Problem solving The terms problems

Problem solving
The terms problems and problem solving occur many disciplines but are perhaps more closely related to mathematics than any other. Over the years much has been written about problems and problem solving giving rise to various schools of thought.
In mathematics education, problem solving has been emphasized since Polya's work in the 1940s. Polya, who is often considered the father of problem solving, describes it as follows
Solving a problem is finding the unknown means to a distinctly conceived end to find a way where no way is known off-hand. For a question to be a problem, it must present a challenge that cannot be resolved by some routine procedure. Problem solving is a process of accepting a challenge and striving to resolve it.
Polya believed that in order to solve a problem a student had first to come to terms with what the problem was really about. Once he or she had gained an insight into the problem only then could a plan for solving it be devised. When the plan had been carried out Polya emphasized the need to look back over the problem in terms of the solution. His four-step problem solving model, which has been used as the basis for many subsequent frameworks, can be summarized as:
1 understand the problem
2 devise a plan
3 carry out the plan
4 look bock
Schoenfeld (1989) discussed problem solving in terms of 'tasks to be solved'. He believes that for problem solving to occur a student must first be motivated to solve the problem and to have no obvious ways to do so. He states that, for any student, a mathematical problem is a task:
♦ in which the student is interested and engaged and for which they wish to obtain a resolution; and
♦ for which the student does not have readily accessible mathematical means by which to achieve that resolution.
Siemon and Booker (1990) have a similar view of problem solving, highlighting the need for the student to want or need to solve the problem individually or as a group and having no immediate means to do so. They go on to describe problem solving as a process of achieving the solution to a problem, often with identifiable beginning, middle and end phases. They state that a problem is a task or situation:
♦ that you want to or need to solve;
♦ that you believe you have some reasonable chance of solving, either individually or in a group; but
♦ for which you or the group have no immediately available solution strategy.
These views on problem solving highlight that a problem is a task for which there is no immediate or obvious solution and problem solving is the process students undertake when engaging with this task. Problem solving involves engaging in tasks for which the solution strategy is not immediately obvious. In order to discover the possible solution students need to draw on their current knowledge and processes and will often develop new knowledge and understandings as they progress towards a solution.
A key issue in problem solving is the idea of no immediately available or obvious solution. A task that is a problem for students in Year 1 would not necessarily be a problem for students in Year 7. Indeed a task that would be a problem for some students in a particular year level would not necessarily be a problem for all students in that year level.
This task would not be a problem for students in Year 7 as most students would be able to think of an immediate solution. Their mathematical knowledge and understanding of the concept of multiplication and multiplication basic facts would lead them immediately to think about 3 fives and come up with the solution of 15 puppies. This task therefore could not be considered a problem for these students.
However, for students in Year 1 who would in most cases not have this mathematical knowledge and understanding, it could well be conceived as a problem for which there is no immediate or obvious solution. In most situations a Year 1 student would need to think about how to solve this problem and would perhaps come up with the possibility of using materials and counting or drawing a diagram and counting.
When selecting problems for a class program it is important to keep in mind the concept of problem solving as a task or situation for which there is no immediate or obvious solution. At times in a classroom students may be 'problem solving' when in fact they are reading 'problems' and immediately knowing what to do to solve this 'problem'. In these situations the activity being undertaken by the students could not in fact be considered to be problem solving.
Strategies and problem solving
In 1965 Polya observed that students needed techniques to help them plan for solutions. This observation provided the catalyst for over two decades of research into the identification and utilization of problem solving strategies. The outcome of this research ultimately led to strategy driven problem solving programs in schools. These programs centered around the teaching of particular strategies such as make a list, work backwards, guess and check, where the strategy rather than the problem was the focus. A strategy was introduced and then the class would solve a variety of problems using the identified strategy.
For example, the strategy of 'make a list' would be taught and then the class would spend time solving problems by making a list. Students did not need to understand or come to terms with problems as they knew immediately each one could be solved using the particular
. This method of teaching problem solving continued for some time until researchers began to notice that students didn't necessarily become more proficient at problem solving in situations outside of the specific lessons. This thinking was not generalised into different situations where the students had to identify the appropriate strategy for themselves.
Research began to focus on problem solving and cognition and the methodology of problem solving. This emphasis led to changes in the nature of the mathematics curriculum itself and strengthened the importance of problem solving in school mathematics. Throughout the 1970s the mathematics community expressed the need for clearer guidelines and a more concise sense of direction. The National Council of Teachers of Mathematics (NCTM, 1980a) responded to these concerns with a document titled Agenda for Action: Recommendations for School Mathematics of the 1980s, which outlined eight explicit recommendations, the first of which was that problem solving should be the focus of school mathematics.
By 1989 the NCTM had taken its earlier recommendation even further and was now stating that problem solving must be integral to all mathematical activities. Problem solving was to be viewed not as a separate topic but as a process that should permeate the entire mathematics program from beginning to end. Viewing problem solving in this way would provide the context in which concepts and processes could be learned. This approach enables mathematical constructs to be grounded in and emerge from students' own solutions to problems that are, to them, real and genuine. Hence, as problem solving as such is an individualized Endeavour, mathematics becomes both functional and meaningful to each individual.
Similar calls were made in Australia. State and territory education departments began to interpret problem solving as a process, placing importance on the procedures and strategies used by the students rather than their answers. Problem solving was often viewed as the central focus of the curriculum and integrated across all mathematical areas.
In 1991 the Australian Education Council published A National Statement on Mathematics for Australian Schools. The purpose of this statement was to provide a framework around which states and territories and thus schools could build their mathematics curriculum. It identifies important components of mathematics education and stales that experiences with problems should he provided to enable students to use a wide range of problem solving strategies across all topics in mathematics. This document is still the central framework for the various syllabuses that have evolved.
Today many educators believe that the most important goal of the study of mathematics is fostering and developing students' abilities to solve problems. Yet, as mentioned, adherence to traditional styles of teaching leads to difficulties with problem solving. For problem solving to be worthwhile it is essential that teachers view it as a valuable, motivating and pedagogically sound approach for introducing, developing and applying concepts and processes.
Small-group instruction, team teaching, learning centers and technology such as computers and calculators have become more common in classrooms. 1 however, this style of teaching is often only conducted after the 'real work' is completed—after the content involving rules and procedures has been taught. It is usually not used as a means of teaching a concept but rather as consolidation or reinforcement. Activities where students arc seen to be talking, interacting and even enjoying themselves are not always accepted as pedagogically sound. Yet, this is often how students learn best—in environments where they can engage in activities that allow exploration, language and socialization from which they can make sense of complex ideas.
Worthwhile problems and building new knowledge
For students to really develop mathematical ways of thinking and number sense it is essential
for good worthwhile problems to be selected for the class program. A teacher needs not only
to select problems for which there are no immediate or obvious solutions but also to select
problems which will consolidate, extend and stimulate mathematical knowledge and
understandings.
When choosing problems for a particular mathematics classroom a teacher needs to thoroughly explore the problem and the possible mathematical ideas which can be brought by the students when working through

Hasil (Bahasa Indonesia) 1:

Pemecahan masalah Istilah masalah dan memecahkan masalah terjadi banyak disiplin tetapi mungkin lebih berhubungan erat dengan matematika daripada yang lain. Selama bertahun-tahun banyak yang telah ditulis tentang masalah dan pemecahan sehingga menimbulkan berbagai sekolah pemikiran.Pendidikan matematika, pemecahan masalah telah ditekankan sejak Polya's bekerja di tahun 1940-an. POLYA, yang sering dianggap sebagai ayah dari pemecahan masalah, menggambarkannya sebagai berikutMemecahkan masalah adalah menemukan cara tidak diketahui untuk akhir yang jelas dipahami untuk menemukan cara dimana ada cara dikenal off-tangan. Sebuah pertanyaan menjadi masalah, itu harus hadir tantangan yang tidak dapat diselesaikan oleh beberapa prosedur rutin. Pemecahan masalah adalah proses menerima tantangan dan berjuang untuk mengatasinya.POLYA percaya bahwa untuk memecahkan masalah mahasiswa harus pertama datang untuk berdamai dengan apa masalahnya adalah benar-benar tentang. Sekali ia telah memperoleh wawasan tentang masalah hanya kemudian dapat menyusun rencana untuk memecahkannya. Kapan rencana telah dilakukan Polya menekankan perlu untuk melihat kembali atas masalah dalam hal solusi. Nya empat langkah pemecahan masalah yang model, yang telah digunakan sebagai dasar untuk banyak kerangka kerja berikutnya, dapat diringkas sebagai:1 memahami masalah2 merancang sebuah rencana3 melaksanakan rencana4 Lihat bockSchoenfeld (1989) dibahas pemecahan masalah dalam hal 'tugas untuk diselesaikan'. Dia percaya bahwa untuk masalah memecahkan terjadi seorang mahasiswa pertama harus termotivasi untuk memecahkan masalah dan tidak ada cara yang jelas untuk melakukannya. Dia menyatakan bahwa, untuk setiap siswa, masalah matematika adalah tugas:♦ yang siswa tertarik dan bergerak dan untuk yang ingin mendapatkan resolusi; dan♦ yang siswa tidak memiliki mudah diakses sarana matematika yang digunakan untuk mencapai resolusi.Siemon dan Booker (1990) memiliki pandangan yang sama masalah memecahkan, menyoroti kebutuhan bagi siswa untuk ingin atau perlu memecahkan masalah secara individual maupun kelompok, dan memiliki tidak berarti segera untuk melakukannya. Mereka melanjutkan untuk menjelaskan pemecahan masalah sebagai proses mencapai solusi untuk masalah, sering dengan diidentifikasi fase awal, pertengahan dan akhir. Mereka menyatakan bahwa masalah adalah tugas atau situasi:♦ yang Anda ingin atau perlu memecahkan;♦ yang Anda percaya Anda memiliki beberapa kesempatan yang wajar untuk menyelesaikan, baik secara perorangan maupun kelompok; Tapi♦ yang Anda atau kelompok tidak memiliki solusi yang segera tersedia strategi.Pandangan ini pada pemecahan masalah menyoroti bahwa masalah adalah tugas yang ada yang tidak ada solusi langsung atau jelas dan pemecahan masalah proses siswa melakukan ketika terlibat dengan tugas ini. Pemecahan masalah melibatkan terlibat dalam tugas-tugas yang strategi solusi ini tidak segera jelas. Untuk menemukan solusi yang mungkin siswa perlu menarik pada pengetahuan dan proses dan akan sering mengembangkan pengetahuan baru dan pemahaman mereka kemajuan menuju solusi.Isu utama dalam pemecahan masalah ini ide tidak ada solusi yang segera tersedia atau jelas. Tugas yang merupakan masalah bagi siswa tahun 1 tidak akan menjadi masalah bagi pelajar Year 7. Memang tugas yang akan menjadi masalah untuk beberapa siswa di tingkat tahun tertentu tidak akan menjadi masalah bagi semua siswa di tingkat tahun itu.Tugas ini tidak akan menjadi masalah bagi pelajar Year 7 seperti kebanyakan siswa akan mampu memikirkan solusi segera. Matematika pengetahuan dan pemahaman tentang konsep perkalian dan fakta-fakta dasar perkalian akan memimpin mereka segera untuk berpikir tentang 3 balita dan datang dengan solusi 15 anak anjing. Tugas ini karena itu tidak dapat dipertimbangkan masalah untuk para mahasiswa.However, for students in Year 1 who would in most cases not have this mathematical knowledge and understanding, it could well be conceived as a problem for which there is no immediate or obvious solution. In most situations a Year 1 student would need to think about how to solve this problem and would perhaps come up with the possibility of using materials and counting or drawing a diagram and counting.When selecting problems for a class program it is important to keep in mind the concept of problem solving as a task or situation for which there is no immediate or obvious solution. At times in a classroom students may be 'problem solving' when in fact they are reading 'problems' and immediately knowing what to do to solve this 'problem'. In these situations the activity being undertaken by the students could not in fact be considered to be problem solving.Strategies and problem solvingIn 1965 Polya observed that students needed techniques to help them plan for solutions. This observation provided the catalyst for over two decades of research into the identification and utilization of problem solving strategies. The outcome of this research ultimately led to strategy driven problem solving programs in schools. These programs centered around the teaching of particular strategies such as make a list, work backwards, guess and check, where the strategy rather than the problem was the focus. A strategy was introduced and then the class would solve a variety of problems using the identified strategy.For example, the strategy of 'make a list' would be taught and then the class would spend time solving problems by making a list. Students did not need to understand or come to terms with problems as they knew immediately each one could be solved using the particular. This method of teaching problem solving continued for some time until researchers began to notice that students didn't necessarily become more proficient at problem solving in situations outside of the specific lessons. This thinking was not generalised into different situations where the students had to identify the appropriate strategy for themselves.Research began to focus on problem solving and cognition and the methodology of problem solving. This emphasis led to changes in the nature of the mathematics curriculum itself and strengthened the importance of problem solving in school mathematics. Throughout the 1970s the mathematics community expressed the need for clearer guidelines and a more concise sense of direction. The National Council of Teachers of Mathematics (NCTM, 1980a) responded to these concerns with a document titled Agenda for Action: Recommendations for School Mathematics of the 1980s, which outlined eight explicit recommendations, the first of which was that problem solving should be the focus of school mathematics.By 1989 the NCTM had taken its earlier recommendation even further and was now stating that problem solving must be integral to all mathematical activities. Problem solving was to be viewed not as a separate topic but as a process that should permeate the entire mathematics program from beginning to end. Viewing problem solving in this way would provide the context in which concepts and processes could be learned. This approach enables mathematical constructs to be grounded in and emerge from students' own solutions to problems that are, to them, real and genuine. Hence, as problem solving as such is an individualized Endeavour, mathematics becomes both functional and meaningful to each individual.Similar calls were made in Australia. State and territory education departments began to interpret problem solving as a process, placing importance on the procedures and strategies used by the students rather than their answers. Problem solving was often viewed as the central focus of the curriculum and integrated across all mathematical areas.In 1991 the Australian Education Council published A National Statement on Mathematics for Australian Schools. The purpose of this statement was to provide a framework around which states and territories and thus schools could build their mathematics curriculum. It identifies important components of mathematics education and stales that experiences with problems should he provided to enable students to use a wide range of problem solving strategies across all topics in mathematics. This document is still the central framework for the various syllabuses that have evolved.Today many educators believe that the most important goal of the study of mathematics is fostering and developing students' abilities to solve problems. Yet, as mentioned, adherence to traditional styles of teaching leads to difficulties with problem solving. For problem solving to be worthwhile it is essential that teachers view it as a valuable, motivating and pedagogically sound approach for introducing, developing and applying concepts and processes.Small-group instruction, team teaching, learning centers and technology such as computers and calculators have become more common in classrooms. 1 however, this style of teaching is often only conducted after the 'real work' is completed—after the content involving rules and procedures has been taught. It is usually not used as a means of teaching a concept but rather as consolidation or reinforcement. Activities where students arc seen to be talking, interacting and even enjoying themselves are not always accepted as pedagogically sound. Yet, this is often how students learn best—in environments where they can engage in activities that allow exploration, language and socialization from which they can make sense of complex ideas.Worthwhile problems and building new knowledge
For students to really develop mathematical ways of thinking and number sense it is essential
for good worthwhile problems to be selected for the class program. A teacher needs not only
to select problems for which there are no immediate or obvious solutions but also to select
problems which will consolidate, extend and stimulate mathematical knowledge and
understandings.
When choosing problems for a particular mathematics classroom a teacher needs to thoroughly explore the problem and the possible mathematical ideas which can be brought by the students when working through