Metacognitive failure in constructing proof and how to scaffold it
Abstract
This research aims to describe the students’ metacognitive failure in constructing proof and the scaffolding support. The participants of this qualitative case study were eight preservice mathematics teachers of six-semester, State University of Malang. We carried out a test about proof construction problems in Abstract Algebra. Then we verified the data using triangulation of constant comparative method from a test and a task-based interview with the stimulated recall. The results indicated two groups of students in proving strategy. Group I performed “appropriate” syntactic strategy and Group II vice versa. Blindness was experienced by the subject that does not recognize errors detection or the ambiguity of the proof. Mirage occurred when the subject recognizes an error detection on the proper strategy or application of a theorem, then is unable to verify the truth of his work. Misdirection appeared when the subject recognizes a lack of progress, then uses an incomplete or irrelevant concept. Vandalism emerged with no progress or detection of errors of the strategy then the subject performs some irrelevant steps to the issue or uses a misconception. Practically, the teachers can use these results for learning innovations in scaffolding-based proof courses. The scaffolding might need some development and application in supporting students to overcome difficulty in proving mathematical sentences.
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Alifiani, A., & Walida, S. E. (2020). Proses Metakognitif Mahasiswa Dalam Mengerjakan Soal Higher Order Thinking Skills Ditinjau Dari Gaya Kognitif. Prima: Jurnal Pendidikan Matematika, 4(2), 84–99. https://doi.org/10.31000/prima.v4i2.2614
Anggoro, B. S., Agustina, S., Komala, R., Komarudin, K., Jermsittiparsert, K., & Widyastuti, W. (2019). An Analysis of Students’ Learning Style, Mathematical Disposition, and Mathematical Anxiety toward Metacognitive Reconstruction in Mathematics Learning Process Abstract. Al-Jabar : Jurnal Pendidikan Matematika, 10(2), 187–200. https://doi.org/10.24042/ajpm.v10i2.3541
Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9(1), 33–52. https://doi.org/10.1007/s10857-006-9005-9
Barbacena, L. B., & Sy, N. R. (2013). Metacognitive Model In Mathematical Problem Solving. BU Faculty E-Journal, 1(1).
Basir, M. A., & Wijayanti, D. (2020, December 14). Strategies to Provide Scaffolding when Teaching Mathematical Reasoning. ICIC 2020, Semarang, Indonesia. https://doi.org/10.4108/eai.27-8-2020.2303266
Biryukov, P. (2014). Metacognitive Aspects of Solving Combinatorics Problems. Journal Mathematics Teaching and Learning.
CadwalladerOlsker, T. (2011). What do we mean by mathematical proof? Journal of Humanistic Mathematics, 1(1), 33–60.
Chytrỳ, V., Říčan, J., Eisenmann, P., & Medová, J. (2020). Metacognitive knowledge and mathematical intelligence—Two significant factors influencing school performance. Mathematics, 8(6), 969.
Denley, P., & Bishop, K. (2010). The potential of using stimulated recall approaches to explore teacher thinking. In Using Analytical Frameworks for Classroom Research. Routledge.
Falloon, G. (2020). From simulations to real: Investigating young students’ learning and transfer from simulations to real tasks. https://bera-journals.onlinelibrary.wiley.com/doi/full/10.1111/bjet.12885
Fatmiyati, N., Triyanto, & Fitriana, L. (2020). Error analysis of undergraduate students in solving problems on ring theory. Journal of Physics: Conference Series, 1465, 012050. https://doi.org/10.1088/1742-6596/1465/1/012050
Garofalo, J., & Lester Jr, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 163–176.
Geiger, V., Stillman, G., Brown, J., Galbriath, P., & Niss, M. (2018). Using mathematics to solve real world problems: The role of enablers. Mathematics Education Research Journal, 30(1), 7–19. https://doi.org/10.1007/s13394-017-0217-3
Goos, M. (2002). Understanding metacognitive failure. The Journal of Mathematical Behavior, 21(3), 283–302.
Goos, M., Galbraith, P., & Renshaw, P. (2000). A money problem: A source of insight into problem solving action. 1–21.
Gusmiyanti, R., Suhendra, S., & Nurlaelah, E. (2018). Improving mathematical problem solving ability of senior high school students through learning with metacognitive scaffolding approach. International Conference on Mathematics and Science Education of Universitas Pendidikan Indonesia, 3, 774–779.
Hamami, Y., & Morris, R. L. (2020). Philosophy of mathematical practice: A primer for mathematics educators. ZDM, 52(6), 1113–1126.
Hanna, G. (2018). Reflections on proof as explanation. In Advances in mathematics education research on proof and proving (pp. 3–18). Springer.
Huda, N., Subanji, Nusantar, T., Susiswo, Sutawidjaja, A., & Rahardjo, S. (2016). University Students’ Metacognitive Failures in Mathematical Proving Investigated Based on the Framework of Assimilation and Accommodation. Educational Research and Reviews, 11(12), 1119–1128.
Huda, N., Sutawidjaja, A., Subanji, & Rahardjo, S. (2018). The errors of metacognitive evaluation on metacognitive failure of students in mathematical problem solving. Journal of Physics: Conference Series, 1008, 012073. https://doi.org/10.1088/1742-6596/1008/1/012073
Hughes, E. M., Lee, J.-Y., Cook, M. J., & Riccomini, P. J. (2019). Exploratory Study of a Self-Regulation Mathematical Writing Strategy: Proof-of-Concept. Learning Disabilities: A Contemporary Journal, 17(2), 185–203.
Ishikawa, T., Toshima, M., & Mogi, K. (2019). How and When? Metacognition and Solution Timing Characterize an “Aha” Experience of Object Recognition in Hidden Figures. Frontiers in Psychology, 0. https://doi.org/10.3389/fpsyg.2019.01023
Kilic, H. (2018). Pre-service Mathematics Teachers’ Noticing Skills and Scaffolding Practices. International Journal of Science and Mathematics Education, 16(2), 377–400. https://doi.org/10.1007/s10763-016-9784-0
Könings, K. D., van Zundert, M., & van Merriënboer, J. J. G. (2019). Scaffolding peer-assessment skills: Risk of interference with learning domain-specific skills? Learning and Instruction, 60, 85–94. https://doi.org/10.1016/j.learninstruc.2018.11.007
Magiera, M. T., & Zawojewski, J. S. (2011). Characterizations of social-based and self-based contexts associated with students’ awareness, evaluation, and regulation of their thinking during small-group mathematical modeling. Journal for Research in Mathematics Education, 42(5), 486–520.
Margulieux, L., & Catrambone, R. (2017). Using Learners’ Self-Explanations of Subgoals to Guide Initial Problem Solving in App Inventor. Proceedings of the 2017 ACM Conference on International Computing Education Research, 21–29. https://doi.org/10.1145/3105726.3106168
Moore, R. C. (2016). Mathematics professors’ evaluation of students’ proofs: A complex teaching practice. International Journal of Research in Undergraduate Mathematics Education, 2(2), 246–278.
Nunokawa, K. (2010). Proof, mathematical problem-solving, and explanation in mathematics teaching. In Explanation and proof in mathematics (pp. 223–236). Springer.
Oliviani, F. N. (2018). Identifikasi dan dampak kesalahan metakognitif siswa SMP dalam pemecahan masalah Persamaan Linear. Dissertation and Thesis Graduate Programme UM.
Ozan, P., Aksoy, E., & Narli, S. (2021). Can the Proof Image Exist in the Absence of the Formal Proof?: Analyses of an Unsuccessful Proving Attempt. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 15(1), 1–31. https://doi.org/10.17522/balikesirnef.843527
Pol, J. van de, Mercer, N., & Volman, M. (2019). Scaffolding Student Understanding in Small-Group Work: Students’ Uptake of Teacher Support in Subsequent Small-Group Interaction. Journal of the Learning Sciences, 28(2), 206–239. https://doi.org/10.1080/10508406.2018.1522258
Puente-Díaz, R., Cavazos-Arroyo, J., & Vargas-Barrera, F. (2021). Metacognitive feelings as a source of information in the evaluation and selection of creative ideas. Thinking Skills and Creativity, 39, 100767. https://doi.org/10.1016/j.tsc.2020.100767
Reiser, B. J. (2004). Scaffolding Complex Learning: The Mechanisms of Structuring and Problematizing Student Work. Journal of the Learning Sciences, 13(3), 273–304. https://doi.org/10.1207/s15327809jls1303_2
Rozak, A. (2018). Proses kegagalan metakognitif pada pemecahan masalah matematika [PhD Thesis]. Universitas Negeri Malang.
Salem, A. A. M. S. (2019). Learning in a sheltered online scaffolding environment (SOSE). Education and Information Technologies, 24(4), 2503–2521. https://doi.org/10.1007/s10639-019-09883-6
Shvarts, A., & Bakker, A. (2019). The early history of the scaffolding metaphor: Bernstein, Luria, Vygotsky, and before. Mind, Culture, and Activity, 26(1), 4–23. https://doi.org/10.1080/10749039.2019.1574306
Sirmaci, N. (2012). Knowledge Level of Undergraduate Students of Mathematics Teaching on Proof Methods. Global Advanced Research Journal of Educational Research and Review (ISSN: 2315-5132), 1(6), 118–123.
Stillman, G. (2011). Applying Metacognitive Knowledge and Strategies in Applications and Modelling Tasks at Secondary School. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical Modelling (pp. 165–180). Springer Netherlands. https://doi.org/10.1007/978-94-007-0910-2_18
Stillman, G. A. (2015). Applications and Modelling Research in Secondary Classrooms: What Have We Learnt? In S. J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 791–805). Springer International Publishing. https://doi.org/10.1007/978-3-319-17187-6_44
Thomas, M. O., de Freitas Druck, I., Huillet, D., Ju, M.-K., Nardi, E., Rasmussen, C., & Xie, J. (2015). Key mathematical concepts in the transition from secondary school to university. The Proceedings of the 12th International Congress on Mathematical Education, 265–284.
Van Der Stuyf, R. R. (2002). Scaffolding as a teaching strategy. 52(3), 5-18.
Varghese, T. (2009). Secondary-Level Student Teachers’ Conceptions of Mathematical Proof. Issues in the Undergraduate Mathematics Preparation of School Teachers, 1.
Wasserman, N., Weber, K., Villanueva, M., & Mejia-Ramos, J. P. (2018). Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis. The Journal of Mathematical Behavior, 50, 74–89.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.
Weber, K., & Alcock, L. (2004). Semantic and Syntactic Proof Productions. Educational Studies in Mathematics, 56(3), 209–234. https://doi.org/doi:10.1023/b:educ.0000040410.57253.a1
Wittmann, E. C. (2021). When is a proof a proof? In Connecting Mathematics and Mathematics Education (pp. 61–76). Springer.
Wood, D., Bruner, J. S., & Ross, G. (1976). THE ROLE OF TUTORING IN PROBLEM SOLVING. Journal of Child Psychology and Psychiatry, 17(2), 89–100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x
Wright, V. (2018). Vygotsky and a Global Perspective on Scaffolding in Learning Mathematics. In J. Zajda (Ed.), Globalisation and Education Reforms: Paradigms and Ideologies (pp. 123–135). Springer Netherlands. https://doi.org/10.1007/978-94-024-1204-8_8
Zackariasson, M. (2019). Encouraging student independence: Perspectives on scaffolding in higher education supervision. Journal of Applied Research in Higher Education, 12(3), 495–505. https://doi.org/10.1108/JARHE-01-2019-0012
Zengin, Y. (2017). The effects of GeoGebra software on pre-service mathematics teachers’ attitudes and views toward proof and proving. International Journal of Mathematical Education in Science and Technology, 48(7), 1002–1022.
Zhao, N., Teng, X., Li, W., Li, Y., Wang, S., Wen, H., & Yi, M. (2019). A path model for metacognition and its relation to problem-solving strategies and achievement for different tasks. ZDM, 51(4), 641–653.
Zimmermann, B. (2016). Improving of mathematical problem-solving: Some new ideas from old resources. In Posing and solving mathematical problems (pp. 83–108). Springer.
DOI: http://dx.doi.org/10.24042/ajpm.v12i2.9590
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