Spatial Geometry Learning in Higher Education Through Flipped Classroom with Metacognitive Scaffolding: A Mixed-Methods Study
DOI:
https://doi.org/10.58524/oler.v5i2.817Keywords:
Spatial Geometry, Flipped Classroom, Metacognitive ScaffoldingAbstract
Mathematics education students continue to struggle with the complex notion of spatial geometry. Therefore, learning approaches should incorporate strong conceptual understanding. This study examined the effectiveness of flipped classroom-based geometry learning combined with metacognitive scaffolding in developing students’ spatial ability. It also identified students’ spatial knowledge needs, types of metacognitive questions posed by the teacher, and forms of scaffolding used. A mixed-methods design was employed, consisting of an exploratory qualitative phase followed by a quasi-experimental phase. The qualitative phase explored key spatial knowledge demands and scaffolding patterns through questionnaires and interviews, while the quasi-experimental phase compared three instructional conditions: flipped classroom with metacognitive scaffolding, flipped classroom without scaffolding, and conventional instruction. Results indicated that students needed four types of spatial geometry knowledge: 3D coordinate representation, geometric transformation, spatial visualization, and angle–distance relationships. Teachers’ metacognitive questions mainly emphasized awareness of understanding, while conceptual scaffolding was more dominant than metacognitive scaffolding. Quantitatively, students in the flipped classroom with metacognitive scaffolding showed greater improvements in spatial ability. These findings suggest integrating self-directed learning and metacognitive support in geometry learning that requires advanced visualization and independent thinking.
References
Borji, V., & Martínez-Planell, R. (2023). On students’ understanding of volumes of solids of revolution: An APOS analysis. Journal of Mathematical Behavior, 70, 101027. https://doi.org/10.1016/j.jmathb.2022.101027
Bragelman, J., Amador, J. M., & Superfine, A. C. (2024). The expertise of novices: A framework for prospective teachers’ noticing of children’s mathematical thinking. Journal of Mathematical Behavior, 74, 101151. https://doi.org/10.1016/j.jmathb.2024.101151
Cevikbas, M., & Kaiser, G. (2020). Flipped classroom as a reform-oriented approach to teaching mathematics. ZDM–Mathematics Education, 52(7), 1291–1305. https://doi.org/10.1007/s11858-020-01191-5
Chang, B. L., Cromley, J. G., & Tran, N. (2016). Coordinating multiple representations in a reform calculus textbook. International Journal of Science and Mathematics Education, 14(8), 1475–1497. https://doi.org/10.1007/s10763-015-9652-3
Cory, B., & Garofalo, J. (2011). Using dynamic sketches to enhance preservice secondary mathematics teachers’ understanding of limits of sequences. Journal for Research in Mathematics Education, 42(1), 65–97. https://www.jstor.org/stable/10.5951/jresematheduc.42.1.0065
Cromley, J. G., Booth, J. L., Wills, T. W., Chang, B. L., Tran, N., Madeja, M., Shipley, T. F., & Zahner, W. (2017). Relation of spatial skills to calculus proficiency: A brief report. Mathematical Thinking and Learning, 19(1), 55–68. https://doi.org/10.1080/10986065.2017.1258614
Cronhjort, M., Filipsson, L., & Weurlander, M. (2018). Improved engagement and learning in flipped-classroom calculus. Teaching Mathematics and Its Applications, 37(3), 113–121. https://doi.org/10.1093/teamat/hrx007
Cuevas-Vallejo, A., Orozco-Santiago, J., & Paz-Rodríguez, S. (2023). A learning trajectory for university students regarding the concept of vector. Journal of Mathematical Behavior, 70, 101044. https://doi.org/10.1016/j.jmathb.2023.101044
David, E. J., Roh, K. H., & Sellers, M. E. (2018). Value-thinking and location-thinking: Two ways students visualize points and think about graphs. Journal of Mathematical Behavior, 52, 1–18. https://doi.org/10.1016/j.jmathb.2018.09.004
Finesilver, C. (2022). Beyond categories: Dynamic qualitative analysis of visuospatial representation in arithmetic. Educational Studies in Mathematics, 110(2), 271–290. https://doi.org/10.1007/s10649-021-10123-3
Fredriksen, H. (2021). Exploring realistic mathematics education in a flipped classroom context at the tertiary level. International Journal of Science and Mathematics Education, 19(2), 377–396. https://doi.org/10.1007/s10763-020-10053-1
Gasteiger, H., Bruns, J., Benz, C., Brunner, E., & Sprenger, P. (2020). Mathematical pedagogical content knowledge of early childhood teachers: A standardized situation-related measurement approach. ZDM–Mathematics Education, 52(2), 193–205. https://doi.org/10.1007/s11858-019-01103-2
Haciomeroglu, E. S. (2015). The role of cognitive ability and preferred mode of processing in students’ calculus performance. Eurasia Journal of Mathematics, Science and Technology Education, 11(5), 1165–1179. https://doi.org/10.12973/eurasia.2015.1400a
Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students’ understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176. https://doi.org/10.1080/10986060903480300
Haciomeroglu, E. S., & LaVenia, M. (2017). Object-spatial imagery and verbal cognitive styles in high school students. Perceptual and Motor Skills, 124(3), 689–702. https://doi.org/10.1177/0031512517698555
Harris, D., Logan, T., & Lowrie, T. (2023). Spatial visualization and measurement of area: A case study in spatialized mathematics instruction. Journal of Mathematical Behavior, 70, 101038. https://doi.org/10.1016/j.jmathb.2023.101038
Hartmann, L. M., Schukajlow, S., Niss, M., & Jankvist, U. T. (2024). Preservice teachers’ metacognitive process variables in modeling-related problem posing. Journal of Mathematical Behavior, 76, 101195. https://doi.org/10.1016/j.jmathb.2024.101195
Hein, K., & Prediger, S. (2024). Scaffolds for seeing, using, and articulating logical structures in proofs: Design research study with high school students. Journal of Mathematical Behavior, 74, 101123. https://doi.org/10.1016/j.jmathb.2023.101123
Helgevold, N., & Moen, V. (2015). The use of flipped classrooms to stimulate students’ participation in an academic course in initial teacher education. Nordic Journal of Digital Literacy, 10(1), 29–42. https://doi.org/10.18261/ISSN1891-943X-2015-01-03
Kosko, K. W. (2020). The multiplicative meaning conveyed by visual representations. Journal of Mathematical Behavior, 60, 100800. https://doi.org/10.1016/j.jmathb.2020.100800
Krawec, J. L. (2014). Problem representation and mathematical problem solving of students of varying math ability. Journal of Learning Disabilities, 47(2), 103–115. https://doi.org/10.1177/0022219412436976
Leron, U., & Paz, T. (2014). Functions via everyday actions: Support or obstacle? Journal of Mathematical Behavior, 36, 126–134. https://doi.org/10.1016/j.jmathb.2014.09.005
Minh, T. K., & Lagrange, J. B. (2016). Connected functional working spaces: A framework for the teaching and learning of functions at upper secondary level. ZDM–Mathematics Education, 48(6), 793–807. https://doi.org/10.1007/s11858-016-0774-z
Montenegro, P., Costa, C., & Lopes, B. (2018). Transformations in the visual representation of a figural pattern. Mathematical Thinking and Learning, 20(2), 91–107. https://doi.org/10.1080/10986065.2018.1441599
Nagle, C., Martínez-Planell, R., & Moore-Russo, D. (2019). Using APOS theory as a framework for considering slope understanding. Journal of Mathematical Behavior, 53, 0–1. https://doi.org/10.1016/j.jmathb.2018.12.003
Pitta-Pantazi, D., Chimoni, M., & Christou, C. (2020). Different types of algebraic thinking: An empirical study focusing on middle school students. International Journal of Science and Mathematics Education, 18(5), 965–984. https://doi.org/10.1007/s10763-019-10003-6
Rach, S., & Ufer, S. (2020). Which prior mathematical knowledge is necessary for study success in the university study entrance phase? International Journal of Research in Undergraduate Mathematics Education, 6(3), 375–403. https://doi.org/10.1007/s40753-020-00112-x
Ramful, A., Lowrie, T., & Logan, T. (2017). Measurement of spatial ability: Construction and validation of the spatial reasoning instrument for middle school students. Journal of Psychoeducational Assessment, 35(7), 709–727. https://doi.org/10.1177/0734282916659207
Rellensmann, J., Schukajlow, S., & Leopold, C. (2017). Make a drawing: Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53–78. https://doi.org/10.1007/s10649-016-9736-1
Robinson, K. M., & LeFevre, J. A. (2012). The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework. Educational Studies in Mathematics, 79(3), 409–428. https://doi.org/10.1007/s10649-011-9330-5
Saleh, M., Charitas, R., Prahmana, I., & Isa, M. (2018). Improving the reasoning ability of elementary school students through Indonesian realistic mathematics education. Journal on Mathematics Education, 9(1), 41–54. https://doi.org/10.22342/jme.9.1.5049.41-54
Scheibling-Sève, C., Pasquinelli, E., & Sander, E. (2020). Assessing conceptual knowledge through solving arithmetic word problems. Educational Studies in Mathematics, 103(3), 293–311. https://doi.org/10.1007/s10649-020-09938-3
Schoonen, R., Van Gelderen, A., Stoel, R. D., Hulstijn, J., & De Glopper, K. (2011). Modeling the development of L1 and EFL writing proficiency of secondary school students. Language Learning, 61(1), 31–79. https://doi.org/10.1111/j.1467-9922.2010.00590.x
Sevinc, S., & Lesh, R. (2022). Preservice mathematics teachers’ conceptions of mathematically rich and contextually realistic problems. Journal of Mathematics Teacher Education, 25(6), 667–695. https://doi.org/10.1007/s10857-021-09512-5
Shaughnessy, M., DeFino, R., Pfaff, E., & Blunk, M. (2021). I think I made a mistake: How do prospective teachers elicit the thinking of a student who has made a mistake? Journal of Mathematics Teacher Education, 24(4), 335–359. https://doi.org/10.1007/s10857-020-09461-5
Soneira, C., González-Calero, J. A., & Arnau, D. (2018). An assessment of the sources of the reversal error through classic and new variables. Educational Studies in Mathematics, 99(1), 43–56. https://doi.org/10.1007/s10649-018-9828-1
Speer, N. M., & Wagner, J. F. (2009). Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education, 40(5), 530–562. https://doi.org/10.5951/jresematheduc.40.5.0530
Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. https://doi.org/10.1007/BF03217474
Tondorf, A., & Prediger, S. (2022). Connecting characterizations of equivalence of expressions: Design research in Grade 5 by bridging graphical and symbolic representations. Educational Studies in Mathematics, 111(3), 399–422. https://doi.org/10.1007/s10649-022-10158-0
Trapman, M., Van Gelderen, A., Van Schooten, E., & Hulstijn, J. (2018). Writing proficiency level and writing development of low-achieving adolescents: The roles of linguistic knowledge, fluency, and metacognitive knowledge. Reading and Writing, 31(4), 893–926. https://doi.org/10.1007/s11145-018-9818-9
Voigt, M., Fredriksen, H., & Rasmussen, C. (2020). Leveraging the design heuristics of realistic mathematics education and culturally responsive pedagogy to create a richer flipped classroom calculus curriculum. ZDM–Mathematics Education, 52(5), 1051–1062. https://doi.org/10.1007/s11858-019-01124-x
Walkington, C., Nathan, M. J., Hunnicutt, J., Washington, J., & Zhou, M. (2024). New kinds of embodied interactions that arise in augmented reality dynamic geometry software. Journal of Mathematical Behavior, 75, 101175. https://doi.org/10.1016/j.jmathb.2024.101175
Xin, Y. P. (2019). The effect of a conceptual model-based approach on additive word problem solving of elementary students struggling in mathematics. ZDM–Mathematics Education, 51(1), 139–150. https://doi.org/10.1007/s11858-018-1002-9
Yao, X. (2022). Representation transformations in transition from empirical to structural generalization. Journal of Mathematical Behavior, 66, 100964. https://doi.org/10.1016/j.jmathb.2022.100964
Yimer, A., & Ellerton, N. F. (2010). A five-phase model for mathematical problem solving: Identifying synergies in pre-service teachers’ metacognitive and cognitive actions. ZDM–International Journal on Mathematics Education, 42(2), 245–261. https://doi.org/10.1007/s11858-009-0223-3
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