Using ChatGPT as a proof assistant in a mathematics pathways course
1
조회
0
다운로드
3
인용
0
참고문헌
키워드
초록
The purpose of this study is to examine the capabilities of ChatGPT as a tool for supporting students in generating mathematical arguments that can be considered proofs. To examine this, we engaged students enrolled in a mathematics pathways course in evaluating and revising their original arguments using ChatGPT feedback. Students attempted to find and prove a method for the area of a triangle given its side lengths. Instead of directly asking students to prove a formula, we asked them to explore a method to find the area of a triangle given the lengths of its sides and justify why their methods work. Students completed these ChatGPT-embedded proving activities as class homework. To investigate the capabilities of ChatGPT as a proof tutor, we used these student homework responses as data for this study. We analyzed and compared original and revised arguments students constructed with and without ChatGPT assistance. We also analyzed student-written responses about their perspectives on mathematical proof and proving and their thoughts on using ChatGPT as a proof assistant. Our analysis shows that our participants’ approaches to constructing, evaluating, and revising their arguments aligned with their perspectives on proof and proving. They saw ChatGPT’s evaluations of their arguments as similar to how they usually evaluate arguments of themselves and others. Mostly, they agreed with ChatGPT’s suggestions to make their original arguments more proof-like. They, therefore, revised their original arguments following ChatGPT’s suggestions, focusing on improving clarity, providing additional justifications, and showing the generality of their arguments. Further investigation is needed to explore how ChatGPT can be effectively used as a tool in teaching and learning mathematical proof and proof-writing.
Abstract
The purpose of this study is to examine the capabilities of ChatGPT as a tool for supporting students in generating mathematical arguments that can be considered proofs. To examine this, we engaged students enrolled in a mathematics pathways course in evaluating and revising their original arguments using ChatGPT feedback. Students attempted to find and prove a method for the area of a triangle given its side lengths. Instead of directly asking students to prove a formula, we asked them to explore a method to find the area of a triangle given the lengths of its sides and justify why their methods work. Students completed these ChatGPT-embedded proving activities as class homework. To investigate the capabilities of ChatGPT as a proof tutor, we used these student homework responses as data for this study. We analyzed and compared original and revised arguments students constructed with and without ChatGPT assistance. We also analyzed student-written responses about their perspectives on mathematical proof and proving and their thoughts on using ChatGPT as a proof assistant. Our analysis shows that our participants’ approaches to constructing, evaluating, and revising their arguments aligned with their perspectives on proof and proving. They saw ChatGPT’s evaluations of their arguments as similar to how they usually evaluate arguments of themselves and others. Mostly, they agreed with ChatGPT’s suggestions to make their original arguments more proof-like. They, therefore, revised their original arguments following ChatGPT’s suggestions, focusing on improving clarity, providing additional justifications, and showing the generality of their arguments. Further investigation is needed to explore how ChatGPT can be effectively used as a tool in teaching and learning mathematical proof and proof-writing.
참고문헌 (56)
- Alcock, L., "Research in collegiate mathematics education VII", pp.93~114 (2010), American Mathematical Society
- Appel, K., "The solution of the four-color-map problem", Scientific American, 237(4), pp.108~121 (1977)
- Arnau, D., "Fundamentals of the design and the operation of an intelligent tutoring system for the learning of the arithmetical and algebraic way of solving word problems", Computers & Education, 63, pp.119~130 (2013)[DOI]
- Avigad, J., "Proof technology in mathematics research and teaching, Series: Mathematics education in the digital era , (Vol. 14)", pp.277~290 (2019), Springer
- Bertot, Y., "Interactive theorem proving and program development: Coq’Art: The calculus of inductive constructions" (2013), Springer Science & Business Media
- Botana, F., "Automated theorem proving in GeoGebra : Current achievements", Journal of Automated Reasoning, 55(1), pp.39~59 (2015)[DOI]
- Bozkurt, A., "Generative AI and prompt engineering: The art of whispering to let the genie out of the algorithmic world", Asian Journal of Distance Education, 18(2), pp.i~vii (2023)
- Brown, T., "Language models are few-shot learners", Proceedings of the 34th International Conference on Neural Information Processing Systems (NIPS ’20 ), pp.1877~1901 (2020), Curran Associates Inc
- Butgereit, L., "Prof Pi: Using WhatsApp bots and GPT-4 for tutoring mathematics in underserved areas", Proceedings of the International Conference on Innovations and Interdisciplinary Solutions for Underserved Areas, pp.278~289 (2023), Springer Nature Switzerland
- Cain, W., "Prompting change : Exploring prompt engineering in large language model AI and its potential to transform education", TechTrends, 68(1), pp.47~57 (2024)[DOI]
- Chazan, D., "High school geometry students’ justification for their views of empirical evidence and mathematical proof", Educational Studies in Mathematics, 24(4), pp.359~387 (1993)[DOI]
- "Evaluating large language models trained on code"[URL]
- Coe, R., "Proof practices and constructs of advanced mathematics students", British Educational Research Journal, 20(1), pp.41~53 (1994)[DOI]
- de Moura, L., "Automated Deduction-CADE-25", pp.378~388 (2015), Springer International Publishing
- de Villiers, M., "The role and function of proof in mathematics", Pythagoras, 24(24), pp.17~24 (1990)
- Epp, S. S., "The role of logic in teaching proof", The American Mathematical Monthly, 110(10), pp.886~899 (2003)[DOI]
- Finnie-Ansley, J., "The robots are coming : Exploring the implications of OpenAI Codex on introductory programming", Proceedings of the 24th Australasian Computing Education Conference, pp.10~19 (2022), Association for Computing Machinery
- First, E, "Baldur: Whole-proof generation and repair with large language models", Proceedings of the 31st ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering, pp.1229~1241 (2023), Association for Computing Machinery
- Fitting, M., "First-order logic and automated theorem proving" (2012), Springer Science & Business Media
- Font, L., "Mathematics Education in the Age of Artificial Intelligence: How Artificial Intelligence Can Serve Mathematical Human Learning", pp.45~76 (2022), Springer International Publishing
- "Improving QED-Tutrix by automating the generation of proofs"[URL]
- Frick, T., "Are we dupes? Limitations of AI systems: What should educators do with them?", TechTrends, 68(1), pp.14~26 (2024)[DOI]
- Frieder, S, "Mathematical capabilities of ChatGPT", Proceedings of the Advances in Neural Information Processing Systems (NeurIPS 2023) Track on Datasets and Benchmarks, 36 (2024)
- "AIED2023Empowering Education with LLMs - the Next-Gen Interface and Content Generation", 2023, ~
- Gonthier, G., "Formal proof–the four-color theorem", Notices of the AMS, 55(11), pp.1382~1393 (2008)
- Hanna, G., "Proof, explanation and exploration : An overview", Educational Studies in Mathematics, 44(1), pp.5~23 (2000)[DOI]
- Hanna, G., "Proof Technology in Mathematics Research and Teaching, Series: Mathematics Education in the Digital Era (Vol. 14)", pp.3~9 (2019), Springer
- Harel, G., "Research in collegiate mathematics education III", pp.234~283 (1998), American Mathematical Society
- Healy, C., "A study of proof conceptions in algebra", Journal for Research in Mathematics Education, 31(4), pp.396~428 (2000)[DOI]
- Hohenwarter, M., "Proof technology in mathematics research and teaching, Series: Mathematics Education in the Digital Era (Vol. 14)", pp.215~236 (2019), Springer
- Jiang, A. Q., "Thor: Wielding hammers to integrate language models and automated theorem provers", Proceedings of the Advances in Neural Information Processing Systems (NeurIPS 2022), 35, pp.8360~8373 (2022)
- Kang, Y., "A study on the didactical application of ChatGPT for mathematical word problem solving", Communications of Mathematical Education, 38(1), pp.49~67 (2024)[DOI]
- "Learning to prove in order to prove to learn" (2005)[URL]
- Knuth, E. J., "Proof as a tool for learning mathematics", Mathematics Teacher, 95(7), pp.486~490 (2002)[DOI]
- Knuth, E. J., "Proof : Examples and beyond", Mathematics Teaching in the Middle School, 15(4), pp.206~211 (2009)[DOI]
- "GPT-4 vs. GPT-3.5: A concise showdown" (2023)[URL]
- Kovács, Z., "Computer based conjectures and proofs in teaching euclidean geometry", Doctor, pp.2015
- Li, Z., "A survey of convolutional neural networks : analysis, applications, and prospects", IEEE Transactions on Nneural Networks and Learning Systems, 33(12), pp.6999~7019 (2021)[DOI]
- Marty, R. H., "Teaching proof techniques", Mathematics in College, (Spring, pp.Summer) (1986), 46~53
- Moore, R. C., "Making the transition to formal proof", Educational Studies in Mathematics, 27, pp.249~266 (1994)[DOI]
- National Academies of Sciences, Engineering, and Medicine, "Artificial intelligence to assist mathematical reasoning:Proceedings of a workshop" (2023), The National Academies Press
- HOL: A proof assistant for higher-order logic, "Isabelle", pp.2002, Nipkow, T.
- Papadopoulos, D., "Transitioning to proof with worked examples" (2016), Drexel University
- Patero, J. L., "Revolutionizing Math Education: Harnessing ChatGPT for student success", International Journal of Advanced Research in Science, Communication and Technology, 3(1), pp.807~813 (2023)[DOI]
- Selden, A., "Proof and proving in mathematics education", pp.391~414 (2012), Springer
- Selden, J., "Unpacking the logic of mathematical statements", Educational Studies in Mathematics, 29(2), pp.123~151 (1995)[DOI]
- Strauss, A. L., "Basics of qualitative research: Grounded theory procedures and techniques" (1990), Sage
- Stylianides, G. J., "Facilitating the transition from empirical argument to proof", Journal for Research in Mathematics Education, 40(3), pp.314~352 (2009)[DOI]
- Van Vaerenbergh, S., "Mathematics education in the age of artificial intelligence:How artificial intelligence can serve mathematical human learning", pp.89~106 (2022), Springer International Publishing
- Vaswani, A., "Attention is all you need", Proceedings of the Advances in Neural Information Processing Systems (NeurIPS 2017), 30, pp.6000~6010 (2017), Curran Associates Inc
- Volmink, J. D., "The nature and role of proof in mathematics education", Pythagoras, 23, pp.7~10 (1990)
- Wardat, Y., "ChatGPT : A revolutionary tool for teaching and learning mathematics", Eurasia Journal of Mathematics, Science and Technology Education, 19(7), pp.em2286~ (2023)[DOI]
- Weber, K., "Student difficulty in constructing proofs : The need for strategic knowledge", Educational Studies in Mathematics, 48(1), pp.101~119 (2001)[DOI]
- Weber, K., "Semantic and syntactic proof productions", Educational Studies in Mathematics, 56, pp.209~234 (2004)[DOI]
- Yang, K., "LeanDojo : Theorem proving with retrieval-augmented language models", Proceedings of the 37th Conference on Neural Information Processing Systems (NIPS ’23 ) (2023)
- Zafrullah, Z., "ChatGPT Open AI : Analysis of mathematics education students learning interest", Journal of Technology Global, 1(1), pp.1~10 (2023)
