Development of ACERA Learning Model Based on Proof Construction Analysis

Deni Hamdani; Cholis Sa’dijah; Subanji Subanji; Sri Subarinah

doi:10.31764/jtam.v7i3.12354

Development of ACERA Learning Model Based on Proof Construction Analysis

Deni Hamdani, Cholis Sa’dijah, Subanji Subanji, Sri Subarinah

Abstract

Proof constructing is the process of justifying a claim using the methods and concepts of proof to produce mathematical proof. Proof constructing is also an aspect of proof, and is often the only way to assess student performance. However, proof construction is still a constant problem (difficulty) for every student. The cause of this difficulty is not only because of the content of proof in textbooks/sources, over-reliance on examples, understanding, underlying logic, and the ability to use proof writing strategies, but also due to the lack of proof discussion activities that train students to understand and answer proof practice questions, give proof reasoning against the proof that has been constructed, and validating own and other colleagues' answers. Thus, this study aims to develop a valid and practical ACERA (Activities, Classroom Discussion, Exercises, Reason, and Audience) learning model and has a potential effect on students' ability to proof construction. This study uses research design research development methods in three stages, namely the preliminary stage, the model development stage and the assessment stage. The research subjects were 23 students of the Mathematics Education study program at the University of Mataram. The development of the ACERA model offers an alternative solution to reduce the difficulty of proof construction, thus enabling this model to have characteristics that are valid, practical and have a potential effect in increasing the productivity of student proof construction.

Keywords

Proof Construction; Activities; Classroom Discussion; Exercises; Reason; Audience.

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References

Antonini, S., & Mariotti, M. A. (2006). Reasoning in an Absurd World: Difficulties With Proof By Contradiction. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, 2(1945), 65–72.

Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM - International Journal on Mathematics Education, 40(3), 401–412. https://doi.org/10.1007/s11858-008-0091-2

Ashton, Z. (2021). Audience role in mathematical proof development. Synthese, 198, 6251–6275. https://doi.org/10.1007/s11229-020-02619-x

Barnard, T. (2000). Why are proofs difficult? The Mathematical Gazette, 84(501), 415–422. https://doi.org/10.2307/3620770

Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE theory to improve students’ graphical understanding of derivative. Eurasia Journal of Mathematics, Science and Technology Education, 14(7), 2947–2967. https://doi.org/10.29333/ejmste/91451

Brown, S. A. (2018). Are indirect proofs less convincing? A study of students’ comparative assessments. Journal of Mathematical Behavior, 49, 1–23. https://doi.org/10.1016/j.jmathb.2016.12.010

Chamberlain Jr, D., & Vidakovic, D. (2017). Developing student understanding: The case of proof by contradiction. Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education, April.

Corcoran, J. (1989). Argumentations and logic. Argumentation, 3(1), 17–43. https://doi.org/10.1007/BF00116415

Davies, B., Alcock, L., & Jones, I. (2020). Comparative judgement, proof summaries and proof comprehension. Educational Studies in Mathematics, 105(2), 181–197. https://doi.org/10.1007/s10649-020-09984-x

Davies, B., & Jones, I. (2022). Assessing Proof Reading Comprehension Using Summaries. International Journal of Research in Undergraduate Mathematics Education, 8(3), 469–489. https://doi.org/10.1007/s40753-021-00157-6

de Villiers, M. (1990). The Role and Function of Proof in Mathematics. Pythagoras, 24, 17–24.

Dubinsky, E. (2001). Using a Theory of Learning in College Mathematics Courses. MSOR Connections, 1(2), 10–15. https://doi.org/10.11120/msor.2001.01020010

Dubinsky, E., Arnon, I., & Weller, K. (2013). Preservice Teachers’ Understanding of the Relation Between a Fraction or Integer and its Decimal Expansion: The Case of 0.9 and 1. Canadian Journal of Science, Mathematics and Technology Education, 13(3), 232–258. https://doi.org/10.1080/14926156.2013.816389

Güler, G. (2016). The Difficulties Experienced in Teaching Proof to Prospective Mathematics Teachers: Academician Views. Higher Education Studies, 6(1), 145. https://doi.org/10.5539/hes.v6n1p145

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23. https://doi.org/10.1023/A:1012737223465

Hoyles, C. (1997). The Curricular Shaping of Students’ Approaches to Proof. For the Learning of Mathematics, 17(1), 7–16.

Ioannou, M. (2017). Exploring Undergraduate Mathematics Students’ Difficulties with the Proof of Subgroup’s Closure under Operation. Eric, 333–340. https://eric.ed.gov/?id=ED589477

Knipping, C. (2008). A method for revealing structures of argumentations in classroom proving processes. ZDM - International Journal on Mathematics Education, 40(3), 427–441. https://doi.org/10.1007/s11858-008-0095-y

Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. Journal of Mathematical Behavior, 53(January), 256–262. https://doi.org/10.1016/j.jmathb.2017.06.002

Lakatos, I. (2015). Proof and Refutations: The Logic of Mathematical Discovery, Edited John Worrall and Elie Zahar (J. Worrall & E. Zahar (eds.); Cambridge). Cambridge Philosophy Classics edition.

Lesseig, K., & Hine, G. (2021). Teaching mathematical proof at secondary school: an exploration of pre-service teachers’ situative beliefs. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1895338

Lew, K., Weber, K., & Mejía Ramos, J. P. (2020). Do Generic Proofs Improve Proof Comprehension? Journal of Educational Research in Mathematics. https://doi.org/10.29275/jerm.2020.08.sp.1.229

Miller, D., Infante, N., & Weber, K. (2018). How mathematicians assign points to student proofs. Journal of Mathematical Behavior, 49, 24–34. https://doi.org/10.1016/j.jmathb.2017.03.002

Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-016-9720-9

Moore, R. C. (1994). Making the Transition to Formal Proof. Educational Studies in Mathematics, 27(3), 249–266.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. In School Science and Mathematics. The National Council of Teachers of Mathematics, Inc.

Pair, J., Strachota, S., & Singh, R. (2021). The Academic Community’s Perceptions of the Two-Column Proof. Mathematics Enthusiast, 18(1–2), 1–30. https://doi.org/10.54870/1551-3440.1521

Quarfoot, D., & Rabin, J. M. (2021). A Hypothesis Framework for Students’ Difficulties with Proof By Contradiction. In International Journal of Research in Undergraduate Mathematics Education. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-021-00150-z

Rabin, J. M., & Quarfoot, D. (2021). Sources of Students’ Difficulties with Proof By Contradiction. International Journal of Research in Undergraduate Mathematics Education, July. https://doi.org/10.1007/s40753-021-00152-x

Reid, D. (2001). Proof, Proofs, Proving and Probing: Research Related To Proof. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 1(May), 360.

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

Selden, A., & Selden, J. (2017). A comparison of proof comprehension, proof construction, proof validation and proof evaluation. Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline, December, 339–345.

Selden, J., & Selden, A. (2009). Understanding the proof construction process. Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education, 2(November), 196–201.

Shaker, H., & Berger, M. (2016). Students’ difficulties with definitions in the context of proofs in elementary set theory. African Journal of Research in Mathematics, Science and Technology Education, 20(1), 80–90. https://doi.org/10.1080/10288457.2016.1145449

Shongwe, B. (2019). Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources. International Electronic Journal of Mathematics Education, 15(2). https://doi.org/10.29333/iejme/6278

Stavrou, S. G. (2014). Common Errors and Misconceptions in Mathematical Proving by Education Undergraduates. IUMPST: The Journal, 1(March), 1–8.

Stylianides, A. J. (2007). Proof and Proving in School Mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288. https://doi.org/10.1080/10986060903253954

Stylianides, G. J., & Stylianides, A. J. (2017). Research-based interventions in the area of proof: the past, the present, and the future. Educational Studies in Mathematics, 96(2), 119–127. https://doi.org/10.1007/s10649-017-9782-3

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. B. (2020). Elementary and Middle School Mathematics: Teaching Developmentally (Tenth Edit). Pearson Education, Inc. http://books.google.com/books?id=MmA6PgAACAAJ

Van den Akker, J., Bannan, B., Kelly, A. E., Nieveen, N., & Plomp, T. (2013). Educational Design Research Educational Design Research. In T. Plomp & N. Nieveen (Eds.), Netherlands Institute for Curriculum Development: SLO. http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=EJ815766

Vidakovic, D., Dubinsky, E., & Weller, K. (2018). APOS Theory: Use of Computer Programs to Foster Mental Constructions and Student’s Creativity. Springer International Publishing. https://doi.org/10.1007/978-3-319-72381-5_18

Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119. https://doi.org/10.1023/A:1015535614355

Weber, K. (2003). Research Sampler 8: students’ difficulties with proof. The Mathematical Association of America: Online, 1, 1–8. http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/research-sampler-8-students-difficulties-with-proof

Weber, K. (2004). A Framework for Describing the Processes That Undergraduate use to Construct Proofs. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4, 42

DOI: https://doi.org/10.31764/jtam.v7i3.12354