Article Sidebar
Main Article Content
Abstract
This exploratory study investigates the teaching and learning of mathematical proofs, specifically focusing on real nalysis proofs, in mathematics education programs at the higher education level in Bengkulu Province. Data were collected through in-depth interviews with lecturers and students, classroom observations, and document analysis of students' assignments. The findings reveal that lecturers employ various teaching strategies, such as active learning, step-by-step explanations, and the use of technology to aid understanding. However, students face significant challenges, including difficulties with abstract thinking, gaps in foundational knowledge, and the complex language of mathematical proofs. Despite these challenges, students reported improvements in their logical reasoning, problem-solving, and self-confidence as they were engaged in the construction mathematical proofs. Classroom observations confirmed that collaborative learning was effective in promoting understanding. Document analysis of students' assignments indicated a range of proficiency levels, with some students struggling to produce clear and logical proofs. The study highlights the importance of mathematical proofs in developing critical thinking skills and analytical abilities. It suggests that more interactive, student-centered teaching methods are necessary to address the challenges students face and improve learning outcomes. These findings provide valuable insights for enhancing teaching practices and supporting students' mastery of mathematical proofs.
Keywords
Article Details
Copyright (c) 2025 Dodi Isran, Agus Susanta, Dewi Rahimah, Fatrima Santri Syafri
This work is licensed under a Creative Commons Attribution 4.0 International License.
This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.
All articles published Open Access will be immediately and permanently free for everyone to read and download. We are continuously working with our author communities to select the best choice of license options, Creative Commons Attribution-ShareAlike (CC BY-NC-SA).
References
- Abbott, S. (2015). Understanding Analysis. Springer New York. https://doi.org/10.1007/978-1-4939-2712-8
- Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129. https://doi.org/10.1007/s10649-008-9149-x
- Alexanderson, G. L., & Polya, G. (1979). Mathematics and Plausible Reasoning: Vol. I: Induction and Analogy in Mathematics. The Two-Year College Mathematics Journal, 10(2), 119. https://doi.org/10.2307/3027025
- Apostol, T. M. (1991). Calculus, Volume 1. John Wiley & Sons.
- Bartle, R. G., & Sherbert, D. R. (2000). Introduction to Real Analysis. John Wiley & Sons, Inc. https://archive.org/details/robertg.bartledonaldr.sherbertintroductiontorealanalysiswiley2000
- B., T. A. A., & Bell, E. T. (1937). Men of Mathematics. The Mathematical Gazette, 21(245), 311. https://doi.org/10.2307/3607740
- Berki, E., & Valtanen, J. (2007). Critical and creative mathematical thinking with practical problem solving skills-A new old challenge. Proceedings of 3rd South-East European Workshop on Formal Methods. Service-Oriented Computing; Teaching Formal Methods, 154–170.
- Čížková, L., & Čížek, P. (2012). Numerical linear algebra. In Handbook of Computational Statistics: Concepts and Methods: Second Edition (Vol. 181, pp. 105–137). Siam. https://doi.org/10.1007/978-3-642-21551-3__5
- Er, Z. (2024). Examination of the Relationship between Mathematical and Critical Thinking Skills and Academic Achievement. Pedagogical Research, 9(1). https://doi.org/https://doi.org/10.29333/pr/14028
- Faizah, S., Nusantara, T., Sudirman, S., & Rahardiİ, R. (2020). Exploring students’ thinking process in mathematical proof of abstract algebra based on Mason’s framework. Journal for the Education of Gifted Young Scientists, 8(2), 871–884. https://doi.org/10.17478/jegys.689809
- Fisher, A., & Scriven, M. (2001). Critical thinking. Cambridge, UK, Cambridge University. https://pgwritinghub.net/wp-content/uploads/2024/09/Critical-thinking-An-introduction.pdf
- Halmos, P. R. (1983). How to Talk Mathematics. In Selecta Expository Writing (Vol. 21, Issue 3, pp. 187–190). https://doi.org/10.1007/978-1-4613-8211-9_17
- Hanna, G. (2002). Mathematical Proof. In Advanced Mathematical Thinking (pp. 54–61). Springer Netherlands. https://doi.org/10.1007/0-306-47203-1_4
- Hanna, G. (2020). Mathematical Proof, Argumentation, and Reasoning. In Encyclopedia of Mathematics Education (pp. 561–566). Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_102
- Hardy, G. H., & Snow, C. P. (1992). A Mathematician’s Apology. In A Mathematician’s Apology. Cambridge University Press. https://doi.org/10.1017/cbo9781139644112
- Hasan, A. K. (2020). A topology on d-algebra via dual stabilizers. Journal of Physics: Conference Series, 1660(1), 012103. https://doi.org/10.1088/1742-6596/1660/1/012103
- Hemmi, K. (2006). Approaching proof in a community of mathematical practice.
- Jamilah, J., & Fadillah, S. (2017). The Use of Algebraic Structure Teaching Materials to Improve Mathematical Proof Skills in Ikip PGRI Pontianak Students [In Bahasa]. Jurnal Pendidikan Matematika Dan IPA, 8(2), 60. https://doi.org/10.26418/jpmipa.v8i2.21178
- Jessup, N. A., Wolfe, J. A., & Kalinec-Craig, C. (2021). Rehumanizing Mathematics Education and Building Community for Online Learning. In Online learning in mathematics education (pp. 95–113). Springer. https://doi.org/10.1007/978-3-030-80230-1_5
- Krantz, S. G. (2017). Discrete Problems. In Essentials of Mathematical Thinking (Vol. 243, pp. 243–267). Chapman and Hall/CRC. https://doi.org/10.1201/9781315116822-11
- Lestari, K. E. (2015). Analysis of students' mathematical proof abilities using an inductive-deductive approach in real analysis courses [in Bahasa]. MENDIDIK: Jurnal Kajian Pendidikan Dan Pengajaran, 1(2), 128–135. https://doi.org/10.30653/003.201512.20
- Macwhinney, B. (2008). Functional Analysis. In A Companion to Cognitive Science (pp. 402–412). Wiley. https://doi.org/10.1002/9781405164535.ch31
- Mitchelmore, M., & White, P. (2004). Abstraction in Mathematics and Mathematics Learning. International Group for the Psychology of Mathematics Education. https://eric.ed.gov/?id=ED489589
- Moreno, R., Ozogul, G., & Reisslein, M. (2011). Teaching with concrete and abstract visual representations: Effects on students’ problem solving, problem representations, and learning perceptions. Journal of Educational Psychology, 103(1), 32–47. https://doi.org/10.1037/a0021995
- Muzangwa, J., & Ogbonnaya, U. (2024). Visualization techniques for proofs: Implications for enhancing conceptualization and understanding in mathematical analysis. Journal of Honai Math, 7(2), 347–362. https://doi.org/10.30862/jhm.v7i2.603
- NCTM. (2000). Principles and standards for school mathematics. Reston, VA. In National Council of Teachers of Mathematics. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ffec290ccfaa3de1d643a09e992349f3f7ff9d4c
- Nonami, K., Hoshiba, K., Nakadai, K., Kumon, M., Okuno, H. G., Tanabe, Y., Yonezawa, K., Tokutake, H., Suzuki, S., Yamaguchi, K., Sunada, S., Takaki, T., Nakata, T., Noda, R., Liu, H., & Tadokoro, S. (2019). Recent R&D Technologies and Future Prospective of Flying Robot in Tough Robotics Challenge. In Disaster Robotics: Results from the ImPACT Tough Robotics Challenge (pp. 77–142). Springer. https://doi.org/10.1007/978-3-030-05321-5_3
- Polya, G. (2004). How to solve it: A new aspect of mathematical method (Vol. 85). Princeton university press.
- Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational Studies in Mathematics, 48(1), 83–99.
- Ross, K. A. (2013). Elementary Analysis. Springer New York. https://doi.org/10.1007/978-1-4614-6271-2
- Rota, G.-C. (1997). Indiscrete Thoughts. In F. Palombi (Ed.), Indiscrete Thoughts. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4781-0
- Russell, B. (2020). The Principles of Mathematics. Routledge. https://doi.org/10.4324/9780203822586
- Stefanowicz, A., Kyle, J., & Grove, M. (2014). Proofs and mathematical reasoning. University of Birmingham. https://math.ucr.edu/~mpierce/teaching/amp-algebra/docs/Stefanowicz-ProofsAndMathematicalReasoning.pdf
- Stroch, J. A., & Börgers, C. (2024). Introduction to Numerical Linear Algebra [Bookshelf]. IEEE Control Systems, 44(1), 79–80. https://doi.org/10.1109/MCS.2023.3329927
- Strogatz, S. H. (2013). The joy of X: a guided tour of math, from one to infinity. Choice Reviews Online, 50(08), 50-4492-50–4492. https://doi.org/10.5860/choice.50-4492
- Syafri, F. S. (2017). Mathematical representation ability and mathematical proof ability [in Bahasa]. JURNAL E-DuMath, 3(1). https://doi.org/10.52657/je.v3i1.283
- Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. https://doi.org/10.1007/BF03217474
- Tao, T. (2006). Solving Mathematical Problems. Oxford University PressOxford. https://doi.org/10.1093/oso/9780199205615.001.0001
- Trefethen, L. N., & Bau, D. (2022). Numerical linear algebra. SIAM. https://epubs.siam.org/doi/pdf/10.1137/1.9781611977165.bm
- Velleman, D. J. (2019). HOW TO PROVE IT: A Structured Approach. In How to Prove It: A Structured Approach. Cambridge University Press. https://doi.org/10.1017/9781108539890
- Watkins, J. J. (2007). Solving mathematical problems: a personal perspective. The Mathematical Intelligencer, 29(3), 60–63. https://doi.org/10.1007/bf02985692
References
Abbott, S. (2015). Understanding Analysis. Springer New York. https://doi.org/10.1007/978-1-4939-2712-8
Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129. https://doi.org/10.1007/s10649-008-9149-x
Alexanderson, G. L., & Polya, G. (1979). Mathematics and Plausible Reasoning: Vol. I: Induction and Analogy in Mathematics. The Two-Year College Mathematics Journal, 10(2), 119. https://doi.org/10.2307/3027025
Apostol, T. M. (1991). Calculus, Volume 1. John Wiley & Sons.
Bartle, R. G., & Sherbert, D. R. (2000). Introduction to Real Analysis. John Wiley & Sons, Inc. https://archive.org/details/robertg.bartledonaldr.sherbertintroductiontorealanalysiswiley2000
B., T. A. A., & Bell, E. T. (1937). Men of Mathematics. The Mathematical Gazette, 21(245), 311. https://doi.org/10.2307/3607740
Berki, E., & Valtanen, J. (2007). Critical and creative mathematical thinking with practical problem solving skills-A new old challenge. Proceedings of 3rd South-East European Workshop on Formal Methods. Service-Oriented Computing; Teaching Formal Methods, 154–170.
Čížková, L., & Čížek, P. (2012). Numerical linear algebra. In Handbook of Computational Statistics: Concepts and Methods: Second Edition (Vol. 181, pp. 105–137). Siam. https://doi.org/10.1007/978-3-642-21551-3__5
Er, Z. (2024). Examination of the Relationship between Mathematical and Critical Thinking Skills and Academic Achievement. Pedagogical Research, 9(1). https://doi.org/https://doi.org/10.29333/pr/14028
Faizah, S., Nusantara, T., Sudirman, S., & Rahardiİ, R. (2020). Exploring students’ thinking process in mathematical proof of abstract algebra based on Mason’s framework. Journal for the Education of Gifted Young Scientists, 8(2), 871–884. https://doi.org/10.17478/jegys.689809
Fisher, A., & Scriven, M. (2001). Critical thinking. Cambridge, UK, Cambridge University. https://pgwritinghub.net/wp-content/uploads/2024/09/Critical-thinking-An-introduction.pdf
Halmos, P. R. (1983). How to Talk Mathematics. In Selecta Expository Writing (Vol. 21, Issue 3, pp. 187–190). https://doi.org/10.1007/978-1-4613-8211-9_17
Hanna, G. (2002). Mathematical Proof. In Advanced Mathematical Thinking (pp. 54–61). Springer Netherlands. https://doi.org/10.1007/0-306-47203-1_4
Hanna, G. (2020). Mathematical Proof, Argumentation, and Reasoning. In Encyclopedia of Mathematics Education (pp. 561–566). Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_102
Hardy, G. H., & Snow, C. P. (1992). A Mathematician’s Apology. In A Mathematician’s Apology. Cambridge University Press. https://doi.org/10.1017/cbo9781139644112
Hasan, A. K. (2020). A topology on d-algebra via dual stabilizers. Journal of Physics: Conference Series, 1660(1), 012103. https://doi.org/10.1088/1742-6596/1660/1/012103
Hemmi, K. (2006). Approaching proof in a community of mathematical practice.
Jamilah, J., & Fadillah, S. (2017). The Use of Algebraic Structure Teaching Materials to Improve Mathematical Proof Skills in Ikip PGRI Pontianak Students [In Bahasa]. Jurnal Pendidikan Matematika Dan IPA, 8(2), 60. https://doi.org/10.26418/jpmipa.v8i2.21178
Jessup, N. A., Wolfe, J. A., & Kalinec-Craig, C. (2021). Rehumanizing Mathematics Education and Building Community for Online Learning. In Online learning in mathematics education (pp. 95–113). Springer. https://doi.org/10.1007/978-3-030-80230-1_5
Krantz, S. G. (2017). Discrete Problems. In Essentials of Mathematical Thinking (Vol. 243, pp. 243–267). Chapman and Hall/CRC. https://doi.org/10.1201/9781315116822-11
Lestari, K. E. (2015). Analysis of students' mathematical proof abilities using an inductive-deductive approach in real analysis courses [in Bahasa]. MENDIDIK: Jurnal Kajian Pendidikan Dan Pengajaran, 1(2), 128–135. https://doi.org/10.30653/003.201512.20
Macwhinney, B. (2008). Functional Analysis. In A Companion to Cognitive Science (pp. 402–412). Wiley. https://doi.org/10.1002/9781405164535.ch31
Mitchelmore, M., & White, P. (2004). Abstraction in Mathematics and Mathematics Learning. International Group for the Psychology of Mathematics Education. https://eric.ed.gov/?id=ED489589
Moreno, R., Ozogul, G., & Reisslein, M. (2011). Teaching with concrete and abstract visual representations: Effects on students’ problem solving, problem representations, and learning perceptions. Journal of Educational Psychology, 103(1), 32–47. https://doi.org/10.1037/a0021995
Muzangwa, J., & Ogbonnaya, U. (2024). Visualization techniques for proofs: Implications for enhancing conceptualization and understanding in mathematical analysis. Journal of Honai Math, 7(2), 347–362. https://doi.org/10.30862/jhm.v7i2.603
NCTM. (2000). Principles and standards for school mathematics. Reston, VA. In National Council of Teachers of Mathematics. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ffec290ccfaa3de1d643a09e992349f3f7ff9d4c
Nonami, K., Hoshiba, K., Nakadai, K., Kumon, M., Okuno, H. G., Tanabe, Y., Yonezawa, K., Tokutake, H., Suzuki, S., Yamaguchi, K., Sunada, S., Takaki, T., Nakata, T., Noda, R., Liu, H., & Tadokoro, S. (2019). Recent R&D Technologies and Future Prospective of Flying Robot in Tough Robotics Challenge. In Disaster Robotics: Results from the ImPACT Tough Robotics Challenge (pp. 77–142). Springer. https://doi.org/10.1007/978-3-030-05321-5_3
Polya, G. (2004). How to solve it: A new aspect of mathematical method (Vol. 85). Princeton university press.
Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational Studies in Mathematics, 48(1), 83–99.
Ross, K. A. (2013). Elementary Analysis. Springer New York. https://doi.org/10.1007/978-1-4614-6271-2
Rota, G.-C. (1997). Indiscrete Thoughts. In F. Palombi (Ed.), Indiscrete Thoughts. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4781-0
Russell, B. (2020). The Principles of Mathematics. Routledge. https://doi.org/10.4324/9780203822586
Stefanowicz, A., Kyle, J., & Grove, M. (2014). Proofs and mathematical reasoning. University of Birmingham. https://math.ucr.edu/~mpierce/teaching/amp-algebra/docs/Stefanowicz-ProofsAndMathematicalReasoning.pdf
Stroch, J. A., & Börgers, C. (2024). Introduction to Numerical Linear Algebra [Bookshelf]. IEEE Control Systems, 44(1), 79–80. https://doi.org/10.1109/MCS.2023.3329927
Strogatz, S. H. (2013). The joy of X: a guided tour of math, from one to infinity. Choice Reviews Online, 50(08), 50-4492-50–4492. https://doi.org/10.5860/choice.50-4492
Syafri, F. S. (2017). Mathematical representation ability and mathematical proof ability [in Bahasa]. JURNAL E-DuMath, 3(1). https://doi.org/10.52657/je.v3i1.283
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. https://doi.org/10.1007/BF03217474
Tao, T. (2006). Solving Mathematical Problems. Oxford University PressOxford. https://doi.org/10.1093/oso/9780199205615.001.0001
Trefethen, L. N., & Bau, D. (2022). Numerical linear algebra. SIAM. https://epubs.siam.org/doi/pdf/10.1137/1.9781611977165.bm
Velleman, D. J. (2019). HOW TO PROVE IT: A Structured Approach. In How to Prove It: A Structured Approach. Cambridge University Press. https://doi.org/10.1017/9781108539890
Watkins, J. J. (2007). Solving mathematical problems: a personal perspective. The Mathematical Intelligencer, 29(3), 60–63. https://doi.org/10.1007/bf02985692