Teaching of mathematics is not just transmitting a mathematical knowledge to students, but helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics. Then, I naturally reach to a question of what is teaching for understanding in mathematics education. First, I will consider the meaning of understanding in mathematics education. It is not easy to define understanding in a brief sentence. It will largely depend upon each of us who has already defined the word unconsciously through the long period of education. R. Skemp said that to understand something means to assimilate it into an appropriate schema, but his definition is not so perfect to accept it as an absolute idea. The notion that understanding in mathematics is making connections between ideas, facts, or procedures looks like being generally accepted by many people, and I am also with this notion in this paper. Of course, there are many different research results about ways of making connections that students construct to create mental network. And talking about the mental network and its function in students' learning and understanding mathematics will be another important topic to be studied. I understood that teaching for understanding is realizing the effective teaching of mathematics. I will try to find some approaches through which teachers can make progress in the process of teaching mathematics for students' understanding. Four different issues which I think as " the big ideas" for the effective teaching will be discussed in this paper.
The first issue is teaching mathematics with strategies. What makes the effective teaching for understanding more complicated? Let us suppose that human mental network is so much hierarchical, and predictable, then teachers might find the answers to the question. But, as we can know easily the structure of human knowledge is just like a spider's web and the inside connections are mutually dependent. So, it is hard for us to understand the inter-relatedness clearly. For example, if all students take the identified four stages of intellectual growth of Piaget as they get older, then teachers' efforts to realize effective teaching would not be as difficult as we know now. Unfortunately, it does not work in reality. Here is a more specific example of students who are in ages 11 to 14 show different levels of ability in understanding a symbolic representation in a place value system. In this case, a teacher should decide of which kind of strategy he can use for the students' understanding of the symbolic representation. If he feels the need of a concrete material to teach the abstract to the students the material would be able to be money, or multibase arithmetic blocks (MAB) by Dienes. Whatever he chooses, it is for proper abstraction of mathematical concept and the students will benefit from it. The students will understand the concept through representations of the materials and interactions with the material. So it is possible that the teacher's strategy will turn out to be effective.
The more important thing is that the teacher should be able to think about strategies in students' place because a strategy that the teacher thinks as the best is not necessarily the best one for the students. It is good to remember what Glaser put about this matter (1984). He told that people continually try to understand and think about the new in terms of what they already know. Students are the same in learning and understanding of mathematics. In deciding a strategy, my experience as a foreign student in the United States provides an important idea. The social and cultural factors should play a role in deciding a strategy in a country like America where the diversity is common to everything. In the above example, if I were a student who are supposed to learn the place value with American monetary system by my teacher the material would not provide me any significant information that I have to obtain from the material. The multibase arithmetic blocks will be more useful to me because of their difference in quantity while the bills which have no physical difference will be more effective to most of American students because of a familiarity to the monetary system in the society for a long time.
The second issue is teaching mathematics with knowledge about the results of educational researches. I think it is very important for teachers to be exposed to current articles, journals, and books in the mathematics education field since they will be able to make the teachers be a good decision maker when they face a problem in teaching of mathematics. I don't want to mean that teachers should wait for researchers to tell them what to do. They rather have to think about the problem which is likely to show up in over class, and take action to solve it. The point is the importance of referring to results of many different researches, and the appropriate use of them at appropriate time and place. For example, Bell's experiment (1967) about the difference in adaptability between that based on a rule and that which results from understanding can be a good example. In the experiment, three groups of students were given a task to find a rule with different teaching methods. Since the differences in teaching methods are believed to derive the big differences in the ratio of finding the rule the experiment by Bell was very impressive to me. The results were:

He found that 75% of the first group were able to adapt to the new task, but only 30% of the second, who did little better than the group with no previous experience (p. 32, Skemp). This experiment guided me to assure the relevance between teaching mathematics and students' understanding.
Just as teachers talk about students' motivation to learn, teachers also need a motivation to improve his pedagogical power by reading many good research results. These kind of activities certainly require teachers to make a conscious effort and spend their time to be familiarized to those materials. The role of a teacher is a facilitator in the effective teaching of mathematics. The teacher should not be a giver of everything who are likely to restrict the development of students, and make the students to be passive takers of their own learning.
The third issue is teaching mathematics within a good educational environment. This issue is deeply related with the idea of being a facilitator because students need an appropriate environment where they can understand a mathematical knowledge in more active ways than before. Most of all, the environment should be able to provide students with many chances of self exploration so that the students can be less dependent learners compared to their past education. The reason I use "less dependent" instead of "independent" which sounds to have the same meaning is that students still need a teacher in their learning and understanding of mathematics. The use of technology and the introduction to a cooperative learning are important areas. It is sure that the availability of calculators and computers will change the meaning of understanding. I hope that the use of as many as technological tools will be able to encourage a spirit of mathematical research by students' hands and in their mind.
I have seen the power of cooperative learning in my own experience in the United States. Teachers should promote students to work with others toward common goals and to be actively involved in doing mathematics. It will be a good challenge for students to listen to the others and compare each one's solution process. Students will be able to feel that mathematics is not necessarily step-by-step any more in the cooperative learning environment. Teachers' role is now really being a facilitator, not being a master who is expected to explain everything to students.
History of mathematics can also be included as part of the good educational environment. In particular, if the way mathematics is presented in school is considered, introducing the history of mathematics can be a good way to give students an interest to learn a new mathematical idea. For example, telling about how Bombelli wrote a symbol to represent the modern notation x2 in 1572 may be good enough to open the students' mind. Moreover, interest in mathematics would be able to increase not only for students, but also for teachers.The final issue will be about teachers' knowledge of mathematics and many other areas other than mathematics. I think the next issue may be very controversial since all the preparation that I have talked so far looks too big for a teacher to posses.
So, the final issue is teaching of mathematics with enough mathematical knowledge and the relationship between knowledge of mathematics and teaching effectiveness. If I just imagine a teacher who asks a question to which he knows the answer himself, and his students who answer the question and are evaluated by the teacher, then the depth of the teacher's mathematical knowledge will be really important in the effective teaching of mathematics. Though a research shows that no significant statistical educational relationship appears to exist between knowledge of mathematics and teaching effectiveness (Cooney, 1994) , I think, teachers need to have a firm mathematical knowledge in a way that is consistent enough with the way that students are expected to learn and understand mathematics.
It was not easy for me to decide these four issues and devise their content since teaching for understanding is so broad that it could cover every single issue raised in the mathematics education field. But, I could have much time to think about the role of teachers for the effective teaching of mathematics and the importance of students' learning and understanding. Teacher should be able to lead students to become aware of mathematics and create a good learning environment where the students can experience and reflect their full abilities of doing mathematics.
Finally, I want to draw a picture which shows the relationship of understanding between a teacher and students. I hope this will be able to demonstrate us the equal importance of the pedagogical power of teachers and mathematical power of students.

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