The topic that I have chosen for the third final project is a general form of a philosophy from EMT 705. The focus will be on an effective implementation of EMT 705 class to many issues in mathematics education and a substantial planning of a class instruction. I hope that I will be able to focus on the role of a mathematics teacher and the way of organizing an instruction and curriculum with a knowledge from this course. I will try to supplement some issues from our discussion with more information. In particular, I will discuss some other new points of view which are not addressed in EMT 705 class, but I think that are important enough to be remembered. There will be a Class Design section at the end of paper which is planned for a mathematics teacher to teach a series of mathematical concept by the way of ideas in the Discussion section. I intend to show a connection in some mathematical topics which may be looked with no connection one another. The reason why I devise the Class Design section is not to escape from a risk of making this final paper just a file of good words without action. In fact, I supposed that I was a high school mathematics teacher when I designed the Class Design section. The basis of writing this paper is that the characteristic of mathematics lies not so much in its theoretical methods, but in the substantial material and the objects with which it deals.

First thing that I have to think about mathematics education in school is a fact that regrettably, school education has been usually only attempting to obtain a knowledge of elementary facts and methods from teaching of mathematics. Unfortunately, students who are objectives of mathematics education have confessed their difficulty in getting access to learning of mathematics. What do we need to know to make some changes in this sad reality? Let us refer to the following by A. G. Howson and J.P. Kahane (1990). They identified four components of 'mathematical culture':
(a) a knowledge of elementary facts and methods,
(b) the development of a certain way of thinking and of approaching problems,
(c) some knowledge of the history of mathematical concepts and theories,
(d) some knowledge of recent developments.

I think that I have already an answer to the question that I posed in the above. We need a shared recognition within the mathematics education society to realize the four components by A. G. Howson and J.P. Kahane. If a mathematics teacher considers the four components of 'mathematical culture', then he (or she) has to think of a need to develop other objectives in his (or her) class design except for transmitting a knowledge of elementary facts and methods. He (or she) should bring as many topics as psychology of mathematics, history of mathematics, using technology in teaching of mathematics, gender issues and its relatedness, and possible ideas from and outside of mathematics education. Since we talked about problem solving, effective teaching of mathematics, application of science and other outside of mathematics into mathematics, and so on through the discussions of EMT 705 we are comparably familiar with (b). In fact, I think that we have spend quite amount of time to solve problems from not considering (b). But, we didn't talk much about (c) some knowledge of the history of mathematical concepts and theories, and (d) some knowledge of recent developments.
Histories of mathematical concepts and theories will be able to enlighten students to know how human endeavors have kept on trying to find something new. This will show that history of mathematics is an endless process of human efforts.
When it comes to talking about some knowledge of recent developments, it is important to know that they are not necessarily curriculum-driven. That is to say, they may not have a deep relationship with mathematics which students have to learn in their school. But, for example, talking about a young mathematician who claimed to solve the Fermat's Last Theorem will intrigue students' interest to learn more about mathematics. A simple illustration of the Fermat's Last Theorem and its connection to a class design will be showed in CLASS DESIGN section of this paper.
Teachers need to use various educational materials to change students' attitude toward mathematics. Once attitudes have been changed, then learning mathematics may be significantly affected.
Second issue is a thought of where most of people learn mathematics and the role of a mathematics teacher. It is obvious that the only contact that most of people ever have with mathematics is at school. Isn't it enough for teaching of mathematics to have a change in its feature? Then, who have to take major responsibility for the change? I think that it will be mathematics teachers. If a mathematics teacher think that most people leave school knowing almost nothing about mathematics, then he (or she) have to change his teaching of mathematics. I think that a good initial way of the change is to introduce some important mathematicians and their works. For example, instead of just teaching the Pythagorean Theorem in geometry a teacher should be able to make a connection to ancient people who tried to figure the principle out. Teaching of the Pythagorean Theorem is going to be introduced in CLASS DESIGN section to make a connection between learning mathematics and a knowledge of the history of mathematical concepts and theories.
Third issue that we have to think about is a possible influence of mass media on mathematics education community. We have not considered the power of mass media as a good transmitant of mathematics to students, parents, business people and politicians. This idea is related with popularization of mathematics in a broad sense. How can we make people know about new movements happening in teaching of mathematics? Most of all, what we have to do is persuade people out side of school understand an effort to change the system of education, and its implementation. As population of mathematics diversifies, we need to find good medium to send messages from the mathematics educational society. The characteristics of mass media will be able to transmit the diversity of mathematics without loss of interest from general population. Books and magazines, newspapers, radio, TV and films, and exhibitions can be good medium.
But, we should understand the characteristics of each medium and make the best of them. Even though we can not expect students to learn significant mathematics from books, and magazines the students may be able to start reading books on mathematics, not making specifically mathematical paper, but matching mathematics with philosophy, culture, music and art. We have seen the possibility of using newspapers for teaching of mathematics since newspapers are full of mathematical topics for students' exploration. Not only do mathematics teachers use newspapers for obtaining meaningful mathematical information, but also do newspapers play a role of communicating current issue of mathematics to the public. Just as scientific and technological advances are reported as big events, mathematical topics and related educational issues should be disclosed to the public.
The influence of radio may seem to be unclear because of the limitation of transmitting ideas only with verbal representation. But, one that that we should notice is that radio can force students to visualize. When we talked about the communication problem in teaching and learning of mathematics we could know that various way of communication should be taught to students. Listening is one of important issues in communication as well as Reading, writing, and speaking are, and the use of radio can be an effective way for students to have a different access to learning of mathematics.
I am very fond of watching a science-oriented TV program about anthropology, ancient building, dinosaurs, plant, animals, sea, and all kinds of nature program. I think that most of students have their favorite type of programs and films. Why don't we make a TV program or films about mathematics? For example, in Korea, we have an educational channel and mathematics are treated in a broad way and have many general audience. I understand that there is a little chance of selling the idea of mathematics program, but it must be challenged because we may chose an opposite way of interpretation and say that watching TV itself is fun. Funny characters will be generalized so that students can enjoy mathematics.
Fourth issue is that we didn't talk about using games and puzzles in teaching and learning of mathematics. There exist many research results about the effectiveness of using games and puzzles in mathematics education. Of course, not all people do agree with the idea, but, there are significant facts that we can bring games and puzzles in mathematics education. Most of all, games and puzzles can bring considerable interest and have the capacity to encourage students to explore the problems with mathematical thought or reasoning. In particular, the environment of games and puzzles provide many chances for cooperative learning among students. If a teacher wants to see an influence of using games and puzzles in a view point of mathematics, he (or she) may not see a direct result right after giving students some games or puzzles. But I think that they will serve to foster mathematical thought.
Fermat's last theorem which will be introduced in my Class Design section as an extension of a series of mathematical concept can be also viewed as an example of a game. Because there have been uncountable open problems of an amusing and attractive appearance that are possibly as easy to state and as difficult to solve in the history of mathematics. For a game or puzzle to be a good mathematical medium it is important for the game or puzzle to have some characteristics so that students can challenge with their imagination.
The final issue will be about problems from gender issues. Though we have talked a lot about gender and its influence on teaching of mathematics, I think, it is still difficult for teachers to take an action not to have problems from gender-related situation. According to Bridget Arvold, Pamela Turner, and Thomas J. Cooney from Mathematics Teacher (p. 328 April, 1996) inequities in the classroom can easily threaten a productive learning environment and a teacher's awareness of equity issues presents a significant challenge. It is again ended up with giving another responsibility to mathematics teachers. What can they do? I think that gathering information on gender equity from their classroom teaching will be the first step. This issue can be related to teaching as profession. Once a teacher starts analyzing his ( or her ) own class and engaging him(or her) in helping students learn, then the reward of his (or her) action will be back to him(her) as a form of confidence about his(or her) role in school. There is a good saying from R. Brown and T. Porter about profession. "The difference between a professional and an amateur is that an amateur can do things, in many cases as well as a professional, but a professional also knows how he (or she) does things." A teacher is a significant person to all the students in a classroom, and he(or she) doesn't have to forget this fact.

The main flaw of this paper can be understood in a way of popularization of mathematics. In particularly, it is evident that mathematics education community alone will not be able to achieve all the goals. We need wide range of support from other areas of society. Finally, let us think about the well-known cycle happening in mathematics education. Just imagine that the quality of mathematics teachers and that of teaching mathematics is lower. Then, we have come up with a fact that the image of mathematics from students is poor. Finally, there is a shortage of those wishing to teach, and we end up with the low quality of mathematics teacher and that of students' mathematical ability. This bad cycle should be broken away by all the efforts from people who care about the future of mathematics education in our school. It is not difficult to define what mathematics is about since there are so many issues to be addressed. But, it is evident that we need other tools for real mathematics and teaching and learning of mathematics. I hope that these five issues can be realized in mathematics education community.


The mathematical concept starts from a simple situation. I will be treating a triangle with my students. In particular, three sides of a triangle and their relationship will be considered to make a connection to the famous Pythagorean Theorem. That is to say, inequality from the sum of three sides of a triangle will be addressed with actual measuring and comparing of students' activity. Then, as a natural transition of human curiosity grows, the activity goes to a question of finding positive numbers which satisfies the Pythagorean Theorem. The theorem is going to be viewed as an extension from a linear inequality of three variables to a quadratic equation with three variables. Here is a significant bias for me to choose the Pythagorean Theorem from many current mathematical topics in secondary school. It is because most of mathematics teachers may agree that the Pythagorean theorem is one of the most important properties in geometry in secondary school. In fact, the Curriculum and Evaluation Standards (NCTM 1989) stressed the importance of the theorem (p. 113). I thought that mathematics teachers should check that how meaningfully students can understand the theorem. To make an effective learning environment many issues from EMT 705 will be supplemented to teaching of the Pythagorean Theorem.
Then, I will suggest another connection to the Fermat's Last Theorem of xn + yn= zn . We have known the famous story about the proof of the theorem. And it was declared by a mathematician that the theorem was proved completely. Though a deep mathematical understanding can not place among secondary school mathematics classroom it would be very helpful for a teacher to introduce many possible issues raised around the Fermat's Last Theorem. I hope that students will be able to learn a meaningful knowledge.

(1) x+y=z in a basic Algebraic way of thinking.
If a teacher wants to teach an equation x+y=z with three unknown variables, then he (or she) may have to give some restrictions on x, y, and z so that students can obtain any solution. Let's say that the restriction is that all variables should be integers. (-1, 1, 0), (1, 1, 2) or (2,3,5) can be talked by students. What happens if the variables should be positive integers? Then some answers from the previous situation will be deleted. No matter how different sets of solutions are among students they don't seem to have a problem in getting an answer. Of course, though the domain of x, y, and z are extended to complex numbers the problem will be fairly accessible to students.

(2) Is x+y=z possible in a area of Geometry? ( a triangle approach).
Now, let us bring this x+y=z problem in the area of Geometry. Let students make an arbitrary triangle with thread, wire, or any material that they like to play with. Then have them measure the each side of the triangle. Can you obtain a triangle satisfying the equation x+y=z with x, y, and z are the sides of a triangle? No! Students will obtain only the case of x+y>z. The following are some examples. First, a triangle which looks like a right triangle is drawn. Let us measure the each side of the triangle. The longest side is less than the sum of the other two ( 7.9 < 7.2 + 3.1 ).

In the next example, a triangle looks more like an equilateral triangle. How about the relationship between the sides of the triangle? It does not satisfy the equation, either. The relationship of the sides of the triangle is still an inequality ( 6.4 < 5.0 + 4.8 ).

The intuition says that the triangle should be thinner so that it has a long side compared with two other sides. Let's consider the next one and make a calculation with three sides of the triangle.

Students will find out that they still have an inequality and any triangle doesn't seem to fit the equation x+y=z with the three sides of a triangle. What makes this feature? The teacher can explain now about the components of a triangle as three sides, three points, and three angle. He (or she ) can continue to explain the angles of a triangle if he (or she) wants to give a deep reason to the impossibility of the problem.
Students will understand that there will be no triangle which can satisfies the equation. Even though the idea can not understood by exploring a small number of triangles the main idea of this problem is for a teacher to show the role of some restrictions which can be applied to the world of mathematics.

(3) Why don't you try to bring the Pythagorean theorem now to students?
Now, I want to extend the linear equation x+y=z into the quadratic equation a2 + b2= c2. There is a reason to use a, b, and c as the variables instead of using x, y, and z. It is to make an easy connection to the sides of a right triangle and the Pythagorean theorem. Before starting teaching the theorem, I will talk about the background of the Pythagorean theorem. A teacher will tell about a fact that the Pythagorean theorem was known more than a thousand years before Pythagoras. How does the theorem have his name on it? Just as many other mathematical theorems did take some special name for the memory of a person who contribute much to a theorem, the Pythagorean theorem had the same history. Students will be able to trace back to 2000 B.C. when the Egyptians realized that 42 + 32= 52 , though there is no evidence that the Egyptians knew or could prove the right angle property of the figure involved.
Can students find (a, b, c) satisfying the quadratic equation a2 + b2= c2 ?

They might not feel difficult in getting an answer set since they have already leaned the Pythagorean theorem and remember the numbers like (1, 1, ), (3,4, 5) or (5, 12, 13). But, students have to start randomly with a number just as they did with the linear equation, x+y=z, then they will find out that getting an answer is not as easy as they think. We can omit the case of (1, 1, ) as a solution by restricting the domain of variables to integers for now.
Here is another problem which has to be solved in teaching of the Pythagorean theorem in high school. I could notice the same type of mistake from my students in Korea. Does it just exist in students' mind as an equation of three symbols as "a2 + b2= c2 "? Students may not handle even simple computations using the formula. Or, they may do not know that c should be the hypotenuse of a triangle, and b and c two legs of the triangle in the formula. A teacher will be able to check this by offering "c2 + b2= a2 " instead of "a2 + b2= c2" to his students. My experience with second grade students in WooSuck Girls' High School (Korea, 1994) said that about 50% of students had a difficulty when I changed the formula into a different format. The students seemed to think that c should always be the hypotenuse. So, I had to spend quite amount of time for some remedial class. This problem is explained in a way of the problems of mathematical representation and its internalization to students. What does a mathematical representation mean to students? How do students interpret a mathematical formula which is given by their teacher? Teachers will be able to understand any mathematical concept in students' point of view.

(4) What is new in Fermat's last theorem, xn + yn= zn from the previous process?
How is it interesting to know that that mathematicians seriously attacked, in general terms, the problem of finding positive integer solutions to the equation xn + yn= zn only after 17 century A.D. while the history of Pythagorean theorem goes back to 1600 B.C.? Because the students in this Class Design may think of the connection between Pythagorean theorem and Fermat's last theorem.
I think that a main reason of the fame of Fermat's last theorem is in the simplicity of the problem itself. The theorem seems simple enough to find the answer and that's what many people have been trying to prove the theorem. That is to say, the theorem has an attractive appearance. But, as the history of mathematics shows us the process of finding the way of proof was a long way to human being. This theorem itself is not an appropriate topic to secondary school students, and a teacher can not ask his (or her ) students to find positive integer solutions to the equation xn + yn= zn. But, a teacher may present the theorem and its related historical stories. I remember when I had fun reading a note of Fermat ( p. 79 Historical Topics for the Mathematics Classroom, NCTM).
" To divide a cube into two cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it. "

Not only does a teacher can introduce the stories about Fermat and his note, but also can he (or she) talk about a young mathematician who claimed to prove the Fermat's Last Theorem. Teachers will be able to bring as many as topics from outside of mathematics to their classroom.


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