This unit will be developed for middle school students. As I develop this unit and make use of it with my students it will enhance their ability to solve problems while developing math skills in the four basic math operations in problem solving.
Problem solving is the process by which students acquire knowledge, skills, habits, values and attitudes. They must learn to read mathematics in order to become able to use math in everyday life. Like skills, problem solving must be practiced; when it is practiced it becomes far less difficult. Some problem solving techniques at times are so very important, they should be the focus of instruction.
Problem solving is a challenge for the majority of our students. Real life situations, applications and interdisciplinary connections are a part of everyday lessons to be learned. In most math classes, there is a wide range of pupil ability. Pupils do not mature at the same rate, and they vary in both the ability to learn and the level of learning attained. Furthermore, different pupils learn in different ways. Instructors are faced daily with the problem of dealing with the many variables existent in their classes.
Other resources are needed to augment the existing curriculum and my unit on problem solving will aid the curriculum in that area. The most important aspect of problem solving is the quality of thinking it engenders among students. Clarity of thought is aided by precision; the development of precise language is more than a matter of memorizing mathematical words and phrases. Students should be led to think about concepts and then generalize about them. Systematic attention should be given to the development of problem solving skills as they relate to word or story.
There are several ways to present problems other than to refer to textbooks and workbooks. Problems can be written on the chalkboard, thus inviting a discussion. An overhead projector may be used when presenting a problem, it can be presented either in it’s entirety or line by line to focus students attention and pinpoint parts in case there is difficulty in understanding the problem.
Problem solving is the process by which students acquire knowledge, skills, habits, values and attitudes. Problem solving includes a variety of learning experiences and it takes place outside the classroom as well as inside the classroom. It involves both learning and teaching. Some students learn by teaching themselves and they also learn with the help of other people, such as parents and teachers. Problem solving is as old as humanity. People have always needed problem solving in order to survive. Today, our society must become skilled in government, industry, commerce, the arts, and agriculture in order to survive. A society can not survive without being skilled in problem solving, especially in math.
Problem solving is more important today than ever before. It helps students acquire skills that are needed in everyday activities. It also gives them training that they may need to prepare for a career or job. It is also important because it helps students get more out of life, increases their knowledge and understanding of the universe, and helps students acquire skills that make their lives more interesting and enjoyable. Problem solving provides skills needed to participate in sports, play a musical instrument, or paint a picture and helps them to adjust to change.
Problem solving in and out of the classroom makes learning a delightful adventure. Problem solving is for improving the lives of others as well as our own and leaving the community and the world a better place for it. Mathematicians identify sources of change, distinguish between patterns, and seek multiple representations through mathematical language, such as numeric, graphic, verbal, and symbolic to express what it transpires.
The art of problem solving is the developed ability over a period of time. The information and activities is this unit will be designated to offer students opportunities to enrich their problem experiences.
It is important to realize that students cannot be expected to use strategies that are unfamiliar to them. Problem solving skills are learned just like other skills are learned. Students need to be exposed to a wide variety of problems so that they can try out new strategies and practice using them.
Learners will become familiar with problem solving as they relate to everyday life. This unit will allow students to relate to real life situations. The activities in this unit fosters quantitative thinking in the learner, which will lead the students to develop interest, objectivity, attitudes and problem solving skills. It is my intent that these skills will grow with adequate use, that is, by having certain thought patterns recur in a wide range of problem solving. The adequate use will enhance the required amount of application of a concept necessary to insure its future availability, and in this way the students will really become mathematically literate in problem solving.
The middle and high school learner is an entity in himself / herself with unpredictable reactions to problems and personal situations. The learner should see that in problem solving the thought processes help to analyze the factors involved when something needs to be done as well as to organize the chosen factors in a problem in order to bring about a satisfactory outcome. The learners are introduced to problems that are common. It is my hope that by solving some of the problems the learners will acquire some of the skills and understandings that will be needed in the future. There are understandings to be mastered. There is insight to be gained. One of the purposes is to guide the learners thinking to help make decisions. It is hoped that the learners will not only develop problem solving skills, but take a critical look at himself / herself in this ever changing and complex society of ours where problems of all kinds do exist.
Problem solving is more than the ability to compute. It is believed that the ability to solve problems develops automatically from the mastery of computational skills. This is not true, problem solving is itself a skill that has to be taught.
When the learner is confronted with a problem situation whose solution is not known, the student must rely on his or her problem solving skills. One problem solving skill is the ability to pick out the important facts that are given and disregard the facts that are not pertinent. Another skill is to recognize what information is missing and how to find it. Another is the ability of the learner to recognize similarities between problems which are being solved and have been solved.
Problems can be solved in a variety of ways, yet not one of these ways is appropriate to the solution of all problems. However it is helpful to have at least a general framework within which students can organize their efforts.
There are several techniques, methods and strategies used in problem solving. Problem solving involves trying several ways to solve the problem before deciding on which to use. It involves putting together the facts that the learner has with mathematics that he or she knows in such a way that the result is a solution that was unknown to begin with. A model of a problem can be a picture worth five hundred words, a sketch, a scale drawing, a chart or graph. It’s whatever helps the student. There are times when information is missing and the skill is the ability to recognize what additional information is necessary.
Problem solving situations are likely to be representative of those which the learner will face sometime in the future. Problem solving will give the students experiences to help them in their thinking and decision making processes. The main thrust of problem solving should be toward the development of logical thinking. Taking the numbers out of problems seems to be one way of doing this without the negative reaction that usually accompanies mathematical problems.
Applications of mathematics to problems from the daily lives of pupils give added depth of understanding. In making and applications, the essential problem is to choose the appropriate mathematical structure for the application. As students study and analyze a given situation to determine the most appropriate mathematical structure, they learn to solve real life problems, and also broaden and expand their knowledge of the mathematical structure involved. Problem solving, in its broadest sense, includes the way one approaches any mathematical idea as well as the way he or she approaches a practical problem stated in words.
Learning is an individual matter. However, it often takes place in a group situation. Students learn by doing, thinking, discussing, and it is important that problem solving be the kind that invites students to participate, to do, to think and to respond. Students should be challenged to react, to reason, and come to their own conclusions.
Students should be led to explore, experiment, and analyze unusual problem solving within the four basic mathematical operations, such as addition, subtraction, multiplication, and division. Problem solving, in it’s broadest sense, will include the way the learners approach any mathematical idea as well as the way they approach any practical problem stated in words and the writing of story problems. No mathematical problem solving idea is completely developed at any one time. Each problem solving idea is introduced at an early level and is expanded at different levels. Numerous exercises should be provided for fast and slow learners. Some adaptation is desirable. Each instructor should make use of problem solving ideas suggested.
Applications of mathematical problem solving ideas from the daily lives of students can give added depth of understanding. In making applications, the most important problem is to create an appropriate mathematical structure for the application. As they study and analyze a given problem to determine the most appropriate mathematical structure, the students learn to solve life problems, and also broaden and deepen their knowledge of the mathematical structure involved. Emphasis should always be given to the relationship between the situation and the appropriate mathematical structure.
Another important aspect of problem solving is motivation. Motivation for the study of problem solving comes from a variety of resources. One important resource of motivation is the inherent interest of mathematical ideas themselves. Many students study with enthusiasm when they are allowed to explore the why and how of the mathematical situations. Students are also motivated by working with special activities that require the application of mathematical principles, by exploring such topics as early ways of computing or primitive ways of measuring and by seeing how the mathematics that they study are applied to their lives and others.
Another motivating factor, and one that should not be overlooked, is the success in learning. A learning student is a motivated student. When the student’s experiences in math lead to success and achievement, the learner has excellent motivation for more and further study.
Problem solving ability develops over a long period of time and grows with experience in solving a variety of problems in many different ways. Students must learn to be flexible and make use of a variety of methods, techniques and strategies.
If a problem is more complex, the strategy for solving it may not be immediately apparent. Problem solving requires some degree of creativity on the part of the problem solver. The problem solver can be the actor by acting out the problem or by devising a plan to solve the problem.
Problem solving can be accomplished by using a four step or five step problem solving plan.
A Four Step Plan Involves:
1. Understand the problem
a. what are the facts?
b. what do you need to know?
2. Make a plan
a. what do you do to solve the problem?
3. Show the work
a. do the arithmetic
4. Interpret the answer
a. Is the answer reasonable?
b. Does it answer the question?
A Five Step Problem Solving Plan Includes:
1. Reading the problem carefully, understanding what it says, and reading it more than once
2. Ask questions like the following
a. what is asked for?
b. what facts are given?
3. What facts are not needed to solve the problem?
4. What facts are needed to solve the problem?
a. will a sketch or diagram help?
5. Determine which operation or operations can be used to solve the problem, carry out the operations carefully and give the answer.
There are several ways to present problems other than to refer to textbooks and workbooks. Problems can be written on the chalkboard, thus inviting a discussion. An overhead projector may be used when presenting a problem; you may project the entire problem at once on each individual part pinpointing any difficulty in the understanding of the problem.
Duplicated masters is another way of presenting a problem where by each student have its own individual sheet. Some students may find its easier to refer to a problem in this way. It also eliminates possible student errors in copying a problem from a chalkboard or an overhead projector.
Oral presentation is another way of presenting a problem. This can be done by the teacher reading the problem aloud while the students read it softly or by having a student so the same.
Students need help in solving problems. Giving students too much help will leave them with nothing to do. Too little help will cause frustration. The right amount of help will allow students to experience the challenge of a problem and the pleasure of finding its solution.
Students may need to test survival strategies before an appropriate one is found. Once the problem is solved the students think that’s the end, but they should be encouraged to take a second look and consider the reasonableness of the of their answers.
1. Help students increase their thinking skills and decision making process.
2. Apply questions to decision making process.
3. Students will understand that there are certain conditions under which they must work in order to bring about a desirable and valued solution.
4. Students will become familiar with and understand the language that is used in problem solving.
5. Evaluate the process and consequences.