For hundreds of years, x has been t

For hundreds of years, x has been the go-to symbol for the unknown quantity in mathematical equations. So who started this practice?

Algebra was born in the Middle East, during the Golden Age of medieval Islamic civilization (750 to 1258 AD), and its early form can be seen in the work of Muhammad Al-Khwarizmi and his 9th century book, Kitab al-jabr wal-muqabala (al-jabr later morphing into algebra in English). During this heyday, Muslim rule and culture had expanded onto the Iberian Peninsula, where the Moors encouraged scholarship in the sciences and math.

So what does this have to do with the letter "x" in math? In a recent TED talk, the director of The Radius Foundation, Terry Moore, posited that the the use of "x" in this way began with the inability of Spanish scholars to translate certain Arabic sounds, including the letter sheen (or shin). According to Moore, the word for "unknown thing" in Arabic is al-shalan, and it appeared many times in early mathematical works. (For example, you might see "three unknown things equals 15," with the "unknown thing" then being 5.)

The principle problem with Moore's explanation is that there is no direct documented evidence to support it. More speculatively, people translating the works would not care about phonetics, but the meaning of the words. So whether they had a "sh" or not one would think would be irrelevant. Despite the lack of direct evidence and flaws in the argument, it nonetheless remains a very popular origin theory, even among many academics. (Do a quick Google search and you'll find many a PhD in mathematics parroting this theory.)

The 1909-1916 edition of Webster's Dictionary, among others, also puts forth a similar theory, although stating that the Arabic word for the singular "thing," "shei," was translated into the Greek "xei," and later shortened to x. Dr. Ali Khounsary also notes that the Greek word for unknown, xenos, also begins with x, and the convention could simply have been born of an abbreviation. But here, again, we have a lack of any direct documented evidence to support these theories.

As for a documented theory, we turn to the great philosopher and mathematician, René Descartes (1596-1650). It's entirely possible Descartes did not come up with the practice of using "x" for an unknown, perhaps borrowing it from someone else, but at least as far as documented evidence that has survived to today goes, he seems to be the creator of the practice, as noted by the OED and the phenomenal work by Florian Cajori, A History of Mathematical Notations (1929). At the least, Descartes' helped popularize the practice.

Specifically, in his landmark work, La Géométrie (1637), Descartes solidified the movement to symbolic notation by instituting the convention of using the lowercase letters at the beginning of the alphabet for known quantities (e.g., a, b and c) and using those at the end of the alphabet for unknown quantities (e.g., z, y and x).

Why? And why x more than y, and z for unknowns? Nobody knows. It has been speculated that the prominence of x being used more than y and z for unknowns in this work had to do with typesetting; one story goes that it was Descartes' printer who suggested x be the principle unknown in La Géométrie because it was the letter least used and so the one he had more letter blocks available to use. Whether this is true or not, Descartes used the x to be an unknown at least as early as 1629 in various manuscripts, well before La Géométrie. And, indeed, it would seem he had not come to any hard rules on x, y, and z indicating unknowns; in some manuscripts from this time, he actually used x, y, and z to represent known quantities, casting even further doubt on the supposed "unknown thing" translation theories listed above.

So, in the end, by all appearances, Descartes simply arbitrarily chose the letters to represent different things in his works as was convenient and it just so happened in his landmark work, La Géométrie, he decided the specific variable nomenclature, perhaps, on a whim.

Whatever the case, as with Descartes' notation for powers (x3), after the publication of La Géométrie, the use of x as a principle unknown (as well as the more general tradition of a, b, c = knowns and x, y, z = unknowns) gradually caught on. And the rest, as they say, is mathematical history.

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The Origin of the Hashtag Symbol

Bonus Facts:

The equal sign ("=") was invented in 1557 by Welsh mathematician Robert Recorde, who was fed up with writing "is equal to" in his equations. He chose the two lines because "no two things can be more equal."
Other early symbols used to represent unknowns in mathematics before Descartes' landmark work include Benedetto of Florence's 1463 Trattato di praticha d'arismetrica where he uses the Greek letter rho; Michael Stifel's 1544 Arithmetic integra where he uses q (for quantita) as well as A, B, C, D, and F; Francois Vieta's late 16th century nomenclature where vowels are used as unknowns and consonants are used as constants, among others. (Incidentally, if you're curious: What Makes a Vowel a Vowel and a Consonant a Consonant?)
In modern English, x is the third least used letter, occurring in only about 0.15% of all words. The least used letters are q and z.
The word "algorithm" comes from none other than al-Khwarizmi's name. If you distort the name slightly when you say it, you'll get the connection.
The mathematical volume of a pizza is pizza. How does that work you say? Well if z = radius of the pizza and a = the height then Π * radius2 * height = Pi * z * z * a = Pizza.
As mentioned, La Géométrie was a ground-breaking work. In it, Descartes introduced the idea that eventually became known as Cartesian coordinates; this included the ideas of two perpendicular lines called axes, naming the horizontal one x and the vertical axis y, and also designating the point of intersection as the origin. Descartes is also credited with one of the most famous lines in all of Western thought – Cognito ergo sum (I think, therefore I am.)
That said, while Descartes is famous for the notion of "I think, therefore I am," he was not the first to express such an idea. For instance, Aristotle said something similar in Nicomachean Ethics, "But if life itself is good and pleasant… and if one who sees is conscious that he sees, one who hears that he hears, one who walks that he walks and similarly for all the other human activities there is a faculty that is conscious of their exercise, so that whenever we perceive, we are conscious that we perceive, and whenever we think, we are conscious that we think, and to be conscious that we are perceiving or thinking is to be conscious that we exist…" Of course, "I think, therefore I am" is a lot more succinct.
Muhammad Al-Khwarizmi was one of the first directors of the House of Wisdom in Bagdad. Having supervised the translations of important Indian and Greek mathematical and astronomical works, Al-Khwarizmi became an advocate for the adoption of the Indian numeric system (1-9 plus 0) and is the father of algebra. With the publication of The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi introduced using abstract analysis in problem solving (although with words, rather than symbolic notation). He also introduced the algebraic method of reducing (rewriting the expression to ever simpler, but equivalent, forms), as well as that of balancing (doing the same things to each side of the equation – again to make it simpler).
The Programme for International Student Assessment (PISA) assesses the competencies of 15-year-olds in 65 countries and economies, including in math. For 2012, the country/economy with the highest scores in math was Shanghai-China, which was closely followed by Singapore, Hong Kong-China, Chinese Taipei and Korea. Notably, Canada ranked 13th, Australia 19th, Ireland 20th and the United Kingdom 26th. The United States' kids ranked 36th. In fact, according to PISA, the performance of one of our highest-scoring states, Massachusetts, was so low, it was as if those students had two fewer years of mathematical education than the students in Shanghai-China. PISA also noted that although the U.S. spends more per student than most countries, this doesn't translate into performance. In 2012, per-student spending in the U.S. was listed at $115,000, while in the Slovak Republic, a country that performed at the same level, they spend only $53,000 per student.
It should be noted of the PISA's results, though, that they are drastically over simplified. For instance, as noted in a report by Dr. Martin Carnoy of Stanford and Richard Rothstein of the Economic Policy Institute, American students actually perform better than the much higher ranked Finland in algebra in general, but worse in fractions. Further, when you normalize the results between the countries adjusting for the relative poverty of the students taking the PISA tests, the U.S performs significantly better, ranking 6th in reading and 13th in mathematics, a huge jump in both categories. They further note in their report What Do International Tests Really Show About U.S. Student Performance? that when you divide the kids based on family wealth, the actual gap in performance isn't so stark between countries, with a not insignificant portion of the ultimate ranking of each nation being based on how many impoverished vs. middle class vs. wealthy students are taking the tests. For reference, about 40% of the schools the PISA used in the U.S.'s sample had more than 50% of their students eligible for free lunch.
Despite their results being oversimplified, the

Hasil (Bahasa Indonesia) 1:

Selama ratusan tahun, x telah masuk ke simbol untuk jumlah yang tidak diketahui dalam persamaan matematika. Jadi yang mulai praktek ini?Aljabar dilahirkan di Timur Tengah, selama zaman keemasan peradaban Islam abad pertengahan (750 untuk 1258 m), dan bentuk awal dapat dilihat dalam karya Muhammad Al-Khwarizmi dan bukunya abad ke-9, Kitab al-jabr wal-muqabala (al-jabr kemudian berubah menjadi aljabar dalam bahasa Inggris). Selama masa kejayaan ini, kekuasaan Islam dan budaya telah diperluas ke Semenanjung Iberia, di mana orang-orang Moor mendorong beasiswa dalam ilmu pengetahuan dan matematika.Jadi, apa ini harus dilakukan dengan huruf "x" dalam matematika? Dalam pembicaraan TED hari, Direktur Yayasan Radius, Terry Moore, mengemukakan bahwa penggunaan "x" cara ini dimulai dengan ketidakmampuan para sarjana Spanyol untuk menerjemahkan suara tertentu Arab, termasuk surat kemilau (atau shin). Menurut Moore, adalah kata untuk "hal yang tidak diketahui" dalam bahasa Arab al-shalan, dan itu muncul beberapa kali dalam karya matematika awal. (Sebagai contoh, Anda mungkin melihat "tiga hal yang tidak diketahui sama dengan 15," dengan hal yang"tidak diketahui" kemudian menjadi 5.)The principle problem with Moore's explanation is that there is no direct documented evidence to support it. More speculatively, people translating the works would not care about phonetics, but the meaning of the words. So whether they had a "sh" or not one would think would be irrelevant. Despite the lack of direct evidence and flaws in the argument, it nonetheless remains a very popular origin theory, even among many academics. (Do a quick Google search and you'll find many a PhD in mathematics parroting this theory.)The 1909-1916 edition of Webster's Dictionary, among others, also puts forth a similar theory, although stating that the Arabic word for the singular "thing," "shei," was translated into the Greek "xei," and later shortened to x. Dr. Ali Khounsary also notes that the Greek word for unknown, xenos, also begins with x, and the convention could simply have been born of an abbreviation. But here, again, we have a lack of any direct documented evidence to support these theories.As for a documented theory, we turn to the great philosopher and mathematician, René Descartes (1596-1650). It's entirely possible Descartes did not come up with the practice of using "x" for an unknown, perhaps borrowing it from someone else, but at least as far as documented evidence that has survived to today goes, he seems to be the creator of the practice, as noted by the OED and the phenomenal work by Florian Cajori, A History of Mathematical Notations (1929). At the least, Descartes' helped popularize the practice.Specifically, in his landmark work, La Géométrie (1637), Descartes solidified the movement to symbolic notation by instituting the convention of using the lowercase letters at the beginning of the alphabet for known quantities (e.g., a, b and c) and using those at the end of the alphabet for unknown quantities (e.g., z, y and x).Why? And why x more than y, and z for unknowns? Nobody knows. It has been speculated that the prominence of x being used more than y and z for unknowns in this work had to do with typesetting; one story goes that it was Descartes' printer who suggested x be the principle unknown in La Géométrie because it was the letter least used and so the one he had more letter blocks available to use. Whether this is true or not, Descartes used the x to be an unknown at least as early as 1629 in various manuscripts, well before La Géométrie. And, indeed, it would seem he had not come to any hard rules on x, y, and z indicating unknowns; in some manuscripts from this time, he actually used x, y, and z to represent known quantities, casting even further doubt on the supposed "unknown thing" translation theories listed above.So, in the end, by all appearances, Descartes simply arbitrarily chose the letters to represent different things in his works as was convenient and it just so happened in his landmark work, La Géométrie, he decided the specific variable nomenclature, perhaps, on a whim.Whatever the case, as with Descartes' notation for powers (x3), after the publication of La Géométrie, the use of x as a principle unknown (as well as the more general tradition of a, b, c = knowns and x, y, z = unknowns) gradually caught on. And the rest, as they say, is mathematical history.If you liked this article, you might also enjoy: Where the Ampersand Symbol and Name Came From Where the Dollar Sign Comes From How "XOXO" Came to Mean "Hugs and Kisses" Did Newton Really Have an Apple Fall on His Head, Inspiring Him to Come Up with His Theory on Gravity? The Origin of the Hashtag SymbolBonus Facts: The equal sign ("=") was invented in 1557 by Welsh mathematician Robert Recorde, who was fed up with writing "is equal to" in his equations. He chose the two lines because "no two things can be more equal." Other early symbols used to represent unknowns in mathematics before Descartes' landmark work include Benedetto of Florence's 1463 Trattato di praticha d'arismetrica where he uses the Greek letter rho; Michael Stifel's 1544 Arithmetic integra where he uses q (for quantita) as well as A, B, C, D, and F; Francois Vieta's late 16th century nomenclature where vowels are used as unknowns and consonants are used as constants, among others. (Incidentally, if you're curious: What Makes a Vowel a Vowel and a Consonant a Consonant?) In modern English, x is the third least used letter, occurring in only about 0.15% of all words. The least used letters are q and z. The word "algorithm" comes from none other than al-Khwarizmi's name. If you distort the name slightly when you say it, you'll get the connection. The mathematical volume of a pizza is pizza. How does that work you say? Well if z = radius of the pizza and a = the height then Π * radius2 * height = Pi * z * z * a = Pizza. As mentioned, La Géométrie was a ground-breaking work. In it, Descartes introduced the idea that eventually became known as Cartesian coordinates; this included the ideas of two perpendicular lines called axes, naming the horizontal one x and the vertical axis y, and also designating the point of intersection as the origin. Descartes is also credited with one of the most famous lines in all of Western thought – Cognito ergo sum (I think, therefore I am.) That said, while Descartes is famous for the notion of "I think, therefore I am," he was not the first to express such an idea. For instance, Aristotle said something similar in Nicomachean Ethics, "But if life itself is good and pleasant… and if one who sees is conscious that he sees, one who hears that he hears, one who walks that he walks and similarly for all the other human activities there is a faculty that is conscious of their exercise, so that whenever we perceive, we are conscious that we perceive, and whenever we think, we are conscious that we think, and to be conscious that we are perceiving or thinking is to be conscious that we exist…" Of course, "I think, therefore I am" is a lot more succinct. Muhammad Al-Khwarizmi was one of the first directors of the House of Wisdom in Bagdad. Having supervised the translations of important Indian and Greek mathematical and astronomical works, Al-Khwarizmi became an advocate for the adoption of the Indian numeric system (1-9 plus 0) and is the father of algebra. With the publication of The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi introduced using abstract analysis in problem solving (although with words, rather than symbolic notation). He also introduced the algebraic method of reducing (rewriting the expression to ever simpler, but equivalent, forms), as well as that of balancing (doing the same things to each side of the equation – again to make it simpler). The Programme for International Student Assessment (PISA) assesses the competencies of 15-year-olds in 65 countries and economies, including in math. For 2012, the country/economy with the highest scores in math was Shanghai-China, which was closely followed by Singapore, Hong Kong-China, Chinese Taipei and Korea. Notably, Canada ranked 13th, Australia 19th, Ireland 20th and the United Kingdom 26th. The United States' kids ranked 36th. In fact, according to PISA, the performance of one of our highest-scoring states, Massachusetts, was so low, it was as if those students had two fewer years of mathematical education than the students in Shanghai-China. PISA also noted that although the U.S. spends more per student than most countries, this doesn't translate into performance. In 2012, per-student spending in the U.S. was listed at $115,000, while in the Slovak Republic, a country that performed at the same level, they spend only $53,000 per student. It should be noted of the PISA's results, though, that they are drastically over simplified. For instance, as noted in a report by Dr. Martin Carnoy of Stanford and Richard Rothstein of the Economic Policy Institute, American students actually perform better than the much higher ranked Finland in algebra in general, but worse in fractions. Further, when you normalize the results between the countries adjusting for the relative poverty of the students taking the PISA tests, the U.S performs significantly better, ranking 6th in reading and 13th in mathematics, a huge jump in both categories. They further note in their report What Do International Tests Really Show About U.S. Student Performance? that when you divide the kids based on family wealth, the actual gap in performance isn't so stark between countries, with a not insignificant portion of the ultimate ranking of each nation being based on how many impoverished vs. middle class vs. wealthy students are taking the tests. For reference, about 40% of the schools the PISA used in the U.S.'s sample had more than 50% of their students eligible for free lunch. Despite their results being oversimplified, the

Hasil (Bahasa Indonesia) 2:

Selama ratusan tahun, x telah menjadi simbol go-to untuk kuantitas yang tidak diketahui dalam persamaan matematika. Jadi yang memulai praktek ini? Aljabar lahir di Timur Tengah, selama Golden Age peradaban Islam abad pertengahan (750-1258 M), dan bentuk awal dapat dilihat dalam karya Muhammad Al-Khwarizmi dan buku abad ke-9 nya, Kitab al-jabr wal-muqabala (al-jabr kemudian morphing menjadi aljabar dalam bahasa Inggris). Selama masa kejayaan ini, aturan dan budaya Muslim telah diperluas ke Semenanjung Iberia, di mana Moor mendorong beasiswa dalam ilmu dan matematika. Jadi, apa ini harus dilakukan dengan huruf "x" dalam matematika? Dalam pembicaraan TED baru-baru ini, direktur The Radius Foundation, Terry Moore, mengemukakan bahwa penggunaan "x" dengan cara ini dimulai dengan ketidakmampuan ulama Spanyol untuk menerjemahkan tertentu Arab suara, termasuk surat kemilau (atau shin). Menurut Moore, kata untuk "hal yang tidak diketahui" dalam bahasa Arab adalah al-Shalan, dan ternyata berkali-kali dalam karya matematika awal. (Sebagai contoh, Anda mungkin akan melihat "tiga hal yang tidak diketahui sama dengan 15," dengan "hal yang tidak diketahui" kemudian menjadi 5.) Masalah Prinsip dengan penjelasan Moore adalah bahwa tidak ada bukti yang terdokumentasi langsung untuk mendukungnya. Lebih spekulatif, orang menerjemahkan karya-karya tidak akan peduli fonetik, tetapi arti dari kata-kata. Jadi apakah mereka memiliki "sh" atau tidak orang akan berpikir akan menjadi tidak relevan. Meskipun kurangnya bukti langsung dan kelemahan dalam argumen, itu tetap tetap teori asal yang sangat populer, bahkan di antara banyak akademisi. (Lakukan pencarian Google cepat dan Anda akan menemukan banyak PhD dalam matematika menirukan teori ini.) Edisi 1909-1916 dari Webster Dictionary, antara lain, juga mengemukakan teori yang sama, meskipun menyatakan bahwa kata Arab untuk tunggal " hal, "" shei, "diterjemahkan ke dalam bahasa Yunani" Xei, "dan kemudian disingkat menjadi x. Dr Ali Khounsary juga mencatat bahwa kata Yunani untuk diketahui, xenos, juga dimulai dengan x, dan konvensi bisa hanya lahir dari singkatan. Tapi di sini, sekali lagi, kita memiliki kekurangan bukti terdokumentasi langsung untuk mendukung teori ini. Adapun teori didokumentasikan, kita beralih ke filsuf besar dan matematika, Rene Descartes (1596-1650). Sangat mungkin Descartes tidak datang dengan praktek menggunakan "x" untuk diketahui, mungkin meminjam dari orang lain, tapi setidaknya sejauh didokumentasikan bukti yang bertahan sampai hari ini berjalan, ia tampaknya menjadi pencipta praktek, seperti dicatat oleh OED dan karya fenomenal oleh Florian Cajori, A History of Matematika Notasi (1929). Setidaknya, Descartes membantu mempopulerkan praktek. Secara khusus, dalam karya monumentalnya, La Géométrie (1637), Descartes dipadatkan gerakan notasi simbolik dengan membentuk konvensi menggunakan huruf kecil di awal alfabet untuk jumlah dikenal ( misalnya, a, b, dan c) dan menggunakan mereka pada akhir alfabet untuk jumlah yang tidak diketahui (misalnya, z, y, dan x). Mengapa? Dan mengapa x lebih dari y, dan z untuk diketahui? Tidak ada yang tahu. Telah berspekulasi bahwa keunggulan x yang digunakan lebih dari y dan z untuk diketahui dalam pekerjaan ini harus dilakukan dengan typesetting; satu cerita berlanjut bahwa itu adalah printer Descartes yang menyarankan x menjadi dikenal prinsip di La Géométrie karena itu surat yang paling sering digunakan dan yang ia punya surat blok lagi yang tersedia untuk digunakan. Apakah ini benar atau tidak, Descartes menggunakan x untuk menjadi diketahui setidaknya sejak 1629 di berbagai manuskrip, baik sebelum La Géométrie. Dan, memang, tampaknya ia tidak datang ke aturan keras pada x, y, dan z menunjukkan tidak diketahui; dalam beberapa naskah dari waktu ini, ia benar-benar digunakan x, y, dan z untuk mewakili jumlah dikenal, keraguan lebih jauh tentang seharusnya "hal yang tidak diketahui" teori terjemahan yang tercantum di atas. Jadi, pada akhirnya, dengan semua penampilan, Descartes hanya sewenang-wenang memilih huruf untuk mewakili hal yang berbeda dalam karya-karyanya seperti nyaman dan itu hanya kebetulan dalam karya monumentalnya, La Géométrie, ia memutuskan nomenklatur variabel tertentu, mungkin, pada kehendak. Apapun kasus, seperti dengan notasi Descartes 'untuk kekuasaan (x3), setelah penerbitan La Géométrie, penggunaan x sebagai diketahui prinsip (serta tradisi yang lebih umum dari a, b, c = knowns dan x, y, z = tidak diketahui) secara bertahap tertangkap. . Dan sisanya, seperti kata mereka, adalah sejarah matematika Jika Anda menyukai artikel ini, Anda mungkin juga menikmati: Dimana Ampersand Simbol dan Nama Datang Dari mana Dollar Masuk Datang Dari Bagaimana "XOXO" Datang ke Mean "Hugs and Kisses" Apakah Newton Benar-benar Memiliki Apple Jatuh di kepala-Nya, Inspiring Dia Datang Up dengan Teori-Nya di Gravity? Asal Usul Hashtag Simbol Bonus Fakta: Tanda sama dengan ("=") diciptakan pada 1557 oleh matematikawan Welsh Robert Recorde, yang muak dengan tulisan "sama dengan" dalam persamaan nya. Dia memilih dua baris karena "tidak ada dua hal bisa lebih sama." simbol awal lain yang digunakan untuk mewakili diketahui dalam matematika sebelum Descartes 'tengara kerja termasuk Benedetto of Florence 1463 Trattato di praticha d'arismetrica mana ia menggunakan huruf Yunani rho; Michael Stifel di 1544 Arithmetic integra di mana ia menggunakan q (untuk quantita) serta A, B, C, D, dan F; Akhir abad ke-16 nomenklatur Francois Vieta di mana vokal digunakan sebagai diketahui dan konsonan digunakan sebagai konstanta, antara lain. (Kebetulan, jika Anda penasaran: Apa yang Membuat vokal vokal dan konsonan yang merupakan konsonan?) Dalam bahasa Inggris modern, x adalah huruf ketiga paling banyak digunakan, terjadi hanya sekitar 0,15% dari semua kata. Surat-surat yang paling sedikit digunakan adalah q dan z. Kata "algoritma" berasal dari tidak lain dari nama al-Khwarizmi. Jika Anda mengubah nama sedikit ketika Anda mengatakan itu, Anda akan mendapatkan koneksi. Volume matematika pizza adalah pizza. Bagaimana pekerjaan yang Anda katakan? Nah jika z = jari-jari pizza dan = tinggi maka Π * radius2 * height = Pi * z * z * a = Pizza. Seperti disebutkan, La Géométrie adalah karya tanah-melanggar. Di dalamnya, Descartes memperkenalkan gagasan yang akhirnya dikenal sebagai koordinat Cartesian; ini termasuk ide-ide dari dua garis tegak lurus yang disebut sumbu, penamaan horisontal satu x dan sumbu vertikal y, dan juga menunjuk titik persimpangan sebagai asal. Descartes juga dikreditkan dengan salah satu baris paling terkenal di seluruh pemikiran Barat - Cognito ergo sum ​​(. Saya berpikir, maka saya) Yang mengatakan, sementara Descartes terkenal dengan gagasan "Aku berpikir, maka aku," dia bukan yang pertama untuk mengekspresikan gagasan seperti itu. Misalnya, Aristoteles mengatakan hal serupa di Nicomachean Ethics, "Tetapi jika kehidupan itu sendiri baik dan menyenangkan ... dan jika orang yang melihat sadar bahwa ia melihat, orang yang mendengar bahwa ia mendengar, orang yang berjalan bahwa ia berjalan dan sama untuk semua kegiatan manusia lainnya ada fakultas yang sadar latihan mereka, sehingga setiap kali kita merasakan, kita sadar bahwa kita merasakan, dan setiap kali kita berpikir, kita sadar bahwa kita berpikir, dan sadar bahwa kita memahami atau berpikir adalah menjadi sadar bahwa kita ada ... "Tentu saja," Aku berpikir, maka aku "adalah jauh lebih ringkas. Muhammad Al-Khawarizmi adalah salah satu direktur pertama dari House of Wisdom di Bagdad. Setelah diawasi terjemahan dari karya penting matematika dan astronomi India dan Yunani, Al-Khwarizmi menjadi advokat untuk adopsi sistem numerik India (1-9 ditambah 0) dan merupakan ayah dari aljabar. Dengan diterbitkannya The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi memperkenalkan menggunakan analisis abstrak dalam pemecahan masalah (meskipun dengan kata-kata, bukan notasi simbolik). Dia juga memperkenalkan metode aljabar mengurangi (menulis ulang ekspresi pernah sederhana, tapi setara, bentuk), serta yang menyeimbangkan (melakukan hal yang sama untuk setiap sisi persamaan - lagi untuk membuatnya lebih sederhana). Program untuk International Student Assessment (PISA) menilai kompetensi dari 15-year-olds di 65 negara dan ekonomi, termasuk dalam matematika. Untuk tahun 2012, negara / ekonomi dengan skor tertinggi dalam matematika adalah Shanghai-China, yang diikuti oleh Singapura, Hong Kong-China, China Taipei dan Korea. Khususnya, Kanada peringkat ke-13, Australia 19, Irlandia 20 dan Inggris 26. Amerika Serikat 'anak-anak peringkat ke-36. Bahkan, menurut PISA, kinerja salah satu negara kami tertinggi mencetak, Massachusetts, sangat rendah, seolah-olah para siswa memiliki dua tahun lebih sedikit pendidikan matematika dibandingkan siswa di Shanghai-China. PISA juga mencatat bahwa meskipun AS menghabiskan lebih per siswa daripada kebanyakan negara, ini tidak diterjemahkan ke dalam kinerja. Pada tahun 2012, per-siswa belanja di AS tercatat di $ 115.000, sedangkan di Republik Slovakia, negara yang dilakukan pada tingkat yang sama, mereka hanya menghabiskan $ 53.000 per siswa. Perlu dicatat dari hasil PISA, meskipun, bahwa mereka yang drastis selama disederhanakan. Misalnya, seperti yang tercantum dalam laporan Dr. Martin Carnoy dari Stanford dan Richard Rothstein dari Economic Policy Institute, mahasiswa Amerika benar-benar melakukan lebih baik daripada jauh lebih tinggi peringkat Finlandia dalam aljabar pada umumnya, tapi lebih buruk dalam pecahan. Selanjutnya, ketika Anda menormalkan hasil antara negara-negara disesuaikan dengan kemiskinan relatif dari siswa mengambil tes PISA, AS melakukan secara signifikan lebih baik, peringkat ke-6 dalam membaca dan 13 dalam matematika, sebuah lompatan besar dalam kategori baik. Mereka lebih perhatikan dalam laporan mereka Apa Pengujian International Sungguh Tampilkan Tentang AS Kinerja Mahasiswa? bahwa ketika Anda membagi anak-anak berdasarkan kekayaan keluarga, kesenjangan yang sebenarnya kinerja tidak begitu mencolok antar negara, dengan porsi yang tidak signifikan dari peringkat tertinggi dari setiap negara yang didasarkan pada berapa banyak yang miskin vs kelas menengah vs siswa kaya mengambil tes. Untuk referensi, sekitar 40% dari sekolah yang PISA digunakan dalam sampel AS memiliki lebih dari 50% dari siswa mereka memenuhi syarat untuk makan siang gratis. Meskipun hasil mereka yang disederhanakan, yang