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