Rigorous Mathematical Thinking: Conceptual Knowledge and Reasoning in the Case of Mathematical Proof

Siska Firmasari, Tatang Herman, Eris Fanny Firdaus

Abstract

This study aims to analyze in-depth students' conceptual knowledge and reasoning when solving problems using mathematical proof as a rigorous mathematical thinking paradigm. The research uses a qualitative method with a case study approach that analyzes the mathematical proof ability of nine students who represent different cognitive functions from each level of rigorous mathematical thinking. The results showed that each level of rigorous mathematical thinking meant other indicators according to their ability to master conceptual knowledge and implement mathematical ideas through reasoning. This research has an impact on the treatment that the teacher must give in determining the learning model and evaluation instrument that can raise students' conceptual knowledge and reasoning.

Penelitian ini bertujuan untuk menganalisis secara mendalam pengetahuan konseptual dan penalaran siswa ketika memecahkan masalah menggunakan pembuktian matematis sebagai paradigma berpikir matematis yang ketat. Penelitian ini menggunakan metode kualitatif dengan pendekatan studi kasus yang menganalisis kemampuan pembuktian matematis sembilan siswa yang mewakili fungsi kognitif yang berbeda dari setiap tingkat pemikiran matematis yang teliti. Hasil penelitian menunjukkan bahwa setiap tingkat berpikir matematis yang teliti berarti indikator lain sesuai dengan kemampuannya untuk menguasai pengetahuan konseptual dan mengimplementasikan ide-ide matematika melalui penalaran. Penelitian ini berdampak pada perlakuan yang harus diberikan guru dalam menentukan model pembelajaran dan instrumen evaluasi yang dapat meningkatkan pengetahuan konseptual dan penalaran siswa.

Keywords

Rigorous Mathematical Thinking, Conceptual Knowledge, Reasoning, Mathematical Proving

Full Text:

PDF

References

Agustyaningrum, N., Hanggara, Y., Husna, A., Abadi, A. M., & Mahmudii, A. (2019). An analysis of students’ mathematical reasoning ability on abstract algebra course. International Journal of Scientific and Technology Research, 8(12), 2800–2805.

Al-Mutawah, M. A., Thomas, R., Eid, A., Mahmoud, E. Y., & Fateel, M. J. (2019). Conceptual understanding, procedural knowledge and problem-solving skills in mathematics: High school graduates work analysis and standpoints. International Journal of Education and Practice, 7(3), 258–273. https://doi.org/10.18488/journal.61.2019.73.258.273

Alpi, K. M., & Evans, J. J. (2019). (2019) Distinguishing case study as a research method from.pdf. 107(January), 1–5.

Arshad, M. N., Atan, N. A., Abu, M. S., Abdullah, A. H., & Mokhtar, M. (2017). Improving the reasoning skills of students to overcome learning difficulties in additional mathematics. Man in India, 97(17), 41–52. https://doi.org/10.37134/jsml.vol5.3.2017

Aulia, E. T., & Fitriyani, H. (2019). Implementasi Pendekatan Rigorous Mathematical Thinking (Rmt) Untuk Meningkatkan Kemampuan Pemecahan Masalah Siswa. JOURNAL of MATHEMATICS SCIENCE and EDUCATION, 1(2), 28–42. https://doi.org/10.31540/jmse.v1i2.300

CadwalladerOlsker, T. (2011). What Do We Mean by Mathematical Proof? Journal of Humanistic Mathematics, 1(1), 33–60. https://doi.org/10.5642/jhummath.201101.04

Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344–377. https://doi.org/10.1016/j.dr.2014.10.001

Darling-Hammond, L., Flook, L., Cook-Harvey, C., Barron, B., & Osher, D. (2020). Implications for educational practice of the science of learning and development. Applied Developmental Science, 24(2), 97–140. https://doi.org/10.1080/10888691.2018.1537791

Dayat Hidayat, Ahmad Wachidul Kohar, Nina Rinda Prihartiwi, Husni Mubarok, & Abebayehu Yohannes. (2021). Design of Learning Activities using Rigorous Mathematical Thinking (RMT) Approach in Application of Derivatives. IJORER : International Journal of Recent Educational Research, 2(1), 111–120. https://doi.org/10.46245/ijorer.v2i1.75

Erdem, E., & Gürbüz, R. (2014). an Analysis of Seventh-Grade Students’ Mathematical Reasoning. Cukurova University Faculty of Education Journal, 44(1), 123. https://doi.org/10.14812/cufej.2015.007

Esterhuizen, S. (2014). Improving some cognitive functions, specifically executive functions in grade R learners. South African Journal of Childhood Education, 4(1), 28. https://doi.org/10.4102/sajce.v4i1.181

Fuady, I. (2016). Dharmakarya: Jurnal Aplikasi Ipteks untuk Masyarakat ISSN 1410 - 5675. Jurnal Aplikasi Ipteks Untuk Masyarakat, 5(1), 34–37.

Gilmore, C., Clayton, S., Cragg, L., McKeaveney, C., Simms, V., & Johnson, S. (2018). Understanding arithmetic concepts: The role of domain-specific and domain-general skills. PLoS ONE, 13(9), 1–20. https://doi.org/10.1371/journal.pone.0201724

Gürbüz, R., & Erdem, E. (2016). Relationship between mental computation and mathematical reasoningi. Cogent Education, 3(1). https://doi.org/10.1080/2331186X.2016.1212683

Hamami, Y. (2014). Mathematical rigor, proof gap and the validity of mathematical inference. Philosophia Scientiae, 18(1), 7–26. https://doi.org/10.4000/philosophiascientiae.908

Hanna, G., & Knipping, C. (2020). Proof in Mathematics Education, 1980-2020: An Overview. Journal of Educational Research in Mathematics, 30(SP1), 1–13. https://doi.org/10.29275/jerm.2020.08.sp.1.1

Harrison, H., Birks, M., Franklin, R., & Mills, J. (2017). Case study research: Foundations and methodological orientations. Forum Qualitative Sozialforschung, 18(1). https://doi.org/10.17169/fqs-18.1.2655

Dollo, A. (2018). Mathematical Reasoning Analysis in Solving Limit Functions Problems of Students in Mathematics Education of Pare-pare Muhammadiyah University. Mathematics Education Journals, 2(1), 59–65.

Hurrell, D. (2021). Conceptual Knowledge OR Procedural Knowledge or Conceptual Knowledge and Procedural Knowledge: Why the Conjunction is Important to Teachers. Australian Journal of Teacher Education, 46(2), 57–71. https://doi.org/10.14221/ajte.2021v46n2.4

Jon, R. (2013). Procedural and Conceptual Knowledge : Exploring the Gap Between Knowledge Type and Knowledge Quality. https://doi.org/10.1080/14926156.2013.784828

Jones, M., & McLean, K. (2018). Personalising learning in teacher education. Personalising Learning in Teacher Education, 1–237. https://doi.org/10.1007/978-981-10-7930-6

Kinard, J. T. (2006). Creating rigorous mathematical thinking: A dynamic that drives mathematics and science conceptual development. Erdélyi Pszichológiai Szemle, Spec Iss2, 251–266.

Kinard, J. T., & Kozulin, A. (2008). Rigorous mathematical thinking: Conceptual formation in the mathematics classroom. Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom, 1–209. https://doi.org/10.1017/CBO9780511814655

Kinard, J. T., & Kozulin, A. (2015). Review : WHAT MATHEMATICS SHOULD KIDS LEARN , AND HOW SHOULD THEY LEARN IT ? Author ( s ): Alan H . Schoenfeld Review by : Alan H . Schoenfeld Published by : University of Illinois Press Stable URL : http://www.jstor.org/stable/27784415 Your use of the JS. 122(3), 417–420.

Kollosche, D. (2021). Styles of reasoning for mathematics education. Educational Studies in Mathematics, 107(3), 471–486. https://doi.org/10.1007/s10649-021-10046-z

Kozulin, A. (2002). Sociocultural Theory and the Mediated Learning Experience. School Psychology International, 23(1), 7-35.

Krantz, S. G. (2007). The History and Concept of Mathematical Proof. The History and Concept of Mathematical Proof, 36.

Letuna, N. A., Natalia, D., & Resureicao, A. A. da. (2020). The Influence of Rigorous Mathematical Thinking (RMT) Learning Pathway to Junior High School Students’Conceptual Understanding. Pancaran Pendidikan, 9(4), 45–54. https://doi.org/10.25037/pancaran.v9i4.314

Lingefjärd, T., & Hatami, R. (2020). The beauty of abstraction in mathematics. Policy Futures in Education, 18(4), 467–482. https://doi.org/10.1177/1478210319895104

Lövdén, M., Fratiglioni, L., Glymour, M. M., Lindenberger, U., & Tucker-Drob, E. M. (2020). Education and Cognitive Functioning Across the Life Span. Psychological Science in the Public Interest, 21(1), 6–41. https://doi.org/10.1177/1529100620920576

Lyons, C. (2014). Relationships between Conceptual Knowledge and Reasoning about Systems: Implications for Fostering Systems Thinking in Secondary Science. ProQuest LLC.

Maidiyah, E., Anwar, N., Zaura, B., Harnita, F., Pendidikan, J., & Kuala, U. S. (2021). Kreano, Jurnal Matematika Kreatif-Inovatif, 12(2), 276–287.

Noto, M. S., Priatna, N., & Dahlan, J. A. (2019). Mathematical proof: The learning obstacles of pre-service mathematics teachers on transformation geometry. Journal on Mathematics Education, 10(1), 117–125. https://doi.org/10.22342/jme.10.1.5379.117-126

Nugraheni, Z., Budiyono, B., & Slamet, I. (2018). Upgrading geometry conceptual understanding and strategic competence through implementing rigorous mathematical thinking (RMT). Journal of Physics: Conference Series, 983(1), 0–5. https://doi.org/10.1088/1742-6596/983/1/012121

Nurjanah, Dahlan, J. A., & Wibisono, Y. (2020). The Effect of Hands-On and Computer-Based Learning Activities on Conceptual Understanding and Mathematical Reasoning. International Journal of Instruction, 14(1), 143–160. https://doi.org/10.29333/IJI.2021.1419A

Rashid, Y., Rashid, A., & Warraich, M. A. (2019). Case Study Method : A Step-by-Step Guide for Business Researchers. 18, 1–13. https://doi.org/10.1177/1609406919862424

Rocess, G. A. P., & Ecurrent, D. E. E. P. R. (2018). L Earning To T Reat S Epsis With M Ulti -O Utput. 2017, 1–10.

Rocha, H. (2019). Mathematical proof: From mathematics to school mathematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377(2140), 1–12. https://doi.org/10.1098/rsta.2018.0045

Saleh Haji. (2019). NCTM’s Principles and Standards for Developing Conceptual Understanding in Mathematics. Journal of Research in Mathematics Trends and Technology, 1(2), 56–65. https://doi.org/10.32734/jormtt.v1i2.2836

Stefanowicz, A., Kyle, J., & Grove, M. (2014). Proofs and Mathematical Reasoning. University of Birmingham, September, 0–49.

Syamsuri, Marethi, I., & Mutaqin, A. (2018). Understanding on strategies of teaching mathematical proof for undergraduate students. Cakrawala Pendidikan, 37(2), 282–293. https://doi.org/10.21831/cp.v37i2.19091

Tanujaya, B., Mumu, J., & Margono, G. (2017). The Relationship between Higher Order Thinking Skills and Academic Performance of Student in Mathematics Instruction. International Education Studies, 10(11), 78. https://doi.org/10.5539/ies.v10n11p78

Wilkinson, L. C., Bailey, A. L., & Maher, C. A. (2018). Students’ Mathematical Reasoning, Communication, and Language Representations: A Video-Narrative Analysis. ECNU Review of Education, 1(3), 1–22. https://doi.org/10.30926/ecnuroe2018010301

DOI: https://doi.org/10.15294/kreano.v13i2.34536

Refbacks

  • There are currently no refbacks.


Bookmark and Share