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Enhancing Problem-Solving Skills in Mathematics

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# 1. Introduction to Problem-Solving in Mathematics

This book is about strategies for developing problem-solving skills. Despite the prevalence of the importance of problem-solving skills, there is no general agreement as to what problem solving is. Problem-solving is a complex process. It is a problem-solving process in which one must rely on one’s understanding of mathematics and reasoning. According to literary sources, problem-solving involves four main aspects, namely, task definition, planning, carrying out the plan, and looking back. Depending on the situation, these aspects do not necessarily appear in this order. Good problem solvers utilize heuristics to guide the problem-solving process. However, not all the problems require one to utilize all aspects of the problem-solving process and heuristics. Some aspects are more relevant than others to solve the problem at hand. Some heuristics are more relevant than others to help determine a plan for the solution. Without utilizing the heuristics, the problem solution process may sometimes turn out futile.

1. Introduction This book is about enhancing problem-solving skills in mathematics. Knowing mathematics is more than knowing the facts, definitions, theorems, and procedures. It includes knowing how to apply this knowledge to solve problems. The goal of every mathematics teacher is to teach students to solve problems. Problem-solving skills are the most important skills in all of mathematics. The purpose of all courses in mathematics—elementary school mathematics, algebra, geometry, trigonometry, calculus, and undergraduate mathematics programs—is the development of the problem-solving skills of students. Only by solving problems can the students confirm their understanding of old knowledge and constantly develop new knowledge. The impressiveness of a mathematical result lies not in the result itself, but in the processes of how the result has been achieved. In this perspective, the ability to solve problems and the ability to think and reason mathematically are essential to every educated person.

# 2. Key Strategies and Techniques for Solving Math Problems

One of these techniques, the use of ungraded small group homework assignments, speaks directly to the intellectual freedom aspect. These are simply a small number of questions, perhaps two from each section covered in class work, that can usually be answered in a few sentences. The students are required to discuss the questions to establish a consensus among the members of each group. At the beginning of the next class meeting, a spokesperson from each group is called upon at random to explain the consensus reached on the written query of his or her group to the rest of the class. The questions are never graded. Their purpose is to help overcome the stage fright that the fear of public disclosure often induces and the tendency to quit effort under disengagement. The students know that if only members of their group understand the question, either their response will never be called upon or that they may request help from their peers in reading, checking, and discussing the question in the photograph.

If a structure can be instituted that makes quizzes relatively stress-free, students begin to develop confidence in moving on to new material, secure in the knowledge that they can always relearn (or in most cases, for the first time, learn) what they need to know when required. Not having to achieve perfection on all homework problems or to anticipate what will be learned and tested during the final period is particularly conducive to promoting a creative learning environment. From an understanding of logically related generality and an appreciation of using and applying the creative process, problem-solving ability naturally evolves. Other strategies and techniques can also be used to decrease anxiety and encourage discussion, of course.

We will discuss the effectiveness of diverse instructional strategies in promoting problem-solving skills in teaching mathematics. For students who are not used to thinking on their feet, many problem-solving situations give rise to an extremely high level of anxiety because the outcome is unpredictable. This anxiety may become acute under the prolonged pressure of a classroom examination. One remarkably simple procedure to reduce this anxiety is to conduct frequent, brief, relatively unstressful, written quizzes. The instructor has made it clear to the students that during the final examination at the end of each of the sections, they will be expected to solve twelve of the problems that appeared in quizzes given over that section, without access to their notes or textbooks. After any quiz, the instructor prepares a key to the problems and the students also provide accurate keys.

# 3. Practical Applications and Examples in Various Mathematical Fields

The concepts of “sufficient”, “high”, “low”, “doubled”, “half”, etc. are familiar to the child as an inheritance of his experiences with objects of the real world. It can be used as a vivify example when dealing with “increasing or decreasing in 100%, 50% or 10%”. Helping the child to see the practical way in which both properties of addition are approached offered opportunity to understand why he can estimate results of accounting operations, respecting the order of magnitude of the numbers and, although inductively, realize some relation between “near numbers’ operation result” and “scissor-and-cut operations”. It is a fact that art, in a broad sense, can offer us material for developing these and many other precise questions. Raising problems and teaching mathematics work only this way.

The domain of mathematics does not reside solely in scientific knowledge and art, nor in reality, in examples of everyday usage and problems, from markets to cleaning the house, to commercial transactions or daily home expenses. However, it is these examples or problems that may make relation toward different stages of the mathematics teaching discipline possible. More specifically, linked to the school teaching of mathematics and beginning in the very first years, this relation with concrete facts showing practical uses, without ruling out the possibility of taking into consideration examples on specialized texts or popular market leaflets which should be checked, may enable the child to have the experience of “completing a cycle”. This should happen all the way through basic education, offering construction bases for more abstract content approached later.

How can practical applications be introduced from the first years of school? Firstly, let us analyze examples of practically applied problems inserted into a few contents of the three initial years of mathematics. Secondly, let us then pay more attention to the concept of sufficiency and working with real levels of “solving” in analogies.

# 4. Advanced Problem-Solving Approaches and Tools in Mathematics

When working on a mathematical problem, the student has to work in hypothetical statements, operate instances and use reasoning that is active. With that in mind, the problem-solver, competent in visual representations and interactions in mathematics, makes use of technological tools that he or she selects after identifying the desired mathematical results and after analyzing the representations that can satisfy or not own emerging questions. A judicious choice and manipulation of the didactic tools selected and of the software of mathematics make it possible to optimize the search for solutions to problems which the students of future secondary teaching have set for themselves.

Problem-solving plays an essential role both in the education of mathematics teachers and in the interest that teachers have in the subject generally. This paper uses some problems and examples in order to explore the part played by problem-solving in learning, and to look at the theory of problem-solving. The statement of the problem-solving method, associated with the mathematician George Pólya and others, is also presented.

# 5. Conclusion and Tips for Continued Improvement

Helping students to develop the metacognitive skills needed in order to arrive at a correct answer to a mathematical problem on a test is just one of several important educational objectives. Ultimately, our goal is to have students who can think critically about life’s challenges, formulate and solve problems, and effectively share and apply the results of their problem-solving to real-world situations. Developing these skills takes time and effort, but every step you take with your students will bring them one step closer to true success in the classroom and in life.

In this paper, we review many techniques and strategies for helping students to become better problem solvers. After reading the paper, you should have a better understanding of the many dimensions of the problem-solving skill set, the variety of instructional techniques that can be employed to develop them in the mathematics classroom, and evidence from educational research about the effectiveness of these strategies. As a result, you will be in a position to better prepare students to meet the challenges of the 21st century.

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