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