An analysis of gifted students’ proof and reasoning process based on the Toulmin model

Derya Zengin; Menekşe Seden Tapan Broutin

doi:10.29333/iejme/17050

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Derya Zengin ¹ ^* , Menekşe Seden Tapan Broutin ²

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¹ Bursa Halil İnalcık Science and Art Center, TÜRKIYE ² Bursa Uludağ University, TÜRKIYE^* Corresponding Author

Abstract

This study aims to examine the mathematical reasoning and proof processes of gifted students within the framework of Toulmin’s Argumentation Model. Grounded in the Rich Teaching Model, the instructional design included seven differentiated lesson plans. The research was conducted with two 9th-grade students in a Science and Art Centre over 20 weeks during the 2022–2023 academic year. Employing a design-based research methodology and a qualitative case study approach, the study focused on a non-routine problem from the product phase of the first lesson plan. This task was examined in depth through micro-level case analysis. Data were analyzed using descriptive methods aligned with Toulmin’s model components. Findings indicate that students used diverse representations, models, and notations to make sense of abstract mathematical concepts and integrated dynamic tools such as GeoGebra into their reasoning. Nonetheless, challenges with formal notation were observed. The study underscores the value of structured, technology-supported instruction in enhancing gifted learners’ mathematical thinking.

Keywords

gifted students
Toulmin model
mathematical proof

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 20, Issue 4, November 2025, Article No: em0855

https://doi.org/10.29333/iejme/17050

Publication date: 01 Oct 2025

Online publication date: 16 Sep 2025

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References

Altun, M. (2010). Mathematics teaching. Aktüel Alfa Publishing.
Anderson, L. W., & Krathwohl, D. R. (Eds.). (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom's taxonomy of educational objectives. Longman
Anderson-Pence, K. L., & Moyer-Packenham, P. S. (2016). Students’ use of mathematical justifications during a dynamic geometry exploration. Journal of Computers in Mathematics and Science Teaching, 35(2), 171-194.
Arzarello, F., Sabena, C., & Robutti, O. (2011). Cognitive unity and the dynamic dragging tool: A case study. ZDM–Mathematics Education, 43(3), 325-336. https://doi.org/10.1007/s11858-011-0323-4
Aydoğdu, M. Z., & Keşan, C. (2016). Geometry problem-solving strategies of 9th-grade gifted students. Journal of Research in Education and Teaching, 5(2), 48-55. http://www.jret.org/FileUpload/ks281142/File/07.mustafa_zeki_aydogdu.pdf
Aygün, Y. İ. (2019). An investigation of mathematical thinking processes of diagnosed and non-diagnosed gifted students in different learning environments [Master’s thesis, Amasya University].
Baig, S., & Halai, A. (2006). Learning mathematical rules with reasoning. Eurasia Journal of Mathematics, Science and Technology Education, 2(2), 15-39. https://doi.org/10.12973/ejmste/75451
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
Bergqvist, T., & Norqvist, M. (2022). Creative and algorithmic reasoning: The role of strategy choices in practice and test. NOMAD Nordic Studies in Mathematics Education, 27(1), 5-25. https://doi.org/10.7146/nomad.v27i1.149181
Berland, L. K., & Reiser, B. J. (2009). Making sense of argumentation and explanation. Science Education, 93(1), 26-55. https://doi.org/10.1002/sce.20286
Betts, G. T., & Kercher, J. K. (1999). The autonomous learner model: Optimizing ability. Sopris West.
Cisneros, C. (2022). Digital tools to enhance interdisciplinary mathematics teaching across STEM disciplines. In Proceedings of the International Conference on STEM Education (pp. 163-176). Springer. https://doi.org/10.1007/978-3-031-29800-4_16
Conner, A. M., & Singletary, L. M. (2021). Teacher support for argumentation: An examination of beliefs and practice. Journal for Research in Mathematics Education, 52(2), 213-247. https://doi.org/10.5951/jresematheduc-2020-0250
Creswell, J. W. (2016). Qualitative inquiry and research design: Choosing among five approaches (3rd ed., S. B. Demir, Trans.). Siyasal Publishing.
Çetinkaya, Ç., & Erden, M. (2017). Differentiated instruction for gifted students: Teachers’ perspectives. Education and Science, 42(189), 155-180. https://doi.org/10.15390/EB. 2017.6839
Çitil, M. (2018). Evaluation of educational policies for gifted students in Turkey. Journal of National Education, 47(1), 143-172. https://dergipark.org.tr/en/pub/milli egitim /issue/ 40518/480017
Davaslıgil, Ü. (2004). Gifted children. Çocuk Vakfı Publishing.
Demiray, E., Işıksal-Bostan, M., & Saygı, E. (2023). How argumentation relates to the formal proof process in geometry. International Electronic Journal of Mathematics Education, 18(3), Article em0741. https://doi.org/10.29333/iejme/13214
Dinamit, D. (2020). Investigation of gifted students’ mathematical proof processes [Master’s thesis, Aydın Adnan Menderes University].
Dinçer, S. (2011). Analysis of discussions in undergraduate mathematics courses according to the Toulmin model [Master’s thesis, Hacettepe University].
Driver, R., Newton, P., & Osborne, J. (2000). Establishing the norms of scientific argumentation in classrooms. Science Education, 84(3), 287-312. https://doi.org/10.1002/(SICI)1098-237X(200005)84:3%3C287::AID-SCE1%3E3.0.CO;2-A
Durand-Guerrier, V. (2020). Logic and proof: A challenge for mathematics education research. ZDM Mathematics Education, 52(6), 1073-1085. https://doi.org/10.1007/s11858-020-01162-z
Elballah, K., Alkhalifah, N., Alomari, A., & Alghamdi, A. (2024). Enhancing cognitive dimensions in gifted students through future problem-solving enrichment programs: A meta-analysis of studies from 2010 to 2023. Discover Sustainability, 5, Article 248. https://doi.org/10.1007/s43621-024-00470-5
Eraky, A., Leikin, R., & Hadad, B.-S. (2022). Relationships between general giftedness, expertise in mathematics, and mathematical creativity are associated with pattern generalization tasks in different representations. Gifted Education International, 38(1), 3-22. https://doi.org/10.1177/27527263221093427
Fabio, R. A., Croce, A., & Calabrese, C. (2023). Critical thinking in ethical and neutral settings in gifted children and non-gifted children. Children, 10(1), Article 74. https://doi.org/10.3390/children10010074
Faizah, S., Rahmawati, N. D., & Murniasih, T. R. (2021). Investigasi struktur argumen mahasiswa dalam pembuktian aljabar berdasarkan skema Toulmin [Investigating the structure of students' arguments in algebraic proofs based on the Toulmin scheme]. Jurnal Program Studi Pendidikan Matematika, 10(3), 1466-1476. https://doi.org/10.24127/ajpm.v10i3.3781
Feldhusen, J., & Kolloff, P. B. (1986). The Purdue three-stage enrichment model for gifted education at the elementary level. In J. S. Renzulli (Ed.), Systems and models for developing programs for the gifted and talented (pp. 153-179). Creative Learning Press.
García-Martínez, I., Gutiérrez Cáceres, R., Luque de la Rosa, A., & León, S. P. (2021). Analyzing educational interventions with gifted students: A systematic review. Children, 8(5), Article 365. https://doi.org/10.3390/children8050365
Healy, L., & Hoyles, C. (2001). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235-256. https://doi.org/10.1023/A:1013305627916
Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73-122. https://doi.org/10.1207/s1532690xci2401_2
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Information Age Publishing & National Council of Teachers of Mathematics.
Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164-192. https://doi.org/10.2307/30034955
Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop et al. (Eds.), Second international handbook of mathematics education (pp. 323-349). Springer. https://doi.org/10.1007/978-94-010-0273-8_11
Hwang, J. (2024). Developing an instrument to measure Korean pre‑service teachers’ understanding of language as an epistemic tool in mathematics education. School Science and Mathematics, 64(2), 89-109. https://doi.org/10.1111/ssm.18309
Inglis, M., & Mejía-Ramos, J. P. (2019). Linguistic conventions of mathematical proof writing at the undergraduate level: Mathematicians' and students' perspectives. Journal for Research in Mathematics Education, 50(2), 121-155. https://doi.org/10.5951/jresematheduc.50.2.0121
Jiménez-Aleixandre, M. P., Rodríguez, A. B., & Duschl, R. A. (2000). “Doing the lesson” or “doing science”: Argument in high school genetics. Science Education, 84(6), 757-792. https://doi.org/10.1002/1098-237X(200011)84:6%3C757::AID-SCE5%3E3.0.CO;2-F
Jones, K. (2000). Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies İn Mathematics, 44(1-2), 55-85. https://doi.org/10.1023/A:1012789201736
Kanbur Teker, B., & Argün, Z. (2022). Üstün yetenekli öğrencilerin geometri öğrenme alanında akıl yürütme becerilerinin incelenmesi [An investigation of reasoning skills in the field of geometry learning among gifted students]. Gazi Journal of Educational Sciences, 8(2), 306-332. https://doi.org/10.30855/gjes.2022.08.02.008
Kaptan, F., & Kuşakçı, F. (2002). The effect of brainstorming technique on student creativity in science teaching. In Proceedings of the 5th National Congress on Science and Mathematics Education (pp. 16-18). Middle East Technical University.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academies Press.
Knuth, E. J., Choppin, J. M., & Bieda, K. N. (2009). Middle school students' production of mathematical justifications. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). Routledge. https://doi.org/10.4324/9780203882009-9
Komatsu, K., & Jones, K. (2022). Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning. Educational Studies in Mathematics, 109(3), 567-591. https://doi.org/10.1007/s10649-021-10086-5
Kovács, Z., Recio, T., & Vélez, M. P. (2024). Showing proofs, assessing difficulty with GeoGebra Discovery. Electronic Proceedings in Theoretical Computer Science (EPTCS), 398, 43-52. https://doi.org/10.4204/EPTCS.398.8
Kozlowski, J. S., & Chamberlin, S. A. (2019). Raising the bar for mathematically gifted students through creativity-based mathematics instruction. Gifted and Talented International, 34(1-2), 79-90. https://doi.org/10.1080/15332276.2019.1690954
Küçük-Demir, B. (2014). An investigation of preservice elementary mathematics teachers’ argumentation development levels and their epistemological beliefs [Master’s thesis, Marmara University].
Laina, V. (2025). Employing digital tools for collaborative and flexible mathematical proving. Digital Experiences in Mathematics Education, 11, 247-261. https://doi.org/10.1007/s40751-024-00157-6
Le Roux, K., Adler, J., & Setati, M. (2004). Reflecting on argumentation in mathematics classrooms: An analysis of learners’ claims and their justification. Pythagoras, 59, 6-13. https://doi.org/10.4102/pythagoras.v0i59.89
Lee, K. H. (2005). Mathematically gifted students' geometrical reasoning and informal proof. In L. C. Helen, & L. V. Jill (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 3, pp. 241-248). PME.
Lincoln, Y. S., & Guba, E. G. (1986). But is it rigorous? Trustworthiness and authenticity in naturalistic evaluation. New Directions for Evaluation, 1986(30), 73-84. https://doi.org/10.1002/ev.1427
Mahdiyyah, N. S., & Susanah. (2022). Analisis argumen matematika siswa SMA ditinjau dari gaya kognitif visualizer-verbalizer [Analysis of high school students' mathematical arguments in terms of the visualizer-verbalizer cognitive style]. Jurnal Ilmiah Pendidikan Matematika, 11(1), 80-96. https://doi.org/10.26740/mathedunesa.v11n1.p80-96
Maker, C. J. (1982). Curriculum development for the gifted. Aspen Systems Corporation.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutierrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 173-204). Sense Publishers. https://doi.org/10.1163/9789087901127_008
Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Pearson Education.
McNeill, K. L., Lizotte, D. J., Krajcik, J., & Marx, R. W. (2006). Supporting students’ construction of scientific explanations by fading scaffolds in instructional materials. The Journal of the Learning Sciences, 15(2), 153-191. https://doi.org/10.1207/s15327809jls1502_1
Merriam, S. B. (2013). Qualitative research: A guide to design and implementation (S. Turan, Trans.). Nobel Publishing.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Sage.
Ministry of National Education [MoNE]. (2017). 9th grade mathematics modeling activity book. https://ogmmateryal.eba.gov.tr/panel/upload/etkilesimli/kitap/beceri_temelli/9/matematik-modelleme/index.html
Özçakır, B., Özdemir, D., & Kıymaz, Y. (2020). Effects of dynamic geometry software on students’ geometric thinking regarding probability of giftedness in mathematics. International Journal of Contemporary Educational Research, 7(2), 48-61. https://doi.org/10.33200/ijcer.664985
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66, 23-41. https://doi.org/10.1007/s10649-006-9057-x
Pitta-Pantazi, D., Christou, C., Kontoyianni, K., & Kattou, M. (2011). A model of mathematical giftedness: Integrating natural, creative, and mathematical abilities. Canadian Journal of Sciennce, Mathematics and Technology Education, 11(1), 39-54. https://doi.org/10.1080/14926156.2011.548900
Porter, L. (2005). Gifted young children (2nd ed.). Open University Press.
Pramesti, P., & Rosyidi, A. H. (2020). Profil argumentasi siswa dalam memecahkan masalah PISA-like berdasarkan model Toulmin [Student argumentation profile in solving Pisa-like problems based on the Toulmin model]. Jurnal Riset Pendidikan Dan Inovasi Pembelajaran Matematika (JRPIPM), 3(2), 92-101. https://doi.org/10.26740/jrpipm.v3n2.p92-101
Ramos, D., & Camacho-Machín, M. (2021). Flexible problem-solving strategies in mathematically talented students: A case study using dynamic geometry. International Journal of Science and Mathematics Education, 19(3), 523-541. https://doi.org/10.1007 /s10763-020-10093-5
Reid, D. A., & Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Sense Publishers. https://doi.org/10.1163/9789460912467
Reis, S. M., Renzulli, S. J., & Renzulli, J. S. (2021). Enrichment and gifted education pedagogy to develop talents, gifts, and creative productivity. Education Sciences, 11(10), Article 615. https://doi.org/10.3390/educsci11100615
Reis, S. M., & Renzulli, J. S. (2010). Is there still a need for gifted education? An examination of current research. Learning and Individual Differences, 20(4), 308-317. https://doi.org/10.1016/j.lindif.2009.10.012
Reuter, T. (2023). Exploratory mathematical argumentation in classroom discourse. Educational Studies in Mathematics, 112(1), 87–106. https://doi.org/10.1007/s10649-022-10199-5
Rudenko¸ I., Bystrova, N., Smirnova, Z., Vaganova, O., & Kutepov, M. (2021). Modern technologies in working with gifted students. Propósitos y Representaciones, 9, Article e818. https://doi.org/10.20511/pyr2021.v9nSPE1.818
Sak, U. (2011). Gifted children: Characteristics, identification, and education. Vize Publishing.
Sampson, V., Grooms, J., & Walker, J. P. (2011). Argument‐driven inquiry as a way to help students learn how to participate in scientific argumentation and craft written arguments: An exploratory study. Science Education, 95(2), 217-257. https://doi.org/10.1002/sce.20421
Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36. https://doi.org/10.2307/30034698
Simon, S., Erduran, S., & Osborne, J. (2006). Learning to teach argumentation: Research and development in the science classroom. International Journal of Science Education, 28(2-3), 235-260. https://doi.org/10.1080/09500690500336957
Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics? Journal of Secondary Gifted Education, 17(1), 20-36. https://doi.org/10.4219/jsge-2005-389
Sriraman, B., Haavold, P., & Lee, K. (2013). Mathematical creativity and giftedness: A commentary on and review of theory, new operational views, and ways forward. ZDM, 45(2), 215-225. https://doi.org/10.1007/s11858-013-0494-6
Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behaviour, 21(4), 459-490. https://doi.org/10.1016/S0732-3123(02)00145-1
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. https://psycnet.apa.org/record/2007-06531-004
Stylianides, A. J., Stylianides, G. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237-266). National Council of Teachers.
Stylianides, G. J., Stylianides, A. J., & Moutsios-Rentzos, A. (2023). Proof and proving in school and university mathematics education research: A systematic review. ZDM Mathematics Education 56, 47-59. https://doi.org/10.1007/s11858-023-01518-y
Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematics problem solving. Educational Studies in Mathematics, 76(3), 265-280. https://doi.org/10.1007/s10649-010-9273-2
Suartha, I. N., Setiawan, I. G. A. N., & Sudiatmika, A. A. R. (2020).Toulmin's argument pattern in the science learning process at SMP Negeri 1 Amlapura. Jurnal Imiah Pendidikan Dan Pembelajaran, 1-11. https://doi.org/10.23887/jipp.v4i1.24151
Tomlinson, C. A. (2014). The differentiated classroom: Responding to the needs of all learners (2nd ed.). ASCD.
Tomlinson, C. A., Kaplan, S. N., Renzulli, J. S., Purcell, J., Leppien, J., & Burns, D. (2002). The parallel curriculum: A design to develop high potential and challenge high-ability learners. Corwin Press.
Toulmin, S. E. (2003). The uses of argument (Updated ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511840005
Trpin, A. (2024). Teaching gifted students in mathematics: A literature review. International Journal of Childhood Education, 5(1). https://doi.org/10.33422/ijce.v5i1.498
Umah, U., Asari, A. R., & Sulandra, I. M. (2016). Covariational reasoning argumentation structure of class VIII MTSN 1 Kediri students. Jurnal Matematika Dan Pendidikan Matematika, 1(1). https://doi.org/10.26594/jmpm.v1i1.498
Urhan, S., & Bülbül, A. (2016). The relationship between argumentation and mathematical proof processes. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 10(1), 351-373. https://doi.org/10.17522/nefefmed.00387
Vargas-Montoya, M., Gimenez, J., & Tkacheva, N. (2023). Only gifted students benefit from ICT use at school in mathematics learning. Education and Information Technologies,29(7),8301-8326. https://doi.org/10.1007/s10639-023-12136-2
Vatandaş, B. (2022). An investigation of proof processes of gifted 8th grade students [Master’s thesis, Adnan Menderes University].
Verallo, E. R., & Cajandig, A. J. S. (2024). Enhancing critical thinking skills in geometry through the guided discovery approach with GeoGebra. International Journal of Studies in Education and Science, 5(4), 416-431. https://doi.org/10.46328/ijses.110
Wang, F., & Hannafin, M. J. (2005). Design-based research and technology-enhanced learning environments. Educational Technology Research and Development, 53(4), 5-23. https://doi.org/10.1007/BF02504682
Weber, K. (2021). Sources of students' difficulties with proof by contradiction. International Journal of Research in Undergraduate Mathematics Education, 7(3), 386-408. https://doi.org/10.1007/s40753-021-00152-x
Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227-236). National Council of Teachers of Mathematics.
Yao, X. (2020). Unpacking learners’ growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames. The Journal of Mathematical Behaviour, 60, Article 100803. https://doi.org/10.1016/j.jmathb.2020.100803
Yıldırım, A., & Şimşek, H. (2013). Qualitative research methods in social sciences (9th ed.). Seçkin Publishing.
Ysseldyke, J., Tardrew, S., Betts, J., Thill, T., & Hannigan, E. (2004). Use of an instructional management system to enhance math instruction of gifted and talented students. Journal for the Education of the Gifted, 27(4), 293-310. https://doi.org/10.4219/jeg-2004-319
Zohar, A., & Nemet, F. (2002). Fostering students’ knowledge and argumentation skills through dilemmas in human genetics. Journal of Research in Science Teaching, 39(1), 35-62. https://doi.org/10.1002/tea.10008
Zulainy, F., Rusdi, R., & Marzal, J. (2021). Pengembangan lembar kerja peserta didik berbasis realistic mathematics education untuk meningkatkan kemampuan argumentasi peserta didik [Development of student worksheets based on realistic mathematics education to improve students' argumentation skills]. Jurnal Pendidikan Matematika, 5(1), 812-828. https://doi.org/10.31004/cendekia.v5i1.440

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APA

Zengin, D., & Tapan Broutin, M. S. (2025). An analysis of gifted students’ proof and reasoning process based on the Toulmin model. International Electronic Journal of Mathematics Education, 20(4), em0855. https://doi.org/10.29333/iejme/17050

Vancouver

Zengin D, Tapan Broutin MS. An analysis of gifted students’ proof and reasoning process based on the Toulmin model. INT ELECT J MATH ED. 2025;20(4):em0855. https://doi.org/10.29333/iejme/17050

AMA

Zengin D, Tapan Broutin MS. An analysis of gifted students’ proof and reasoning process based on the Toulmin model. INT ELECT J MATH ED. 2025;20(4), em0855. https://doi.org/10.29333/iejme/17050

Chicago

Zengin, Derya, and Menekşe Seden Tapan Broutin. "An analysis of gifted students’ proof and reasoning process based on the Toulmin model". International Electronic Journal of Mathematics Education 2025 20 no. 4 (2025): em0855. https://doi.org/10.29333/iejme/17050

Harvard

Zengin, D., and Tapan Broutin, M. S. (2025). An analysis of gifted students’ proof and reasoning process based on the Toulmin model. International Electronic Journal of Mathematics Education, 20(4), em0855. https://doi.org/10.29333/iejme/17050

MLA

Zengin, Derya et al. "An analysis of gifted students’ proof and reasoning process based on the Toulmin model". International Electronic Journal of Mathematics Education, vol. 20, no. 4, 2025, em0855. https://doi.org/10.29333/iejme/17050