Jenis-Jenis Data

Sebelum mempelajari statistik lebih lanjut, terlebih dahulu kita harus mengenal jenis-jenis data.

Berdasarkan cara memperolehnya
1. Data Primer yaitu data yang diperoleh langsung dari objek penelitian.
    contoh : data hasil wawancara/ quosioner
2. Data Sekunder yaitu data yang diperoleh secara tidak langsung dari objek penelitian.
    contoh : penelitian yang menggunakan data dari BPS

Berdasarkan Sebaran Data

1. Data internal, yaitu data yang menggambarkan situasi dan kondisi pada suatu organisasi secara internal.
    contoh : data pegawai
2. Data eksternal, yaitu data yang menggambarkan situasi dan kondisi diluar organisasi.
    contoh : data persebaran penduduk

Berdasarkan Jenis Data
1. Data Kuantitatif, yaitu data yang dipaparkan dalam bentuk angka-angka.
    contoh : data tinggi badan siswa
2. Data Kualitatif, yaitu data yang dipaparkan dalam bentuk kata-kata yang mengandung makna.
    contoh : data persepsi konsumen terhadap kemasan suatu produk.

Berdasarkan Sifat Data

1. Data Diskrit, yaitu data yang nilainya berupa bilangan asli (data yang diperoleh dari hasil perhitungan).
    contoh : nilai rupiah dari waktu ke waktu
2. Data Kontinu, yaitu data yang nilainya ada pada suatu interval tertentu (data yang diperoleh dari hasil pengukuran)
    contoh : data tinggi badan, berat badan.

Berdasarkan Waktu Pengumpulan

1. Data Cross-section, yaitu data yang menunjukkan titik waktu tertentu.
    contoh : laporan keuangan per 31 Desember 2010
2. Data Time-series/berkala, yaitu data yang menggambarkan sesuatu dari waktu ke waktu (periodik).
    contoh : data perkembangan nilai tukar dollar Amerika terhadap rupiah dari tahun 2006-2010

Mathematics is an art

"Mathematics is an art"... 

Do you agree with the statement??

I'm totally agree!!

Matematika adalah suatu seni. Salah seorang dosenku pernah berkata bahwa matematika adalah suatu seni untuk menghilangkan rumus. Jadi adalah salah ketika kita selalu menghubungkan matematika dengan perhitungan yang rumit. Rumus diciptakan untuk mempermudah perhitungan, bukan mempersulit, walaupun terkadang tidak semua permasalahan harus selalu dikerjakan dengan rumus (mengingat keterbatasan daya ingat kita).

Hal pertama yang perlu kita perhatikan dalam menyelesaikan permasalahan matematika adalah menemukan trik yang tepat. Tidak ada gunanya jika kita hafal berbagai macam rumus tapi tidak tahu kapan kita harus menggunakan rumus tersebut, pada kasus apa, manfaatnya apa? Sehingga jangan terburu-buru menghitung soal matematika. Jangan sampai kita sudah bersusah payah menghitung, padahal ada cara yang lebih simpel dengan tingkat ketelitian yang lebih tinggi. Tipsnya adalah : mengawali proses pengerjaan soal matematika (perhitungan) dengan mengamati bentuk soal, sebisa mungkin gunakan cara-cara dasar untuk menyederhanakan perhitungan. Manfaatkan sifat-sifat dasar seperti komutatif, asosiatif, memfaktorkan dan sebagainya untuk mempermudah perhitungan!!

Semoga bermanfaat!!

The Research of Mathematics

Introduction

As we know, the nature of mathematics consist of formal/axiomatic/pure mathematics, applied mathematics, and school mathematics.

1. Formal mathematics/axiomatic mathematics/pure mathematics
Mathematics is a deductive system consists of definitions, axioms, and theorems in which there is no contradiction inside. It is very easy to establish mathematical system step by step like make a definition then use the axiom and theorem, proof the theorem. The substance of formal mathematics such as numbers theory, group theory, ring theory, field theory, Euclidian Geometry, Non Euclidian Geometry, etc.

2. Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. The are a lot of applied mathematics in this word, in optic, mechanic, astronomy, engineering, etc. The most simple example is the use of pythagoras’ theorem to make a right angle (for example when someone wants to make a building,etc).

3. School mathematics/ concret mathematics/ real mathematics.
School mathematics is really different with formal mathematics. In school mathematics we just focus in mathematics phenomenon. We must transform the mathematics phenomenon with its abstraction and idealization to the student’s mindset. So, the student just need to aware about the characteristic of the object. It is need awareness and intention from the student. For example, if given a cube (three dimentional object), in school mathematics we just need to transform about the its shape and the length of its sides in the student’s mindset. We no need to transform about the material that we use to make the cube, the color of the cube, etc. It is not necessary. So, the important think in school mathematics is about how to transform the real object to the abstraction in the student’s mind.
According to Ebbute Straker (1995), school mathematics is about :
- pattern / relationship
- problem solving
- investigation
- communication
To identify mathematics problem we need mathematics knowledge, mathematics system, and mathematics characteristic. The three aspects above can we get easily if we have a will, attitude, knowledge, skill, and experience.
 
Purposes

The aim of the research of mathematics is to examine and develop mathematics. If we want to be a mathematician we must doing the research of mathematics to develop our subject. If we do many research in mathematics, automatically it will improve our knowledge and experience, beside that, we can develop mathematics as well as we can. It is very useful to the future, especially in sciences.

Method

There are a lot of methods that we can use in the research of mathematics, such:
- by analyze the data,
- by collect data/ literature,
- deductive method /syntetic method, etc.
And in this paper, I use some of the methods above that is collect the data (literature) and analyze it.

Discussion

From the three branchs of mathematics above I will focus in more spesific case. I will focus in applied mathematics. According to me, applied mathematics is very useful in the real world. As we know, one of mathematical attitude is applicable. So it means that mathematics be able to be put to practice use (both practical and theoretical aspects).
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis), and applied probability.
In this case, I will give an example about applied mathematics, that is, the application in which first order linear differential equation be a model to determination of the time of death. In the investigation of a homicide or accidental death it is often important to estimate the time of death. So, we need the mathematical way to approach this problem.
From the experimental observations it is known that, to an accuracy satisfactory in many circumstances, the surface temperature of the object changes at a rate proportional to the difference between the temperature of the object and that of the surrounding environtment(the ambient temperature). This is known as Newton’s law of cooling. Thus, if Ɵ(t) is the temperature of the object at time t, and T is the constant ambient temperature, then Ɵ must satisfy the linear differential equation
dƟ/dt=-k(Ɵ-T ) ...........(i)
where k > 0 is a constant of proportionality. The minus sign equation (i) above is due to the fact that if the object is warmer than its surrounding (Ɵ>T), then it will become cooler with time. Thus dƟ/dt <> 0.
Now suppose that at time t = 0 a corpse is discovered, and that its temperature is measured to be Ɵ0. We assume that at the time of death td the body temperature Ɵd had the normal value of 98.6oF for 37oC. If we assume that equation (i) is valid in this situation, then our task is determine td.
The solution of equation (i) subject to the initial condition Ɵ(0)= Ɵ0 is
Ɵ(t) = T +( Ɵ0 - T )e-kt ..........(ii)
However,the cooling rate k that appears in this expression is as yet unknown. We can determine k by making a second measurement of the body’s temperature at some later time t1; suppose that Ɵ= Ɵ1 when t = t1. By substituting these value in equation (ii) we find that
Ɵ1 = T +( Ɵ0 - T )e-kt1
hence k = - 1/t_1 ln (θ_1- T)/(θ_2- T) .......(iii)
where Ɵ0, Ɵ1 , T, and t1 are known quantities.
Finally, to determine td we substitute t = td and Ɵ= Ɵd in equation(ii) and solve for td. We obtain
td = - 1/k ln (θ_d- T)/(θ_0- T) ............(iv)
where k is given by equation (iii).
For example, suppose that the temperature of the corpse is 85 oF when discovered and 74 oF two hours later, and that the ambient temperature is 68 oF. Then, from equation(iii)
k = - 1/2 ln (74-68)/(85-68)≈0,5207 hr^(-1)
and from equation(iv)
td = - 1/0,5207 ln (98.6-68)/(85-68)≈-1,129hr
Thus we conclude that the body was discovered approximately 1hr, 8min after death.
 
Conclusion

From the example above, we should realize that applied mathematics is very useful in our life. The application in which first order linear differential equation be a model to determination of the time of death. In the investigation of a homicide or accidental death it is often important to estimate the time of death. This method is necessary to predict how long the body death (the time of death). It just one example of applied mathematics. In fact (especially the application of first order differential equation), can also be used to analyze a number of other financial situations, including annuities, mortgages, and automobile loans among others.

References

http://kirk.math.twsu.edu/appliedmath.html
http://en.wikipedia.org/wiki/Applied_mathematics
Microsoft® Encarta® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
E.B.William, and C.D.Richard. Elementary Differential Equations and Boundary Value Problems. New York: John Wiley & Sons,Inc. 1997.

ASSIGNMENT 6

PROBLEMS

1. Find the approximation value of √ 5 according to Heron’s method!

2. Determine the Mac Laurin series of function; f(x) = ln(x+1)

3. Find the integration factor of y’=e^2x + y – 1 and solve those differential equation!

4. Find the solution of ∫ (x-7)/(x²-x-12) dx

5. Find the solution of ∫ e^2x sinx dx

ANSWERS

1. Given √ n with n=5, to find the approximation value; at first we have to change n be multiplication form that is n = 2a → a = n/2 so, we find a = 2.5

Then, we must subtitute it to Heron’s formula;

a₁ = (a+2)/2 , so we have a₁ = 2.25

a₂ = (a₁ + (n:a₁))/2 , so we have a₂ = 2.23611

a₃ = (a₂ + (n:a₂))/2 , so we have a₂ = 2.23606

From the calculation above, we can conclude that the approximation value of √ 5 according to Heron’s method is equal to 2.23606.

2. The basic form of Mac Laurin series function is

p(x) = f(0) + f’(0)x + f’’(0)x²/2! + f’’’(0)x³/3! + …+ f ^k (0)x^k/k!

Given a function that is;

f(x) = ln(x+1) then f(0) = 0

f’(x) = 1/ (x+1) then f’(0) = 0

f’’(x) = -1/(x+1)² then f’’(0) = 1

f’’’(x) = 2/(x+1)³ then f’’’(0) = 2

f’^v (x) = -6/(x+1)4 then f’^v (0) = -6

f^k(x)=(-1)^k+1 (k-1)! / (x+1)^k then f ^k(0)=(-1)^k+1 (k-1)!

Substitute those value above to the Mac Laurin basic form series.

So, the Mac Laurin form series of function f(x) = ln(x+1) is equal to

p(x) = x – x²/3 + x³/3 – x⁴/4 + … + (-1)^k+1 x^k/k!

3. Given a differential equation y’= e^2x + y - 1

Step 1 : find the integration factor

Remember the formula; if given y’+p(t)=q(t) the integration factor is µ(t) = exp ∫ p(t) dt

So, y’= e^2x + y – 1 ↔ y’ - y= e^2x – 1

µ(x) = exp ∫ -1 dx = e^-x

We have found the integration factor.

Step 2: find the solution of these differential equation

y’ - y= e^2x – 1

Multiplying each parts of the equation with the integration factor

y’e^-x – ye^-x = e^x – e^-x

[ye^-x ]’ = ∫ (e^x – e^-x) dx

ye^-x = e^x + e^-x+c

y=e^2x + 1 + ce^x

So the solution is y=e^2x + 1 + ce^x

4. Given ∫ (x-7)/(x²-x-12) dx

The form above is one of the partial integration of a function which have different linear factor.

To find the solutions, we must change the function to be partial fraction;

∫ (x - 7)/(x²-x-12) dx = ∫ (x-7)/(x – 4)(x + 3) dx

Then

(x -7)/(x – 4)(x + 3) = (A/(x – 4)) + (B/(x + 3))

↔ (x -7)/(x – 4)(x + 3) = (A(x + 3) + (B(x - 4))/(x – 4)(x + 3)

↔ x - 7 = A(x + 3) + B(x – 4)

↔ x - 7 = Ax + 3A + Bx – 4B

↔ x - 7 = (A + B)x + (3A – 4B)

From the last equation above we find that (A+B) = 1 and (3A - 4B) = -7

With substitution or elimination method we will find that A = -3/7and B = 10/7

So, the solution is:

∫ (x-7)/(x – 4)(x + 3) dx

= ∫ (-3/7)/(x – 4) dx + ∫ (10/7)/(x + 3) dx

= (-3/7) ∫ (x – 4) dx + (10/7) ∫ (x + 3) dx

= (-3/7) ln │x - 4│+ (10/7) ln│x + 3│+ c

5. Remember that d(uv) = u dv + v du ↔ uv = ∫ u dv + ∫ v du

So, we have a formula that ∫ u dv =uv - ∫ v du

Given ∫ e^2x sinx dx = I

Choose;

u = e^2x

du = 2e^2x dx

dv = sin x dx

v = - cos x

Remember the formula above;

∫ u dv = uv - ∫ v du

∫ e^2x sinx dx = -e^2x cos x + 2 ∫ e^2x cos x dx

I = -e^2x cos x + 2 ∫ e^2x cos x dx

Seen that ∫ e^2x cos x dx is have a same form with ∫ e^2x sinx dx.

So, we will do the same procedure;

Choose;

u = e^2x

du = 2e^2x dx

dv = cos x dx

v = sin x

Subtitute to the formula, then we have;

∫ e^2x cos x dx = e^2x sin x - 2 ∫ e^2x sin x dx

Subtitute to the previous equation, we have;

I = -e^2x cos x + 2 ∫ e^2x cos x dx

I = -e^2x cos x + 2 (e^2x sin x - 2 ∫ e^2x sin x dx)

I = -e^2x cos x + 2e^2x sin x - 4 ∫ e^2x sin x dx

I = -e^2x cos x + 2e^2x sin x - 4I + c

5I = -e^2x cos x + 2e^2x sin x + c

5I = (2 sin x – cos x) e^2x + c

I = (1/5) (2 sin x – cos x) e^2x + c

So, the solution of ∫ e^2x sinx dx =(1/5) (2 sin x – cos x) e^2x + c

Book Review

MATHEMATICS FOR JUNIOR HIGH SCHOOL YEAR VIII
By MARSIGIT


PREFACE

First of all, we be grateful to Allah because we still given ability to review Mathematics for School Junior High School year VIII. We would like to thank to Mr. Marsigit who give us opportunity to review Mathemathics For Junior High School year VIII.
There are some benefit that we get while review Mathematics for Junior High School year VIII such as remember to grammar, know more about the different between curriculum mathemathics now and 6 years ago, etc.
We hope this review book is useful. It is intended to students and teachers of Junior High School as an opinion before buy mathematics book.
To release of this review book has been made possible due to the assistance and contributions of various people who cannot mention one by one. To all who involved in this preparation of this review book, I would like to express my high appreciation and gratitude.
Comments and suggestions to improve the contents of this review book are always welcome.

CONTENT

There are 7 chapters in this book. All of them are used to learn Mathemathics for Junior High School year VIII. The material divided into 2 units, that is:
UNIT I : Algebra
Chapters:
a. Algebra and its Applications.
b. The relation and Functions.
c. The equations of a Straight Line.
d. The System of Linear Equations in two variables.
UNIT II : Geometry and Measurement
Chapters:
a. The Pythagorean Theorem.
b. A Circle.
c. Polyhedral.
I think that the content of this book is complete enough. There is a previous in every chapter that make the readers know what will they learn in that chapter. All of the chapters are arranged by some subtitle. It is complete with the definitions, examples (and its problem solving), and more exercise. To remember the material or some formula there is an “be remember” in some page in every chapter. It’s content some formula or material briefly. So, it can help the reader (especially the students) to remember the materials of every study easily.
Beside the exercise of every subtitle, there is an exercise in the end of every chapter that content of some mathematics problems suitable to the material of every chapter. There are 20 multiple choices problems and 5 essays. At the end of every unit, there is an evaluation. This book also be completed with “final evaluation” at the last part. Student must solve 30 multiple choices problems and 10 essays.
Generally, this book is very good and interest for student year VIII. Its content is suitable to the curriculum of study in Junior High School in Indonesia at this moment (suitable to KTSP). The problem in the exercise and evaluations are realistic problems. We can find the problems in our daily activities. So, it makes the student can imagine and solve the problem easier.
The excellence of this books is because it is a bilingual books. So, there are two languages (Indonesian and English Language) in one book. Every Indonesian page translate to English directly. It is very useful, because with reading this books, students not only can improve their mathematics skill and knowledge but also develop their English. I think this book is suitable to some school to be world class school in Indonesia (school with international school standardization).
Over all, the content of this book is complete enough and simple (not too difficult to understand all of material mathematics on it). The point plus of this book is about it bilingual. So, I recommend this book to all of student year VIII in Indonesia. I hope it can help you to learn mathematics easier.


PROBLEM

In this book, there are seven chapter. There are exercise in each chapter. In this book also existed three evaluations : evaluation 1 available after 5 chapter, evaluation 2 available after last two-chapter and evaluation final available in the end. Evaluation was made to examine how far the students understand the substances which are in this book.
In this book there are the problem examples which are clarified in each chapter. The problem examples which are given consist of substances which are stated. In examples is given many examples problem solving which is enable the students to choose the easiest way. there are problem solving. After given examples, there are exercises for students. The form of exercises is varied and still consist all of the substances. Method of problem solving is stated and sequential.


INFORMATION

Book mathematics for Junior High School VIII by Marsigit give complete information about its chapter, so that easy to understand. Reasons information of this book easy to understand are:
a. Each chapter explained in detail that is by sub chapter.
b. Presented with bilingual that is in Indonesian and English language, so English people also can study it.
c. Each sub chapter explained theoretically, given example exercise, and exercise. So, after understand explained fill chapter, then can understand example exercise, so can finish exercise. In this book there are seven chapters, each chapter explained in detail became subchapter.
d. Explained fill chapter not out from studied problem.
e. There are chapter explained with picture (more use picture). For example in chapter six is circle, this book more use picture.
To limiting reader to understand information of this book, depended from reader. But, in general this book present information in detail.


INTEREST

Mathematics for Junior High School year VIII which is written by Mr. Marsigit is different from the other mathematics book commonly. This book is presented in bilingual (Indonesia’s language and English). We can get many advantages by learn this book such as get mathematics knowledge. The students of Junior high school year VIII also can increase vocabullary owing to mathematics.
This book is suitable used to the students or teachers of international school standardization and students which want to continue their studies abroad. On the other hand, this book will inspire them to study better.
Not all people like reading book with black-white color. Some prefer like reading full color book. Mathematics junior high school year VIII as a full book that will stimulate students more interest to study. Talk about the substance, this book explain the substances early. In the end of chapter, there is exercise which consist of 30 multiple choices and 10 essays. The purpose is students can apply what has been him learn. This book has some of ancient marhematician story such Al-Khwarizmi.

My answer to Ika Indriyati

There are some definitions, samples, and explanations about Math (asked by IKA INDRIYATI) that I wrote in English as result by reading the dictionary, reading some of English books, download internet, reading Mr.Marsigit's blog, and also follow Mr.Marsigit's class (English1 subject lesson).

By : ARTIKA KRISTIANINGRUM (NIM. 08305141029)

1. Luas daerah lingkaran dalam : area of an interior circle
Luas : area, extent, broad, scope
Lingkaran : circle
Lingkaran dalam : interior circle ; inscribed circle

2. Pangkat tiga : cubed
Definition : result of multiplying a number by itself twice
Example : 3 cubed is 27

3. Pangkat : (Math) degree, power
Example :empat pangkat lima = four to the 5th power ; dipangkatkan tujuh : raised to the 7th power

3. Teknik pengintegralan : integration tehnique
Definition :
Integration is a important concept in mathematics, specifically in the field of calculus and, more broadly, mathematically analysis.
Given a function f of a real variable x and an interval [a,b] of the real line, the integral is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b.
Teknik : tehnique
Integral : integral
Pengintegralan : integration

4. Limas segiempat beraturan : regular quadrangle pyramid
Definition :
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is conic solid with polygonal base. Pyramid can have from three to a virtual unlimited amount of sides.
Limas : pyramid
Segiempat : quadrangle
Beraturan : regular

5. Prisma tegak : right prism
Definition : Right prism is a prism in which the joining edges and faces are perpendicular to the base faces.
Prisma : prism
Tegak : vertical; right

6. Ekuivalen dengan : equivalent with
Definiton : that has the same function, importance, etc.
Example : x + 1 = 3 equivalent with x = 2

7. Turunan fungsi : derivative function
Turunan : derivative

8. Matrik eselon baris : row echelon matrix

9. Garis bagi sudut : bisector angle
Garis bagi : bisector

10. Sudut sepihak : unilateral angle
Sepihak : unilateral

11. Saling penyiku : complementary
Definition :
a pair of angles are complementary if the sum of their measures is 90 degrees.
Penyiku : complementary

12. Berpotongan tegak lurus dengan bidang : cutting plane perpendicular
Berpotongan : cutting
Tegak lurus : upright, perpendicular
Bidang : plane

13. Sudut di antara dua bidang : dihedral planes

14. Mencari tinggi bangun kerucut : look for (calculate) the altitude of cone
Definition:
Cone is a three-dimensional geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex.
Kerucut : cone
Tinggi; ketinggian : altitude

15. Segiempat tak beraturan : irregular quadrangle
Tak beraturan : irregular
Segiempat : quadrangle

16. Segitiga siku-siku : right-angle triangle
Segitiga : triangle
siku-siku : right

17. Jika dan hanya jika saling berpelurus : if suplementary each other
Berpelurus : suplementary

18. Luas bidang segiempat : area of rectangle

19. Berpikir secara matematika dan logis : think mathematically and logically
Definiton : logic is science or method of organized reasoning (sensible reasoning)
Berpikir : think
Secara matematika : mathematically
Logis : logically

20. Himpunan semesta dari : universal set of
Definition:
Set is a collection of distinct objects, considered as an object in its own right.
Himpunan : set
semesta : universe

21. Tali busur : chord
Busur : arc

22. Juring lingkaran : section of a circle

23. Sudut kelling lingkaran : cricumference angle of a circle
Keliling lingkaran : circumference of a circle

24. Sudut dalam berseberangan : interior alternates angles
Berseberangan : alternate

25. Sudut berlainan pihak : other side of angle
Pihak : side

26. Irisan 2 bidang sejajar : the section of two parallel planes
Irisan : slice; section
sejajar : parallel

27. Menggambar garis asimtot : draw asymptot line
Menggambar : draw

28. Sudut lurus : straight angle

29. Sumbu simetri : axis of symmetry
Sumbu : axis
Symmetry : simetri
Definition :
The axis of symmetry of two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical.
Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image.


I think that's all of my answer to Ika Indriyati's questions.
I hope it can help Ika to learn and communicate mathematics in English.
If there are any mistakes in my answer (opinion), I'm so sorry.
I promise I will study hard to complete and develop my knowledge about mathematic especially in English.
I'm very glad if somebody who read my article can develop his/her knowledge.. and also can correction of my article ( thank you so much).

My reference :
Glosarium Matematika by Djati Kerami and Ellya Iswati.
http://en.wikipedia.org/wiki/

Introduction to English 1

First, let me to introduce myself. My complete name is Artika Kristianingrum, but just call me Tika. now, I'm a student of Mathematics program at Yogyakarta State University. February 17th, 2009 was my first time to have an English lecture. Dr.Marsigit was the name of my English lecturer. He was very friendly and humorist person. After the introduction, Mr. Marsigit taught us about "communicate Mathematics in English"...

Now,I will retell you about it.......

Our principal purpose in study English at Mathematics program is to know "How to communicate mathematics in English?"
So, to reach our purpose above, we must do some steps;
1. High spirit and motivation;
Have high spirit and motivation are difficult things, but it can be easier if we usually pray to God and make a god perception in our life.
2. God behaviour and attitude;
to support our purpose, we must have a good behaviour and good attitude.
3. Knowledge;
With our knowledge (of communication, of technology, of English, of Mathematics), we know how to learn effectively, we know the tools that useful to learn mathematics, etc.
4. Develop our skill to commnicate mathematics in English;
With our skill ( to hear, to read, to write, to speak), we will have many experience such as international conference and debate.
5. Use our knowledge to others;
Our life will be valuable if we can use our knowledge to others.
6. Have a good role in international networking.

If we can do the six steps above, Mr.Marsigit sure that we can be a professional and we will reach our principal purpose. Be a good mathematician!!! The next English lecture ( February 24th, 2009)As adult, we should have responsibility and be an independent learner to find out the reference about our lecture. There are many resources of mathematics that we can access.

What really the meaning of mathematics?

Who have a right to define mathematics?

People who have high motivation coming from,
People who have high spirit comng from,
People who have good understanding of mathematics coming from,
People who have skill coming from,........
They have a difference define about mathematic.
So, don't be worry to express our idea of mathematics.


According to Prof.Katagiri (Tokyo, Japan) : if we talk about math, we can talk about math thinking ;
that is ;
1. math attitude,
2. math method,
3. math content,
The three components related each others.
We can't have math content without math method although we have a good attitude.

And the really meaning of mathematics is.........

Math is part of our life ;

Math is ourself ;

Math is our mind