THE POWER OF CATEGORY AND NETWORKING

In the 1771, Immanuel Kant was explain four category of thinking.
1.Category
2.Quality
3.Quantity
4.Relation

In Kant's philosophy, a category is a pure concept of the understanding. A Kantian category is a characteristic of the appearance of any object in general, before it has been experienced. A category is an attribute, property, quality, or characteristic that can be predicated of a thing.
“The Categories do not provide knowledge of individual, particular objects. Any object, however, must have Categories as its characteristics if it is to be an object of experience. It is presupposed or assumed that anything that is a specific object must possess Categories as its properties because Categories are predicates of an object in general. An object in general does not have all of the Categories as predicates at one time. For example, a general object cannot have the qualitative Categories of reality and negation at the same time. Similarly, an object in general cannot have both unity and plurality as quantitative predicates at once. The Categories of Modality exclude each other. Therefore, a general object cannot simultaneously have the Categories of possibility/impossibility and existence/non–existence as qualities.”
(http://en.wikipedia.org/w/index.php?title=Category_%28Kant%29&action=edit)
A quality is an attribute or a property. Attributes are ascribable, by a subject, whereas properties are possessible. Some philosophers assert that a quality cannot be defined. In contemporary philosophy, the idea of qualities and especially how to distinguish certain kinds of qualities from one another remains controversial. (http://en.wikipedia.org/w/index.php?title=Quality_%28philosophy%29&action=edit)
In the wikipedia, we can find that,
“Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity. Being a fundamental term, quantity is used to refer to any type of quantitative properties or attributes of things. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. One form of much, muchly is used to say that something is likely to happen. A small quantity is sometimes referred to as a quantulum.”
(http://en.wikipedia.org/w/index.php?title=Quantity&action=edit)
Another information is said that,
“A Relation of Ideas, in the Human sense, is the type of knowledge that can be characterized as arising out of pure conceptual thought and logical operations (in contrast to a Matter of Fact). In a Kantian philosophy, it is equivalent to the analytic a priori. It is also closely coincident with the so-called Truths of Reason of Leibniz, which are defined as those statements whose denials are self-contradictory.”
(http://en.wikipedia.org/w/index.php?title=Relation_of_Ideas&action=edit)

Person’s attention always begin by awareness. Sourced by Wikipedia,
“Awareness is the state or ability to perceive, to feel, or to be conscious of events, objects or sensory patterns. In this level of consciousness, sense data can be confirmed by an observer without necessarily implying understanding. More broadly, it is the state or quality of being aware of something. In biological psychology, awareness is defined as a human's or an animal's perception and cognitive reaction to a condition or event.” (http://en.wikipedia.org/w/index.php?title=Awareness&action=edit)
When we notice at mathematics, we will find that there are two kind objects.
1. Abstraction
2. Idealisation
Abstraction is about shape and measurements, then idealisation is assuming that everything is perfect.
Covert awareness is the knowledge of something without knowing it. A philosopher, Edmund Husserl said that in the Hermeneutics Phenomenology, the truly knowledge is not mind invention, but an existence of awareness. From this argument, we can take abstraction and idealisation into a place, called as Epoche.
Awareness forms a basic concept of the theory and practice of Gestalt therapy. The theory of Gestalt is according to deduction method. So, its possible to grows bottom-up and top-down thinking.
Katagiri define three categories of mathematical thinking:
1. Mathematical Attitudes
2. Mathematical Thinking Related to Mathematical Methods
3. Mathematical Thinking Related to Mathematical Contents
Then, there are three categories of the nature of school mathematics:
1. Pattern
2. Problem Solving
3. Investigation
4. Communication
When we analyse student’s mathematical thinking in the framework of the nature of school mathematics, we can find that there are some agreements of them.
The nature of school mathematics
pattern Problem solving Investigation Communication
Attitudes    
Methods    
contents    

This activity show that there is a network between students mathematical thinking and the nature of school mathematics. This activity also proof the power of category and networking in the mathematics.

REFERENCES
http://en.wikipedia.org/w/index.php?title=Category_%28Kant%29&action=edit (Accessed 14th Dec ’09, 10:21)
http://en.wikipedia.org/w/index.php?title=Quality_%28philosophy%29&action=edit (Accessed 14th Dec ’09, 10:21)
http://en.wikipedia.org/w/index.php?title=Quantity&action=edit (Accessed 14th Dec ’09, 10:21)
http://en.wikipedia.org/w/index.php?title=Relation_of_Ideas&action=edit(Accessed 14th Dec ’09, 10:21)
http://en.wikipedia.org/w/index.php?title=Awareness&action=edit(Accessed 14th Dec ’09, 10:21)
http://pbmmatmarsigit.blogspot.com/ (Accessed: 10/27/09)
http://marsigitpsiko.blogspot.com/2008/12/psikologi-siswa-belajar- maematika.html Accessed: 10/12/09)

SMALL RESEARCH: Mathematical Thinking of Students Learn Mathematics

A. BACKGROUND
Generally, people think that mathematics is a very difficult subject. How come????
In the first, every child likes mathematics indeed. Let we look out the kindergarten student. They look very enthusiast when their teacher introduce about basic geometry shapes like a circle, triangle, rectangle, square, rhombus, kite, or even three dimensional shape like a ball, cube, pyramid, and another geometry shapes.
The other example is when their teacher ask them to learn the number one to ten, we can see that they are very antucias to spell and create the shape.
But, when the learning goes to the higher levels, they start to judge that mathematics is difficult subject.

B. BASIC THEORIES
Before we talk about ”mathematical thinking”, I think we have to review about what is the meaning of mathematics.
In the 2nd semester of English class in the mathematics education at Yogyakarta State University, Dr. Marsigit said that mathematics always has a lot of meaning. There is different types/characteristics between school-mathematics and university-mathematics. In the university, we learn pure an applied mathematics. Mathematics is the body of knowledge and mathematics is the king of knowledge.
Then, what is the meaning of mathematical thinking???
Shikgeo Katagiri (2004) said that Mathematical thinking is like an attitude, as in it can be expressed as a state of “attempting to do” or “working to do” something. It is not limited to results represented by actions, as in “the ability to do”, or “could do”, or “couldn’t do” something (http://pbmmatmarsigit.blogspot.com/)
Katagiri define that there are three types of mathematical thinking:
1. Mathematical Attitudes
2. Mathematical Thinking Related to Mathematical Methods
3. Mathematical Thinking Related to Mathematical Contents
Mathematical attitudes means always have questions (curious), consistent, critical, and regard.
Then there are a lot of mathematical methods:
 Deduction (general to specific)
 Induction
 Incomplete induction
 Syllogism
 Logic
 Proof (direct/indirect proof)
The contents of mathematics are that objects. There are two objects of mathematics:
1. Abstraction
2. Idealisation
Abstraction is about shape and measurement. Then, idealisation means to assume that everything is perfect.

C. THE RESEARCH
Subject : Caca Handika
Grade : 2nd Grade of Elementary School
Problem :

When Caca start to introduce into multiplication operation, he got some difficulties. He couldn't imagine how to calculate
Solving : His father taught him by using some marbles. His father asked him to move 5 marbles into another dish as 4 times. Then, he asked his son to count the marbles.
Finally, Caca start to count from 1-5, then 6-10, then 11-15, and 16-20.

D. ANALYSIS
For 2nd grade of elementary school student, multiplication is assumed as difficult thing. Without using some daily life models, mathematics learning become more difficult.
In this case, Caca’s father just directed his son to find the solution by using the marbles, then the constructor is Caca. By counting 1-5, then 6-10, then 11-15, and 16-20, the student (Caca) is admitted to construct his mathematical thinking. By using the marbles as the models, Caca become more antucias to learn.
Constructivist activity in the mathematics is famous as “contextual method”. Here, the students allowed to construct their think by using their logic.
In this case, three types of mathematical thinking by Katagiri can be applied.
1. Mathematical Attitudes
In this case, the mathematical attitudes can be shown by the enthusiasm, curiosity, and motivation.
2. Mathematical Methods
In this case, Caca was introduced into contextual method by constructing his mathematical thinking by himself.
3. Mathematical Contents
The contents of mathematics are that objects. In this case, the object is multiplication operation.

E. CLOSING
From those research, we can learn that using the right method will makes the students more understand the materies. Beside that, by using some models, the students become more enthusiasm and the teaching-learning activity’s aim can be reached.
The other conclusion is we can find the fact that in the mathematics teaching-learning activities, students always use mathematical thinking.

F. REFERENCES
http://pbmmatmarsigit.blogspot.com/
http://marsigitpsiko.blogspot.com/2008/12/psikologi-siswa-belajar- maematika.html

HOW TO UNCOVER PSYCHOLOGICAL PHENOMENA

Do you ever heard the word “traumatic”???
What do you think about ”it”????
Traumatic is undefined term communication problem caused by disorderly. Communication is characterized by translation. There are a sender, a receiver, and the constraint. It’s very possible that misunderstanding will be appear around communication process. So that, traumatic can be solved by communication.
When we talk about translation, indirectly, we talk about hermeneutics theory. In the hermeneutics, dynamic spiral are flexible and contextual. Flexible means there are a motif, a subject, and an object. Whereas, contextual means match to the time and to the dimension.
Educational facts show that there is a relation between “traumatic” and “the students”. Recently, teacher always become the subject of education then student become the object. Teacher always build traumatic around the students by giving formula without those implementation. So, we have to change those paradigma and we have to revitalize that assumption.
The wrong transfered concept often break student’s assumption in learning mathematics. Aperseption appear from the sense. Aperseption forming a sensation, then its forming perception. Finally,the rotation is aperseption--->sensation---> perception---> concept. Concept will be transfer to our student. So, we can change the student’s assumption by concept that will be explain.
As we have known, most of students will be “trauma” when heard the word “mathematics”. They think that mathematics is something terrible.
Mathematics is sudent’s mind. When we think it’s dfficult, it will be difficult. But, if we think it’s interesting and challenging, we will enjoy to learn mathematics.
So, the real challenge is: “How to erase the student’s trauma???”
As we talk before, traumatic can be solved by communication. So, with the good communication between the teacher and the student, the learning mathematics trauma will be disappear.

Mathematical Thinking and Scientific Works

What do you know about mathematical thinking and scientific work????

What do you know about scientific paper, mathematics research, electric jurnal, and arrange some books??

Right!!! They are the examples of scientific works. There are unlimited scientific works and there are some definitions in learning mathematics. School mathematics always explain to students that math is a such pattern, problem solving, investigation activity, and mathematics mean communication. In the other hand, pure math more formal and has characteristics. It’s has acsiomatics. According to the mathematicians the scient of acsiomatics thinking establish by deductive methode, consisting of concept,definition, theorem, acsioma, procedore to proof thorem. Math is a system. To establish mathematics system should has assumtion, usually can be a concept/definition. Mathematics has a grond, there are definition and acsiom.

Before define something we need to have a clear picture about the conceptÞobjectsÞideas (in our mind)Þfrom existance and the possibility.

Ideas appear in our mind spontantly. It’s abstract, can’t be touch, can’t be manipulated. Books is a sample of concrete object. It’s can be touch and can be bring.

To get mathematical thinking, we need idealisation and assumtion.

Idealisation; assumtion to get absolutely object.

Assumtion very important in mathematics.

Abstraction; just to observe a certain characteristics

Let we observe a paper. There are a lot of characteristics of paper. It’s fluent and etc. But in mathematics, we just observe the shape and the size.

Mathematical thinking always consistant, allowed the first rules to meet the procedure or the princip. Mathematical thinking also logic, it’s build up from daily logic and formal logic.

Let we observe the number of six and seven, it’s can differentciate, which is big and small. We get the value that six is less tahan seven. This sample talk about order and relationship. So, the nature of mathematics is relationship or join.

Math operation use operated like plus, minus, and etc. It’s used like at arithmatics operation, addition, substraction, division, power, exponent, multiplication, and etc.

The other form is, if…then…statement.

The sample of mathematics sentence is proposition. How to conclude?

premisÞthesisÞ antithesis (contradictory) and then hipethesis.

Scientific work have to impersonal. The second rule is has a standard/criteria. For example, reference: bibliography of scientific paper. It’s has to allowed etical code and free from plagiarisme.

So, let we start to be mathematical thinking and produce more scientific works….

Definition, Theorem, Rules, Example, Application, and It's problem

ROM I

1. To proof that the square roots of two is an irrational number, first we must assume that the square roots of two is a rational number.
2. To show/indicate that the sum angle’s of triangle is equal to one hundred and eighty degree, first, we can make a straight line. Then, sketch a parallel line that equivalent with one of triangle side above that straight line. The next step, sketch the other side at the line and make sure that the line intersect both of the line. From this sketch, we can see that this picture build up angles that equal to the vertex of triangle. As we know that the sum angle of straight line always one hundred and eighty degree, so we can conclude that the sum angle of triangle is one hundred and eighty degree.
3. How we are able to get phi????

To proof how we get the value of phi, we can use the perimeter of a circle. As we know that the formula to find the perimeter of the circle is two times phi times the radius of the circle.

Prepare a wire with then makes a circle shape with define radius. Stretch that wire and observe the length. From this problem, we got the length as the perimeter of circle and that radius, so we can easily get the value of phi by dividing the perimeter by two times that radius.

4. To find the area of region bounded by the graph of y equal x square and y equal to x plus two:

The first step is sketch the graph. Then find the intersection points both of the graph. That intersection point can be the bound of the area. Use integration to solve this problem.

We get the intersection point by combine both of the equation, x square equal to w plus two and we get the bound that x equal negative one and x equal two. So, the integration is define integral of x plus two minus x square dx from x equal negative one to x equal two. From the integration, we get the value of nine second or four point five, so the area of the region is four point five unit of region.

5. How we are able to determine the intersection point between the circle x square plus y square equal to twenty and y equal to x plus one.

To make us more easy, we can sketch the graph before. So, we can see that the line intersect the circle at two points.

The next step is combine both of the equation by substitution methode. And at the last, we have an equation two x square plus two x minus nineteen equal to oh. Find the value of x and then substitute at the equation and we’ll get the value of y too. So, the intersection points, is x point y.

4.

5.

ROM II

• A Kite

Everybody have ever seen a kite. But, what do you know about mathematics kite?

Based on definition, a kite is a four-sided which one of its diagonal joining to the other diagonal axis. A theorem said that if a quadrangular ABCD is a kite and diagonal AC joining to the axis of diagonal BD, then the line AB congruent with AD and the line CB congruent with CD.This theorem shows that the diagonals of a kite have perpendicular intersection. The line AC cutting the segmebt BD and make the right angel. It’s shows that a kite has one folded-simetry and the line AC as the axis.

The other theorem said that if a kite which the close angels are straightly, then it’s called a rhombus. From this theorem, we know that to find the region of a kite has the same rules to find rhombus area. We must times a half with the length of the first diagonal and then times with the second diagonal.

• Factorisation

When we talk bout factorisation, it’ll remind us with the prime numbers, the least common multiple, and the great common multiple.

A theorem said that every whole numbers which more than one are devided by a prime number, so every positive whole number which more than one is a prime number or those number are multiplication from some prime numbers.

The singular factorisation theorem said that factorisation of a positive whole numbers which more than one from the prime number is singular, except the rotation of that factors. This theorem become the base theorem at arithmatics.

An interesting theorem from Euclid said if the quantity of the prime number unlimited. The interesting point which located at the proof of this theorem is the forming of N positive whole number as the result of multiplication all prime numbers plus one.

• Permutation

One problem which have to thoght and evaluated by statisticians is the influence of probable factor which connected to some cases. This problem included to the branch of mathematics called as probability theory.

Sometimes we have a population which has a sample. For exampler, if we’ll find how many formation that probable formed by six persons to seat round a rond table or how probable if we take two lottery-tickets from twenty tickets. The different formation is called permutation.

Permutation is a formation which formed by all or a part from gathering things. The sum of permutation ‘n’ different things is n factorial. The sum of permutation caused by the take ‘r’ things from ‘n’ different things is n factorial over n minus r in bracket factorial. And the sum of permutation ‘n’ different things formed at a round is n minus one in bracket factorial. This kind called round permutation. It’s usually used at the bridge games.

Reflection of Video

As the agent of change, we have to learn more and more. We know that study not only transfer of knowledge from the teacher to the students, but also can be done in the other ways. A film / a video can be learning alternative media. Students have to analyze and then reflect in their daily lifes. There are some videos that I’ve seen.

Ø Video I

This film talking about how the teacher motivate his students. The teacher tell themabout W. Shakespeare, someone who write an interesting story. He talk very exciting and takes the students attention.

He jump on the table. He said that we don’t have to be afraid to make some mistakes. We must try to find our own voice. We must begin and find new ground. We must break out! We must look at something in different way. In the last part, his action has done by his students.

Ø Video II

This video grounded at the stage round by a lot of audiens. The host let a little boys to raise on the stage. The children ask the audiens whether they believe at him again and again. He said that he do anything, because anything, cause they believe at him. He is very confident.

Ø Video III

It is an exciting video. This scene offered in a rap music by two college students. It’s shows the high spirits of mathematicians. “What you know about math?” This question is given for us. What we know about math? A significant figure, limit, trigonometry, math B, curves, exponent to decline, integral, square and cubic equation, plus, minus, multiplycation, and overing (+ - x ¸).

“What you know about math?” I know all about math….

Ø Video IV

This video shows how to solve the differential equation.

First, we must find y =f (x), satisfies the equation for values x & y.

Dy = 4x dx, integrate

, try to get dependent variable y all by itself

dy = 4x² dx , integrate the whole equation, integrate

, represent the infinite family of solution curve

, infinite of identical curve

Ø Video V

There are some problem study in a linear equation.

1) X-5=3

To solve this problem, first step is makes one part become 0 by adding the left part and the right one with -3. So, the pattern being X-5-3=3-3. This form more simple in X-8=0. So, the value of X=8.

2) 7=4a-1

First we must add both of parts by 1, and the pattern is 7+1=4a-1+1. The simple pattern: 8=4a. The right part is multiplication, so the next step is over 8 by 4 or multiply both of the part by ¹/₄, and we get the value of a=2.

3) ²/₃X=8

First, we have to change the part with the x variable has the coefficient of 1 by multiply both of the parts with ³/₂. So, x=8׳/₂ , and we get the value of X=12.

4) 5-2x=3x+1

Add both of the part by -3x. The form being 5-5x=1

Add both of the part by -5, so the pattern become -5x=-4

Thew next step is multiply both of the part by -¹/_5.

-¹/₅ (-5x)= -¹/₅ (-4), so we get that x=⁴/₅.

5) 3-5(2m-5)=-2

For this problem, use the same rules with the problems before. Combines the number in front of the bracket.

3-10m+25=-2, use the same rules and we get 30=10m. So, m=3. The problem is solved.

Ø Video VI

This video show how to find the logarithm pattern by the simple pattern before. We know that log_xA=B is equal to x^B=A. And C(log_xA)=B×C is equal to x^BC=A^C , so log_x(A^C)=BC.

We also know that C×log_xA=log_xA^C and log_xA+log_xB=log_xA×B

So, if log_xA=l => x^l = A

log_x B=m => x^m=B

and log_x A/B=n => xⁿ =A/B = x^l/xⁿ = x^l-n

and we get the new logfarithm is log_xA/B= log_xA-log_xB

My Reflection in Learning English

Every body always has a certain ability. It is important for us to know our movement in learning something. Is it increase or decrease? There are a lot of ways to check our ability. It can helps us to find the smartest way to overcome them.

Reflection being one of them. If we heard the world reflection, we’ll always imagine a mirror. We’ll see our silhouette. We will see the best part and also the worst part. And then we will keep the best and correct the worst one.

In learning English, reflection is the smart way to developt our ability. There is an excited story related to reflection. Yesterday, when my class, my mathematics class get a reflection test, when we have to mention mathematics words in English, we got a lot of difficulty. Even, we heard and we used those terminology, its hard for us to said them in English. There was a small number of students could answer more than a half. Its so terrible I think. So, I hope that the test can realizang us, to learn more and more, not only in Indonesian language, but also in English one.

If we comparing our nation to the other one, we’ll see, how left we are!!

So, its important for us to reflect ourself to the other nation, we can learn from them, put the good things and leave the bad one. So, I hope, one by one, step by step, we’ll move in the front formation in the world. Keep our spirit and we’ll be the best…