CC221104 Practice in Introduction to Geometry

Hiroshima University Syllabus

Japanese

Academic Year 2024Year School/Graduate School School of Education

Lecture Code CC221104 Subject Classification Specialized Education

Subject Name 幾何学概論演習

Subject Name
（Katakana）キカガクガイロンエンシュウ

Subject Name in
English Practice in Introduction to Geometry

Instructor TERAGAITO MASAKAZU

Instructor
(Katakana) テラガイト　マサカズ

Campus Higashi-Hiroshima Semester/Term 2nd-Year,  Second Semester,  4Term

Days, Periods, and Classrooms (4T) Weds1-4：EDU K102

Lesson Style Seminar Lesson Style
(More Details) 　

Lecture-oriented, Slides

Credits 2.0 Class Hours/Week 　 Language of Instruction J : Japanese

Course Level 2 : Undergraduate Low-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Students of Faculty of Education

Keywords metric space, topological space

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students)
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Integrated Arts and Sciences
（Knowledge and Understanding）
・Knowledge and understanding of the importance and characteristics of each discipline and basic theoretical framework.
（Abilities and Skills）
・The ability and skills to specify necessary theories and methods for consideration of issues.

Secondary School Mathematics Education
（Knowledge and Understanding）
・To understand basic knowledge of teaching contents of mathematic education.
（Abilities and Skills）
・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.

Class Objectives
/Class Outline This course will provide students with lectures on set theory (introduction to mathematics in the 1st semester) which is the basis of mathematics in university education, followed by the lectures on the basics of metric space and topological space, with an aim to generalize and abstract space concepts in secondary education.
Although you are education major, it is not desirable to graduate without learning the concept of ‘topology’, which is the basis of almost all areas of mathematics.
A topological space is highly abstract idea.
This course will provide the students of the Faculty of Education, who tend to seek images in learning mathematics, with an opportunity to experience abstraction so that they can expand their view. The importance of having logical thinking because you cannot image will be emphasized in the lectures.

Class Schedule Lesson1: Review from Set Theory (Set)

Lesson2: Review of Set Theory (Mapping and Equivalence relation )

Lesson3: Metric Space (Distance Function)

Lesson4: Distance Function ( Specific Examples)

Lesson5: Open Set

Lesson6: Closed Set

Lesson7: Continuous Mapping between Metric Spaces

Lesson8: Convergence of Sequences

Lesson9: Topology
Lesson10: Interior, Exterior, Frontier, and Closure

Lesson11: Continuous Mapping betweeness Topological Spaces

Lesson12: Relative Topology and Subspace

Lesson13: Hausdorff Space

Lesson14: Compact Space

Lesson15: Connectivity

There will be a final exam. No make-ups for this exam will be given. If you cannot take a final exam, you will have to give up credits.

If you hear the term ‘ set’ which is just an aggregation of points, you think that there are no relations between the two points and points are isolated. However if you hear the term ‘ space’, you can image some sorts of distance or relations. The essence of this is what we call ‘topology’. In this course, we will first learn metric space with essence of xy-plane and xyz-space that are covered in secondary mathematics education, and then learn more abstract topological space. There will a simple quiz to check the students’ learning level at the beginning of every week. Data of finished quizzes will be available through moodle provided by the Media Center.

Text/Reference
Books,etc. Lecture style. A textbook will be used. The detail will be given in the first lesson.
There are a number of reference books about General Topology.
Students should read those books. You can find those books in the library.．

PC or AV used in
Class,etc. 　

(More Details) Slides

Learning techniques to be incorporated 　

Suggestions on
Preparation and
Review Lesson1-2: Briefly review minimum required knowledge of set theory.
It is desirable that you have taken ‘ Introduction to Mathematics ‘ in Semester 1. Regardless of whether you have taken the course or not, you should review sets, intersection, union, complement,  de Morgan's laws, family of sets, mapping, image, inverse image, equivalence relation, quotient set, and cardinality before the lesson starts.

After that, the lesson will be carried out based on the textbook. Preparation is not required. You should review the lessons thoroughly. If there is anything not clear to you, you should try to resolve them by all means before the next lesson.

Lesson3-4：Learn metric space. You have been familiar with the concept of distance since you were elementary school students. In the lessons, we will start with rethinking the concept of distance strictly from the aspect of mathematics, which allows you to introduce the concept of distance in various sets.

Lesson5-6: Learn the concept of open sets and closed sets that play a basic role in metric space. You should understand that finding out the property of a family of open sets, rather than that of individual open set, will be a motive to introduce topological spaces.

Lesson7: Learn continuous mapping between metric spaces. This is generalized concept of continuous functions by the Epsilon Delta Method that you studied in the 1st year. You should not be obsessed with the term ‘ continuous’ because this is a theory of metric spaces, which is invisible world.

Lesson8: Generalize a convergence of sequences on a real number line into the concept of a convergence of point sequences in metric space. This is a natural generalization of the Epsilon Delta method you learned in the 1st year.

Lesson9-10: Introduce topological space. This is an idea of axioms of topology using properties of family of open sets in metric space in an opposite way. Topology is invisible and is a ‘ structure’ added on a set.

Lesson11: Study continuous mapping and homeomorphisms.

Lesson12: A topology that is naturally induced to a subset in a topological space is called relative topology.

Lesson13: Hausdorff space is a topological space with sufficient open sets

Lesson14: It is often said that compactness is difficult to understand. A compact space does not mean having an open covering that consists of a finite number of open sets．
Make sure you do not make a mistake in this point. Whatever an open covering is given, finite subcovering of the open covering can be selected.

Lesson15: Lastly, study ‘connectivity’ meaning topological spaces are connected. The term, connected, is often used in our daily life. However, connectivity that is used to define topological spaces is described in terms of open sets.

Requirements Special Subject (Geometry)
You should review what you learned after every lesson. This course requires the fundamental knowledge of sets and mappings. Students should study by themselves and/or review those items before the first lesson starts. There will be a quiz about those in the first lesson. Students should sit in the front rows. I don’t need students who sit in the back rows by choice.

Grading Method Students will be evaluated based on final exam: 100%

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message The contents of the lesson cannot be unique to the Faculty of Education since this is a fundamental area of mathematics. General Topology that is covered in this course has already established. Unless an extremely unusual theory is constructed, there will be no difference in contents between the one offered by Dept. of Mathematics, Faculty of Science and Science, Technology and Society Education Course of Faculty of Education as a matter of course. However, this course will often talk about comparison in teaching of graphic concept between elementary education and secondary education as a course offered by Faculty of Education. For example, the instructor will talk about metric space that is behind simple questions such as ‘ What is a circle?’ ‘Why is a circle round ?’ ‘ Is it a circle because it has a round shape?’ Since many of the students wish to become a teacher, students’ active participation in class such as sharing opinions and asking questions is encouraged. Students should not get used to participate in lessons in a passive manner if they want to become a teacher.
Every year, some students struggle in understanding the course. It seems that they do not understand the concept of family of sets, which is a set of sets, in the first place. You will eventually understand it if you do not think about it alone and are willing to discuss thoroughly with your friends without giving up.
The grade of this course will be referred to when screening students for ‘Study of Instructional Materials in Geometry’ in Semester 6.

Other Please refer to the following websites:
http://mathworld.wolfram.com/TopologicalSpace.html
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hausdorff.html
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html
"

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2024Year	School/Graduate School	School of Education
Lecture Code	CC221104	Subject Classification	Specialized Education
Subject Name	幾何学概論演習
Subject Name （Katakana）	キカガクガイロンエンシュウ
Subject Name in English	Practice in Introduction to Geometry
Instructor	TERAGAITO MASAKAZU
Instructor (Katakana)	テラガイト　マサカズ
Campus	Higashi-Hiroshima	Semester/Term	2nd-Year, Second Semester, 4Term
Days, Periods, and Classrooms	(4T) Weds1-4：EDU K102
Lesson Style	Seminar	Lesson Style (More Details)
Lecture-oriented, Slides
Credits	2.0	Class Hours/Week		Language of Instruction	J : Japanese
Course Level	2 : Undergraduate Low-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Students of Faculty of Education
Keywords	metric space, topological space
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Integrated Arts and Sciences （Knowledge and Understanding）・Knowledge and understanding of the importance and characteristics of each discipline and basic theoretical framework. （Abilities and Skills）・The ability and skills to specify necessary theories and methods for consideration of issues. Secondary School Mathematics Education （Knowledge and Understanding）・To understand basic knowledge of teaching contents of mathematic education. （Abilities and Skills）・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.
Class Objectives /Class Outline	This course will provide students with lectures on set theory (introduction to mathematics in the 1st semester) which is the basis of mathematics in university education, followed by the lectures on the basics of metric space and topological space, with an aim to generalize and abstract space concepts in secondary education. Although you are education major, it is not desirable to graduate without learning the concept of ‘topology’, which is the basis of almost all areas of mathematics. A topological space is highly abstract idea. This course will provide the students of the Faculty of Education, who tend to seek images in learning mathematics, with an opportunity to experience abstraction so that they can expand their view. The importance of having logical thinking because you cannot image will be emphasized in the lectures.
Class Schedule	Lesson1: Review from Set Theory (Set) Lesson2: Review of Set Theory (Mapping and Equivalence relation ) Lesson3: Metric Space (Distance Function) Lesson4: Distance Function ( Specific Examples) Lesson5: Open Set Lesson6: Closed Set Lesson7: Continuous Mapping between Metric Spaces Lesson8: Convergence of Sequences Lesson9: Topology Lesson10: Interior, Exterior, Frontier, and Closure Lesson11: Continuous Mapping betweeness Topological Spaces Lesson12: Relative Topology and Subspace Lesson13: Hausdorff Space Lesson14: Compact Space Lesson15: Connectivity There will be a final exam. No make-ups for this exam will be given. If you cannot take a final exam, you will have to give up credits. If you hear the term ‘ set’ which is just an aggregation of points, you think that there are no relations between the two points and points are isolated. However if you hear the term ‘ space’, you can image some sorts of distance or relations. The essence of this is what we call ‘topology’. In this course, we will first learn metric space with essence of xy-plane and xyz-space that are covered in secondary mathematics education, and then learn more abstract topological space. There will a simple quiz to check the students’ learning level at the beginning of every week. Data of finished quizzes will be available through moodle provided by the Media Center.
Text/Reference Books,etc.	Lecture style. A textbook will be used. The detail will be given in the first lesson. There are a number of reference books about General Topology. Students should read those books. You can find those books in the library.．
PC or AV used in Class,etc.
(More Details)	Slides
Learning techniques to be incorporated
Suggestions on Preparation and Review	Lesson1-2: Briefly review minimum required knowledge of set theory. It is desirable that you have taken ‘ Introduction to Mathematics ‘ in Semester 1. Regardless of whether you have taken the course or not, you should review sets, intersection, union, complement, de Morgan's laws, family of sets, mapping, image, inverse image, equivalence relation, quotient set, and cardinality before the lesson starts. After that, the lesson will be carried out based on the textbook. Preparation is not required. You should review the lessons thoroughly. If there is anything not clear to you, you should try to resolve them by all means before the next lesson. Lesson3-4：Learn metric space. You have been familiar with the concept of distance since you were elementary school students. In the lessons, we will start with rethinking the concept of distance strictly from the aspect of mathematics, which allows you to introduce the concept of distance in various sets. Lesson5-6: Learn the concept of open sets and closed sets that play a basic role in metric space. You should understand that finding out the property of a family of open sets, rather than that of individual open set, will be a motive to introduce topological spaces. Lesson7: Learn continuous mapping between metric spaces. This is generalized concept of continuous functions by the Epsilon Delta Method that you studied in the 1st year. You should not be obsessed with the term ‘ continuous’ because this is a theory of metric spaces, which is invisible world. Lesson8: Generalize a convergence of sequences on a real number line into the concept of a convergence of point sequences in metric space. This is a natural generalization of the Epsilon Delta method you learned in the 1st year. Lesson9-10: Introduce topological space. This is an idea of axioms of topology using properties of family of open sets in metric space in an opposite way. Topology is invisible and is a ‘ structure’ added on a set. Lesson11: Study continuous mapping and homeomorphisms. Lesson12: A topology that is naturally induced to a subset in a topological space is called relative topology. Lesson13: Hausdorff space is a topological space with sufficient open sets Lesson14: It is often said that compactness is difficult to understand. A compact space does not mean having an open covering that consists of a finite number of open sets． Make sure you do not make a mistake in this point. Whatever an open covering is given, finite subcovering of the open covering can be selected. Lesson15: Lastly, study ‘connectivity’ meaning topological spaces are connected. The term, connected, is often used in our daily life. However, connectivity that is used to define topological spaces is described in terms of open sets.
Requirements	Special Subject (Geometry) You should review what you learned after every lesson. This course requires the fundamental knowledge of sets and mappings. Students should study by themselves and/or review those items before the first lesson starts. There will be a quiz about those in the first lesson. Students should sit in the front rows. I don’t need students who sit in the back rows by choice.
Grading Method	Students will be evaluated based on final exam: 100%
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message	The contents of the lesson cannot be unique to the Faculty of Education since this is a fundamental area of mathematics. General Topology that is covered in this course has already established. Unless an extremely unusual theory is constructed, there will be no difference in contents between the one offered by Dept. of Mathematics, Faculty of Science and Science, Technology and Society Education Course of Faculty of Education as a matter of course. However, this course will often talk about comparison in teaching of graphic concept between elementary education and secondary education as a course offered by Faculty of Education. For example, the instructor will talk about metric space that is behind simple questions such as ‘ What is a circle?’ ‘Why is a circle round ?’ ‘ Is it a circle because it has a round shape?’ Since many of the students wish to become a teacher, students’ active participation in class such as sharing opinions and asking questions is encouraged. Students should not get used to participate in lessons in a passive manner if they want to become a teacher. Every year, some students struggle in understanding the course. It seems that they do not understand the concept of family of sets, which is a set of sets, in the first place. You will eventually understand it if you do not think about it alone and are willing to discuss thoroughly with your friends without giving up. The grade of this course will be referred to when screening students for ‘Study of Instructional Materials in Geometry’ in Semester 6.
Other	Please refer to the following websites: http://mathworld.wolfram.com/TopologicalSpace.html http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hausdorff.html http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html "
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

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