Geometry is the basic and important part of school mathematics. Among numerous theorems in mathematics, the Pythagorean theorem is one of the most important basic theorems, and has been generalized in many ways of forms. As is well-known, the Pythagorean theorem is systematically instructed and stud...
Geometry is the basic and important part of school mathematics. Among numerous theorems in mathematics, the Pythagorean theorem is one of the most important basic theorems, and has been generalized in many ways of forms. As is well-known, the Pythagorean theorem is systematically instructed and studied with numerous applications in the geometry classes in the middle school mathematics. We know that the Pythagorean theorem has numerous equivalent theorems, e.g., the Heron theorem, Pappus theorem, and the parallelogram law which are very useful in the plane geometry.
The main purposes of this thesis are to develop new teaching-learning materials of the Pythagorean triples for the mathematics gifted students' learning to enhance their mathematical thinking abilities, and using those materials, we will analyze the mathematical thinking characteristics of mathematics gifted students' shown in the process of creative problem solving, and find what has to be set in place of teacher’s teaching method and program. For this study, we choose two mathematics gifted students who are taking an enrichment mathematics course at the education institute for the gifted students. Then we observe and interview with the students how to find and handle many expressions on the algorithm and characteristics of Pythagoreans triples with g-gap. This kind of teaching and studying method and materials on the Pythagoreans theorem unit are very helpful and instructive for the gifted students' learning to enhance their mathematical thinking abilities.
The contents of this thesis are organized as follows:
First, in the chapter of literature studies, we survey the historical developments of the Pythagorean theorem, and next show the equivalences of the Pythagorean theorem with the Heron theorem, Pappus theorem, and the parallelogram law, and next introduce some generalizations and the various proofs of those theorems.
In the main chapter of this thesis, we give an algebraic-geometric proof for the Pythagorean triples, and investigate the algorithm for generating Pythagorean triples with -gap. And we give several forms of teaching-learning materials on the Pythagorean triples with 1-gap. This kind of new teaching-learning method and materials in the Pythagorean theorem unit are very helpful and instructive for the excellent or talented the third year of middle school students' learning to enhance their mathematical thinking abilities.
As conclusions of this thesis, we can obtain the following:
First, from the intuitive insight into problem-solving skills, mathematics gifted students know the fast way of expressing the core contents which they should solve.
Secondly, mathematics gifted students have a tendency to generalize and improve several ways of problem solving in general.
Thirdly, mathematics gifted students enjoy learning of self-driven ways, and they also prefer to prove by mathematical reasoning which is worthy of being very high achievements.
Finally, we suggest two ways of desirable teaching-learning methods for further researches of teaching the Pythagorean theorem in the class of mathematics gifted student.
Geometry is the basic and important part of school mathematics. Among numerous theorems in mathematics, the Pythagorean theorem is one of the most important basic theorems, and has been generalized in many ways of forms. As is well-known, the Pythagorean theorem is systematically instructed and studied with numerous applications in the geometry classes in the middle school mathematics. We know that the Pythagorean theorem has numerous equivalent theorems, e.g., the Heron theorem, Pappus theorem, and the parallelogram law which are very useful in the plane geometry.
The main purposes of this thesis are to develop new teaching-learning materials of the Pythagorean triples for the mathematics gifted students' learning to enhance their mathematical thinking abilities, and using those materials, we will analyze the mathematical thinking characteristics of mathematics gifted students' shown in the process of creative problem solving, and find what has to be set in place of teacher’s teaching method and program. For this study, we choose two mathematics gifted students who are taking an enrichment mathematics course at the education institute for the gifted students. Then we observe and interview with the students how to find and handle many expressions on the algorithm and characteristics of Pythagoreans triples with g-gap. This kind of teaching and studying method and materials on the Pythagoreans theorem unit are very helpful and instructive for the gifted students' learning to enhance their mathematical thinking abilities.
The contents of this thesis are organized as follows:
First, in the chapter of literature studies, we survey the historical developments of the Pythagorean theorem, and next show the equivalences of the Pythagorean theorem with the Heron theorem, Pappus theorem, and the parallelogram law, and next introduce some generalizations and the various proofs of those theorems.
In the main chapter of this thesis, we give an algebraic-geometric proof for the Pythagorean triples, and investigate the algorithm for generating Pythagorean triples with -gap. And we give several forms of teaching-learning materials on the Pythagorean triples with 1-gap. This kind of new teaching-learning method and materials in the Pythagorean theorem unit are very helpful and instructive for the excellent or talented the third year of middle school students' learning to enhance their mathematical thinking abilities.
As conclusions of this thesis, we can obtain the following:
First, from the intuitive insight into problem-solving skills, mathematics gifted students know the fast way of expressing the core contents which they should solve.
Secondly, mathematics gifted students have a tendency to generalize and improve several ways of problem solving in general.
Thirdly, mathematics gifted students enjoy learning of self-driven ways, and they also prefer to prove by mathematical reasoning which is worthy of being very high achievements.
Finally, we suggest two ways of desirable teaching-learning methods for further researches of teaching the Pythagorean theorem in the class of mathematics gifted student.
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