Komunikasi Melalui Interaksi Sosial:

Membina dan Memperkembangkan Pengetahuan Matematik

 
 

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SINOPSIS

Artikel ini menerangkan secara ringkas bagaimana berkomunikasi dapat membina dan memperkembangkan pengetahuan matematik dalam kalangan murid-murid sekolah rendah. Keperluan berkomunikasi dalam bilik darjah melalui interaksi sosial yang dirangsang oleh guru matematik dikatakan dapat membantu pelajar menguasai kemahiran membaca, menulis, mendengar, memikir secara kreatif dan berkomunikasi tentang masalah, yang mana akan memperkembang dan memperdalamkan pemahaman pelajar-pelajar tentang matematik. Artikel ini  turut menyentuh tentang bentuk perancangan pengajaran matematik yang seharusnya dihasilkan agar perkara ini dapat diterjemahkan secara praktikal dalam bilik darjah dan selari dengan pendekatan dan teori pembelajaran konstruktivisme.

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1.0 Pengenalan

Persoalan yang selalu timbul dalam proses pengajaran dan pembelajaran matematik ialah: 'Bagaimanakah hendak menghasilkan perancangan pengajaran matematik yang berkesan?'

Soalan ini bukan satu soalan yang boleh dijawab dengan mudah. Dalam era konstruktivisme ini, sebagai seorang guru matematik, dia harus tahu bagaimana pelajar-pelajar memperolehi atau membina dan memperkembangkan pengetahuan matematik, dan cara-cara yang boleh menggalakkan proses membina pengetahuan matematik di dalam minda pelajar.

Konstruktivisme merupakan satu pendekatan dalam pengajaran dan pembelajaran. Dalam pendekatan ini murid dianggap telah mempunyai idea yang tersendiri tentang sesuatu konsep yang belum dipelajari. Idea tersebut mungkin benar atau tidak.

Konstruktivisme melibatkan lima fasa, iaitu:

  1. Guru meneroka pengetahuan sedia ada murid pada permulaan sesuatu pelajaran melalui soal jawab  atau ujian.

  2. Guru menguji idea atau pendirian murid melalui aktiviti yang mencabar idea atau pendiriannya.

  3. Guru membimbing murid menstruktur semula idea.

  4. Guru memberi peluang kepada murid mengaplikasikan idea baru yang telah diperoleh untuk menguji kesahihannya.

  5. Guru membimbing murid membuat refleksi dan perbandingan idea lama dengan idea yang baru diperoleh.

Pendapat yang dikemukakan oleh ahli-ahli konstruktivisme ialah seseorang individu membina pengetahuan dalam mindanya melalui proses-proses menghubungkaitkan maklumat baru dengan pengetahuan yang lama atau sedia ada.

Unsur-unsur konstruktivisme telah lama dipraktikkan dalam kaedah pengajaran dan pembelajaran di peringkat sekolah, maktab dan universiti tetapi tidak begitu ketara dan tidak ditekankan.

Mengikut kefahaman konstruktivisme, ilmu pengetahuan sekolah tidak boleh dipindahkan daripada guru kepada guru dalam bentuk yang serba sempurna. Murid perlu membina sesuatu pengetahuan itu mengikut pengalaman masing-masing. Pembelajaran adalah hasil daripada usaha murid itu sendiri dan guru tidak boleh belajar untuk murid. Blok binaan asas bagi ilmu pengetahuan sekolah ialah satu skema iaitu aktiviti mental yang digunakan oleh murid sebagai bahan mentah bagi proses renungan dan pengabstrakan. F

ikiran murid tidak akan menghadapi realiti yang wujud secara terasing dalam persekitaran. Realiti yang diketahui murid adalah realiti yang dia bina sendiri. Murid sebenarnya telah mempunyai satu set idea dan pengalaman yang membentuk struktur kognitif terhadap persekitaran mereka.

"Mengikut pendapat ahli konstruktivisme, pengetahuan konsep tidak dapat dipindah dari seorang kepada seorang yang lain, tetapi mesti dibina oleh setiap pelajar berasaskan sepenuhnya kepada pengalamannya."

"Satu konsep matematik boleh ditakrifkan sebagai corak asas yang menghubungkaitkan set-set objek atau tindakan-tindakan antara satu sama lain dan pengajaran konsep-konsep matematik merupakan satu usaha yang kompleks"

Pengajaran pengetahuan matematik selalu dihubungkaitkan dengan pengajaran konsep matematik. Mengikut Souviney (1989), satu konsep matematik boleh ditakrifkan sebagai corak asas yang menghubungkaitkan set-set objek atau tindakan-tindakan antara satu sama lain dan pengajaran konsep-konsep matematik merupakan satu usaha yang kompleks.

Beliau mengatakan setiap pelajar mempunyai set pengalaman dan kebolehan yang unik untuk menyelesaikan setiap tugasan pembelajaran. Dengan ini, guru matematik memainkan peranan yang penting dalam perancangan pengajaran yang berkesan untuk membantu pelajar membina dan mengembangkan pengetahuan matematik.

2.0 Pembinaan dan Perkembangan Pengetahuan Matematik dalam Minda Pelajar

Semenjak kebelakangan ini, hasil-hasil kajian tentang proses pembelajaran matematik telah menunjukkan bahawa pengetahuan matematik adalah dibina dan dikembangkan dalam minda seseorang individu itu oleh dirinya sendiri.

Mengikut Piaget (dalam Souviney, 1989), semua pengetahuan baru boleh difahami hanya apabila dikaitkan dengan yang sedia ada. Beribu-ribu pengurusan struktur atau skema dikembangkan di sepanjang hayat seseorang. Melalui proses interaktif asimilasi dan akomodasi, individu-individu berusaha mencapai keseimbangan yang bersepadu dan mengalami peringkat-peringkat perkembangan itu. Seseorang individu dikatakan akan mengasimilasikan apa yang baru diketahui dengan apa yang sudah diketahui, untuk mencapai pemahaman. Pengalaman lama akan berubah secara beransur-ansur, atau yang dikenali sebagai akomodasi, disebabkan oleh pengalaman baru ini. Ahli psikologi Rusia, Lev Vygotsky (dalam Souviney, 1989) pula mengatakan operasi mental adalah dirangsangkan melalui interaksi sosial yang aktif dengan rakan sebaya dan orang dewasa yang lebih berterampilan. Operasi-operasi ini akan diserap ke dalam minda seseorang dan menukar menjadi sesuatu yang diperlukannya. Beliau juga membahaskan bahawa pengajaran berkesan ialah apabila pelajar bekerjasama melibatkan diri dalam aktiviti dalam suasana yang menyokong pembelajaran dan menerima bimbingan yang berpatutan dari guru. Guru berperanan mengorganisasikan interaksi untuk membantu kanak-kanak menyelesaikan tugasan pembelajaran.

 

 

Kajian yang dijalankan ke atas kanak-kanak di tadika oleh Baroody & Ginsburg (1990), menunjukkan bahawa kanak-kanak sendiri membina pengetahuan matematik yang tidak formal sebelum mereka mengikuti kelas formal di sekolah. Hasil kajian ini boleh kita lihat dalam aktiviti harian kanak-kanak yang belum mengikuti pendidikan formal. Semasa bermain, kanak-kanak dalam golongan ini selalunya bersua dengan istilah-istilah matematik seperti ‘lebih tinggi’, ‘lebih rendah’, ‘segitiga’, ‘bulat’, ‘dua’, ‘tiga’, dan sebagainya. Proses-proses pembelajaran yang tidak formal tentang pengetahuan matematik, seperti ukuran, ruang, bentuk geometri, dapat dikatakan berlaku dalam situasi sedemikian.

kanak-kanak

"Semasa bermain, kanak-kanak bersua dengan pelbagai istilah matematik dalam persekitarannya, Ini telah membina pengetahuan matematik tidak formal yang bermakna, menarik dan berguna kepada mereka."

Dengan menggunakan istilah matematik sedemikian semasa berinteraksi dengan rakannya, matematik tidak formal dibina dalam minda kanak-kanak kerana ia berguna kepada mereka atau bermakna bagi mereka. Mengikut Baroody & Ginsburg (1990), pengetahuan matematik tidak formal ini dibina dan diperkembangkan oleh kanak-kanak kerana ia bermakna, menarik dan berguna kepada mereka, dan perasaan ingin tahu yang ada pada kanak-kanak mendesak mereka untuk menjadikan persekitaran bermakna, dan mempunyai keupayaan untuk menguruskannya.

Steffe (1990) pula mengatakan bahawa mengikut pendapat ahli konstruktivisme, pengetahuan konsep tidak dapat dipindah dari seorang kepada seorang yang lain, tetapi mesti dibina oleh setiap pelajar berasaskan sepenuhnya kepada pengalamannya.

Jadi, pembelajaran matematik berlaku apabila kanak-kanak berinteraksi dengan persekitarannya yang juga termasuk rakan sebaya dan guru. Pengalaman seseorang kanak-kanak itu yang merupakan asas kepada pembinaan pengetahuan matematik dalam minda juga berhubungkait dengan persekitarannya. Dalam proses membesar, kanak-kanak memperoleh pengalaman melalui proses berinteraksi dengan persekitarannya iaitu melakukan pemerhatian, mendengar, bercakap, menyentuh, merasa, meniru dan sebagainya.

Kad permainan matematik My Rummy

 

My Rummy adalah permainan kad secara berkumpulan bagi kanak-kanak sekolah rendah yang menekankan aspek interaksi dengan rakan sebaya dan guru sebagai pembimbing. Bercakap, mendengar dan mencongak operasi mental adalah antara elemen yang menjadi ciri utama permainan kad My Rummy.

3.0 Komunikasi melalui Interaksi Sosial - Membina dan memperkembangkan Pengetahuan Matematik

Komunikasi melalui interaksi sosial berperanan penting dalam membina pengetahuan matematik dalam minda pelajar. Interaksi sosial sebenarnya merupakan salah satu ciri persekitaran semula jadi yang dialami oleh individu-individu yang normal. Bermula dari peringkat awal persekolahan lagi, guru harus mewujudkan komunikasi yang berbentuk interaksi sosial dalama kalangan pelajar dengan pelajar, pelajar dengan guru dalam proses pengajaran dan pembelajaran matematik. Dengan berbuat sedemikian guru dapat membantu kanak-kanak yang mulai mengikuti pendidikan formal ini memperlengkapkan serta memperbaiki pengetahuan matematik yang tidak formal yang telah terbina sebelum ini.

Mengikut Ginsburg & Baron (1993), satu pendekatan yang dikatakan berguna haruslah yang boleh merangsangkan, secara spontan, minat dan penglibatan kanak-kanak dalam persekitaran yang semula jadi dan menolong mereka memperkembangkan dan melengkapi pengetahuan matematik tidak formal itu.

Koehler & Prior (1993: 281-282) menegaskan bahawa interaksi guru dan pelajar adalah penting dengan mengatakan,

"Most would agree that teaching and learning could occur without texts, blackboards, or manipulative, but we maintain that the learning process would exist for only a very few students if classroom interaction with teachers and peers were eliminated. Teacher-student interactions are indeed the heartbeat of the teaching-learning process."

Petikan di atas menyatakan ramai orang yang setuju bahawa pengajaran dan pembelajaran boleh berlaku tanpa buku teks, papan tulis, atau bahan manipulatif, tetapi proses pembelajaran hanya akan wujud bagi beberapa orang pelajar sahaja sekiranya interaksi pelajar dengan guru dan rakannya dihapuskan. Interaksi pelajar dengan guru dan rakan sebayanya merupakan ‘denyutan nadi’ proses pengajaran dan pembelajaran.

Oleh yang demikian, interaksi sosial di antara guru dan pelajar, pelajar dan pelajar, secara individu atau berkumpulan kecil merupakan salah satu proses komunikasi yang harus diwujudkan dalam bilik darjah bagi pengajaran dan pembelajaran matematik.

Dalam pembelajaran matematik, cara-cara untuk berkomunikasi idea-idea matematik melalui interaksi sosial ialah melukis atau menulis perwakilan, bercakap, menanya, memberi komen, mengkritik, membukti, memberi penjelasan, memberi pendapat, mendengar dan sebagainya.

Mengikut NCTM (1989), perwakilan melibatkan perterjemahan satu masalah atau idea kepada satu bentuk yang baru, yang selalunya melibatkan gambarajah, simbol, tatatanda. Manakala apabila kanak-kanak dalam kumpulan kecil berbincang dan menyelesaikan masalah, mereka boleh mengaitkan bahan yang mereka tahu dengan istilah matematik yang mungkin mereka tidak biasa lihat atau dengar.

Mengikut teori psikologi, kanak-kanak mempunyai sifat yang aktif dan suka bergaul, yang mana mendorong kanak-kanak berkomunikasi dengan orang lain. Dengan berkomunikasi, kanak-kanak berpeluang menjelaskan pemikiran dan mempertajamkan pemikiran mereka. Aktiviti seperti menerokai, menyiasat, menghuraikan dan menerangkan idea matematik mempromosikan komunikasi. Soalan berbentuk penyiasatan dan bimbingan boleh menggalakkan kanak-kanak berfikir dan menerangkan pemikiran mereka secara lisan atau bertulis, membolehkan mereka lebih memahami idea-idea yang mereka sampaikan, seperti yang dikemukakan oleh NCTM (1989:24),

"Interacting with classmates helps children construct knowledge, learn other ways to think about ideas, and clarify their own thinking. Writing about mathematics, such as describing how a problem was solved, also helps clarify their thinking and develop deeper understanding. Reading children's literature about mathematics, and eventually text material, also is an important aspect of communication that needs more emphasis in the K-4 curriculum."

Perbualan berikut adalah di antara seorang kanak-kanak tadika dengan gurunya setelah guru itu mengajar tentang konsep 'olahan tolak' dan perwakilan simbolnya. Kanak-kanak itu di tanya oleh gurunya apa yang beliau faham dengan " 6 - 2 = 4 ":

  • Guru : "Apa yang anda faham dengan "6 - 2 = 4"? Cuba anda bercerita."

  • Kanak-kanak :" ...Oh, mula-mula saya ada enam biji gula-gula, lepas itu saya makan dua biji. Jadi saya masih ada empat biji lagi. ..."

  • Kad permainan matematik My Rummy

    Bahasa matematik yang terdapat pada kad permainan My Rummy misalnya "What is the balance if we take away 3 from 4?" seperti di atas amat sesuai dengan pengalaman kanak-kanak.

    Daripada perbualan ini seseorang guru itu boleh melihat bagaimana kanak-kanak 'mengkonkritkan' simbol yang abstrak ke dalam makna yang sesuai dengan pengalamannya.

    Seterusnya, salah seorang ahli konstruktivisme, Von Glasersfeld (1990) berpendapat bahawa pengetahuan matematik bukanlah dibina secara terasing dari perkara-perkara lain. Setiap abstraksi yang dibuat oleh individu, ke atas perkara yang berkaitan dengan pengalaman, adalah terkawal oleh interaksi sosial dan kolaborasi dan komunikasi yang dibuat olehnya dengan ahli kumpulannya yang mana beliau dibesarkan bersama. Tiada individu boleh mengelakkan daripada mewujudkan persesuaian yang berkaitan dengan domain persetujuan persekitaran sosial. Domain persetujuan yang perlu dipenuhi oleh seseorang individu itu ialah ahli-ahli matematik, guru dan orang dewasa yang lain.

    Dalam kehidupan harian, kita sentiasa dikehendaki membuat rundingan dalam mengatasi masalah. Tujuan rundingan adalah untuk mencapai persetujuan di antara dua pihak atau lebih dalam proses interaksi sosial. Jadi, kemahiran membuat rundingan perlu dikuasai oleh pelajar-pelajar sebelum mereka meninggalkan sekolah. Untuk menghasilkan rundingan yang menyakinkan orang lain, seseorang itu haruslah mengumpulkan sebarang maklumat yang berkenaan dan membentuk hujah-hujah yang sesuai. Contoh berikut mengilustrasikan satu proses rundingan di dalam situasi pengajaran dan pembelajaran matematik di bilik darjah.

    Guru :"Bolehkah anda tolong cikgu kira jawapan bagi 240 x 22 ?"

    (Selepas lebih kurang 30 saat)

    Pelajar A :" Cikgu, jawapannya ialah 5280."

    Pelajar-pelajar lain :

    "Cepatnya engkau kira! Betul tak jawapan anda itu?"

    Guru :"Boleh anda tunjukkan penyelesaiannya? "

    Pelajar A :"Boleh! " (Lihat penyelesaian berikut yang ditulis olehnya.)

    240 x 22 = 4800 + 480 = 240 x 20  diikuti dengan 240 x 2 = 5280

    Pelajar-pelajar lain :"Betullah jawapannya. Oh, macam ini rupanya!"

    Dalam situasi ini, penyelesaian yang ditunjukkan oleh pelajar A adalah berlainan dengan yang lazim dilakukan oleh pelajar lain. Namun, pelajar A dapat merundingcarakan penyelesaiannya untuk diterima oleh kawan sebayanya dengan mengemukakan hujah-hujah yang logik untuk mempertahankan penyelesaiannya.

    Dalam proses pengajaran dan pembelajaran seperti di atas, seseorang itu (pelajar-pelajar lain) akan membina atau menyusun semula pengetahuan yang baru diperolehi dengan yang sedia ada dan membentuk pengetahuan yang baru. Proses komunikasi melalui interaksi sosial dalam pembelajaran matematik memerlukan pelajar membuat perundingan yang mana membolehkan pengetahuan matematik dibina dan perkembangkan dalam mindanya.

    Mengikut Blumer (1969) dan Bauersfeld (1988), peluang-peluang bagi kanak-kanak membina pengetahuan matematik wujud apabila mereka berinteraksi dengan guru dan rakan sebayanya. Pembinaan matematik yang dihasilkan oleh kanak-kanak dikatakan bukan wujud secara tersendiri. Sebaliknya, pembinaan-pembinaan itu terkawal oleh kewajipan masing-masing untuk membentuk interpretasi yang boleh disesuaikan dengan pembinaan ahli dalam komuniti bilik darjah.

    Cobb (in press) (dalam Cobb, Wood & Yackel, 1990) pula mengatakan dalam komunikasi berciri matematik, makna-makna dirundingcarakan. Peranan komunikasi melalui interaksi sosial dalam pembinaan dan memperkembangkan pengetahuan matematik pelajar juga dikemukakan oleh Davidson (1990) dalam menyatakan cara-cara bagaimana pembelajaran koperatif kumpulan kecil boleh membantu mengatasi masalah pelajar seperti perasaan kecewa, takut kepada matematik, mengelak matematik dan lain-lain lagi.

    "Pembelajaran koperatif kumpulan kecil boleh membantu mengatasi masalah pelajar seperti perasaan kecewa, takut kepada matematik, mengelak matematik dan lain-lain lagi."

    Beliau mengatakan:

     
    • kumpulan kecil dapat memberi sokongan sosial untuk mempelajari matematik,

    • interaksi kumpulan boleh membantu semua ahli kumpulan mempelajari konsep-konsep dan strategi penyelesaian masalah,

    • dalam memperbincangkan penyelesaian-penyelesaian yang dikemukakan pelajar boleh memujuk antara satu sama lain dengan argumen yang logik,

    • pelajar-pelajar boleh memperbincangkan kelebihan penyelesaian-penyelesaian yang dikemukakan,

    • pelajar-pelajar dalam kumpulan boleh bantu antara satu sama lain untuk menguasai fakta dan prosedur-prosedur pengira yang perlu dalam konteks permainan, memahami masalah-masalah.

    • seseorang itu belajar melalui dengan bercakap, mendengar, menerangkan, dan melakukan proses berfikir secara bersendirian dan juga bersama-sama orang lain,

    • dalam kumpulan, pelajar-pelajar boleh mengatasi masalah yang mencabar yang mungkin di luar keupayaan individu.

    Maher & Alston (1990) juga membincangkan kepentingan interaksi sosial dengan mengatakan bahawa persekitaran baru diperlukan untuk mengadakan peluang membina struktur yang lebih kukuh. Mereka juga mengatakan situasi yang membolehkan guru-guru dan pelajar-pelajar memperluaskan pengetahuan mereka, dan berinteraksi dengan orang lain dalam proses perundingan sosial, mengenai fahaman yang diperoleh dari pengalaman tersebut, adalah diperlukan untuk perkembangan yang berterusan. Pendapat ini juga dapat diilustrasikan dalam contoh mengira jawapan bagi 240 x 22, di mana pelajar-pelajar lain mungkin dapat menggunakan cara pelajar A untuk situasi yang lain, atau bagi mereka yang lebih kreatif boleh menggunakan cara itu sebagai batu loncatan untuk menghasilkan cara yang lain. Dalam proses berinteraksi dengan rakan sebaya dan guru, pelajar-pelajar akan membina pengetahuan baru dan memperkembangkan pengetahuan sedia ada, seperti yang dinyatakan oleh NCTM (1991:34),

    "Students must talk, with one another as well as in response to the teacher...When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community."

    Komunikasi memainkan satu peranan yang penting dalam membantu kanak-kanak membina pengetahuan mereka. Melalui komunikasi, kanak-kanak membina pertalian antara fahaman tak formal dan intuitif dengan bahasa matematik iaitu tatatanda, simbol, persetujuan dan istilah matematik yang sering dikaitkan sebagai abstrak. Komunikasi memainkan peranan utama dalam membantu kanak-kanak menghubungkaitkan antara perwakilan idea matematik yang dalam bentuk fizikal, simbol, lisan, mental dan lain-lain lagi

    4.0 Kesimpulan

    Berkomunikasi melalui interaksi sosial yang wujud di bilik darjah boleh membantu pelajar menguasai kemahiran membaca, menulis, mendengar, memikir secara kreatif dan berkomunikasi tentang masalah, yang mana akan memperkembang dan memperdalamkan pemahaman pelajar-pelajar tentang matematik. Dalam proses pembelajaran matematik yang boleh mewujudkan interaksi sosial, sering melibatkan proses rundingan. Proses rundingan akan membantu pelajar melihat bagaimana rakan sebayanya memahami sesuatu konsep, dan secara langsung skema dalam mindanya berubahsuai dan memperkembangkan pengetahuan sedia ada. Di samping itu, proses berunding boleh mempertajam dan memperdalamkan lagi pemikiran seseorang. Akhir sekali, jika kita menerima premis bahawa pengetahuan adalah dibina dan mempunyai hubungkait yang rapat dengan alam sekitar, salah satu fokus guru matematik ialah mewujudkan suasana yang menggalakkan komunikasi melalui interaksi sosial berciri matematik yang bertujuan dalam bilik darjah bagi proses pengajaran dan pembelajaran matematik.

    Rujukan

     

    Baroody, A.J. & Ginsburg, H.P. (1990). Children's learning: A cognitive view. In R.B.

    Davis, C. A. Maher & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (m.s. 51-64). Reston, VA: National Council of Teachers of Mathematics, Inc..

    Bauersfed, H. (1988). Interaction, construction and knowledge : Alternative persperctive for mathematics education. In T. Cooney & D. Grouws (Eds.), Effective mathematics teaching (m.s 27 – 46). Reston, VA : NCTM.

    Blumer, H. (1969). Symbolic interactionism Englewood Clliffs, NJ : Prentice – Hall.

     Cobb, P. , Wood, T. & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers. In R.B. Davis, C. A. Maher & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (m.s. 125-146).

    Reston, VA: National Council of Teachers of Mathematics, Inc..

    Davidson, N. (1990). Small-group cooperative learning in mathematics. In T.J. Cooney & C.R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (m.s. 52-61). Reston, VA: National Council of Teachers of Mathematics, Inc..

    Ginsburg, H.P. & Baron, J. (1993) . Cognition: Young childen's construction of mathematics. In R.J. Jensen (Ed.), Research ideas for the classroom: Early childhood mathematics (m.s. 3-21). New York: Macmillan Publishing Company for NCTM.

    Koehler, M.S. & Prior, M. (1993). Classroom interactions: The heartbeat of the teaching/learning process. In D.T. Owens (ed.), Research ideas for the classroom: Middle grades mathematics (m.s. 280-298). New York: Macmillan Publishing Company for NCTM.

    Maher, C.A. & Alston, A. (1990). Teacher development in mathematics in a constructivist framework. In R.B. Davis, C.A. Maher & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (m.s. 147-166). Reston, VA: National Council of Teachers of Mathematics, Inc..

    National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

    National Council of Teachers of Mathematics (1991). Professional standards for Souviney, R.J. (1989). Learning to teach mathematics. Columbus, Ohio: Merril Publishing Company.

    RUJUKAN WIKIPEDIA

    Constructivism (learning theory)

    Constructivism is a psychological theory of knowledge (epistemology) [1] which argues that humans generate knowledge and meaning from their experiences. Constructivism is not a specific pedagogy, although it is often confused with Constructionism, an educational theory developed by Seymour Papert. Piaget's theory of Constructivist learning has had wide ranging impact on learning theories and teaching methods in education and is an underlying theme of many education reform movements. Research support for constructivist teaching techniques has been mixed, with some research supporting these techniques and other research contradicting those results.

    Formalization of the theory of constructivism is generally attributed to Jean Piaget, who articulated mechanisms by which knowledge is internalized by learners. He suggested that through processes of accommodation and assimilation, individuals construct new knowledge from their experiences. When individuals assimilate, they incorporate the new experience into an already existing framework without changing that framework. This may occur when individuals' experiences are aligned with their internal representations of the world, but may also occur as a failure to change a faulty understanding; for example, they may not notice events, may misunderstand input from others, or may decide that an event is a fluke and is therefore unimportant as information about the world. In contrast, when individuals' experiences contradict their internal representations, they may change their perceptions of the experiences to fit their internal representations. According to the theory, accommodation is the process of reframing one's mental representation of the external world to fit new experiences. Accommodation can be understood as the mechanism by which failure leads to learning: when we act on the expectation that the world operates in one way and it violates our expectations, we often fail, but by accommodating this new experience and reframing our model of the way the world works, we learn from the experience of failure, or others' failure.

    It is important to note that constructivism is not a particular pedagogy. In fact, constructivism is a theory describing how learning happens, regardless of whether learners are using their experiences to understand a lecture or following the instructions for building a model airplane. In both cases, the theory of constructivism suggests that learners construct knowledge out of their experiences. However, Constructivism is often associated with pedagogic approaches that promote active learning, or learning by doing.

    Constructivist learning intervention

    The nature of the learner

    The learner as a unique individual

    Social constructivism views each learner as a unique individual with unique needs and backgrounds. The learner is also seen as complex and multidimensional. Social constructivism not only acknowledges the uniqueness and complexity of the learner, but actually encourages, utilises and rewards it as an integral part of the learning process (Wertsch 1997).

    The importance of the background and culture of the learner

    Social constructivism encourages the learner to arrive at his or her version of the truth, influenced by his or her background, culture or embedded worldview. Historical developments and symbol systems, such as language, logic, and mathematical systems, are inherited by the learner as a member of a particular culture and these are learned throughout the learner's life. This also stresses the importance of the nature of the learner's social interaction with knowledgeable members of the society. Without the social interaction with other more knowledgeable people, it is impossible to acquire social meaning of important symbol systems and learn how to utilize them. Young children develop their thinking abilities by interacting with other children, adults and the physical world. From the social constructivist viewpoint, it is thus important to take into account the background and culture of the learner throughout the learning process, as this background also helps to shape the knowledge and truth that the learner creates, discovers and attains in the learning process (Wertsch 1997).

    The responsibility for learning

    Furthermore, it is argued that the responsibility of learning should reside increasingly with the learner (Von Glasersfeld 1989). Social constructivism thus emphasizes the importance of the learner being actively involved in the learning process, unlike previous educational viewpoints where the responsibility rested with the instructor to teach and where the learner played a passive, receptive role. Von Glasersfeld (1989) emphasizes that learners construct their own understanding and that they do not simply mirror and reflect what they read. Learners look for meaning and will try to find regularity and order in the events of the world even in the absence of full or complete information.

    The motivation for learning

    Another crucial assumption regarding the nature of the learner concerns the level and source of motivation for learning. According to Von Glasersfeld (1989) sustaining motivation to learn is strongly dependent on the learner’s confidence in his or her potential for learning. These feelings of competence and belief in potential to solve new problems, are derived from first-hand experience of mastery of problems in the past and are much more powerful than any external acknowledgment and motivation (Prawat and Floden 1994). This links up with Vygotsky’s "zone of proximal development" (Vygotsky 1978) where learners are challenged within close proximity to, yet slightly above, their current level of development. By experiencing the successful completion of challenging tasks, learners gain confidence and motivation to embark on more complex challenges.

    The role of the instructor

    Instructors as facilitators

    According to the social constructivist approach, instructors have to adapt to the role of facilitators and not teachers (Bauersfeld, 1995). Where a teacher gives a didactic lecture which covers the subject matter, a facilitator helps the learner to get to his or her own understanding of the content. In the former scenario the learner plays a passive role and in the latter scenario the learner plays an active role in the learning process. The emphasis thus turns away from the instructor and the content, and towards the learner (Gamoran, Secada, & Marrett, 1998). This dramatic change of role implies that a facilitator needs to display a totally different set of skills than a teacher (Brownstein 2001). A teacher tells, a facilitator asks; a teacher lectures from the front, a facilitator supports from the back; a teacher gives answers according to a set curriculum, a facilitator provides guidelines and creates the environment for the learner to arrive at his or her own conclusions; a teacher mostly gives a monologue, a facilitator is in continuous dialogue with the learners (Rhodes and Bellamy, 1999). A facilitator should also be able to adapt the learning experience ‘in mid-air’ by using his or her own initiative in order to steer the learning experience to where the learners want to create value.

    The learning environment should also be designed to support and challenge the learner's thinking (Di Vesta, 1987). While it is advocated to give the learner ownership of the problem and solution process, it is not the case that any activity or any solution is adequate. The critical goal is to support the learner in becoming an effective thinker. This can be achieved by assuming multiple roles, such as consultant and coach.

    According to the social constructivist approach, instructors have to adapt to the role of facilitators and not teachers (Bauersfeld, 1995). Where a teacher gives a didactic lecture which covers the subject matter, a facilitator helps the learner to get to his or her own understanding of the content. In the former scenario the learner plays a passive role and in the latter scenario the learner plays an active role in the learning process. The emphasis thus turns away from the instructor and the content, and towards the learner (Gamoran, Secada, & Marrett, 1998). This dramatic change of role implies that a facilitator needs to display a totally different set of skills than a teacher (Brownstein 2001). A teacher tells, a facilitator asks; a teacher lectures from the front, a facilitator supports from the back; a teacher gives answers according to a set curriculum, a facilitator provides guidelines and creates the environment for the learner to arrive at his or her own conclusions; a teacher mostly gives a monologue, a facilitator is in continuous dialogue with the learners (Rhodes and Bellamy, 1999). A facilitator should also be able to adapt the learning experience ‘in mid-air’ by using his or her own initiative in order to steer the learning experience to where the learners want to create value.

    The learning environment should also be designed to support and challenge the learner's thinking (Di Vesta, 1987). While it is advocated to give the learner ownership of the problem and solution process, it is not the case that any activity or any solution is adequate. The critical goal is to support the learner in becoming an effective thinker. This can be achieved by assuming multiple roles, such as consultant and coach.

    The nature of the learning process

    Learning is an active, social process

    Social constructivist scholars view learning as an active process where learners should learn to discover principles, concepts and facts for themselves, hence the importance of encouraging guesswork and intuitive thinking in learners (Brown et al.1989; Ackerman 1996). In fact, for the social constructivist, reality is not something that we can discover because it does not pre-exist prior to our social invention of it. Kukla (2000) argues that reality is constructed by our own activities and that people, together as members of a society, invent the properties of the world.

    Other constructivist scholars agree with this and emphasize that individuals make meanings through the interactions with each other and with the environment they live in. Knowledge is thus a product of humans and is socially and culturally constructed (Ernest 1991; Prawat and Floden 1994). McMahon (1997) agrees that learning is a social process. He further states that learning is not a process that only takes place inside our minds, nor is it a passive development of our behaviours that is shaped by external forces and that meaningful learning occurs when individuals are engaged in social activities.

    Vygotsky (1978) also highlighted the convergence of the social and practical elements in learning by saying that the most significant moment in the course of intellectual development occurs when speech and practical activity, two previously completely independent lines of development, converge. Through practical activity a child constructs meaning on an intrapersonal level, while speech connects this meaning with the interpersonal world shared by the child and her/his culture.

    Dynamic interaction between task, instructor and learner

    A further characteristic of the role of the facilitator in the social constructivist viewpoint, is that the instructor and the learners are equally involved in learning from each other as well (Holt and Willard-Holt 2000). This means that the learning experience is both subjective and objective and requires that the instructor’s culture, values and background become an essential part of the interplay between learners and tasks in the shaping of meaning. Learners compare their version of the truth with that of the instructor and fellow learners in order to get to a new, socially tested version of truth (Kukla 2000). The task or problem is thus the interface between the instructor and the learner (McMahon 1997). This creates a dynamic interaction between task, instructor and learner. This entails that learners and instructors should develop an awareness of each other's viewpoints and then look to own beliefs, standards and values, thus being both subjective and objective at the same time (Savery 1994).

    Some studies argue for the importance of mentoring in the process of learning (Archee and Duin 1995; Brown et al. 1989). The social constructivist model thus emphasizes the importance of the relationship between the student and the instructor in the learning process.

    Some learning approaches that could harbour this interactive learning include reciprocal teaching, peer collaboration, cognitive apprenticeship, problem-based instruction, web quests, anchored instruction and other approaches that involve learning with others.

    Collaboration among learners

    Learners with different skills and backgrounds should collaborate in tasks and discussions in order to arrive at a shared understanding of the truth in a specific field (Duffy and Jonassen 1992).

    Most social constructivist models, such as that proposed by Duffy and Jonassen (1992), also stress the need for collaboration among learners, in direct contradiction to traditional competitive approaches. One Vygotskian notion that has significant implications for peer collaboration, is that of the zone of proximal development. Defined as the distance between the actual developmental level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance or in collaboration with more capable peers, it differs from the fixed biological nature of Piaget's stages of development. Through a process of 'scaffolding' a learner can be extended beyond the limitations of physical maturation to the extent that the development process lags behind the learning process (Vygotsky 1978).

    Learning by teaching (LdL) as constructivist method

    Main article: Learning by teaching
    If students have to present and train new contents with their classmates, a non-linear process of collective knowledge-construction will be set up.

    The importance of context

    The social constructivist paradigm views the context in which the learning occurs as central to the learning itself (McMahon 1997).

    Underlying the notion of the learner as an active processor is "the assumption that there is no one set of generalised learning laws with each law applying to all domains" (Di Vesta 1987:208). Decontextualised knowledge does not give us the skills to apply our understandings to authentic tasks because, as Duffy and Jonassen (1992) indicated, we are not working with the concept in the complex environment and experiencing the complex interrelationships in that environment that determine how and when the concept is used. One social constructivist notion is that of authentic or situated learning, where the student takes part in activities which are directly relevant to the application of learning and which take place within a culture similar to the applied setting (Brown et al. 1989). Cognitive apprenticeship has been proposed as an effective constructivist model of learning which attempts to "enculturate students into authentic practices through activity and social interaction in a way similar to that evident, and evidently successful, in craft apprenticeship" (Ackerman 1996:25).

    Assessment

    Holt and Willard-Holt (2000) emphasize the concept of dynamic assessment, which is a way of assessing the true potential of learners that differs significantly from conventional tests. Here the essentially interactive nature of learning is extended to the process of assessment. Rather than viewing assessment as a process carried out by one person, such as an instructor, it is seen as a two-way process involving interaction between both instructor and learner. The role of the assessor becomes one of entering into dialogue with the persons being assessed to find out their current level of performance on any task and sharing with them possible ways in which that performance might be improved on a subsequent occasion. Thus, assessment and learning are seen as inextricably linked and not separate processes (Holt and Willard-Holt 2000).

    According to this viewpoint instructors should see assessment as a continuous and interactive process that measures the achievement of the learner, the quality of the learning experience and courseware. The feedback created by the assessment process serves as a direct foundation for further development.

    • The selection, scope and sequencing of the subject matter

    • Knowledge should be discovered as an integrated whole

    Knowledge should not be divided into different subjects or compartments, but should be discovered as an integrated whole (McMahon 1997; Di Vesta 1987).

    This also again underlines the importance of the context in which learning is presented (Brown et al. 1989). The world, in which the learner needs to operate, does not approach one in the form of different subjects, but as a complex myriad of facts, problems, dimensions and perceptions (Ackerman 1996).

    Engaging and challenging the learner

    Learners should constantly be challenged with tasks that refer to skills and knowledge just beyond their current level of mastery. This will capture their motivation and build on previous successes in order to enhance the confidence of the learner (Brownstein 2001). This is in line with Vygotsky’s zone of proximal development which can be described as the distance between the actual developmental level (as determined by independent problem-solving) and the level of potential development (as determined through problem-solving under adult guidance or in collaboration with more capable peers) (Vygotsky 1978).

    Vygotsky (1978) further claimed that instruction is good only when it proceeds ahead of development. Then it awakens and rouses to life an entire set of functions which are in the stage of maturing, which lie in the zone of proximal development. It is in this way that instruction plays an extremely important role in development.

    In order to fully engage and challenge the learner, the task and the learning environment should reflect the complexity of the environment that the learner should be able to function in at the end of learning. Learners must not only have ownership of the learning or problem-solving process, but of the problem itself (Derry 1999).

    Where the sequencing of subject matter is concerned, it is the constructivist viewpoint that the foundations of any subject may be taught to anybody at any stage in some form (Duffy and Jonassen 1992). This means that instructors should first introduce the basic ideas that give life and form to any topic or subject area, and then revisit and build upon these repeatedly. This notion has been extensively used in curricula.

    It is also important for instructors to realize that although a curriculum may be set down for them, it inevitably becomes shaped by them into something personal which reflects their own belief systems, their thoughts and feelings about both the content of their instruction and their learners (Rhodes and Bellamy 1999). Thus, the learning experience becomes a shared enterprise. The emotions and life contexts of those involved in the learning process must therefore be considered as an integral part of learning. The goal of the learner is central in considering what is learned (Brown et al. 1989; Ackerman 1996).

    The structuredness of the learning process

    It is important to achieve the right balance between the degree of structure and flexibility that is built into the learning process. Savery (1994) contends that the more structured the learning environment, the harder it is for the learners to construct meaning based on their conceptual understandings. A facilitator should structure the learning experience just enough to make sure that the students get clear guidance and parameters within which to achieve the learning objectives, yet the learning experience should be open and free enough to allow for the learners to discover, enjoy, interact and arrive at their own, socially verified version of truth.

    Pedagogies based on constructivism

    In fact, there are many pedagogies that leverage constructivist theory. Most approaches that have grown from constructivism suggest that learning is accomplished best using a hands-on approach. Learners learn by experimentation, and not by being told what will happen. They are left to make their own inferences, discoveries and conclusions. It also emphasizes that learning is not an "all or nothing" process but that students learn the new information that is presented to them by building upon knowledge that they already possess. It is therefore important that teachers constantly assess the knowledge their students have gained to make sure that the students' perceptions of the new knowledge are what the teacher had intended. Teachers will find that since the students build upon already existing knowledge, when they are called upon to retrieve the new information, they may make errors. It is known as reconstruction error when we fill in the gaps of our understanding with logical, though incorrect, thoughts. Teachers need to catch and try to correct these errors, though it is inevitable that some reconstruction error will continue to occur because of our innate retrieval limitations.

    In most pedagogies based on constructivism, the teacher's role is not only to observe and assess but to also engage with the students while they are completing activities, wondering aloud and posing questions to the students for promotion of reasoning (DeVries et al., 2002). (ex: I wonder why the water does not spill over the edge of the full cup?) Teachers also intervene when there are conflicts that arise; however, they simply facilitate the students' resolutions and self-regulation, with an emphasis on the conflict being the students' and that they must figure things out for themselves. For example, promotion of literacy is accomplished by integrating the need to read and write throughout individual activities within print-rich classrooms. The teacher, after reading a story, encourages the students to write or draw stories of their own, or by having the students reenact a story that they may know well, both activities encourage the students to conceive themselves as reader and writers.

    Specific approaches to education that are based on constructivism include:

    Constructionism

    An approach to learning developed by Seymour Papert and his colleagues at MIT in Cambridge, Massachusetts. Papert had worked with Piaget at the latter's Institute in Geneva. Papert eventually called his approach "constructionism." It included everything associated with Piaget's constructivism, but went beyond it to assert that constructivist learning happens especially well when people are engaged in constructing a product, something external to themselves such as a sand castle, a machine, a computer program or a book. This approach is greatly facilitated by the ready availability of powerful 'constructing' applications on personal computers. Promoters of the use of computers in education see an increasing need for students to develop skills in Multimedia literacy in order to use these tools in constructivist learning.

    • Reciprocal Learning (each one teach one)

    • Procedural Facilitations for Writing

    • Cognitive Tutors

    • Cognitively Guided Instruction

    A research and teacher professional development program in elementary mathematics created by Thomas P. Carpenter, Elizabeth Fennema, and their colleagues at the University of Wisconsin-Madison. Its major premise is that teachers can use students' informal strategies (i.e., strategies students construct based on their understanding of everyday situations, such as losing marbles or picking flowers) as a primary basis for teaching mathematics in the elementary grades.

    • Anchored Instruction (Bransford et al)

    • Problems and approaches to solutions are embedded in a narrative environment.

    • Cognitive Apprenticeship (Collins et al)

    • Learning is achieved by integration into a specific implicit and explicit culture of knowledge.

    • Cognitive Flexibility (Sprio et al)

    Constructive alignment (Biggs and Tang, 2007), a constructivist approach to curriculum design, in which the learning activities spelled out in the intended learning outcomes are built into the teaching methods and assessment tasks.

    Pragmatic Constructivism (Müller, Klaus 2001)

    The silent way, a constructivist approach to foreign language teaching and learning developed by Caleb Gattegno who worked with Piaget before WWII and in the late 1940s.

    Research and evidence supporting constructivism

    Hmelo-Silver, Duncan, & Chinn cite several studies supporting the success of the constructivist problem-based and inquiry learning methods. For example, they describe a project called GenScope, an inquiry-based science software application. Students using the GenScope software showed significant gains over the control groups, with the largest gains shown in students from basic courses. [2]

    Hmelo-Silver et al also cite a large study by Geier on the effectiveness of inquiry-based science for middle school students, as demonstrated by their performance on high-stakes standardized tests. The improvement was 14% for the first cohort of students and 13% for the second cohort. This study also found that inquiry-based teaching methods greatly reduced the achievement gap for African-American students.[2]

    Guthrie et al (2004) compared three instructional methods for third-grade reading: a traditional approach, a strategies instruction only approach, and an approach with strategies instruction and constructivist motivation techniques including student choices, collaboration, and hands-on activities. The constructivist approach, called CORI (Concept-Oriented Reading Instruction), resulted in better student reading comprehension, cognitive strategies, and motivation.[3]

    Jong Suk Kim found that using constructivist teaching methods for 6th graders resulted in better student achievement than traditional teaching methods. This study also found that students preferred constructivist methods over traditional ones. However, Kim did not find any difference in student self-concept or learning strategies between those taught by constructivist or traditional methods.[4]

    Doğru and Kalender compared science classrooms using traditional teacher-centered approaches to those using student-centered, constructivist methods. In their initial test of student performance immediately following the lessons, they found no significant difference between traditional and constructivist methods. However, in the follow-up assessment 15 days later, students who learned through constructivist methods showed better retention of knowledge than those who learned through traditional methods.[5]

     

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