Article Sidebar

Download
PDF
Statistic
Read Counter : 491 Download : 205

Downloads

Download data is not yet available.

Main Article Content

Abstract


The need to develop consistent theoretical frameworks for the teaching and learning of discrete mathematics, specifically of graph theory, has attracted the attention of the researchers in mathematics education. Responding to this demand, the scope of the Van Hiele model has been extended to the field of graphs through a proposal of four levels of reasoning whose descriptors need to be validated according to the structure of this model. In this paper, the validity of these descriptors has been approached with a theoretical analysis that is organized by means of the so-called processes of reasoning, which are different mathematics abilities that students activate when solving graph theory problems: recognition, use and formulation of definitions, classification, and proof. The analysis gives support to the internal validity of the levels of reasoning in graph theory as the properties of the Van Hiele levels have been verified: fixed sequence, adjacency, distinction, and separation. Moreover, the external validity of the levels has been supported by providing evidence of their coherence with the levels of geometrical reasoning from which they originally emerge. The results thus point to the suitability of applying the Van Hiele model in the teaching and learning of graph theory.

Keywords

Graph Theory Levels of Reasoning Processes of Reasoning Van Hiele Model

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
González, A., Gavilán-Izquierdo, J. M., Gallego-Sánchez, I., & Puertas, M. L. (2022). A theoretical analysis of the validity of the Van Hiele levels of reasoning in graph theory. Journal on Mathematics Education, 13(3), 515–530. https://doi.org/10.22342/jme.v13i3.pp515-530
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. 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
  2. Biggs, N. L. (2003). Discrete mathematics (2nd ed.). Oxford University Press.
  3. Blanco, R., & García-Moya, M. (2021). Graph theory for primary school students with high skills in mathematics. Mathematics, 9(13), 1567. https://doi.org/10.3390/math9131567
  4. Carbonneaux, Y., Laborde, J. M., & Madani, R. M. (1996). CABRI-Graph: A tool for research and teaching in graph theory. In F. J. Brandenburg (Ed.), Graph Drawing. GD 1995. Lecture Notes in Computer Science (Vol. 1027, pp. 123-126). Springer. https://doi.org/10.1007/BFb0021796
  5. 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
  6. Educational Studies in Mathematics Editors (2002). Reflection on educational studies in mathematics: The rise of research in mathematics education. Educational Studies in Mathematics, 50(3), 251-257. https://doi.org/10.1023/A:1021259630296
  7. Ferrarello, D., & Mammana, M. F. (2018). Graph theory in primary, middle, and high school. In E. W. Hart & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research. ICME-13 Monographs (pp. 183-200). Springer. https://doi.org/10.1007/978-3-319-70308-4_12
  8. Fielder, D. C., & Dasher, B. J. (1968). Some Classroom Uses of Ambits in Teaching Graph Theory. IEEE Transactions on Education, 11(2), 104-107. https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4320356
  9. 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
  10. Godino, J. D., Aké, L. P., Gonzato, M., & Wilhelmi, M. R. (2014). Niveles de algebrización de la actividad matemática escolar. Implicaciones para la formación de maestros [Algebrization levels of school mathematics activity. Implication for primary school teacher education]. Enseñanza de las Ciencias 32(1), 199-219. https://doi.org/10.5565/rev/ensciencias.965
  11. 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), article em1990. https://doi.org/10.29333/ejmste/11020
  12. 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. https://www.uv.es/Angel.Gutierrez/archivos1/textospdf/GutJai98.pdf
  13. Hart, E. W., & Sandefur, J. (Eds.) (2018). Teaching and learning discrete mathematics worldwide: Curriculum and research. Springer. https://doi.org/10.1007/978-3-319-70308-4
  14. 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
  15. Isoda, M. (1996). The development of language about function: An application of van Hiele levels. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 105-112). University of Valencia. https://math-info.criced.tsukuba.ac.jp/isoda_lab/isoda_pdf/unk.0000.00.07.pdf
  16. 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.
  17. Khalil, E., Dai, H., Zhang, Y., Dilkina, B., & Song, L. (2017). Learning combinatorial optimization algorithms over graphs. In U. von Luxburg & I. Guyon (Eds.), Advances in Neural Information Processing Systems, (pp. 6351–6361). https://dl.acm.org/doi/pdf/10.5555/3295222.3295382
  18. Lodder, J. (2014) Networks and spanning trees: The juxtaposition of Prüfer and Borůvka, PRIMUS, 24 (8), 737-752. https://doi.org/10.1080/10511970.2014.896835
  19. Llorens-Fuster, 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
  20. Medová, J., Páleníková, K., Rybanský, Ľ., & Naštická, Z. (2019). Undergraduate students’ solutions of modeling problems in algorithmic graph theory. Mathematics, 7(7), 572. https://doi.org/10.3390/ math7070572
  21. Milková, E. (2014). Puzzles as excellent tool supporting graph problems understanding. Procedia- Social and Behavioral Sciences 131, 177-181. https://doi.org/ 10.1016/j.sbspro.2014.04.100
  22. Moala, J. G. (2021). Creating algorithms by accounting for features of the solution: The case of pursuing maximum happiness. Mathematics Education Research Journal, 33(2), 263-284. https://doi.org/10.1007/s13394-019-00288-9
  23. Navarro, M. A., & Pérez-Carreras, P. (2006). Constructing a concept image of convergence of sequences in the van Hiele framework. CBMS Issues in Mathematics Education, 13, 61-98. https://doi.org /10.1090/cbmath/013
  24. 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
  25. Ouvrier-Buffet C. (2020) Discrete mathematics teaching and learning. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 227-233). Springer. https://doi.org/10.1007/978-3-030-15789-0_51
  26. Ouvrier-Buffet, C., Meyer, A., & Modeste, S. (2018). Discrete mathematics at university level. Interfacing mathematics, computer science and arithmetic. In V. Durand Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of INDRUM 2018 – second conference of the international network for didactic research in university mathematics (pp. 255-264). University of Agder. https://hal.archives-ouvertes.fr/hal-01849537/document
  27. Roa-Fuentes, S., & Oktaç, A. (2010). Construcción de una descomposición genética: Análisis teórico del concepto transformación lineal [Constructing a genetic decomposition: Theoretical analysis of the linear transformation concept]. Revista Latinoamericana de Investigación en Matemática Educativa, 13(1), 89-112. http://www.scielo.org.mx/pdf/relime/v13n1/v13n1a5.pdf
  28. Rosen, K. H. (2019). Discrete Mathematics and its Applications (8th ed.). McGraw Hill.
  29. Sánchez-Torrubia, G., Torres-Blanc, C., & Giménez-Martínez, V. (2008). An eMathTeacher tool for active learning Fleury’s algorithm. International Journal Information Technologies and Knowledge, 2, 437-442. http://sci-gems.math.bas.bg/jspui/bitstream/10525/213/1/ijitk02-5-p07.pdf
  30. Santos-Trigo, M., & Barrera-Mora, F. (2007). Contrasting and looking into some mathematics education frameworks. The Mathematics Educator, 10(1), 81-106. https://www.uaeh.edu.mx/investigacion/icbi/LI_EconomiaFinanzasMat/Barrera_Mora/Santos-Barrera-2007.pdf
  31. Schoenfeld, A. H. (2000). Purposes and methods of research in mathematics education. Notices of the AMS, 47(6), 641-649. https://www.ams.org/notices/200006/fea-schoenfeld.pdf
  32. Silver, E., & Herbst, P. (2007). The role of theory in mathematics education scholarship. In F. Lester (Ed.), Second handbook of research in mathematics teaching and learning (pp. 39-67). Information Age.
  33. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. CDASSG Project. https://files.eric.ed.gov/fulltext/ED220288.pdf
  34. Van Hiele, P. M. (1986). Structure and insight. A theory of mathematics education. Academic Press.