My Small Research on the Psychology of Teaching Learning Mathematics

Observation’s result about Mathematical Method, Mathematical Attitude, And Mathematical Content Hold by Katagiri’s version, in teaching Learning Mathematic about Learning Area of Triangle, Rectangular, and persegi to Grade IV of Elementary School Year 2009.
Observer: Ety Antikasari

Prof. Katagiri from Japan when his presentation in Sapporo-Tokyo in said that mathematics is mathematical thinking consist three aspects, and I conclude that:
1. Mathematical attitude.
It is very important in determining student’s behavior in mathematical thinking. A good attitude is always asks to be perfection. Mathematical attitude is very important affective factor in determining student’s behavior in mathematical thinking depend on how interested, they are in problem solving on the lesson. Student’s expectation that mathematic will be useful and their personal attitude such as confidence, persistence, and organization are mentioned by Stacey (2006) as some of the skill and abilities required from problem solving.
2. Mathematical method.
Why we prove that? We can study from the specifics to general is called inductive thinking, from general to specifics is called deductive thinking, logical mathematics, or syllogism is one choice to it. We also use preposition. What is preposition? Preposition is sentence which consists true or false. We use conjunction, disjunction, implication (if then), or if then if. Beside that, we use direct or indirect prove.
3. Mathematical content.
Mathematical contents include ideas of set or unit operation, algorithm, or approximation. These can be compared to mathematical skill (as well as estimation mental computation) or deep mathematical knowledge as stated in Stacey’s requirement from problem solving.
Interview’s Result with one of the student grade IV of Elementary School:
A. Mathematical Method
1. What are the students do abstraction?
Abstraction is search similarity to get form or characteristic is very simple can make object of mathematical thinking. For example: with abstraction, let circle just is learning about just form and size does not color, material, prize, an characteristic others.
Answer:
- The students do abstraction to model of triangle, rectangular, and square.
- The students do abstraction that a rectangular is formed from fours line which first line and second line have same in long, but third line and fourth line also have same in long. For example: in every day, we can get model of rectangular such surface of table, surface of book, black or white board
- The students do abstraction that a square is formed from fours line which the fourth line have same in long. For example, the thing in our life which have form as square such paper which side has same in long.
- The students do abstraction that triangle is formed from three points was related with lines so when the student have two points, they can not make a triangle. For example, the thing in our life which has form as triangle is three wheel pictures on cover or packet cement. In teaching learning, the students do abstraction from triangle with unite tip of forefinger our hand and tip of our thumb.
2. How step or process to do abstraction and what term is used to it?
Answer:
The method to do abstraction:
• The students take instrument or thing in our life every day which has form such rectangular such book, paper; square such the paper has same in long of side it; triangle such picture on cover cement Gresik with merk “tiga roda”.
• The students use paper, pencil, eraser, ruler, and scissor to make form of rectangular, square, and triangle then they direct to me. So the form of triangle, rectangular, and square; the students must understanding it.
• The students use term on unsure that square has point side, side, and area of square. Rectangular has side, point side, and area of rectangular. Triangle also has base, high, side, point of side, and area of triangle.
• The students give definition of model with our sentences every day so more easily to understanding. Square is model was formed from fours line which the fourth line have same in long Square is such floor of home so it side has same. Rectangular is model was formed from fours line which first line and second line have same in long, but third line and fourth line also have same in long Rectangular such surface of door which second side same and second side other also same. Triangle is formed from three points was related with lines so when the student have two points, they can not make a triangle such picture on “pagoda pastilus” candy.
3. What the students do to represent it so easy to understanding?
Answer:
- The students represent area of rectangular with long time width so notasion to it that p x l
- The students represent area of square with side time side so notasion to it that s x s
- The students represent area of triangle with (base time heigh) divide two so notasion to it that (a x t) :2
4. What are the students do idealization?
Answer: Yes, so every rectangular, square and triangle, are formed from straight line there are not crooked lines. The student has assumption that to make rectangular, square, and triangle must use straight lines.
B. Mathematical Attitude
Mathematic learn with students honesty, consistent, absolute, etc. We can see it when the students do examination from their teacher. That time, I give my students question as examination in end month. The students imitate student other’s answer so I see that the student does trust. From observation it, we can see that the student have not honesty so the student must trust and believe with themselves. The student’s assumption that mathematic is difficult, so when they do exercise of mathematic although just several minute but they were bored and feel lazy to do it.
Observer : “Every time, how long you study mathematic?”
Student : “Average approximately an hour sister.”
Observer : “Why just an hour? We know that mathematic need much time to do exercise or solve problem?”
Student : “ Oh no,,,,,,mathematich is difficult so when we long time to it we will was bored.”
Observer : “Ok, I will give you exercise.”
Student : “I like do question if it from yourself does not from my book.”
From dialog above, we can give conclusion that the student assumption that mare and more difficult mathematic to learn it. The student like exercise question from our then from book. The students often do not hear my explanation, but far-out when I give some question to the student, they can give answer with true. They do not hear explanation but in end explanation, they ask several questions and in my opinion their question is brilliant. The student looks for apathetic but they give critical and active when there are several question. Sometimes, along teaching learning in class, the student request to rest because they tired with mathematic.
C. Mathematical Content
On above discussion about triangle, rectangular, and square so this explanation it, I more emphasize on comprehension student with my concept about:
 What is rectangular, square, and triangle?
 How form from rectangular, square, and triangle?
When the student understanding about two problem above so I will explanation what is side, point of side to square; base and high for triangle; long and width for rectangular. I will start to count area of rectangular, square, and triangle when the student understanding unsure of rectangular, square, and triangle. The student can learn to use formula if they can write it with notation. For side of square can be notation s, for long of rectangular can be notation p, for width of rectangular can be notation l, for base of triangle can be notation a, for high of triangle can be notation t. The students have assumption that mathematic is very difficult so I more why the student love with mathematic. I more emphasize so that the student few material but understanding than they much material but don’t understanding with it. Useless if I give much material but my student zero about it. So, I teach with several methods in this bottom:
1. Giving material so in every meeting, I more explain concept.
2. When the material reach finish, I will repeat or just review it to remember again so does not forgot.
3. Exercise question and I discussion the question with student.
4. Do evaluation pass trough examination in
Conclusion:
So, mathematical thinking has three component are mathematical method, mathematical attitude, and mathematical content. Mathematical thinking is very important to teaching learning mathematic. Every student has different mathematical thinking to understanding what mathematic. Different student has different paradigm, so a teacher has to can identify characteristic their student.

The Power of Category and Networking

If was was asked about what is category? So, what is our answer? Many people can give several definitions about category. But hold by Kant, 1771 that in our mind there is a something and we can called are category and relation. In category, we must can to chose to do and we must enter so save in epoche’s house and which still phenomena form. 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. Such a category is not a classificatory division, as the word is commonly used. It is, instead, the condition of the possibility of objects in general, that is, objects as such, any and all objects, not specific objects in particular. Meaning of category that the word comes from the katēgoria, meaning that which can be said, predicated, or publicly declared (describe) and asserted (explain), about something. A category is an attribute (characteristic), property, quality, or characteristic that can be predicated of a thing. Kant said, "I remark concerning the categories that their logical employment consists in their use as predicates of objects." So Kant called it ontological predicates.
For example, we will discus about mathematics from definition of mathematics, characteristic of mathematics, and type of mathematics. There are difference between school mathematics and university mathematics. In university mathematics, study pure mathematics and applied mathematics. But in school mathematics, study general mathematics. The object of mathematics is abstract while abstract is our mine. There are two steps to get mathematics concept are abstraction and idealization. Abstraction is search similarity to get simple form and simple characteristic so it will to can be object of mathematical thinking. When we will discuss about teaching and learning of mathematics in front of class so our students will take notice our. Let, we will describe a cube so they will think about shape and size. When we think about cube, in our memory or our brain, there are many points which mixed in our mind such think our family in home or our lovely. Consequently, we must make a grouping everything which enters in our mind because it is very important to ours. When we can not grouping points it so our mind will was boxes. Everyone has to priority or accentuate several points on their mind. Brain always was used human to think, work, and study so was trained to grouping everything. If our brain over long time to rest, it means vacuum to think so it will cause we lazy to grouping everything.
”What is mathematical thinking?” Prof. Katagiri from Japan when his presentation in Sapporo-Tokyo in said that mathematics is mathematical thinking consist three aspects, and I conclude that:
1. Mathematical attitude.
It is very important in determining student’s behavior in mathematical thinking. A good attitude is always asks to be perfection. Mathematical attitude is very important affective factor in determining student’s behavior in mathematical thinking depend on how interested, they are in problem solving on the lesson. Student’s expectation that mathematic will be useful and their personal attitude such as confidence, persistence, and organization are mentioned by Stacey (2006) as some of the skill and abilities required from problem solving.
2. Mathematical method.
Why we prove that? We can study from the specifics to general is called inductive thinking, from general to specifics is called deductive thinking, logical mathematics, or syllogism is one choice to it. We also use preposition. What is preposition? Preposition is sentence which consists true or false. We use conjunction, disjunction, implication (if then), or if then if. Beside that, we use direct or indirect prove.
3. Mathematical content.
Mathematical contents include ideas of set or unit operation, algorithm, or approximation. These can be compared to mathematical skill (as well as estimation mental computation) or deep mathematical knowledge as stated in Stacey’s requirement from problem solving.
In Ebbut and Straker, 1995 the nature of school mathematics has four aspects are pattern or relation, problem solving, investigation, and communication. From Katagiri’s idea and Ebbut’s idea so we can write it in a table in this bellow:
pattern problem solving investigation communication
atitude √ √ √ √
method √ √ √ √
content √ √ √ √
If the people think from top to down so we use reference such book, journal, or research report. We have a reference to grouping everything in our memory. So, for example when we will write essay with title “Student’s Mathematical Thinking in the Framework of The Nature of School Mathematics of Geometry” we has reference so we can grouping everything. We must to think extensive, intensive, and routine as a spiral dynamic. The people must group everything in our mind so they are habitual to it.