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Strategies and problem solvingIn 19
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.
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.
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Strategi dan pemecahan masalahPada tahun 1965 Polya mengamati bahwa para siswa butuh teknik untuk membantu mereka berencana untuk solusi. Pengamatan ini disediakan katalis untuk lebih dari dua dekade penelitian ke dalam identifikasi dan pemanfaatan pemecahan strategi. Hasil dari penelitian ini pada akhirnya mengarah pada strategi yang didorong pemecahan masalah program di sekolah. Program-program ini berpusat di sekitar pengajaran strategi tertentu seperti membuat daftar, bekerja mundur, rasa dan memeriksa, dimana strategi daripada masalah adalah fokus. Strategi yang diperkenalkan dan kemudian kelas akan memecahkan berbagai masalah yang menggunakan strategi diidentifikasi.Misalnya, strategi 'membuat daftar' akan diajarkan dan kemudian kelas akan menghabiskan waktu memecahkan masalah dengan membuat daftar. Siswa tidak perlu memahami atau datang untuk berdamai dengan masalah saat mereka tahu segera masing-masing dapat diselesaikan dengan menggunakan tertentu. Metode pengajaran pemecahan masalah berlanjut selama beberapa waktu sampai peneliti mulai menyadari bahwa siswa tidak perlu menjadi lebih mahir pada pemecahan masalah dalam situasi di luar pelajaran tertentu. Pemikiran ini tidak generalised ke dalam situasi yang berbeda yang mana siswa harus mengidentifikasi strategi yang tepat untuk diri mereka sendiri.Penelitian mulai fokus pada pemecahan masalah dan kognisi dan metodologi pemecahan masalah. Penekanan ini menyebabkan perubahan dalam sifat kurikulum matematika itu sendiri dan memperkuat pentingnya pemecahan dalam matematika sekolah. Selama 1970-an masyarakat matematika menyatakan perlunya untuk pedoman lebih jelas dan lebih ringkas rasa arah. Nasional Dewan guru matematika (NCTM, 1980a) menanggapi kekhawatiran ini dengan dokumen berjudul Agenda Aksi: rekomendasi untuk matematika sekolah pada 1980-an, yang diuraikan rekomendasi eksplisit delapan, yang pertama adalah bahwa pemecahan masalah harus menjadi fokus dari matematika sekolah.Pada tahun 1989 NCTM telah mengambil rekomendasi sebelumnya lebih jauh dan sekarang menyatakan bahwa pemecahan masalah harus menjadi bagian integral semua kegiatan matematika. Pemecahan masalah adalah untuk melihat bukan sebagai topik yang terpisah tetapi sebagai suatu proses yang harus menyerap seluruh matematika program dari awal sampai akhir. Melihat pemecahan masalah dengan cara ini akan memberikan konteks di mana konsep-konsep dan proses bisa dipelajari. Pendekatan ini memungkinkan konstruksi matematika didasarkan pada dan muncul dari siswa sendiri solusi untuk masalah yang, mereka, nyata dan asli. Oleh karena itu, sebagai pemecahan masalah seperti itu upaya individual, matematika menjadi fungsional dan bermakna bagi setiap individu.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.
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