Mathematical Thinking from Student in Senior High School

Mathematical Thinking and How to Teach It
By Shigeo Katagiri
Translated of the rewritten version from Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. Meijitosyo Publishers, Tokyo. Copyright of English version has CRICED, University of Tsukuba. All rights reserved.

The Aim of Education and Mathematical Thinking
1. The Aim of Education: Scholastic Ability From the Perspective of “Cultivating Independent Persons”
The aim of school education is described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.”
This guideline is a straightforward expression of the preferred aim of education.

2. The Scholastic Ability to Think and Make Judgments Independently Is Mathematical Thinking
The most important ability that arithmetic and mathematics courses need to cultivate in order to instill in students this ability to think and make judgments independently is mathematical thinking. This is why cultivation of this “mathematical thinking” has been an objective of arithmetic and mathematics courses in Japan since the year 1950. Unfortunately, however, the teaching of mathematical thinking has been far from adequate in reality.
Deciding the operation clearly determines the meaning of each computation, and decides what must be done based on that meaning. This is why “the ability to clarify the meanings of addition, subtraction, multiplication, and division and determine operations based on these meanings” is the most important ability required for computation.

3. The Hierarchy of Scholastic Abilities and Mathematical Thinking
As the previous discussion makes clear, there is a hierarchy of scholastic abilities. * The ability to memorize methods of formal calculation and to carry out these calculation.
* The ability to understand the rules of calculation and how to carry out formal calculation.
* The ability to understand the meaning of each operation, to decide which operations to use based on this understanding, and to solve simple problems
* The ability to form problems by changing conditions or abstracting situations
* The ability to creatively make problems and solve them
The higher the level, the more important it is to cultivate independent thinking in individuals. To this end, mathematical thinking is becoming even more and more necessary.

The Importance of Teaching Mathematical Thinking
As we found in the previous chapter, the method of thinking is the center of scholastic ability. In arithmetic and mathematics courses, mathematical thinking is the center of scholastic ability. However, in Japan, in spite of the fact that the improvement of mathematical thinking was established as a goal more than 45 years ago, the teaching of mathematical thinking is by no means sufficient.

Mathematical thinking allows for:
(1) The Driving Forces to Pursue Knowledge and Skills
Mathematics involves the teaching of many different areas of knowledge, and of many skills. If children are simply taught to “use some knowledge or skill” to solve problems, they will use that knowledge or skill. In this case, however, children will not realize why they are being told to use such a knowledge or skill. Also, when new knowledge or skills are required for problem solving and students are taught what skill to use, they will be able to use that skill to solve the problem, but they will not know why this skill must be used. Students will therefore fail to understand why the new skill is good.
What is important is “how to realize” which previously learned knowledge and skills should be used. It is also important to “sense the necessity of” and “perceive the need or desirability of using” new knowledge and skills.
Therefore, it is necessary for something to act as a drive towards the required knowledge and skills. Children first understand the benefits of using knowledge and skills when they possess and utilize such a drive. This leads them to fully acquire the knowledge and skills they have used.

(2) Achieving Independent Thinking and the Ability to Learn Independently
Possession of this driving force gives students an understanding of how to learn by themselves.
Cultivating the power to think independently will be the most important goal in education from now on, and in the case of arithmetic and mathematics courses, mathematical thinking will be the most central ability required for independent thinking. By mastering this skill even further, students will attain the ability to learn independently.

4. Mathematical Thinking is the Key Ability
The ability to write the problem as a formula, using multiplication and addition Possession of this understanding and skills, however, is not enough to solve the problem. An additional, more powerful ability is necessary

The Meaning of Mathematical Thinking and How to Teach It Characteristics of Mathematical Thinking
Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist.

( It is taken from http://marsigit English.blogspot.com )


My Research
After I observe the children who live near my grandmother’s house, I get some information. The children aren’t come from rich family so it is different from the other family generally. The parents have no enough money to enter their children in to school. But, they hope them can be a success people. Spirit of the children must be grown up because they are lazy to study. They have to study hard. They didn’t study if their parents don’t ask them. Their achievement in the school isn’t too good. When they are given some questions about mathematics, they still have a wrong answer. I will try to explain the problem solving but they didn’t understand. Usually the children spare their time for playing with their friends. It’s good but if they play routine it will be disturb their concentrating of study. Because of that the parents must to ask their children to study in order that their children to be a success people.

The second children who I research, she is my neighbor. She is student in one of senior high school in Yogyakarta. Although she is already student in senior high school, she can’t distributive count in mathematics. Should distributive count have studied in junior high school. I ask her with some question.
1. What must be sum between positive and negative number?
2. What kind of method seems likely to work?
3. What kinds of rules seem to be involved?
4. Is there something else with a similar meaning (properties)?
From her answer, I know that she still doesn’t understand with my questions and she can’t solve her problem about mathematics.

The Power of Category and Net Working

Power is a measure of an entity's ability to control the environment around itself, including the behavior of other entities. In our mind, we have different category and perception between one people to another. Because of that there are many perceptions. Based of Kant’s opinion, in our mind there are four items that is qualitative, quantitative, category, and relationship. Qualitative means depend on characteristics. Quantitative means depend on amount. Category means a way to look. Relationship means relation with other people. The most important in our mind is awareness. We must aware when we think something and before we doing something in order that we don’t have any mistakes. We have to know the intention about the function of our activities. The example is the teacher teaches in a class, that is a phenomenon. When the teacher gives some information about the lesson, he must really know what he said.

In mathematics there are two categories, that are abstract and idealization. Abstract means it’s not entered to our mind it’s only about form and size. Idealization means it’s necessary to be entered because it’s really perfect categories with the characteristics. WE have to understand about the category in our mind. Which is category to be entered to epoche house and which is category is still phenomenon. The phenomenon in order to be ‘science of teaching learning of mathematics’. The category has to be placed in correct position. We also necessary mind from top to down and we need some theories and references such as book, journal, research, and report to learn more about knowledge. There are some people have mind from down to up. Kinds of mind are extensive, routine, and intensive. Extensive means the mind from op to down. Routine means the mind is constant. Intensive means the mind from right to left.

Science of teaching learning of mathematics will be focus in student mathematical thinking so it’s necessary to read some theories or references.

How to Uncover The Psychological Phenomenon

Everyone have good side and bad side. Important for us to know and to understand about bad side. So, we need reflection from our self or the other person. They have different assumption about us. Event in past time sometimes make a traumatic. Traumatic there are positive traumatic and negative traumatic. Positive traumatic make us to do better and positive thinking about every event. Negative traumatic make me hopeless and always have negative thinking. Reflection make to uncover in order to be careful next time. Every event has different sensation. From the sensation emerge the perception. The perception is situation to think something and also emerge the value of result what we think. Because of that we must to rein our perception because it makes us to do wrong. After the perception emerge, we have to know the concept.