Matematika di Mata Paul Lockhart
Malam itu saya berdebat dengan adik saya mengenai matematika. Menurut adik saya, matematika itu seni. Matematika itu indah. Dan people should learn mathematics for the sake of mathematics bukan hanya sekedat hitungan kering seperti yang ada di sekolah. Dia lalu cerita membayangkan titik. Kalau sekilas hanya terlihat sebagai titik. Lama-lama titik itu membesar, terlihar seperti lingkaran. Kita kira itu lingkaran ternyata bukan, saat dilihat lebih jelas lagi ternyata merupakan bola. Dengan matematika ia bisa bermain dengan imajinasinya. “It’s fun!” katanya.
Halah, dia ngomong apa lagi? “Dasar matematikawan!” batinku dalam hati. Adik saya memang punya basis matematika yang kuat. Dia sempat belajar matematika saat kuliah. Maksudku, jurusan matematika murni (pure mathematics). Saya sendiri menyukai matematika, tapi kesukaan itu bukan tiba-tiba datang karena saya memang menyukai matematika. Saya mulai menyukai matematika pertama kalinya saat saya belajar fisika. Ternyata matematika bisa digunakan untuk memahami fisika. Ada aplikasi praktisnya!
Adik saya ngotot bahwa tanpa aplikasi praktis matematika tetap menarik. Saya tidak setuju. Saya ingat ketika kuliah saya sempat belajar mengenai deret fourier saat pelajaran matematika teknik. Benar-benar membosankan. Tapi suatu hari deret fourier menjadi menarik, yakni saat saya menggunakannya saat belajar pelajaran getaran mesin. Deret fourier ternyata ada aplikasinya, bisa digunakan untuk memahami getaran mesin! Menyenangkan! Waktu menemani seseorang untuk tes kesehatan menggunakan ultrasound, sang dokter menunjukkan sebuah grafik yang ternyata menggunakan deret fourier. Bagi saya matematika menjadi bermakna ketika ada aplikasi praktisnya. Mungkin karena saya cenderung menyebut diri saya praktisi. Adik saya, menyukai matematika dari sisi yang berbeda dari saya. Dia menyukai filosofinya, keindahannya dan sebagainya. Saya kemudian baru sadar bahwa hal yang sama, misalnya matematika, bisa memiliki arti (meaning) yang berbeda bagi setiap orang, tergantung dari latar belakangnya, pengalamannya, dan sebagainya. Meanings depends on how we perceive the world around us. Different people have different ways of seeing things, including about mathematics.
Tapi saya beruntung. Adik saya begitu penasaran untuk membuat saya tercerdaskan! Dia ingin memberi saya pemahaman yang lebih mendalam mengenai matematika. Dia memaksa saya membaca beberaoa hal mengenai matematika, misalnya kisah mengenai sejarah angka ‘nol’ dan bagaimana ide angka ‘nol’ bisa menggemparkan dunia. Dia juga meminta saya membaca buku karangan Paul Lockhart berjudul Mathematician’s Lament. Di dalam buku tersebut ada banyak kritik mengenai pendidikan matematika di sekolah. Kali ini saya ingin sedikit sharing mengenai buku yang terakhir ini.
Ternyata sang penulis punya pandangan yang sama dengan adik saya, menurut sang penulis :
“The first thing to understand is that mathematics is an art. The difference between math and other arts, such as music and painting, is that our culture does not recognize it as such.” (Lockhart, 2009, h. 22)
Sang penulis memandang matematika sebagai seni dan bedanya dengan seni lain adalah bahwa di dalam budaya kita, music dan melukis dipandang sebagai seni tetapi matematika tidak. Penulis melanjutkan pendapatnya sebagai berikut :
“Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science – perhaps they help the scientists with their formulas, or feed big numbers into computers for some reasons or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is a bit as mind-blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than any poetry, art, or music… Mathematics is the purest of arts, as well as the most misunderstood.” (Lockhart, 2009, h. 22 - 23)
Katanya, matematika memang membantu para ilmuwan tapi bukan itu esensi matematika. Matematika adalah ilmu seni yang paling murni. Ah, dasar matematikawan!
Sang penulis lalu member contoh sederhana mengenai sebuah bentuk (yang sudah sering kita pelajari di sekolah). Katanya :
“… For example, if I’m in the mood tp think about shapes – and I often am – I might imagine a triangle inside a rectangular box :
I wonder how much of the box the triangle takes uo – two thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor I am talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior purpose here. I’m just playing. That’s what math is – wondering, playing, amusing yourself with your imagination… The mathematical question is about the imaginary triangle inside the box. The edges are perfect becauseI want them to be – that is the sort of object I prefer to think about. This is a major theme in mathematics : things are what you want them to be, there is no reality to get in your way. ” (Lockhart, 2009, h. 24 - 25)
Lalu lanjutnya :
“In the case of the triangle in the box, I do see something simple and pretty :
I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space3 inside the triangle as outside. That must mean that the triangle must take up exactly half the box!
… Now where did this idea of mine come from? How did I know to draw the line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck” (Lockhart, 2009, h. 26)
Dia menyayangkan bagaimana keindahan matematika, direduksi menjadi fakta-fakta yang perlu diingat dan prosedur yang harus diikuti. Menurutnya :
“… It is so heart breaking to see what is being done to mathematics in school. This rich and fascinating adventure of imagination has been reduced to a set sterile facts to be memorized and procedures to be followed.
… “The area of a triangle is equal to one-half its base times its height” Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative art. There is even no problem anymore. The question has been asked and answered at the same time – there is nothing left for the students to do.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and its place just as learning a vocabulary does – it helps to create richer, more nuanced works of art. But it’s not the fact that triangles take up half of their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to the creative breakthroughs in other problems – something a mere statement of fact can never give you. ” (Lockhart, 2009, h. 27 - 28)
Menurut Lockhart, matematika di dalam kelas sangat kering karena kita sering berkonsentrasi pada pertanyaan ‘apa’ dan bukan ‘kenapa’. Menurutnya matematika adalah seni menjelaskan. Tanpa membiarkan siswa mengemukakan berbagai masalah matematika yang ditemukannya sendiri, membiarkan mereka melakukan berbagai penemuan matematika, merasa frustasi, terinspirasi, mencoba menjelaskan dan memperlihatkan bukti-bukti yang mereka temukan, maka menurutnya tidak ada cukup matematika di dalam kelas matematika kita (there is the lack of mathematics in our mathematics classroom). Selengkapnya, diuraikan sebagai berikut :
“By concentrating on ‘what’ and leaving out ‘why’, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself that gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny the students the opportunity to engage in this activity – to pose their own problems, to make their own conjectures and discoveries, to be wrong and frustrated, to have an inspiration, and to cobble together their own explanations and proofs – you deny them mathematics itself. So no, I’m not complaining about the present facts and formulas in our mathematics classes, I’m complaining about the lack of ‘mathematics’ in our mathematics classes.”
Membaca buku tersebut, saya mendapat tambahan insight mengenai bagaimana seorang Lockhart memandang matematika, yang berbeda dengan saya yang memandang matematika sebagai alat yang membantu dalam berbagai aplikasi dalam kehidupan sehari-hari (dan fisika tentunya!). Bagi Lockhart matematika adalah seni (bukan hanya bermanfaat karena aplikasi praktisnya), matematika adalah bermain dengan imajinasi yang tidak bisa dihalangi oleh apa yang realita yang dilihat mata (is a major theme in mathematics [is that] things are what you want them to be, there is no reality to get in your way. ),matematika adalah merasa terinspirasi yang bisa membantu kita menemukan penyelesaian untuk berbagai masalah lain (inspire other beautiful ideas and lead to the creative breakthroughs in other problems) , dan matematika adalah seni menjelaskan (the art of explanation). Itu pendapat Paul Lockhart. Bagaimana dengan anda?
Sumber :
Lockhart, Paul. (2009). Mathematician’s Lament : How School Cheats Us Out Our Most fascinating and Imaginative Art Form New York : Bellevue
Halah, dia ngomong apa lagi? “Dasar matematikawan!” batinku dalam hati. Adik saya memang punya basis matematika yang kuat. Dia sempat belajar matematika saat kuliah. Maksudku, jurusan matematika murni (pure mathematics). Saya sendiri menyukai matematika, tapi kesukaan itu bukan tiba-tiba datang karena saya memang menyukai matematika. Saya mulai menyukai matematika pertama kalinya saat saya belajar fisika. Ternyata matematika bisa digunakan untuk memahami fisika. Ada aplikasi praktisnya!
Adik saya ngotot bahwa tanpa aplikasi praktis matematika tetap menarik. Saya tidak setuju. Saya ingat ketika kuliah saya sempat belajar mengenai deret fourier saat pelajaran matematika teknik. Benar-benar membosankan. Tapi suatu hari deret fourier menjadi menarik, yakni saat saya menggunakannya saat belajar pelajaran getaran mesin. Deret fourier ternyata ada aplikasinya, bisa digunakan untuk memahami getaran mesin! Menyenangkan! Waktu menemani seseorang untuk tes kesehatan menggunakan ultrasound, sang dokter menunjukkan sebuah grafik yang ternyata menggunakan deret fourier. Bagi saya matematika menjadi bermakna ketika ada aplikasi praktisnya. Mungkin karena saya cenderung menyebut diri saya praktisi. Adik saya, menyukai matematika dari sisi yang berbeda dari saya. Dia menyukai filosofinya, keindahannya dan sebagainya. Saya kemudian baru sadar bahwa hal yang sama, misalnya matematika, bisa memiliki arti (meaning) yang berbeda bagi setiap orang, tergantung dari latar belakangnya, pengalamannya, dan sebagainya. Meanings depends on how we perceive the world around us. Different people have different ways of seeing things, including about mathematics.
Tapi saya beruntung. Adik saya begitu penasaran untuk membuat saya tercerdaskan! Dia ingin memberi saya pemahaman yang lebih mendalam mengenai matematika. Dia memaksa saya membaca beberaoa hal mengenai matematika, misalnya kisah mengenai sejarah angka ‘nol’ dan bagaimana ide angka ‘nol’ bisa menggemparkan dunia. Dia juga meminta saya membaca buku karangan Paul Lockhart berjudul Mathematician’s Lament. Di dalam buku tersebut ada banyak kritik mengenai pendidikan matematika di sekolah. Kali ini saya ingin sedikit sharing mengenai buku yang terakhir ini.
Ternyata sang penulis punya pandangan yang sama dengan adik saya, menurut sang penulis :
“The first thing to understand is that mathematics is an art. The difference between math and other arts, such as music and painting, is that our culture does not recognize it as such.” (Lockhart, 2009, h. 22)
Sang penulis memandang matematika sebagai seni dan bedanya dengan seni lain adalah bahwa di dalam budaya kita, music dan melukis dipandang sebagai seni tetapi matematika tidak. Penulis melanjutkan pendapatnya sebagai berikut :
“Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science – perhaps they help the scientists with their formulas, or feed big numbers into computers for some reasons or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is a bit as mind-blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than any poetry, art, or music… Mathematics is the purest of arts, as well as the most misunderstood.” (Lockhart, 2009, h. 22 - 23)
Katanya, matematika memang membantu para ilmuwan tapi bukan itu esensi matematika. Matematika adalah ilmu seni yang paling murni. Ah, dasar matematikawan!
Sang penulis lalu member contoh sederhana mengenai sebuah bentuk (yang sudah sering kita pelajari di sekolah). Katanya :
“… For example, if I’m in the mood tp think about shapes – and I often am – I might imagine a triangle inside a rectangular box :
I wonder how much of the box the triangle takes uo – two thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor I am talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior purpose here. I’m just playing. That’s what math is – wondering, playing, amusing yourself with your imagination… The mathematical question is about the imaginary triangle inside the box. The edges are perfect becauseI want them to be – that is the sort of object I prefer to think about. This is a major theme in mathematics : things are what you want them to be, there is no reality to get in your way. ” (Lockhart, 2009, h. 24 - 25)
Lalu lanjutnya :
“In the case of the triangle in the box, I do see something simple and pretty :
I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space3 inside the triangle as outside. That must mean that the triangle must take up exactly half the box!
… Now where did this idea of mine come from? How did I know to draw the line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck” (Lockhart, 2009, h. 26)
Dia menyayangkan bagaimana keindahan matematika, direduksi menjadi fakta-fakta yang perlu diingat dan prosedur yang harus diikuti. Menurutnya :
“… It is so heart breaking to see what is being done to mathematics in school. This rich and fascinating adventure of imagination has been reduced to a set sterile facts to be memorized and procedures to be followed.
… “The area of a triangle is equal to one-half its base times its height” Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative art. There is even no problem anymore. The question has been asked and answered at the same time – there is nothing left for the students to do.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and its place just as learning a vocabulary does – it helps to create richer, more nuanced works of art. But it’s not the fact that triangles take up half of their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to the creative breakthroughs in other problems – something a mere statement of fact can never give you. ” (Lockhart, 2009, h. 27 - 28)
Menurut Lockhart, matematika di dalam kelas sangat kering karena kita sering berkonsentrasi pada pertanyaan ‘apa’ dan bukan ‘kenapa’. Menurutnya matematika adalah seni menjelaskan. Tanpa membiarkan siswa mengemukakan berbagai masalah matematika yang ditemukannya sendiri, membiarkan mereka melakukan berbagai penemuan matematika, merasa frustasi, terinspirasi, mencoba menjelaskan dan memperlihatkan bukti-bukti yang mereka temukan, maka menurutnya tidak ada cukup matematika di dalam kelas matematika kita (there is the lack of mathematics in our mathematics classroom). Selengkapnya, diuraikan sebagai berikut :
“By concentrating on ‘what’ and leaving out ‘why’, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself that gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny the students the opportunity to engage in this activity – to pose their own problems, to make their own conjectures and discoveries, to be wrong and frustrated, to have an inspiration, and to cobble together their own explanations and proofs – you deny them mathematics itself. So no, I’m not complaining about the present facts and formulas in our mathematics classes, I’m complaining about the lack of ‘mathematics’ in our mathematics classes.”
Membaca buku tersebut, saya mendapat tambahan insight mengenai bagaimana seorang Lockhart memandang matematika, yang berbeda dengan saya yang memandang matematika sebagai alat yang membantu dalam berbagai aplikasi dalam kehidupan sehari-hari (dan fisika tentunya!). Bagi Lockhart matematika adalah seni (bukan hanya bermanfaat karena aplikasi praktisnya), matematika adalah bermain dengan imajinasi yang tidak bisa dihalangi oleh apa yang realita yang dilihat mata (is a major theme in mathematics [is that] things are what you want them to be, there is no reality to get in your way. ),matematika adalah merasa terinspirasi yang bisa membantu kita menemukan penyelesaian untuk berbagai masalah lain (inspire other beautiful ideas and lead to the creative breakthroughs in other problems) , dan matematika adalah seni menjelaskan (the art of explanation). Itu pendapat Paul Lockhart. Bagaimana dengan anda?
Sumber :
Lockhart, Paul. (2009). Mathematician’s Lament : How School Cheats Us Out Our Most fascinating and Imaginative Art Form New York : Bellevue
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