Transforming Students’ Pseudo-Thinking Into Real Thinking in Mathematical Problem Solving
This exploratory and descriptive study aims to theoretically promote the schema of pseudo-thinking processes in mathematical problem-solving by studen.
- Pub. date: August 15, 2023
- Pages: 477-491
- 371 Downloads
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This exploratory and descriptive study aims to theoretically promote the schema of pseudo-thinking processes in mathematical problem-solving by students. The participants in this study were 36 eighth graders and one math teacher. The researchers collected the data using tests and interviews. The results showed that the structure of pseudo-thinking based on the processes of assimilation and accommodation is theoretically composed of five hierarchical components, namely (a) the structure of the problem, (b) the structure of the subject's thinking, (c) the analytic process, (d) the integration of structures or substructures, and (e) the complete integration of structures. When the subject integrates incomplete substructures into existing thinking schemes, assimilation or accommodation becomes imperfect, resulting in cognitive disequilibrium. The results of such a thought process are called pseudo-thinking. Pseudo-thinking processes can be refined and improved into actual thinking processes through reflection and scaffolding. Assimilation and accommodation occur through defragmentation or organization to rearrange the internal schema so that full structural integration occurs. In the end, the subject experiences cognitive equilibrium so that it becomes an actual student thought process.
Keywords: Assimilation and accommodation, mathematical thinking, pseudo-thinking, structured thinking.
References
Adler, R. H. (2022). Trustworthiness in qualitative research. Journal of Human Lactation, 38(4), 598–602. https://doi.org/10.1177/08903344221116620
Basir, M. A., Waluya, S. B., Dwijanto, & Isnarto. (2022). How students use cognitive structures to process information in the algebraic reasoning? European Journal of Educational Research, 11(2), 821–834. https://doi.org/10.12973/eu-jer.11.2.821
Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22
Breive, S. (2020). Student–teacher dialectic in the co-creation of a zone of proximal development: An example from kindergarten mathematics. European Early Childhood Education Research Journal, 28(3), 413–423. https://doi.org/10.1080/1350293X.2020.1755498
Cowan, N. (2014). Working memory underpins cognitive development, learning, and education. Educational Psychology Review, 2, 197–223. https://doi.org/10.1007/s10648-013-9246-y
Creswell, J. W. (2014). Research design: Qualitative, quantitative, and mixed methods approaches. SAGE Publications.
Evans, J. S. B. T., & Stanovich, K. E. (2013). Dual-process theories of higher cognition: Advancing the debate. Perspectives on Psychological Science, 8(3), 223–241. https://doi.org/10.1177/1745691612460685
Fernández, C., Sánchez-Matamoros, G., Valls, J., & Callejo, M. L. (2018). Mirar profesionalmente el pensamiento matemático de los estudiantes: Caracterización, desarrollo y contextos [Noticing students’ mathematical thinking: Characterization, development and contexts]. Avances de Investigacion En Educacion Matematica, (13), 39–61. https://doi.org/10.35763/aiem.v0i13.229
Garner, B. K. (2012). Getting to got it: Helping struggling students learn how to learn. Association for Supervision and Curriculum Development (ASCD).
Gavaz, H. O., Yazgan, Y., & Arslan, Ç. (2021). Non-routine problem solving and strategy flexibility: A quasi-experimental study. Journal of Pedagogical Research, 5(3), 40–54. https://doi.org/10.33902/jpr.2021370581
Hanfstingl, B., Arzenšek, A., Apschner, J., & Gölly, K. I. (2022). Assimilation and accommodation. European Psychologist, 26(4), 320-337. https://doi.org/10.1027/1016-9040/a000463
Hasanah, E., Suyatno, S., Maryani, I., Al Badar, M. I., Fitria, Y., & Patmasari, L. (2022). Conceptual model of differentiated-instruction (DI) based on teachers’ experiences in Indonesia. Education Sciences, 12(10), Article 650. https://doi.org/10.3390/educsci12100650
Holland, P. C. (2008). Cognitive versus stimulus-response theories of learning. Learning and Behavior, 36, 227–241. https://doi.org/10.3758/LB.36.3.227
Hurst, C., & Hurrell, D. (2020). Multiplicative thinking: ‘Pseudo-procedures’ are enemies of conceptual understanding. International Electronic Journal of Mathematics Education, 15(3), em0611. https://doi.org/10.29333/iejme/8567
Ifenthaler, D., Masduki, I., & Seel, N. M. (2011). The mystery of cognitive structure and how we can detect it: Tracking the development of cognitive structures over time. Instructional Science, 39, 41–61. https://doi.org/10.1007/s11251-009-9097-6
Joubish, M. F., & Khurram, M. A. (2011). Cognitive development in Jean Piaget’s work and its implications for teachers. World Applied Sciences Journal, 12(8), 1260–1265. http://www.idosi.org/wasj/wasj12(8)/20.pdf
Kiryak, Z., Candaş, B., & Özmen, H. (2021). Investigating preservice science teachers’ cognitive structures on environmental issues. Journal of Science Learning, 4(3), 244–256. https://doi.org/10.17509/jsl.v4i3.30366
Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2, Article 105. https://doi.org/10.1186/2193-1801-2-105
Kusmaryono, I. (2018). Analysis of sudents’ incorrect answers in a mathematical test: An insight on students’ learning based on SOLO taxonomy and error analysis. Jurnal Pengajaran MIPA, 23(1), 1–8. https://bit.ly/3Of5OVT
Kusmaryono, I., Jupriyanto, & Kusumaningsih, W. (2021). Construction of students’ mathematical knowledge in the zone of proximal development and zone of potential construction. European Journal of Educational Research, 10(1), 341–351. https://doi.org/10.12973/eu-jer.10.1.341
Kusmaryono, I., Ubaidah, N., & Basir, M. A. (2020). The role of scaffolding in the deconstructing of thinking structure: A case study of pseudo-thinking process. Infinity Journal, 9(2), 247–262. https://doi.org/10.22460/infinity.v9i2.p247-262
Lester, J. N., Cho, Y., & Lochmiller, C. R. (2020). Learning to do qualitative data analysis: A starting point. Human Resource Development Review, 19(1), 94–106. https://doi.org/10.1177/1534484320903890
Marsigit, M., Retnawati, H., Apino, E., Santoso, R. H., Arlinwibowo, J., Santoso, A., & Rasmuin, R. (2020). Constructing mathematical concepts through external representations utilizing technology: An implementation in IRT course. TEM Journal, 9(1), 317–326. https://doi.org/10.18421/TEM91-44
Miles, M. B., Huberman, A. M., & Saldaña, J. (2019). Qualitative data analysis: A methods sourcebook (4th ed.). SAGE Publications.
Nepal, B. (2016). Relationship between mathematical thinking and mathematics achievement. IOSR Journal of Research & Method in Education, 6(6), 46–49. https://bit.ly/44DbEWe
Newen, A. (2015). What are cognitive processes ? An example-based approach approach. Synthese, 194, 4251-4268. https://doi.org/10.1007/s11229-015-0812-3
Nowell, L. S., Norris, J. M., White, D. E., & Moules, N. J. (2017). Thematic analysis: Striving to meet the trustworthiness criteria. International Journal of Qualitative Methods, 16(1), 1–13. https://doi.org/10.1177/1609406917733847
Piaget, J. (1964). Part I: Cognitive development in children: Piaget development and learning. Journal of Research in Science Teaching, 2(3), 176–186. https://doi.org/10.1002/tea.3660020306
Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. Journal of Education, 196(2), 1–38. https://doi.org/10.1177/002205741619600202
Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM - Mathematics Education, 49, 307–322. https://doi.org/10.1007/s11858-017-0864-6
Simatwa, E. M. W. (2010). Piaget’s theory of intellectual development and its implication for instructional management at pre-secondary school level. Education Research and Reviews, 5(7), 366–371. https://bit.ly/413cZUK
Stahl, A. N., & King, J. R. (2020). Expanding approaches for research:Understanding and using trustworthiness in qualitative research. Journal of Developmental Education, 44(1), 26–28. https://bit.ly/3n7dZbK
Subanji, & Nusantara, T. (2016). Thinking process of pseudo construction in mathematics concepts. International Education Studies Journal, 9(2), 17-31. https://doi.org/10.5539/ies.v9n2p17
Suranata, K., Rangka, I. B., Ifdil, I., Ardi, Z., Susiani, K., Prasetyaningtyas, W. E., Daharnis, D., Alizamar, A., Erlinda, L., & Rahim, R. (2018). Diagnosis of students zone proximal development on math design instruction: A rasch analysis. Journal of Physics: Conference Series, 1114, Article 012034. https://doi.org/10.1088/1742-6596/1114/1/012034
Suresh, P. L., & Raju, K. N. (2022). Study of test for significance of pearson ’ s correlation coefficient. International Journal of Science and Research, 11(10), 164–166. https://bit.ly/3NNzwjf
Taber, K. S. (2011). Constructivism as educational theory: Contingency in learning, and optimally guided instruction. Nova Science Publishers Inc.
Taherdoost, H. (2016). Sampling methods in research methodology: How to choose a sampling technique for research. International Journal of Academic Research in Management, 5(2), 18–27. https://doi.org/10.2139/ssrn.3205035
Tsang, S., Royse, C. F., & Terkawi, A. S. (2017). Guidelines for developing, translating, and validating a questionnaire in perioperative and pain medicine. Saudi Journal of Anaesthesia, 11(Suppl 1), S80–S89. https://doi.org/10.4103/sja.SJA_203_17
Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34, 97–129. https://doi.org/10.1023/A:1002998529016
Wibawa, K. A., Nusantara, T., Subanji, & Parta, I. N. (2018). Defragmentation of student’s thinking structures in solving mathematical problems based on CRA framework. Journal of Physics: Conference Series, 1028, Article 12150. https://doi.org/10.1088/1742-6596/1028/1/012150
Yeong, M. L., Ismail, R., Ismail, N. H., & Hamzah, M. I. (2018). Interview protocol refinement: Fine-tuning qualitative research interview questions for multi-racial populations in Malaysia. The Qualitative Report, 23(11), 2700–2713. https://doi.org/10.46743/2160-3715/2018.3412
Yilmaz, E. (2019). Cognitive structure determination of prospective science teacher via word association test. Asian Journal of Education and Training, 5(3), 422–428. https://doi.org/10.20448/journal.522.2019.53.422.428
Yilmaz, K. (2011). The cognitive perspective on learning: Its theoretical underpinnings and implications for classroom practices. The Clearing House: A Journal of Educational Strategies, Issues and Ideas, 84(5), 204–212. https://doi.org/10.1080/00098655.2011.568989
Zhiqing, Z. (2015). Assimilation, accommodation, and equilibration: A schema-based perspective on translation as process and as product. International Forum of Teaching and Studies, 11(1-2), 84–89. https://bit.ly/4345azR