Pembelajaran Matematika Pembelajaran Matematika Berbasiskan Sifat AlamiahnyaBerbasiskan Sifat Alamiahnya

Dr. rar. Net.Dr. rar. Net. Muhammad Farchani RosyidMuhammad Farchani Rosyid(Fisikawan UGM dan (Fisikawan UGM dan

Orang Tua Siswa SDIT Alam Nurul) Orang Tua Siswa SDIT Alam Nurul)

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

1 + 2 + 3 + 4 + …= ?1 + 2 + 3 + 4 + …= ?

Mengukur keliling bumiMengukur keliling bumi

Jika A sembarang himpunan, maka Jika A sembarang himpunan, maka A. A.

Eksternalisme :Eksternalisme :objek-objek matematik tidak gayut pada objek-objek matematik tidak gayut pada realitas realitas objek-objek matematik memiliki objek-objek matematik memiliki eksistensi tersendiri di alam eksistensi tersendiri di alam eksternal/alam ideaeksternal/alam ideaPertanyaan : mengapa beberapa fenomena Pertanyaan : mengapa beberapa fenomena alamiah memperlihatkan pola-pola matematis?alamiah memperlihatkan pola-pola matematis?

Internalisme :Internalisme :objek-objek matematik gayut pada objek-objek matematik gayut pada realitas yang dapat dicerap oleh realitas yang dapat dicerap oleh pancaindera melalui eksperimen, pancaindera melalui eksperimen, pengamatan dan abstraksipengamatan dan abstraksi

Dua aliran besar matematika :Dua aliran besar matematika :- Eksternalisme (platonik/rasionalisme)- Eksternalisme (platonik/rasionalisme)- Internalisme (aristotelian/empirisme)- Internalisme (aristotelian/empirisme)

Adakah suatu himpunan yang memuat dirinya sendiri?Adakah suatu himpunan yang memuat dirinya sendiri?Ada. Ada. Andaikan Andaikan AA adalah himpunan yang beranggotakan adalah himpunan yang beranggotakan semuasemua himpunan. Karena himpunan. Karena AA adalah juga himpunan, maka adalah juga himpunan, maka A A berada di berada di AA. Artinya, . Artinya, AA memuat dirinya sendiri. memuat dirinya sendiri.

Sekarang, andaikan Sekarang, andaikan SS adalah himpunan yang adalah himpunan yang beranggotakan semua himpunan yang tidak memuat beranggotakan semua himpunan yang tidak memuat dirinya sendiri. dirinya sendiri. Jika Jika B B B B, maka , maka BB SS..

Jika Jika B B B B, maka , maka BB S.S. Bagaimana dengan himpunan Bagaimana dengan himpunan S S itu sendiri?itu sendiri?

Jika Jika S S SS, maka , maka S S SS..Jika Jika S S S S, maka , maka S S S S..

S S S S jika dan hanya jika jika dan hanya jika S S SS..

Aliran-aliran modern : Aliran-aliran modern : (respon atas paradoks yang muncul dalam teori bilangan (respon atas paradoks yang muncul dalam teori bilangan dan antinomi pada teori himpunan)dan antinomi pada teori himpunan)

- logisisme- logisisme (Gottlob Frege) : (Gottlob Frege) : gagasan-gagasan matematik tersubordinasi oleh gagasan-gagasan matematik tersubordinasi oleh logika (kebenaran matematika ditentukan oleh logika (kebenaran matematika ditentukan oleh

bentuk bentuk proposisi)proposisi)- intuisionisme/konstruktivisme- intuisionisme/konstruktivisme (Brouwer) :(Brouwer) :

gagasan-gagasan matematik harus dapat gagasan-gagasan matematik harus dapat dikonstruksi dikonstruksi dari bilangan aslidari bilangan asli- formalisme- formalisme (David Hilbert) : (David Hilbert) :

gagasan-gagasan matematik berawal dari intuisi gagasan-gagasan matematik berawal dari intuisi berdasarkan atas objek-objek yang setidak-berdasarkan atas objek-objek yang setidak-

tidaknya tidaknya memiliki wakilan dalam pikiran manusia. memiliki wakilan dalam pikiran manusia. Karakterisasi Karakterisasi gagasan-gagasan matematis melalui gagasan-gagasan matematis melalui aksioma yang aksioma yang formal (ingat : problem Hilbert nomor 6)formal (ingat : problem Hilbert nomor 6)

Persepsi tentang matematika yang dimiliki oleh Persepsi tentang matematika yang dimiliki oleh para guru sangat berpengaruh pada cara para guru sangat berpengaruh pada cara pembelajaran/pengajaran matematikapembelajaran/pengajaran matematika

Cara mengajar para guru ternyata lebih Cara mengajar para guru ternyata lebih ditentukan oleh persepsi mereka tentang ditentukan oleh persepsi mereka tentang matematika daripada oleh keyakinan mereka matematika daripada oleh keyakinan mereka akan cara mengajar yang paling baik.akan cara mengajar yang paling baik.

Mathematics problems have one and only one right Mathematics problems have one and only one right answer.answer.

Mathematics is facts and rules with one way to get the Mathematics is facts and rules with one way to get the right answer. You find the rule and get the answer. right answer. You find the rule and get the answer. Usually, the rule to use is the one your teacher just taught Usually, the rule to use is the one your teacher just taught you.you.

You don’t need to understand why the rules work.You don’t need to understand why the rules work. If you don’t solve a problem in five minutes, then you’ll If you don’t solve a problem in five minutes, then you’ll

never solve it. Give up.never solve it. Give up. Only geniuses discover or create mathematics, so if you Only geniuses discover or create mathematics, so if you

forget something, you’ll never be able to figure it out on forget something, you’ll never be able to figure it out on your own.your own.

Mathematics problems have little to do with the real world. Mathematics problems have little to do with the real world. In the real world, do what make sense. In mathematics, In the real world, do what make sense. In mathematics, follow the rules.follow the rules.

Mathematics is arithmeticMathematics is arithmetic

Student’s View of mathematics (Schoenfeld, 1992) :Student’s View of mathematics (Schoenfeld, 1992) :

Parent’s View of mathematics (Schoenfeld, 1992) :Parent’s View of mathematics (Schoenfeld, 1992) :

Mathematics is about numbers and arithmetic, Mathematics is about numbers and arithmetic, unbending accuracy and infallible rules.unbending accuracy and infallible rules.

The students should know the basics.The students should know the basics. Mathematics is an innate ability. Mathematics is Mathematics is an innate ability. Mathematics is

difficult, and so, students should not be difficult, and so, students should not be expected to do too much.expected to do too much.

Teacher’s View of mathematicsTeacher’s View of mathematics

Richard Skemp (1976) : there are two effectively different Richard Skemp (1976) : there are two effectively different subjects being taught under the same name subjects being taught under the same name “mathematics”.“mathematics”.

1.1. Instrumental MathematicsInstrumental MathematicsIt consists of a limited number of rules without reasonsIt consists of a limited number of rules without reasons

2.2. Relational MathematicsRelational MathematicsIt is knowing both what to do and why. It involves It is knowing both what to do and why. It involves building up conceptual structures or schemas from building up conceptual structures or schemas from which a learner can produce an unlimited number of which a learner can produce an unlimited number of rules to fit an unlimited sets of situation.rules to fit an unlimited sets of situation.

Comprehensive View of Comprehensive View of Mathematics :Mathematics : Mathematics is not arithmetic. Mathematics is not arithmetic. Mathematics is problem posing and problem Mathematics is problem posing and problem

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

Mathematics is the activity of finding and Mathematics is the activity of finding and studying patterns and relationships.studying patterns and relationships.

Mathematical activity includes perceiving, Mathematical activity includes perceiving, describing, discriminating, classifying, and describing, discriminating, classifying, and explaining patterns everywhere explaining patterns everywhere in number, data, in number, data, and space, and even in patterns themselves.and space, and even in patterns themselves.

Mathematics is a language.Mathematics is a language.

Mathematics is also used to communicate about Mathematics is also used to communicate about patterns.patterns.

Mathematics is a way of thinking and a tool Mathematics is a way of thinking and a tool for thinking.for thinking.

Mathematics is a changing body of Mathematics is a changing body of knowledge, an ever-expanding collection of knowledge, an ever-expanding collection of related ideas.related ideas.

Mathematics is doing mathematics.Mathematics is doing mathematics.The process of ‘doing’ mathematics is far more than just The process of ‘doing’ mathematics is far more than just calculation or deduction; it involves observation of calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results.patterns, testing of conjectures, and estimation of results.

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

Mathematics is useful for everyone.Mathematics is useful for everyone.

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.

Yang perlu ditekankan :Yang perlu ditekankan :

Seeking solutions, not just memorizing Seeking solutions, not just memorizing procedures.procedures.

Exploring patterns, not just memorizing Exploring patterns, not just memorizing formulas.formulas.

Formulating conjectures, not just doing Formulating conjectures, not just doing exercise.exercise.

7.7. Jumlah nilai sudut keseluruhan sebuah segitiga adalah Jumlah nilai sudut keseluruhan sebuah segitiga adalah 180180º. Maksudnya, untuk sembarang segitiga seperti º. Maksudnya, untuk sembarang segitiga seperti gambar di bawah ini, gambar di bawah ini, a a + + bb + + cc = = 180180º.º.

Berapakah jumlah nilai sudut keseluruhan sembarang Berapakah jumlah nilai sudut keseluruhan sembarang segilima? segilima? Berapa jumlah nilai sudut keseluruhan sembarang Berapa jumlah nilai sudut keseluruhan sembarang segiduabelas?segiduabelas?

a

b

c

Strategi Pembelajaran yang cocokStrategi Pembelajaran yang cocok

Socratic Questioning :Socratic Questioning :1. The teacher directs children’s discovery through a series 1. The teacher directs children’s discovery through a series of leading questionsof leading questions2. A large- or small-group activity is conducted under direct 2. A large- or small-group activity is conducted under direct teacher guidance.teacher guidance.

Pattern Searching :Pattern Searching :1. The teacher presents several worked examples of the 1. The teacher presents several worked examples of the pattern to be discussed.pattern to be discussed.2. Students work individuals or in small groups.2. Students work individuals or in small groups.3. Students discover a “rule” for the pattern and then test it.3. Students discover a “rule” for the pattern and then test it.

Group Thinking and LaboratoryGroup Thinking and Laboratory1. The teacher presents a problem that causes the 1. The teacher presents a problem that causes the students to use experience to discover the new students to use experience to discover the new mathematical idea. It may require exploration and mathematical idea. It may require exploration and activity.activity.

2. Students exchange their ideas to synthesize 2. Students exchange their ideas to synthesize and develop new ideas.and develop new ideas.

3. A variety of procedures are developed.3. A variety of procedures are developed.

4. Manipulative material are often used.4. Manipulative material are often used.

5. Often individual work is done first, then group 5. Often individual work is done first, then group work.work.