I. M. Gelfand :

1. Pembelajaran Matematika Berbasiskan Sifat AlamiahnyaDr. rar. Net.Muhammad Farchani Rosyid(Fisikawan UGM dan Orang Tua Siswa SDIT Alam Nurul)

2. I. M. Gelfand : “… Let us begin with the last question : What is mathematics? …I have mentioned the closeness between the style of mathematics and the style of classical music or poetry. I was happy to find the following four common features : first, beauty; second, simplicity; third, exactness; fourth, crazy ideas. The combination of these four things : beauty, exactness, simplicity, and crazy ideas is just the heart of mathematics, …”

3. 1 + 2 + 3 + 4 + …= ? Mengukur keliling bumi Jika A sembarang himpunan, maka   A.

4. Dua aliran besar matematika : - Eksternalisme (platonik/rasionalisme) - Internalisme (aristotelian/empirisme) • Eksternalisme : objek-objek matematik tidak gayut pada realitas objek-objek matematik memiliki eksistensi tersendiri di alam eksternal/alam idea Pertanyaan : mengapa beberapa fenomena alamiah memperlihatkan pola-pola matematis? • Internalisme : objek-objek matematik gayut pada realitas yang dapat dicerap oleh pancaindera melalui eksperimen, pengamatan dan abstraksi

5. Adakah suatu himpunan yang memuat dirinya sendiri? Ada. Andaikan Aadalah himpunan yang beranggotakan semua himpunan. Karena A adalah juga himpunan, maka A berada di A. Artinya, A memuat dirinya sendiri. • Sekarang, andaikan S adalah himpunan yang beranggotakan semua himpunan yang tidak memuat dirinya sendiri. Jika B  B, maka B S. Jika B  B, maka B S. • Bagaimana dengan himpunan S itu sendiri? Jika S  S, maka S  S. Jika S  S, maka S  S.  S  S jika dan hanya jika S  S.

6. Aliran-aliran modern : (respon atas paradoks yang muncul dalam teori bilangan dan antinomi pada teori himpunan) - logisisme (Gottlob Frege) : gagasan-gagasan matematik tersubordinasi oleh logika (kebenaran matematika ditentukan oleh bentuk proposisi) - intuisionisme/konstruktivisme(Brouwer) : gagasan-gagasan matematik harus dapat dikonstruksi dari bilangan asli - formalisme(David Hilbert) : gagasan-gagasan matematik berawal dari intuisi berdasarkan atas objek-objek yang setidak-tidaknya memiliki wakilan dalam pikiran manusia. Karakterisasi gagasan-gagasan matematis melalui aksioma yang formal (ingat : problem Hilbert nomor 6)

7. Persepsi tentang matematika yang dimiliki oleh para guru sangat berpengaruh pada cara pembelajaran/pengajaran matematika • Cara mengajar para guru ternyata lebih ditentukan oleh persepsi mereka tentang matematika daripada oleh keyakinan mereka akan cara mengajar yang paling baik.

8. Student’s View of mathematics (Schoenfeld, 1992) : • Mathematics problems have one and only one right answer. • Mathematics is facts and rules with one way to get the right answer. You find the rule and get the answer. Usually, the rule to use is the one your teacher just taught you. • You don’t need to understand why the rules work. • If you don’t solve a problem in five minutes, then you’ll never solve it. Give up. • Only geniuses discover or create mathematics, so if you forget something, you’ll never be able to figure it out on your own. • Mathematics problems have little to do with the real world. In the real world, do what make sense. In mathematics, follow the rules. • Mathematics is arithmetic

9. Parent’s View of mathematics (Schoenfeld, 1992) : • Mathematics is about numbers and arithmetic, unbending accuracy and infallible rules. • The students should know the basics. • Mathematics is an innate ability. Mathematics is difficult, and so, students should not be expected to do too much.

10. Teacher’s View of mathematics Richard Skemp (1976) : there are two effectively different subjects being taught under the same name “mathematics”. • Instrumental Mathematics It consists of a limited number of rules without reasons • Relational Mathematics It is knowing both what to do and why. It involves building up conceptual structures or schemas from which a learner can produce an unlimited number of rules to fit an unlimited sets of situation.

11. Comprehensive View of Mathematics : • Mathematics is not arithmetic. • Mathematics is problem posing and problem solving. Meaningful problems take a long time to pose as well as to solve. They stimulate curiosity about mathematics, not just about the answer to a problem. They engage a variety of students’ ideas and skills. They lead students to thinking about how the world work from a mathematical point of view and to think about how mathematics itself works. They open up discussion to a variety of contributions from multiple participants.

12. Mathematics is the activity of finding and studying patterns and relationships. Mathematical activity includes perceiving, describing, discriminating, classifying, and explaining patterns everywhere in number, data, and space, and even in patterns themselves. • Mathematics is a language. Mathematics is also used to communicate about patterns.

13. Mathematics is a way of thinking and a tool for thinking. • Mathematics is a changing body of knowledge, an ever-expanding collection of related ideas. • Mathematics is doing mathematics. The process of ‘doing’ mathematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results.

14. Mathematics is a path to independent thinking. Mathematics is an area in which even young children can pose and solve a problem and have confidence that the solution is correct not because the teacher says it is, but because its inner logic is so clear. • Mathematics is useful for everyone.

15. Di antara kata-kata sifat dalam bahasa Indonesia terdapat kata-kata seperti “pendek”, “terdiri dari banyak suku kata” dan beberapa kata sifat lain yang dapat diterapkan untuk mensifati kata-kata itu sendiri : ─“Pendek” itu pendek. ─ “Terdiri dari banyak suku kata” terdiri dari banyak suku kata. Kata-kata semacam ini dimasukkan ke dalam kelompok kata sifat yang homologis. Tetapi, ada pula kata-kata semacam “biru”, “sedih” dan beberapa kata sifat yang lain yang tidak dapat digunakan untuk mensifati kata-kata itu sendiri : ─“Biru” itu tidak biru. ─“Sedih” tidak sedih. Kata-kata terakhir ini digolongkan sebagai kata sifat yang heterologis. Bagaimana dengan kata “heterologis” itu sendiri? ☺ Jika “heterologis” itu heterologis, maka “heterologis” itu homologis. ☺ Jika “heterologis” itu homologis, maka “heterologis” itu heterologis.

16. Yang perlu ditekankan : • Seeking solutions, not just memorizing procedures. • Exploring patterns, not just memorizing formulas. • Formulating conjectures, not just doing exercise.

17. b c a • Jumlah nilai sudut keseluruhan sebuah segitiga adalah 180º. Maksudnya, untuk sembarang segitiga seperti gambar di bawah ini, a + b + c = 180º. Berapakah jumlah nilai sudut keseluruhan sembarang segilima? Berapa jumlah nilai sudut keseluruhan sembarang segiduabelas?

18. Socratic Questioning : 1. The teacher directs children’s discovery through a series of leading questions 2. A large- or small-group activity is conducted under direct teacher guidance. Pattern Searching : 1. The teacher presents several worked examples of the pattern to be discussed. 2. Students work individuals or in small groups. 3. Students discover a “rule” for the pattern and then test it. Strategi Pembelajaran yang cocok

19. Group Thinking and Laboratory 1. The teacher presents a problem that causes the students to use experience to discover the new mathematical idea. It may require exploration and activity. 2. Students exchange their ideas to synthesize and develop new ideas. 3. A variety of procedures are developed. 4. Manipulative material are often used. 5. Often individual work is done first, then group work.