Overview of the research by students.

What the authors introduced in this article is only a small part of the results they got. They have already presented their ideas in References [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and [11], and there are a lot of results that they have not yet published.

The authors are sure that the research by high school students is a very good way of teaching how to be creative, and to make students creative is one of the most important things in mathematics educaiton.

5.1. Prerequisite of the successful research by students.
First we are going to discuss the prerequisite of the successful research by students.

Students need someone who has the experience of research of mathematics. If the teacher is a mathematician, the condition of the research is quite good. One of the authors (Miyadera) is a mathematician who has published papers in mathematics magazines, and he could help students how to do the research.

In most of high schools it is difficult for students to find a high school teacher is a mathematician, but if their teacher has a friend who is a mathematician, then their teacher can teach students how to do a research of mathematics by the help of the friend.

There are many mathematician who are interested in the education of high school students. If they read this article and realize that the research of mathematics is possible for high school students, they will help students in the research of mathematics.

It is better for students to have a good computer algebra system. A good computer algebra system can give them a very powerful tool to explore the world of mathematics. By using a computer algebra students can do many kind of experiments that were impossible even for the best mathematicians in the past.

With computer algebra systems you can calculate very big numbers. You can expand out products. You can use many tools of calculus, too.

Usually it takes much less time to make a program with computer algebra systems than computer languages such as C, Java or Pascal. In the research of mathematics it is very important to make a program in a short time and do an experiment with it.

In the research the authors used the computer algebra system Mathematica, but there are other good computer algebra systems, too.

If you do not afford to buy expensive systems, you can use computer systems that are free to use.

Most of the computer algebra systems with good interface are quite expensive, and most of open source computer algebra systems are a little bit difficult to use, but these open source softwares are quite powerful systems.

A good problem to study is very important, and teachers have to look for good problems. Combinatorics are especially good for students. Many types of puzzles that have mathematical background are good, too.

If you use Mathematica, then Reference [12] can be a very inspiring book to use in your research of mathematics. Most of the mathematics topics in this book can be taught to high school students, and the authors of the book show the reader how to do the research.

5.1. What you need to do as a teacher.
Once you give students interesting problems, you have to wait and see. Some students may begin to do something strange, but it is very important not to stop them. The power of a computer algebra system plus the power of fresh mind of students may produce things beyond your imagination. Therefore one very important thing is to believe in the possibility of students.

For example the teacher (Miyadera) did not find the generalization of the Russian roulette game interesting until students made Figure 3.

When students get an interesting idea, it is better to make a program using computer algebra at once and study it carefully. Sometimes a small comment of a student can lead to a very interesting theory.

When students find new facts, teachers should teach them how to make a theory out of facts. Sometimes it is better to generalize , and in other times it is better to make it more concrete.

When students and teachers find a new fact and prove it, teachers should find a good conference or a magazine to present their work. This is a very important thing. Once students learn the joy of presenting new ideas to other people, they can work very hard.