Eighth-grades students’ mental models in solving a number pattern problem

Abstract

This study aims to explore all the types of students' mental models of number patterns. The study used a qualitative approach with an explorative type. The subjects used to characterize the student's mental models in this study were 46 eighth grade students in Indonesia. To reveal the subjects’ mental model, they were asked to solve the number pattern problem and were interviewed. For ensuring the validity and reliability of the research results, triangulation technique was used by comparing the results of video recording interviews and written test results. The study showed that in solving the problem of number patterns given, there were 4 types of mental models. They were formal direct mental model, formal indirect mental model, synthetic direct mental model, and synthetic indirect mental model. What we found in this study shows that some students have different mental models to solve the problem. Hence, in future teachers must introduce various strategies to solve the problem and conduct learning that can enrich students' mental models.

Keywords

• mental model
• problem solving
• students characterization
• number pattern

References

1. Allen, L. G. (2001). Teaching mathematical induction: An alternative approach. Mathematics Teacher, 94(6), 500-504.
2. Anwar, & Rofiki, I. (2018). Investigating students’ learning trajectory: A case on triangle. Journal of Physics: Conference Series, 1088(1), 012021. https://doi.org/10.1088/1742-6596/1088/1/012021
3. 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/jresematheduc.45.2.0194
4. Botzer, G., & Yerushalmy, M. (2008). Embodied semiotic activities and their role in the construction of mathematical meaning of motion graphs. International Journal of Computers for Mathematical Learning, 13(3), 1-12. https://doi.org/10.1007/s10758-008-9133-7
5. Carifio, J. (2015). Updating, modernizing, and testing Polya’s theory of [mathematical] problem solving in terms of current cognitive, affective, and information processing theories of learning, emotions, and complex performances. Journal of Education and Human Development, 4(3), 105–117. https://doi.org/10.15640/jehd.v4n3a12
6. Chiou, G. L. (2013). Reappraising the relationships between physics students’ mental models and predictions: An example of heat convection. Physical Review Special Topics-Physics Education Research, 9(1), 010119. https://doi.org/10.1103/PhysRevSTPER.9.010119
7. Chittleborough, G. (2004). The role of teaching models and chemical representations in developing students' mental models of chemical phenomena (Doctoral dissertation). Curtin University.
8. Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In the Teaching and Learning of Mathematics at University Level (pp. 275-282). Springer, Dordrecht.

9. Fazio, C., Battaglia, O. R., & Di Paola, B. (2013). Investigating the quality of mental models deployed by undergraduate engineering students in creating explanations: The case of thermally activated phenomena. Physical Review Special Topics-Physics Education Research, 9(2), 020101. https://doi.org/10.1103/PhysRevSTPER.9.020101
10. Font, V., Trigueros, M., Badillo, E., & Rubio, N. (2016). Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA. Educational Studies in Mathematics, 91(1), 107–122. https://doi.org/10.1007/s10649-015-9639-6
11. Golfashani, N. (2003). Understanding reliability and validity in qualitative research. The Qualitative Report, 8(4), 597–607. https://nsuworks.nova.edu/tqr/vol8/iss4/6
12. Goodnough, K. C., & Hung, W. (2008). Engaging teachers’ pedagogical content knowledge: Adopting a nine-step problem-based learning model. Interdisciplinary Journal of Problem-Based Learning, 2(2), 10–13. https://doi.org/10.7771/1541-5015.1082
13. Greca, I. M., & Moreira, M. A. (2002). Mental, physical, and mathematical models in the teaching and learning of physics. Science Education, 86(1), 106–121. https://doi.org/10.1002/sce.10013
14. Halim, N. D. A., Ali, M. B., Yahaya, N., & Said, M. N. H. M. (2013). Mental model in learning chemical bonding: A preliminary study. Procedia-Social and Behavioral Sciences, 97, 224–228. https://doi.org/10.1016/j.sbspro.2013.10.226
15. Hartikainen, S., Rintala, H., Pylväs, L., & Nokelainen, P. (2019). The concept of active learning and the measurement of learning outcomes: A review of research in engineering higher education. Education Sciences, 9(4), 9–12. https://doi.org/10.3390/educsci9040276
16. Hidayah, I., Pujiastuti, E., & Chrisna, J. E. (2017). Teacher’s stimulus helps students achieve mathematics reasoning and problem solving competences. Journal of Physics: Conference Series, 824(1), 012042. https://doi.org/10.1088/1742-6596/755/1/011001
17. Hillman, W. (2003). Learning how to learn: Problem based learning. Australian Journal of Teacher Education, 28(2), 1-10. https://doi.org/10.14221/ajte.2003v28n2.1
18. Jansoon, N., Coll, R. K., & Somsook, E. (2009). Understanding mental models of dilution in Thai students. International Journal of Environmental and Science Education, 4(2), 147–168.
19. Kusdinar, U., Sukestiyarno, Isnarto, & Istiandaru, A. (2017). Krulik and Rudnik model heuristic strategy in mathematics problem solving. International Journal on Emerging Mathematics Education, 1(2), 205-210. https://doi.org/10.12928/ijeme.v1i2.5708
20. Lindblom, S. (2019). The role of research-based evidence in cultivating quality of teaching and learning in higher education. Uniped, 42(01), 106-110. https://doi.org/10.18261/issn.1893-8981-2019-01-08
21. Mathison, S. (1988). Why Triangulate. Educational researcher, 17(2), 13-17.
22. Ministry of Education and Culture. (2016). The management of national education in Year 2014/2015 at a Glance. http://publikasi.data.kemdikbud.go.id/uploadDir/isi_6549DA84-7A7F-44B5-AD22-829B1F002A4F_.pdf
23. Mulligan, J. T., Mitchelmore, M. C., English, L. D., & Robertson, G. (2010). Implementing a Pattern and Structure Mathematics Awareness Program (PASMAP) in Kindergarten. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp.793-803). MERGA.
24. NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
25. Nite, S. (2017). Using Polya’s problem solving process in the mathematics classroom to prepare for taks. In G. D. Allen, & A. Ross. (Eds), Pedagogy and Content in Middle and High School Mathematics (pp.233-235). SensePublishers, Rotterdam. https://doi.org/10.1007/978-94-6351-137-7_51
26. Rogers, M. A. P., Cross, D. I., Gresalfi, M. S., Trauth-Nare, A. E., & Buck, G. A. (2011). First year implementation of a project-based learning approach: The need for addressing teachers’ orientations in the era of reform. International Journal of Science and Mathematics Education, 9(4), 893-917.
27. Prayekti, N., Nusantara, T., Sudirman, & Susanto, H. (2019). Students ’ mental model in solving the patterns of generalization problem. IOP Conf. Series: Earth and Environmental Science, 243. https://doi.org/10.1088/1755-1315/243/1/012144
28. Putra, H. D., Herman, T., & Sumarmo, U. (2017). Development of student worksheets to improve the ability of mathematical problem posing. International Journal on Emerging Mathematics Education, 1(1), 1-10. https://doi.org/10.12928/ijeme.v1i1.5507
29. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70. https://doi.org/10.1207/s15327833mtl0501_02
30. Radford, L., & Peirce, C. S. (2006, March). Algebraic thinking and the generalization of patterns: A semiotic perspective. In Proceedings of the 28th conference of the international group for the psychology of mathematics education, North American chapter (Vol. 1, pp. 2-21).
31. Resnik, M. D. (1997). Mathematics as a Science of Patterns. Oxford University Press.
32. Rofiki, I., & Santia, I. (2018). Describing the phenomena of students’ representation in solving ill-posed and well-posed problems. International Journal on Teaching and Learning Mathematics, 1(1), 39–50. https://doi.org/https://doi.org/10.18860/ijtlm.v1i1.5713
33. Suryani, A. I., Anwar, Hajidin, & Rofiki, I. (2020). The practicality of mathematics learning module on triangles using GeoGebra. Journal of Physics: Conference Series, 1470(1), 012079. https://doi.org/10.1088/1742-6596/1470/1/012079
34. Tikekar, V. G. (2009). Deceptive patterns in mathematics. International Journal Mathematic Science Education, 2(1), 13-21.
35. Vogel, R. (2005). Patterns - a fundamental idea of mathematical thinking and learning. ZDM - International Journal on Mathematics Education, 37(5), 445–449. https://doi.org/10.1007/s11858-005-0035-z
36. Vosniadou, S. (1994). Capturing and modeling the process of conceptual change. Learning and Instruction, 4, 4–9. https://doi.org/10.1016/0959-4752(94)90018-3
37. Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive psychology, 24(4), 535-585.
38. Wenglinsky, H. (2002). How schools matter: The link between teacher classroom practices and students academic performance. Education Policy Analysis Archives, 10(12), 1–30.
39. Widodo, S. A., Darhim, D., & Ikhwanudin, T. (2018). Improving mathematical problem solving skills through visual media. Journal of Physics: Conference Series, 948(1), 012004. https://doi.org/10.1088/1742-6596/948/1/012004