This study aimed to describe the shift in students’ mathematical arguments from working on a proof individually to working on it in a group. This descriptive-exploratory study used a qualitative approach. Nine undergraduate students from a private university in Jombang, Indonesia, in the 5th semester were the subjects of the study. They were selected because they had already learned the concepts of math logic, argumentation, the theory of numbers, and analysis. Hence, they were ready to construct arguments in dialogue and non-dialogue forms. A task of argumentation and an interview were used to collect data. The study had several stages. First, the students solved the given argumentation task individually. Second, they had to make small groups of 3 members and discuss the same task. The result found that four of them had a complementary shift in their arguments, while the other five students had a reconstructive shift in their views. The complementary change happened because they reconsidered their initial thoughts, complementing their previous thinking structure. On the other hand, the reconstructive shift occurred due to group discussion (i.e., dialogue), which made them reconstruct or even entirely change their previous thoughts. Furthermore, they initially used inductive reasoning and then shifted their reasoning to a deductive one.

**Keywords**: Mathematical Argumentation, Proving, Reconstruct, Complementary shift, Reconstructive shift

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[32] Fischbein E, Kedem I. Proof and Certitude in The Development of Mathematical Thinking. In Sixth Annual Conference of the International Group for the Psychology of Mathematics Education. Antwerp; 1982.

[33] Martin WG, Harel G. Proof Frames of Preservice Elementary Teachers. J Res Math Educ. 1989;20(1):41–51.

[34] Tristanti LB, Sutawidjaja A, As’ari AR, Muksar M. Modelling Student Mathematical ArgumentationWith Structural-Intuitive and Deductive Warrantto Solve Mathematics Problem. in Proceeding of International Conference on Educational Research and Development (ICERD, 2015). pp. 130–139.

[35] Tall D. Introducing Three Worlds of Mathematics. Learn. Math. 2004;23(3):29–33.

[36] 36. Boero P. Argumentation and Mathematical Proof: A Complex, Productive, Unavoidable Relationship in Mathematics and Mathematics Education. Int. Newsl. Teach. Learn. Math. Proof. 1999;7(8).

[37] Boero P, Garuti R, Lemut E, Mariotti AM. Challenging the Traditional School Approach to Theorems: A Hypothesis about the Cognitive Unity of Theorems. Puig L, Gutiérrez A (Eds.). Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education. 1996;2:113-120.

[38] Whitenack J, Yackel E. Making Mathematical Argumens in the Primary Grades: The importance of Explaining and Justifying Ideas. Teach Child Math. 2002;8(9):524–7.

[39] Zhu X, Simon HA. Learning Mathematics from Examples and By Doing. Cogn Instr. 1987;4(3):137–66.

[40] Klauer KJ. Teaching Inductive Reasoning: Some Theory and Three Experimental Studies. Learn Instr. 1996;6(1):37–57.

[41] Tristanti TLB, Nusantara. Identifying Students’ Mathematical Argumentation Competence in Solving Cubes and Pyramid Problems. Journal of Physics: Conference Series. 2021:1933(1). Available: https://files.eric.ed.gov/fulltext/EJ1341588.pdf

[42] Tristanti LB, Nusantara T. The Advantage and Impact of CIRC-Typed and Problem- Based Cooperative Learning Models on Students’ Mathematical Argument. In: 2nd International Conference on Education and Technology (ICETECH 2021). 2022;172– 178.

[43] Tristanti LB, Nusantara T. The Influence of Infusion Learning Strategy on Students’ Mathematical Argumentation Skill. Int J Instr. 2022;15(2):277–92.

[44] Sriraman B, Umland K. Argumentation in Mathematics. Encycl. Math. Educ. 2014;1995: 44–46. https://doi.org/10.1007/978-94-007-4978-8.

[45] Rudhumbu N. A Gender-Based Analysis of Classroom Interaction Practices: The Effect Thereof on University Students’ Academic Performance. Int. J. Learn. Teach. Educ. Res. 2022;21(5):22–45.

[46] Tong DH, Loc NP, Uyen BP, Giang LT. Developing The Competency of Mathematical Modelling: A Case Study of Teaching The Cosine and Sine Theorems. Int. J. Learn. Teach. Educ. Res. 2019;18(11):18–37.

[47] Lin PJ. The development of students’ mathematical argumentation in a primary classroom. Educ Real. 2018;43(3):1171–92.

[48] Osborne J. The Role Of Argument. Science Education,” in Research and the Quality of Science Education. Dordrecht: Springer; 2005. pp. 367–80.

[2] Kim H. Secondary Teachers’ Views about Proof and Judgements on Mathematical Arguments. Res Math Educ. 2022;25(1):65–89.

[3] Netti S, Nusantara T, Subanji S, Abadyo A, Anwar L. Subanji, Abadyo, and L. Anwar, “The Failure to Construct Proof based on Assimilation and Accommodation Framework from Piaget,”. Int Educ Stud. 2016;9(12):12–22.

[4] Weber K, Tanswell FS. Instructions And Recipes In Mathematical Proofs. Educ Stud Math. 2022;111(1):1–15.

[5] Wittmann EC. When is a proof a proof? Connecting mathematics and mathematics education. Cham: Springer; 2021. pp. 61–76.

[6] Wu H. The Role of Euclidean geometry in High School. J Math Behav. 1996;15(3):221– 37.

[7] Kuhn D, Udell W. The development of argument skills. Child Dev. 2003;74(5):1245– 60.

[8] Cerbin B. “The nature and development of informal reasoning skills in college students,” Pap. Present. Twelfth Natl. Inst. Issues Teach. Learn. Teaching Crit. Think. Campus Pract. Emerg. Connect; 1988. pp. 1–17.

[9] Cho KL, Jonassen DH. The effects of argumentation scaffolds on argumentation and problem solving. Educ Technol Res Dev. 2002;50(3):5–22.

[10] Hoyles C, Küchemann D. Students’ understanding of logical implication. Educ Stud Math. 2002;51(3):193–223.

[11] Alexandre J, Pilar M, Munoz P, Cristina A. Cuadrado, Virginia. Expertise, Argumentation and Scientific Practice: a Case Study about Environmental Education in the 11th Grade. in annual meeting of the national association for research in science teaching (NARST). New Orleans, LA; 2000, pp. 1–21.

[12] Krummheuer G. “The Narrative Character of Argumentative Mathematics Classroom Interaction in Primary Education,” Eur. Res. Math. Educ., vol. Group. 1999;4(I):331–41.

[13] Tristanti LB. The Process Of Thinking By Prospective Teachers Of Mathematics In Making Arguments. J. Educ. Learn. 2019;13(1):17–24.

[14] Verheij HB. Evaluating Argumens based on Toulmin’s Scheme. Argumentation. 2005;19(3):347–71.

[15] Toulmin S. The uses of argument. UK: Cambridge University Press; https://doi.org/10.1017/CBO9780511840005.

[16] Walton DN. What is reasoning? What is an argument? J Philos. 1990;87(8):399–419.

[17] Tristanti LB, Sutawidjaja A, As’ari AR, Muksar M. Types of Warrant in Mathematical Argumentations of Prospective-Teacher. Int. J. Sci. Eng. Investig. 2017;6(68):96–101.

[18] Conner AM. Student Teachers’ Conceptions of Proof and Facilitation of Argumentation in Secondary Mathematics Classrooms. The Pennsylvania State University; 2007.

[19] Inglis M, Mejia-Ramos JP, Simpson A. Modelling mathematical argumentation: the importance of qualification. Educ Stud Math. 2007;66(1):3–21.

[20] Durand-Guerrier V. Which Notion of Implication is the Right One? From Logical Considerations to a Didactic Perspective. Educ Stud Math. 2003;53(1):5–34.

[21] Krummheuer G. Methods for Reconstructing Processes of Argumentation and Participation in Primary Mathematics Classroom Interaction. Approaches to Qualitative Research in Mathematics Education. Dordrecht: Springer; 2015. pp. 51– 74.

[22] Tristanti LB, Sutawidjaja A, Asâ AR, Muksar M. The Construction of Deductive Warrant Derived from Inductive Warrant in Preservice-Teacher Mathematical Argumentations. Educ Res Rev. 2016;11(17):1696–708.

[23] Fuat F, Nusantara T, Hidayanto E, Irawati S. The Exploration Of Argument Scheme Expression In Students’ Proof Construction. Int. J. Sci. Technol. Res. 2020;9(01):2369–72.

[24] Alcock L, Weber K. Referential and Syntactic Approaches to Proving: Case Studies from a Transition-to-Proof Course. Res. Coll. Math. Educ. 2010;VII:93–114.

[25] Alcock L, Weber K. Proof Validation In Real Analysis: Inferring And Checking Warrants. J Math Behav. 2005;24(2):125–34.

[26] Magiera MT, Zawojewski JS. Characterizations of Social-Based and Self-Based Contexts Associated with Students’ Awareness, Evaluation, and Regulation of Their Thinking During Small- Group Mathematical Modeling. J. Res. Math. Educ. 2011;42(5):486– https://epublications.marquette.edu/mscs_fac/85/ https://doi.org/10.5951/jresematheduc.42.5.0486.

[27] Harel G. “The Development of Mathematical Induction as a Proof Scheme : A Model for DNR-Based Instruction,” Learn. Teach. Number Theory. J Math Behav. 2001;:185– 212.

[28] Hodds M, Alcock L, Inglis M. Self-Explanation Training Improves Proof Comprehension. J Res Math Educ. 2014;45(1):62–101.

[29] Inglis M, Alcock L. Expert and Novice Approaches to Reading Mathematical Proofs. J Res Math Educ. 2012;43(4):358–90.

[30] Selden A, Selden J. Validations of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument Proves a Theorem? J Res Math Educ. 2003;34(1):4–36.

[31] Weber K. Mathematics Majors’ Perceptions of Conviction, Validity, and Proof. Math Think Learn. 2010;12(4):306–36.

[32] Fischbein E, Kedem I. Proof and Certitude in The Development of Mathematical Thinking. In Sixth Annual Conference of the International Group for the Psychology of Mathematics Education. Antwerp; 1982.

[33] Martin WG, Harel G. Proof Frames of Preservice Elementary Teachers. J Res Math Educ. 1989;20(1):41–51.

[34] Tristanti LB, Sutawidjaja A, As’ari AR, Muksar M. Modelling Student Mathematical ArgumentationWith Structural-Intuitive and Deductive Warrantto Solve Mathematics Problem. in Proceeding of International Conference on Educational Research and Development (ICERD, 2015). pp. 130–139.

[35] Tall D. Introducing Three Worlds of Mathematics. Learn. Math. 2004;23(3):29–33.

[36] 36. Boero P. Argumentation and Mathematical Proof: A Complex, Productive, Unavoidable Relationship in Mathematics and Mathematics Education. Int. Newsl. Teach. Learn. Math. Proof. 1999;7(8).

[37] Boero P, Garuti R, Lemut E, Mariotti AM. Challenging the Traditional School Approach to Theorems: A Hypothesis about the Cognitive Unity of Theorems. Puig L, Gutiérrez A (Eds.). Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education. 1996;2:113-120.

[38] Whitenack J, Yackel E. Making Mathematical Argumens in the Primary Grades: The importance of Explaining and Justifying Ideas. Teach Child Math. 2002;8(9):524–7.

[39] Zhu X, Simon HA. Learning Mathematics from Examples and By Doing. Cogn Instr. 1987;4(3):137–66.

[40] Klauer KJ. Teaching Inductive Reasoning: Some Theory and Three Experimental Studies. Learn Instr. 1996;6(1):37–57.

[41] Tristanti TLB, Nusantara. Identifying Students’ Mathematical Argumentation Competence in Solving Cubes and Pyramid Problems. Journal of Physics: Conference Series. 2021:1933(1). Available: https://files.eric.ed.gov/fulltext/EJ1341588.pdf

[42] Tristanti LB, Nusantara T. The Advantage and Impact of CIRC-Typed and Problem- Based Cooperative Learning Models on Students’ Mathematical Argument. In: 2nd International Conference on Education and Technology (ICETECH 2021). 2022;172– 178.

[43] Tristanti LB, Nusantara T. The Influence of Infusion Learning Strategy on Students’ Mathematical Argumentation Skill. Int J Instr. 2022;15(2):277–92.

[44] Sriraman B, Umland K. Argumentation in Mathematics. Encycl. Math. Educ. 2014;1995: 44–46. https://doi.org/10.1007/978-94-007-4978-8.

[45] Rudhumbu N. A Gender-Based Analysis of Classroom Interaction Practices: The Effect Thereof on University Students’ Academic Performance. Int. J. Learn. Teach. Educ. Res. 2022;21(5):22–45.

[46] Tong DH, Loc NP, Uyen BP, Giang LT. Developing The Competency of Mathematical Modelling: A Case Study of Teaching The Cosine and Sine Theorems. Int. J. Learn. Teach. Educ. Res. 2019;18(11):18–37.

[47] Lin PJ. The development of students’ mathematical argumentation in a primary classroom. Educ Real. 2018;43(3):1171–92.

[48] Osborne J. The Role Of Argument. Science Education,” in Research and the Quality of Science Education. Dordrecht: Springer; 2005. pp. 367–80.