In our daily life, we will and always face the problem. Problem like our friend, that can make us be wise person, better person, and strong person to face uncertain real life. Problem as our rival when we can’t solve it and make us stress. Let’s study about solving our problem, by know the definition and step to solve it. Problem solving is a mental process which is the concluding part of the larger problem process that includes problem finding and problem shaping where problem is defined as a state of desire for the reaching of a definite goal from a present condition that either is not directly moving toward the goal, is far from it or needs more complex logic for finding a missing description of conditions or step toward the goal. Problem solving is a mental process that involves discovering, analyzing, and solving problems. Problem solving is a process and skill that you develop over time to be used when needing to solve immediate problems in order to achieve a goal. The ultimate goal of problem solving is to overcome obstacles and find a solution that best resolves the issue. Problem solving has two major domains: mathematical problem solving and personal problem solving where, in the second, some difficulty or barrier is encountered.
A seven-step problem solving cycle
There are many different ways to solve a problem, however all ways involve a series of steps.
The following is a seven-step problem solving model:
Step 1. Identify the problem
Try to identify and name the problem so that you can find an appropriate solution.
Step 2. Explore the problem
When you are clear about what the problem is you need to think about from different angles. You can ask yourself questions such as:
- How is this problem affecting me?
- How is it affecting others?
- Who else experiences this problem?
- What do they do about it?
- Seeing the problem in different ways is likely to help you find an effective solution.
Step 3. Set goals
Working out your goals is a vital part of the problem solving process.
Step 4. Look at Alternatives
When you have decided what your goals is you need to look for possible solutions. The more possible solutions you find the more likely it is that you will be able to discover an effective solution. You can brain-storm (the purpose of it is to collect together a long list of possibilities) for ideas.
Step 5. Select a possible solution
From the list of possible solutions you can sort out which are most relevant to your situation and which are realistic and manageable. You can do this by predicting the outcomes for possible solutions and also checking with other people what they think the outcomes may be. When you have explored the consequences, you can use this information to identify the solution which is most relevant to you and is likely to have the best outcomes for your situation.
Step 6. Implement a possible solution
Once you have selected a possible solution you are ready to put it into action. You will need to have energy and motivation to do this because implementing the solution may take some time and effort. You can prepare yourself to implement the solution by planning when and how you will do it, whether you talk with others about it, and what rewards you will give yourself when you have done it.
Step 7. Evaluate
Just because you have implemented the best possible solution, you may not have automatically solved your problem, so evaluating the effectiveness of your solution is very important. You can ask yourself (and others) :
- How effective was that solution?
- Did it achieve what I wanted?
- What consequences did it have on my situation?
If the solution was successful in helping you solve your problem and reach your goal, then you know that you have effectively solved your problem. If you feel dissatisfied with the result, then you can begin the steps again.
That is steps to solve our daily life problem, in Mathematics we also need it to make that subject become fun lesson and make students love it. There are many researchers about it, but I just take two of them.
First, from Mr. Marsigit and friends (Yogyakarta State University) in “Lesson Study : Promoting Student Thinking on the Concept of Least Common Multiple (LCM) Through Realistic Approach in the 4th Grade of Primary Mathematics Learning.” Mathematics in primary school have function to encourage the students to think logically, analytically, systematically, critically, creatively, and be able to collaborate with others. Those competencies are needed for the students to access and employ information to preserve their live. Mathematical thinking is defined as students’ activities to communicate mathematical ideas in involves the using of symbols, tables, diagrams, and other sources hope the students are able to solve their problems. Contextual and realistic approaches are recommended to be developed by the teachers to encourage mathematical thinking in primary schools. Hope students can step-by step learn and master mathematics enthusiastically. To make their teaching learning of primary mathematics more effective, teachers also need to develop resources such as information technology, teaching aids, and other media.
Specifically, the primary mathematics curriculum outlines the aims of teaching learning of mathematics are as follows:
1. To understand the concepts of mathematics, to explain the relationships among them and to apply them to solve the problems accurately and efficiently.
2. To develop thinking skills to learn patterns and characteristics of mathematics, to manipulate them in order to generalize, to proof and to explain ideas and mathematics propositions.
3. To develop problems solving skills which cover understanding the problems, outlining mathematical models, solving them and estimating the outcomes.
4. To communicate mathematics ideas using symbols, tables, diagrams and other media.
5. To develop appreciations of the uses of mathematics in daily life, curiosity, consideration, and willingness to learn mathematics as well as tough and self-confidence.
As the Program for International Student Assessment (PISA) that concerned with the capacities of students to analyze, reason, and communicate ideas effectively as they pose, formulate and interpret mathematics in a variety of situations. Realistic Mathematics Education (RME) in the schema of Lesson Study let the teachers to improve instructional approach from traditional to progressive one. Isoda, M. (2006), outlined that mathematical thinking is open ideas; thus, it is very difficult to discuss its development without having a window to discuss. When we focus on each lesson, we easily focus on specific knowledge and skills (Understanding), and easily forget to develop Attitude, Mathematical Thinking and Representation. Freudenthal's view on mathematics (Freudenthal, 1991). Two of his important points of views are mathematics must be connected to reality (Mathematics must be close to children and be relevant to everyday life situations) and mathematics as human activity (Mathematics education organized as a process of guided reinvention, where students can experience a similar process compared to the process by which mathematics was invented, the meaning of invention is steps in learning processes while the meaning of guided is the instructional environment of the learning process). Two types of mathematization which were formulated explicitly in an educational context by Treffers, 1987, in Zulkardi, 2006, are horizontal and vertical mathematization. In horizontal mathematization, the students come up with mathematical tools which can help to organize and solve a problem located in a real-life situation. Vertical mathematization is the process of reorganization within the mathematical system itself. The process of reinvention in which both the horizontal and vertical mathematization take place in order to develop basic concepts of mathematics or formal mathematical language.
The learning process starts from contextual problems. Using activities in the horizontal mathematization, for instance, the student gains an informal or a formal mathematical model. By implementing activities such as solving, comparing and discussing, the student deals with vertical mathematization and ends up with the mathematical solution. Then, the student interprets the solution as well as the strategy which was used to another contextual problem.
The mathematics lessons that will be designed in Realistic Approach should represents the characteristics of how the students do matematization. The way of embed these characteristics into the lesson plan components can be seen in the following diagram (Zulkardi, 2006):
Activities:
1. Introduction
2. Describing prepared contextual problems (problems situated in reality as follow):
Since the early of the year 2006, Shinta has two activities i.e. swimming and gardening. She is periodically going to swim once a week and gardening every 8 days, as shown in the following calendar:
Question:
When Shinta is going for swimming and gardening on the same days?
3. Developing (group discussion)
4. Reason and explanation (presentation)
5. Conclusion (homework)
6. Closing
The search in this lesson study is to uncover the idea of mathematics as a human activity that is stressed in realistic approach. Teacher organized the class as a process of guided reinvention (De Lange, 1996, in Zulkardi, 2006) that is to step in learning LCM by developing instructional environment example let the students to freely choose and develop their methods and aids to solve the problems. The teacher let the students to work individually and in group to perform horizontal mathematization and then anticipating the structure to more formal raise mathematization activities.
The second is from Shigeo Katagiri (University of Tsukuba) in Mathematical Thinking and How to Teach It. The Aim of School Education described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.” Of course, not every child will be able to act independently at the same level, but each school child must be able to act independently according to his or her own capabilities. To this end, teaching methods that focus on the individual are important. Mathematical thinking allows for: (1) An understanding of the necessity of using knowledge and skills, (2) Learning how to learn by oneself, and the attainment of the abilities required for independent learning. Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist.
People will love something if they can enjoy it. Same condition with students, they will love Mathematics if they enjoy it. Many people said that Mathematics is subject that terrify, make stress, etc. They said like that because when they studied Mathematics, they were not enjoyed it. So, make them uncomfortable. By listening people answer about why they dislike Mathematics, we as the teacher in the next decade need method to minimize that problem and show that Mathematics is the universal science and it is the ground of other sciences. Let’s study about Lesson Study especially Mathematics in other country.
Established in 1989, the Asia-Pacific Economic Cooperation, or APEC, is the premier forum for facilitating economic growth, cooperation, trade and investment in the Asia-Pacific region. Decisions made within APEC are reached by consensus and commitments are undertaken on a voluntary basis. APEC is composed of 21 member economies - Australia; Brunei Darussalam; Canada; Chile; People's Republic of China; Hong Kong, China; Indonesia; Japan; Republic of Korea; Malaysia; Mexico; New Zealand; Papua New Guinea; Peru; The Republic of the Philippines; The Russian Federation; Singapore; Chinese Taipei; Thailand; United States of America; and Viet Nam.
Specialists on APEC Lesson Study: What is going on in each economy?
1) Max Stephens, University of Melbourne, Australia
Elementary schools in several Australian States, for example Victoria and New South Wales, are exploring Lesson study as a means of teacher professional development and school improvement. At present, the main focus appears to be on shared planning of lessons on particular topics, with some emphasis on the researcher phase.
2) Francisco Cerda Bonomo, St. Thomas University, Chile
We began the project on elementary schools with a problem solving strategy focused on mathematical thinking and mathematics communication. The lesson study methodology is one of the preferred we consider in the training of future teachers.
3) CHENG Chun Chor Litwin, Hong Kong Institute of Education, Hong Kong, China
The most important component of mathematics teaching is good mathematics content, without good content, lesson study could not improve any classroom teaching. This lesson study is conducted in two primary schools in Hong Kong. The challenges of this lesson study are to locate good teaching points in learning fractions multiplication and division through conceptual understanding rather than rote operation. The teaching is at a primary 4 classroom, usually multiplication and division of fraction is taught at primary 5. Taking an investigative approach, children understand the concept of fraction multiplication and division by repeated subtraction (same denominator) and extend the understanding to different denominators.
4) Marsigit, The State University of Yogyakarta, Indonesia
The specific objectives of activities were: to develop instrument and equipment for teaching learning process, to develop teaching method and model for teaching learning process, to develop teaching material for teaching learning process, and to develop teaching evaluation for teaching learning process. Temporary results of Lesson Studies activities indicated that there were improvements in teaching learning practice of primary and secondary mathematics together with the innovations in approaches and media of learning to bring good results for students.
5) Soledad Ulep, University of Philippines, Philippines
In mathematics education, lesson study was first introduced in the Philippines in April 2006. A group consisting of 4 grade 8 mathematics teachers and the department head of a school was formed to demonstrate how lesson study could be used to enable them to adapt in their own classes good teaching practices. These practices were modeled in a teacher training program conducted by the University of the Philippines National Institute for Science and Mathematics Education Development (UPNISMED) in which the teachers participated. Through lesson study, the teachers encounter valuable and new experiences like collaboratively developing lessons to teach mathematics through problem solving. Formulating open-ended questions and anticipating various student responses deepen their mathematical understanding.
Based on Specialists on APEC Lesson Study, we know that each country have different result about Lesson Study (the method for the teacher). In Australia, several Elementary schools are exploring Lesson study as a means of teacher professional development and school improvement. In Chile, the lesson study methodology is one of the preferred that consider in the training of future teachers. Then in Hong Kong, the challenges of the lesson study are to locate good teaching points in learning fractions multiplication and division through conceptual understanding rather than rote operation. Different in Indonesia, the specific objectives of Lesson Studies activities were : to develop instrument and equipment for teaching learning process, to develop teaching method and model for teaching learning process, to develop teaching material for teaching learning process, and to develop teaching evaluation for teaching learning process. Temporary results of Lesson Studies activities indicated that there were improvements in teaching learning practice of primary and secondary mathematics together with the innovations in approaches and media of learning to bring good results for students. And in Philippines, through lesson study, the teachers encounter valuable and new experiences like collaboratively developing lessons to teach mathematics through problem solving. Formulating open-ended questions and anticipating various student responses deepen their mathematical understanding.
From the different result we can conclude that the activities from the Lesson Study are teachers collaboratively plan a lesson, teach the lesson, and review the lesson before teaching the lesson to a different class. At present, Lesson Study is focused on particular mathematical topics. Like in Chile, the teacher began the project on elementary schools with a problem solving strategy focused on mathematical thinking and mathematics communication. In Hong Kong, The teaching is at a primary 4 classroom, usually multiplication and division of fraction is taught at primary 5. Taking an investigative approach, children understand the concept of fraction multiplication and division by repeated subtraction (same denominator) and extend the understanding to different denominators.
The last, hope the teacher in the next decade can make students love Mathematics by using many method based on research of many observers.
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