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