Organization
Faculty of Education Professor
Position
Professor
External link
NAKAGAWA Masaki
Degree
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Doctor ( Kyoto University )
Research Interests
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シューベルトカルキュラス
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チャウ環
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等質空間
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Lie群
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旗多様体
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コホモロジー
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homogeneous spaces
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Lie groups
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cohomology
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flag manifolds
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Chow rings
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Schubert calculus
Research Areas
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Natural Science / Geometry
Education
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Kyoto University 大学院理学研究科 数学・数理解析専攻(数学)博士後期課程
2000.4 - 2002.3
Country： Japan
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Kyoto University 大学院理学研究科 数学・数理解析専攻(数学)修士課程
1998.4 - 2000.3
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Kyoto University 理学部 数学系
1994.4 - 1998.3
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Kyoto University 工学部 機械系学科
1991.4 - 1994.3
Research History
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Okayama University Graduate School of Education Associate Professor
2012.4 - 2021.3
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Kagawa National College of Technology Department of General Education Associate Professor
2011.10 - 2012.3
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Kagawa National College of Technology Department of General Education Lecturer
2009.4 - 2011.9
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Takamatsu National College of Technology 一般教育課 Lecturer
2003.4 - 2009.3
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JSPS (Japan Society for the Promotion of Science) Fellow
2001.4 - 2003.3
Professional Memberships
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Mathematical Society of Japan
2007.9
Papers
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The universal factorial Hall–Littlewood P- and Q-functions Reviewed International journal
Masaki Nakagawa, Hiroshi Naruse
FUNDAMENTA MATHEMATICAE 2023.8
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Darondeau–Pragacz formulas in complex cobordism Reviewed
Masaki Nakagawa, Hiroshi Naruse
Mathematische Annalen 2021.5
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On Heron's problem Reviewed
Masaki Nakagawa
( 27 ) 1 - 20 2021.3
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The mod 2 cohomology ring of the classifying space of the exceptional Lie group E_6 Reviewed
Masaki Kameko, Masaki Nakagawa, Tetsu Nishimoto
Proceedings of Japan Academy, Series A Mathematical Sciences 95 ( 9 ) 91 - 96 2019.11
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Universal Gysin formulas for the universal Hall--Littlewood functions Reviewed
Masaki Nakagawa, Hiroshi Naruse
Contemporary Mathematics 708 201 - 244 2018.6
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Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur P- and Q-functions Reviewed
Masaki Nakagawa, Hiroshi Naruse
Advanced Studies in Pure Mathematics 71 337 - 417 2016.12
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The integral cohomology ring of E_8/T Reviewed
Masaki Nakagawa
Proceedings of Japan Academy, Series A Mathematical Sciences 86 ( 3 ) 64 - 68 2010.3
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A description based on Schubert classes of cohomology of flag manifolds Reviewed
Masaki Nakagawa
Fundamenta Mathematicae 199 ( 3 ) 273 - 293 2008.2
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The Chern classes of the Hermitian symmetric space EVII Reviewed
Masaki Nakagawa
Far East Journal of Mathematical Sciences 20 ( 3 ) 283 - 308 2006.3
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Cohomology operations in the space of loops on the exceptional Lie group E_6 Reviewed
Masaki Nakagawa
Journal of Mathematics of Kyoto University 44 ( 1 ) 43 - 53 2004.3
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The space of loops on the exceptional Lie group E_6 Reviewed
Masaki Nakagawa
Osaka Journal of Mathematics 40 ( 2 ) 429 - 448 2003.6
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The mod 2 cohomology ring of the symmetric space EVI Reviewed
Masaki Nakagawa
Journal of Mathematics of Kyoto University 41 ( 3 ) 535 - 556 2001.9
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The integral cohomology ring of E_7/T Reviewed
Masaki Nakagawa
Journal of Mathematics of Kyoto University 41 ( 2 ) 303 - 321 2001.6
Research Projects
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職業科学者コミュニティの価値と面白さを擬似体験できる科学探究ゲームの原理の開発
Grant number：24K06413 2024.04 - 2028.03
日本学術振興会 科学研究費助成事業 基盤研究(C)
稲田 佳彦, 篠原 陽子, 中川 征樹, 李 キョンウォン, 清田 哲男
Grant amount：\4550000 （ Direct expense: \3500000 、 Indirect expense：\1050000 ）
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Equivariant Schubert calculus for p-compact groups
Grant number：23K03092 2023.04 - 2026.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
中川 征樹, 西本 哲, 鳥居 猛, 奥山 真吾
Grant amount：\3510000 （ Direct expense: \2700000 、 Indirect expense：\810000 ）
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Research on topological models for combinatorial Hopf algebras
2018.04 - 2021.03
日本学術振興会 科学研究費補助金 基盤研究(C)
Masaki Nakagawa, Hiroshi Naruse
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Development of learning materials with sensation by using virtual reality technology - in the case of three dimensional figures -
Grant number：15K00921 2015.04 - 2019.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Irie Takashi
Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）
The learning materials, which allow learners instinctive handling of three dimensional figures in the virtual reality space and offer them the characteristics of those figures through visual and kinematic sense, were developed by using a haptic device.
(1) The material to experience the characteristics of basic three dimensional figures. (2) The material to experience the characteristics of regular polyhedrons and semi-regular polyhedrons. (3) The material to experience the transition of regular polyhedrons by truncation. In all materials, learners were able to contact the figures, hold them, move them, rotate them, and observe them from three directional view. -
Research on generalized cohomology of flag varieties and Schur functions and their variants
2015.04 - 2018.03
日本学術振興会 科学研究費補助金 基盤研究(C)
Masaki Nakagawa
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Combinatorics of Schubert calculus and its application
Grant number：25400041 2013.04 - 2016.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
NARUSE Hiroshi, IKEDA Takeshi, NAKAGAWA Masaki, ISHIKAWA Masao, HAGIWARA Manabu
Grant amount：\3770000 （ Direct expense: \2900000 、 Indirect expense：\870000 ）
We defined good polynomial representative of torus equivariant Schubert class in K-theory of flag varieties of the classical groups. We also give combinatorial formula for them. As an application of equivariant Schubert calculus we give a proof of the hook formula, which gives the number of standard tableaux on a given shape of partition. We also give a generalization of the hook formula and its equivariant K-theory version. By alanogy with this formula we get a solution to the Casselman's problem related to the representation of p-adic algebraic groups using some techinques of Schubert calculus.
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Research on the loop spaces on Lie groups by combinatorial methods
2012.04 - 2015.03
日本学術振興会 科学研究費補助金 基盤研究(C)
Masaki Nakagawa
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Morava K-theory of the exceptional Lie group and flag manifold
Grant number：24540102 2012.04 - 2015.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
NISHIMOTO Tetsu, MIMURA Mamoru, NAKAGAWA Masaki
Grant amount：\2080000 （ Direct expense: \1600000 、 Indirect expense：\480000 ）
In order to know the property of the fibre bundle, there is a way to calculate the characteristic classes. The most important fibre bundle is the universal bundle, and to determine its characteristic classes is equivalent to calculate the cohomology of the classifying space of the structure group. The spectral sequence is the tool to calculate the cohomology of the classifying space. This time, I calulated the E_2-term of the spectral sequence convergence to the mod 3 cohomology of the classifying spaces of the exceptional Lie groups E_7 and E_8. Moreover, I calculated the invariant ring of the Weyl group of E_7 which acts on the mod 3 cohomology of the classifying space of the maximal torus.
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Schubert calculus on flag varieties and its application
Grant number：21540104 2009.04 - 2012.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
NAKAGAWA Masaki, MIMURA Mamoru, NARUSE Hiroshi, IKEDA Takeshi, NISHIMOTO Tetsu, KAJI Shizuo
Authorship：Principal investigator Grant type：Competitive
(1) We determined the ring structure of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact simply-connected simple exceptional Lie group of rank 8 and T its maximal torus. We also identified the Schubert classes which generate this ring by means of the divided difference operators. Using this result, we were able to determine the Chow ring of the corresponding complex algebraic group E_8.
(2) Using the localization technique and the GKM description of the torus equivariant cohomology rings of homogeneous spaces, we computed the torus equivariant cohomology ring of the flag variety G_2/B and the complex quadric Q_n explicitly, where G_2 denotes the complex Lie group of type G_2 and B a Borel subgroup.
(3) Extending the result due to Kono-Kozima, we showed that the Pontrjagin ring of the based loop space of the infinite symplectic group Sp(resp. infinite orthogonal group SO) is isomorphic to the ring of Schur P-(resp. Q) functions -
Schubert geometry and special polynomials
Grant number：20540053 2008 - 2010
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
IKEDA Takeshi, NARUSE Hiroshi, OHMOTO Toru, NAKAGAWA Masaki
Grant amount：\4550000 （ Direct expense: \3500000 、 Indirect expense：\1050000 ）
For the flag variety of classical Lie groups, we introduced special polynomials representing the Schubert classes in torus equivariant cohomology ring. We established some fundamental properties for these polynomials. In order to extend this result to torus equivariant K-theory, we introduced special family of polynomials (K-theoretic Q-and P-functions) representing the Schubert classes in the torus equivariant K-theory of classical Grassmannian varieties. We also developed combinatorics of these polynomials. In particular, we discussed a relationship to excited Young diagrams and shifted set-valued tableaux. We introduced a Robinson-Schensted type algorithm for these tableaux. As an application of this algorithm, we proved Pieri type formulas for the K-theoretic Q-functions.
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On the characteristic classes and The Chow ring of the exceptional Lie groups
Grant number：20540099 2008 - 2010
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
NISHIMOTO Tetsu, MIMURA Mamoru, NAKAGAWA Masaki
Grant amount：\3380000 （ Direct expense: \2600000 、 Indirect expense：\780000 ）
In order to determine the cohomology of the classifying space of the exceptional Lie group, the action of the cohomology operation and the characteristic classes induced by the representation, we made some programs using some computer algebra system. We compute the cohomology of the classifying space of some exceptional Lie groups applying the programs.
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Invariants of the Weyl groups of exceptional Lie groups and their applications
Grant number：18540106 2006.04 - 2009.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Masaki Nakagawa
Grant type：Competitive
ワイル群の不変量を用いたリー群の等質空間のコホモロジーの表示をもとに, 旗多様体, 対称空間, リー群上のループ空間などの位相幾何学的側面を研究する.
Class subject in charge
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Learning for Sustainability Ⅰ （2024academic year） 1st semester - 火7～8
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Learning for Sustainability Ⅱ （2024academic year） Second semester - 火7～8
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Secondary Mathematics Content Construction （2024academic year） 1st and 2nd semester - 金5～6
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Figure Ⅰ （2024academic year） 3rd and 4th semester - 火1～2
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Elementary Geometry Ⅰ （2024academic year） 1st semester - 水1～2
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Elementary Geometry Ⅰ （2024academic year） 1st semester - 水1～2
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Elementary Geometry Ⅱ （2024academic year） Second semester - 水1～2
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Elementary Geometry Ⅱ （2024academic year） Second semester - 水1～2
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Advanced Geometry BⅠ （2024academic year） 1st semester - 月5～6
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Advanced Geometry BⅡ （2024academic year） Second semester - 月5～6
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Advanced Geometry CⅠ （2024academic year） Fourth semester - 月5～6
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Advanced Geometry CⅡ （2024academic year） Fourth semester - 月7～8
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Analytical Geometry Ⅰ （2024academic year） Third semester - 木7～8
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Analytical Geometry Ⅰ （2024academic year） Third semester - 木7～8
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Analytical Geometry Ⅱ （2024academic year） Fourth semester - 木7～8
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Analytical Geometry Ⅱ （2024academic year） Fourth semester - 木7～8
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Set Theory and General TopologyⅠ （2024academic year） 1st semester - 金5～6
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Set Theory and General TopologyⅡ （2024academic year） Second semester - 金5～6
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Secondary Mathematics Education Methodology Development（AdvancedⅠ） （2024academic year） Fourth semester - 月1～2
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Secondary Mathematics Education Methodology Development （AdvancedⅡ） （2024academic year） Fourth semester - 月3～4
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Secondary Mathematics Education Methodology Development B (1) （2024academic year） Fourth semester - 月1～2
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Secondary Mathematics Education Methodology Development B (2) （2024academic year） Fourth semester - 月3～4
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Secondary Education Mathematics Lesson Development(AdvancedⅠ) （2024academic year） Fourth semester - 月1～2
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Secondary Education Mathematics Lesson Development(AdvancedⅡ) （2024academic year） Fourth semester - 月3～4
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Creativity and Diversity Challenge Ⅰ （2024academic year） Third semester - 金1～2
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Creativity and Diversity Challenge Ⅱ （2024academic year） Third semester - 金7～8
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Creativity and Diversity Challenge Ⅲ （2024academic year） Fourth semester - 金7～8
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Creativity and Diversity Challenge Ⅳ （2024academic year） 1st semester - 金7～8
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Coordinate geometry II (1) （2024academic year） Third semester - 木7～8
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Coordinate geometry II (2) （2024academic year） Fourth semester - 木7～8
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Introduction to projective geometry (1) （2024academic year） 1st semester - 月5～6
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Introduction to projective geometry (2) （2024academic year） Second semester - 月5～6
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Differential geometry of curves and surfaces (1) （2024academic year） Fourth semester - 月5～6
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Differential geometry of curves and surfaces (2) （2024academic year） Fourth semester - 月7～8
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Math for math teachers （2024academic year） Summer concentration - その他
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Special Studies in Educational Science(Mathematics IC): Seminar （2024academic year） 1st semester - 木5,木6
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Special Studies in Educational Science(Mathematics IIC): Seminar （2024academic year） Second semester - 木5,木6
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Special Studies in Educational Science(Geometry IA) （2024academic year） 1st semester - 火5,火6
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Special Studies in Educational Science(Geometry IB) （2024academic year） Second semester - 火5,火6
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Special Studies in Educational Science(Geometry IIA) （2024academic year） Third semester - 火5,火6
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Special Studies in Educational Science(Geometry IIB) （2024academic year） Fourth semester - 火5,火6
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Project Research in Educational Science （2024academic year） 1st-4th semester - その他
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Introduction to Calculus （2024academic year） Third semester - 木1～2
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Arithmetic Basic Content （2024academic year） Second semester - 水3～4
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Arithmetic Basic Content （2024academic year） 1st semester - 水3～4
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Arithmetic Content Construction （2024academic year） Fourth semester - 水3～4
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Arithmetic content composition theoryⅠ （2024academic year） 1st semester - 木1～2
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Arithmetic content composition theoryⅠ （2024academic year） Second semester - 木1～2
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Arithmetic Content Teaching （2024academic year） Second semester - 水3～4
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Arithmetic Content Teaching （2024academic year） 1st semester - 水3～4
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Set Theory and General Topology (1) （2024academic year） 1st semester - 金5～6
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Set Theory and General Topology (2) （2024academic year） Second semester - 金5～6
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Learning for Sustainability Ⅰ （2023academic year） 1st semester - 火7～8
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Learning for Sustainability Ⅱ （2023academic year） Second semester - 火7～8
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Secondary Education Mathematics Content Construction Ⅰ （2023academic year） Summer concentration - その他
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Secondary Mathematics Content Construction （2023academic year） 1st and 2nd semester - 金5～6
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Figure Ⅰ （2023academic year） 3rd and 4th semester - 火1～2
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Elementary Geometry Ⅰ （2023academic year） 1st semester - 木1～2
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Elementary Geometry Ⅱ （2023academic year） Second semester - 木1～2
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Advanced Geometry AⅠ （2023academic year） 1st semester - 月5～6
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Advanced Geometry AⅡ （2023academic year） Second semester - 月5～6
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Analytical Geometry Ⅰ （2023academic year） Third semester - 木1～2
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Analytical Geometry Ⅱ （2023academic year） Fourth semester - 木1～2
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Set Theory and General TopologyⅠ （2023academic year） 1st semester - 金5～6
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Set Theory and General TopologyⅡ （2023academic year） Second semester - 金5～6
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Secondary Mathematics Education Methodology Development（AdvancedⅠ） （2023academic year） Fourth semester - 月1～2
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Secondary Mathematics Education Methodology Development （AdvancedⅡ） （2023academic year） Fourth semester - 月3～4
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Secondary Education Mathematics Lesson Development(AdvancedⅠ) （2023academic year） Fourth semester - 月1～2
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Secondary Education Mathematics Lesson Development(AdvancedⅡ) （2023academic year） Fourth semester - 月3～4
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Coordinate geometry II (2) （2023academic year） Fourth semester - 木1～2
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Math for math teachers （2023academic year） Summer concentration - その他
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Seminar in Teaching Profession Practice (Secondary school A) （2023academic year） 1st-4th semester - 水7～8
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Special Studies in Educational Science(Mathematics IC): Seminar （2023academic year） 1st semester - 木5,木6
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Special Studies in Educational Science(Mathematics IIC): Seminar （2023academic year） Second semester - 木5,木6
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Special Studies in Educational Science(Geometry IA) （2023academic year） 1st semester - 火5,火6
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Special Studies in Educational Science(Geometry IB) （2023academic year） Second semester - 火5,火6
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Special Studies in Educational Science(Geometry IIA) （2023academic year） Third semester - 火5,火6
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Special Studies in Educational Science(Geometry IIB) （2023academic year） Fourth semester - 火5,火6
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Project Research in Educational Science （2023academic year） 1st-4th semester - その他
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Arithmetic Basic Content （2023academic year） 1st semester - 水3～4
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Arithmetic Basic Content （2023academic year） Second semester - 水3～4
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Arithmetic Content Construction （2023academic year） Fourth semester - 水3～4
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Arithmetic Content Teaching （2023academic year） 1st semester - 水3～4
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Secondary Education Mathematics Content Construction Ⅰ （2022academic year） Third semester - 金3,金4
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Elementary Geometry Ⅰ （2022academic year） 1st semester - 木1,木2
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Elementary Geometry Ⅱ （2022academic year） Second semester - 木1,木2
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Advanced Geometry BⅠ （2022academic year） 1st semester - 月5～6
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Advanced Geometry BⅡ （2022academic year） Second semester - 月5～6
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Advanced Geometry CⅠ （2022academic year） Fourth semester - 月5～6
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Advanced Geometry CⅡ （2022academic year） Fourth semester - 月7～8
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Analytical Geometry Ⅰ （2022academic year） Third semester - 木1,木2
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Analytical Geometry Ⅱ （2022academic year） Fourth semester - 木1,木2
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Set Theory and General TopologyⅠ （2022academic year） 1st semester - 木7,木8
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Set Theory and General TopologyⅡ （2022academic year） Second semester - 木7,木8
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Secondary Mathematics Education Methodology Development（AdvancedⅠ） （2022academic year） Fourth semester - 月1～2
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Secondary Mathematics Education Methodology Development （AdvancedⅡ） （2022academic year） Fourth semester - 月3～4
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Secondary Mathematics Education Methodology Development B (1) （2022academic year） Fourth semester - 月1,月2
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Secondary Mathematics Education Methodology Development B (2) （2022academic year） Fourth semester - 月3,月4
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Secondary Education Mathematics Lesson Development(AdvancedⅠ) （2022academic year） Fourth semester - 月1～2
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Secondary Education Mathematics Lesson Development(AdvancedⅡ) （2022academic year） Fourth semester - 月3～4
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Approaches to Education （2022academic year） 1st semester - 火1～2
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Coordinate geometry II (1) （2022academic year） Third semester - 木1,木2
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Coordinate geometry II (2) （2022academic year） Fourth semester - 木1,木2
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Introduction to projective geometry (1) （2022academic year） 1st semester - 月5,月6
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Introduction to projective geometry (2) （2022academic year） Second semester - 月5,月6
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Differential geometry of curves and surfaces (1) （2022academic year） Fourth semester - 月5,月6
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Differential geometry of curves and surfaces (2) （2022academic year） Fourth semester - 月7,月8
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Euclidean Geometry I (1) （2022academic year） 1st semester - 木1,木2
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Euclidean Geometry I (2) （2022academic year） Second semester - 木1,木2
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Math for math teachers （2022academic year） Summer concentration - その他
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Special Studies in Educational Science(Mathematics IC): Seminar （2022academic year） 1st semester - 木5,木6
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Special Studies in Educational Science(Mathematics IIC): Seminar （2022academic year） Second semester - 木5,木6
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Special Studies in Educational Science(Geometry IA) （2022academic year） 1st semester - 火5,火6
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Special Studies in Educational Science(Geometry IB) （2022academic year） Second semester - 火5,火6
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Special Studies in Educational Science(Geometry IIA) （2022academic year） Third semester - 火5,火6
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Special Studies in Educational Science(Geometry IIB) （2022academic year） Fourth semester - 火5,火6
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Project Research in Educational Science （2022academic year） 1st-4th semester - その他
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Elementary Mathematics (GeometryⅠ) （2022academic year） Third semester - 金1,金2
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Elementary Mathematics (GeometryⅡ) （2022academic year） Fourth semester - 金1,金2
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Elementary Geometry (1) （2022academic year） Third semester - 金1,金2
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Elementary Geometry (2) （2022academic year） Fourth semester - 金1,金2
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Arithmetic Content Construction （2022academic year） Fourth semester - 水3～4
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Set Theory and General Topology (1) （2022academic year） 1st semester - 木7,木8
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Set Theory and General Topology (2) （2022academic year） Second semester - 木7,木8
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Secondary Education Mathematics Content Construction Ⅰ （2021academic year） Third semester - 金3,金4
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Elementary Geometry Ⅰ （2021academic year） 1st semester - 木1,木2
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Elementary Geometry Ⅱ （2021academic year） Second semester - 木1,木2
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Advanced Geometry AⅠ （2021academic year） 1st semester - 月5～6
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Advanced Geometry AⅡ （2021academic year） Second semester - 月5～6
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Analytical Geometry Ⅰ （2021academic year） Third semester - 木1,木2
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Analytical Geometry Ⅱ （2021academic year） Fourth semester - 木1,木2
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Set Theory and General TopologyⅠ （2021academic year） 1st semester - 木7,木8
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Set Theory and General TopologyⅡ （2021academic year） Second semester - 木7,木8
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Secondary Mathematics Education Methodology Development（AdvancedⅠ） （2021academic year） Fourth semester - 月1～2
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Secondary Mathematics Education Methodology Development （AdvancedⅡ） （2021academic year） Fourth semester - 月3～4
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Secondary Mathematics Education Methodology Development B (1) （2021academic year） Fourth semester - 月1,月2
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Secondary Mathematics Education Methodology Development B (2) （2021academic year） Fourth semester - 月3,月4
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Secondary Education Mathematics Lesson Development(AdvancedⅠ) （2021academic year） Fourth semester - 月1～2
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Secondary Education Mathematics Lesson Development(AdvancedⅡ) （2021academic year） Fourth semester - 月3～4
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Approaches to Education （2021academic year） 1st semester - 火1～2
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Coordinate geometry II (1) （2021academic year） Third semester - 木1,木2
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Coordinate geometry II (2) （2021academic year） Fourth semester - 木1,木2
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Geometry of Patterns (1) （2021academic year） 1st semester - 月5～6
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Geometry of Patterns (2) （2021academic year） Second semester - 月5～6
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Euclidean Geometry I (1) （2021academic year） 1st semester - 木1,木2
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Euclidean Geometry I (2) （2021academic year） Second semester - 木1,木2
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Math for math teachers （2021academic year） Summer concentration - その他
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Special Studies in Educational Science(Mathematics IC): Seminar （2021academic year） 1st semester - 木5,木6
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Special Studies in Educational Science(Mathematics IIC): Seminar （2021academic year） Second semester - 木5,木6
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Special Studies in Educational Science(Geometry IA) （2021academic year） 1st semester - 火5,火6
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Special Studies in Educational Science(Geometry IB) （2021academic year） Second semester - 火5,火6
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Special Studies in Educational Science(Geometry IIA) （2021academic year） Third semester - 火3,火4
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Special Studies in Educational Science(Geometry IIB) （2021academic year） Fourth semester - 火3,火4
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Project Research in Educational Science （2021academic year） 1st-4th semester - その他
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Elementary Mathematics (GeometryⅠ) （2021academic year） Third semester - 金1,金2
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Elementary Mathematics (GeometryⅡ) （2021academic year） Fourth semester - 金1,金2
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Elementary Geometry (1) （2021academic year） Third semester - 金1,金2
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Elementary Geometry (2) （2021academic year） Fourth semester - 金1,金2
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Arithmetic Content Construction （2021academic year） Fourth semester - 水3～4
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Set Theory and General Topology (1) （2021academic year） 1st semester - 木7,木8
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Set Theory and General Topology (2) （2021academic year） Second semester - 木7,木8
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Secondary Education Mathematics Content Construction Ⅰ （2020academic year） Third semester - 金3,金4
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Elementary Geometry Ⅰ （2020academic year） 1st semester - 木1,木2
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Elementary Geometry Ⅱ （2020academic year） Second semester - 木1,木2
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Analytical Geometry Ⅰ （2020academic year） Third semester - 木1,木2
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Analytical Geometry Ⅱ （2020academic year） Fourth semester - 木1,木2
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Set Theory and General TopologyⅠ （2020academic year） 1st semester - 木7,木8
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Set Theory and General TopologyⅡ （2020academic year） Second semester - 木7,木8
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Secondary Mathematics Education Methodology Development B (1) （2020academic year） Fourth semester - 月1,月2
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Secondary Mathematics Education Methodology Development B (2) （2020academic year） Fourth semester - 月3,月4
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Approaches to Education （2020academic year） 1st semester - 火1,火2
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Coordinate geometry II (1) （2020academic year） Third semester - 木1,木2
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Coordinate geometry II (2) （2020academic year） Fourth semester - 木1,木2
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Introduction to projective geometory (1) （2020academic year） 1st semester - 月5,月6
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Introduction to projective geometory (2) （2020academic year） Second semester - 月5,月6
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Differential geometry of curves and surfaces (1) （2020academic year） Fourth semester - 月5,月6
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Differential geometry of curves and surfaces (2) （2020academic year） Fourth semester - 月7,月8
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Euclidean Geometry I (1) （2020academic year） 1st semester - 木1,木2
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Euclidean Geometry I (2) （2020academic year） Second semester - 木1,木2
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Introduction to Calculus （2020academic year） Third semester - 木3,木4
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Math for math teachers （2020academic year） Summer concentration - その他
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Seminar in Teaching Profession Practice (Secondary school A) （2020academic year） 1st-4th semester - 水7,水8
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Seminar in Teaching Profession Practice (Elementary school) （2020academic year） 1st-4th semester - 水7,水8
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Seminar in Teaching Profession Practice (Elementary school) （2020academic year） 1st-4th semester - 水7,水8
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Seminar in Teaching Profession Practice (Elementary school) （2020academic year） 1st-4th semester - 水7,水8
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Seminar in Teaching Profession Practice (Elementary school) （2020academic year） 1st-4th semester - 水7,水8
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Special Studies in Educational Science(Mathematics IC): Seminar （2020academic year） 1st semester - 木5,木6
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Special Studies in Educational Science(Mathematics IIC): Seminar （2020academic year） Second semester - 木5,木6
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Special Studies in Educational Science(Geometry IA) （2020academic year） 1st semester - 火5,火6
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Special Studies in Educational Science(Geometry IB) （2020academic year） Second semester - 火5,火6
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Special Studies in Educational Science(Geometry IIA) （2020academic year） Third semester - 火3,火4
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Special Studies in Educational Science(Geometry IIB) （2020academic year） Fourth semester - 火3,火4
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Project Research in Educational Science （2020academic year） 1st-4th semester - その他
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Elementary Mathematics (GeometryⅠ) （2020academic year） Third semester - 金1,金2
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Elementary Mathematics (GeometryⅡ) （2020academic year） Fourth semester - 金1,金2
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Elementary Geometry (1) （2020academic year） Third semester - 金1,金2
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Elementary Geometry (2) （2020academic year） Fourth semester - 金1,金2
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Introduction to Linear Algebra （2020academic year） Fourth semester - 木3,木4
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Set Theory and General Topology （2020academic year） 1st and 2nd semester - 木7,木8
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Set Theory and General Topology (1) （2020academic year） 1st semester - 木7,木8
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Set Theory and General Topology (2) （2020academic year） Second semester - 木7,木8