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Articles

Proof levels of graph theory students under the lens of the Van Hiele model

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Pages 1938-1956 | Received 08 Feb 2022, Published online: 12 Oct 2022

References

  • Abdullah, A. H., & Zakaria, E. (2013). The effects of Van Hiele’s phases of learning geometry on students’ degree of acquisition of Van Hiele levels. Procedia - Social and Behavioral Sciences, 102, 251–266. https://doi.org/10.1016/j.sbspro.2013.10.740
  • Aravena, M., Gutiérrez, A., & Jaime, A. (2016). Estudio de los niveles de razonamiento de Van Hiele en alumnos de centros de enseñanza vulnerables de educación media en Chile [study of Van Hiele levels of reasoning in students from vulnerable secondary schools in Chile]. Enseñanza de las Ciencias, 34(1), 107–128. https://doi.org/10.5565/rev/ensciencias.1664
  • Arnal-Bailera, A., & Oller-Marcén, A. M. (2017). Formación del profesorado y demostración matemática. Estudio exploratorio e implicaciones [teacher training and mathematical proof. Exploratory study and implications]. Bolema, 31(57), 135–157. https://doi.org/10.1590/1980-4415v31n57a07
  • Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
  • Biggs, J., & Collis, K. (1982). Evaluating the quality of learning: The SOLO taxonomy. Academic Press.
  • Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the Van Hiele levels of development on geometry. Journal for Research in Mathematics Education, 17(1), 31–48. https://doi.org/10.2307/749317
  • Cartier, L. (2008). Le graphe comme outil pour enseigner la preuve et la modélisation [The graph as a tool for teaching proof and modelling] [Doctoral Dissertation, Université Joseph Fourier]. HAL Theses. https://tel.archives-ouvertes.fr/tel-00416598v2/document.
  • Chen, Y., Senk, S. L., Thompson, D. R., & Voogt, K. (2019). Examining psychometric properties and level classification of the Van Hiele geometry test using CTT and CDM frameworks. Journal of Educational Measurement, 56(4), 733–756. https://doi.org/10.1111/jedm.12235
  • Costa, G., D’Ambrosio, C., & Martello, S. (2014). Graphsj 3: A modern didactic application for graph algorithms. Journal of Computer Science, 10(7), 1115–1119. https://doi.org/10.3844/jcssp.2014.1115.1119
  • DeBellis, V., & Rosenstein, J. (2004). Discrete mathematics in primary and secondary schools in the United States. ZDM: The International Journal on Mathematics Education, 36(2), 46–55. https://doi.org/10.1007/BF02655758
  • Demiray, E., & Işiksal, I. (2017). An investigation of pre-service middle school mathematics teachers’ ability to conduct valid proofs, methods used, and reasons for invalid arguments. International Journal of Science and Mathematics Education, 15(1), 109–130. https://doi.org/10.1007/s10763-015-9664-z
  • Durand-Guerrier, V., Hochmuth, R., Nardi, E., & Winsløw, C. (2021). Research and development in university mathematics education: Overview produced by the international network for didactic research in university mathematics. Routledge.
  • Ferrarello, D., & Mammana, M. F. (2018). Graph theory in primary, middle, and high school. In W. Hart, & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research (pp. 183–200). Springer.
  • Fraenkel, J. R., & Wallen, N. E. (1996). How to design and evaluate research. Mc. Graw-Hill.
  • Gavilán-Izquierdo, J. M., & González, A. (2016). Investigación sobre el concepto de grafo a través del modelo de Van Hiele [research on the concept of graph through the Van Hiele model]. In J. A. Macías, A. Jiménez, J. L. González, M. T. Sánchez, P. Hernández, C. Fernández, F. J. Ruiz, T. Fernández, & A. Berciano (Eds.), Investigación en educación matemática XX (p. 597). SEIEM.
  • Geschke, A., Kortenkamp, U., Lutz-Westphal, B., & Materlik, D. (2005). Visage – Visualization of algorithms in discrete mathematics. ZDM Mathematics Education, 37(5), 395–401. https://doi.org/10.1007/s11858-005-0027-z
  • González, A., Gallego-Sánchez, I., Gavilán-Izquierdo, J. M., & Puertas, M. L. (2021). Characterizing levels of reasoning in graph theory. Eurasia Journal of Mathematics, Science and Technology Education, 17(8), 1–16. https://doi.org/10.29333/ejmste/11020
  • Grenier, D., & Payan, C. (1999). Discrete mathematics in relation to learning and teaching proof and modelling. In I. Schwank (Ed.), Proceedings of CERME 1 (Vol. 1, pp. 143–155). Forschungsinstitut fuer Mathematikdidaktik.
  • Gutiérrez, A., & Jaime, A. (1995). Towards the design of a standard test for the assessment of the students reasoning in geometry. In L. Meira, & D. Carraher (Eds.), Proceedings of the 19th PME conference (Vol. 3, pp. 11–18). PME.
  • Gutiérrez, A., & Jaime, A. (1998). On the assessment of the Van Hiele levels of reasoning. Focus on Learning Problems in Mathematics, 20(2-3), 27–46.
  • Gutiérrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the Van Hiele levels. Journal for Research in Mathematics Education, 22(3), 237–251. https://doi.org/10.2307/749076
  • Gutiérrez, A., Pegg, J., & Lawrie, C. (2004). Characterization of students’ reasoning and proof abilities in 3-dimensional geometry. In M. J. Høines, & A. B. Fuglestad (Eds.), Proceedings of the 28th PME conference (Vol. 2, pp. 511–518). PME.
  • Hazzan, O., & Hadar, I. (2005). Reducing abstraction when learning graph theory. Journal of Computers in Mathematics and Science Teaching, 24(3), 255–272. http://citeseerx.ist.psu.edu/viewdoc/download?doi = 10.1.1.96.5275&rep = rep1&type = pdf.
  • Heinze, A., Anderson, I., & Reiss, K. (2004). Discrete mathematics and proof in the high school. ZDM: The International Journal on Mathematics Education, 36(2), 44–45. https://doi.org/10.1007/BF02655757
  • Hoffer, A. (1983). Van Hiele based research. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 205–227). Academic Press.
  • Huerta, M. P. (1999). Los niveles de Van Hiele y la taxonomía SOLO: Un análisis comparado, una integración necesaria [The Van Hiele levels and the SOLO taxonomy: A compared analysis, a necessary integration]. Enseñanza de las Ciencias, 17(2), 291–310. https://doi.org/10.5565/rev/ensciencias.4093
  • Jaime, A., & Gutiérrez, A. (1990). Una propuesta de fundamentación para la enseñanza de la geometría: El modelo de Van Hiele [A proposal for the foundation for the teaching of geometry: The Van Hiele model]. In S. Llinares, & V. M. Sánchez (Eds.), Teoría y práctica en educación matemática (pp. 295–384). Alfar.
  • Jaime, A., & Gutiérrez, A. (1994). A model of test design to assess the Van Hiele levels. In J. P. Ponte, & J. F. Matos (Eds.), Proceedings of the 18th PME conference (Vol. 3, pp. 41–48). PME.
  • Jaramillo, C. M. (2000). La noción de serie convergente desde la óptica de los niveles de Van Hiele [The notion of convergence through the optic of the Van Hiele levels] [Doctoral dissertation, Universitat Politèctnica de València]. Universitat Politèctnica de València Thesis Archive.
  • Lee, M. Y. (2015). The relationship between pre-service teachers’ geometric reasoning and their Van Hiele levels in a geometer’s sketchpad environment. Korean Mathematical Education Journal Series D: Mathematics Education Research, 19(4), 229–245. https://doi.org/10.7468/jksmed.2015.19.4.229
  • Leon, N., Modeste, S., & Durand-Guerrier, V. (2020). Récurrence et récursivité: Analyses de preuves de chercheurs dans une perspective didactique à l’interface mathématiques-informatique [recurrence and recursion: Analyzes of research evidence from a didactic perspective at the interface between mathematics and computer science]. In T. Hausberger, M. Bosch, & F. Chellougui (Eds.), INDRUM 2020 (pp. 368–377). University of Carthage and INDRUM.
  • Llorens, J. L., & Pérez-Carreras, P. (1997). An extension of Van Hiele’s model to the study of local approximation. International Journal of Mathematical Education in Science and Technology, 28(5), 713–726. https://doi.org/10.1080/0020739970280508
  • Manero, V., & Arnal-Bailera, A. (2021). Understanding proof practices in pre-service mathematics teachers in geometry. Mathematics Teaching Research Journal, 13(3), 99-130.
  • Mayberry, J. (1983). The van Hiele levels of geometric thought in undergraduate preservice teachers. Journal for Research in Mathematics Education, 14(1), 58-69. https://doi.org/10.2307/748797
  • Medová, J., Páleníková, K., Rybanský, L., & Naštická, Z. (2019). Undergraduate students’ solutions of modeling problems in algorithmic graph theory. Mathematics, 7(7), 572–587. https://doi.org/10.3390/math7070572
  • Navarro, M., & Pérez-Carreras, P. (2006). Constructing a concept image of convergence of sequences in the Van Hiele framework. In A. Selden, F. Hitt, & G. Harel (Eds.), Research in collegiate mathematics education VI (pp. 61–98). American Mathematical Society. https://doi.org/10.1090/cbmath/013
  • Niman, J. (1975). Graph theory in the elementary school. Educational Studies in Mathematics, 6(3), 351–373. https://doi.org/10.1007/BF01793617
  • Nisawa, Y. (2018). Applying Van Hiele’s levels to basic research on the difficulty factors behind understanding functions. International Electronic Journal of Mathematics Education, 13(2), 61–65. https://doi.org/10.12973/iejme/2696
  • O'Leary-Kelly, S. W., & Vokurka, R. J. (1998). The empirical assessment of construct validity. Journal of Operations Management, 16(4), 387–405. https://doi.org/10.1016/S0272-6963(98)00020-5
  • Pandiscio, E. A., & Knight, K. C. (2010). An investigation into the Van Hiele levels of understanding geometry of preservice mathematics teachers. Journal of Research in Education, 20(1), 45–53.
  • Perdikaris, S. C. (2004). The problem of transition across levels in the Van Hiele theory of geometric reasoning. Philosophy of Mathematics Education Journal, 18, Research paper.
  • Perdikaris, S. C. (2011). Using fuzzy sets to determine the continuity of the Van Hiele levels. Journal of Mathematical Sciences and Mathematics Education, 6(3), 81–86.
  • Pirie, S., & Kieren, T. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7–11.
  • Rosen, K. H. (2019). Discrete mathematics and its applications (8th ed.). McGraw Hill.
  • Rosenstein, J. G. (2018). The absence of discrete mathematics in primary and secondary education in the United States … and why that is counterproductive. In E. W. Hart, & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research. ICME-13 monographs (pp. 21–40). Springer. https://doi.org/10.1007/978-3-319-70308-4_2
  • Santoso, E. B. (2018). Mathematics classroom activities based on some topics in graph theory to develop critical thinking of primary and secondary school students. International Journal of Indonesian Education and Teaching, 2(2), 154–160. https://doi.org/10.24071/ijiet.2018.020207
  • Senk, S. L. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education, 20(3), 309–321. https://doi.org/10.2307/749519
  • Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10(3), 145–166. https://doi.org/10.1007/s10857-007-9034-z
  • Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of research of mathematics teaching and learning (pp. 495–511). Macmillan.
  • Torgerson, W. S. (1967). Theory and methods of scaling. Willey.
  • Uğurel, I., Moralı, H. S., Yiğit-Koyunkaya, M., & Karahan, Ö. (2015). Pre-service secondary mathematics teachers’ behaviors in the proving process. EURASIA Journal of Mathematics, Science and Technology Education, 12(2), 203–231. https://doi.org/10.12973/eurasia.2016.1523a
  • Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry (ERIC Document Reproduction Service N° ED 220 288). ERIC.
  • Van Hiele, P. M. (1957). De problematiek van het inzicht [The problem of insight] [Unpublished doctoral dissertation]. University of Utrecht.
  • Van Hiele-Geldof, D. (1957). De didaktiek van de Meetkunde in de eerste kaas van het V.H.M.O. [The didactics of geometry in the lowest class of Secondary School] [Unpublished doctoral dissertation]. University of Utrecht.
  • Van Hiele, P. M. (1986). Structure and insight. A theory of mathematics education. Academic Press.
  • Voskoglou, M. G. (2017). Managing the uncertainty in the Van Hiele levels of geometric reasoning. American Journal of Educational Research, 5(2), 109–113.
  • Wang, S., & Kinzel, M. (2014). How do they know it is a parallelogram? Analysing geometric discourse at Van Hiele level 3. Research in Mathematics Education, 16(3), 288–305. https://doi.org/10.1080/14794802.2014.933711

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