COLLECTIONS OF ARTICLES: 2009

Minggu, 22 Februari 2009

PENYUSUNAN PERANGKAT PEMBELAJARAN DALAM KEGIATAN LESSON STUDY

Suhadi, S.Pd., M.Pd.
A. Pendahuluan
Penyelenggaraan proses belajar mengajar (PBM) menuntut guru untuk menguasai isi atau materi bidang studi yang akan diajarkan serta wawasan yang berhubungan dengan materi tersebut. Selain itu guru juga harus memiliki kompetensi pedagogik, sehingga guru dapat memainkan perannya sebagai fasilitator bagi pembelajaran siswanya. Sebagai penyelenggara PBM guru juga harus dapat mengembangkan sikap positif siswa dan dapat merespon ide-ide mereka. Guru harus dapat menerapkan inovasi-inovasi baru dalam pendidikan khususnya inovasi pembelajaran di kelas sebagaimana yang telah direkomendasikan para pakar pendidikan agar dapat memenuhi tuntutan kurikulum.
Melalui lesson study, guru dapat mengamati pelaksanaan pembelajaran—yang diteliti (research lesson) dan juga dapat mengadopsi pembelajaran sejenis setelah mengamati respon siswa yang tertarik dan termotivasi untuk belajar dengan cara seperti yang dilaksanakan pada kegiatan lesson study ini. Pengamatan terhadap pelaksanaan pembelajaran ini dapat dilakukan melalui pengamatan langsung terhadap pembelajaran yang diteliti maupun melalui laporan tertulis, video, ataupun forum diskusi untuk berbagi pengalaman dengan kolega. Sehingga dengan adanya Lesson study, guru dapat memperbaiki mutu pengajarannya di kelas serta meningkatkan keprofesionalannya.
Melalui Lesson study, guru dapat secara kolaboratif berupaya menterjemahkan tujuan dan standar pendidikan ke alam nyata di kelas. Kolaborasi yang dilakukan bertujuan untuk merancang pembelajaran sedemikian rupa sehingga siswa dapat mencapai kompetensi dasar yang diharapkan akan mereka kuasai. Dalam kolaborasi ini, guru-guru yang tergabung dalam kelompok lesson study berupaya merancang suatu skenario pembelajaran yang memperhatikan kompetensi dasar, pengembangan kebiasaan berpikir ilmiah, dan strategi pembelajaran yang digunakan sehingga siswa dapat memperoleh pengetahuan tertentu yang terkait dengan materi yang dibelajarkan. Guru-guru dalam kelompok lesson study juga harus membuat perangkat-perangkat lain yang diperlukan dalam pembelajaran seperti LKS, panduan guru (teaching guide), media pembelajaran, instrumen evaluasi pembelajaran.

B. Sikap yang Diperlukan Sebelum Memulai Kegiatan Lesson study
Untuk dapat memulai kegiatan lesson study maka di perlukan perubahan dari dalam diri guru sehingga memiliki sikap sebagai berikut:
1. Semangat introspeksi terhadap apa yang sudah dilakukan selama ini terhadap proses pembelajaran. Hal ini dapat dilakukan misalnya dengan mengajukan pertanyaan terhadap diri sendiri dengan pertanyaan seperti:
• Apakah saya sudah melakukan tugas sebagai guru dengan baik?
• Apakah pembelajaran yang saya lakukan telah sesuai dengan kompetensi yang diharapkan akan dicapai siswa?
• Apakah saya telah membuat siswa merasa jenuh dengan pembelajaran saya?
• Adakah strategi-strategi lain yang lebih baik yang bisa digunakan untuk melaksanakan pembelajaran ini selain strategi yang biasa saya gunakan?
• Apakah ada alternatif kegiatan belajar lain yang juga cocok untuk pembelajaran ini?
• Adakah media pembelajaran yang lebih baik yang dapat dipakai untuk pembelajaran ini selain media pembelajaran yang biasa saya gunakan?
• Mengapa siswa saya tidak termotivasi untuk mengikuti pembelajaran dari saya?
• Apakah selama ini saya telah menggunakan instrumen evaluasi yang tepat?
• Dan lain-lain.
Serangkaian pertanyaan itu yang harus dijawab dengan jujur oleh setiap guru yang ingin terlibat/dilibatkan dalam kegiatan lesson study. Jawaban terhadap pertanyaan-pertanyaan tersebut di atas tentu akan mendorong guru pada proses pencarian cara untuk menyempurnakan kekurangan-kekurangan PBM-nya selama ini.
2. Keberanian membuka diri untuk dapat menerima saran dari orang lain untuk peningkatan kualitas diri.
3. Keberanian untuk mengakui kesalahan diri sendiri.
4. Keberanian untuk mau mengakui dan memakai ide orang lain yang baik.
5. Keberanian memberikan masukan yang jujur dan penuh penghormatan

C. Pembuatan Perangkat Pembelajaran
Jika guru yang terlibat dalam kegiatan lesson study sudah memiliki atau menyadari pentingnya sikap-sikap di atas, maka langkah selanjutnya adalah memfokuskan kegiatan lesson study dengan cara menyepakati tema permasalahan dan pembelajaran yang akan diangkat dalam kegiatan. Kemudian kelompok lesson study dapat membuat perencanaan pembelajaran yang akan dilakukan. Perencanaan pembelajaran ini dituangkan dalam bentuk perangkat pembelajaran dan lembar instrumen observasi pengumpulan data PBM.
Penyusunan lembar observasi untuk mengumpulkan data PBM merupakan suatu elemen penting lesson study yang didasarkan pada rencana pembelajaran yang disusun. Lembar observasi ini akan memandu pengamat untuk memperhatikan aspek-aspek khusus yang menjadi fokus kegiatan lesson study. Pengumpulan data dari hasil observasi PBM ini biasanya terkait dengan suasana kelas, ketercapaian tujuan pembelajaran, keterlaksanaan langkah-langkah pembelajaran yang telah direncanakan, hambatan-hambatan yang muncul saat PBM berlangsung, antusiasme siswa, dsb.
Perangkat pembelajaran adalah sejumlah bahan, alat, media, petunjuk dan pedoman yang akan digunakan dalam proses pembelajaran atau digunakan pada tahap tindakan (do) dalam kegiatan lesson study. Karena lesson study adalah kegiatan yang direncanakan, dilakukan dan dinilai bersama oleh kelompok, maka perlu disadari betul bahwa keberhasilan dan kegagalan PBM adalah tanggung jawab bersama semua anggota kelompok. Oleh karena itu tujuan utama penyusunan perangkat pembelajaran adalah agar segala sesuatu yang telah direncanakan bersama dapat tercapai.
Pembelajaran merupakan suatu proses untuk mengembangkan potensi siswa, baik potensi akademik, potensi kepribadian dan potensi sosial ke arah yang lebih baik menuju kedewasaan. Dalam proses ini diperlukan perangkat pembelajaran yang disusun dan dipilih sesuai dengan kompetensi yang akan di kembangkan. Pada dasarnya perangkat pembelajaran lesson study tidak berbeda dengan perangkat pembelajaran yang biasa disiapkan oleh masing-masing guru di sekolah. Namun karena pembelajaran dalam program lesson study dirancang untuk keperluan peningkatan pembelajaran yang inovatif dan melibatkan kelompok guru serta dimungkinkan untuk dijadikan sebagai ajang penelitian tindakan kelas, maka dalam perencanaannya perangkat pembelajaran harus disusun bersama (kelompok guru), secara seksama, sistematis dan terukur.
Seperti telah disebutkan sebelumnya, pembuatan perangkat pembelajaran dan lembar observasi ini harus dilakukan secara kolaboratif oleh seluruh peserta program lesson study. Urun pendapat, berbagi pengalaman, dan diskusi dengan dilandasi komitmen untuk melakukan inovasi dan memperbaiki kualitas pembelajaran mutlak diperlukan.

D. Beberapa Dasar Pemikiran Penyusunan Perangkat Pembelajaran dalam Lesson Study
Berikut ini dipaparkan beberapa dasar pemikiran yang harus diperhatikan dalam penyusunan suatu perangkat pembelajaran dalam kegiatan lesson study:
1. Kompetensi dasar yang akan di kembangkan
Dalam kurikulum KTSP guru dituntut untuk mempunyai kreativitas lebih dalam merancang pembelajaran, agar kompetensi dasar yang telah ditetapkan dapat tercapai. Ada tiga aspek dalam kompetensi dasar untuk siswa SMP yang harus dicapai, yaitu kompetensi akademik meliputi penguasaan konsep dan metode keilmuan, kompetensi pribadi yang menyangkut perkembangan etika dan moral, serta kompetensi sosial. Ketiga kompetensi ini dikembangkan dalam proses pembelajaran, oleh karena itu harus nampak dalam perangkat pembelajaran, mulai dari rencana pembelajaran sampai evaluasi proses pembelajaran.

2. Karakteristik materi pelajaran atau pokok bahasan
Setiap materi pelajaran mempunyai sifat masing masing. Materi IPA akan berbeda dengan matematika, atau bahasa. Matematika dengan sifat materinya yang abstrak memerlukan perangkat pembelajaran yang mampu membuat lebih kongkrit. Sedangkan materi IPA yang umumnya gejalanya dapat diindera , memerlukan perangkat pembelajaran yang membuat anak mampu mengungkap gejala alam yang ada dan menganalisisnya menjadi suatu pengertian atau konsep yang utuh. Perangkat pembelajaran dalam rangka kongkritisasi persoalan maupun dalam rangka konseptualisasi fakta perlu disusun dengan mempertimbangkan kaidah keilmuan masing-masing agar hasil belajar yang akan diperoleh siswa tidak menyimpang dari kaidah keilmuan yang berlaku. Dalam rangka lesson study hendaknya guru mampu memilih dan mengorganisasi materi pelajaran dan mengemasnya sebagai bahan ajar sebagai salah satu perangkat pembelajaran. Dalam hal ini guru hendaknya tahu persis esensi dari materi pelajaran tersebut (materi esensial) agar tidak mengalami kesulitan dalam menyusun perangkat pembelajaran.

3. Karakteristik subyek didik
Subyek didik dalam proses pembelajaran pada hakekatnya adalah pribadi yang kompleks yang berbeda antara satu dengan lainnya. Walaupun mereka ada dalam kelas yang sama namun kenyataannya dalam banyak hal mereka berbeda. Variabel subyek didik yang perlu dipertimbangkan dalam menyusun perangkat pembelajaran adalah: (1) tingkat perkembangan kognitifnya; (2) gaya belajarnya; (3) lingkungan sosial budayanya; (4) keterampilan motoriknya; (5) dan lain-lain. Seringkali perangkat pembelajaran yang dibuat tidak dapat dipergunakan secara optimal karena saat membuatnya, guru mengabaikan karakteristik subyek didik. Dalam pembelajaran untuk lesson study perubahan perilaku siswa ini menjadi fokus perhatian. Seorang guru model dalam tahap refleksi (see) sesudah pembelajaran akan menguraikan/menyampaikan tentang semua kondisi yang dia ciptakan untuk membelajarkan siswa., sesuai dengan program pengembangan yang di rencanakan. Hal ini sangat penting karena refleksi para observer tidak di tujukan kepada penampilan guru (subyektif), tetapi lebih tertuju pada cara guru mengelola kegiatan pembelajaran dan aktifitas belajar siswa (obyektif).

4. Pemilihan model pembelajaran
Setiap model pembelajaran yang dipilih dalam perencanaan pembelajaran mencerminkan urutan pembelajaran yang terjadi . Urutan pembelajaran model deduktif misalnya akan berbeda dengan urutan pembelajaran model induktif, model kooperatif, atau model pembelajaran langsung. Demikian juga dengan model- model pembelajaran yang lain. Pilihan model pembelajaran ini akan mewarnai penyusunan perangkat pembelajaran, terutama dalam penyusunan skenario pembelajaran dan penyusunan lembar kegiatan siswa. Dalam pelaksanaan lesson study penetapan model pembelajaran, terutama yang inovatif diharapkan mampu mengubah paradigma pembelajaran dari pola pembelajaran yang terpusat pada guru menjadi pola pembelajaran yang menekankan pada keterlibatan murid, baik dalam mengekplorasi gejala, memecahkan masalah maupun dalam proses pembangunan konsep, ecara kooperatif di dalam kelompok, maupun secara individu.

5. Karakteristik lingkungan sekitar sekolah
Lingkungan sekolah sebenarnya sangat potensial sebagai sumber belajar. Banyak hal yang dapat dipelajari siswa dari lingkungannya, baik yang terkait dengan matematika, bahasa, IPA maupun mata pelajaran lainnya. Kemampuan anak mengekplorasi lingkungan merupakan bekal penting untuk dapat memecahkan masalah yang timbul di masyarakat, terutama jika kita memilih pendekatan Contextstual Teaching Learning ( CTL). Pengembangan kecakapan hidup bagi siswa SMP dapat dimulai dari lingkungan sekolah.. Perangkat pembelajaran yang memungkinkan anak belajar di luar kelas mempunyai karakteristik yang agak berbeda dengan perangkat pembelajaran di dalam kelas. Dalam proses pembelajaran di luar kelas siswa lebih leluasa mengekpresikan dirinya, sehingga perangkat evaluasi pembelajaran terutama evaluasi afektif lebih mudah untuk diimplementasikan. .

6. Alokasi Waktu
Alokasi waktu yang tersedia untuk kegiatan lesson study juga penting untuk diperhatikan dalam perencanaan yang dituangkan dalam perangkat pembelajaran agar pelaksanaan lesson study benar-benar efektif dan tidak berakibat sebaliknya. Perlu diingat bahwa bgaimanapun waktu merupakan salah satu faktor pembatas utama dalam PBM.

E. Perangkat Pembelajaran yang Disusun
Perangkat pembelajaran yang disusun dalam tahap perencanaan (plan) suatu kegiatan lesson study meliputi:
1. Rencana Pembelajaran
Adapun komponen rencana pembelajaran adalah:
a. Standar kompetensi dan kompetensi dasar, dalam hal ini kita harus memilih dari kurikulum
b. Pokok bahasan, dipilih dari kurikulum
c. Indikator, disusun sendiri oleh kelompok guru dan dijabarkan dari standar kompetensi.
d. Model Pembelajaran, dipilih sesuai penekanan kompetensi dan materi.
e. Skenario pembelajaran, berisi urutan aktivitas pembelajaran siswa dan mencerminkan pilihan model Pembelajaran.
f. Urutan Metode Pembelajaran, disesuaikan dengan aktivitas siswa dan model pembelajaran.
g. Media pembelajaran, dipilih dan di urutkan sesuai skenario pembelajaran.
h. Instrumen evaluasi meliputi kognitif, afektif dan psikomotorik

2. Lembar Kerja Siswa ( LKS)
Berisi langkah- langkah kegiatan belajar siswa. LKS yang di susun dapat bersifat panduan tertutup yang dapat dikerjakan siswa, sesuai dengan tuntunan yang ada, atau dapat juga LKS yang bersifat semi terbuka. LKS model ini memberi peluang bagi siswa untuk mengembangkan kreativitasnya, walaupun masih ada peranan guru dalam memberikan arahan. LKS dapat juga berupa modul pembelajaran. LKS model apapun yang di susun harus mampu memberikan panduan agar siswa dapat belajar dengan benar, baik dari segi proses keilmuan maupun dalam memperoleh konsep.

3. Teaching Guide (Panduan Guru )
Dalam Lesson study perencanaan dibuat oleh kelompok guru, namun pelaksanaannya tetap di lakukan oleh seorang guru. Agar apa yang di rencanakan sesuai dengan yang dilaksanakan, maka perlu adanya pedoman/petunjuk guru. Panduan guru ini biasanya berisi bagaimana guru harus mengorganisasi siswa, mengunakan LKS, memimpin diskusi sampai bagaimana guru harus mengevaluasi.

4. Media Pembelajaran
Media pembelajaran yang dipergunakan dalam proses pembelajaran dapat berupa perangkat lunak seperti : lembar transparansi, gambar, CD maupun perangkat keras seperti : OHP, LCD, VCD Player, piranti demonstrasi ataupun piranti ekperimen.

“Lesson study melibatkan banyak orang, dalam kaitannya dengan manajemen waktu dan media pembelajaran, maka guru harus benar- benar melakukan uji waktu sebelum tampil, apalagi jika menggunakan perangkat untuk demonstrasi atau eksperimen.”

5. Instrumen Evaluasi
Instrumen evaluasi meliputi :
a. Evaluasi kognitif untuk melihat daya serap anak terhadap materi yang di pelajari
b. Evaluasi afektif untuk melihat perubahan perilaku, etika, nilai- nilai (value) pada siswa
c. Evaluasi psikomotorik untuk mengetahui keterampilan siswa dalam melakukan pekerjaan.
Instrumen ini disusun baik dalam bentuk instrumen test maupun non test

F. Bahan Rujukan:
Hidayati, S., Listyani. E. & Warsono. 2006. Penyusunan Perangkat Pembelajaran Lesson Study. Makalah disajikan dalam Pelatihan Lesson StudyBagi Guru Berprestasi dan Pengurus MGMP MIPA Seluruh Indonesia, PPPG Kesenian Yogyakarta, tanggal 26 Nopember – 10 Desember.

Richardson, J. 2007. Lesson Study, Teacher Learn How To Improve Instruction. National Staf Depelovment Council. (Online). http://www.nsdc.org di akses 23 Mei 2008).

Sukirman. 2006. Peningkatan Keprofesionalan Guru Melalui Lesson Study. Makalah disajikan dalam Pelatihan Lesson StudyBagi Guru Berprestasi dan Pengurus MGMP MIPA Seluruh Indonesia, PPPG Kesenian Yogyakarta, tanggal 26 Nopember – 10 Desember 2006

Minggu, 11 Januari 2009

LANGKAH-LANGKAH PEMBELAJARAN MATEMATIKA

Dalam mengembangkan kreatifitas dan kompetensi siswa, maka guru hendaknya dapat menyajikan pembelajaran yang efektif dan efisien, sesuai dengan kurikulum dan pola piker siswa। Dalam mengajarkan matematika, guru harus memahami bahwa kemampuan setiap siswa berbeda-beda, serta tidak semua siswa menyenangi mata pelajaran matematika.

Konsep-konsep pada kurikulum matematika SD dapat dibagi menjadi tiga kelompok besar, yaitu penanaman konsep dasa, pemahaman konsep, dan pembinaan ketrampilan. Memang, tujuan akhir pembelajaran matematika di SD ini yaitu agar siswa terampil dalam menggunakan berbagai konsep matematika dalam kehidupan sehari-hari. Akan tetapi, untuk menuju tahap ketrampilan tersebut harus melalui langkah-langkah benar yang sesuai dengan kemampuan dan lingkungan siswa. Berikut ini penjabaran pembelajaran yang ditekankan pada konsep-konsep matematika.

1. Penanaman konsep dasar (penanaman konsep), yaitu pembelajaran suatu konsep baru matematika, ketika siswa belum pernah mempelajari konsep tersebut. Kita dapat mengetahui konsep ini dari isi kurikulum, yang cirikan dengan kata mengenal. Pembelajaran penanaman konsep dasar merupakan jembatan yang harus dapat menghubungkan kemampuan kognitif siswa yang konkrit dengan konsep baru matematika yang abstrak. Dalam kegiatan pembelajaran konsep dasar ini, media atau alat peraga diharapkan dapat digunakan untuk membantu kemampuan pola piker siswa.

2. Pemahaman konsep, yaitu pembelajaran lanjutan dari penanaman konsep, yang bertujuan agar siswa lebih memahami suatu konsep matematika. Pemahaman konsep terdiri dari atas dua pengertian. Pertama, merupakan kelanjutan dari pembelajaran penanaman konsep dalam satu pertemuan. Sedangkan kedua, pembelajaran pemahaman konsep dilakukan pada pertemuan yang berbeda, tetapi masih merupakan lanjutan dari penanaman konsep. Pada pertemuan tersebut, penanaman konsep dianggap sudah disampaikan pada pertemuan sebelumnya, disemester atau kelas sebelumnya.

3. Pembinaan ketrampilan, yaitu pembelajaran lanjutan dari penanaman konsep dan pemahaman konsep. Pembelajaran pembinaan ketrampilan bertujuan agar siswa lebih terampil dalam menggunakan berbagai konsep matematika. Seperti halnya pada pemahaman konsep, pembinaan ketrampilan juga teratas dua pengertian. Pertama, merupakan kelanjutan dari pembelajaran penanaman konsep dan pemahaman konsep dalam satu pertemuan. Sedangkan kedua, pembelajaran pembinaan ketrampilan dilakukan pada pertemuan yang berbeda, tapi masih merupakan lanjutan dari penanaman dan pemahaman konsep. Pada pertemuan tersebut, penanaman dan pemahaman konsep dianggap sudah disampaikan pada pertemuan sebelumnya, disemester atau kelas sebelumnya.

Selasa, 06 Januari 2009

Bagaimana Melakukan Diskusi Matematika dengan Siswa?

Oleh: Marsigit
Jika seorang guru ingin melakukan diskusi matematika dengan siswa maka pastikanlah apakah diskusi akan dilakukan secara klasikal, pada kelompok atau secara individual. Berdasarkan pengalaman penulis, maka secara umum dapat diberikan semacam pedoman bagaimana seorang guru seyogyanya melakukan diskusi:
1. dengarkanlah apa yang mereka katakan
2. usahakan suasana bersifat mendukung diskusi
3. usahakan suasana berpikir merdeka
4. usahakan agar siswa mempunyai inisiatif
5. usahakan agar siswa mempunyai rasa percaya diri
6. gunakan kata-kata yang mudah dipahami oleh siswa
7. usahakan hubungan yang baik antara guru dan siswa
8. berusaha mengetahui ide/gagasan siswa
9. menampung ungkapan siswa sebagai informasi
10. utarakan balikan secara bertahap
11. gunakan ilustrasi untuk mempermudah komunikasi
12. menghargai setiap ide yang disampaikan oleh siswa
13. usahakan agar diskusi bersifat efaktif dan bermanfaat.
14. memahami bahwa siswa memerlukan penjelasan
15. fokus pada pertanyaan siswa/penanya (jangan Jawa: disambi)
16. memikirkan hal-hal yang terkait dengan pertanyaan.
17. memikirkan solusi jawaban
18. bersifat jujur dan terusterang
19. berpikir positif (tidak prejudice) terhadap si penanya
20. tidak mempunyai kepentingan ganda misalnya mengajukan pertanyaan yang sulit untuk tujuan menghukum
21. gunakan referensi-referensi untuk mencari solusi.
22. kembangkan sikap menghargai dan empati
23. hindari sikap menang sendiri/otoriter
24. letakkan persoalan pada jejaring konsep yang lebih luas
25. usahakan agar siswa senang melakukannya kembali
26. jika memungkinkan libatkan siswa yang lainnya
27. menyadari manfaat jawaban terbuka
28. menyadari bahwa pertanyaan siswa adalah awal dari pengetahuannya
29. Jangan memberi jawaban yang memojokkan siswa sehingga mereka merasa malu
30. memperhatikan alokasi waktu dan konteks sekitar

Pada pembelajaran tradisional guru terpaksa sibuk mengontrol siswanya

Oleh: Marsigit
Seperti kita ketahui bersama bahwa pembelajaran tradisional mempunyai ciri-ciri menggunakan metode tunggal yaitu ekspositori dengan delivery method, memposisikan guru sebagai pelaku utama dan siswa terposisikan sebagai peserta didik yang pasif. Dengan asumsi ingin memberi bekal materi sebanyak-banyaknya kepada siswa, maka pada pembelajaran tradisional, guru terpaksa melakukan berbagai kegiatan kontrol agar siswa bersikap kooperatif dan memperhatikan guru. Kontrol dilakukan melalui berbagai cara bahkan jika perlu ketika guru mengajukan pertanyaan sekalipun. Hal ini disebabkan karena belum dipahaminya paradigma pendidikan sebagai kebutuhan siswa dan tidak adanya skema untuk itu. Di samping itu guru juga belum mampu mengembangkan skema pembelajaran untuk melayani berbagai macam kebutuhan akademik siswa. Berikut contoh cuplikan berbagai aterensi yang ditemukan penulis yang menggambarkan bagaimana guru melakukan kontrol terhadap siswa melalui directed teaching:

Aterensi 1:
Wahyu ngantuk ya ?
Nah tadi sudah makan belum ?
Sudah.
Banyak nonton T.V. ya ?
Nah berapa coba ?

Aterensi 2:
Bagaimana cara penyelesaiannya ?
Penyelesaiannya bagaimana ?
Bagaimana Anto ?
YANG LAIN DIAM.

Aterensi 3:
Sudah apa belum ?
Nah kalau sudah diteliti dulu dari nomor satu sampai nomor sepuluh
AYO PRIHANTO KAMU SUDAH SELESAI ?
SUDAH DITELITI ?
AYO DITELITI LAGI
COBA SAMPAI BETUL
TIDAK HANYA DILIHAT LHO YA PRIHANTO

Aterensi 4:
Ayo Candra duduknya bagaimana ?
Hayo yang tertib Rona, ayo Rona duduknya yang bagus.
Nanti supaya betul semua

Aterensi 5:
Nol dapat dikurang lima ?
Sebetulna dapat tapi ini kurang ya to ?
Jadi barang tidak ada dikurangi lima, maka ...
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The Role of Cognitive Development Theory for Mathematics Education

By. Marsigit
Theories about how children think and learn have been put forward and debated by philosophers, educators and psychologists for centuries; however, the contemporary thinking about education, learning and teaching is not 'brand new'; certain theories have been absorbed and transformed over time or translated into modern terms; and, some of them become prominent and influential (Wood, 1988).There is no question at all to the fact that anything related to the term 'cognitive development' is greatly embeded to the work of two greatest figures of developmental psychology in twentieth century, Jean Piaget and Lev Vygotsky. Piaget's influence on the primary mathematics curriculum and on research developmental psychology has been immense; Vygotsky's work has been gaining in influence over the past ten years. Traditionally, primary education has looked to child development and psychology for theoretical guidance and underpining (Gipps, 1994); Piaget's positive contribution, however, was both to start a theoretical debate about young children's intellectual development and to encourage the close observation of children; Vygotsky, the Russian psychologist, has given us a number of crucial insights into how children learn, of which to have particular consequences for classroom.
Observing child's behaviours when she interacts with surrounding objects or people, may be the starting point to discuss about the mechanisms of her cognitive development. In the interactions she may look at the object, take hold of it, listen to the sound or talk to the people; more than just these, she may also categorize, memorize or even make the plan for a certain activity. Such behaviour is taken for granted, much is automatic, yet for it happen at all requires the utilization of complex cognitive processes (Turner, 1984). By perceiving or attending to the visual and auditory surroundings, she may keep these in her mind. Her recognition of the functions of the objects, for example that the chair has the function for sitting, is related to the using of her memories and her developing the concepts of a 'chair'. Cognitive processes underlie the ability to solve problems, to reason and to learn. Implicitly, the above proposition lead that the term of 'cognitive development' is associated with the development of the processes and the content to which these processes are applied. Behaviourists characterize internal processes by associating them with the 'stimulus-response'. The reason why a person gives a particular response to a particular stimulus was thought to be either because the two were associated in some way, that is, the response was 'conditioned', or because the appearance of this response had been rewarded previously (Turner, 1984). Information-processing approach assumes that a person who perceives stimuli, stores it, retrieves it, and uses it (ibid, 5); information is transformed in various ways at certain stages in its processing.
Piaget (1969) admitted that any explanation of the child's development must take into consideration two dimensions: an ontogenetic dimension and a social dimension (in the sense of the transmission of the successive work of generations). Piaget used a biological metaphor and characterized mathematical learning as a process of conceptual reorganization. At the heart of Piaget's theory is the idea of structure; cognitive development, and in particular the emergence of operational thought, is characterized in term of the emergence of new logical or logico-mathematical structures. Further, Light states that Piaget's theory has a functional aspect, concerned with intelligence as adaptation, with assimilation, accommodation and equilibration; his main contribution and influence lay in his structural account of cognition.
Central to Piaget's view of the child is the assumption that the child actively constructs his own ways of thinking through his interactions with the environment (ibid, p.216). Piaget used observations of his own children to formulate some aspects of the development of intelligence. Absolutists view mathematical truth as absolute and certain; and, progressive absolutists view that value is attached to the role of the individual in coming the truth (Ernest, 1991). They see that humankind is seen to be progressing, and drawing nearer to the perfect truths of mathematics and mathematics is perceived in humanistic and personal terms and as a language (ibid, p.182). Piaget provides 'a license for calling virtually anything a child does education (McNamara, 1994); moreover, an analysis of the development of the progressive movement in the UK suggests that it was only after child-centered methods were established in some schools that educationists turned to psychologists such as Piaget to provide a theoretical justification for classroom practice. The other foundation for a number studies in Psychology, in which Piaget played a prominent part, seems to be influenced greatly by Durkheim's assumption, as Luria cited that the basic processes of the mind are not manifestations of the spirit's inner life or the result of natural evolution, but rather originated in society. And Vygotsky stressed on using socio-cultural as the process by which children appropriate their intellectual inheritance.
Reference:
Ernest, P.,1991, The Philosophy of Mathematics Education, London : The Falmer Press.
Gipps, C., 1994, 'What we know about effective primary teaching' in Bourne, J., 1994, Thinking Through Primary Practice, London : Routledge.
McNamara, D., 1994, Classroom pedagogy and primary practice, London : Routledge.
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Turner, J.,1984, Cognitive Development and Education, London: Methuen.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Wood, D., 1988, How Children Think & Learn, Oxford: Basil Blackwell.

Indikator Guru Matematika yang Profesional

Oleh : Marsigit
(Dirangkum dari berbagai sumber)
1. Memanfaatkan dan mengembangkan lingkungan belajar matematika
2. Mengembangkan sumber-sumber belajar matematika
3. Melibatkan siwa dalam kegiatan apersepsi pembelajaran matematika
4. Berhasil mempromosikan motivasi siswa belajar matematika
5. Mengembangkan pembelajaran matematika secara klasikal
6. Mengembangkan pembelajaran matematika secara diskusi kelompok
7. Mengembangkan pembelajaran matematika dalam pelayanan individu
8. Menghubungkan matematika dengan keperluan lain dalam mata pelajaran lain
9. Mengembangkan sturktur pbm matematika
10. Mengembangkan skenario interaksi pbm matematika
11. Mengembangkan skenario pencapaian kompetensi matematika
12. Mengembangkan skenario kegiatan matematika siswa
13. Mengembangkan penilaian berbasis kelas
14. Melakukan kegiatan refleksi pbm matematika
15. Meneliti pbm matematika yang diselenggarakannya
16. Menjadi pengembang kurikulum matematika sekaligus silabusnya
17. Mengembangkan media pembelajaran matematika
18. Mengembangkan alat peraga matematika
19. Mampu menyusun buku text pelajaran matematika
20. Mampu mengembangkan berbagai macam LKS
21. Menyelenggarakan pbm matematika berdasarkan suatu teori baik metode maupun teori belajar siswa
22. Melakukan refleksi pbm matematika
23. Mampu memanfaatkan dan mengembangkan ICT untuk pbm matematika misalnya pemenfaatan BLOG
24. Melakukan inovasi pbm matematika secara kontinue dan konsisten
25. Mengembangkan pelayanan terhadap kebutuhan belajar matematika siswa termasuk kesulitan-kesulitannya.
26. Mampu bekerjasama dengansesama guru dalam memperbaiki pbm matematikanya
27. Mampu mengkomunikasikan problematika pbm matematika kepada orang lain.
28. Secara ikhlas dan terbuka menerima kritik dan saran dai orang lain tentang kekurangan dalam menyelenggarakan pbm matematika.
29. Selalu berusaha menjadi inisiator atau orang terdepan dalam mengembangkan pbm matematika yang inovatif.
30. Memandang bahwa kagiatan mengajar juga merupakan bagian dari mengisi dan mengamalkan ibadahnya.
31. Merasa bertanggungjawab dan konsisten kepada semua level kepentingan, mulai diri sendiri, siswa, teman guru, sekolah, Kepala Sekolah, dst.
32. Mampu menulis karya-karya atau artikel di penerbitan, koran, majalah atau jurnal mengenai aspek pengembangan pbm matematika yang diselenggarakannya.
33. Aktif mengikuti kegiatan-kegiatan MGMP, seminar, lokakarya dan lesson study bidang pbm matematika.
34. Terlibat aktif di sekolah dalam bidang pengembangan pbm matematika
35. Kegiatan mengajar matematika menjadi kegiatan dan prioritas utamanya
36. Menguasai konten matematika
37. Menyadari dan mampu mengimplementasikan pbm matematika pada tataran kualitas yang lebih tinggi (kualitas ke dua)

REFLECTION ON THE TEACHING OF “THE MULTIPLICATION ALGORITM OF THE 3rd GRADE OF PRIMARY SCHOOL” THROUGH VTR

By Marsigit
OVERVIEW
Recently study in Indonesia indicated that the use of VTR (Video Tape Recorder) in the teacher training program was perceived by the teachers as good and useful. There is a higher frequency to use the VTR to promote teachers’ professional development in Japan and in developed countries; however, in Indonesia, it pops up like a jack-in-the-box. VTR for teacher education and reform movement in Mathematics Education, specifically for developing lesson study has some benefits as: a) short summary of the lesson with emphasis on major problems in the lesson, b) components of the lesson and main events in the class, and, c) possible issues for discussion and reflection with teachers observing the lesson (Isoda, M., 2006).
Katagiri, S (2004) listed the types of mathematical thinking as mathematical attitudes, mathematical thinking related to mathematical methods, and mathematical thinking related to mathematical contents. This identification of mathematical thinking by Katagiri can be the starting point to reflect any mathematics teaching learning process at school as for to reflect the teaching of “the multiplication algoritm of the 3rd grade of primary school” by Mr. Hideyuki Muramoto; then, the VTR of this lesson will be targeted for the series of activities: observation and reflection.
Characterizing The Lesson from Lesson Plan
The preliminary characteristics of Muramoto’s teaching are stipulated in the Lesson Plan such as follows:
a. Theme : Third grade mathematics lessons that foster students’ ability to use what they learned before to solve problems and make connections in order to solve problems in new learning situations
b. Method: Teaching “the Multiplication algorithm (1)” in a way that develops students who can use what they learned before to solve problems in new learning situations by making connections.
c. Goals of the Unit: To be able to think about how to carry out the calculation of a 2-digit number x a 1-digit number by using what was previously learned about multiplication (mathematical thinking).
d. Scenario of Teaching
1) Developing teaching that help students to become aware of the connection between what they learned before and what they are learning now and use previously learned knowledge to overcome obstacles in a new situation.
2) Connections between previously learned knowledge and new learning
3) Representing a problem situation with diagrams based on the idea of “how many times as much as a unit quantity” consistently and helping students to understand the situation and solution of the problem more clearly.
4) Developing lessons that incorporate this idea and help students to use the diagram to think
Characterizing the Lesson through VTR
a. The problem of video taping
- The quality of pictures are relatively good
- The single camera made the limitation of landscaping the class
- The small caption in the screen helps to catch more the picture of the class
b. The components of the lesson
- The whole class teaching has reduced the complexity of class interaction into the simple or linear pattern of interaction between teacher and students.
- Highlighting the certain ideas from certain student has ignored the other students’ ideas.
- Highlighting the certain aspect of mathematical thinking of a certain students endanger the total management of the class.
c. Encouraging and uncovering students’ mathematical thinking
- Teacher’s effort in encouraging and uncovering students’ mathematical thinking were effective enough.
- Teacher’s effort in serving individual students has not been effective yet.
- Some of the students were able to perform mathematical thinking
- Teacher was able to achieve the goal of the lesson
- Mathematical thinking of a certain students can be a model for others.
- Different students, in the same allocation of time, did similar problems by employing different methods to cultivate the similar results.
- Students’ discussion among themselves has not emerged yet.
- Students’ involvements in classroom management were still limited.
- Teacher has effectively employed the proper teaching aids.
CONCLUSION
The conclusion of the paper highlights some problems as follows:
- The problem of the reduction of the complexity of classroom interaction into the simple or linear pattern between teacher and his students.
- The problem of landscaping the whole classroom activities
- The negative correlation between focusing a certain aspect of students thinking and reducing the variant of their learning contexts.
- The problem of the pattern for the relation for promoting individual needs and the whole classroom management.
- The problem of the gap amongst teachers’ effort (including methods and media), students’ findings and the concept/understanding/rational of the vertical way of calculating 23 times 3.
- The problem of matching the theory of the concept of mathematical thinking and the factual condition of students’ mathematical thinking.
- The problem of mathematical thinking of the lower achievement students.
- The problem of exploring intrinsic, extrinsic and systemic of mathematical thinking.
REFERENCE:
Isoda, M. (2006). Reflecting on Good Practices via VTR Based on a VTR of Mr.
Tanaka's lesson `How many blocks? Draft for APEC-Tsukuba Conference in
Tokyo, Jan 15-20, 2006
Marsigit, (2006), Lesson Study: Promoting Student Thinking On TheConcept Of Least
Common Multiple (LCM) Through Realistic Approach In The 4th Grade Of
Primary Mathematics Teaching, in Progress report of the APEC project: “Colaborative Studies on Innovations for Teaching and Learning Mathematics in Diferent Cultures (II) – Lesson Study focusing on Mathematical Thinking -”, Tokyo: CRICED, University of Tsukuba.
Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. in Progress
report of the APEC project: “Colaborative Studies on Innovations for Teaching
and Learning Mathematics in Diferent Cultures (II) – Lesson Study focusing on
Mathematical Thinking -”, Tokyo: CRICED, University of Tsukuba

The Role of Cognitive Development Theory for Mathematics Education

Ritual Mathematics

By Marsigit
In Javanese society all aspect of daily life or activities can be viewed as underpine from religious perspectives. It is not surprising that, for example, in a certain village, the people in one week get invitation to come to ritual activities for more than seven. Some of the important activities are the celebration of the 'fifth' and the 'thirty fifth' baby birthday, the ritual feast to mark some one's death in the 'seventh day', 'forthyth day', 'a hundreth day' and 'a thousanth day'. The problem is the people, specifically the man who are responsible of carrying out the ritual meal, should exactly decide that the death has long been whether seven days, forthy days, one hundreth days, or one thousand days. They do it well and they learn it for generations. The people, who are concerned about it, sometimes involves in the dialog informally to justify whether the counting the numbers of tha days is right or wrong. Most of them are relatively correct. They just use simple formula which is ussualy spoken and not ever written, e.g. 'nomosarmo', 'norosarmo', nonemsarmo', etc.
They use three kinds of numbers system at the same time : 10 based numbers system, 7 based numbers system and 5 based numbers system, in the frame work of position system. Position system for numbers was found by Indian and Javanese people knew it before the Europen because it directly was brought by Indian to Indonesia. They use Numbers System Basis 10 when they should decide the duration of time; 35 days, 40 days, 100 days and 1000 days. They use Numbers System Basis 7 (Week system) when they use the name orderly of the days : Sunday, Monday, Tuesday, Wednesday, Thuersday, Friday and Saturday. They use Numbers System Basis 5 (Dino Pasaran system) when Javanese people have been using system 'Dino Pasaran' for along time before Western system of callendar came to Indonesia. In this system there are only five days orderly in a cycle of period of time called 'Pasar'. Those are 'Legi', 'Pahing', 'Pon', 'Wage', 'Kliwon'. One day in this system is equals to one day in Week system, that is 24 hours. Thus one Dino Pasaran has five days, two Dino Pasaran has 10 days, three Dino Pasaran has 15 days, etc.
Mathematically, for all of the system numbers, the notation of numbers 'm' can be written as polynomial from 'b' such as follows : m = a0 bn-1 + a1 bn-2 + a2 bn-3 + ... + an-1 b + an
where, b is any numbers greater than 1, and a is the basis. According to this formula, we can write any number at any system using the same pattern i.e. for basis 10, basis 7 and basis 5. In Basis 10 we have the numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In Basis 7 we have the numbers : 0, 1, 2, 3, 4, 5, 6
We can match the name of the day with those numbers orderly : 0 = Sunday 1 = Monday; 2 = Tuesday; 3 = Wednesday; 4 = Thuersday; 5 = Friday and 6 = Saturday. In Basis 5 we have the numbers : 0, 1, 2, 3, 4. We can match the name of the day in Dino Pasaran with those numbers orderly : 0 = Legi, 1 = Pahing, 2 = Pon, 3 = Wage, 4 = Kliwon. How can we write 35 of Basis 10 into Basis 7 ? The following is the formula 3510 = 5 x 71 + 0 x 70 = 507. How can we write 100 of Basis 10 into Basis 7 ? The formula is 10010 = 2 x 72 + 0 x 71 + 2 x 70 = 2027. How can we write 1000 of Basis 10 into Basis 7 ? The formula is 100010 = 2 x 73 + 6 x 72 + 2 x 71 + 6 x 70 = 26267
How can we write 35 of Basis 10 into Basis 5? The following is the formula 3510 = 7 x 51 + 0 x 50 = 705. How can we write 100 of Basis 10 into Basis 5? Thai is 10010 = 4 x 52 + 0 x 51 + 0 x 50 = 4005 How can we write 1000 of Basis 10 into Basis 5 ? That is 100010 = 1 x 54 + 3 x 53 + 0 x 52 + 0 x 51 + 0 x 50 = 130005
Javanese people always have two unseparated name for the day. For example : Sunda Legi, Sunday Kliwon, Monday Wage, Friday Pahing, Saturday Kliwon, Saturday Pon, etc. It is clear that the numbers of combinations is 35 names. If this day is Monday Pahing then the duration up to on the next Monday Pahing is 7 x 5 = 35 days. Thus when somebody wish to celebrate the 35 th of his son's birthday he just waiting for the next day with the same name. It is easy and he need not to do with mathematics at all in his mind. Forthyth days ritual feast to mark some one's death by transforming 4010 = 557 and 4010 = 805. Numbers 55 is ended by 5; it mean that 'the fourty days duration of time' will begin on the day i and ended on the day 5 th of Week System. Thus, if now is Sunday then forty days to come will be the fifth day from Sunday that is Thuersday. Javanese people called 'five' as 'limo' or briefly 'mo'. Number 80 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then it will be 40 days on the next Thuersday Kliwon.
If this day is Friday Legi then it will be 40 days on the next Tuesday Legi. If this day is Wednesday Pahing then it will be 40 day on the next Sunday Pahing, etc. Javanese people just called 'no mo; sar mo' that mean : no = dino = day; mo = limo = five; and sar = Pasar = Basis 5. Thus, no mo means 'the fifth day of Week System' sar mo means 'the fifth day of Pasar System'
For one hundred days ritual feast to mark some one's death as I described that :10010 = 2027 and 10010 = 4005. Number 202 is ended by 2; its mean that 'the one hundred days duration of time' will begin on the day i and ended on the day 2 th of Week System . Thus, if now is Sunday then forty days to come will be the 2 nd day from Sunday that is Monday. Javanese people called 'two' as 'loro' or briefly 'ro'. Number 400 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then the next 100 days will be on the next Monday Kliwon. If this day is Friday Legi then the next 100 days will be on Saturday Legi. If this day is Wednesday Pahing then the next 100 day will be on Sunday Pahing, etc. Javanese people just called 'no ro; sar mo' that mean : no = dino = day; ro = loro = dua; and sar = Pasar = Basis 5. Thus, no ro means 'the second day of Week System' sar mo means 'the fifth day of Pasar System'. One thousand days ritual feast to mark some one's death is calculated by the same way they use the formula 'no nem; sar mo' that mean 'the sixth day of Week Syatem and the fifth day of Pasar System'

Piaget's Work and its Relevance to Mathematics Education

By Marsigit
Piaget's theory of intellectual development focuses on two central aspects of the progressive view of childhood; first, on the centrality of children's experience, especially physical interaction with the world; second, on the unfolding logic of children's thought, which differs from that of the adult (Ernest, 1991). Piaget proposed four major stages of intellectual development: (1). the sensori-motor stage (birth to 1 1/2 to 2 year), (2). the pre-operational stage (2 to 7 years), (3). the concrete operational stage (7 to 12 years), (4). the formal operational stage (12 to 15 years and up). The characteristic in which a remarkably smooth succession of stages, until the moment when the acquired behaviour presents seems to be recognizes as 'intelligence' (Piaget and Inhelder, 1969); there is a continuous progression from spontaneous movements and reflexes to acquired habits and from the latter to intelligence. They further stated that this mechanism is one of association, a cumulative process by which conditionings are added to reflexes and many other acquisitions to the conditioning themselves. They then regarded that every acquisition, from the simplest to the most complex, is a response to external stimuli, that is a response whose associative character expresses a complete control of development by external connections. They described that this mechanism consists in assimilation, that reality data are treated or modified in such a way as to become incorporated into structure of subject.
In moving from the sensory-motor stage to operational thought, several things must occur during the preoperational period (Becker, et al., 1975); there must be a speeding in thought or actions, there must be an expansion of the contents and scope of what can be thought; and there must be concern not only with the results of action but also with understanding the processes by which a result is achieved. Piaget and Inhelder (1969) outlined that there are three levels in the transition from action to operation; at the ages of two or three there is a sensory-motor level of direct action upon reality; after seven or eight there is the level of the operations in which concern transformations of reality by means of internalized actions that are grouped into coherent and reversible systems; and between these two level there is another level obviously represents an advance over direct action in which the actions are internalized by means of the semiotic function and characterized by new and serious obstacles. Toward the end of the preoperational stage, the basis for logico-mathematical thinking has been laid in the use of language, but the child is still far from reaching operational thought (Becker, et al., 1975).
During the years between two and seven the child learns much about the physical world; some of this is spontaneous, while other is deliberately taught by parents and teachers; despite the many intellectual feats of the period, children do not reason in a logical or a fully mathematical way. Children's thinking in the pre-operational period is characterized by what Piaget called moral realism as well as animism and egocentrism (ibid, p.102). Animism is the failure to adopt one stance towards inanimate objects and another towards oneself; moral realism is the consequence of viewing morality in one sense only; egocentrism is the consequence of the child's taking only one perspective; and the child achieves the next stage of intellectual development when at last he can consider a situation from several different aspects - in other words, he can de-centre (ibid, p.103). After many experiments, Piaget and his colleagues concluded that there is a sequence of development for each of the conservations; each experience requires that the child must judge whether the two things are still the same or are different when the entities is transformed in appearance by being changed in shape or transferred to another receptacle (Sutherland, 1992). It has been shown that children in the period of concrete operation can perform the mental operation of reversibility and can attend to several aspects of a situation at once (ibid, p.109).
For Piaget, an operation is a mental action (Becker, et al., 1975) that usually occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation; an operation is said to be concrete if it can be used only with concrete referent rather than hypothetical referents. The first obstacle to operations (Piaget and Inhelder, 1969), then, is the problem of mentally representing what has already been absorbed on the level of action. In the concrete operational stage, thinking shows many characteristics of mature logic, but it is restricted to dealing with the 'real' (Becker, et al., 1975). The second obstacle to this stage is that on the level of representation (Piaget and Inhelder, 1969); achieving this systematic mental representation involves constructive processes analogous to those which take place during infancy; the transition from an initial state in which everything is centered on the child's own body and actions to a decentered state. The third obstacle is related to the complexity of the using of language and the semiotic function involving more than one participant.
Formal operations involve thinking in terms of the formal propositions of symbolic logic and mathematics or in terms of principles of physics (Becker, 1975); one can deal with the hypothetical and one can deal with operations on operations. Piaget studied the development of logical thinking in adolescence and reflective abstraction, that very human capacity to be aware of one's own thoughts and strategies. Piaget assert that the basis of all learning is the child's own activity as he interacts with his physical and social environment; the child's mental activity is organized into structures and related to each other and grouped together in the pattern of behaviour (Adler, 1968). Piaget also asserts that mental activity is a process of adaptation to the environment which consists of two opposed but inseparable processes, assimilation and accommodation (ibid, p. 46). The child does not interact with his physical environment as an isolated individual but as part of a social group; as he progresses from infancy to maturity, his characteristic ways of acting and thinking are changed several times as new mental structure emerge out of the old ones modified by accumulated accommodations (ibid, p.46).
Piaget found that there is a time lag between the development of a child's ability to perceive a thing and the development of this ability to form a mental image of that thing when it is not perceptually present (Adler, 1968). The development of the child's concepts of space, topological notions, such as proximity, separation, order, enclosure, and continuity, arise first; projective and Euclidean notions arise later; and his grasp of order relation and cardinal number grow hand in hand in the concept of numbers (ibid, p.51). Piaget also asserts that a child progresses through the four major stages of mental growth is fixed; but, his rate of progress is not fixed; and, the transition from one stage to the next can be hastened by enriched experience and good teaching (ibid, p.53). Based on all the above propositions, some of their implications for the mathematics teaching in the primary school can be asserted. Piaget maintained that internal organization determines how people respond to external stimuli and that this determines man's unique 'model of functioning' which is invariant or unchangeable (Turner, 1984); a person attempts to make sense the environmental stimulus by using his existing structure or by assimilating or accommodating it.
The structure and their component schemes were said to change over time through the process of equilibration; if a subject finds that her present schemes are inadequate to cope with a new situation which has arisen in the environment so that she cannot assimilate the new information, she will be drawn, cognitively, into disequilibrium (ibid, p.8). Given these fundamental postulates of Piaget's theory : internal organization, invariant functions, variant structures, equilibration and organism/environment interaction; what then are the implications for mathematics education in the primary mathematics school ?
References:
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.

Vygotsky's Work and Its Relevance to Mathematics Education

By. Marsigit
One of the most fundamental assumptions that guided Vygotsky's attempt to reformulate psychology was that in order to understand the individual, one must first understand the social relations in which the individual exists (Wertsch, 1985); Vygotsky argued that the social dimension of consciousness is primary in time and in fact, the individual dimension of consciousness is derivative and secondary. Thus, to explain the psychological, we must look not only at individual but also at external world in which that individual life has developed (Tharp and Gallimore, 1988). The first key feature of Vygotsky's theory is that of internalization. The process by which the social becomes the psychological is called internalization (Tharp and Gallimore, 1988); the individual's plane of consciousness is formed in structures that are transmitted to the individual by others in speech, social interaction, and the processes of cooperative activity; thus, individual consciousness arises from the actions and speech of others. Wertsch (1985) listed that Vygotsky's account of internalization is grounded in four major points : (1) internalization is a process wherein an internal plane of consciousness is formed; (2) the external reality at issue is a social interactional one; (3) the specific mechanism at issue is the mastery of external sign forms; and (4) the internal plane of consciousness takes on a 'quasi-social' nature because of its origins. In the beginning of the transformation to the intramental plane, the child need not understand the activity as the adult understands, need not be aware of its reason or of its articulation with other activities (Tharp and Gallimore, 1988); all that is needed is performance, through assisting interaction; through this process, the child acquires the plane of consciousness of the natal society and is socialized, acculturated, made human.
The second key feature of Vygotsky's theory is that of the zone of proximal development; this refers to the gap that exists for children between what they can do alone and what they can do with help from someone more knowledgeable or skilled than themselves (Gipps, 1994). Vygotsky introduced the notion of zone of proximal development in an effort to deal with two practical problems in educational psychology (Wertsch, 1985): the assessment of children's intellectual abilities and the evaluation of instructional practices. He argued that it is just as crucial, if not more so, to measure the level of potential development as it is to measure the level of actual development; existing practices were such that 'in determining the mental age of a child with the help of tests we almost always are concerned with the actual level of development' (ibid, p. 68). Vygotsky argued that the zone of proximal development is a useful construct concerns processes of instruction (ibid, p. 70); instruction and development do not directly coincide, but represent two processes that exist in very complex interrelationships.
Assisted performance defines what a child can do with help, with the support of the environment, of others, and of the self (Tharp and Gallimore, 1988); the transition from assisted performance to unassisted performance is not abrupt. They present problems through the zone of proximal development (ZPD) in a model of four stages; (1) the stage where performance is assisted by more capable others (Stage I); (2) the stage where performance is assisted by the self (Stage II); (3) the stage where performance is developed, automatized, and 'fossilized' (Stage III); and (4) the stage where de-automatization of performance leads to recursion back through the ZPD (Stage IV). Tharp and Gallimore (1988) outlined the propositions of the problems in the Stage I as follows : (1) before children can function as independent agents, they must rely on adults or more capable peers for outside regulation of task performance; (2) the amount and kind of outside regulation a child requires depend on the child's age and the nature of the task; (3) the child may have a very limited understanding of the situation, the task, or goal to be achieved; at this level, the parent, teacher, or more capable peer offers directions or modeling, and the child's response is acquiescent or imitative; (4) only gradually does the child come to understand the way in which the parts of an activity relate to one another or to understand the meaning of the performance; (5) ordinarily, the understandings develops through conversation during the task performance; (6) the child can be assisted by questions, feedback, and further cognitive structuring that is such assistance of performance has been described as scaffolding, by Wood, Bruner, and Ross (1976); (7) the various means of assisting performance are indeed qualitatively different; (8) a child's initial goal might be to sustain a pleasant interaction or to have access to some attractive puzzle items, or there might be some other motive that adults cannot apprehend; (9) the adult may shift to a subordinate or superordinate goal in response to ongoing assessment of the child's performance; (10) the task of Stage I is accomplished when the responsibility for tailoring the assistance, tailoring the transfer, and performing the task itself has been effectively handed over to the learner; this achievement is gradual, with progress occuring in fits and starts.
For the Stage II, Tharp and Gallimore (1988), outlined the propositions: (1) the child carries out a task without assitance from others; however, this does not mean that the performance is fully developed or automatized; (2) the relationships among language, thought, and action in general undergo profound rearrangements - ontogenetically, in the years from infancy through middle childhood; (3) control is passed from the adult to the child speaker, but the control function remains with the overt verbalization; the transfer from external to internal control is accomplished by transfer of the manipulation of the sign from others to the self; (4) the phenomenon of self-directed speech reflects a development of the most profound significance; self-control may be seen as a recurrent and efficacious method that bridges between help by others and fully automated, fully developed capacities; (5) for children older than 6 years, semantic meaning efficiently mediates performance; (6) for children, a major function of self-directed speech is self-guidance; this remains true throughout lifelong learning. For the Stage III, Tharp and Gallimore (1988), outlined the propositions: (1) once all evidence of self-regulation has vanished, the child has emerged from the ZPD into the development stage for the task; (2) the task execution is smooth and integrated; it has been internalized and automatized; (3) assistance from the adult or the self is no longer needed; indeed assistance would now be disruptive; (4) it is at this stage that self-consciousness itself is detrimental to the smooth integration of all task components; (5) this is a stage beyond self-control and beyond social control; (6) performance here is no longer developing; it is already developed. For the Stage IV, Tharp and Gallimore (1988), outlined the propositions: (1) there will be a mix of other-regulation, self-regulation, and automatized processes; (2) once children master cognitive strategies, they are not obliged to rely only on internal mediation; (3) enhancement, improvement, and maintenance of performance provide a recurrent cycle of self-assistance to other-assistance; (4) de-automatization and recursion occur so regularly that they constitute a Stage IV of the normal development process; after de-automatization, if the capacity is to be restored, then the developmental process must become recursive.
References:
Wertsch, J.V.,1985, Vygotsky and The Social Formation of Mind,London : Harvard University Press.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

The Metaphysic of Science

To be reviewed from many sources by Marsigit
Steven Kreis , 2001 in his Lectures on Modern European Intellectual History elaborated Giambattista Vico position in his “The New Science (1725)” some notions of the ontology and or metaphysic of science. Accordingly, Vico, G (1725) stated that Science or metaphysic, studying the common nature of nations in the light of divine providence, discovers the origins of divine and human institutions among the gentile nations, and thereby establishes a system of he natural law of the gentes, which proceeds with the greatest equality and constancy through the three ages which the Egyptians handed down to us as the three periods through which the world has passed up to their time; these are (1) The age of the gods, in which the gentiles believed they lived under divine governments, and everything was commanded them by auspices and oracles, which are the oldest institutions in profane history. (2) The age of the heroes, in which they reigned everywhere in aristocratic commonwealths, on account of a certain superiority of nature which they held themselves to have over the plebs. (3) The age of men, in which all men recognized themselves as equal in human nature, and therefore there were established first the popular commonwealths and then the monarchies, both of which are forms of human government.
Next, Vico (1725), as cited by Kreis (2001) indicated that peoples who have reached the point of premeditated malice, when they receive this last remedy of providence and are thereby stunned and brutalized, are sensible no longer of comforts, delicacies, pleasures, and pomp, but only of the sheer necessities of life. And the few survivors in the midst of an abundance of the things necessary for life naturally become sociable and, returning to the primitive simplicity of the first world of peoples, are again religious, truthful, and faithful; thus providence brings back among them the piety, faith, and truth which are the natural foundations of justice as well as the graces and beauties of the eternal order of God.
Meanwhile, Thales (1999) in SAINTS argued that there is no reason why modern religion shouldn't incorporate the latest discoveries of Psychical Research or Metaphysics; from the time of Aristotle (300 BCE) to the time of Galileo (1600 CE), nearly 2000 years, the worldview, the background of all thought, was that of Aristotle and Ptolemy. It made a large distinction between the heavens (i.e. stars, the moon, the sun, and the planets) and earth. Earth was made of four elements, earth, air, fire, and water, and was mutable and perishable. The heavenly bodies were made of a fifth element (quintessence) which was immutable, imperishable and eternal. Thus, the correct translation of this metaphor is "realm of the imperishable," or "realm of the quintessence."
Thales said that it is not unusual for religions to begin with the mystical teachings of the founder to a small circle of disciples; as the religion develops it is not unusual for it to absorb elements from other religions over the centuries (syncretism) and to incorporate fantastic fairy tales, which may incorporate some symbolic truth (mythology). According to him, the religions who gave up worldly concerns and went off into the desert as seekers of the illumination of fire often succeeded, and when they returned to the world (or when the world came to them), they were not only holy and wise, but they also had "miraculous" powers, such as healing, or walking on water. The miracles of one age are the science of the next. The age of faith passes, and the age of spiritual science begins.
Bryan Appleyard, 1992, in “Understanding the Present: Science and the Soul of Modern Man” clarified that Western science is not simply a neutral method of acquiring knowledge but that it is ‘a metaphysic like any other.’; the foundations of this metaphysic were laid by Galileo, for his discovery was that one of the most effective ways of understanding the world ‘is to pretend that we do not exist.’ He, further indicated that it is the history of science in which he traces the development of physics from Plato and Aristotle through Thomas Aquinas to Galileo, Descartes and Newton and their modern descendants; modern science gradually emerges not as the embodiment of reason but as a form of worldly mysticism whose zeal for accumulating knowledge about the inanimate and the non-human, and whose ‘rational’ commitment to technological power and material wealth has almost completely obscured its radical anti-humanism.
However, Appleyard, B., (1992), pointed out that the contradictions between science and religion are absolutely and irresolvable conflict; he, then stated that the most obvious problem here is that Islam developed directly out of the Judaeo-Christian tradition and shares much of its world-view with Judaism – whose prophets Muslims revere. On the other hand, according to him, at the same time modern science was the almost exclusive creation of zealous Christians who were seeking not to escape their faith but to confirm and magnify it. Descartes, Newton and Robert Boyle, to name but three representative figures, all believed they had triumphantly succeeded through their science in bearing witness to the majesty and rationality of God.
Appleyard, B., (1992), explained that one reaction to the failure to escape is for us all to throw up our hands and loudly proclaim our belief in the reality and complexity of the human soul in the hope that by doing so we can triumph over science; while, the other reaction is to think more carefully, more sensitively and more systematically about the very aspects of human reality which science has traditionally neglected. He, then concluded that only if we do this is it possible that our intellectual culture may yet triumph over its own history, and over the spiritual extremism which shaped modern rationalism and bequeathed to us a contempt for the ‘human element’ whose religious origins we too readily forget.
REFERENCE
Appleyard, B., 1992, “Understanding The Present: Science And The Soul Of Modern Man”: Picador
Giambattista Vico in Kreis, S., 2001, “The New Science: Lectures On Modern European Intellectual History”, The History Guide
Iranzo, V., 1995, “ Epistemic Values In Science” : Sorites
Katz, M, 2004, “Value Science Can Change The World (And Be Changed By It)” : Cristina Lafont
Meer, J.M.V.D., 1995, “The Struggle Between Christian Theism,
Metaphysical Naturalism And Relativism: How To Proceed In Science?”, Ontarion: Pascal Centre, Redeemer College
Wikipedia, The Free Encyclopedia.,
Wilson, F.L., 1999, “Plato, Science And Human Values”, Rochester Institute Of Technology: Physics Teacher.Org

Indonesian Philosophy among the World’s Crises

By Marsigit
Since the crisis, began in February 1998 and to be continues in 2008 as massive layoffs and the collapse of the country's multi dimensional aspects; there are a great euphoria of all the nations components, independent organizations of the workers, peasants, students, intellectuals or any other sector of the masses, to reflect the past and the present and expect of the future life. It of course, including to reflect the role of Indonesian philosophy “Pancasila” as the ideology of the nation. As it was stated in preamble, the 1945 constitution sets forth the Indonesian philosophy as the embodiment of basic principles of an independent Indonesian state. Indonesian philosophy consists of two Sanskrit words." Panca "meaning five, and " Sila " meaning principle. It comprises five inseparable and interrelated principles. They are: Belief in the One and Only God ; Just and Civilized Humanity ; The Unity of Indonesia ; Democracy Guided by the Inner Wisdom in the Unanimity Arising Out of Deliberations Amongst Representatives, and Social Justice for all Indonesian people.
Kaelan, 2002, in his “Pancasila” elaborated that the first principle of Indonesian philosophy cannot be separated from the value of religious in Indonesia due to the fact that Indonesian philosophy is not only such kind of contemplation but also a philosophical and political consensus. Indonesian philosophy is the resources of the value for running the country; therefore, the essence of first principle of Indonesian philosophy, which is characterized as abstract and universal, should meet with the operational, moral and legal aspects of the nations. For Indonesian people, God is understood not only as the Supreme Being but also a the creator of the cosmos, he must also be understood as a supernatural being and as a supernatural cause; God created man in his own image by giving human beings immaterial intellects and, with that, also free will is a further indication that in the course of human affairs the totally unpredictable is present.
This principle requires that human beings be treated with due regard to their dignity as God's creatures; it emphasizes that the Indonesian people do not tolerate physical or spiritual oppression of human beings by their own people or by any other nations. It is that they are all human, all members of one species, called *Homo sapiens*, and all having the same natural and thereby the same specific attributes that differentiate them from the members of all other species. In all other respects, any two human beings may be found unequal, one having more of a certain human attribute than another, either as the result of native endowment or of individual attainment; however, this second principle taught that (1) that all human beings are equal in respect of their common humanity, and (2) that all human beings are also unequal, one with another, in a wide variety of respects in which they differ as individual members of the human species.
Kaelan, 2002, noted that Indonesian philosophy develops “monodualism democracy” in which people make decision-making through deliberations, or musyawarah, to reach a consensus, or mufakat; the democracy that right must always be exercised with a deep sense of responsibility to God Almighty according to ones own conviction and religious belief with respect for humanitarian values of man's dignity and integrity, and with a view to preserving and strengthening national unity and the pursuit of social justice. He also stated that this fourth principle of Indonesian philosophy consists of three aspects of philosophical ideal: political democracy, socio-economical democracy and reaching the concencus.
This principle of social justice means for the equitable spread of welfare to the entire population, not in a static but in a dynamic and progressive way. This means that all the country's natural resources and the national potentials should be utilized for the greatest possible good and happiness of the people. Social justice implies protection of the weak. But protection should not deny them work. On the contrary, they should work according to their abilities and fields of activity. Protection should prevent willful treatment by the strong and ensure the rule of justice. These are the sacred values of Indonesian philosophy which, as a cultural principle should always be respected by every Indonesian because it is now the ideology of the state and the life philosophy of the Indonesian people. Amidst the world’s crises it may useful for Indonesian people to reflects their contemporary life in order to do the best in the future.
REFERENCE
Kaelan, 2002, “Filsafat Indonesian philosophy: Pandangan Hidup Bangsa Indonesia”, Yogyakarta: Paradigma

The Implication Of Piaget's Work to Mathematics Education

By Marsigit
Especially attractive to workers in mathematics education were Piaget's conceptions that children's intellectual development progresses through well-defined stages, that children develop their concepts through interaction with the environment, and that for most of the primary years most children are in the stage of concrete operations. Further, she stated that in mathematics education, it was a natural consequence of belief in Piaget's theory about the central role of interaction with objects that, when they learn mathematics, children should be expected to work practically, alone with their apparatus, and to work out mathematical concepts for themselves. Piaget proposes that children create their knowledge of the world; however, he also argued that in the creation and unfolding of their knowledge, children are constrained by absolute conceptual structures, especially those of mathematics and logic; thus, Piaget accepts an absolutist view of knowledge, especially mathematics (ibid, p.185). For primary children (7 to 12 years) in which Piaget called they are in the stage of concrete operation, mental action occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation (Becker, 1975); thinking shows many characteristics of mature logic, but it is restricted to dealing with the real; an eight-year old, for example, has no trouble ordering a set of sticks according to height, but might fail to solve the problem.
In order to draw out the explicit implications of Piaget's work for mathematics teaching in primary school for the children who are at concrete-operational stage, it is useful to divide this stage into three stages : early concrete-operational (7-9 years), middle concrete-operational (10-12 years) and late concrete-operational (13-15 years). The children at early concrete-operational and middle concrete-operational will be discuss in the following. The child at early concrete-operational stage is confined to operations upon immediately observable physical phenomena; therefore, he states that the implications of the teaching mathematics may be translated as : (1) both the elements and operations of ordinary arithmetics must be related directly to physically available elements and operations, (2) there should be no more than two elements connected by one operation even with the restriction and the result must be actually closed to avoid the problem of any doubt about the uniqueness of the result, (3) the only notion of inverse is physical, (4) there is no basis for seeking a consistency in relationships with a system of elements selected two at a time and connected by an operation.
The child at middle concrete-operational tends to work with qualitative correspondences, e.g. the closer, the bigger; and thus is still reality bound and not capable of setting up a reliable system based on measurement. For these reason he outlined that its implications for teaching mathematics in the primary school are that : (1) Children begin to work with operations as such but only where uniqueness of result is guaranteed by their experience both with the operations and the elements operated upon; this in effect means two operations closed in sequence with small numbers or one familiar operation using numbers beyond his verified range, e.g. the child can cope with items involving the following types of combinations, (3+8+5) and (475+234); (2) The developing notion of the inverse of an operation tends to be qualitative; children regard substracting as destroying an effect of addition without specifically value of 'y' in y+4=7, they regard 'y' as a unique number to which '4' has been added, substracting '4' happens to destroy the effect of the original addition; (3) a basis exists for the development of a notion of consistency as being a necessary condition for a system of operations but the child tend to recognize the need without being able to give a logical reason for it. Preadolescent child makes typical errors of thinking that are characteristics of his stage of mental growth; the teacher should try to understand these error; and, besides knowing what errors the child usually makes, the teacher should also try to find out why he makes them (Adler, 1968). For these reason, further he suggests that an answer or an action that seems illogical from the teacher point of view on the basis of teacher's extensive experience may seem perfectly logical from the child's point of view on the basis of his limited experience.The teacher can help the child overcome the errors in his thinking by providing him with experiences that expose them as errors and point the way to the correction of the errors.
The child in the pre-operational stage tends to fix his attention on one variable to the neglect of others; to help him overcome this error, provide him with many situations (Adler, 1968). Due to the fact that a child's thinking is more flexible when it is based on reversible operations, the teacher should teach them pairs of inverse operations in arithmetic together, and teach that subtraction and addition nullify each other, and multiplication and division nullify each other (ibid, p.58). As Piaget summed up in Copeland (1979), 'numerical addition and subtraction become operations only when they can be composed in the reversible construction which is the additive group of integers, apart from which there can be nothing but unstable intuition'. Adler (1968) also suggested that physical action is one of the basis of learning; to learn effectively, the child must be a participant in events; to develop his concepts of numbers and space, for example, he needs to touch things, move them, turn them, put together or take them apart. Children should have many experiences in sorting common classroom materials, working with concrete shapes and sizes and colors, and discussing all sorts of relationship; this activities provides a basis for determining in a clearly defined way what progress children are making in their ability to realize, as well as copy, basic spatial distinction; and, most children will be ready at first-grade level to learn the basic shapes.
Since there is a lag between perception and the formation of a mental image, the teacher needs to reinforce the developing mental image with frequent use of perceptual data, for example, let him see the addition once more as a succession of motions on the number line when the child falters in the addition of integers (Adler, 1968). What the mind 'represents' may be and often is different from what is 'seen' or 'felt' by small children (Copeland, 1979); the teachers need to know the stages through which children go in developing the ability to consider geometric ideas. In order that the students are ready to learn a new concept, the teacher should examine the mastering of student's prerequisites concept; Piaget's theories suggested that learning was based on intellectual development and occured when the child had available the cognitive structures necessary for assimilating new information (Leder, 1992). In his teaching, the teacher needs to look at the way pupils go about their work and not just at the products; he also needs to listen to pupil's ideas and try to understand their reasoning and discuss the problems so that pupils reveal their ways of thinking. These activities are actually in the framework of teacher's method of assessing students' thinking.
Piaget developed his 'clinical method' as a way of exploring the development of children's understanding, and employed observation along with interview as a means of accessing children's views of the world (Conner, 1991). He is the pioneers who advocated using observations of children in real situation; the observation, that is more than just looking, serves a useful assessment purpose involves : looking at the way pupils go about their work and not just at the products, listening to pupils' ideas and trying to understand their reasoning, and discussing problems so that pupils reveal their ways of thinking (ibid, pp. 50-51). The observation were used to support the hypothesis that the children, at a certain stage, were discriminating between 'means' and 'ends' (Becker, et al.,1975). In this interview, the answers of students at various ages are then analyzed to see how properties of 'mental structures' change with age (ibid, p. 218). The justification of whether the Piaget's idea of assessment is practical or not in the classroom practice depends much on : the philosophy of mathematics education in which we start to do so, the characteristics of his paradigm of cognitive development or student's competence, the capability of the teacher ; and, in general, this is dependent on its interpretation.
References:
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

The Implication of Vigotsky’s Work to Mathematics Education

By Marsigit
Proper organisation of the learning is the key factor in the pedagogic processes described by Vygotsky in which the teacher holds for the responsibility of the child's learning. This implies careful diagnostic assessment of the child's existing category system and appropriate sequencing of learning experiences to move the child from that point towards the next defined curricular goal (Galloway and Edwards, 1991). The process of generalization indicates the abstraction of rules and the beginnings of the development of internal consciousness and higher cognitive functioning (Evans, 1986); through this process the curriculum is changed and developed to meet the needs of the pupils more fully. By concentrating on the analysis of the overall process of education, Vygotsky sees teachers occupying a didactic role. He defined intelligence as the capacity to learn from instruction (Sutherland, 1992); it implies that the teacher should guide her pupils in paying attention, concentration and learning effectively; the teacher should scaffold a pupil to competence in any skill. Vygotsky places the teacher firmly alongside the child in a process of jointly constructing meaning and so emphasises the importance of language and communication in the construction of an understanding of the world (Galloway and Edwards, 1991).The teacher's role then is to make the classroom as rich an interactive learning community as she or he can and through language to lead children into new zones of proximal development (Gipps, 1994); and he suggested that instruction is most effective when it is addressed to the child's zone of proximal development (Blenkin & Kelly, 1984). Internalization of the learning is demonstrated through the ability to transfer the learning to new situation (Evans, 1986). Vygotsky proposed that every specific state of a pupil's development is characterized by an actual development level and a level of potential development (Hoyles, 1987); the pupil is not able to exploit the possibilities at the latter level on her own, but can do so with educational support; thus, teaching should provide 'scaffolding' for voyaging into the next level of intended learning. Hoyles (1987) concluded that the ideas of providing 'scaffolding' leads on to think about this model of teaching which does not necessarily lead to conflict between the learner's autonomy and pedagogic guidance.
It is important to note here that Vygotsky at one time acknowledge the operation of societal or social institutional forces; Vygotsky and Mead studied social processes in small group interaction in terms of interpersonal dynamics and communication. As emphasized by Vygotsky (1978), the social context affects development at both the institutional and material level, as well as the interpersonal level. In development, children adapt their cognitive and social skills to the particular demands of their culture through practice in particular activities; children learn to use physical and conceptual tools provided by the culture to handle the problems of importance in routine activities (ibid, p. 328). Study after study has documented the absence in classrooms of the fundamental tool for the teaching of children: assistance provided by more capable others that is responsive to goal-directed activities (Tharp and Gallimore, 1988). To provide assistance in the ZPD, the teacher must be in close touch, sensitive and accurate in assisting. There should be opportunities for assisted performance, for using of small groups and for the maintenance of a positive classroom atmosphere that will increase independent task involvement of students, new material and technology with which students can interact independent of the learner (ibid, p. 58).
The explicit implication from above propositions for the teaching of primary mathematics is that the children need to actively engage with mathematics, posing as well as solving problems, discussing the mathematics embedded in their own lives and environment as well as broader social context (Ernest, 1991). The appropriate of teaching, as he suggested, may include a number of components : genuine discussion, both student-student and student-teacher, since learning is the social construction of meaning; cooperative groupwork, project-work and problem solving for confidence, engagement and mastery; autonomous projects, exploration, problem posing and investigative work, for creativity, student self-direction and engagement through personal relevance; learner questioning of course contents, pedagogy and the modes of assessment used, for critical thinking; and, socially relevant materials, projects and topics, including race, gender and mathematics, for social engagements and empowerment. Related to the resources of teaching, Ernest (1991) suggested that due to the learning should be active, varied, socially engaged and self-regulating, the theory of resources has three main components : (1) the provision of a wide variety of practical resources to facilitate the varied and active teaching approaches; (2) the provision of authentic material, such as newspaper, official statistics, and so on for socially relevant and socially engaged study and investigation; and (3) the facilitation of student self-regulated control and access to learning resources.
When cognitive change is considered as much a social as an individual process, new question arise about when and how to track or measure change (Newman, et al., 1989). This is about the role of assessment in the process of instructional interaction. In the 'dynamic assessment', derives from a particular interpretation of Vygotsky's zone of proximal development (ZPD), the ZPD provides a very interesting alternative to the traditional standardized test (Newman, et al., 1989). For Vygotsky, assessment which focuses only on a child's actual level of attainment or development is incomplete and gives only a partial picture. Instead of giving the children a task and measuring how well they do or how badly they fail, one can give the children the task and observe how much and what kind of help they need in order to complete the task successfully; in this approach the child is not assessed alone; rather, the social system of the teacher and child is dynamically assessed to determined how far along it had progressed. Assessment tasks and outcomes should be open to pupil discussion, scrutiny and negation where appropriate, and student choice for topic for investigation and project-work (Ernest, 1991). Further, he suggested that the content of assessment tasks, such as projects and examination questions, should include socially embedded mathematical issues, requiring critical thinking about the social role of mathematics.
Within the ZPD, and suggest that clarification and communication of purpose, aims, and expectations are central to strategy for self-assessment; the variation in assistance to the child that Tharp and Gallimore describe permeate this account of development activities as assessment itself is treated as a performance. He found that, by interviewing the children in the six classes aged between 5 and 9, pupils self-assessment provide the basis for development activities with the clarification of purposes, aims, and expectation through the use of long-term aims and short-term target. Tharp and Gallimore's model provides a framework for developing the ways in which children can be encouraged to assess their own progress; the clarification and evaluation of targets become a zone in which each child's performance is assisted by their teacher (ibid, p.236); as they become involved in their own assessment they gradually take over the task and complement the wide range of skills and talents with each child begins school. So the purpose of mathematics education should be enable students to realize, understand, judge, utilize and sometimes also perform the application of mathematics in society, in particular to situations which are of significance to their private, social and professional lives (Niss, 1983, in Ernest, 1991). Accordingly, the curriculum should be based on project to help the pupil's self-development and self-reliance; the life situation of the learner is the starting point of educational planning; knowledge acquisition is part of the projects; and social change is the ultimate aim of the curriculum (Ernest, 1991).
References:
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

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More Link to Mathematics Teaching Resources

Bagaimana Melakukan Diskusi Matematika dengan Siswa?

Pada pembelajaran tradisional guru terpaksa sibuk mengontrol siswanya

The Role of Cognitive Development Theory for Mathematics Education

Indikator Guru Matematika yang Profesional

REFLECTION ON THE TEACHING OF “THE MULTIPLICATION ALGORITM OF THE 3rd GRADE OF PRIMARY SCHOOL” THROUGH VTR

The Role of Cognitive Development Theory for Mathematics Education

Ritual Mathematics

Piaget's Work and its Relevance to Mathematics Education

Vygotsky's Work and Its Relevance to Mathematics Education

The Metaphysic of Science

Indonesian Philosophy among the World’s Crises

The Implication Of Piaget's Work to Mathematics Education

The Implication of Vigotsky’s Work to Mathematics Education

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