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.

Mathematical Thinking and Scientific Work

Mathematical can’t liberate from problem solving and number as mean of communication. Everyone can descriptive what about mathematics and also mathematicians have different definition about mathematics. Different mathematician said that mathematics is a science of axiomatic thinking established with deductive method consisting of definition, theorem, axiom, role, etc.

We usually have assumption when we solve some mathematics problem. It’s a concept or definition by theory and also it can built system of mathematics. We need a clear people as an object of mathematics. An object of mathematics must be clear. For examples are pen, paper, table, necktie, etc.

We can say that object of mathematics is our idea, because all of thing in our mind. An object of mathematics is Meta mathematics that is thinking about mathematics.

Two ways to get an object of mathematics (concrete thing) that is
1. Use idealization method
There are someone said that idealization method is assumption to get absolutely. Assumption all of thing is perfect. We put example a flat plane, no things in the world is really as a flat plane. Some examples for concrete thing that is she is a beautiful girl form of thing, size of thing, in the reality there are more than million characteristic of something. So, we know that object of mathematics not all can be touched, but the characteristic absolutely because it is only use an assumption.
2. Abstraction
In abstraction we only use a characteristic.
Just to learn a certain characteristic

 Thinking about mathematics must consistent.
Agree with first agreement reached that is the definition.
If a proof not consistent, the statement will be wrong.

 Mathematical logic
In mathematical logic catch with everything, we can difference easily which is small which is bigger, which is prime number which is perfect number, etc. For example six less than seven based the number but if we write number six bigger than number of seven, it will be six bigger than seven. If there are some random number that is eleven, four, seven it’s can be write in a series. We may write the first is four the second is seven and the third is eleven. In reality of mathematics that is relation. We can put for example that is what is the relation of nineteen and twenty seven? We can answer the question, nineteen plus eight is twenty seven or twenty seven minus eight is nineteen.
And also about relationship two sets. If there are two sets A and B, is it a relationship?
 Mathematical Operation
In mathematical operation usually use arithmetic that is addition, subtract, multiplication, and division. The statement is logic if the statement can be proofed. For example eleven plus nine is twenty. The other is radix, square root, square, power, exponent, etc.

 True or False Statement
We can know mathematics sentence, we can use the truth table. It is learned of logic of mathematics. The function is conjunction, disjunction, implication, and double implication. There is conclusion from the premise. For example the first premise if you are student of university then the second premise the next week go to campus so the conclusion from the premise is You are go to campus.

Thesis is sentence show truth.
For example is seven is odd number and also prime number.
Antithesis is opposite thesis.
For example seven is not odd number.
Hypothesis is not permanent assumption.
For example x, we can’t know x odd number or even number. The equipment is presumption.

Scientific work has some characteristics,
1. Impersonal
Not relation with personal.
Personal, for example is say OK, good luck
2. Have a standard or criteria
Scientific work must get a standard because it’s fixed work.
3. Objective
The reference is got by text book, proposal, or depend on.

Reflection of video

Video 1 => Tell us about teacher and his students when they are in class. The teacher had a good idea for his students. The teacher came to class, so it made his students stop to speak anymore. Without good manners, he ordered one of his students go up to the teacher’s table. I remember that his name is victor. Then the teacher ordered him to look at something. After it, he also ordered to the other students to done it. The teacher said, “look at thing at different way, you must find your wrong voice”. From the video I think that the teacher has method to come up the spirit for his student to study and be a hero.

Video 2 => There were a boy run then he climbed the stage. On the stage he showed a poem. He expressed the sentence of the poem, so it made the spectators cheer. The spectators feel happy with that show. One of the sentences is ‘do you believe in me?’ the spectators answer yes spontaneous. The boy who showed on the stage not only shows his poem, but also he gives a spirit or statement that mathematics isn’t difficult. He ordered the spectators and us to believe in our self that we can solve the problems of mathematics.

Video 3 => There were two boys played rap music. They showed about mathematics functions. Some of them are ln of x. They expressed a simple mathematics. The others were exponent equation, the graphic of sinus, and integrate. One of boys said, ’what you know about math?’ and the other said that ‘I know all about math’ and also they showed the operation from symbol of mathematics +,-,x,: (plus, minus, times, and divide).

Video 4 => Solving differential equations, the boy and the girl expressed part of the mathematics. The girl explained about algebra. Y is the function of x. She had a method to memorize the mathematics function, she wrote that function on the roasted bread.

The boy explained about integrates. He said, ‘Integrate of dy dx equal integrate of four x square, dy dx times dx equal 4xsquare times dx to find y. integrate of dy means y and integrate of four three x to the point of three plus c.

Video 5 => Professor explained method to solve the mathematics equation.

1. x minus five equal three. How the value of x? He continues it, each parts equalize with plus of five, x minus five plus five equal three plus five. So we can say that x equal eight.

2. Seven equal four times a minus one, same method from number one, seven plus one equal four times a minus one plus one, eight equal four times a, equalize with times one four, one four times eight equal one four times four times a, so a equal two.

3. Two three times x equal eight, three two times two three times x equal three times eight, x equal twelve.

4. Five minus two times x equal three times x plus one, equalize with minus negative three times x, five minus five times x equal one, equalize again with negative five, negative five times x equal negative four, negative one five times negative five times x equal negative one five times negative four, x equal four five.

5. Three minus five times (two times x minus five) equal negative two, three minus negative ten times m plus twenty five equal negative two, negative ten times m plus twenty eight equal negative two, negative ten times m equal negative thirty, equalize with time negative one ten, m equal three

6. Twelve times (one two times x plus one four) equal twelve times (one three times x plus five four), six times x plus three equal four times x plus fifteen, two times x plus negative twelve equal zero, two times x equal twelve, x equal six.

7. One hundred times (zero point three five times x minus zero point two) equal one hundred times ( zero point one five times x plus zero point one), thirty five times x minus twenty equal fifteen times x plus ten, twenty times x equal thirty, x equal three two.

Video 6 => About logarithm, logarithm base x from a times b equal logarithm base x from a plus logarithm base x from b. Logarithm base x fro a to the power of b equal b times logarithm base x from a. Logarithm base x from a divide b equal logarithm base x from a minus logarithm base x from b.

Reflection

Reflection

Thursday, March 12^th 2009 we done the first English test. No estimate if the day held a test. I feel confuse with the question, because the university-level instructor give some questions from Indonesian to English about mathematics. Owh…

English isn’t easy for me, because since formerly I don’t like English. I don’t know why. Whereas, my friend joy with English.

I hope the next test will be better than the past.

The meaning of mathematics

The object of mathematics,

1. Abstraction

Abstraction isn’t same with abstract. Abstract mean something not concrete, then abstraction mean characteristic and descriptive. The example, we think about “5”, it is green or it is from wood or it is beautiful, etc. But not all from the example is used. In mathematics we only use the value.

2. Idealism / idealization

It means art. Art to make question of exercises.

Assume that needle is sharp and pointed, in the reality nothing is sharp and pointed.

Define of mathematics

Structure of system mathematics built in deductive. Consist of formula, define, axiom, theorem, rule, pattern, etc.

The characteristic of mathematics is logic and consistent. Consistent mean one formula to another formula.

Prof. Kayastesi from Melbourne said that mathematics consists of two.

1. Conjecture

To think or to predict

2. Convince

To communicate the result

Prof. Katagiri from Japan said that mathematics consist of three.

1. Mathematical attitude

If someone doesn’t like the attitude, the reality is someone has attitude.

Good attitude mean positive thinking, consistent, critical, and careful.

2. Mathematical method

How to get mathematics or how to learn mathematics

The meaning of mathematics based from philosophy, so that one another to another is different to assume the meaning of mathematics.

Study of mathematics in inductive. First meeting is know, second is know by heart, etc. The example of study of mathematics in inductive is logic, the specific of logic is silogism. There are mathematical sentence, direct proof, inverse proof, and also inductive mathematics.

3. Mathematical content

Realistic of mathematics consist of two.

1. Horizontal mathematics

Daily mathematics and concrete

2. Vertical mathematics

Abstract

an introduction to english 2

An Introduction to English II

As we know that English is international language, so very important to learn more about English. If we meet someone or some people who is not live in Indonesia, we usually talk in English because they exactly can’t Indonesian. We must be learn English seriously. When we looking for a job, we will easy to get it.

To be a competent person :

1. Motivation

To learn English, we usually have a motivation and a target. Motivation can be high spirit. To get the maximal target, we really learn English continue and practice it. Also we should be aware that motivation can be got from our parents, friends, and job. We don’t forget to have knowledge in order that we more love English.

2. Attitude / Behavior

We must be focus when we study our love lesson (may be English) because we will get it easily and we can understand fast.

3. Understand

a. Responsibility

Important for us to understand the lesson because if we have an English task we can respond it easily. May be if the formula ‘after modal is V1’ we not only be written but also understand to use.

2. Communication

Communication such us read, talk, write, hear, and also translate.

4. Skill

We can take CD to learn English by self. We also follow TOEFL.

5. Experience

We can percept where the formula use because we use the formula regularly. If we read the English book, we like hear music, or write English we can respond the mean quickly.