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