The future society needs a creative and interdisciplinary person. The creative and interdisciplinary person means a creative man who can create something new with a variety of thoughts. For this, ability to fuse various fields of knowledge and experience, and to deal with a new situation are needed. In this thesis, we will develop the educational programs and materials for the solutions of equation, apply them in a classroom and analyze the results. First of all, solutions of the equation are largely divided into three categories ― the geometrical perspective, the analytical geometrical perspective and the algebraic perspective.
First, the solution of the equation in the geometrical perspective is defined as obtaining a root by using the construction. It is largely divided into an essential method and a complementary method. An essential method is a Euclidean construction using only a compass and a ruler without scale. It has a solution using the concept of ratio and a solution using the concept of area. In an essential method, we can solve the quadratic equation. But, it is difficult to solve the equation of higher order than quadratic equation and we can not get an imaginary root. To overcome the limitations of this Euclidean construction, there are complementary methods using a neusis construction and an approximate construction. These complementary methods allow us to know the needs of graphs and coordinates. That is, a solution in the new perspective is required in order to overcome the limitations of the solution in the geometrical perspective.
Second, the solution of the equation in the analytical geometrical perspective is defined as a way to obtain the solution of the equation using a one-to-one correspondence with a coordinate plane. The solution is to correspond the equation to the graph, and find the intersection of graphs. It is possible to solve equation in -degree in analytical geometrical perspective. But we can not get an imaginary root and the root we found with this perspective is an approximate value. The need for a new point of view arises to overcome the limitation. That is, the algebraic perspective is required to obtain a solution by operating the equation directly.
Third, the solution of equation in the algebraic perspective is defined as a method based on Lagrange’s general theory of algebraic equation. Beyond the limits of the geometrical perspective, we can get an imaginary root as well as real roots. Also, it is possible to get the exact root, not just an approximate value. The solution of the equation in this perspective is to find all the permutations of the roots and a resolvent(the function of the roots). And in this perspective, we solve the equation by creating a new auxiliary equation of lower order from the original equation, using the relationship between the root and the coefficient. Finally, the roots of the original equation can be obtained by using the value of new auxiliary equation.
In this study, various solutions of equation were didactically organized to develop educational materials. Classes were given to four students. And this study contains ten hours of classes designed with three perspectives. The solutions of the equation in each point of view is divided into three phases : (step 1) to understand the issues, (step 2) to explore solutions and execution, (step 3) to justify.
In the geometrical perspective, the overall method was the same for each student. But detailed construction sequence and construction times were different. In the step of justifying, there were a student who logically explained with mathematical expressions and a student who directly checked the root by substituting the value of the expression. In the process of solving cubic equations, students felt the limitations of the construction and naturally recognized the need for a solution in a new perspective.
In the analytical geometrical perspective, students solved the same equation in many ways. And it was possible to observe various approaches in the justifying step, such as by assigning a number to determine or describing in writing. But students recognized that you can not find the imaginary root with the analytical geometrical perspective. They felt the need for a solution in an algebraic perspective.
The core of the solutions of equation in the algebraic perspective is symmetry. Students understood the characteristics of a resolvent and noticed the symmetry in a resolvent. Eventually the students understood the fact that the symmetry is a key to solve the equation.
According to the results of the survey conducted after the class, students understood the limitations of each perspective. Since in the algebraic perspective avails us to find an imaginary root, students selected the algebraic perspective as the best method. And most importantly, students said a symmetry and a resolvent are the most important things in the solutions of the equations. But the most interesting thing was that students selected the various perspectives.
The solutions of the equation In different perspectives help students experience the diversity of thoughts to interpret a situation in a variety of views. And solutions provide the opportunity to enhance the confidence. This study also gives the systematic experience of equation by the didactical organization, not just the limited solutions of the equation dealt in school mathematics. Thus, this study can be applied to educational materials to realize the diversity in mathematics. It will require ongoing follow-up studies on the subject.