Mathematical word problems that contain a constant in the course of mathematics of primary school in Ukraine

High level. The student unmistakably develops a plan for solving the problem, justifies the arithmetic actions with the help of which they solve this problem. The learner creates and solves inverted problems; establishes a correspondence between the numbers that result from the solution of the problem and the given numbers, checks the correctness of the solution of the task. The student is able to determine different methods of solving on their own and to separate the most rational one. It is easy for them to summarize how to accomplish a specific problem. He/she reveals the flexibility of thinking by switching easily from one method to another.

In determining the level of completion of each problem-solving action, the researcher focused on the above criteria and the corresponding indicators for each criterion.

Tab. 1. Distribution of students of experimental and control classes by levels of formation of abilities to solve math problems containing a constant.

The students who have demonstrated the low level perceive the math problems superficially and incompletely. They can neither distinguish between the condition and the problem of the mathematical problem or identify the object of the math problem, or separate the numerical data and the unknown number of the math problem. They distinguish external, often non-essential elements of the math problems, cannot and do not try to imagine the course of solving the math problem. They cannot perform a short math problem record or schematic drawing. Immediately after reading the math problem, they begin to "solve"; they choose numerical data and arithmetic action at random. Before solving the math problem, they cannot make an estimation. Even if the math problem is solved, the students are having difficulty in formulating the answer. They "don't see" different methods of solving, even when the teacher points to them.

The students who have demonstrated the average level show the following: their perception of the math problem is accompanied by its analysis; they distinguish the condition and the question, the object of the math problem, numerical data and the unknown number. They seek to understand the math problem, to separate the data and the unknown number, but can only establish separate links between them. They are able to highlight keywords and write a short math problem record or perform a schematic drawing, but they are having difficulty estimating the expected result. The students don not relate the math problem with the previously studied ones, therefore do not recognize the typical math problem, and as a result do not update the way of its solution, resorting to considerations from the question of the math problem to the numerical data - analysis. They find different methods of solving the math problem that depends on their experience in solving similar math problems.

The students who have demonstrated the sufficient level can analyze the math problem. They are able to identify a group of proportional values that describe the situation of the math problem, and also form an auxiliary model of the math problem. Based on a complete comprehensive analysis of the math problem, the students determine the connections between the numerical data and the unknown number and estimate the expected result. After performing the auxiliary math problem model, they correlate the math problem with the previously studied ones and recognize the type of the math problem and update the plan for its solution. They formulate and solve the inverse math problems, as well as establish the correspondence between the numbers obtained because of solving the math problem and the given numbers.

High level. Immediately after reading the math problem, the students determine its type and actualize the generalized way of its solution. They are able to independently see different methods of solving and point out the most rational. When analyzing the situation, the pupils freely reject insignificant and unnecessary elements in terms of its requirement.

The developed and experimentally tested teaching methods of solving mathematic word problems that contain a constant value have been implemented in a series of mathematics textbooks and exercise books by S. Skvortsova and O. Onopriienko [19].

Conclusions

Solving mathematical word problems has a significant place in the structure of the content of elementary mathematics education. The result of teaching mathematics in the primary school in Ukraine is the general ability of students to solve mathematical word problems, as well as the ability to solve math problems of certain types (mathematical problems on finding the fourth proportional, on proportional division, on finding the unknown by two differences, on double reduction to one, on collaborative work and on motion). Achieving this result is possible provided that the theoretically grounded methodical system of teaching primary school students to solve the mathematical word problems is implemented.

In the experimental study we have tested the method of generalization of mathematical structures and methods of solving mathematical word problems that contain a constant value. We have implemented an approach that consists in solving a system of interrelated math problems, namely, the transformation of the math problem on finding the fourth proportional into the math problem on proportional division; the transformation of the math problem on proportional division into the math problem on finding the unknowns by two differences. Moreover, at each stage, we offer the comparing of the mathematical structures of the math problem of each pair, the defining of common and distinctive features in them, and investigate the influence of differences on the method of solving the math problem. Finally, after solving two pairs of math problems, students are asked to compare all the three math problems and to investigate the effect of differences on the way mathematical problems are solved. The considered method of generalization should be an integral part of the method of forming skills to solve the math problems of certain types. After its implementation in the experimental class, 56% of pupils demonstrated a sufficient level of abilities in solving math problems of these types.

The prospects to further research are seen in the development of drilling exercises for generalizing mathematical structures and methods for solving math problems on processes and checking their effectiveness experimentally.

References

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