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:

(first rule with understanding) out of twelve (75%)

(first rule without understanding) out of ten (30%)

(no previous knowledge) out of twelve (17%)

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.

Pedagotical Power Mathematical Power

**References**

Aichele, D.B., Coxford, A.F. (1994). Professional Development for teachers
of mathematics.

Reston, VA. : The national council of teachers of mathematics, Inc.

Cooney, T.J. , Grouws, D.A., Jones, D. (1988). Effective mathematics teaching.
Reston. VA. :

Lawrence erlbaum associates, National council of teachers of mathematics.

Coxford, A.F., House, P.A. (1995). Connecting mathematics across the curriculum.
Reston, VA.

Crosswhite, F.J., Reys, R.E. (1977). Organizing for mathematics instruction.
Reston, VA. : The

national council of teachers of mathematics, Inc.

Reisman, K.F. (1981). Teaching mathematics, methods and content. (2nd ed.).
Boston, MA. :

Houghton mifflin company.

Resnick, L.B. (1981). The psychology of mathematics for instruction. Hillsdale,
NF. : Lawrence

erlbaum associates.

Simmons, M. (1993). The effective teaching of mathematics. New York. NY.
: Longman.

Skemp, R. (1987). The psychology of learning mathematics. (expanded american
ed.). Hillsdale,

NJ. : Lawrence erlbaum associates, Inc.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation
standards for

school mathematics. Reston, VA.

National Council of Teachers of Mathematics. (1989). Professional standards
for teaching

mathematics. Reston, VA.

National Council of Teachers of Mathematics. Handbook of research on mathematics
teaching and learning. Reston, VA.

National Council of Teachers of Mathematics. (1993). Historical topics for
the mathematics for the mathematics classroom. Reston, VA.