SMALL RESEARCH ABOUT
MATHEMATICAL THINKING AND HOW TO TEACH IT

In a few weeks, I made a research about mathematical thinking in student of junior high school. The object is my sister, she is in the second class of junior high school in Magelang. I gave her some test about mathematic and perceived how she solve the test. I didn’t persuade her with my idea to solve the test but I let her to think herself with her idea to solve the problem.
According to Shikgeo Katagiri (2004) 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.”
The example of the problem:
1. The area of isosceles triangle is 240 cm2 . How is the circumference of the triangle if the high is 24 cm?
Answer :
Area = 240 cm2
High = 24 cm
Area = ½ x base x high
240 = ½ x base x 24
= ½ x base
10 = ½ x base
- ½ x base = -10
Base = -10 : – ½
Base = 20
According the Pythagoras equation,
(The hypotenuse)2 = 242 + (1/2 x 20)2
= 242 + 102
= 576 + 100
= 676
The hypotenuse =
= 26
The circumference = sum of the side
= 26 + 26 + 20
= 72

2. The area of rhombus is 960 cm2 . How is the circumference of the triangle if the one of the hypotenuse is 60 cm?
Answer :
The area of rhombus = 960 cm2
the diagonal 1 = 60 cm
The area of rhombus = ½ x diagonal 1 x diagonal 2
960 = ½ x 60 x diagonal 2
960 = 30 x diagonal 2
= diagonal 2
diagonal 2 = 32

According the Pythagoras equation,
(The hypotenuse)2 = 302 + (1/2 x 32)2
= 302 + 162
= 900 + 256
= 1156
The hypotenuse =
= 34
The circumference = 4 x the side of hypotenuse
= 4 x 34
= 136 cm
In the two problem of mathematic that I gave for her, my sister can did it well. She can differentiating the types of the problem. Students should have the ability to reach the type of solution shown above independently.
This is a desirable scholastic ability that includes the following aims:

• Clearly grasp the meaning of operations, and decide which operations to use based on this understanding
• Functional thinking
• Analogical thinking
• Expressing the problem with a better formula
• Reading the meaning of a formula
• Economizing thought and effort (seeking a better solution)
The ability to use “mathematical thinking” is even more important than knowledge and skill, because it enables to drive the necessary knowledge and skill.
Shikgeo Katagiri (2004)
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.
1. The Driving Forces to Pursue Knowledge and Skills
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. Mathematical thinking acts as this drive.
2. Achieving Independent Thinking and the Ability to Learn Independently
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.
3. Mathematical Thinking is the Key Ability
It is evident that mathematical thinking serves an important purpose in providing the ability to solve problems on one’s own as described above, and that this is not limited to this specific problem.

List of Types of Mathematical Thinking
1. Mathematical Attitudes
2. Mathematical Thinking Related to Mathematical Methods
3. Mathematical Thinking Related to Mathematical Contents

Reference : www.powermathematic.blogspot.com (24 November 2009)

The Power of Category and Networking

To be a good mathematic teacher, we must attention on science of teaching and learning mathematic. The based of attention is awareness. In the science of teaching and learning mathematics, there is mathematic education phenomena. Shikgeo Katagiri (2004) said that there are three types of mathematical thinking :
I. Mathematical Attitudes
1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
1) Attempting to have questions
2) Attempting to maintain a problem consciousness
3) Attempting to discover mathematical problems in phenomena
2. Attempting to take logical actions
1) Attempting to take actions that match the objective
2) Attempting to establish a perspective
3) Attempting to think based on the data that can be used, previously
learned items, and assumptions
3. Attempting to express matters clearly and succinctly
1) Attempting to record and communicate problems and results
clearly and succinctly
2) Attempting to sort and organize objects when expressing them
4. Attempting to seek better things
1) Attempting to raise thinking from the concrete level to the abstract
level
2) Attempting to evaluate thinking both objectively and subjectively,
and to refine thinking
3) Attempting to economize thought and effort
II. Mathematical Thinking Related to Mathematical Methods
1. Inductive thinking
2. Analogical thinking
3. Deductive thinking
4. Integrative thinking (including expansive thinking)
5. Developmental thinking
6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
7. Thinking that simplifies
8. Thinking that generalizes
9. Thinking that specializes
10. Thinking that symbolize
11. Thinking that express with numbers, quantifies, and figures
III. Mathematical Thinking Related to Mathematical Contents
1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)
2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3. Attempting to think based on the fundamental principles of expressions (Idea of expression)
4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5. Attempting to formalize operation methods (Idea of algorithm)
6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7. Focusing on basic rules and properties (Idea of fundamental properties)
8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)
A mathematic teacher must learn about category. Category is difference of phenomena and epoche. We can learn about phenomena in the process of teaching and learning mathematic to be a science. The phenomena can be a science with use theory, reference, books, journal and research. The process to product a science from phenomena called a synthesis.
The instrument to synthesis are :
a. Observation : check list
b. Question : from student
c. Questioner : teacher

How to Uncover Psychological Phenomena

The world is traumatic.Traumatic means a situation where is no clarification.Traumatic is the kind of communication problem.
There are two kind of traumatic :
1. Positive traumatic : motivation, happiness
2. Negative traumatic : otoriter

To solve traumatic, we can use communication. The quantity of well communication are flexible and contextual.

A lot of mathematics teacher, make their students traumatic. They are assume that their students is the object of study. We must revitalize those assumption.
There are a lot of theory to revitalize it like :
1. Structure of teaching
• Introduction
• Process
• Closing remoute

2. The schema of interaction
• Classical
• Group discussion
• Individual

3. The nature of student learn mathematics
• Need motivation
• Individual
• Need collaboration
• Learn in context

4. The nature of school mathematics
• Pattern and relation
• Problem solving
• Investigation
• Communication

The use of those theory is contextual,depend with the place and the time or situation. A teacher must know the characteristic of their students and make a comfortable condition to learn.

Mathematic Thinking and Scientific Work

Mathematics Thinking and Scientific Work


Pure mathematics is about formal mathematics. There is thinking about axiomatic. From the pure mathematics, the definition of mathematics are about deductive method, concept, theorem, axiomatic, and approve.
The object of mathematics is your idea in your mind. It can not be manipulated. From the object of mathematics, there is turn up meta mathematics. Meta mathematics is thinking about mathematics.
There are two ways to get mathematics object from the konkret object :
1. Idealisation :
Asumtion that the object is perfect. In mathematics, we must asumted that the object of mathematics is perfect.
Example : We asumted that the plate is absolutely plain. The real, there is no something perfect in the world. The absolutely perfect just god.
2. Abstraction :
Abstraction is just learn some characteristic that we need from the object.
Example : There are much characteristic from the paper, but in mathematics we just learn about the shape and the size.

Mathematics thinking are about :
1. Consistence
If we find the wrong from the solve of the problem, the solve can not be applied.
2. Logic
Logic is begin from the daily life
3. Order
4. Relationship
5. Mathematics operation :
Example : Addition, substraction, multiplication, division, root, etc.
6. If than statement
7. How to get conclusion
8. Thesis, antithesis, hypothesis
Thesis : sentences that show the true statement
Antithesis : statement that contradiction with thesis
Hypothesis : the temporary thesis

Scientific Work
The characteristic of scientific work are :
1. Impersonal :
Doesn’t related with personal problem
2. Have a standard or criteria : ethical code, write condition
The example of scientific work are : report, proposal, text book, paper
The content of scientific paper are :
• Abstract
• Discussion
• Conclution/recommendation
• Reference
• Introduction

Reflection of Video

Reflection of Video



Video 1 : Dead Poets Society
If you look something, you must think from the different way. From this, you can find the solution about your problem. The change is you. You can choose your way and your think. You can choose what you want and how to get it.

Video 2 : Believed
We must believe in our self. We must believe that we can do anything, dream anything and get anything. We can get what we want if we believe in our self. Believe is very important in our life. From that, we can get motivation to do anything and power to do it.

Video 3 :
What you know about math?
Math is about significant figure. The examples are trigonometry, exponent, geometry, algebra etc. math is also about equation like plus, minus, over and times.

Video 4 : Types Differential Equation
To solve differential equation is like integrate in calculus.
To find y = f(x)
We must usually get dependent y

Video 5 :

1. x – 5 = 3
To solve this problem, we can plus x – 5 with 5 and plus 3 with 5 too. So we can get that x is 3 plus 5 equals 8.
2. 7 = 4a – 1
To solve this problem, we can plus 7 with 1 and plus 4a – 1 with 1 too. From this, we can get that 4a equals 8. To get the value of a, we can times 4a with 1/4 and times 8 with 1/4 too. So, we can get that the value of a is 2.
3. 2/3x = 8
To solve this problem, we can times 2/3 x with 3/2 and times 8 with 3/2 too. From this, we can find that the value of x is 8 times 3/2 equals 12.
4. 5 – 2x = 3x + 1
To solve this problem, we must eliminate 3x. To do that, we can plus 3x + 1 with -3x and plus 5 – 2x with 3x too. We can get that 5 – 5x = 1. After this, we eliminate 5. To do that, we can plus 5 – 5x with -5 and plus 1 with -5 too. We can get that -5x = -4. To get the value of x, we can over -4 with -5. So, we can find that the value of x is 4/5.
5. 3 – 5(2m – 5) = 2
To solve this problem, we must times 5(2m – 5) first. From this we can get that
3 – 10m + 25 = -2
-10m + 28 =
-10m = -30
To get the value of m, we can over -30 with -10. So we can get that the value of m is 3.
6x + 3 = 4x + 15
6x – 4x = 15
2x = 12
To get the value of x, we can over 12 with 2. So, we can get that the value of x is 6.
6. 0,35x – 0,7 = 0,15x + 0,1
To solve this question, we must times 0,35x – 0,7 with 100 and times 0,15x + 0,1 with 100 too. We can get
35x – 70 = 15x + 10
35x – 15x = 10 + 70
20x = 80
To get the value of x, we can over 80 with 20. So, we can get that the value of x is 4.

Video 6 :
In video 6, there are a formula about logarithm and the law about log
• The formula is :
Based a log b equals c is a based c equals b
• The law about log are :
1. log A + log B = log (AB)
2. logA – log B = log (A/B)

My reflection in Learning English

My Reflection in Learning English


In 12 March 2009, Mathematic Education in Yogyakarta State University has reflection in learning English. Reflection was like a test. My lecture, Mr. Marsigit gave us 50 questions about vocabulary in English. The vocabulary was about mathematic. We were very confused because we didn’t know if that day Mr. Marsigit would give us a test.

Mr. Marsigit gave us 1 hour to did the test. After we done the test, Mr. Marsigit instructed us to changed the answer with other friends to checked the answer. After that, Mr. Marsigit instructed us to report the mark of this test to him. I was very sad because my mark was very bad. From 50 questions the false are 37 and the true just 13. My friends were also had a bad mark. Mr. Marsigit also sad because our bad values.

Mr. Marsigit instructed us to improve our competent in learning English. He said that to improve our competent, we must read, read, and read to get much knowledge. If we just study in class it didn’t enough to learned English. We must learned alone to get knowledge about English. We could access internet, and bought TOEFL CD to improve our competent in English.

Although I couldn’t speak English well, I promised to repair it. Mr. Marsigit gave us motivation to study English. He said that we must enjoyed and happy to learned and talk in English. If we scared before, we’ll not able in English. We didn’t have to be afraid to made some mistakes in our speech.

We must convidence to spoke English because English was international language in the world. We didn’t afraid to try spoke in English. We could try speak English in the daily life like in our family, friends and community.

English was important in our life moreover in this global area. Much school in Indonesia opened the International school. The learning in that school use bilingual language. Because of that, we from Mathematic Education in Yogyakarta States University must can speak English to be a good teacher in International school.

The Meaning of Mathematic

The Meaning of Mathematic


Mathematic has a lot of definition. The definition of mathematic in junior high school, senior high school, and in the university is different. Mathematic in university is pure mathematic. Mathematic in the university is pure mathematic and applied mathematic. There are two object in mathematic:
1. Abstraction
Abstraction is different with abstract.
Abstract is not real, we can’t touch but we can think that.
Abstraction in mathematic just about value.
Example : 5<6,>4
2. Idealisation
There is no real in the world. The real is in your mind (assumption)
The characteristic of mathematic is logic and consistent.
Profesor Kayestesi from Melbourne University said that the characteristic of mathematic are:
1. conjecture : can to predicate
2. convince : certain, sure
Mathematic consist of 3 aspects these are:
1. Mathematical attitude
2. Mathematical method
3. Mathematical content

 Good attitude in mathematic is always have a question, consistent, and critically.
 Mathematical method is how to learn mathematic
Example :
• Deduction : from general to specific
• Induction : from specific to general
• Syllogism : withdrawal of conclusion pursuant to premis
• Logical method : mathematical sentences like conjunction ,if that statement
• How to prove : direct, indirect
• Incomplete induction
 Content is the object of mathematic

The definition of mathematic from realistic
1. horizontal mathematic : daily mathematic (the object is real)
2. vertical mathematic : abstract mathematic (university mathematic)