Construction of framework for describing students' activity of proofs and refutations in mathematical learning

• Project/Area Number: 24730730
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Education on school subjects and activities
• Research Institution: Shinshu University
• Principal Investigator: KOMATSU Kotaro 信州大学, 学術研究院教育学系, 准教授 (40578267)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 数学教育 / 証明 / 論駁 / 反例 / ラカトシュ / 発見法的規則 / 演繹的推量による内容の増加

Outline of Final Research Achievements

This study addresses mathematical activity of proofs and refutations, aiming to construct a framework for describing such activity by students in mathematical learning. For that purpose, I examined Proofs and Refutations (Lakatos, 1976), observing and analysing mathematical lessons in secondary schools. This study constructs a framework for describing students’ reaction to a global counterexample, a counterexample that refutes a statement, and to a local counterexample, a counterexample that refutes a proof.

Research Products

• [Journal Article] Invention of new statements for counterexamples (2014)
  • Author(s): Komatsu, K., & Sakamaki, A.
  • Journal Title: Proceedings of the Joint Meeting of PME 38 and PME-NA 36 Volume: 4 Pages: 17-24
  • NAID: 120007101000
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] 「円周角と中心角の関係の活用」において課題探究として証明することの授業化 (2014)
  • Author(s): 小松孝太郎・牧野圭介
  • Journal Title: 日本数学教育学会誌数学教育 Volume: 96(9) Pages: 26-29
  • NAID: 110009857083
  • Acknowledgement Compliant
• [Journal Article] Rethinking the discovery function of proof within the context of proofs and refutations (2014)
  • Author(s): Komatsu, K., Tsujiyama, Y., & Sakamaki, A.
  • Journal Title: International Journal of Mathematical Education in Science and Technology Volume: 45(7) Issue: 7 Pages: 1053-1067
  • DOI: 10.1080/0020739x.2014.902135
  • NAID: 120007100998
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Proof problems with diagrams: An opportunity for experiencing proofs and refutations (2014)
  • Author(s): Komatsu, K., Tsujiyama, Y., Sakamaki, A., & Koike, N.
  • Journal Title: For the Learning of Mathematics Volume: 34(1) Pages: 36-42
  • NAID: 120007100999
  • Peer Reviewed
• [Journal Article] Invention of new statements for counterexamples (2014)
  • Author(s): Komatsu, K., & Sakamaki, A.
  • Journal Title: Proceedings of the Joint Meeting of the International Group for the Psychology of Mathematics Education (PME 38) and the North American Chapter of the Psychology of Mathematics Education (PME-NA 36) Volume: -
  • NAID: 120007101000
  • Peer Reviewed
• [Journal Article] Principles of task design to foster proofs and refutations in mathematical learning: Proof problem with diagram (2013)
  • Author(s): Komatsu, K., & Tsujiyama, Y.
  • Journal Title: Task Design in Mathematics Education: Proceedings of ICMI Study 22 Volume: - Pages: 471-479
  • Peer Reviewed
• [Journal Article] Principles of task design to foster proofs and refutations in mathematical learning: Proof problem with diagram (2013)
  • Author(s): Komatsu, K., & Tsujiyama, Y.
  • Journal Title: Proceedings of the 22nd ICMI Study Conference: Task Design in Mathematics Education Volume: -
  • Peer Reviewed
• [Journal Article] Potentials for spatial geometry curriculum development with three-dimensional dynamic geometry software in lower secondary mathematics (2012)
  • Author(s): Miyazaki, M., Chino, K., Katoh, R., Arai, H., Ogihara, F., Oguchi, Y., Morozumi, T., Kon, M., & Komatsu, K.
  • Journal Title: The International Journal for Technology in Mathematics Education Volume: 19(2) Pages: 73-79
  • Peer Reviewed
• [Journal Article] Lakatos’ heuristic rules as a framework for proofs and refutations in mathematical learning: Local-counterexample and modification of proof (2012)
  • Author(s): Komatsu, K.
  • Journal Title: Proceedings of the 12th International Congress on Mathematical Education Volume: - Pages: 2838-2847
  • NAID: 120007100930
  • Peer Reviewed
• [Presentation] 学校数学における証明と論駁の活動を促すデザイン実験の構想 (2014)
  • Author(s): 小松孝太郎・辻山洋介
  • Organizer: 日本数学教育学会第47回秋期研究大会
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-11-08
• [Presentation] 中学校第３学年における課題探究として証明することの授業化：第３学年の小単元「円周角と中心角の関係の活用」 (2014)
  • Author(s): 小松孝太郎・牧野圭介
  • Organizer: 日本科学教育学会第38回年会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2014-09-15
• [Presentation] 課題探究として証明することのカリキュラム開発：中学校数学科第三学年の領域「数と式」及び「図形」における学習の構想 (2013)
  • Author(s): 永田潤一郎・小松孝太郎・中川裕之
  • Organizer: 日本数学教育学会第1回春期研究大会
  • Place of Presentation: 筑波大学東京キャンパス
• [Presentation] 学校数学における図が付された証明問題の可能性：ラカトシュの演繹的推量に焦点を当てて (2013)
  • Author(s): 小松孝太郎・辻山洋介・坂巻主太・小池徳男
  • Organizer: 日本科学教育学会第37回年会
  • Place of Presentation: 三重大学
• [Presentation] 学校数学における反例や負例への対応に関する一考察 (2013)
  • Author(s): 小松孝太郎
  • Organizer: 日本数学教育学会第46回秋期研究大会
  • Place of Presentation: 宇都宮大学
• [Presentation] Lakatos’ heuristic rules as a framework for proofs and refutations in mathematical learning: Local-counterexample and modification of proof (2012)
  • Author(s): Komatsu, K.
  • Organizer: The 12th International Congress on Mathematical Education
  • Place of Presentation: 韓国・ソウル
• [Book] 算数・数学教育における証明指導の改善 (2014)
  • Author(s): 小松孝太郎
  • Total Pages: 279
  • Publisher: 東洋館出版社