소수와 소인수분해는 우리나라 중학교 학교수학(school mathematics)에서 제1차 교육과정부터 빠지지 않고 나오는 교육내용이다. 따라서 본 연구에서는 이런 중요성을 가진 소수와 소인수분해의 수학적 배경지식을 강화하고자 첫째, 소수와 소인수분해가 나오게 된 대수적(algebra) 배경을 먼저 알아본다. 둘째, 이때 필요에 의해 나온 개념들을 잘 살려 교육내용을 편성․개선한다면 어떤 점을 강조해서 지도해야 하는지에 대한 ...
소수와 소인수분해는 우리나라 중학교 학교수학(school mathematics)에서 제1차 교육과정부터 빠지지 않고 나오는 교육내용이다. 따라서 본 연구에서는 이런 중요성을 가진 소수와 소인수분해의 수학적 배경지식을 강화하고자 첫째, 소수와 소인수분해가 나오게 된 대수적(algebra) 배경을 먼저 알아본다. 둘째, 이때 필요에 의해 나온 개념들을 잘 살려 교육내용을 편성․개선한다면 어떤 점을 강조해서 지도해야 하는지에 대한 방향을 제안하고자 한다. 즉, 필요에 의해 발생한 대수학적 내용과 흐름이 녹아있는 교육내용을 담고자 하였다.
소수와 소인수분해는 우리나라 중학교 학교수학(school mathematics)에서 제1차 교육과정부터 빠지지 않고 나오는 교육내용이다. 따라서 본 연구에서는 이런 중요성을 가진 소수와 소인수분해의 수학적 배경지식을 강화하고자 첫째, 소수와 소인수분해가 나오게 된 대수적(algebra) 배경을 먼저 알아본다. 둘째, 이때 필요에 의해 나온 개념들을 잘 살려 교육내용을 편성․개선한다면 어떤 점을 강조해서 지도해야 하는지에 대한 방향을 제안하고자 한다. 즉, 필요에 의해 발생한 대수학적 내용과 흐름이 녹아있는 교육내용을 담고자 하였다.
As far as 2,000 years ago, the ancient Greeks understood all the numbers can be represented as multiplications of prime numbers, and many mathematicians have paid great effort on prime decompositions of numbers. This is not because prime numbers and primary decomposition are the only intriguing aspe...
As far as 2,000 years ago, the ancient Greeks understood all the numbers can be represented as multiplications of prime numbers, and many mathematicians have paid great effort on prime decompositions of numbers. This is not because prime numbers and primary decomposition are the only intriguing aspects of mathematics but people have known that primary decompositions are the critical elements of the modern crypto system and that prime numbers are important as the basic unit composing numbers in multiplication. With their importance, since the 1st Educational Curriculum, prime numbers and primary decompositions have taken important parts in middle school mathematics of Korea. This study focuses on providing sufficient background knowledge for prime numbers and primary decompositions. Above all, this study reviews the introduction of prime numbers and primary decompositions on the aspects of algebra. Secondly, the concepts and ideas, which appeared on the process of their introduction, are carefully checked to provide appropriate suggestions for the better revisions and developments for mathematics education. That is, the researcher tried to present educational contents which shows the ideas of algebra and the process in which these ideas appear as well. Chapter Ⅱ, providing the algebraic background of prime numbers and primary decompositions, can be summarized as follows. When we want the information of a random set , this can be identified by linking to an already known set. This means, we can identify the structure of by transferring the operation of the known set to using the onto function and one-to-one function. In this process the known set is , as this is a typical ‘ring’ and with additional conditions turns into a ‘field.’ The structure of , however, cannot be identified only through a field, because it takes only ‘0’ and itself as its ‘ideal.’ So we need , which calls for studies on various kinds of ideals. Now, the onto function which shows the correspondence between a set of integers and set is defined as ‘’, and the relation ‘’, which is formed by function is defined as: when , is . Here ‘’ becomes a equivalence relation and a partition is created in . Then a set of equivalence class, , appears. Both and are isomorphic here. That shows the structure of set and are identical and this is the process of identifying the structure of . Now the operations ‘+’ and ‘․’ need to be moved to through , and in this process ‘ideal’ plays a critical role. Euclidean Division Algorithm is the key to prove that the ideal takes the form of PID as . When the ideal has the structure of , is , and needs to be decomposed as to identify the structure of . Here comes the necessity for the ideas of prime numbers and primary decompositions, and the research on the structure of ‘lattice’, which is the relationship between ideals. So the ideas of lattice, PID, Euclidean Division Algorithm, one-to-one function, and onto function have come in as algebraic background for prime numbers and primary decompositions. Chapter Ⅲ elaborates on the problems with background ideas in the current educational curriculum and suggests possible revisions and improvements. The following are the distinctive parts in Chapter Ⅲ. Firstly, UFD is the basis for the greatest common divisors and least common multiples. This, however, cannot provide proper explanation for the greatest common divisors and least common multiples of ‘0’ and negative integers. So it is suggested that we teach the relative structures of ‘lattices’ in relations with sets, integers, and flat surfaces to help students enlarge their idea span. Only the lessons with sets deal with this relative structures of lattices currently. Secondly, Euclidean Division Algorithm, which proved that the ideal of is PID, surely deserves greater importance in textbooks, so a variety of educational contents need to be introduced to help recognize the importance of it. Introducing the idea of ‘rational number’ and ‘irrational number’ through Euclidean Division Algorithm is suggested as an example educational content. Thirdly, the fundamental role of functions is to transfer the structures, but this role is not fully explained. An effective way to introduce this basic role of functions is suggested using various examples of multiplication of matrix and symmetric difference, etc. With this new approach, students have chances to understand the background knowledge for these new operations, which in turn help them appreciate the necessity for the introduction of the operations instead of forcing them mechanical calculations. This study may find its meaning in providing contents, which appear when necessary, based the core aspects like prime numbers and primary decompositions, not just presenting the encyclopedic contents of mathematics education.
As far as 2,000 years ago, the ancient Greeks understood all the numbers can be represented as multiplications of prime numbers, and many mathematicians have paid great effort on prime decompositions of numbers. This is not because prime numbers and primary decomposition are the only intriguing aspects of mathematics but people have known that primary decompositions are the critical elements of the modern crypto system and that prime numbers are important as the basic unit composing numbers in multiplication. With their importance, since the 1st Educational Curriculum, prime numbers and primary decompositions have taken important parts in middle school mathematics of Korea. This study focuses on providing sufficient background knowledge for prime numbers and primary decompositions. Above all, this study reviews the introduction of prime numbers and primary decompositions on the aspects of algebra. Secondly, the concepts and ideas, which appeared on the process of their introduction, are carefully checked to provide appropriate suggestions for the better revisions and developments for mathematics education. That is, the researcher tried to present educational contents which shows the ideas of algebra and the process in which these ideas appear as well. Chapter Ⅱ, providing the algebraic background of prime numbers and primary decompositions, can be summarized as follows. When we want the information of a random set , this can be identified by linking to an already known set. This means, we can identify the structure of by transferring the operation of the known set to using the onto function and one-to-one function. In this process the known set is , as this is a typical ‘ring’ and with additional conditions turns into a ‘field.’ The structure of , however, cannot be identified only through a field, because it takes only ‘0’ and itself as its ‘ideal.’ So we need , which calls for studies on various kinds of ideals. Now, the onto function which shows the correspondence between a set of integers and set is defined as ‘’, and the relation ‘’, which is formed by function is defined as: when , is . Here ‘’ becomes a equivalence relation and a partition is created in . Then a set of equivalence class, , appears. Both and are isomorphic here. That shows the structure of set and are identical and this is the process of identifying the structure of . Now the operations ‘+’ and ‘․’ need to be moved to through , and in this process ‘ideal’ plays a critical role. Euclidean Division Algorithm is the key to prove that the ideal takes the form of PID as . When the ideal has the structure of , is , and needs to be decomposed as to identify the structure of . Here comes the necessity for the ideas of prime numbers and primary decompositions, and the research on the structure of ‘lattice’, which is the relationship between ideals. So the ideas of lattice, PID, Euclidean Division Algorithm, one-to-one function, and onto function have come in as algebraic background for prime numbers and primary decompositions. Chapter Ⅲ elaborates on the problems with background ideas in the current educational curriculum and suggests possible revisions and improvements. The following are the distinctive parts in Chapter Ⅲ. Firstly, UFD is the basis for the greatest common divisors and least common multiples. This, however, cannot provide proper explanation for the greatest common divisors and least common multiples of ‘0’ and negative integers. So it is suggested that we teach the relative structures of ‘lattices’ in relations with sets, integers, and flat surfaces to help students enlarge their idea span. Only the lessons with sets deal with this relative structures of lattices currently. Secondly, Euclidean Division Algorithm, which proved that the ideal of is PID, surely deserves greater importance in textbooks, so a variety of educational contents need to be introduced to help recognize the importance of it. Introducing the idea of ‘rational number’ and ‘irrational number’ through Euclidean Division Algorithm is suggested as an example educational content. Thirdly, the fundamental role of functions is to transfer the structures, but this role is not fully explained. An effective way to introduce this basic role of functions is suggested using various examples of multiplication of matrix and symmetric difference, etc. With this new approach, students have chances to understand the background knowledge for these new operations, which in turn help them appreciate the necessity for the introduction of the operations instead of forcing them mechanical calculations. This study may find its meaning in providing contents, which appear when necessary, based the core aspects like prime numbers and primary decompositions, not just presenting the encyclopedic contents of mathematics education.
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