be solved, its role in human learning is very important. So, with its help, a theoretical question is illustrated - its content is studied, clarified. Through simple exercises, which are carried out according to the pattern that the theory gives, the assimilation of the studied fact is achieved. The task and its solution form the students' ability to navigate in new situations,collect information to perform other tasks or study new sections of science, as well as knowledge of reality.
Objectives for learning with tasks
A task is a tool used in teaching, designed to interest and motivate students, to form in them the concept of a mathematical model. Correctly delivered, it reveals modern teaching methods, since its solution serves many learning purposes. For example, tasks (grade 7) can be used when studying a new topic or for monitoring (self-control) knowledge, developing interest in mathematics. Most importantly, they serve to familiarize the student with search and creative activities, to develop his thinking and logic.
Problem and solution
Decision occurs in four stages:
- Comprehension of the conditions of the task, as well as its individual components.
- Building a solution plan.
- Practicing the plan and all its details.
- Final verification of the solution, revision in order to assimilate the material, identifying what may be useful in the future when mastering other tasks.
To get the right solution, you need to clearly imagine the whole situation proposed in the problem. We need 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. Mathematics of the problem puts forward those that are solved by logical thinking, the scheme allows you to visually see the right direction.
Systemhints
To optimally activate the mental activity of students, it is recommended to use a didactic technique called "Hint System". This technique consists of secondary tasks or questions that give the right direction to the thought flow, making the search for a solution orderly. Solving tasks requires combinational abilities, that is, the ability to make the right choice in conditions of knowledge overload. This search and selection must be purposeful. The choice will be made much faster and easier if we turn to a suitable analogy. For example, you can ask the question: “Where has something similar been seen before?” Using the analogy method when solving tasks, it is recommended to change their wording. It is best to use 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 solving directs students on the right path, develops the emergence of fruitful ideas when drawing up a solution plan.
Methods for solving mathematical problems
Since a problem is a question (a situation) that needs to be solved, finding the correct answer to a mathematical problem means identifying a sequence of mathematical statements that are applied in order to derive the correct result. To date, there are several methods for solving mathematical problems:
- Arithmetic. The answer is found by performing mathematical operations on the numbers that are given in the task. Yes, one and the samethe same problem can often be solved using different arithmetic methods that differ in the logic of reasoning.
- Algebraic. The answer is found by compiling and solving the equation. First, the quantities are distinguished and a relationship is established between them, then variables are introduced, denoting them with letters, they make up an equation with their help and solve it. After that, the solution is checked and the answer is recorded.
- Combined. This method includes both arithmetic and algebraic problem solving methods.
Summing up
A mathematical problem is a problematic situation that is solved by using mathematical techniques that require certain skills and knowledge. Tasks are divided into simple and compound, depending on the number of actions. When solving a task involves the use of only one action, we are talking about a simple task. In the case of using more than two actions, we will talk about compound tasks. But both of them can be solved in several ways.
Solving one task in different ways is very useful, because in this case, different mental operations begin their work, such as analysis, generalization, comparison, and others. This, in turn, has a positive effect on the development of mathematical thinking in students. To solve the task correctly, 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 should always remember that any task is solvable, it is necessaryonly find the right path using the knowledge, skills and abilities that come with learning.
