How to find the Purpose of Life?

Sekedar berbagi ilmu dari seorang dosen tentang bagaimana mencari tujuan hidup. Ini salah satu catatan lama yang mungkin berguna bagi yang membutuhkan. Bagi yang masih belum menemukan tujuan hidupnya semoga tulisan sedikit ini dapat membantu.

Untuk memperoleh tujuan hidup yang sesuai, pertama kita harus mengetahui terlebih dahulu apa kelebihan dan kekurangan kita. Hal ini dapat menunjukan seberapa jauh kita mengenal diri kita sendiri. Kemudian coba deh susun daftar, apa yang ingin/dapat kita lakukan dalam jangka waktu 1 jam, 1 minggu, 1 bulan, dan seterusnya.

Setelah itu coba pikirkan, motif dominan kita itu apa? Apakah kekuasaan, persahabatan, atau prestasi? ciri-ciri tiap orang dengan motif dominan yang berbeda itu juga beda loh...misalnya orang yang motif dominannya persahabatan. Dia cenderung untuk berkumpul bersama teman2, supel, pendengar yang baik, dll. Kemudian orang yang motif dominannya kekuasaan, orang dengan motif ini cenderung pandai bicara, egois, suka mengatur, dll.

Meskipun motif kekuasaan mendapat pandangan negatif yang cukup besar, tak selamanya seperti itu. Seorang pemimpin yang baik, pasti bisa mengatur anak buahnya dengan baik kan? Selanjutnya orang yang motif dominannya prestasi, orang dengan motif ini cenderung untuk serius, punya rasa tanggung jawab yang besar, pandai mengatur waktu, dsb.

Nah sekarang coba pikirkan motif dominan apa yang ada dalam diri anda?

Number Sense

Have you ever heard about number sense? What's the meaning and how to use it?

Karena sedikit artikel yang mengulasnya dalam bahasa indonesia. Kali ini saya menggunakan bahasa indonesia, by the way saat ini saya dalam tahap peningkatan kemampuan bahasa inggris di sebuah lembaga. Semoga nanti membawa manfaat.

Number sense merupakan sebuah intuisi yang baik mengenai bilangan dan hal yang terkait dengan bilangan. Hal ini dapat didefinisikan secara luas sebagai pemahaman makna bilangan dan pemahaman hubungan anratbilangan (Malofeeva, Day, Saco, Young, & Ciancio, 2004). Kemampuan number sense meliputi kemampuan mengenal bilangan, mengidentifikasi nilai bilangan dan memahami bagaimana mengaplikasikannya dalam berbagai macam cara, seperti berhitung, perhitungan mental, pengukuran atau estimasi/penaksiran. (http://surfnetparents.com/math_glasarry.html).

Number sense dihasilkan oleh suatu approach/pendekatan yang dikembangkan di Belanda sejak tahun 1970-an; yaotu RME (Realistic Mathematics Education) oleh Institut Freudenthal. matematika sebagai aktivitas manusia berarti manusia harus diberi kesempatan untuk menemukan kembali ide dan konsep matematika dengan bimbingan orang dewasa (Gravemeijer, 1994). Upaya ini dilakukan melalui penjelajahan berbagai situasi dan persoalan realistik. Realistik dalam hal ini dimaksudkan tidak mengacu pada realitas tetapi pada sesuatu yang dibayangkan siswa. Prinsip penemuan kembali dapat diinspirasi oleh prosedur-prosedur pemecahan informal, sedangkan proses penemuan kembali menggunakan konsep matematisasi.

Number sense merupakan kesadaran akan jumlah, memberi makna pada bilangan dan mampu menghubungkannya dengan makna yang berbeda. Siswa menemukan cara yang lebih mudah dalam menghadapi suatu bilangan, dan dapat mengerti bahwa sebuah bilangan dapat direpresentasikan dengan cara yang berbeda.

Terdapat dua model dimensi number sense yaitu kemampuan dasar (Basic skill) dan aritmatika konvensional (Conventional arithmetic). Jordan, kaplan, Olah, dan Locuniak (2006) menggambarkan bahwa dimensi kemampuan dasar meliputi kemampuan berhitung, mengenali bilangan, mengetahui bilangan, perhitungan nonverbal, penaksiran, dan pola bilangan. Dimensi yang lain yaitu atirmatika konvensional meliputi, kombinasi bilangan dan soal cerita. Aspek penting tentang number sense dinyatakan oleh Jordan dan rekan-rekannya yang dirangkum dalam 5 elemen kunci number sense anak (Jordan, et al, 2006 : 154).

1. Berhitung : mengetahui korespondensi satu-satu, urutan yang stabil, pengetahuan tentang prinsip kardinalitas dan mengetahui barisan berhitung.
2. Pengetahuan bilangan : mendeskripsikan dan mengkoordinasikan besaran dan membuat perbandingan besaran bilangan.
3. Transformasi bilangan : transformasi bilangan dengan penjumlahan, pengurangan dan perhitungan dalam konteks verbal dan nonverbal, perhitungan dengan atau tanpa bantuan (fisik atau verbal).
4. Penaksiran : memperkirakan besarnya himpunan dan menggunakan referensi titik.
5. Pola bilangan : meniru pola bilangan, memperbesar pola bilangan dan melihat hubungan antarbilangan.

Referensi :
Malofeeva, E., J., Saco, X., Young L., & Ciancio, D. 2004. Construction and Evaluation of A Number Sense Test With Head Start children. Jurnal Pendidikan Psikologi, 96(4), 648-659.

Gravemeijer, K. P. E. 1994. Developing Realistic Mathematics Education. Netherlands. Utrecht : Freudenthal institute.

Jordan, N. C., Kaplan, D., Olah, A. N., & Locuniak, M. N. 2006. Number Sense Growth in Kindergarten: A Longitudinal Investigation of Children At Risk For Mathematics Difficulties. Jurnal Penelitian Perkembangan Anak, 77(1), 153-175.

The Foundations of Mathematics

What is the foundations of mathematics?
when i look at the foundations of mathematics books 1st edition (by Thomas Q. Sibley). The first chapter is about laguange, logic and set. The second is about proof.
If the foundations is like that, what's the reason?
I though, foundation is something that make it strong and enough to grew up. The principals are quite similar with the building of a house and all the materials support it to stand up. Language, logic, set, proof is essential part. We can't share mathematics idea without language. Logic is about studying the correct reason, it's true or false. Mathematics investigations is just about true or false, so i think it's relevan, since it was developed a long time ago by Aristotles. Set is the most fundamental concept in mathematics, it's about how to collect a distinct objects. It's also relevan. At the end, i agree with Mr. Sibley.

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.
From the philosophy it self. There are three school that very influent the mathematicians and philosophers at the first time.

1. Platonism: "Platonist, such as Kurt Godel, hold that numbers are abstrack, necessarily existing objects, independent of the human mind."
2. Formalism : "Formalists, such as David Hilbert, hold that mathematics is no more or less than mathematical language. It is simply a series of games..."
3. Intuitionism : "Intuitionist, such us L. E. J. Brouwer, hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them."
   

Reference :
http://en.wikipedia.org/wiki/Foundations_of_mathematics

Mathematics Discoveries

In this time, there are three points that i want to write in this article:
1. The old mathematics discoveries which is still exist today.
2. The old mathematics discoveries which isn't exist today.
3. The new mathematics discoveries which unrelated to old mathematics.
The explanations is below :

1. The Old Mathematics discoveries Which is Still Exist Today

SEMPOA

Why i take sempoa as one of mathematics discoveries? Since it's calculations tool,i thought this is a great discoveries.

Sempoa is one of old calculations tool which is still exist today. Although there were so many invenstions in calculations tool like calculators and computers with many series, but sempoa still has their existance.

Where sempoa comes from is unknow because sempoa used by many cultures in the world. Some researchers said it was on Babylonia and Tiongkok about 2400 SM and 300 SM. In English, sempoa is well-know as abacus. This word used since 1387, comes from latin "abakos" which is taken from greek "abax" means calculation tables.

Why sempoa still exist today?
I used to watch a live program in one of tv station about someone who expert using sempoa. There were three persons that gave calculation problems with 12 numbers, example :
278673589835 + 675548770098 + 227648652202 = ??

One person using sempoa while others calculate by arithmetic algorithm. Everything goes right but there something strange with the person who use sempoa, he did't use sempoa at all. He just move his thumb and index finger to calculates the problem on the board. He calculate faster than the others and the answer is right. I think it awesome, since he can do it well and defeated others. It's one of the effect of something called "Image Training" i though. Since sempoa in his mind with perfect shape and he can use it anytime. In this case is by using sempoa in a long time and slowly the sempoa is escape phisicaly, thus sempoa is just imaginations.

Perhaps, this is one of the reason why sempoa still exist today. Honestly i love sempoa too when i was a child, but by the time goes on i'm not expert on it again. Hm... I need more practice. What about you? Do you have ever use sempoa? In Indonesia itself thera so many sempoa clubs, one of the famous is ASMA (Adil Sempoa Mandiri) who win Sempoa National Olympics on August 24 2008.

2. The Old Mathematics Discoveries which isn't Exist Today

I determine to take the contradiction of Euclid fifth's Postulate, I hope it's quite relevan to discussed.

Euclids fifth's Postulate / Parallel Postulate

"If two lines are intersect by a transversal in such a way that the sum of the degree measures of the two interior angels on one side of the transversal is less than 180 degrees, then the two lines meet on that side of the transversal."

Some mathematicians thought that this is not a postulate, since it can proved by another four postulates. There are some mathematicians who has some effort to prove Euclid's postulate. This work begin when Euclid still alive until 1820. Some of them are:

1. Proclus from Alexandria (410-485)
2. Girolamo Saccheri from Italia (1607-1733)
3. karl Frederich Gauss from Jermany (1777-1855)
4. Wolfgang(Farkas) Boylai from Hungaria (1775-1856) and His son Yanos Boylai (1802-1860)
5. Nicolai Ivanoviteh Lobacevsky (1793-1856)

None of them can prove it, :) Euclid is brilliant. But their effort wasn't useless, because it can find another geometry which called Non-Euclides Geometry.

Since this postulate is difficult to understand, there are some mathematicians who tried to change with another equal postulate. Here are some of them :

• Proclus' Axiom : "If a line intersect one of two parallel lines, both of which are coplanar with the original line, then it must intersect the other also".
• Playfair's Axiom : "Through any point in space, there is at most one straight line parallel to a given straight line".

At the end, what happen with the fifth postulate's Euclid? The discovery of non-Euclidean geometry finally allowed mathematicians to open up their study of this familiar round surface along with other, even stranger, surfaces. Then the expansion of Euclid's axioms led the way to examine what might happen if they were to be eliminated or change.


3. The New Mathematics Discoveries Which Unrelated to Old Mathematics

After a long time, at the end, I couldn't found mathematics discoveries which unrelated with the past discoveries. The new theory is developed from the previous axiom or formulas, for example Jean Christopher Yoccoz who recieved the greatest honour for his work on dynamical systems when he was awarded a Fields Medal, the theory of dynamical systems really began with Poincare who was studying the stability of the solar system. Maria Gaetana Agnesi began working on her most important work, Analytical Institutions, dealing with differential and integral calculus.

During the 20th century, there are so many development on mathematics and at the same time, deep discoveries were made about the limitations to mathematics, it was discovered the truth or falsity of all statement formulated. Some of the most profound discoveries is :

1. In a 1990 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics.
2. In 1976, Wolfgang Haken and Kenneth Apple used a computer to prove the four color theorem.
3. In 1931, Kurt Godel found that this was not the case for natural numbers plus both addition and multiplication: this system, know as Peano arithmetic, was in fact incompletable.

There are also new antry areas of mathematics such as mathematical logic, topology, complexity theory, and game theory.

Reference :
http://id.wikipedia.org/wiki/Sempoa
http://turnbull.mcs.st-and.ac.uk/history/Timelines/TimelineG.html
http://www.mathworld.wolfram.com/PlayfairsAxiom.html
http://www.mathworld.wolfram.com/ProclusAxiom.html
http://en.wikipedia.org/wiki/History_of_mathematicians#20th_century
Moeharti Hw, 1986, Modul Materi Pokok : Sistem-Sistem Geometri, Publisher :karunika Jakarta Open University, Jakarta.