tag:blogger.com,1999:blog-70388680807040558762024-03-14T14:44:02.520+07:00Psychology of Mathematics EducationDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-7038868080704055876.post-73124806737851531492009-12-20T09:05:00.000+07:002009-12-20T09:08:01.428+07:00Psikologi Golongan Darah<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CUsers%5Cmarsigit%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CUsers%5Cmarsigit%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CUsers%5Cmarsigit%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> 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gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;} </style> <![endif]--> <p class="MsoNormal" style="line-height: normal;"><b><span style="font-size: 13.5pt; font-family: "Times New Roman","serif";"><a href="http://powermathematics.blogspot.com/2009/12/psikologi-golongan-darah.html"><span style="color: blue;">Psikologi Golongan Darah</span></a> <o:p></o:p></span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><b><span style="font-size: 12pt; font-family: "Times New Roman","serif";"></span></b><span style="font-size: 12pt; font-family: "Times New Roman","serif";">Di referensikan oleh Silfi Yulian dari :
<br />http://www.akhirzaman.info/tanda-akhir-zaman/asal-anda-tahu/1549-asal-anda-tahu-psikologi-golongan-darah
<br />
<br />GOLONGAN DARAH
<br />
<br />Di Jepang, ramalan ttg seseorang lebih ditentukan oleh golongan darah daripada zodiak atau shio. Kenapa? Katanya, golongan darah itu
<br />ditentukan oleh protein-protein tertentu yang membangun semua sel di tubuh kita dan oleh karenanya juga menentukan psikologi kita. Benar apa tidak?
<br />
<br />SIFAT SECARA UMUM :
<br />A : terorganisir, konsisten, jiwa kerja-sama tinggi, tapi selalu cemas (krn perfeksionis) yg kadang bikin org mudah sebel, kecenderungan politik: "destra"
<br />B : nyantai, easy going, bebas, dan paling menikmati hidup, kecenderungan politik: "sinistra"
<br />O : berjiwa besar, supel, gak mau ngalah, alergi pada yg detil, kecenderungan politik: "centro"
<br />AB: unik, nyleneh, banyak akal, berkepribadian ganda, kecenderungan
<br />politik
<br />
<br />BERDASARKAN URUTAN :
<br />Yg paling gampang ngaret soal waktu :
<br />1 B (krn nyantai terus)
<br />2 O (krn flamboyan)
<br />3 AB (krn gampang ganti program)
<br />4 A (krn gagal dalam disiplin)
<br />
<br />Yg paling susah mentolerir kesalahan org :
<br />1 A (krn perfeksionis dan narsismenya terlalu besar)
<br />2 B (krn easy going tapi juga easy judging)
<br />3 AB (krn asal beda)
<br />4 O (easy judging tapi juga easy pardoning)
<br />
<br />Yg paling bisa dipercaya :
<br />1 A (krn konsisten dan taat hukum)
<br />2 O (demi menjaga balance)
<br />3 B (demi menjaga kenikmatan hidup)
<br />4 AB (mudah ganti frame of reference)
<br />
<br />Menurut survey, gol darah yg paling disukai utk jadi teman :
<br />1 O (orangnya sportif)
<br />2 A (selalu on time dan persis)
<br />3 AB (kreatif)
<br />4 B (tergantung mood)
<br />
<br />Kebalikannya, teman yg paling disebelin/tidak disukai:
<br />1 B (egois, easy come easy go, maunya sendiri)
<br />2 AB (double standard)
<br />3 A (terlalu taat dan scrupulous)
<br />4 O (sulit mengalah)
<br />
<br />MENYANGKUT OTAK DAN KEMAMPUAN :
<br />Yg paling mudah kesasar/tersesat
<br />1 B
<br />2 A
<br />3 O
<br />4 AB
<br />
<br />Yg paling banyak meraih medali di olimpiade olah raga:
<br />1 O (jago olah raga)
<br />2 A (persis dan matematis)
<br />3 B (tak terpengaruh pressure dari sekitar. Hampir seluruh atlet judo,
<br />renang dan gulat jepang bergoldar B)
<br />4 AB (alergi pada setiap jenis olah raga)
<br />
<br />Yg paling banyak jadi direktur dan pemimpin :
<br />1 O (krn berjiwa leadership dan problem-solver)
<br />2 A (krn berpribadi "minute" dan teliti)
<br />3 B (krn sensitif dan mudah ambil keputusan)
<br />4 AB (krn kreatif dan suka ambil resiko)
<br />
<br />Yg jadi PM jepang rata2 bergoldar :
<br />O (berjiwa pemimpin)
<br />
<br />Mahasiswa Tokyo Univ pada umumnya bergol darah : B
<br />
<br />Yg paling gampang nabung :
<br />1 A (suka menghitung bunga bank)
<br />2 O (suka melihat prospek)
<br />3 AB (menabung krn punya proyek)
<br />4 B (baru menabung kalau punya uang banyak)
<br />
<br />Yg paling kuat ingatannya :
<br />1 O
<br />2 AB
<br />3 A
<br />4 B
<br />
<br />Yg paling cocok jadi MC : A (kaya planner berjalan)
<br />
<br />MENYANGKUT KESEHATAN :
<br />Yg paling panjang umur :
<br />1 O (gak gampang stress, antibody nya paling joss!)
<br />2 A (hidup teratur)
<br />3 B (mudah cari kompensasi stress)
<br />4 AB (amburadul)
<br />Yg paling gampang gendut
<br />1 O (nafsu makan besar, makannya cepet lagi)
<br />2 B (makannya lama, nambah terus, dan lagi suka makanan enak)
<br />3 A (hanya makan apa yg ada di piring, terpengaruh program diet)
<br />4 AB (Makan tergantung mood, mudah kena anoressia)
<br />
<br />Paling gampang digigit nyamuk : O (darahnya manis)
<br />
<br />Yg paling gampang flu/demam/batuk/pilek
<br />1 A (lemah terhadap virus dan pernyakit menular)
<br />2 AB (lemah thd hygiene)
<br />3 O (makan apa saja enak atau nggak enak)
<br />4 B (makan, tidur nggak teratur)
<br />
<br />Apa yg dibuat pada acara makan2 di sebuah pesta :
<br />O (banyak ngambil protein hewani, pokoknya daging2an)
<br />A (ngambil yg berimbang. 4 sehat 5 sempurna)
<br />B (suka ambil makanan yg banyak kandungan airnya spt soup, soto, bakso dsb)
<br />AB (hobby mencicipi semua masakan, "aji mumpung")
<br />
<br />Yg paling cepat botak :
<br />1 O
<br />2 B
<br />3 A
<br />4 AB
<br />
<br />Yg tidurnya paling nyenyak dan susah dibangunin :
<br />1 B (tetap mendengkur meski ada Tsunami)
<br />2 AB (jika lagi mood, sleeping is everything)
<br />3 A (tidur harus 8 jam sehari, sesuai hukum)
<br />4 O (baru tidur kalau benar2 capek dan membutuhkan)
<br />
<br />Yg paling cepet tertidur
<br />1 B (paling mudah ngantuk, bahkan sambil berdiripun bisa tertidur)
<br />2 O (Kalau lagi capek dan gak ada kerjaan mudah kena ngantuk)
<br />3 AB (tergantung kehendak)
<br />4 A (tergantung aturan dan orario)
<br />
<br />Penyakit yg mudah menyerang :
<br />A (stress, majenun/linglung)
<br />B (lemah terhadap virus influenza, paru-paru)
<br />O (gangguan pencernaan dan mudah kena sakit perut)
<br />AB (kanker dan serangan jantung, mudah kaget)
<br />
<br />Apa yg perlu dianjurkan agar tetap sehat :
<br />A (Krn terlalu perfeksionis maka nyantailah sekali-kali, gak usah terlalu tegang dan serius)
<br />B (Krn terlalu susah berkonsentrasi, sekali-kali perlu serius sedikit,
<br />meditasi, main catur)
<br />O (Krn daya konsentrasi tinggi, maka perlu juga mengobrol santai,jalan-jalan)
<br />AB (Krn gampang capek, maka perlu cari kegiatan yg menyenangkan dan bikin lega).
<br />
<br />Yg paling sering kecelakaan lalu lintas (berdasarkan data kepolisian)
<br />1 A
<br />2 B
<br />3 O
<br />4 AB
<br />
<br />"Nilai manusia, bukan bagaimana ia mati, melainkan bagaimana ia hidup; bukan apa yang diperoleh, melainkan apa yang telah diberikan; bukan apa pangkatnya, melainkan apa yang telah diperbuat dengan tugas yang diberikan Tuhan kepadanya." - Ministry<o:p></o:p></span></p> <p class="MsoNormal"><o:p> </o:p></p> Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com19tag:blogger.com,1999:blog-7038868080704055876.post-477728979197192532009-01-12T22:26:00.006+07:002009-01-12T22:38:09.831+07:00Kualitas Kedua dalam RPP pada Skema Pencapaian Kompetensi 'is never ending effort'<h3 style="font-weight: normal;font-family:times new roman;" class="post-title entry-title"><p style="text-align: justify; font-weight: bold;"><span style="font-size:100%;"><span style="font-size:85%;">Oleh: Euis Kurniawati, SPd</span></span></p><p style="text-align: justify;"><span style="font-size:100%;">Direkomendasikan oleh Dr Marsigit<br /></span></p><p style="text-align: justify;"><span style="font-size:100%;"><span style="font-size:85%;">Pada perkuliahan terakhir hari Rabu (31 Desember 2008) sampailah kami pada suatu kesimpulan akhir tentang materi kuliah Perencanaan Pembelajaran Matematika (PPM). Ternyata apa yang kami harapkan selama ini dari kuliah PPM yaitu berupa 'produk' RPP (Rencana Pelaksanaan Pembelajaran atau Lesson Plan) yang kami bayangkan, yang baku, baik dan benar (<em>adakah?</em>) sesuai kebutuhan di sekolah, yang siap pakai (<em>siap difoto copy oleh rekan-rekan di sekolah/daerah</em>) terhempaskan sedalam-dalamnya ke dasar jurang pemikiran kami yang masih dalam kategori 'mengkhawatirkan'.</span><br /></span></p></h3> <div style="font-family: times new roman;font-family:times new roman;" class="post-body entry-content" > <span style="font-size:100%;"><span xmlns=""><p style="text-align: justify;">RPP yang kami harapkan tersebut adalah RPP Formal, RPP kualitas Pertama. RPP yang hanya menampakkan alur proses pembelajaran, yang bisa diperoleh dari contoh yang sudah ada, bahkan kami bisa saja mencarinya di internet.<br /></p><p style="text-align: justify;">Senyatanya, RPP yang beliau (Dr. Marsigit M.A.) maksudkan, yang beliau harapkan agar kami pahami, adalah RPP dalam ranah kualitas Kedua.<br /></p><p style="text-align: justify;">Dengan segala keterbatasan saya dalam memahami dan memaknai RPP dalam kualitas Kedua, saya mencoba untuk menuangkannya sebagai suatu wujud permulaan pemahaman yang saya miliki, tentu saja jauh dari sempurna.<br /></p><p style="text-align: justify;"><strong>Apa yang dimaksud dengan RPP<em>?<br /></em></strong></p><p style="text-align: justify;">RPP (Rencana Pelaksanaan Pembelajaran) adalah seperangkat persiapan yang dilakukan oleh guru meliputi skenario sebelum, selama, dan setelah kegiatan pembelajaran berlangsung. Sedangkan menurut Dr. Marsigit M.A., RPP pada dasarnya merefleksikan aktivitas yang terjalin antara siswa dan guru.<br /></p><p style="text-align: justify;">Perbedaan nyata antara kegiatan pembelajaran di Indonesia dengan negara lain adalah terletak pada 'persiapan'. Contohnya persiapan (RPP) yang digunakan di negara Jepang berisi tentang hal-hal yang akan dikerjakan oleh siswa. Sedangkan RPP yang digunakan di Indonesia berisi hal-hal yang akan dikerjakan oleh guru. RPP yang biasanya disusun oleh guru di Indonesia, kebanyakan berupa RPP legal formal yang dipergunakan untuk kepentingan karir atau dinas semata belum menyentuh hakikat RPP yang sebenarnya (Nature Leson Plan).<br /></p><p style="text-align: justify;">RPP dapat ditinjau dari berbagai skema, diantaranya: Struktur Pembelajaran (Pendahuluan, Kegiatan Inti, dan Penutup), Skema Pencapaian Kompetensi (Will, Attitude, Knowledge, Skill, dan Experience), Skema Interaksi (Klasikal, Kelompok dan Individua), Skema Variasi Metode (Induksi-Deduksi) Skema Variasi Media atau alat bantu pembelajaran (LKS dan Alat Peraga) dan Skema Variasi Sumber Belajar (Buku Teks, Internet atau Blog dan ICT).<br /></p><p style="text-align: justify;"><strong>Kualitas Kedua dalam RPP<br /></strong></p><p style="text-align: justify;">Kualitas Kedua dalam RPP pada Skema Pencapaian Kompetensi (Will, Attitude, Knowledge, Skill, dan Experience) dapat dipandang sebagai suatu cara dalam memaknai RPP berdasarkan hakikat kompetensi yang ingin dicapai dan ditampilkan dalam RPP.<br /></p><p style="text-align: justify;"><em>1. Will: </em>Kemauan, Kehendak (Senang, gembira)<br /></p><p style="text-align: justify;">Bagaimana upaya yang dilakukan guru dalam merancang suatu RPP yang dapat mengeksplorasi dan membangkitkan kemauan atau kehendak siswa untuk belajar dan terlibat aktif dalam kegiatan pembelajaran secara menyenangkan dan dengan perasaan gembira, jauh dari rasa was-was, cemas, bahkan takut. Bagaimana upaya guru dalam membawa dunia matematika ke dunia anak (siswa). Contohnya: menghargai bahasa ibu dalam pembelajaran matematika (selawe = 25; selangkung = 50).<br /></p><p style="text-align: justify;"><em>2. Attitude: </em>Sikap, Pendirian (Sabar, tepat waktu)<br /></p><p style="text-align: justify;">Bagaimana upaya yang dilakukan guru dalam merancang suatu RPP yang dapat menghidupkan 'ruh' pembelajaran matematika dalam diri anak yaitu berupa keuletan, ketekunan, sifat pantang menyerah, kemampuan mengkomunikasikan ide dan gagasan, kemampuan berpikir kritis dan logis, serta kemampuan mencari solusi dari suatu permasalahan. Sehingga melahirkan suatu sikap atau pendirian yang positif bagi siswa misalnya berupa sifat sabar, tekun, serta tepat waktu (disiplin).<br /></p><p style="text-align: justify;"><em>3. Knowledge: </em>Pengetahuan (Mengetahui)<br /></p><p style="text-align: justify;">Bagaimana upaya guru dalam merancang suatu RPP yang dapat menumbuhkan kesadaran pada diri siswa agar mereka 'mengetahui' eksistensi mereka dalam dunia matematika, pembelajaran matematika dan pendidikan matematika. Bagaimana upaya guru membangkitkan kesadaran siswa akan pentingnya membangun pengetahuan (khususnya matematika) mereka sendiri dengan potensi, cara, dan keunikan mereka masing-masing.<br /></p><p style="text-align: justify;"><em>4. Skill: </em>Keterampilan (Terampil)<br /></p><p style="text-align: justify;">Bagaimana upaya guru dalam merancang suatu RPP yang dapat mengasah potensi siswa dalam hal keterampilan mengemukakan ide/gagasan; keterampilan memilih solusi dari permasalahan matematika; keterampilan menyanggah, menjawab, membuktikan suatu hipotesa; keterampilan menggunakan dan menyusun alat peraga; dan sebagainya. Inti dari semuanya itu adalah agar siswa terampil menggunakan potensi keunikan mereka untuk menghadapi dan mencari solusi (secara cerdas) bagi permasalahan dalam pembelajaran matematika. Sehingga matematika tidak lagi menjadi mata pelajaran yang menakutkan dan membosankan bagi siswa.<br /></p><p style="text-align: justify;"><em>5. Experience: </em>Pengalaman (Kebermaknaan)<br /></p><p style="text-align: justify;">Bukankah pepatah mengatakan bahwa <em>pengalaman adalah guru yang terbaik</em>? Maka sangatlah diperlukakan upaya guru dalam merancang suatu RPP yang dapat menjadikan kegiatan pembelajaran matematika sebagai suatu pengalaman penting, berkesan dan berharga bagi siswa. Pengalaman akan dirasakan penting, berkesan dan berharga bagi seseorang jika ada makna yang mendalam dari pengalaman tersebut. Kebermaknaan dalam belajar akan dicapai mana kala siswa merasa terlibat dan berperan aktif secara menyenangkan dan penuh perasaan gembira. Kebermaknaan dapat tercapai dalam kegiatan pembelajaran yang memberi ruang dan kesempatan sebesar-besarnya bagi siswa untuk mengkonstruksi pengetahuan dengan cara dan potensi unik mereka masing-masing, dengan bimbingan guru (tanpa mendominasi) yang mampu melayani kebutuhan para siswanya yang heterogen.<br /></p><p style="text-align: justify;"><strong>Catatan:<br /></strong></p><p style="text-align: justify;"><em>Tulisan ini belum memuaskan bagi saya. Mungkin karena RPP dalam kualitas Kedua merupakan pengetahuan baru bagi saya yang juga sekaligus memancing rasa penasaran saya untuk dapat merancangnya. InsyaAlloh ke depannya akan diupayakan untuk dapat terealisasi. Saya harus lebih banyak belajar dan terus berusaha, <strong>never ending effort</strong> (meminjam istilah Pak Marsigit).</em></p></span></span></div>Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com5tag:blogger.com,1999:blog-7038868080704055876.post-48385323298812309612009-01-10T00:43:00.002+07:002009-01-10T00:56:09.274+07:00Elegi Seorang Guru Menggapai BatasOleh: Marsigit<br /><br />Mulialah hati, pikiran dan tindakan pedagogik guru, karena tiadalah seorang guru bermaksud memberikan keburukan bagi siswanya. Totalitasnya dia mengidamkan kebaikan dan keberhasilan bahkan keberhasilan tertinggi jika memang mungkin bagi siswanya. Maka hati, pikiran dan jiwanya menyatu menjadi motivasi yang kuat bahkan mungkin SANGAT KUAT untuk mewujudkan tindakan pedagogik : MEMBIMBING, MENGAWASI, MEMBEKALI, MENASEHATI, MEWAJIBKAN dan kalau perlu MENGHUKUM siswanya DEMI KEPENTINGAN SISWA.<br />Ketika rakhmat Nya menghampiri guru sedemikian hingga guru dengan sengaja atau tak sengaja menembus BATAS jiwanya sehingga memperoleh kesempatan MELIHAT DIRINYA dari tempat nun jauh secara mandiri maupun berbantuan orang atau guru atau nara sumber, maka adalah suatu tempat dimana BATAS ITU akan digapainya. Dalam batas itulah guru menemukan FATAMORGANA.<br />Atas fatamorgana itu maka terdengarlah nyanyian AKAR dan RUMPUT yang sayup-sayup sampai kira-kira bait-baitnya begini:<br /><span style="font-style: italic;"><br />-Jangan jangan motivasimu yang sangat KUAT telah MELEMAHKAN INISIATIF nya, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MEMBIMBING telah menjadikannya TERGANTUNG DAN TAK BERDAYA, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MENGAWASI telah menjadikannya mereka merasa menjadi makhluk yang memang selalu perlu diawasi dan dengan demikian identik dengan KEBURUKAN, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MEMBEKALI telah menjadikanmu SOMBONG DAN TAKABUR serta menjadikannya RENDAH DIRI DAN TAK BERDAYA, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MENASEHATI adalah AMBISI DAN EGOMU yang menyebabkan mereka hidup TIDAK BERGAIRAH, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MEWAJIBKAN adalah pertanda HILANGNYA NURANIMU sekaligus HILANGNYA NURANI MEREKA, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu MENGHUKUM adalah SEMPITNYA HIDUPMU dan juga SEMPITNYA HIDUP MEREKA, apakah itu batas yang kau cari?<br />-Jangan-jangan maksudmu DEMI KEPENTINGAN SISWA adalah keadaan HAMPA TAK BERMAKNA atau bahkan MENYESATKAN, apakah itu batas yang kau cari?<br /></span><br /><span style="font-weight: bold;">Orang tua berambut putih datang dan berkata:</span><br />"ITULAH SEBENAR-BENAR BATAS YANG KAU CARI, DAN BATASMU TIDAK LAIN TIDAK BUKAN ADALAH BATAS MEREKA. ANTARA DIRIMU DAN MEREKA ITULAH SEBENAR-BENAR ILMUMU TENTANG DIRIMU DAN TENTANG DIRI MEREKA.<br />JIKA KAMU BELUM MENGETAHUI BATASMU JANGAN HARAP RAHMAT ITU DATANG MENGHAMPIRIMU KEMBALI, TETAPI JIKA TOH KAMU TELAH MENGETAHUI BATASMU MAKA ADALAH KODRAT NYA BAHWA KAMU MASIH HARUS BERJUANG MENGGAPAI RAHMATNYA.<br />NAMUN MAAF SAYA TELAH SALAH UCAP, KARENA SEBENAR-BENARNYA YANG ADA, ADALAH TIADALAH ORANG PERNAH DAN DAPAT MENGGAPAI BATASNYA. UNTUK ITU MARILAH KITA TAWADU' DENGAN KODRAT DAN IRADATNYA. TETAPI JANGAN SALAH PAHAM BAHWA IKHTIARMU MENGGAPAI BATASMU ADALAH JUGA KODRATNYA.<br />AMIN YA ROBBIL ALAMIN"Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com34tag:blogger.com,1999:blog-7038868080704055876.post-54012303512836401132009-01-04T09:02:00.003+07:002009-01-04T17:02:48.119+07:00The Implication of Vygotsky’s Work to Mathematics Education<span style="font-weight: bold;">By Marsigit</span><br /> 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.<br /> 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).<br /> 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.<br /> 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.<br /> 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).<br /><span style="font-weight: bold;">References:</span><br />Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.<br />Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.<br />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.<br />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.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com3tag:blogger.com,1999:blog-7038868080704055876.post-14373131610739119892009-01-04T09:00:00.001+07:002009-01-04T09:00:34.752+07:00The Implication Of Piaget's Work to Mathematics Education<span style="font-weight:bold;">By Marsigit</span><br /> 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.<br /> 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.<br /> 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.<br /> 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.<br /> 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.<br /> 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.<br /><span style="font-weight:bold;">References:</span><br />Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.<br />Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.<br />Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.<br />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.<br />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.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com4tag:blogger.com,1999:blog-7038868080704055876.post-22685371344073701262009-01-02T08:18:00.000+07:002009-01-02T08:19:37.016+07:00Vygotsky's Work and Its Relevance to Mathematics Education<span style="font-weight: bold;">By. Marsigit</span><br /> 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.<br /> 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.<br /> 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.<br /> 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.<br /><span style="font-weight: bold;">References:</span><br />Wertsch, J.V.,1985, Vygotsky and The Social Formation of Mind,London : Harvard University Press.<br />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.<br />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.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com0tag:blogger.com,1999:blog-7038868080704055876.post-38133935037339417692009-01-02T08:16:00.000+07:002009-01-02T08:17:01.525+07:00Piaget's Work and its Relevance to Mathematics Education<span style="font-weight:bold;">By Marsigit</span><br /> 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. <br /> 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).<br /> 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).<br /> 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.<br /> 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).<br /> 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.<br /> 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 ?<br /><span style="font-weight:bold;">References:</span><br />Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.<br />Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.<br />Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com2tag:blogger.com,1999:blog-7038868080704055876.post-10896142810043317452009-01-02T04:09:00.000+07:002009-01-02T04:10:13.083+07:00Bagaimana Melakukan Diskusi Matematika dengan Siswa?Oleh: Marsigit<br />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:<br />1. dengarkanlah apa yang mereka katakan<br />2. usahakan suasana bersifat mendukung diskusi<br />3. usahakan suasana berpikir merdeka<br />4. usahakan agar siswa mempunyai inisiatif<br />5. usahakan agar siswa mempunyai rasa percaya diri<br />6. gunakan kata-kata yang mudah dipahami oleh siswa<br />7. usahakan hubungan yang baik antara guru dan siswa<br />8. berusaha mengetahui ide/gagasan siswa<br />9. menampung ungkapan siswa sebagai informasi<br />10. utarakan balikan secara bertahap<br />11. gunakan ilustrasi untuk mempermudah komunikasi<br />12. menghargai setiap ide yang disampaikan oleh siswa<br />13. usahakan agar diskusi bersifat efaktif dan bermanfaat.<br />14. memahami bahwa siswa memerlukan penjelasan<br />15. fokus pada pertanyaan siswa/penanya (jangan Jawa: disambi)<br />16. memikirkan hal-hal yang terkait dengan pertanyaan.<br />17. memikirkan solusi jawaban<br />18. bersifat jujur dan terusterang<br />19. berpikir positif (tidak prejudice) terhadap si penanya<br />20. tidak mempunyai kepentingan ganda misalnya mengajukan pertanyaan yang sulit untuk tujuan menghukum<br />21. gunakan referensi-referensi untuk mencari solusi.<br />22. kembangkan sikap menghargai dan empati<br />23. hindari sikap menang sendiri/otoriter<br />24. letakkan persoalan pada jejaring konsep yang lebih luas<br />25. usahakan agar siswa senang melakukannya kembali<br />26. jika memungkinkan libatkan siswa yang lainnya<br />27. menyadari manfaat jawaban terbuka<br />28. menyadari bahwa pertanyaan siswa adalah awal dari pengetahuannya<br />29. memperhatikan alokasi waktu dan konteks sekitarDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com9tag:blogger.com,1999:blog-7038868080704055876.post-7251021239839192802009-01-02T04:08:00.001+07:002009-01-02T04:08:49.251+07:00Pada pembelajaran tradisional guru terpaksa sibuk mengontrol siswanyaOleh: Marsigit<br /><br />Pada pembelajaran tradisional, guru terpaksa melakukan berbagai kegiatan kontrol agar siswa bersikap kooperatif dan memperhatikan guru. Hal ini disebabkan karena guru 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:<br /><br /><span style="font-weight: bold;">Aterensi 1:</span><br />Wahyu ngantuk ya ?<br />Nah tadi sudah makan belum ?<br />Sudah.<br />Banyak nonton T.V. ya ?<br />Nah berapa coba ?<br /><br /><span style="font-weight: bold;">Aterensi 2:</span><br />Bagaimana cara penyelesaiannya ?<br />Penyelesaiannya bagaimana ?<br />Bagaimana Anto ?<br />YANG LAIN DIAM.<br /><br /><span style="font-weight: bold;">Aterensi 3:</span><br />Sudah apa belum ?<br />Nah kalau sudah diteliti dulu dari nomor satu sampai nomor sepuluh<br />AYO PRIHANTO KAMU SUDAH SELESAI ?<br />SUDAH DITELITI ?<br />AYO DITELITI LAGI<br />COBA SAMPAI BETUL<br />TIDAK HANYA DILIHAT LHO YA PRIHANTO<br /><br /><span style="font-weight: bold;">Aterensi 4: </span><br />Ayo Candra duduknya bagaimana ?<br />Hayo yang tertib Rona, ayo Rona duduknya yang bagus.<br />Nanti supaya betul semua<br /><br /><span style="font-weight: bold;">Aterensi 5:</span><br />Nol dapat dikurang lima ?<br />Sebetulna dapat tapi ini kurang ya to ?<br />Jadi barang tidak ada dikurangi lima, maka ...<br />SETERUSNYA KITA BAGAIMANA ?<br />Pinjam<br />Pinjam pada...?Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com4tag:blogger.com,1999:blog-7038868080704055876.post-69273502433759096822009-01-02T01:51:00.001+07:002009-01-02T01:51:21.064+07:00The Role of Cognitive Development Theory for Mathematics Education<span style="font-weight:bold;">By. Marsigit </span><br /> 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.<br /> 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.<br /> 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.<br /> 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.<br /><span style="font-weight:bold;">Reference:</span><br />Ernest, P.,1991, The Philosophy of Mathematics Education, London : The Falmer Press.<br />Gipps, C., 1994, 'What we know about effective primary teaching' in Bourne, J., 1994, Thinking Through Primary Practice, London : Routledge.<br />McNamara, D., 1994, Classroom pedagogy and primary practice, London : Routledge.<br />Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.<br />Turner, J.,1984, Cognitive Development and Education, London: Methuen.<br />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.<br />Wood, D., 1988, How Children Think & Learn, Oxford: Basil Blackwell.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com1tag:blogger.com,1999:blog-7038868080704055876.post-6932471261809737252008-12-22T06:26:00.001+07:002008-12-22T06:26:47.884+07:00PENDEKATAN MATEMATIKA REALISTIK PADA PEMBELAJARAN PECAHAN DI SMPOleh: Marsigit<br />Sebagaimana telah kita ketahui, Matematika Realistik menekankan kepada konstruksi dari konteks benda-benda konkrit sebagai titik awal bagi siswa guna memperoleh konsep matematika. Benda-benda konkret dan obyek-obyek lingkungan sekitar dapat digunakan sebagai konteks pembelajaran matematika dalam membangun keterkaitan matematika melalui interaksi sosial. Benda-benda konkrit dimanipulasi oleh siswa dalam kerangka menunjang usaha siswa dalam proses matematisasi konkret ke abstrak. Siswa perlu diberi kesempatan agar dapat mengkontruksi dan menghasilkan matematika dengan cara dan bahasa mereka sendiri. Diperlukan kegiatan refleksi terhadap aktivitas sosial sehingga dapat terjadi pemaduan dan penguatan hubungan antar pokok bahasan dalam struktur pemahaman matematika.<br />Menurut Hans Freudental matematika merupakan aktivitas insani (human activities) dan harus dikaitkan dengan realitas. Dengan demikian ketika siswa melakukan kegiatan belajar matematika maka dalam dirinya terjadi proses matematisasi. Terdapat dua macam matematisasi, yaitu: (1) matematisasi horisontal dan (2) matematisasi vertikal. Matematisasi horisontal berproses dari dunia nyata ke dalam simbol-simbol matematika. Proses terjadi pada siswa ketika ia dihadapkan pada problematika yang kehidupan / situasi nyata. Sedangkan matematisasi vertikal merupakan proses yang terjadi di dalam sistem matematika itu sendiri; misalnya: penemuan strategi menyelesaiakn soal, mengkaitkan hubungan antar konsep-konsep matematis atau menerapkan rumus/temuan rumus.<br />Kita dapat menelaah Bilangan Pecah dalam pembelajaran matematika SMP melalui 2 (dua) sisi yaitu kedudukan formal Bilangan Pecah dalam konteks kurikulum dan silabus, dan kajian substantif bilangan pecah itu sendiri. Di dalam Pedoman Pengembangan KTSP disebutkan bahwa dalam pembelajaran matematika dapat dimulai dengan pengenalan masalah yang sesuai dengan situasi (contextual problem). Dengan mengajukan masalah kontekstual, peserta didik secara bertahap dibimbing untuk menguasai konsep matematika. Tujuan pembelajaran bilangan pecahan di SMP dapat disebutkan sebagai berikut:<br />1. Memecahkan masalah kontekstual dan menemukan konsep bilangan pecah dari masalah kontekstual yang dipecahkan.<br />2. Memahami konsep bilangan pecah, menjelaskan keterkaitan antar konsep dan mengaplikasikan konsep bilangan pecah, secara luwes, akurat, efisien, dan tepat, dalam pemecahan masalah<br />3. Menggunakan penalaran pada pola dan sifat, melakukan manipulasi dan membuat generalisasi tentang bilangan pecah.<br />4. Mengomunikasikan konsep dan penggunaan bilangan pecah<br />5. Memiliki sikap menghargai kegunaan bilangan pecah dalam kehidupan sehari-hari.<br />Standar Kompetensi yang berkaitan dengan pembelajaran pecahan adalah agar siswa memahami sifat-sifat operasi hitung bilangan dan penggunaannya dalam pemecahan masalah. Dengan Materi Pokok berupa Bilangan Bulat dan Bilangan Pecah maka diharapkan dapat dicapai menggunakan 2 (dua) Kompetensi Dasar yaitu: Melakukan operasi hitung bilangan pecahan, dan menggunakan sifat-sifat operasi hitung bilangan pecahan dalam pemecahan masalah.<br />Tipe realistik mempunyai ciri pendekatan buttom-up dimana siswa mengembangkan model sendiri dan kemudian model tersebut dijadikan dasar untuk mengembangkan matematika formalnya. Ada dua macam model yang terjadi dalam proses tersebut yakni model dari situasi (model of situation) dan model untuk matematis (model for formal mathematics). Di dalam realistik model muncul dari strategi informal siswa sebagai respon terhadap masalah real untuk kemudian dirumuskan dalam matematika formal, proses seperti ini sesuai dengan sejarah perkembangan matematika itu sendiri.<br />Berikut merupakan contoh pengembangan Masalah Realistik berkaitan dengan Bilangan Pecahan: Suatu Bahan Diskusi Untuk Para Guru<br />1. Pecahan dan bentuknya<br />Diskusikan seberapa jauh anda dapat menggunakan ilustrasi atau gambar sebagai sarana agar siswa dapat menggali atau menemukan konsep dan bentuk pecahan?<br />2. Pecahan Sederhana<br />Buatlah masalah kontekstual yang dapat menunjang pembelajaran Pecahan Sederhana !<br />3. Membandingkan Pecahan<br />Diskusikan bagaimana mengembangkan alat peragayang cukup memadai agar siswa mampu membandingkan pecahan? Jelaskan bagaimana menggunakannya 5. Mengurutkan Pecahan-pecahan <br />4. Pecahan Desimal<br />Diskusikan adakah suatu proses yang cukup memadai agar siswa mampu memahami pecahan desimal?<br />Penulis dapat menyimpulkan bahwa di dalam pembelajaran Bilangan Pecahan melalui pendekatan Realistik kiranya dapat disimpulkan bahwa:<br />1. Siswa perlu diberi kesempatan untuk menggali dan merefleksikan konsep alternatif tentang ide-ide bilangan pecahan yang mempengaruhi belajar selanjutnya.<br />2. Siswa perlu diberi kesempatan untuk menggali dan memperoleh pengetahauan baru tentang bilangan pecahan dengan membentuk pengetahuan itu untuk dirinya sendiri.<br />3. Siswa perlu diberi kesempatan untuk memperoleh pengetahuan sebagai proses perubahan yang meliputi penambahan, kreasi, modifikasi, penghalusan, penyusunan kembali dan penolakan.<br />4. Siswa perlu diberi kesempatan untuk memperoleh pengetahuan baru tentang bilangan pecahan yang dibangun oleh siswa untuk dirinya sendiri berasal dari seperangkat ragam pengalaman<br />5. Siswa perlu diberi kesempatan untuk memahami, mengerjakan dan mengimplementasikan bilangan pecahan.<br />Guru perlu merevitalisasi diri sehingga:<br />1. Mendudukan dirinya sebagai fasilitator<br />2. Mampu mengembangkan pembelajaran secara interaktif<br />3. Mampu memberikan kesempatan kepada siswa untuk secara aktif.<br />4. Mampu mengembangkan kurikulum dan silabus dan secara aktif mengaitkan kurikulum dengan dunia riil, baik fisik maupun sosial.<br />5. Mampu mengembangkan skenario pembelajaran:<br />a. Skema Interaksi: Klasikal, Diskusi Kelompok, Kegiatan Individu<br />b. Skema Pencapaian Kompetensi: Motivasi, Sikap, Pengetahuan, Skill, dan Pengalaman<br /><br />BAHAN BACAAN<br />......... 2003. The PISA 2003 Assessment Framework- Mathematics, Reading, Science and Problem solving Knowledge and Skill.<br />Koeno Gravemeijer. 1994. Developing Realistics Mathematics Education. Utrecht: CD Press.<br />Marsigit, dkk, 2007, Matematika SMP Kl VII, Bogor: Yudistira<br />Sutarto Hadi. 2002. Effective Teacher Profesional Development for Implemention of Realistic Mathematics Education in Indonesia. Disertasi. Enschede: PrintPartners IpskampDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com9tag:blogger.com,1999:blog-7038868080704055876.post-88086572920636821162008-12-21T23:35:00.001+07:002008-12-21T23:35:34.047+07:00Contoh Hasil Observasi Penelitian Kelas Pembelajaran Matematika<span style="font-weight: bold;">Hasil Pengamatan Kelas Tentang Metode Matematika Menurut Versi Katagiri, Terhadap Aktivitas Belajar Matematika SMP Kelas II Dalam Mempelajari Luas dan Volume Tabung, Bola dan Kerucut, Tahun 2006</span><br /><span style="font-weight: bold;">Peneliti/Pengamat: Marsigit, Mathilda Susanti, Elly Arliani</span><br /><span style="font-weight: bold;font-size:130%;" >a. Metode Matematika Jenis Problem Formation and Comprehension</span><br /><span style="font-weight: bold;">1) Apakah siswa melakukan ABSTRAKSI</span><br />Abstraksi adalah mencari kesamaan-kesamaan untuk memperoleh bentuk atau sifat yang paling sederhana yang akan menjadi obyek Mathematical Thinking<br />Misal Abstraksi:<br />Dengan ABSTRAKSI maka tentang KUBUS, hanya dipelajari tentang UKURAN dan BENTUK nya saja (bukan warna, bahan, harga, dan sifat-sifat yang lain)<br />Jawab:<br />- Siswa melakukan abstraksi terhadap Model Tabung, Bola dan Kerucut<br />- Model Tabung---abstraksi---unsur-unsur tabung: alas, tinggi, selimut, volum tabung, luas selimut.<br />- Model Bola---abstraksi----unsur-unsur bola: jari-jari bola, diameter, selimut bola, volum bola, luas selimut<br />- Model Kerucut---abstraksi---unsur-unsur kerucut: alas kerucut, puncak, tinggi, selimut, volum kerucut<br />- KONSEP KERUCUT dianggap sebagai SEGITIGA<br />- KERUCUT adalah bangun yang berbentuk SEGITIGA<br />- KERUCUT adalah SEGITIGA YANG DI DALAMNYA TERDAPAT RUANG.<br /><span style="font-weight: bold;">2) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan ABSTRAKSI</span><br />Jawab:<br />Cara melakukan Abstraksi:<br />- Pengamatan terhadap MODEL PERAGA <br />- Dengan membandingkan Contoh Benda dalam kehidupan se-hari-<br />- hari---dengan Model Bangun/Alat Peraga---dan Gambar<br />- Istila-istilah yang digunakan untuk melakukan abstraksi:<br />o Mendefinisikan bangun dengan Kalimat sehari-hari: Tabung adalah benda yang bermanfaat untuk menyimpan pensil; Bola adalah benda yang menyerupai jeruk; Kerucut adalah benda yang terdiri dari lingkaran-lingkaran yang semakin ke-atas semakin kecil; Kerucut adalah segitiga yang mempunyai ruang dan lengkungan.<br />o Menggunakan Istilah pada unsure-unsur:<br />- unsur-unsur tabung: alas, tinggi, selimut, volum tabung, luas selimut.<br />- unsur-unsur bola: jari-jari bola, diameter, selimut bola, volum bola, luas selimut<br />- unsur-unsur kerucut: alas kerucut, puncak, tinggi, selimut, volum kerucut<br /><span style="font-weight: bold;">3) Keadaan yang bagaimana siswa melakukan ABSTRAKSI</span><br />Keadaan yang menyebabkan siswa melakukan ABSTRAKSI:<br />- Setelah guru memberi pertanyaan<br />- Setelah guru memberi kesempatan melakukan kegiatan kelompok<br />- Ketika mengerjakan LKS<br /><span style="font-weight: bold;">4) Apakah siswa melakukan IDEALISASI</span><br /><span style="font-weight: bold;">IDEALISASI adalah (1) menganggap sempurna sifat yang ada, atau (2) menetapkan tujuan atau keadaan untuk dicapai Misal : (1)lurus sempurna, datar sempurna, dst; (2) harus begini, harus begitu dsb</span><br />Jawab:<br />- Siswa melakukan Idealisasi Semu ?<br />- Mengarah pada bentuk ideal yang ditentukan guru<br />- Siswa tidak mempermasalahkan Alat peraga yang cacat.<br />- Siswa belum bisa mengkritisi tentang Model Kerucut jika terbuat dari Baja Tebal. Dapat menyimpulkan akibatnya (idealisasi) jika di ajak diskusi oleh Peneliti<br />- Idealisasi pada syarat berlakunya rumus Volume Kerucut= 1/3 Volume tabung, yaitu bahwa DIAMETER ALAS KERUCUT dan ALAS TABUNG harus persis sama; dan TINGGI KERUCUT dan TINGGI TABUNG harus persis sama (siswa mengecek/melakukan idealisasi dengan pengamatan)<br /><span style="font-weight: bold;">5) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan IDEALISASI</span><br />- Dengan cara melakukan konfirmasi kepada guru apakah kegiatan sudah sesuai dengan yang diharapkan guru.<br />- Dengan bertanya kepada siswa yang lain.<br />- Dengan membetulkan pendapat siswa yang lain.<br />- Dengan pengamatan pada MODEL PERAGA<br /><span style="font-weight: bold;">6) Keadaan yang bagaimana siswa melakukan IDEALISASI</span><br />- Jika mengalami keraguan/kesulitan<br />- Jika di tanya guru<br /><span style="font-weight: bold;">7) Apakah siswa menggunakan GAMBAR/Model Matematika/ALAT PERAGA untuk menyatakan gagasan matematika? Jika “ya” sebut dan gambarlah serta sebut pula tujuannya.</span><br />a. Siswa menggunakan alat peraga untuk menyatakan UNSUR-UNSUR TABUNG, BOLA dan KERUCUT<br />b. Menggunakan Tinggi Tabung sebagai LEBAR persegi panjang pembentuk selimut.<br />c. Menggunakan KELILING LINGKARAN ALAS sebagai PANJANG persegi panjang pembentuk selimut.<br /><span style="font-weight: bold;">8) Apakah siswa menggunakan ANGKA/BILANGAN/LAMBANG MATEMATIKA/Model Matematika/ALAT PERAGA untuk menyatakan gagasan matematika? Jika “ya” sebutkanlah dan sebut pula tujuannya.</span><br />Jawab:<br />a. Siswa menyatakan KONSEP KERUCUT dengan KONSEP dan LINGKARAN.<br />b. Siswa mendefinisikan KERUCUT sebagai BANGUN YANG TERDIRI DARI GARIS LURUS, LINGKARAN ATAU KURVA LENGKUNG.<br />c. Siswa mendefinisikan KERUCUT sebagai SEGITIGA YANG BERDIRI DI ATAS LINGKARAN.<br />d. Menyatakan LUAS dengan L, JARI-JARI dengan r, TINGGI dengan t<br />e. Menyatakan LUAS PERMUKAAN TABUNG =2 phi r (r+t)<br />f. Menyatakan PERMUKAAN BOLA dengan 4 phi r^2<br />g. Menyatakan VOLUME KERUCUT dengan 1/3 phi r^2 t<br /><span style="font-weight: bold;">9) Apakah siswa melakukan PENYEDERHANAAN Konsep Matematika?</span><br />Jawab:<br />Siswa melakukan penyederhanaan konsep sbb:<br />a. KONSEP KERUCUT dianggap sebagai SEGITIGA<br />b. KERUCUT adalah bangun yang berbentuk SEGITIGA<br />c. KERUCUT adalah SEGITIGA YANG DI DALAMNYA TERDAPAT RUANG.<br />d. Penyederhanaan RUMUS/PERHITUNGAN :<br />4 P r^2= 4 x 22/7 x 10,5 x 10,5 = …<br /><span style="font-weight: bold;">10) Dengan cara apa saja dan sebutkan ISTILAH/LAMBANG/Model Matematika/Alat Peraga yang digunakan untuk melakukan PENYEDERHANAAN</span><br />Siswa melakukan penyederhanaan konsep dengan cara:<br />a. Pengamatan terhadap MODEL PERAGA<br />b. Melakukan operasi hitung dari penjabaran rumus<br /><span style="font-weight: bold;">11) Keadaan yang bagaimana siswa melakukan PENYEDERHANAAN</span><br />a. Jika diberi pertanyaan<br />b. Jika diberi kesempatan bekerja di dalam kelompok<br /><span style="font-weight: bold;">12) Apakah siswa membuat CONTOH Konsep Matematika?</span><br /><span style="font-weight: bold;">Jika “Ya” sebutkan contoh-contoh itu.</span><br />Jawab:<br />a. LUAS PERMUKAAN TABUNG =2 phi r (r+t)<br />b. PERMUKAAN BOLA dengan 4 phi r^2<br />c. VOLUME KERUCUT dengan 1/3 phi r^2 t<br /><span style="font-weight: bold;">13) Keadaan yang bagaimana siswa mampu membuat CONTOH</span><br /><span style="font-weight: bold;">Keterangan: Contoh Positif, Contoh Negatif, Secara Lisan, Secara Tertulis, Secara Langsung, Secara Tidak Langsung.</span><br /> Jawab:<br /> Ketika di beri pertanyaan<br /> Ketika mengerjakan soal<br /><span style="font-size:130%;"><span style="font-weight: bold;">b. Metode Matematika Jenis Establishing a Perspective</span></span><br /><span style="font-weight: bold;">1. Apakah siswa melakukan ANALOGI terhadap Prosedur/Langkah Matematika? </span><br /><span style="font-weight: bold;">Analogi adalah menerapkan suatu prosedur yang sama untuk keadaan atau tujuan yang berbeda.</span><br />JAwab:<br />ya<br />a. SELIMUT TABUNG di analoginak dengan LUAS PERSEGIPANJANG<br />b. LILITAN BOLA di analogikan dengan LUAS PERMUKAAN BOLA<br />c. MODEL GEOMETRI di analogikan dengan BENDA SEKITAR misal, Kerucut dengan Topi, dsb<br /><span style="font-weight: bold;">2. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan ANALOGI</span><br />Jawab:<br />a. Menutup permukaan Bola<br />b. Menutup permukaan Tabung<br />c. Membuka lagi<br />d. Melilit<br /><span style="font-weight: bold;">3. Keadaan yang bagaimana siswa melakukan ANALOGI</span><br />a. Ketika mengerjakan LKS<br />b. Ketika bekerja di kelompok<br />4. Apakah siswa membuat CATATAN/KETERANGAN terhadap Prosedur/Langkah Matematika?<br /> Jawab:<br /> Ya<br /><span style="font-weight: bold;">5. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan CATATAN/KETERANGAN?</span><br /><span style="font-weight: bold;">Jika “Ya” sebutkan CATATAN/KETERANGAN tersebut.</span><br /> Jawab:<br /> Membetulkan rumus<br /><span style="font-weight: bold;">6. Keadaan yang bagaimana siswa Membuat CATATAN/KETERANGAN</span><br /> Jawab:<br />Setelah memperoleh konfirmasi dari anggota kelompom yang lain atau dari Ibu Guru<br /><span style="font-size:130%;"><span style="font-weight: bold;">c. Metode Matematika Jenis Executing Solutions</span></span><br /><span style="font-weight: bold;">1. Apakah siswa melakukan kegiatan INDUKSI</span><br /><span style="font-weight: bold;">Keterangan: Induksi dapat berupa: menemukan pola, menemukan rumus, </span><br /> Jawab:<br />a. Bagi siswa yang belum tahu melakukan induksi untuk menemukan rumus.<br />b. Bagi siswa yang sudah tahu, melakukan induksi untuk reconfirm rumus yang telah mereka ketahui (induksi semu)<br />c. Induksi dilakukan untuk menemukan rumus<br /><span style="font-weight: bold;">2. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan INDUKSI</span><br /> Jawab:<br />a. Luas permuk Tabung, induksi dilakukan dengan :<br />- Pengamatan model tabung<br />- Manipulasi model Tabung<br />- Menggambar komponen tabung: lingkaran bawah, lingkaran atas, dan bagian tengah tabung<br />- Menentukan luas masing-masing komponen<br />- Menjumlahkan luas masing-masing komponen<br />Catatan: ada siswa melakukan kekeliruan dengan mengalikan luas.<br />b. Luas permuk. Bola, induksi dilakukan dengan:<br />- Pengamatan model Bola<br />- Melakukan lilitan menutup muka setengah Bola dengan tali<br />- Memikirkan bahwa panjang lilitan = luas permukaan setengah bola<br />- Lilitan pada bola digunakan untuk menutup luas daerah lingkaran<br />- Menemukan bahwa Luas permukaan setengah bola = dua kali luas lingkaran<br />- Menemukan bahwa luas permukaan bola = 4 kali luas lingkaran.<br />c. Volume kerucut, induksi dilakukan dengan :<br />- pengamatan model<br />- praktek mengisi penuh tabung dengan volume kerucut<br />- (Terjadi kesalahan prosedur di dalam praktek, karena tidak teliti membaca petunjuk yg dibuat guru)<br /><span style="font-weight: bold;">3. Keadaan yang bagaimana siswa melakukan INDUKSI</span><br /> Jawab:<br />a. Ketika mengerjakan LKS<br />b. Ketika bekerja di dalam kelompok<br /><span style="font-weight: bold;">4. Apakah siswa melakukan kegiatan DEDUKSI</span><br />Keterangan: Deduksi dapat berupa: mengerjakan contoh,<br /> Jawab:<br />a. Bagi siswa yang belum tahu rumusnya, tidak melakukan deduksi untuk menemukan rumus<br />b. Bagi siswa yang sudah tahu rumusnya, melakukan deduksi untuk reconfirm rumus yang telah mereka ketahui<br />c. Mereka semua melakukan deduksi bahwa rumus yang mereka temukan berlaku untuk semua (Tabung, Bola, Kerucut); dan ditunjukkan dengan mengerjakan soal yang diberikan oleh gurunya.<br /><span style="font-weight: bold;">5. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan DEDUKSI</span><br /> Jawab:<br />a. Mengerjakan soal<br /><span style="font-weight: bold;">6. Keadaan yang bagaimana siswa melakukan DEDUKSI</span><br /> Jawab:<br />a. Bekerja di dalam kelompok<br />b. Mengerjakan soal<br />c. Diberi pertanyaan lisan<br /><span style="font-weight: bold;">7. Apakah siswa menggunakan GAMBAR/Model Matematika/Alat Peraga untuk menyelesaikan soal matematika? Jika “ya” sebut dan gambarlah serta sebut pula tujuannya.</span><br /> Jawab:<br />a. Menggunakan Alat Peraga untuk menyelesaikan soal<br />b. Tetapi tanpa alat peraga juga bisa<br /><span style="font-weight: bold;">8. Apakah siswa menggunakan ANGKA/BILANGAN/LAMBANG MATEMATIKA/Model Matematika/Alat Peraga untuk menyelesaikan soal matematika? Jika “ya” sebutkanlah dan sebut pula tujuannya.</span><br /> Jawab:<br />a. Menggunakan Alat Peraga untuk menyelesaikan soal<br />b. Tetapi tanpa alat peraga juga bisa<br /><span style="font-size:130%;"><span style="font-weight: bold;">d. Metode Matematika Jenis Logical Organization</span></span><br /><span style="font-weight: bold;">1) Apakah siswa mempertanyakan KEBENARAN suatu konsep matematika?</span><br /><span style="font-weight: bold;">Jika “Ya” sebutkan KEBENARAN dari konsep-konsep yang mana?</span><br /> Jawab: Ya<br />a. Menanyakan benarkah selimut tabung = bentuk persegi panjang<br />b. Lilitan Bola kepanjangan<br />c. Volume pasir pada Tabung kelebihan/kekurangan (kemudian di jelaskan oleh guru beberapa factor penyebabnya a.l. siswa kurang teliti; ukuran tidak tepat, dan pasir tercampur kerikil, dsb)<br />d. Menanyakan benarkah Volume Tabung = 3 x Volume Kerucut<br />e. Menanyakan benarkah Luas muka bola = 4 kali luas lingkaran<br /><span style="font-weight: bold;">2) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan siswa untuk mencek KEBENARAN konsep matematika?</span><br /> Jawab: Ya<br />Dengan cara mengajukan pertanyaan kepada Guru/ Siswa yang lain<br /><span style="font-weight: bold;">3) Keadaan yang bagaimana siswa melakukan ceking terhadap KEBENARAN suatu konsep?</span><br /> Jawab:<br /> Setelah praktek<br /> Pada diskusi kelompok<br /><span style="font-weight: bold;">4) Apakah siswa menggunakan ISTILAH/LAMBANG/NOTASI/MODEL MATEMATIKA/ALAT PERAGA secara BENAR atau SALAH?</span><br /> Jawab:<br />a. Terjadi kesalahan prosedur di dalam praktek, karena tidak teliti membaca petunjuk yg dibuat guru, misal:<br /> - mengisi tabung lebih dulu<br /> - melilitkan dengan tali yang menumpuk<br />b. Salah dalam menulis rumus<br />c. Salah dalam menemukan rumus.<br /><span style="font-weight: bold;">5) Sebutkan ISTILAH/LAMBANG/NOTASI secara BENAR atau SALAH?</span><br />- Salah dalam menulis rumus Karen Kurang teliti<br />- Salah konsep, mestinya dijumlahkan luasnya, tetapi dikalikan<br />- Salah menulis rumus L permuk tabung = 2 phi r (r + t) benar = 2 phi r^2 t (salah)<br /><span style="font-weight: bold;">6) Keadaan yang bagaimana siswa menggunakan ISTILAH/LAMBANG/NOTASI/MODEL MATEMATIKA/ALAT PERAGA secara BENAR atau SALAH?</span><br />Jawab:<br />Dalam diskusi kelompok dan presentasiDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com0tag:blogger.com,1999:blog-7038868080704055876.post-54410996106692990152008-12-21T22:50:00.001+07:002008-12-21T22:50:57.213+07:00Matematika Ditinjau Dari Berbagai Sudut Pandang<span style="font-weight: bold;">Resensi buku “The Philosophy of Mathematics Education”, karya Paul Ernest</span><br /> <span style="font-weight: bold;">Oleh: Marsigit </span><br /> Para absolutis teguh pendiriannya dalam memandang secara objektif kenetralan matematika, walaupun matematika yang dipromosikan itu sendiri secara implisit mengandung nilai-nilai. Abstrak adalah suatu nilai terhadap konkrit, formal suatu nilai terhadap informal, objektif terhadap subjektif, pembenaran terhadap penemuan, rasionalitas terhadap intuisi, penalaran terhadap emosi, hal-hal umum terhadap hal-hal khusus, teori terhadap praktik, kerja dengan fikiran terhadap kerja dengan tangan, dan seterusnya. Setelah mendaftar macam-macam nilai di atas maka pertanyaannya adalah, bagaimana matematisi berpendapat bahwa matematika adalah netral dan bebas nilai ? Jawaban dari kaum absolutis adalah bahwa niai yang mereka maksud adalah nilai yang melekat pada diri mereka yang berupa kultur, jadi bukan nilai yang melekat secara implisist pada matematika. Diakui bahwa isi dan metode matematika, karena hakekatnya, membuat matematika menjadi abstrak, umum, formal, obyektif, rasional, dan teoritis. Ini adalah hakekat ilmu pengetahuan dan matematika. Tidak ada yang salah bagi yang kongkrit, informal, subyektif, khusus, atau penemuan; mereka hanya tidak termasuk dalam sains, dan tentunya tidak termasuk di dalam matematika (Popper, 1979 dalam Ernest, 1991: 132).<br /> Yang ingin ditandaskan di sini adalah bahwa pandangan kaum absolutis, secara sadar maupun tak sadar, telah merasuk ke dalam matematika melalui definisi-definisi. Dengan perkataan lain, kaum absolutis berpendapat bahwa segala sesuatu yang sesuai dengan nilai-nilai di atas dapat diterima dan yang tidak sesuai tidak dapat diterima. Pernyataan-pernyataan matematika dan bukti-buktinya, yang merupakan hasil dari matematika formal, dipandang dapat melegitimasikan matematika. Sementara, penemuan-penemuan matematika, hasil kerja para matematisi dan proses yang bersifat informal dipandang tidak demikian. Dengan pendekatan ini kaum absolutis membangun matematika yang dianggapnya sebagai netral dan bebas nilai. Dengan pendekatan ini mereka menetapkan kriteria apa yang dapat diterima dan tidak diterima. Hal-hal yang terikat dengan implikasi sosial dan nilai-nilai yang menyertainya, secara eksplisit, dihilangkannya. Tetapi dalam kenyataannya, nilai-nilai yang terkandung dalam hal-hal tersebut di atas, membuat masalah-masalah yang tidak dapat dipecahkan. Hal ini disebabkan karena mendasarkan pada hal-hal yang bersifat formal saja hanya dapat menjangkau pada pembahasan bagian luar dari matematika itu sendiri.<br /> Jika mereka berkehendak menerima kritik yang ada, sebetulnya pandangan mereka tentang matematika yang netral dan bebas nilai juga merupakan suatu nilai yang melekat pada diri mereka dan sulit untuk dilihatnya. Dengan demikian akan muncul pertanyaan berikutnya, siapa yang tertarik dengan pendapatnya ? Inggris dan negara-negara Barat pada umumnya, diperintah oleh kaum laki-laki berkulit putih dari kelas atas. Keadaan demikian mempengaruhi struktur sosial para matematisi di kampus-kampus suatu Universitas, yang kebanyakan didominasi oleh mereka. Nilai-nilai mereka secara sadar dan tak sadar terjabarkan dalam pengembangan matematika sebagai bagian dari usaha dominasi sosial. Oleh karena itu agak janggal kiranya bahwa matematika bersifat netral dan bebas nilai, sementara matematika telah menjadi alat suatu kelompok sosial. Mereka mengunggulkan pria di atas wanita, kulit putih di atas kulit hitam, masyarakat strata menengah di atas strata bawah, untuk kriteria keberhasilan penguasaan pencapaian akademik matematikanya.<br /> Suatu kritik mengatakan, untuk suatu kelompok tertentu, misalnya kelompok kulit putih dari strata atas, mungkin dapat dianggap matematika sebagai netral dan bebas nilai. Namun kritik demikian menghadapi beberapa masalah. Pertama, terdapat premis bahwa matematika bersifat netral. Kedua, terdapat pandangan yang tersembunyi bahwa pengajaran matematika juga dianggap netral. Di muka telah ditunjukkan bahwa setiap pembelajaran adalah terikat dengan nilai-nilai. Ketiga, ada anggapan bahwa keterlibatan berbagai kelompok masyarakat beserta nilainya dalam matematika adalah konsekuensi logisnya. Dan yang terakhir, sejarah menunjukkan bahwa matematika pernah merupakan alat suatu kelompok masyarakat tertentu. Kaum ‘social constructivits’ memandang bahwa matematika merupakan karya cipta manusia melalui kurun waktu tertentu. Semua perbedaan pengetahuan yang dihasilkan merupakan kreativitas manusia yang saling terkait dengan hakekat dan sejarahnya. Akibatnya, matematika dipandang sebagai suatu ilmu pengetahuan yang terikat dengan budaya dan nilai penciptanya dalam konteks budayanya.Sejarah matematika adalah sejarah pembentukannya, tidak hanya yang berhubungan dengan pengungkapan kebenaran, tetapi meliputi permasalahan yang muncul, pengertian, pernyataan, bukti dan teori yang dicipta, yang terkomunikasikan dan mengalami reformulasi oleh individu-individu atau suatu kelompok dengan berbagai kepentingannya. Pandangan demikian memberi konsekuensi bahwa sejarah matematika perlu direvisi.<br /> Kaum absolutis berpendapat bahwa suatu penemuan belumlah merupakan matematika dan matematika modern merupakan hasil yang tak terhindarkan. Ini perlu pembetulan. Bagi kaum ‘social constructivist’ matematika modern bukanlah suatu hasil yang tak terhindarkan, melainkan merupakan evolusi hasil budaya manusia. Joseph (1987) menunjukkan betapa banyaknya tradisi dan penelitian pengembangan matematika berangkat dari pusat peradaban dan kebudayaan manusia. Sejarah matematika perlu menunjuk matematika, filsafat, keadaan sosial dan politik yang bagaimana yang telah mendorong atau menghambat perkembangan matematika. Sebagai contoh, Henry (1971) dalam Ernest (1991: 34) mengakui bahwa calculus dicipta pada masa Descartes, tetapi dia tidak suka menyebutkannya karena ketidaksetujuannya terhadap pendekatan infinitas. Restivo (1985:40), MacKenzie (1981: 53) dan Richards (1980, 1989) dalam Ernest (1991 : 203) menunjukkan betapa kuatnya hubungan antara matematika dengan keadaan sosial; sejarah sosial matematika lebih tergantung kepada kedudukan sosial dan kepentingan pelaku dari pada kepada obyektivitas dan kriteria rasionalitasnya. Kaum ‘social constructivist’ berangkat dari premis bahwa semua pengetahuan merupakan karya cipta. Kelompok ini juga memandang bahwa semua pengetahuan mempunyai landasan yang sama yaitu ‘kesepakatan’. Baik dalam hal asal-usul maupun pembenaran landasannya, pengetahuan manusia mempunyai landasan yang merupakan kesatuan, dan oleh karena itu semua bidang ilmu pengetahuan manusia saling terikat satu dengan yang lain. Akibatnya, sesuai dengan pandangan kaum ‘social constructivist’, matematika tidak dapat dikembangkan jika tanpa terkait dengan pengetahuan lain, dan yang secara bersama-sama mempunyai akarnya, yang dengan sendirinya tidak terbebaskan dari nilai-nilai dari bidang pengetahuan yang diakuinya, karena masing-masing terhubung olehnya.<br /> Karena matematika terkait dengan semua pengetahuan dari diri manusia, maka jelaslah bahwa matematika tidaklah bersifat netral dan bebas nilai. Dengan demikian matematika memerlukan landasan sosial bagi perkembangannya (Davis dan Hers, 1988: 70 dalam Ernest 1991 : 277-279). Shirley (1986: 34) menjelaskan bahwa matematika dapat digolongkan menjadi formal dan informal, terapan dan murni. Berdasarkan pembagian ini, kita dapat membagi kegiatan matematika menjadi 4 (empat) macam, di mana masing-masing mempunyai ciri yang berbeda-beda:<br />a. matematika formal-murni, termasuk matematika yang dikembangkan pada Universitas dan matematika yang diajarkan di sekolah;<br />b. matematika formal-terapan, yaitu yang dikembangkan dalam pendidikan maupun di luar, seperti seorang ahli statistik yang bekerja di industri.<br />c. matematika informal-murni, yaitu matematika yang dikembangkan di luar institusi kependidikan; mungkin melekat pada budaya matematika murni.<br />d. matematika informal-terapan, yaitu matematika yang digunakan dalam segala kehidupan sehari-hari, termasuk kerajinan, kerja kantor dan perdagangan.<br /> Dowling dalam Ernest (1991: 93), berdasar rekomendasi dari Foucault dan Bernstein, mengembangkan berbagai macam konteks kegiatan matematika. Dia membagi satu dimensi model menjadi 4 (empat) macam yaitu : Production (kreativitas), Recontextualization (pandangan guru dan dasar-dasar kependidikan), Reproduction (kegiatan di kelas) dan Operationalization (penggunaan matematika). Dimensi kedua dari pengembangannya memuat 4 (empat) macam yaoitu: Academic (pada pendidikan tinggi), School (konteks sekolah), Work (kerja) dan Popular (konsumen dan masyarakat).<br />Dengan memasukkan berbagai macam konteks matematika, berarti kita telah mengakui tesis D’Ambrosio (1985: 25) dalam ‘ethnomathematics’ nya. Tesis tersebut menyatakan bahwa matematika terkait dengan aspek budaya; secara khusus disebutkan bahwa kegiatan-kegiatan seperti hitung-menghitung, mengukur, mendesain, bermain, berbelanja, dst. Merupakan akar dari pengembangan matematika. Dowling dalam Ernest (1991: 120) mengakui bahwa pandangan demikian memang agak kabur; kecuali jika didukung oleh pembenaran tradisi matematika.<br /><span style="font-weight: bold;">DAFTAR PUSTAKA</span><br />Ernest, P., 1991, The Philosophy of Mathematics Education, London : The Falmer Press.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com2tag:blogger.com,1999:blog-7038868080704055876.post-60274189890557194352008-12-21T08:58:00.000+07:002008-12-21T08:59:12.251+07:00Apakah motivasi itu?<span style="font-weight: bold;">Oleh: Bindra, D and Stewart, J</span><br /><span style="font-weight: bold; font-style: italic;">Di resensikan oleh : Marsigit</span><br />Menurut McDougall, motivasi bersumber dari naluri yang terdalam diri manusia. Secara langsung maupun tidak langsung, tindakan manusia ditentukan oleh unsur naluriahnya. sedangkan menurut Freud, sifat dasar naluriah (instinct) manusia berasal dari energi atau potensi seseorang untuk melakukan sesuatu. Sehingga naluri manusia juga mempunyai obyek atau sasaran. Dalam rangka mencapai obyek itulah maka manusia dikatakan mempunyai tujuan. Perlu diketahui bahwa obyek yang dimaksud tidaklah harus berada diluar diri manusia, tetapi bisa berada dalam diri manusia itu sendiri. Naluri itu sendiri juga mempunyai sumber yang disebut sebagai proses dibawah sadar dari organ atau tubuh manusia. Tidaklah jelas apakah proses ini merupakan hasil dari suatu reaksi kimia atau sesuatu yang dihasilkan atau sesuatu yang menghasilkan energi. Yang jelas diketahui adalah naluri yang berbeda-beda dikendalikan oleh kemampuan jiwa/pikiran untuk suatu kegiatan yang berbeda-beda pula.<br />Lorenz, mengembangkan teori tentang keadaan naluriah berdasarkan model produksi energi. Energi akan dihasilkan berdasarkan adanya stimulus penyebab. Dari sini maka lahirlah teori stimulus-respon. Namun model energi ini telah juga mendapatkan kritik dari Hinde yang menyatakan bahwa teori demikian dapat menyesatkan kita, karena kita pun tidak akan secara jelas bahwa energi yang dimaksud itu merupakan proses atau hasil, energi kegiatan atau energi fisik belaka. Bagaimanapun tentang keadaan naluriah, Tinbergen berusaha untuk mengembangkan teori bagaimana naluri manusia dapat dikoordinasikan secara hirarkhi. Menurutnya, keadaan naluri manusia mempunyai dua aspek, pertama keadaan yang berkaitan dengan keinginan dan kedua, keadaan yang berkaitan dengan pemenuhan kebutuhan.<br />Nissen, mengembangkan teori bahwa naluri manusia merupakan daya dorong untuk melakukan suatu kegiatan. Dia menggolongkan daya dorong menjadi 5 (lima) berdasarkan kategori bagaimana suatu aktivitas akan dilaksanakan: a) jika daya dorong relatif lemah dan tidak mempunyai arah maka seseorang akan melakukan kegiatan secara acak atau random atau tanpa pilih, b) jika daya dorong berasal dari tingkatan yang lebih dalam lagi, maka kita katakan bahwa seseorang telah melakukan aktivitas berdasar nalurinya, c) jika hasil belajar merupakan daya dorong maka akan menghasilkan pola kegiatan yang kompleks, d) jika daya dorong didasarkan atas pertimbangan tertentu maka akan menghasilkan suatu kegiatan yang mempunyai tujuan, dan e) jika tidak termasuk ke empat di atas maka naluri itu cukup dikatakan sebagai “potensi”.<br /><span style="font-weight: bold;">Reference:</span><br />Semua nama yang tersebut dalam paragraph di atas merupakan penulis bagian dari buku “ Motivation” Karya Dalbir Bindra dan Jane Stewart, Penguin Modern Psychology, Penguin Book, LondonDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com2tag:blogger.com,1999:blog-7038868080704055876.post-24800880246149670262008-12-08T07:13:00.001+07:002008-12-08T07:13:52.490+07:00KANT’S CONCEPTS OF MATHEMATICSBy <span style="font-weight: bold;">Marsigit</span><br />Kant argued that mathematics is a pure product of reason, and moreover is thoroughly synthetical.1 Next, the question arises: Does not this faculty, which produces mathematics, as it neither is nor can be based upon experience, presuppose some ground of cognition a priori, 2 which lies deeply hidden, but which might reveal itself by these its effects, if their first beginnings were but diligently ferreted out? 3However, Kant found that all mathematical cognition has this peculiarity: it must first exhibit its concept in a visual intuition and indeed a priori, therefore in an intuition which is not empirical, but pure. Without this mathematics cannot take a single step; hence its judgments are always visual, viz., intuitive; whereas philosophy must be satisfied with discursive judgments from mere concepts, and though it may illustrate its doctrines through a visual figure, can never derive them from it. 4<br />On the other hand, Kant claimed that empirical intuition enables us without difficulty to enlarge the concept which we frame of an object of intuition, by new predicates, which intuition itself presents synthetically in experience; while pure intuition does so likewise, only with this difference, that in the latter case the synthetical judgment is a priori certain and apodeictical, in the former, only a posteriori and empirically certain; because this latter contains only that which occurs in contingent empirical intuition, but the former, that which must necessarily be discovered in pure intuition. 5 The next step, Kant questioned: "How is then it possible to intuit [in a visual form] anything a priori?" ; however, according to Kant, as an intuition is such a representation as immediately depends upon the presence of the object, it seems impossible to intuit from the outset a priori, because intuition would in that event take place without either a former or a present object to refer to, and by consequence could not be intuition.6<br />Kant then argued that the intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodictic and necessary are Space and Time. 7 Accordingly, because mathematics must first have all its concepts in intuition, and pure mathematics in pure intuition, hence, mathematics must construct them. 8 Geometry is based upon the pure intuition of space; and, arithmetic accomplishes its concept of number by the successive addition of units in time; and pure mechanics especially cannot attain its concepts of motion without employing the representation of time. Kant stressed that both representations, however, are only intuitions; for if we omit from the empirical intuitions of bodies and their alterations (motion) everything empirical, or belonging to sensation, space and time still remain, which are therefore pure intuitions that lie a priori at the basis of the empirical. 9 Therefore, Kant concluded that pure mathematics, as synthetical cognition a priori, is only possible by referring to no other objects than those of the senses, in which, at the basis of their empirical intuition lies a pure intuition (of space and of time) which is a priori. Kant claimed that this is possible, because the latter intuition is nothing but the mere form of sensibility, which precedes the actual appearance of the objects, in that it, in fact, makes them possible; and yet this faculty of intuiting a priori affects not the matter of the phenomenon 10<br />Kant illustrated that in ordinary and necessary procedure of geometers, all proofs of the complete congruence of two given figures come ultimately to this that they may be made to coincide; which is evidently nothing else than a synthetical proposition resting upon immediate intuition, and this intuition must be pure, or given a priori, otherwise the proposition could not rank as apodictically certain, but would have empirical certainty only. 11 Kant further claimed that everywhere space has three dimensions, and that space cannot in any way have more, is based on the proposition that not more than three lines can intersect at right angles in one point. Kant argued that drawing the line to infinity and representing the series of changes e.g. spaces travers by motion can only attach to intuition, then he concluded that the basis of mathematics actually are pure intuitions; while the transcendental deduction of the notions of space and of time explains, at the same time, the possibility of pure mathematics. 12<br />In the Remark I, Kant elaborated that pure mathematics, and especially pure geometry, can only have objective reality on condition that they refer to objects of sense. But in regard to the latter the principle holds good, that our sense representation is not a representation of things in themselves but of the way in which they appear to us. Hence it follows, that the propositions of geometry are not the results of a mere creation of our poetic imagination, and that therefore they cannot be referred with assurance to actual objects; but rather that they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing else than the form of all external appearances, and it is this form alone in which objects of sense can be given. (Immanuel Kant Prolegomena To Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Remark 1, 287) Sensibility, the form of which is the basis of geometry, is that upon which the possibility of external appearance depends. Therefore these appearances can never contain anything but what geometry prescribes to them. The space of the geometer would be considered a mere fiction, and it would not be credited with objective validity, because we cannot see how things must of necessity agree with an image of them, which we make spontaneously and previous to our acquaintance with them. 13<br />But if this image, or rather this formal intuition, is the essential property of our sensibility, by means of which alone objects are given to us, and if this sensibility represents not things in themselves, but their appearances: we shall easily comprehend, and at the same time indisputably prove, that all external objects of our world of sense must necessarily coincide in the most rigorous way with the propositions of geometry; because sensibility by means of its form of external intuition, viz., by space, the same with which the geometer is occupied, makes those objects at all possible as mere appearances. 14<br />Because the space of the geometer is exactly the form of sensuous intuition which we find a priori in us, and contains the ground of the possibility of all external appearances (according to their form), and the latter must necessarily and most rigidly agree with the propositions of the geometer, which he draws not from any fictitious concept, but from the subjective basis of all external phenomena, which is sensibility itself. 15<br />Because it would be absurd to base an analytical judgment on experience, as our concept suffices for the purpose without requiring any testimony from experience, Kant concluded that Empirical judgments are always synthetical, e.g. “That body is extended” is a judgment established a priori, and not an empirical judgment. And also, for before appealing to experience, we already have all the conditions of the judgment in the concept, from which we have but to elicit the predicate according to the law of contradiction, and thereby to become conscious of the necessity of the judgment, Kant concluded that which experience could not even teach us.16 According to Kant, Mathematical judgments are all synthetical and he argued that this fact seems hitherto to have altogether escaped the observation of those who have analyzed human reason; it even seems directly opposed to all their conjectures, though incontestably certain, and most important in its consequences. Further he claimed that for as it was found that the conclusions of mathematicians all proceed according to the law of contradiction (as is demanded by all apodictic certainty), men persuaded themselves that the fundamental principles were known from the same law. “This was a great mistake”, he said. He then delivered the reason that for a synthetical proposition can indeed be comprehended according to the law of contradiction, but only by presupposing another synthetical proposition from which it follows, but never in itself.17 To support this argument, Kant started to examined the case of addition 7 + 5 = 12. According to him, it might at first be thought that the proposition 7 + 5 = 12 is a mere analytical judgment, following from the concept of the sum of seven and five, according to the law of contradiction. However, accordingly, if we closely examine the operation, it appears that the concept of the sum of 7+5 contains merely their union in a single number, without its being at all thought what the particular number is that unites them. Therefore, he concluded that the concept of twelve is by no means thought by merely thinking of the combination of seven and five; and analyze this possible sum as we may, we shall not discover twelve in the concept. Kant suggested that first of all, we must observe that all proper mathematical judgments are a priori, and not empirical. According to him, mathematical judgments carry with them necessity, which cannot be obtained from experience, therefore, it implies that it contains pure a priori and not empirical cognitions.18<br />We, then, must go beyond these concepts, by calling to our aid some concrete image [Anschauung], i.e., either our five fingers, or five points (as Segner has it in his Arithmetic), and we must add successively the units of the five, given in some concrete image [Anschauung], to the concept of seven; hence our concept is really amplified by the proposition 7 + 5 = I 2, and we add to the first a second, not thought in it. Ultimately, Kant concluded that arithmetical judgments are therefore synthetical, and the more plainly according as we take larger numbers; for in such cases it is clear that, however closely we analyze our concepts without calling visual images (Anscliauung) to our aid, we can never find the sum by such mere dissection. 19<br />Similarly, Kant argued that all principles of geometry are no less analytical. He illustrated that the proposition “a straight line is the shortest path between two points”, is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality. He claimed that the attribute of shortness is therefore altogether additional, and cannot be obtained by any analysis of the concept; and its visualization [Anschauung] must come to aid us; and therefore, it alone makes the synthesis possible.20 Kant confronted the previous geometers assumption which claimed that other mathematical principles are indeed actually analytical and depend on the law of contradiction. However, he strived to show that in the case of identical propositions, as a method of concatenation, and not as principles, e. g., a=a, the whole is equal to itself, or a + b > a, the whole is greater than its part. He then claimed that although they are recognized as valid from mere concepts, they are only admitted in mathematics, because they can be represented in some visual form [Anschauung].21<br /><br /><span style="font-weight: bold;">Notes</span><br /><span style="font-size:78%;">1. A synthetic proposition is a proposition that is capable of being true or untrue based on facts about the world - in contrast to an analytic proposition which is true by definition. (From Wikipedia, the free encyclopedia)<br />2. A priori knowledge is propositional knowledge that can be had without experience. It is usually contrasted with a posteriori knowledge, which requires experience. Mathematics and logic are usually considered a priori disciplines. The natural and social sciences are usually considered a posteriori disciplines (From Wikipedia, the free encyclopedia)<br />3. Immanuel Kant, Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 6.<br />4. This observation on the nature of mathematics gives us a clue to the first and highest condition of its possibility, which is, that some pure intuition [reine Anschauung] must form its basis, in which all its concepts can be exhibited or constructed, in concreto and yet a priori. If we can locate this pure intuition and its possibility, we may thence easily explain how synthetical propositions a priori are possible in pure mathematics, and consequently how this science itself is possible. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sec.7. para. 281<br />5. Pure intuition [viz., the visualization of forms in our imagination, from which every thing sensual, i.e., every thought of material qualities, is excluded]. Here intuition, being an intuition a priori, is before all experience, viz., before any perception of particular objects, inseparably conjoined with its concept. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 7. para. 281<br />6. Hence it follows: that propositions, which concern this form of sensuous intuition only, are possible and valid for objects of the senses; as also, conversely, that intuitions which are possible a priori can never concern any other things than objects of our senses. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) sec.9 para. 282<br />7. Space is the form of the external intuition of this sensibility, and the internal determination of every space is only possible by the determination of its external relation to the whole space, of which it is a part (in other words, by its relation to the external sense).<br />8. If it proceeded in any other way, it would be impossible to make any headway, for mathematics proceeds, not analytically by dissection of concepts, but synthetically, and if pure intuition be wanting, there is nothing in which the matter for synthetical judgments a priori can be given. Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect.10 para. 283<br />9. Hence they can never be omitted, but at the same time, by their being pure intuitions a priori, they prove that they are mere forms of our sensibility, which must precede all empirical intuition, or perception of actual objects, and conformably to which objects can be known a priori, but only as they appear to us. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sec.10 para. 283.<br />10. Phenomenon is the sense-element in it, for this constitutes that which is empirical), but its form, viz., space and time. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect.11 para 284<br />11. In that case, it could only be said that it is always found to be so, and holds good only as far as our perception reaches. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 12 para 285<br />12. Without some such deduction its truth may be granted, but its existence could by no means be understood, and we must assume "that everything which can be given to our senses (to the external senses in space, to the internal one in time) is intuited by us as it appears to us, not as it is in itself." (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect.12 para. 285<br />13. It would be quite otherwise if the senses were so constituted as to represent objects as they are in themselves. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark I, para 287<br />14. Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark 1 , para. 287<br />15. In this and no other way can geometry be made secure as to the undoubted objective reality of its propositions against all the intrigues of a shallow Metaphysics, which is surprised at them [the geometrical propositions], because it has not traced them to the sources of their concepts. (Immanuel Kant Prolegomena To Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark I, para. 288<br />16. Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )<br />17. Kant suggested that first of all, we must observe that all proper mathematical judgments are a priori, and not empirical. According to him, mathematical judgments carry with them necessity, which cannot be obtained from experience, therefore, it implies that it contains pure a priori and not empirical cognitions. (Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )<br />18. Existence and Reality(Kant e-text reading I: Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2: Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )<br />19. ibid.<br />20. ibid.<br />21. What usually makes us believe that the predicate of such apodictic judgments is already contained in our concept, and that the judgment is therefore analytical, is the duplicity of the expression, requesting us to think a certain predicate as of necessity implied in the thought of a given concept, which necessity attaches to the concept. But the question is not what we are requested to join in thought to the given concept, but what we actually think together with and in it, though obscurely; and so it appears that the predicate belongs to these concepts necessarily indeed, yet not directly but indirectly by an added visualization [Anschauung]. (Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )<br /><br /></span>Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com3tag:blogger.com,1999:blog-7038868080704055876.post-8770560648881054472008-12-08T06:36:00.000+07:002008-12-08T06:43:11.707+07:00What is Mathematical Thinking and Whay is it Important?WHAT IS MATHEMATICAL THINKING<br />AND WHY IS IT IMPORTANT?<br />By:<span style="font-weight: bold;"> Kaye Stacey</span><br />University of Melbourne, Australia<br />INTRODUCTION<br />This paper and the accompanying presentation has a simple message, that<br />mathematical thinking is important in three ways.<br />• Mathematical thinking is an important goal of schooling.<br />• Mathematical thinking is important as a way of learning mathematics.<br />• Mathematical thinking is important for teaching mathematics.<br />Mathematical thinking is a highly complex activity, and a great deal has been written and studied about it. Within this paper, I will give several examples of mathematicalthinking, and to demonstrate two pairs of processes through which mathematicalthinking very often proceeds:<br />• Specialising and Generalising<br />• Conjecturing and Convincing.<br />Being able to use mathematical thinking in solving problems is one of the most thefundamental goals of teaching mathematics, but it is also one of its most elusive goals.<br />It is an ultimate goal of teaching that students will be able to conduct mathematical investigations by themselves, and that they will be able to identify where the mathematics they have learned is applicable in real world situations. In the phrase of the mathematician Paul Halmos (1980), problem solving is “the heart of mathematics”. However, whilst teachers around the world have considerable successes with achieving this goal, especially with more able students, there is always a great need for improvement, so that more students get a deeper appreciation of what it means to think mathematically and to use mathematics to help in their daily and working lives.<br /><span style="font-weight: bold;">MATHEMATICAL THINKING IS AN IMPORTANT GOAL OF SCHOOLING</span><br />The ability to think mathematically and to use mathematical thinking to solve problems is an important goal of schooling. In this respect, mathematical thinking will support science, technology, economic life and development in an economy. Increasingly, governments are recognising that economic well-being in a country is underpinned by strong levels of what has come to be called ‘mathematical literacy’(PISA, 2006) in the population. Mathematical literacy is a term popularised especially by the OECD’s PISA program of international assessments of 15 year old students. Mathematical literacy is the ability to use mathematics for everyday living, and for work, and for further study, and so the PISA assessments present students with problems set in realistic contexts. The framework used by PISA shows that mathematical literacy involves many components of mathematical thinking, including reasoning, modelling and making connections between ideas. It is clear then, that mathematical thinking is important in large measure because it equips students with the ability to use mathematics, and as such is an important outcome of schooling.<br />At the same time as emphasising mathematics because it is useful, schooling needs to give students a taste of the intellectual adventure that mathematics can be. Whilst the highest levels of mathematical endeavour will always be reserved for just a tiny minority, it would be wonderful if many students could have just a small taste of the spirit of discovery of mathematics as described in the quote below from Andrew Wiles, the mathematician who proved Fermat’s Last Theorem in 1994. This problem had been unsolved for 357 years.One enters the first room of the mansion and it’s dark. One stumbles around bumping into furniture, but gradually you learn where each piece of furniture is. Finally, after six months of so, you find the light switch, you turn it on, and suddenly it’s all illuminated.<br />You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them. (Andrew Wiles, quoted by Singh, 1997, p236, 237) At the APEC meeting in Tokyo in January 2006, Jan de Lange spoke in detail about the use of mathematics to equip young people for life, so I will instead focus this paper on two other ways in which mathematical thinking is important.<br /><span style="font-weight: bold;">WHAT IS MATHEMATICAL THINKING?</span><br />Since mathematical thinking is a process, it is probably best discussed through examples, but before looking at examples, I briefly examine some frameworks provided to illuminate mathematical thinking, going beyond the ideas of mathematical literacy. There are many different ‘windows’ through which the mathematical thinking can be viewed. The organising committee for this conference (APEC, 2006) has provided a substantial discussion on this point. Stacey (2005) gives a review of how mathematical thinking is treated in curriculum documents in Australia, Britain and USA. One well researched framework was provided by Schoenfeld (1985), who organised his work on mathematical problem solving under four headings: the resources of mathematical knowledge and skills that the student brings to the task, the heuristic strategies that that the student can use in solving problems, the monitoring and control that the student exerts on the problem solving process to guide it in productive directions, and the beliefs that the student holds about mathematics, which enable or disable problem solving attempts. McLeod (1992) has supplemented this view by expounding on the important of affect in mathematical problem solving.<br />In my own work, I have found it helpful for teachers to consider that solving problems with mathematics requires a wide range of skills and abilities, including:<br />• Deep mathematical knowledge<br />• General reasoning abilities<br />• Knowledge of heuristic strategies<br />• Helpful beliefs and attitudes (e.g. an expectation that maths will be useful)<br />• Personal attributes such as confidence, persistence and organisation<br />• Skills for communicating a solution.<br />Of these, the first three are most closely part of mathematical thinking.<br />In my book with John Mason and Leone Burton (Mason, Burton and Stacey, 1982), we provided a guide to the stages through which solving a mathematical problem is likely to pass (Entry, Attack, Review) and advice on improving problem solving performance by giving experience of heuristic strategies and on monitoring and controlling the problem solving process in a meta-cognitive way. We also identified four fundamental processes, in two pairs, and showed how thinking mathematically very often proceeds by alternating between them:<br />• specialising – trying special cases, looking at examples<br />• generalising - looking for patterns and relationships<br />• conjecturing – predicting relationships and results<br />• convincing – finding and communicating reasons why something is true.<br />I will illustrate these ideas in the two examples below. The first example examines the mathematical thinking of the problem solver, whilst the second examines the mathematical thinking of the teacher. The two problems are rather different – the second is within the mainstream curriculum, and the mathematical thinking is guided by the teacher in the classroom episode shown. The first problem is an open problem, selected because it is similar to open investigations that a teacher might choose to use, but I hope that its unusual presentation will let the audience feel some of the mystery<br />and magic of investigation afresh.<br /><span style="font-weight: bold;">MATHEMATICAL THINKING IS IMPORTANT AS A WAY OF LEARNING MATHEMATICS</span><br />In this section, I will illustrate these four processes of mathematical thinking in the context of a problem that may be used to stimulate mathematical thinking about numbers or as an introduction to algebra. If students’ ability to think mathematically is an important outcome of schooling, then it is clear that mathematical thinking must feature prominently in lessons.Number puzzles and tricks are excellent for these purposes, and in the presentation I will use a number puzzle in a format of the Flash Mind Reader, created by Andy Naughton and published on the internet (HREF1). The Flash Mind Reader does not look like a number puzzle. Indeed its creator writes: We have been asked many times how the Mind Reader works, but will not publish that information on this website. All magicians […] do not give away how their effects work.The reason for this is that it spoils the fun for those who like to remain mystified and when you do find out how something works it's always a bit of a let-down. If you are really keen to find out how it works we suggest that you apply your brain and try to work it out on paper or search further afield. (HREF1). As with many other number tricks, an audience member secretly chooses a number (and a symbol), a mathematical process is carried out, and the computer reveals the audience member’s choice. In this case, a number is chosen, the sum of the digits is subtracted from the number and a symbol corresponding to this number is found from a table. The computer then magically shows the right symbol. The Flash Mind Reader is too difficult to use in most elementary school classes, the target of this conference, but I have selected it so that my audience of mathematics education experts can experience afresh some of the magic and mystery of numbers. As the group works towards a solution, we have many opportunities to observe mathematical thinking in action. Through this process of shared problem solving as we investigate the Flash Mind Reader, I hope to make the following points about mathematical thinking. Firstly, when people first see the Flash Mind Reader, mathematical explanations are far from their minds. Some people propose that it really does read minds, and they may try to test their theory by not concentrating hard on the number that they choose. Others hypothesise that the program exerts some psychological power over the person’s choice of number. Others suggest it is only an optical illusion, resulting from staring at the screen. This illustrates that a key component of mathematical thinking is having a disposition to looking at the world in a mathematical way, and an attitude of seeking a logical explanation. As we seek to explain how the Flash Mind reader works, the fundamental processes of thinking mathematically will be evident. The most basic way of trying to understand a problem situation is to try the Flash Mind Reader several times, with different numbers and different types of numbers. This helps us understand the problem (in this case, what is to be explained) and to gather some information. This is a simple example of specialising, the first of the four processes of thinking mathematically processes. As we enter more deeply into the problem, specialising changes its character. First we may look at one number, noting that if 87 is the number, then the sum of its digits is 15 and 87 – 15 is 72.<br />Beginning to work systematically leads to evidence of a pattern:<br />87 8 + 7 = 15 87 – 15 = 72<br />86 8 + 6 = 14 86 – 14 = 72<br />85 8 + 5 = 13 85 – 13 = 72<br />84 8 + 4 = 12 84 – 12 = 72<br />and a cycle of experimentation (which numbers lead to 72?, what do other numbers lead to?) and generalising follows. Of course, at this stage it is important to note the value of working with the unclosed expressions such as 8+7 instead of the closed 15, because this reveals the general patterns and reasons so much better. Working with the unclosed expression to reveal<br />structure is an admirable feature of Japanese elementary education.<br />87 87 – 7 = 80 80 – 8 = 72<br />86 86 – 6 = 80 80 – 8 = 72<br />85 85 – 5 = 80 80 – 8 = 72<br />84 84 – 4 = 80 80 – 8 = 72<br />It is also worthwhile noting at this point, that although we are working with a specific example, the aim here is to see the general in the specific. This generalising may lead to a conjecture that the trick works because all starting numbers produce a multiple of 9 and all multiples of 9 have the same symbol. But this conjecture is not quite true and further examination of examples (more specialising) finally identifies the exceptions and leads to a convincing argument. In school, we aim for students to be able to use algebra to write a proof, but even before they have this skill, they can be produce convincing arguments. An orientation to justify and prove (at an appropriate level of formality) is important throughout school. If students are to become good mathematical thinkers, then mathematical thinking<br />needs to be a prominent part of their education. In addition, however, students who have an understanding of the components of mathematical thinking will be able to use these abilities independently to make sense of mathematics that they are learning. For example, if they do not understand what a question is asking, they should decide themselves to try an example (specialise) to see what happens, and if they are oriented to constructing convincing arguments, then they can learn from reasons rather than rules. Experiences like the exploration above, at an appropriate level build these dispositions.<br /><span style="font-weight: bold;">MATHEMATICAL THINKING IS ESSENTIAL FOR TEACHING MATHEMATICS.</span><br />Mathematical thinking is not only important for solving mathematical problems and for learning mathematics. In this section, I will draw on an Australian classroom episode to discuss how mathematical thinking is essential for teaching mathematics. This episode is taken from data collected by Dr Helen Chick, of the University of Melbourne, for a research project on teachers’ pedagogical content knowledge. For other examples, see Chick, 2003; Chick & Baker, 2005, Chick, Baker, Pham & Cheng, 2006a; Chick, Pham & Baker, 2006b). Providing opportunities for students to learn about mathematical thinking requires considerable mathematical thinking on the part of teachers. The first announcement for this conference states that a teacher requires mathematical thinking for analysing subject matter (p. 4), planning lessons for a specified aim (p. 4) and anticipating students’ responses (p. 5).These are indeed key places where mathematical thinking is required. However, in this section, concentrate on the mathematical thinking that is needed on a minute by minute basis in the process of conducting a good mathematics lesson. Mathematical thinking is not just in planning lessons and curricula; it makes a difference to every minute of the lesson.The teacher in this classroom extract is in her fifth year of teaching. She stands out in Chick’s data as one of the teachers in the sample exhibiting the deepest pedagogical content knowledge (Shulman, 1986, 1987). Her pupils are aged about 11 years, and are in Grade 6. This lesson began by reviewing ideas of both area and perimeter. We will examine just the first 15 minutes.The teacher selected an open and reversed task to encourage investigation and mathematical thinking. Students had 1cm grid paper and were all asked to draw a rectangle with an area of 20 square cm. This task is open in the sense that there are multiple correct answers, and it is ‘reversed’ when it is contrasted to the more common task of being given a rectangle and finding its area. The teacher reminded students that area could be measured by the number of grid squares inside a shape. In terms of the processes of mathematical thinking, the teacher at this stage is ensuring that each student is specialising. They are each working on a special case, and coming to know it well, and this will provide an anchor for future discussions and generalisations. I make no claim that the teacher herself analyses this move in this way. As the teacher circulated around the room assisting and monitoring students, she came to a student who asked if he could draw a square instead of a rectangle. In the dialogue which follows, the teachers’ response highlighted the definition of a rectangle, and she encouraged the student to work from the definition to see that a square is indeed a rectangle.<br />S: Can I do a square?<br />T: Is a square a rectangle?<br />T: What’s a rectangle?<br />T: How do you get something to be a rectangle? What’s the definition of a rectangle?<br />S: Two parallel lines<br />T: Two sets of parallel lines … and …<br />S: Four right angles.<br />T: So is that [square] a rectangle?<br />S: Yes.<br />T: [Pause as teacher realises that student understands that the square is a rectangle, but<br />there is a measurement error] But has that got an area of 20?<br />S: [Thinks] Er, no.<br />T: [Nods and winks]<br />Other responses to this student would have closed down the opportunity to teach him about how definitions are used in mathematics. To the question “Can I do a square?”, she may have simply replied “No, I asked you to draw a rectangle” or she might have immediately focussed on the error that led the student to ask the question. Instead she saw the opportunity to develop his use of definitions. When the teacher realised that the student had asked about the square because he had made a measurement error, she judged that this was within the student’s own capability to correct, and so she simply indicated that he should check his work. In the next segment, a student showed his 4 x 5 rectangle on the overhead projector, and the teacher traced around it, confirmed its area is 20 square cm and showed that multiplying the length by the width can be used instead of counting the squares, which many students did. In this segment, the teacher demonstrated that reasoning is a key component of doing mathematics. She emphasised the mathematical connections between finding the number of squares covered by the rectangle by repeated addition (4 on the first row, 4 on the next, …) and by multiplication. In her classroom, the formula was not just a rule to be remembered, but it was to be understood. The development of the formula was a clear example of ‘seeing the general in a special case’. The formula was developed from the 4 x 5 rectangle in such a way that the generality of the argument was highlighted. The teacher paid further attention to generalisation and over-generalisation at this point, when a student commented: ‘That’s how you work out area – you do the length times the width’. The teacher seized on this opportunity to address students’ tendency to over-generalise, and teased out, through a short class discussion, that LxW only works for rectangles.<br />S1: That’s how you work out area -- you do the length times the width.<br />T: When S said that’s how you find the area of a shape, is he completely correct?<br />S2: That’s what you do with a 2D shape.<br />T: Yes, for this kind of shape. What kind of shape would it not actually work for?<br />S3: Triangles.<br />S4: A circle.<br />T: [With further questioning, teases out that LxW only applies to rectangles]<br />In the next few minutes, the teacher highlighted the link between multiplication and area by asking students to make other rectangles with area 20 square centimetres. Previously all students had made 4 x 5 or 2 x 10, but after a few minutes, the class had found 20 x 1, 1 x 20, 10 x 2, 2 x 10, 4 x 5 and 5 x 4 and had identified all these side lengths as the factors of 20. Making links between different parts of the mathematics curriculum characterises her teaching. Then, in another act of generalisation, the teacher begins to move beyond whole numbers:<br />T: Are there any other numbers that are going to give an area of 20? [Pauses, as if<br />uncertain. There is no response from the students at first]<br />T: No? How do we know that there’s not?<br />S: You could put 40 by 0.5.<br />T: Ah! You’ve gone into decimals. If we go into decimals we’re going to have heaps,aren’t we?<br />After these first 15 minutes of the lesson, the students found rectangles with an area of 16 square centimetres and the teacher stressed the important problem solving strategy of working systematically. Later, in order to contrast the two concepts of area and perimeter, students found many shapes of area 12 square cm (not just rectangles) and determined their perimeters.<br />Even the first 15 minutes of this lesson show that considerable mathematical thinking on behalf of the teacher is necessary to provide a lesson that is rich in mathematical thinking for students. We see how she draws on her mathematical concepts, deeply understood, and on her knowledge of connections among concepts and the links between concepts and procedures. She also draws on important general mathematical principles such as<br />• working systematically<br />• specialising – generalising: learning from examples by looking for the<br />general in the particular<br />• convincing: the need for justification, explanation and connections<br />• the role of definitions in mathematics.<br />Chick’s work analyses teaching in terms of the knowledge possessed by the teachers. She tracks how teachers reveal various categories of pedagogical content knowledge (Shulman, 1986) in the course of teaching a lesson. In the analysis above, I viewed the lesson from the point of view of the process of thinking mathematically within the lesson rather than tracking the knowledge used. To draw an analogy, in researching a students’ solution to a mathematical problem, a researcher can note the mathematical content used, or the researcher can observe the process of solving the problem. Similarly, teaching can be analysed from the “knowledge’ point of view, or analysed from the process point of view. For those us who enjoy mathematical thinking, I believe it is productive to see teaching mathematics as another instance of solving problems with mathematics. This places the emphasis not on the static knowledge used in the lesson asabove but on a process account of teaching. In order to use mathematics to solve a problem in any area of application, whether it is about money or physics or sport or engineering, mathematics must be used in combination with understanding from the area of<br />application. In the case of teaching mathematics, the solver has to bring together expertise in both mathematics and in general pedagogy, and combine these two domains of knowledge together to solve the problem, whether it be to analyse subject matter, to create a plan for a good lesson, or on a minute-by-minute basis to respond to students in a mathematically productive way. If teachers are to encourage mathematical thinking in students, then they need to engage in mathematical thinking throughout the lesson themselves.<br /><span style="font-weight: bold;">References</span><br />APEC –Tsukuba (Organising Committee) (2006) First announcement. InternationalConference on Innovative Teaching of Mathematics through Lesson Study. CRICED, University of Tsukuba.<br />Chick, H. L. (2003). ‘Pre-service teachers’ explanations of two mathematical concepts’Proceedings of the 2003 conference, Australian Association for Research in Education.From: http://www.aare.edu.au/03pap/chi03413.pdf<br />Chick, H.L. and Baker, M. (2005) ‘Teaching elementary probability: Not leaving it to chance’, in P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (eds.) Building Connections: Theory, Research and Practice. (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia), MERGA, Sydney, pp. 233-240.<br />Chick, H.L., Baker, M., Pham, T., and Cheng, H. (2006a) ‘Aspects of teachers’ pedagogical content knowledge for decimals’, in J. Novotná, H. Moraová, M. Krátká, & N.Stehlíková (eds.), Proc. 30th conference e International Group for the Psychology of Mathematics Education, PME, Prague, Vol. 2, pp. 297-304.<br />Chick, H.L., Pham, T., and Baker, M. (2006b) ‘Probing teachers’ pedagogical content knowledge: Lessons from the case of the subtraction algorithm’, in P. Grootenboer, R.Zevenbergen, & M. Chinnappan (eds.), Identities, Cultures and Learning Spaces (Proc.29th annual conference of Mathematics Education Research Group of Australasia),MERGA, Sydney, pp. 139-146.<br />Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519– 524.HREF1 CyberGlass Design - The Flash Mind Reader. http://www.cyberglass.biz Accessed 28 November 2006.<br />Mason, J. Burton, L. and Stacey, K. (1982) Thinking Mathematically. London: Pearson.(Also available in translation in French, German, Spanish, Chinese, Thai (2007))<br />McLeod, D.B. (1992) Research on affect in mathematics education: a reconceptualisation.In D.A. Grouws, Ed., Handbook of research on mathematics teaching and learning, (pp.575–596).New York: MacMillan, New York.<br />PISA (Programme for International Student Assessment) (2006) Assessing Scientific,Reading and Mathematical Literacy. A Framework for PISA 2006. Paris: OECD.<br />Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando: Academic Press.<br />Shulman, L.S. (1986) Those who understand: Knowledge growth in teaching, EducationalResearcher 15 (2), 4-14.<br />Shulman, L.S. (1987) Knowledge and teaching: Foundations of the new reform, HarvardEducational Review 57(1), 1-22.<br />Singh, S. (1997) Fermat’s Enigma, New York: Walter<br />Stacey, K. & Groves, S. (1985) Strategies for Problem Solving. Lesson Plans forDeveloping Mathematical Thinking. Melbourne: Objective Learning Materials.<br />Stacey, K. & Groves, S. (2001) Resolver Problemas: Estrategias. Madrid: Lisbon.<br />Stacey, Kaye (2005) The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behavior 24, pp 341 – 350.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com3tag:blogger.com,1999:blog-7038868080704055876.post-52738693824459607262008-12-08T06:09:00.000+07:002008-12-08T06:10:59.228+07:00Mathematical Thinking and How to Teach It?Mathematical Thinking and How to Teach It<br /><span style="font-weight: bold;">By Shigeo Katagiri</span><br />Translated of the rewritten version from Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. Meijitosyo Publishers, Tokyo. Copyright of English version has CRICED, University of Tsukuba. All rights reserved.<br />Table of Contents<br />Chapter 1 The Aim of Education and Mathematical Thinking<br />Chapter 2 The Importance of Teaching to Cultivate Mathematical Thinking<br /> 2.1 The Importance of Teaching Mathematical Thinking<br /> 2.2 Example 1: How Many Squares are There?<br />Chapter 3 The Meaning of Mathematical Thinking and How to Teach It<br /> 3.1 Characteristics of Mathematical Thinking<br /> 3.2 Substance of Mathematical Thinking<br />List of Types of Mathematical Thinking<br />I. Mathematical Attitudes<br />II. Mathematical Thinking Related to Mathematical Methods<br />III. Mathematical Thinking Related to Mathematical Contents<br />Chapter 4 Detailed Discussion of Mathematical Thinking Related to Mathematical Methods<br />Chapter 5 Detailed Discussion of Mathematical Thinking Related to Mathematical Substance<br />Chapter 6 Detailed Discussion of Mathematical Attitudes Chapter 7 Questions for Eliciting Mathematical Thinking<br /><span style="font-weight: bold;">Chapter 1</span><br /><span style="font-weight: bold;">The Aim of Education and Mathematical Thinking</span><br /><span style="font-weight: bold;">1. The Aim of Education: Scholastic Ability From the Perspective of “Cultivating Independent Persons”</span><br />School-based education must be provided to achieve educational goals. “Scholastic ability” becomes clear when one views the aim of school-based education.<br />The Aim of School Education<br />The aim of school education is described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.”<br />This guideline is a straightforward expression of the preferred aim of education.<br />The most important ability that children need to gain at present and in the future, as society, science, and technology advance dramatically, is not the ability to correctly and quickly execute predetermined tasks and commands, but rather the ability to determine for themselves what they should do, or what they should charge themselves with doing.<br />Of course, the ability to correctly and quickly execute necessary tasks is also necessary, but from now on, rather than adeptly imitating the skilled methods or knowledge of others, the ability to come up with one’s own ideas, no matter how small, and to execute one’s own independent, preferable actions (ability full of creative ingenuity) will be most important. This is why the aim of education from now on is to instill the ability (scholastic ability) to take these kinds of actions. Furthermore, this is something that must be instilled in every individual child or student. From now, it will be of particular importance for each individual school child to be able to act independently (rather than the entire class acting independently as a unit). Of course, not every child will be able to act independently at the same level, but each school child must be able to act independently according to his or her own capabilities. To this end, teaching methods that focus on the individual are important.<br /><span style="font-weight: bold;">2. The Scholastic Ability to Think and Make Judgments Independently Is Mathematical Thinking</span><br />– Looking at Examples –<br />The most important ability that arithmetic and mathematics courses need to cultivate in order to instill in students this ability to think and make judgments independently is mathematical thinking. This is why cultivation of this “mathematical thinking” has been an objective of arithmetic and mathematics courses in Japan since the year 1950. Unfortunately, however, the teaching of mathematical thinking has been far from adequate in reality.<br />One sign of this is the assertion by some that “if students can do calculations, that is enough.”<br />The following example illustrates just how wrong this assertion is:<br />Example: “Bus fare for a trip is 4,500 yen per person. If a bus that can seat 60 people is rented out, however, this fare is reduced by 20% per person. How many people would need to ride for it to be a better deal to rent out an entire bus?”<br />This problem is solved in the following manner:<br />Solving Method When a bus is rented – One person’s fee: 4,500×0.8=3,600<br />In the case of 60 people: 3,600×60=216,000<br />With individual tickets, the number of people that can ride is<br />216,000÷4,500=48 (people).<br />Therefore, it would be cheaper to rent the bus if more than 48 people ride.<br />Sixth-graders must be able to solve a problem of this level. Is it sufficient, however, to solve this problem just by being able to do formal calculation (calculation on paper or mental calculation, or the use of an abacus or calculator)? Regardless of how skilled a student is at calculation on paper, and regardless of whether or not a student is allowed to use a calculator at will, these skills alone are not enough to solve the problem. The reason is that before one calculates on paper or with a calculator, one must be able to make the judgment “what numbers need to be calculated, what are the operations that need to be performed on those numbers, and in what order should these operations be performed?” If a student is not able to make these judgments, then there’s not much point in calculating on paper or with a calculator. Formal calculation is a skill that is only useful for carrying out commands such as “calculate this and this” (a formula for calculation) once these commands are actually specified. Carrying out these commands is known as “deciding the operation.” Therefore, “deciding the operation” for oneself in order to determine which command is necessary to “calculate this and this” is an important skill that is indispensable for solving problems.<br />Deciding the operation clearly determines the meaning of each computation, and decides what must be done based on that meaning. This is why “the ability to clarify the meanings of addition, subtraction, multiplication, and division and determine operations based on these meanings” is the most important ability required for computation.<br />Actually, there is something more important – in order to correctly decide which operations to use in this way, one must be able to think in the following manner “I would like to determine the correct operations, and to do so, I need to recall the meanings of each operation, and think based on these meanings.” This thought process is one kind of mathematical thinking.<br />Even if a student solves the group discount problem as described above, this might not be sufficient to conclude that he truly understood the problem. This is why it is important to “change the conditions of the problem a little” and “consider whether or not it is still possible solve the problem in the same way.” These types of thinking are neither knowledge nor a skill. They are “functional thinking” and “analogical thinking.”<br />For instance, let’s try changing one of the conditions by “changing the bus fare from 4,500 yen to 4,000 yen.” Calculating again as described above results in an answer of 48 people (actually, a better way of thinking is to replace 4,500 above with 4,000 – this is analogical thinking). In this way, one should gain confidence in one’s method of solving the problem, as one realizes that the result is the same: 48 people.<br />The above formulas are expressed in a way that is insufficient for students in fourth grade or higher. It is necessary to express problems using a single formula whenever possible. When these formulas are converted into a single formula based on this thinking, this is the result:4,500×(1-0.2) ×60÷4,500<br />When viewed in this form, it becomes apparent that the formula is simply 60×(1-0.2)<br />What is important here is the idea of “reading the meaning of this formula.” This is important “mathematical thinking regarding formulas.” Reading the meaning of this formula gives us: full capacity × ratio<br />For this reason, even if the bus fare changes to 4,000 yen, the formula 60×0.8=48 is not affected. Furthermore, if the full capacity is 50 persons and the group discount is 30%, then regardless of what the bus fare may be, the problem can always be solved as “50×0.7=35; the group rate (bus rental) is a better deal with 35 or more people.” This greatly simplifies the result, and is an indication of the appreciation of mathematical thinking, namely “conserving cogitative energy” and “seeking a more beautiful solution.”<br /><span style="font-weight: bold;">Students should have the ability to reach the type of solution shown above independently. This is a desirable scholastic ability that includes the following aims:</span><br />• Clearly grasp the meaning of operations, and decide which operations to use based on this understanding<br />• Functional thinking<br />• Analogical thinking<br />• Expressing the problem with a better formula<br />• Reading the meaning of a formula<br />• Economizing thought and effort (seeking a better solution)<br />Although this is only a single example, this type of thinking is generally applicable. In other words, in order to be able to independently solve problems and expand upon problems and solving methods, the ability to use “mathematical thinking” is even more important than knowledge and skill, because it enables to drive the necessary knowledge and skill.<br />Mathematical thinking is the “scholastic ability” we must work hardest to cultivate in arithmetic and mathematics courses.<br /><span style="font-weight: bold;">3. The Hierarchy of Scholastic Abilities and Mathematical Thinking</span><br />As the previous discussion makes clear, there is a hierarchy of scholastic abilities. When related to the above discussion, and limited to the area of computation (this is the same as in other areas, and can be generalized), these scholastic abilities enable the following (from lower to higher levels):<br />1. The ability to memorize methods of formal calculation and to carry out these calculation<br />2. The ability to understand the rules of calculation and how to carry out formal calculation<br />3. The ability to understand the meaning of each operation, to decide which operations to use based on this understanding, and to solve simple problems<br />4. The ability to form problems by changing conditions or abstracting situations<br />5. The ability to creatively make problems and solve them<br />The higher the level, the more important it is to cultivate independent thinking in individuals. To this end, mathematical thinking is becoming even more and more necessary.<br /><span style="font-weight: bold;">Chapter 2</span><br /><span style="font-weight: bold;">The Importance of Teaching to Cultivate Mathematical Thinking</span><br /><span style="font-weight: bold;">2.1 The Importance of Teaching Mathematical Thinking</span><br />As we found in the previous chapter, the method of thinking is the center of scholastic ability. In arithmetic and mathematics courses, mathematical thinking is the center of scholastic ability. However, in Japan, in spite of the fact that the improvement of mathematical thinking was established as a goal more than 45 years ago, the teaching of mathematical thinking is by no means sufficient.<br />One of the reasons that teaching to cultivate mathematical thinking does not tend to happen is, teachers are of the opinion that students can still learn enough arithmetic even if they don’t teach in a way to cultivate the students’ mathematical thinking. In other words, teachers do not understand the importance of mathematical thinking.<br />The second reason is that, in spite of the fact that mathematical thinking was established as a goal, teachers do not understand what it really is. It goes without saying that teachers cannot teach what they themselves do not understand.<br />Therefore, we shall start out by explaining how important the teaching of mathematical thinking is.<br />A simple summary follows.<br />Mathematical thinking allows for:<br />(1) An understanding of the necessity of using knowledge and skills<br />(2) Learning how to learn by oneself, and the attainment of the abilities required for independent learning<br /><span style="font-weight: bold;">(1) The Driving Forces to Pursue Knowledge and Skills</span><br />Mathematics involves the teaching of many different areas of knowledge, and of many skills. If children are simply taught to “use some knowledge or skill” to solve problems, they will use that knowledge or skill. In this case, however, children will not realize why they are being told to use such a knowledge or skill. Also, when new knowledge or skills are required for problem solving and students are taught what skill to use, they will be able to use that skill to solve the problem, but they will not know why this skill must be used. Students will therefore fail to understand why the new skill is good.<br />What is important is “how to realize” which previously learned knowledge and skills should be used. It is also important to “sense the necessity of” and “perceive the need or desirability of using” new knowledge and skills.<br />Therefore, it is necessary for something to act as a drive towards the required knowledge and skills. Children first understand the benefits of using knowledge and skills when they possess and utilize such a drive. This leads them to fully acquire the knowledge and skills they have used.<br />Mathematical thinking acts as this drive.<br /><span style="font-weight: bold;">(2) Achieving Independent Thinking and the Ability to Learn Independently</span><br />Possession of this driving force gives students an understanding of how to learn by themselves.<br />Cultivating the power to think independently will be the most important goal in education from now on, and in the case of arithmetic and mathematics courses, mathematical thinking will be the most central ability required for independent thinking. By mastering this skill even further, students will attain the ability to learn independently.<br />The following specific example serves to clarify this point further.<br />2.2 Example: How Many Squares are There? This instructional material is appropriate for fourth-grade students.<br />1. The Usual Lesson Process<br />This is usually taught in the following way (T refers to the teacher, and C the children):<br />T: There are both big and small squares here. Let’s count how many squares there are in total.<br />T: (When the children start counting) First, how many small squares are there? C: 25.<br />T: Which squares are the second smallest?<br />C: (Indicate the squares using two by two segments)<br />T: Count the number of those squares.<br />T: Which squares are the next biggest size, and how many are there?<br />The questions continue in this manner in order of size. In each case, the teacher asks one child the number, and then asks another child if this number is correct. Alternatively, the teacher might recognize the correctness of the number, and comment “yes, that’s the right number.”<br />The teacher has the children count squares in order of size, and then has the children add the numbers together to derive the grand total.<br />2. Problems with This Method<br />a) When the teacher instructs children to count squares based on size, the children do not realize for themselves that they should sort the squares into groups. As a result, the children do not understand the need to sort, or the thinking behind sorting.<br />b) The number of squares of each size is determined either by the majority of the children’s answers, or based on the teacher’s approval. These methods are not the right way of determining the correct answer. Correctness must be determined based on solid rationales.<br />c) Also, if instruction regarding this problem ends this way, children will only know the answer to this particular problem. The important things they must grasp, however, are what to focus upon in general, and how to think about problems of this nature.<br />Teachers should, therefore, follow the following teaching method:<br />3. Preferred Method<br />(1) Clarification of the Problem – 1<br />The teacher gives the children the previous diagram. T: How many squares are there in this diagram?<br />C: 25 (many children will probably answer this easily).<br />Some children will probably respond with a larger number.<br />The children come up with the answer 25 after counting just the smallest squares. Those who think the number is higher are also considering squares with more than one segment per side. This is the source of the issue, which is not about the correct answer, but the vagueness of the mathematical problem.<br />The teacher should then have the children discuss “which squares they are counting when they arrive at the number 25,” and inform them that “this problem is vague and does not clearly state which squares need to be counted.” The teacher concludes by clarifying the meaning of the problem, saying “let’s count all the squares, of every different size.”<br />(2) Clarification of the Problem – 2<br />First, the teacher lets all the children count the squares independently. Various answers will be given when the teacher asks for totals, or the children may become confused while counting. The children will realize that most of them (or all of them) have failed to count correctly. It is then time to think of a way of counting that is a little better and easier (this becomes a problem for the children to solve).<br />(3) Realizing the Benefit of Sorting The children will realize that the squares should be sorted and counted based on size. The teacher has the children count the squares again, this time sorting according to size.<br />(4) Knowing the Benefit of Encoding<br />Once the children are finished counting, the teacher asks them to give their results. At this point, when the teacher asks “how many squares are there of this size, and how many squares are there of that size...” he/she will run into the problem of not being able to clearly indicate “which size.”<br />At this point, naming (encoding) each square size should be considered. It is important to make sure that the children realize that calling the squares “large, medium, and small” is not preferable because this naming system is limited. However, the children learn that naming the squares in the following way is a good system, as they state each number.<br />Squares with 1 segment 25<br />Squares with 2 segment 16<br />Squares with 3 segment 9<br />Squares with 4 segment 4<br />Squares with 5 segment 1<br />Total 55<br />(5) Judging the Correctness of Results More Clearly, Based on Solid Rationale The correctness or incorrectness of these numbers must be elucidated, so have one child count the squares again in front of the entire class. The student will probably count the squares while tracing each one, as shown to the right. This will result in a messy diagram, and make it hard to tell which squares are being counted. Tracing each square is inconvenient, and will make the students feel their counting has become sloppy.<br />(6) Coming up with a More Accurate and Convenient Counting Method<br />There is a counting method that does not involve tracing squares. Have the students discover that they can count the upper left vertex (corner) of each square instead of tracing, in the following manner: place the pencil on the<br />upper left vertex and start to trace each square in one’s head, without moving the pencil from the vertex.<br />By using this system, it is possible to count two-segment squares as shown in the diagram to the right, by simply counting the upper left vertices of each square. This counting method is easier and clearer.<br />This method takes advantage of the fact that “squares and upper left vertices are in a one-to-one relationship.” In other words, in the case of two-segment squares, once a square is selected, only one vertex will correspond to that square’s upper left corner. The flip side of this principle is that once a point is selected, if that point corresponds to the upper left corner of a square, then it will only correspond to a single square of that size. Therefore, while sorting based on size, instead of counting squares, one can also count the upper left corners.<br />Instead of counting squares, this method “uses a functional thinking by counting the easy-to-count upper left vertices, which are functionally equivalent to the squares (in a one-to-one relationship).”<br />(7) Expressing the Number of Squares as a Formula<br />When viewed in this fashion, the two-segment squares shown in the diagram have the same number as a matrix of four rows by four columns of dots. When one realizes that this is the same as 4×4, it becomes apparent that the total number of squares is as follows: 5×5+4×4+3×3+2×2+1×1 (A)<br />Students will understand that it’s a good idea to think of ways to devise different expression methods, and to express problems as formulas.<br />(8) Generalizing<br />This makes generalization simple. In other words, consider what happens when “the segment length of the original diagram is increased by 1 to a total of 6.” All one needs to do is to add 6×6 to formula (A) above. Thus, the thought process of trying to generalize, and the attempt to read formulas is important.<br />(9) Further Generalization<br />For instance (for students in fifth grade or higher), when this system is applied to other diagrams, such as a diagram constructed entirely of rhombus, how will this change the formula? (Answer: It will not change the above formula at all.)<br />By generalizing to see the case of parallelograms (as long as the counting involves only parallelograms that are similar to the smallest parallelogram, the diagram can be seen in the same way), the true nature of the problem becomes clear.<br /><span style="font-weight: bold;">4. Mathematical Thinking is the Key Ability </span><br />Here What kind of ability is required to think in the manner described above? First, what knowledge and skills are required? The requirements are actually extremely simple: Understanding the meaning of “square,” “vertex,” “segment,” and so on The ability to count to around 100<br />The ability to write the problem as a formula, using multiplication and addition Possession of this understanding and skills, however, is not enough to solve the problem. An additional, more powerful ability is necessary. This ability is represented by the underlined parts above, from (1) to (9):<br />Clarification of the Meaning of the Problem<br />Coming up with an Convenient Counting Method<br />Sorting and Counting<br />Coming up with a Method for Simply and Clearly Expressing How the Objects Are Sorted Encoding<br />Replacing to Easy-to-Count Things in a Relationship of Functional Equivalence<br />Expressing the Counting Method as a Formula<br />Reading the Formula Generalizing<br />This is mathematical thinking, which differs from simple knowledge or skills.<br />It is evident that mathematical thinking serves an important purpose in providing the ability to solve problems on one’s own as described above, and that this is not limited to this specific problem. Therefore, the cultivation of a number of these types of mathematical thinking must be the aim of this class.<br /><span style="font-weight: bold;">Chapter 3</span><br /><span style="font-weight: bold;">The Meaning of Mathematical Thinking and How to Teach It Characteristics of Mathematical Thinking</span><br />Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist. Let us consider the characteristics of mathematical thinking.<br /><span style="font-weight: bold;">1. Focus on Sets</span><br />Mathematical thinking is like an attitude, as in it can be expressed as a state of “attempting to do” or “working to do” something. It is not limited to results represented by actions, as in “the ability to do,” or “could do” or “couldn’t do” something.<br />For instance, the states of “working to establish a perspective” and “attempting to analogize, and working to create an analogy” are ways of thinking. If, on the other hand, one has no intention whatsoever of creating an analogy, and is told to “create an analogy,” he/she might succeed in doing so due to having the ability to do so, but this does not mean that he/she consciously thought in an analogical manner.<br />In other words, mathematical thinking means that when one encounters a problem, one decides which set, or psychological set, to use to solve that problem.<br /><span style="font-weight: bold;">2. Thinking Depends on Three Variables</span><br />In this case, the type of thinking to use is not determined by the problem or situation. Rather, the type of thinking to use is determined by the problem (situation), the person, and the approach (strategy) used. In other words, the way of thinking depends on three variables: the problem (situation), the person involved, and the strategy.<br />Two of these involve the connotative understanding of mathematical thinking. There is also denotative understanding of the same.<br /><span style="font-weight: bold;">3. Denotative Understanding</span><br />Concepts are made up of both connotative and denotative components. One method which clarifies the “mathematical thinking” concept is a method of clearly expressing connotative “meaning.” Even if the concept of mathematical thinking is expressed with words, as in “mathematical thinking is this kind of thing,” this will be almost useless when it comes to teaching, because even if one understands the sentences that express this meaning, this does not mean that they will be able to think mathematically.<br />Instead of describing mathematical thinking this way, it should be shown with concrete examples. At a minimum, doing this allows for the teaching of the type of thinking shown. In other words, mathematical thinking should be captured denotatively.<br /><span style="font-weight: bold;">4. Mathematical Thinking is the Driving Force </span>Behind Knowledge and Skills Mathematical thinking acts as a guiding force that elicits knowledge and skills, by helping one realize the necessary knowledge or skills for solving each problem. It should also be seen as the driving force behind such knowledge and skills.There is another type of mathematical thinking that acts as a driving force for eliciting other types of even more necessary mathematical thinking. This is referred to as the “mathematical attitude.”<br /><span style="font-weight: bold;">3.2 Substance of Mathematical Thinking</span><br />It is important to achieve a concrete (denotative) grasp of mathematical thinking, based on the fundamental thinking described in section 3.1. Let us list the various types of mathematical thinking.<br />First of all, mathematical thinking can be divided into the following three categories:<br />II. Mathematical Thinking Related to Mathematical Methods<br />III. Mathematical Thinking Related to Mathematical Contents<br />Furthermore, the following acts as a driving force behind the above categories: I. Mathematical Attitudes<br />Although the necessity of category I was mentioned above, further consideration as described below reveals the fact that it is appropriate to divide mathematical thinking into II and III. Mathematical thinking is used during mathematical activities, and is therefore intimately related to the contents and methods of arithmetic and mathematics. Put more precisely, a variety of different methods is applied when arithmetic or mathematics is used to perform mathematical activities, along with various types of mathematical contents. It would be accurate to say that all of these methods and types of contents are types of mathematical thinking. It is because of the ways of thinking that the existence of these methods and types of contents has meaning. Let us focus upon these types of contents and methods as we examine mathematical thinking from these two angles.<br />For this reason, three logical categories can be derived.<br />Specific details are provided below.<br /><span style="font-weight: bold;">List of Types of Mathematical Thinking</span><br /><span style="font-weight: bold;">I. Mathematical Attitudes</span><br />1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself<br />(1) Attempting to have questions<br />(2) Attempting to maintain a problem consciousness<br />(3) Attempting to discover mathematical problems in phenomena<br />2. Attempting to take logical actions<br />(1) Attempting to take actions that match the objectives<br />(2) Attempting to establish a perspective<br />(3) Attempting to think based on the data that can be used, previously learned items, and assumptions<br />3. Attempting to express matters clearly and succinctly<br />(1) Attempting to record and communicate problems and results clearly and succinctly<br />(2) Attempting to sort and organize objects when expressing them<br />4. Attempting to seek better things<br />(1) Attempting to raise thinking from the concrete level to the abstract level<br />(2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking<br />(3) Attempting to economize thought and effort<br /><span style="font-weight: bold;">II. Mathematical Thinking Related to Mathematical Methods</span><br />1. Inductive thinking<br />2. Analogical thinking<br />3. Deductive thinking<br />4. Integrative thinking (including expansive thinking)<br />5. Developmental thinking<br />6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)<br />7. Thinking that simplifies<br />8. Thinking that generalizes<br />8. Thinking that specializes<br />9. Thinking that symbolize<br />10. Thinking that express with numbers, quantifies, and figures<br /><span style="font-weight: bold;">III. Mathematical Thinking Related to Mathematical Contents</span><br />1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)<br />2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)<br />3. Attempting to think based on the fundamental principles of expressions (Idea of expression)<br />4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)<br />5. Attempting to formalize operation methods (Idea of algorithm)<br />6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)<br />7. Focusing on basic rules and properties (Idea of fundamental properties)<br />8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)<br />9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)<br />Once the student has written part of the number table, he/she can induce that “it is possible to move from one multiple of 8 to another by going down one row, and then left two columns.” When stated the same way for multiples of 4, it is also possible to induce that “it is possible to move from one multiple of 4 to another by going down one row, and then left two columns.”<br />Then considering “why it is possible to make this simple statement” and “whether or not it is still possible to state this for numbers over 99, and why this is the case” is deductive thinking. Next, consider what to base an explanation of this on. One will realize at this point that it is possible to base this on how the number table is created. This is also deductive thinking, and is based upon the following.<br />Since this number table has 10 numbers in each row, “going one position to the right increases the number by one, and going one position down increases the number by ten.” Based upon this, it is evident that going down one position always adds 10, and going left two positions always subtracts 2. Combining both of these moves always results in an increase of 8 (10-2=8). Therefore, if one adds 8 to a multiple of 4 (or a multiple of 8), the result will always be a multiple of 4 (8). This explains what is happening.<br />By achieving results with one’s own abilities in this way, it is possible to gain confidence in the correctness of one’s conclusion, and to powerfully assert this conclusion. Always try to explain the truth of what you have induced, and you will feel this way. Also, think about general explanations based on clear evidence (the creation of the number table). This is deductive thinking.<br />Example 2: Deductive thinking is not just used in upper grades, but is used in lower grades as well.<br />Assume that at the start of single-digit multiplication in 3rd grade, the problem “how many sheets of paper would you need to hand out 16 sheets each to 8 children” is presented. When the children respond with “16×8,” the teacher could run with this response and say, “all right, let’s consider how to find the answer to this.”<br />This is not adequate, however. The students must be made to thoroughly understand the fundamental reasoning behind the solution. It is important that students independently consider “why this is the way the problem is solved.”<br />The child will probably explain the problem by saying that “in this problem, eight 16s are added: 16+16+16+16+16+16+16+16.” This is based on the meaning of multiplication<br />Chapter 7<br /><span style="font-weight: bold;">Questions for Eliciting Mathematical Thinking</span><br />Teaching should focus on mathematical thinking. Teachers need to first think of how they can help children appreciate and gain the ability to use mathematical thinking. When children get stuck, rather than helping them directly with useful knowledge and skills, teachers must prepare a way to teach the mathematical thinking required to elicit the knowledge and skill and moreover to teach the attitude that leads to this thinking methods. Also, this assistance must be of a general nature, and must be applicable to many different situations. Assistance should take a form that is frequently helpful when one focuses upon it. This is because this kind of assistance is useful in many different situations. By repeatedly providing it, a student can grow accustomed to this type of mathematical thinking. This kind of assistance is not something taught directly, but something that should be used by children themselves to overcome problems. Therefore, this assistance should take the form of questions.<br />It goes without saying that the goal of teaching based on these kinds of questions is for children to gain the ability to ask these questions of themselves, and to learn how to think for themselves.<br />Questions related to mathematical thinking and attitudes must be posed based on a perspective of what kinds of questions must be asked. This must be considered in advance. Questions must be created so that the problem solving process elicits mathematical thinking and attitudes. The following offer a list of question analyses designed to cultivate mathematical thinking, based on a consideration of these kinds of questions. In other words, this question analysis list is comprised of questions derived from the main types of mathematical thinking used at each stage of the problem solving process.<br />The A questions on this list deal with mathematical attitudes, with the stage indicated as “A11” and so on. Questions related to mathematical thinking related to mathematical methods are marked with M, and questions related to mathematical ideas are marked with I. Types of thinking corresponding to the question are given in parentheses ( ).<br />[List of Questions Regarding Mathematical Thinking]<br /><span style="font-weight: bold;"><problem></problem></span><br /><span style="font-weight: bold;">Questions Regarding Mathematical Attitudes</span><br />A11 What kinds of things (to what extent) are understood and usable? (Clarifying the problem)<br />A12 What is needed to understand, and can this be stated clearly? (Clarifying the problem)<br />A13 What kinds of things (from what point) are not understood? What does one want to find? (Clarifying the problem)<br />A14 Does anything seem strange? (A questioning attitude)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Methods</span><br />M11 What is the same? What is shared? (Abstraction)<br />M12 Clarify the meaning of the words and use them by oneself. (Abstraction)<br />M13 What (conditions) are important? (Abstraction)<br />M14 What types of situations are being considered? What types of situations are being proposed? (Idealization)<br />M15 Use figures (numbers) for expression. (Diagramming, quantification)<br />M16 Replace numbers with simpler numbers. (Simplification)<br />M17 Simplify the conditions. (Simplification)<br />M18 Give an example. (Concretization)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Contents</span><br />I11 What must be decided? (Functional)<br />I12 What kinds of conditions are not needed, and what kinds of conditions are not included? (Functional)<br /><span style="font-weight: bold;"><establishing></establishing></span><br /><span style="font-weight: bold;">Questions Regarding Mathematical Attitudes</span><br />A21 What kind of method seems likely to work? (Perspective)<br />A22 What kind of result seems to be possible? (Perspective)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Methods</span><br />M21 Is it possible to do this in the same way as something already known? (Analogy)<br />M22 Will this turn out the same thing as something already known? (Analogy)<br />M23 Consider special cases. (Specialization)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Contents</span><br />I21 What should one consider this based on (what unit)? (Units, sets)<br />I22 What seems to be the approximate result? (Approximation)<br />I23 Is there something else with a similar meaning (properties)? (Expressions, operations, properties)<br /><span style="font-weight: bold;"><executing></executing></span><br /><span style="font-weight: bold;">Questions Regarding Mathematical Attitudes</span><br />A31 Try using what is known (what will be known). (Logic)<br />A32 Are you approaching what you seek? (Logic)<br />A33 Can this be said clearly? (Clarity)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Methods</span><br />M31 What kinds of rules seem to be involved? Try collecting data. (Induction)<br />M32 Think based on what is known (what will be known). (Deduction)<br />M33 What must be known before this can be said? (Deduction)<br />M34 Consider a simple situation (using simple numbers or figures). (Simplification)<br />M35 Hold the conditions constant. Consider the case with special conditions. (Specialization)<br />M36 Can this be expressed as a figure? (Diagramming)<br />M37 Can this be expressed with numbers? (Quantification)<br /><span style="font-weight: bold;">Questions Regarding Thinking Related to Ideas</span><br />I31 Think based on units (points, etc.). (Units)<br />I32 What unit (what scope) should be used for thinking? (Units, sets)<br />I33 Think based on the meaning of words (words used to express methods, or methods themselves). (Expressions, operations, properties)<br />I34 Try following a predetermined procedure (calculations). (Algorithms)<br />I35 What is this (formula or symbol) expressing? (Formulas, expressions)<br />I36 Can I express this in a formula? (Formulas)<br /><span style="font-weight: bold;"><logical></logical></span><br />Questions Regarding Mathematical Attitudes<br />A41 Why is this (always) correct? (Logical)<br />A42 Can this be said more accurately? (Accuracy)Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com3tag:blogger.com,1999:blog-7038868080704055876.post-41601713690832817822008-12-07T21:45:00.000+07:002008-12-07T21:50:06.642+07:00Bagaimana kita mengetahui kemampuan siswa?Berikut kami nukilkan dari Pedoman KTSP, bagaimana seorang guru matematika dapat mengetahui kemampuan matematika para siswanya?<br /><br />v Tes tulis, dapat berupa tes esai/uraian, pilihan ganda, isian, menjodohkan dan sebagainya.<br />v Tes lisan, yaitu berbentuk daftar pertanyaan.<br />v Tes unjuk kerja, dapat berupa tes identifikasi, tes simulasi, dan uji petik kerja produk, uji petik kerja prosedur, atau uji petik kerja prosedur dan produk.<br />v Penugasan, seperti tugas proyek atau tugas rumah.<br />v Observasi yaitu dengan menggunakan lembar observasi.<br />v Wawancara yaitu dengan menggunakan pedoman wawancara<br />v Portofolio dengan menggunakan dokumen pekerjaan, karya, dan atau prestasi siswa.<br />v Penilaian diri dengan menggunakan lembar penilaian diriDr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com5tag:blogger.com,1999:blog-7038868080704055876.post-23475453726825153312008-12-07T21:18:00.000+07:002008-12-07T21:22:39.307+07:00Hakekat Matematika Sekolah dan Siswa Senang Belajar Matematika?Agar siswa menyenangi matematika maka menurut (Ebbut and Straker, 1995) guru tidak seyogyanya menggunakan definisi matematika aksiomatis, melainkan mendefinisikan matematika sebagai matematika sekolah. Berikut adalah hakekat matematika sekolah menurut mereka:<br /><span style="font-weight: bold;">• Matematika merupakan kegiatan penelusuran pola dan hubungan</span><br /> - memberi kesempatan siswa untuk melakukan kegiatan penemuan dan penyelidikan pola-pola untuk menentukan hubungan.<br /> - memberi kesempatan kepada siswa untuk melakukan percobaan denga berbagai cara.<br /> - mendorong siswa untuk menemukan adanya urutan, perbedaan, perbandingan, pengelompokan, dsb.<br /> - mendorong siswa menarik kesimpulan umum.<br /> - membantu siswa memahami dan menemukan hubungan antara pengertian satu dengan yang lainnya.<br />•<span style="font-weight: bold;"> Matematika adalah kreativitas yang memerlukan imajinasi, intuisi dan penemuan</span><br /> - mendorong inisiatif dan memberikan kesempatan berpikir berbeda.<br /> - mendorong rasa ingin tahu, keinginan bertanya, kemampuan menyanggah dan kemampuan memperkirakan.<br /> - menghargai penemuan yang diluar perkiraan sebagai hal bermanfaat dari pada menganggapnya sebagai kesalahan.<br /> - mendorong siswa menemukan struktur dan desain matematika.<br /> - mendorong siswa menghargai penemuan siswa yang lainnya.<br /> - mendorong siswa berfikir refleksif.<br /> - tidak menyarankan penggunaan suatu metode tertentu.<br /><span style="font-weight: bold;">• Matematika adalah kegiatan problem solving</span><br /> - menyediakan lingkungan belajar matematika yang merangsang timbulnya persoalan matematika.<br /> - membantu siswa memecahhkan persoalan matematika menggunakan caranya sendiri.<br /> - membantu siswa mengetahui informasi yang diperlukan untuk memecahkan persoalan matematika.<br /> - mendorong siswa untuk berpikir logis, konsisten, sistematis dan mengembangkan sistem dokumentasi/catatan.<br /> - mengembangkan kemampuan dan ketrampilan untuk memecahkan persoalan.<br /> - membantu siswa mengetahui bagaimana dan kapan menggunakan berbagai alat peraga/media pendidikan matematika seperti : jangka, kalkulator, dsb.<br /><span style="font-weight: bold;">• Matematika merupakan alat berkomunikasi</span><br /> - mendorong siswa mengenal sifat matemaika.<br /> - mendorong siswa membuat contoh sifat matematika.<br /> - mendorong siswa menjelaskan sifat matematika.<br /> - mendorong siswa memberikan alasan perlunya kegiatan matematika.<br /> - mendorong siswa membicarakan persoalan matematika.<br /> - mendorong siswa membaca dan menulis matematika.<br /> - menghargai bahasa ibu siswa dalam membicarakan matematika.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com4tag:blogger.com,1999:blog-7038868080704055876.post-18515675826278323082008-12-07T21:01:00.000+07:002008-12-07T21:16:06.326+07:00Psikologi Siswa Belajar MatematikaMenurut Ebbutt and Straker (1995):<br /><span style="font-weight: bold;">Siswa akan mempelajari matematika dengan senang jika mempunyaiu motivasi, maka implikasi pandangan ini bagi guru matematika di sekolah adalah bahwa guru perlu melakukan hal-hal sebagai berikut:</span><br />- menyediakan kegiatan yang menyenangkan<br />- memperhatikan keinginan mereka<br />- membangun pengertian melalui apa yang mereka ketahui<br />- menciptakan suasana kelas yang mendudukung dan merangsang belajar<br />- memberikan kegiatan yangsesuai dengan tujuan pembelajaran<br />- memberikan kegiatan yang menantang<br />- memberikan kegiatan yang memberikan harapan keberhasilan<br />- menghargai setiap pencapaian siswa<br /><br /><span style="font-weight: bold;">Siswa mempelajari matematika dengan cara yang berbeda dan dengan kecepatan yang berbeda pula. Tiap siswa memerlukan pengalaman tersendiri yang terhubung dengan pengalaman di waktu yang lampau dan tiap siswa mempunyai latar belakang sosial-ekonomi-budaya yang berbeda-beda. Oleh karena itu:</span><br />- guru perlu berusaha mengetahuai kelebihan dan kekurangan para siswanya.<br />- merencanakan kegiatan yang sesuai dengan tingkat kemampuan siswa<br />- membangun pengetahuan dan ketrampilan siswa baik yang dia peroleh di sekolah maupun di rumah.<br />- merencanakan dan menggunakan catatan kemajuan siswa (assessment).<br /><br /><span style="font-weight: bold;">Siswa memerlukan teman dalam mempelajari matematika:</span><br />- belajar dalam kelompok dapat melatih kerjasama.<br />- belajar secara klasikal memberikan kesempatan untuk saling bertukar gagasan.<br />- memberi kesempatan kepada siswa untuk melakukan kegiatannya secara mandiri.<br />- melibatkan siswa dalam pengambilan keputusan tentang kegiatan yang akan dilakukannya.<br />- mengajarkan bagaimana cara belajar<br /><br /><span style="font-weight: bold;">Siswa memerlukan konteks dan situasi yang berbeda-beda dalam belajarnya</span><br />- menyediakan dan menggunakan berbagai alat peraga<br />- belajar matematika diberbagai tempat dan kesempatan.<br />- menggunakan matematika untuk berbagai keperluan.<br />- mengembangkan sikap menggunakan matematika sebagai alat untuk memecahkan problematika baik di sekolahan maupun di rumah.<br />- menghargai sumbangan tradisi, budaya dan seni dalam pengembangan matematika.<br />- memabantu siswa merefleksikan kegiatan matematikanya.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com12tag:blogger.com,1999:blog-7038868080704055876.post-11203124497880828232008-12-07T20:44:00.000+07:002008-12-07T20:45:22.987+07:00Problem Solving Matematika: Hakekat dan Pembelajarannya?Oleh: Marsigit<br />Persoalan matematika secara garis besar dapat dibagi dua yaitu : persoalan yang berhubungan dengan kehidupan sehari-hari dan persoalan matematika. Persoalan yang dimaksud adalah persoalan yang memerlukan matematika untuk pemecahannya. Misal : Berapa lama waktu yang diperlukan untuk menempuh perjalanan suatu jarak tertantu ?; Berapa harga suatu satuan barang tertentu?; dsb. Matematika di sini diperlukan sebagai alat dan bukan sebagai tujuan.Persoalan matematika menekankan pada aspek matematikanya dan proses untuk menyelesaikannya. Proses dan hasil sama-sama diperhatikan dan dikembangkan dalam persoalan matematika. Guru perlu memperhatikan bagaimana persoalan dapat diperluas dan hasilnya dapat ditarik kesimpulan umumnya ? Persoalan yang sering menarik perhatian siswa misalnya : Bagaimana anda dapat mendapatkan bilangan 0 sd 20 hanya dengan menggunakan bilangan 4 ? Misalnya 8 diperoleh dari 4 + 4, 16 diperoleh dari 4 x 4, dst. Bagaimana halnya dengan bilangan yang lain ?<br />Ketrampilan proses dalam pemecahan masalah matematika diperlukan meliputi : - penalaran (reasoning), - organisasi (organising), - pengelompokan (classifing), dan - identifikasi pola (recognising pattern). Siswa yang berhasil memecahkan persoalan matematika adalah siswa yang :- yakin akan kemampuannya, - mau mencoba berbagai cara, dan - mempunyai keingintahuan yang tinggi. Guru dapat mendapatkan persoalan matematika dari berbagai sumber : - melalui dialog dengan para siswanya;- melalui taman sejawat;- melalui orang tua murid;- melalui buku pegangan guru;- melalui pertanyaan murid;- melalui sumber yang lain.<br />Bagaimana menciptakan lingkungan/suasana yang kondusif untuk kegiatan pembelajaran matematika dari aspek problem solving? Berikut merupakan beberapa saran: - mengamati situasi dan keinginan siswa, - memberikan pertanyaan kepada siswa dan memintanya untuk menjawab, - mendorong siswa menggunakan berbagai macam cara, - membuat contoh sederhana, melibatkan siswa dan mengembangkannya, - membuat teka-teki, - mengajukan pertanyaan : Bagaimana jika ... ? Apakah mungkin untuk ...? Berapa banyak cara berbeda ?<br />- dan ketika mereka selesai bekerja, tanyakan kepada mereka : Apakah anda telah mendapatkan semua kemungkinan jawaban ? dan Bagaimana anda tahu ? dst.<br />Jika guru ingin memulai aktivitas problem solving dan merupakan hal baru di kelas, anda dapat memulainya dari yang sederhana. Jika anda mempunyai beberapa persoalan, berilah kesempatan kepada siswa untuk menentukan pilihannya. Anda dapat memberikan pertanyaan dan mengamati kegiatan siswa dalam pemecahan soalnya. Kegiatan problem solving di kelas memerlukan suasana yang menunjang misalnya : suasana kelas; struktur kelas; lingkungan fisik kelas; sumber ajar; kemampuan guru; kemampuan siswa.<br />Kapan kegiatan problem solving dilakukan? Untuk kegiatan problem solving diperlukan waktu yang agak longgar. Kegiatan Problem Solving dengan Kerja Sama. Kerjasama antar siswa akan terwujud jika guru mengembangkan sikap saling menghargai dan komunikasi satu dengan yang lainnya. Manfaat kerjasama dalam pemecahan persoalan adalah untuk : mencoba cara yang berbeda; mengembangkan sikap fleksibel dan menyesuaikan dengan yang lain; mencari alternatif cara jika suatu cara tidak bekerja; membandingkan satu cara dengan yang lainnya; memperoleh kejelasan pengertiannya melalui saran/pendapat orang lain; saling memberikan semangat untuk menyelesaikan persoalannya.<br />Bagaimana mengembangkan Langkah-langkah Problem Solving?<br />Dalam menyelesaikan persoalan matematika baik guru maupun siswa perlu mengembangkan prosedur atau langkah penyelesaiannya. Langkah-langkah berikut mungkin merupakan sebagaian yang mungkin terjadi dalam kegiatan problem-solving : - memahami pokok persoalan, - mendiskusikan alternatif pemecahannya, - memecah persoalan utama menjadi bagian-bagian kecil, - menyederhanakan persoalan, - menggunakan pengalaman masa lampau dan menggunakan intuisi untuk menemukan alternatif pemecahannya. - mencoba berbagai cara dengan mengajukan pertanyaan : Marilah kita coba yang ini dan lihatlah apa yang terjadi ?- bekerja secara sistematis, - mencatat apa yang terjadi, - mengecek hasilnya dengan mengulang kembali langkah-langkahnya, - mencoba memahami persoalan yang lain.<br />Bagaimana mengembangkan strategi pemecahan masalah?<br />Pertanyaan-pertanyaan berikut mungkin membantu guru mendorong siswa memecahkan persoalan matematika dengan baik (berhasil) :- Apakah persoalan cocok bagi siswa anda ? - Apakah mungkin mendiskusikan persoalan dengan siswa tanpa banyak memberi penjelasan ? - Apakah tersedia cukup sumber ajar untuk penyelesaian persoalan ? - Bagaimana mendorong siswa mengembangkan gagasannya ? - Bagaimana mendorong siswa melakukan kegiatan problem solving ? - Bagaimana mendorong siswa melakukan kegiatan problem solving untuk persoalan yang lebih luas dan lebih kompleks ?<br />Bagaimana mengevaluasi kegiatan Problem Solving?<br />Ada beberapa pertanyaan yang perlu diklarifikasi misalnya: - Seberapa jauh siswa menyelesaikan persoalan ? - Apakah siswa menyelesaikan seluruhnya atau sebahagian ?- Apakah penyelesaian bersifat umum atau hanya berlaku untuk kasus tertentu saja?- Apakah dapat ditemukan jawaban yang lebih singkat ? - Apakah terdapat penyelesaian yang lebih baik/menarik ? - Apakah siswa mempunyai alternatif jawaban ? - Seberapa baikkah penyelesaian dapat dijelaskan ? - Dapatkah siswa yang lain memahami penyelesaiannya ? - Apakah siswa telah mengujinya ?Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com2tag:blogger.com,1999:blog-7038868080704055876.post-2138210510458757852008-12-07T20:20:00.000+07:002008-12-07T20:21:27.748+07:00Guru Sebagai Seorang PenelitiOleh : Marsigit<br />Penelitian oleh guru bertujuan untuk meningkatkan kualitas pembelajaran. Penelitian oleh guru dapat dilakukan menggunakan studi kasus atau lebih memfokuskan dan merefleksikan siatuasi pembelajaran oleh guru yang sudah berpengalaman. Dalam penelitian ini, guru sebagai seorang peneliti, terlibat dalam aktivitas kelas dalam refleksi gaya mengajarnya.<br />Namun, secara rinci terdapat beberapa penekanan yang berbeda dalam penelitian yang dilakukan oleh guru. Seorang guru peneliti dapat melakukan penelitian kelas untuk menganalisis dan meningkatkan aspek gaya mengajarnya. Guru lain dapat melakukannya untuk mempelajari ketrampilan mengajar tertentu untuk siswa dengan kemampuan tertentu. Guru yang lainnya lagi dapat menyelidiki aspek penggunaan model-model pembelajaran.<br />Alasan mengapa seorang guru perlu meneliti di kelas pembelajarannya misal karena alasan: a. Profesionalisme, b. Inovasi pendidikan, c. Filsafat pendidikan. Asumsi agar seorang guru mampu menyelenggarakan penelitian di kelasnya, antara lain adalah: a. Guru yang bersifat terbuka cenderung lebih mudah menerima pembaharuan, b. Guru yang bersifat terbuka lebih mudah menerima saran/kritik.c. Guru yang bersifat terbuka lebih mudah melakukan penelitian.d. Guru yang bersifat terbuka lebih mampu merefleksikan gaya mengajarnya.e. Guru yang bersifat terbuka lebih toleran terhadap siswa dan koleganya.f. Kegiatan penelitian melatih guru bersifat terbuka.<br />Kegiatan penelitan yang dilakukan oleh seorang guru dapat meliputi :• Identifikasi masalah, • Klarifikasi masalah, • Identifikasi konteks, • Penjelasan fakta, • Menetapkan langkah-langkah, • Mengembangkan langkah-langkah<br />Adapun asumsi lain yang harus dipenuhi agar seorang guru mampu mengadakan penelitian kegiatan pembelajarannya adalah : a. Mengajar adalah pekerjaan utama guru, b. Pengumpulan data tidak terlalu banyak menyita waktu guru<br />c. Metode dan pendekatan penelitian dipilih yang tepat, d. Permasalahan penelitian harus merupakan bagian dari permasalahan mengajarnya, e. Memperhatikan system yang melingkupinya, f. Memerlukan iklim yang menunjang.<br />g. Kepastian follow up.<br />Penelitian yang dilakukan oleh seorang guru tidak harus dimulai dengan merumuskan masalah. Yang diperlukan adalah sikap guru peneliti yang merasa perlu mengadakan perbaikkan. Pengembangan fokus dapat dilakukan dengan menjawab pertanyaan-pertanyaan :a. Apa yang terjadi dalam pembelajaran sekarang ? b. Pada aspek mana saya guru merasa terdapat masalah ?c. Apa yang dapat guru lakukan terhadap masalah tersebut? Secara lebih khusus, penelitian yang dilakukan oleh guru dapat dimulai dari pernyataan-pernyataan berikut :<br />• Saya ingin memperbaiki tentang ....<br />• Beberapa rekan guru menyoroti tentang ...<br />• Apa yang dapat saya lakukan untuk merubah situasi ?<br />• Saya merasa terganngu oleh ...<br />• Saya mempunyai gagasan untuk mencobanya di kelas.<br />• Bagaimana ketrampilan ini ... diterapkan di.... kepada ...?<br />• dst.<br /><br />Fokus dapat diarahkan kepada sibelajar dengan menjawab pertanyaan-pertanyaan :<br />• Apa yang telah dan sedang dikerjakan siswa ?<br />• Apa yang telah mereka palajari ?<br />• Seberapa manfaatkah yang telah mereka pelajari ?<br />• Apa yang telah saya lakukan untuk mereka ?<br />• Apa yang telah saya pelajari dan saya persiapkan untuk mereka ?<br />• Apa yang akan saya lakukan sekarang ?Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com0tag:blogger.com,1999:blog-7038868080704055876.post-14819803774939877452008-12-07T19:56:00.001+07:002008-12-07T19:58:08.300+07:00UPAYA SISTEMATIS MENDEKATI KEBENARANOleh: Marsigit<br />Binatang mempunyai pengetahuan tetapi hanya terbatas untuk kelangsungan hidupnya (survive). Manusia mengembangkan pengetahuannya menjadi kebudayaan untuk memberi makna dalam hidupnya dan mencapai tujuan tertentu yang lebih tinggi dari pada sekedar kelangsungan hidupnya. Manusia dapat mengembangkan pengetahuannya kerena manusia mempunyai akal pikir dan mempunyai bahasa.<br />Penalaran merupakan proses berpikir dalam menarik kesimpulan yang berupa pengetahuan untuk memperoleh kebenaran. Penalaran merupakan salah satu sifat dari manusia. Sifat yang lain adalah merasa, bersikap dan bertindak. Kebenaran bersifat relatif dan berbeda-beda, oleh karena itu kegiatan penalaran untuk mendapatkannya juga berbeda-beda. Proses berfikir untuk mendapatkan kebenaran mempunyai sifat logis dan analitik. Dengan kata lain, kegiatan penalaran merupakan proses berpikir logis. Konotasi logis bersifat jamak sehingga dapat menimbulkan kekacauan penalaran. Salah satu sebabnya adalah tentang kekonsistensiannya. Penalaran yang bersifat analitik menyandarkan diri kepada suatu analisis tertentu, yaitu berdasarkan suatu langkah-langkah tertentu.<br />Perasaan merupakan suatu penarikan kesimpulan yang tidak sepenuhnya berdasarkan penalaran. Intuisi juga merupakan kegiatan berfikir yang tidak berdasar penalaran dan bersifat non-analitik. Jadi cara berpikir masyarakat kita dapat digolongkan menjadi dua yaitu cara berpikir analitik yang berupa penalaran logis dan cara berpikir non-analitik yang berupa intuisi dan perasaan.<br />Kebenaran agama diperoleh melalui kedua kebenaran di atas; dan pengetahuan yang diperoleh dapat berupa wahyu atau petunjuk yang bersifat personal dan subjektif. Sedangkan kebenaran yang diperoleh dari cara berpikir analitis bersifat logis, impersonal dan objektif. Penalaran demikian sering disebut penalaran ilmiah, yang merupakan gabungan dari penalaran deduktif dan induktif.<br />Untuk memperoleh pengetahuan yang berdasar kebenaran maka proses berpikir harus dilakukan melalui cara tertentu. Suatu penarikan kesimpulan harus bersifat sahih atau valid. Untuk memperoleh kesahihan kesimpulan dapat dilakukan menggunakan dua pendekatan yaiti logika induktif atau logika deduktif atau gabungan diantaranya.<br />Paling tidak terdapat dua macam teori kebenaran yaitu teori keherensi dan teori korespondensi. Teori keherensi menyatakan bahwa kebenaran harus konsisten dengan kebenaran sebelumnya yang dianggap benar. Sedangkan teori korespondensi menyatakan bahwa pengetahuan adalah benar jika berhubungan dengan objek yang dituju. Sedangkan teori pragmatis menyatakan bahwa kebenaran suatu pernyataan perlu diukur dengan kriteria apakah pernyataan tersebut bersifat fungsional dalam kehidupan praktis atau tidak.<br />Pengetahuan merupakan segenap apa yang kita ketahui tentang suatu obyek tertentu, termasuk kedalamnya adalah ilmu. Jadi ilmu merupakan bagian dari pengetahuan. Tiap pengetahuan pada dasarnya menjawab jenis pertanyaan tertentu yang diajukan. Kita harus mengetahuai jawaban apa saja yang mungkin ada. Jawaban bagi pengetahuan dapar bersifat ontologis (menanyakan apanya), bersifat epistemologis (menanyakan bagaimananya) dan bersifat aksiologis (menanyakan untuk apanya). Pengetahuan berfungsi untuk menjelaskan, meramalkan ataupun mengontrol.<br />Metode ilmiah merupakan prosedur untuk mendapatkan pengetahuan yang disebut sebagai ilmu. Metode merupakan prosedur yang bersifat sistematis. Sedangkan metodologi merupakan suatu pengkajian dari aturan-aturan dalam metodenya. Epistemologi merupakan pembahasan menganai bagaimana kita mendapatkan pengetahuan itu.Ilmiah merupakan ekspresi tentang cara bekerja pikiran. Metode ilmiah harus memenuhi sifat keherensi dan sifat korespondensi. Penjelasan sementara mengenai suatu kebenaran disebut hipotesis. Diperlukan langkah-langkah empiris yang bersifat rasional untuk memperoleh kebanaran hipotesis. Hipotesis merupakan dugaan atau jawaban sementara sebagai petunjuk jalan untuk mendapatkan jawabannya.<br />Metode ilmiah yang menggunakan perumusan hipotesis sering disebut sebagai proses logiko-hipotetiko-verivikatif. Jika metode menggunakan pendekatan deduksi maka metodenya sering disebut logiko-deduktio-verivikatif. Atau juga kita kenal logiko-induktio-verivikatif. Kerangka berpikir logiko-hipotetiko-verivikatif pada dasarnya meliputi langkah-langkah : perumusan masalah, penyusunan kerangka berfikir dalam pengujian hipotesis, perumusan hipotesis, pengujian hipotesis, dan penarikan kesimpulan.Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com0tag:blogger.com,1999:blog-7038868080704055876.post-34868079449362336882008-11-30T18:06:00.001+07:002008-11-30T18:06:55.773+07:00Pengumuman/AnnouncementKepada yang kubanggakan semua follower dan para mahasiswa/mahasiswi, untuk kepentingan penningkatan performa dan komunikasi international saya berkeinginan untuk menampilkan Blog-blog dari follower di link yang sesuai, namun saya belum menemukan caranya. Sementara yang saya tempuh adalah melakukan blocking bagi setiap blog yang muncul di halaman muka. Untuk itu saya sarankan agar anda mendaftarkan untuk jadi follower bukan pada alamat utama. Alamat utama Blog ini adalah http://powermathematics.blogspot.com/, melainkan daftarkan ke alamat blog sesuai dengan kriteria. Untuk Sejarah di http://sejarahmatematika.blogspot.com/; untuk Psikology di http://marsigitpsiko.blogspot.com/; untuk Guru dan Pembelajaran Matematika di http://pbmmatmarsigit.blogspot.com/, sedang untuk Filsafat, Blognya sedang dalam proses pembuatan. Jika anda telah menjadi follower di blog utama dan sekarang mengalami pemblokiran, saya sarankan anda membuat askes follower Blog anda sesuai dengan kriteria di atas. Pengumuman tambahan tentang Comment, agar digunakan betul-betul untuk komentar dengan bahasa pendek dan jelas syukur in English. Comment bukan untuk mengirimkan tugas. Tugas anda silahkan anda taruh di Blog masing-masing. Untuk itu maka mulai sekarang saya harus menyeleksi setiap Comment yang layak ditampilkan. Demikian harap maklum. Dan jangan lupa saya selalu membanggakan bagi anda yang telah berpartisipasi dalam Blog ini. Selamat buat anda semua. Mohon maaf kiranya, semua dalam rangka peningkatan mutu. (Marsigit)Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com2tag:blogger.com,1999:blog-7038868080704055876.post-35647077825753572382008-11-27T09:39:00.001+07:002008-11-27T09:41:53.462+07:00We invite people to communitae on psychologyHerewith , I invite people to write your experiences of psychology, psychology of education, and psychology of teaching learning of mathematics.(Marsigit)Dr. Marsigit, M.Ahttp://www.blogger.com/profile/13234941863013770920noreply@blogger.com5