RESEARCH PAPER
Unpacking mental models, strategies, and schemas pre-service mathematics teacher in solving maximum rectangular areas
 
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1
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang, Jawa Timur, INDONESIA
 
2
Universitas Lampung, Kota Bandar Lampung, INDONESIA
 
 
Online publication date: 2023-06-21
 
 
Publication date: 2023-08-01
 
 
EURASIA J. Math., Sci Tech. Ed 2023;19(8):em2309
 
KEYWORDS
ABSTRACT
A mental model is an essential element that influences the quality of problem-solving. This study explores mental models, strategies, and schemes that are active in solving the maximum rectangular area problem. The approach used in this study is a descriptive qualitative approach. A total of four pre-service mathematics teachers as subjects were selected from 108 prospective subjects involved in the study. Data collection was carried out by providing task sheets for solving the most significant rectangular area problem and semi-structured interviews. Data reduction, data presentation, and conclusion are three stages in analyzing the research data. This study indicates three mental models for pre-service mathematics teachers in solving maximum rectangular area problems: initial, adaptive, and formal. Each has different strategies and schemes that are active while solving problems. The results of this study imply that mental models influence the quality of the problem solving process until results are obtained.
 
REFERENCES (55)
1.
Baguley, T., & Payne, S. J. (2000). Long-term memory for spatial and temporal mental models includes construction processes and model structure. Quarterly Journal of Experimental Psychology Section A: Human Experimental Psychology, 53(2), 479-512. https://doi.org/10.1080/713755....
 
2.
Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194-245. https://doi.org/10.5951/jresem....
 
3.
Calderon, S., Ask, K., Mac Giolla, E., & Granhag, P. A. (2019). The mental representation of true and false intentions: A comparison of schema-consistent and schema-inconsistent tasks. Cognitive Research: Principles and Implications, 4(1), 29. https://doi.org/10.1186/s41235....
 
4.
Chi, M. T. H. (2006). Laboratory methods for assessing experts’ and novices’ knowledge. In K. A. Ericsson, N. Charness, P. J. Feltovich, & R. R. Hoffman (Eds.), The Cambridge handbook of expertise and expert performance (pp. 167-184). Cambridge University Press. https://doi.org/10.1017/CBO978....
 
5.
Chinnapan, M. (1998). Schemas and mental models in geometry problem solving. Educational Studies in Mathematics, 36, 201-217. https://doi.org/10.1023/A:1003....
 
6.
Convertino, M., Munoz-Carpena, R., & Murcia, C. (2016). “Reading the minds” for quantitative sustainability: Assessing stakeholder mental models via probabilistic text analysis. In J. Zhang, L. F. Luna-Reyes, T. A. Pardo, & D. S. Sayogo (Eds.), Information, models, and sustainability (pp. 21-38). Springer. https://doi.org/10.1007/978-3-....
 
7.
Creswell, J. W., & Creswell, J. D. (2017). Research design: Qualitative, quantitative, and mixed methods approaches. SAGE.
 
8.
Duit, R., & Treagust, D. F. (2003). Conceptual change: A powerful framework for improving science teaching and learning. International Journal of Science Education, 25(6), 671-688. https://doi.org/10.1080/095006....
 
9.
English, L. D. (2013). Mathematical reasoning: Analogies, metaphors, and images. Routledge. https://doi.org/10.4324/978020....
 
10.
Gentner, D., & Gentner, D. R. (1983). Flowing waters or teeming crowds: Mental models of electricity. In D. Gentner, & A. L. Stevens (Eds.), Mental models (pp. 99-129). Lawrence Erlbaum Associates.
 
11.
Gentner, D., & Stevens, A. L. (1983). Mental models. Lawrence Erlbaum Associates.
 
12.
Gogus, A. (2013). Evaluating mental models in mathematics: A comparison of methods. Educational Technology Research and Development, 61(2), 171-195. https://doi.org/10.1007/s11423....
 
13.
Greefrath, G., Oldenburg, R., Siller, H. S., Ulm, V., & Weigand, H. G. (2020). Basic mental models of integrals: Theoretical conception, development of a test instrument, and first results. ZDM-Mathematics Education, 53, 649-661. https://doi.org/10.1007/s11858....
 
14.
Greefrath, G., Oldenburg, R., Siller, H. S., Ulm, V., & Weigand, H. G. (2022). Mathematics students’ characteristics of basic mental models of the derivative. Journal für Mathematik-Didaktik [Journal for Mathematics Didactics], 44, 143-169. https://doi.org/10.1007/s13138....
 
15.
Halford, G. S. (1993). Children’s understanding: The development of mental models. Lawrence Erlbaum Associates.
 
16.
Hasemann, K. (1986). Mathematische Lernprozesse: Analysen mit kognitionstheoretischen Modellen [Mathematical learning processes: Analyzes with cognitive-theoretical models] Vieweg. https://doi.org/10.1007/978-3-....
 
17.
Hester, K. S., Robledo, I. C., Barrett, J. D., Peterson, D. R., Hougen, D. P., Day, E. A., & Mumford, M. D. (2012). Causal analysis to enhance creative problem-solving: Performance and effects on mental models. Creativity Research Journal, 24(2-3), 115-133. https://doi.org/10.1080/104004....
 
18.
Hoosain, E. (2004). What are mathematical problems? Humanistic Mathematics Network Journal, 1(27), 1-8. https://doi.org/10.5642/hmnj.2....
 
19.
Ifenthaler, D., Pirnay-Dummer, P., & Spector, J. M. (2008). Understanding models for learning and instruction: Essays in honor of Norbert M. Seel. Springer. https://doi.org/10.1007/978-0-....
 
20.
Johnson, R. B., & Christensen, L. (2020). Education research: Quantitative, qualitative, and mixed approaches. SAGE.
 
21.
Johnson-Laird, P. N. (1983). Mental models. Cambridge University Press.
 
22.
Johnson-Laird, P. N., & Byrne, R. M. J. (1993). Précis of deduction. Behavioral and Brain Sciences, 16(2), 323-383. https://doi.org/10.1017/S01405....
 
23.
Jones, N. A., Ross, H., Lynam, T., Perez, P., & Leitch, A. (2011). Mental models: An interdisciplinary synthesis of theory and methods. Ecology and Society, 16(1), 46. https://doi.org/10.5751/ES-038....
 
24.
Justi, R., & Gilbert, J. (2000). History and philosophy of science through models: Some challenges in the case of ‘the atom.’ International Journal of Science Education, 22(9), 993-1009. https://doi.org/10.1080/095006....
 
25.
Kahneman, D., & Tversky, A. (1982). Judgement under uncertainty: Heuristics and biases. Cambridge University Press. https://doi.org/10.1017/CBO978....
 
26.
Kildan, A. O., Altan, M., & Ahi, B. (2013). Mental models of school for preschool children. European Journal of Educational Research, 2(2), 97-105. https://doi.org/10.12973/eu-je....
 
27.
Lin, J. W. (2017). A cross-grade study validating the evolutionary pathway of student mental models in electric circuits. EURASIA Journal of Mathematics, Science and Technology Education, 13(7), 3099-3137. https://doi.org/10.12973/euras....
 
28.
Loarces, R. G., Fernández, G., & González, F. (2019). Evolución de los modelos mentales sobre fosilización tras el proceso de enseñanza-aprendizaje [Evolution of mental models on fossilization after the teaching-learning process]. Revista Eureka Sobre Enseñanza y Divulgación de Las Ciencias [Eureka Magazine on Teaching and Dissemination of Sciences], 16(2), 1-14. https://doi.org/10.25267/Rev_E....
 
29.
Mayr, E., Schreder, G., Smuc, M., & Windhager, F. (2016). Looking at the representations in our mind. In Proceedings of the 6th Workshop on Beyond Time and Errors on Novel Evaluation Methods for Visualization (pp. 96-103). https://doi.org/10.1145/299390....
 
30.
Meela, P., & Yuenyong, C. (2019). The study of grade 7 mental model about properties of gas in science learning through model based inquiry (MBI). AIP Conference Proceedings, 2081, 030028. https://doi.org/10.1063/1.5094....
 
31.
Mikkilä-Erdmann, M., Penttinen, M., Anto, E., & Olkinuora, E. (2008). Constructing mental models during learning from science text: Eye tracking methodology meets conceptual change. In D. Ifenthaler, P. Pirnay-Dummer, & J. M. Spector (Eds.), Understanding models for learning and instruction: Essays in honor of Norbert M. Seel (pp. 63-80). Springer. https://doi.org/10.1007/978-0-....
 
32.
Miles, M. B., Huberman, A. M., & Saldana, J. (2014). Qualitative data analysis. SAGE.
 
33.
Moutinho, S., Moura, R., & Vasconcelos, C. (2017). Contributions of model-based learning to the restructuring of graduation students’ mental models on natural hazards. EURASIA Journal of Mathematics, Science and Technology Education, 13(7), 3043-3068. https://doi.org/10.12973/euras....
 
34.
Norman, D. A., Gentner, D., & Stevens, A. L. (1976). Comments on learning schemata and memory representation. In R. Dillon, & R. Sternberg (Eds.), Cognition and instruction (pp. 177-196). Lawrence Erlbaum Associates.
 
35.
Norman, D. A., Gentner, D., & Stevens, A. L. (1983). Some observations on mental models. In Mental models (pp. 7-14). Psychology Press.
 
36.
Oleson, K. E., Sims, V. K., Chin, M. G., Lum, H. C., & Sinatra, A. (2010). Developmental human factors: Children’s mental models of computers. Proceedings of the Human Factors and Ergonomics Society Annual Meeting, 54(19), 1450-1453. https://doi.org/10.1177/154193....
 
37.
Piaget, J. (1954). The construction of reality in the child. Routledge. https://doi.org/10.1037/11168-....
 
38.
Posamentier, A. S., & Krulik, S. (2008). Problem-solving strategies for efficient and elegant solutions. Corwin Press.
 
39.
Prayekti, N., Nusantara, T., Sudirman, S., & Susanto, H. (2019). Students’ mental model in solving the patterns of generalization problem. IOP Conference Series: Earth and Environmental Science, 243(1), 012144. https://doi.org/10.1088/1755-1....
 
40.
Radvansky, G., & Zacks, J. M. (2012). Event perception. WIREs Cognitive Science, 2(6), 608-620. https://doi.org/10.1002/wcs.13....
 
41.
Riemer, V., & Schrader, C. (2019). Mental model development in multimedia learning: Interrelated effects of emotions and self-monitoring. Frontiers in Psychology, 10, 1-19. https://doi.org/10.3389/fpsyg.....
 
42.
Rouse, W. B., & Morris, N. M. (1986). On looking into the black box: Prospects and limits in the search for mental models. Psychological Bulletin, 100(3), 349-363. https://doi.org/10.1037/0033-2....
 
43.
Saçkes, M. (2015). Kindergartners’ mental models of the day and night cycle: Implications for instructional practices in early childhood classrooms. Educational Sciences: Theory & Practice, 15(4), 997-1006. https://doi.org/10.12738/estp.....
 
44.
Spangenberg, E. D., & Pithmajor, A. K. (2020). Grade 9 mathematics learners’ strategies in solving number-pattern problems. EURASIA Journal of Mathematics, Science and Technology Education, 16(7), em1862. https://doi.org/10.29333/EJMST....
 
45.
Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503-518. https://doi.org/10.1016/J.LEAR....
 
46.
Stamovlasis, D., Papageorgiou, G., Tsitsipis, G., Tsikalas, T., & Vaiopoulou, J. (2018). Illustration of step-wise latent class modeling with covariates and taxometric analysis in research probing children’s mental models in learning sciences. Frontiers in Psychology, 9, 1-20. https://doi.org/10.3389/fpsyg.....
 
47.
Stark, R. (2003). Conceptual change: From a cognitive or situated perspective? Zeitschrift für Padagogische Psychologie [Journal of Educational Psychology], 17(2), 133-144. https://doi.org/10.1024//1010-....
 
48.
Sternberg, R. J., & Sternberg, K. (2016). Cognitive psychology. Cengage Learning.
 
49.
Subanji, S., & Nusantara, T. (2016). Thinking process of pseudo construction in mathematics concepts. International Education Studies, 9(2), 17. https://doi.org/10.5539/ies.v9....
 
50.
Turk, C., Kalakan, H., Kiroglu, K., & Iskeleli, N. O. (2015). Elementary school students’ mental models about formation of seasons: A cross sectional study. Journal of Education and Learning, 5(1), 7. https://doi.org/10.5539/jel.v5....
 
51.
Utami, A. D., Sa’dijah, C., Subanji, & Irawati, S. (2018). Six levels of Indonesian primary school students’ mental model in comprehending the concept of integer Anita. International Journal of Instruction, 11(4), 29-44. https://doi.org/10.12973/iji.2....
 
52.
van Ments, L., & Treur, J. (2022). Dynamics, adaptation and control for mental models: A cognitive architecture. In J. Treur, & L. van Ments (Eds.), Mental models and their dynamics, adaptation, and control (pp. 3-26). Springer. https://doi.org/10.1007/978-3-....
 
53.
Vandenbosch, B., & Higgins, C. (1996). Information acquisition and mental models: An investigation into the relationship between behavior and learning. Information Systems Research, 7(2), 198-214. https://doi.org/10.1287/isre.7....
 
54.
Yaagoubi, R., Edwards, G., Badard, T., & Mostafavi, M. A. (2012). Enhancing the mental representations of space used by blind pedestrians, based on an image schemata model. Cognitive Processing, 13(4), 333-347. https://doi.org/10.1007/s10339....
 
55.
Zwaan, R. A. (2016). Situation models, mental simulations, and abstract concepts in discourse comprehension. Psychonomic Bulletin & Review, 23(4), 1028-1034. https://doi.org/10.3758/s13423....
 
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