Characterizing Levels of Reasoning in Graph Theory

Antonio González; Inés Gallego-Sánchez; José María Gavilán-Izquierdo; María Luz Puertas

doi:10.29333/ejmste/11020

Full Text (PDF)

Antonio González ¹ ^* , Inés Gallego-Sánchez ¹ , José María Gavilán-Izquierdo ¹ , María Luz Puertas ²

More Detail

¹ Department of Didactics of Mathematics, Universidad de Sevilla, SPAIN² Department of Mathematics, Universidad de Almería, SPAIN^* Corresponding Author

Abstract

This work provides a characterization of the learning of graph theory through the lens of the van Hiele model. For this purpose, we perform a theoretical analysis structured through the processes of reasoning that students activate when solving graph theory problems: recognition, use and formulation of definitions, classification, and proof. We thus obtain four levels of reasoning: an initial level of visual character in which students perceive graphs as a whole; a second level, analytical in nature in which students distinguish parts and properties of graphs; a pre-formal level in which students can interrelate properties; and a formal level in which graphs are handled as abstract mathematical objects. Our results, which are supported by a review of the literature on the teaching and learning of graph theory, might be very helpful to design efficient data collection instruments for empirical studies aiming to analyze students’ thinking in this field of mathematics.

Keywords

levels of reasoning
van Hiele model
processes of reasoning
graph theory

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2021, Volume 17, Issue 8, Article No: em1990

https://doi.org/10.29333/ejmste/11020

Publication date: 24 Jun 2021

Article Views: 2399

Article Downloads: 1818

Open Access Disclosures References How to cite this article

Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Aires, A. P., Campos, H., & Poças, R. (2015). Raciocínio geométrico versus definição de conceitos: a definição de quadrado com alunos de 6º ano de escolaridade [Geometric reasoning vs definition of concepts: The definition of square with 6th grade students]. Revista Latinoamericana de Investigación en Matemática Educativa, 18(2), 151-176. https://doi.org/10.12802/relime.13.1821
Alsina, C. (2011). Mapas del metro y redes neuronales. La teoría de grafos [Subway maps and neural networks. Graph theory]. RBA.
Armah, R. B., & Kissi, P. S. (2019). Use of the van Hiele theory in investigating teaching strategies used by college of education geometry tutors. Eurasia Journal of Mathematics, Science and Technology Education, 15(4), em1694. https://doi.org/10.29333/ejmste/103562
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
Bajo, J. M., Gavilán-Izquierdo, J. M., & Sánchez-Matamoros, G. (2019). Caracterización del esquema de sucesión numérica en estudiantes de Educación Secundaria Obligatoria [Characterization of the numeric sequence schema among Compulsory Secondary Education students]. Enseñanza de las ciencias, 37(3), 149-167. https://doi.org/10.5565/rev/ensciencias.2673
Beineke, L. W., & Wilson, R. J. (1997). Graph connections: Relationships between graph theory and other areas of mathematics. Oxford University Press.
Biggs, N. L. (2003). Discrete mathematics (2nd ed.). Oxford University Press.
Bleeker, C., Stols, G., & Van Putten, S. (2013). The relationship between teachers’ instructional practices and their learners’ level of geometrical thinking. Perspectives in Education, 31(3), 66-78.
Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE theory to improve students’ graphical understanding of derivative. Eurasia Journal of Mathematics, Science and Technology Education, 14(7), 2947-2967. https://doi.org/10.29333/ejmste/91451
Brito, L. P., Almeida, L. S., & Osório, A. J. M. (2020). Reasoning abilities and learning math: A Möbius strip? International Electronic Journal of Mathematics Education, 15(2), em0565. https://doi.org/10.29333/iejme/6259
Bruckler, F. M., & Stilinović, V. (2008). Graph theory as a method of improving chemistry and mathematics curricula. In B. Sriraman, C. Michelsen, A. Beckmann, & V. Freiman (Eds.), Proceedings of the 2nd International Symposium on Mathematics and its Connections to the Arts and Sciences (pp. 117-126). Print & Sign.
Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31-48. https://doi.org/10.2307/749317
Cañadas, M. C., Deulofeu, J., Figueiras, L., Reid, D. A., & Yevdokimov, O. (2008). Perspectivas teóricas en el proceso de elaboración de conjeturas e implicaciones para la práctica: Tipos y pasos [Theoretical perspectives in the process of producing conjectures and implications for practice: Types and steps]. Enseñanza de las Ciencias, 26(3), 431-444. https://doi.org/10.5565/rev/ensciencias.3753
Cartier, L. (2008). Le graphe comme outil pour enseigner la preuve et la modélisation [Graphs as tools for teaching proof and modeling]. [Doctoral thesis, Institut Fourier, Grenoble, France]. HAL archives-ouvertes.fr. https://tel.archives-ouvertes.fr/tel-00416598v1
Chaphalkar, R., & Wu, K. (2020). Students’ reasoning about variability in graphs during an introductory statistics course. International Electronic Journal of Mathematics Education, 15(2), em0580. https://doi.org/10.29333/iejme/7602
Chinn, P.Z. (1993). Discovery-method teaching in graph theory. Annals of Discrete Mathematics, 55, 375-384. https://doi.org/10.1016/S0167-5060(08)70402-3
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
Derrible, S., & Kennedy, C. (2011). Applications of graph theory and network science to transit network design. Transport reviews, 31(4), 495-519. https://doi.org/10.1080/01441647.2010.543709
Díaz-Levicoy, D. (2010). Determinación de niveles Van Hiele en alumnos de primer año medio sobre la transformación isométrica de simetría [Determination of Van Hiele levels in first year middle school students on the isometric transformation of symmetry]. Revista investigaciones en educación, 10(2), 65-87.
Do, N. V., Nguyen, H. D., & Mai, T. T. (2018). Intelligent educational software in discrete mathematics and graph theory. Frontiers in Artificial Intelligence and Applications, 303, 925-938.
Ferrarello, D. (2017). Graphs in primary school: Playing with technology. In G. Aldon, F. Aldon, L. Bazzini, & U. Gellert (Eds.), Mathematics and Technology. Advances in Mathematics Education (pp. 143-169). Springer. https://doi.org/10.1007/978-3-319-51380-5_8
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
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
Gibson, J. P. (2012). Teaching graph algorithms to children of all ages. Proceedings of the 17th Annual ACM Conference on Innovation and Technology in Computer Science Education (pp. 34-39). ACM. https://doi.org/10.1145/2325296.2325308
González, A., & Gavilán-Izquierdo, J. M. (2017). Analizando el reconocimiento de grafos a través del modelo de Van Hiele [Analyzing the recognition of graphs through the Van Hiele model]. In Federación Española de Sociedades de Profesores de Matemáticas (Ed.), VIII Congreso Iberoamericano de Educación Matemática. Libro de actas (pp. 286-293). FESPM.
González, A., Muñoz-Escolano, J. M., & Oller-Marcén, A. M. (2019). Presencia de la teoría de grafos en la enseñanza de grado en España [Presence of graph theory in undergraduate teaching in Spain]. In J. M. Marbán, M. Arce, A. Maroto, J. M. Muñoz-Escolano, & Á. Alsina (Eds.), Investigación en Educación Matemática XXIII (p. 622). SEIEM.
Gutiérrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry. Structural Topology, 18, 31-48.
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.
Hart, E. (2008). Vertex-edge graphs: An essential topic in high school geometry. The Mathematics Teacher, 102(3), 178-185. https://doi.org/10.5951/MT.102.3.0178
Hart, E. W., & Martin, W. G. (2018). Discrete mathematics is essential mathematics in a 21st century school curriculum. In E. W. Hart & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research. ICME-13 Monographs (pp. 3-19). Springer. https://doi.org/10.1007/978-3-319-70308-4_1
Hart, E. W., Kenney, M. J., DeBellis, V. A., & Rosenstein, J. G. (2008). Navigating through discrete mathematics in grades 6 to 12. National Council of Teachers of Mathematics.
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
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
Hershkowitz, R. (1989). Visualization in geometry—Two sides of the coin. Focus on Learning Problems in Mathematics, 11(1-2), 61-76.
Hoffer, A. (1988). Geometry and visual thinking. In T. R. Post (Ed.), Teaching mathematics in grades K-8: Research based methods (pp. 233-261). Allyn & Bacon.
Hokor, E. K. (2020). Pre-service teachers’ probabilistic reasoning in constructivist classroom. Pedagogical Research, 5(2), em0053. https://doi.org/10.29333/pr/7838
Jaime, A. (1993). Aportaciones a la interpretación y aplicación del modelo de Van Hiele: la enseñanza de las isometrías del plano. La evaluación del nivel de razonamiento [Contributions to the interpretation and application of the Van Hiele model: The teaching of plane isometries. The evaluation of the level of reasoning] [Doctoral dissertation, Universitat de València].
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.
Kasyanov, V. N. (2001). Support tools for graphs in computer science education. In T. Okamoto, R. Hartley, T. Kinshuk, & J. P. Klus (Eds.), Proceedings of the IEEE International Conference on Advanced Learning Technologies (pp. 307-308). IEEE Computer Society. https://doi.org/10.1109/ICALT.2001.943930
Kolman, P., Zach, P., & Holoubek, J. (2013). The development of e-learning applications solving problems from graph theory. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 61(7), 2311-2316. https://doi.org/10.11118/actaun201361072311
Laborde, C., & Capponi, B. (1994). Cabri-Géomètre constituant d’un milieu pour l’apprentissage de la notion de figure géométrique [Cabri-Géomètre constituting a medium for learning the concept of geometric figure]. Recherches en Didactique des Mathématiques, 14(1/2), 165-210.
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
Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1/2), 87-125. https://doi.org/10.1023/A:1012785106627
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, Ľ., & Naštická, Z. (2019). Undergraduate students’ solutions of modeling problems in algorithmic graph theory. Mathematics, 7(7), 572. https://doi.org/10.3390/math7070572
Menéndez, A. (1998). Una breve introducción a la teoría de grafos [A brief introduction to graph theory]. Revista SUMA, 28, 11-26. https://redined.mecd.gob.es/xmlui/handle/11162/13526
Milková, E. (2009). Constructing knowledge in graph theory and combinatorial optimization. WSEAS Transactions on Mathematics, 8(8), 424-434.
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
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
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
Oller-Marcén, A., & Muñoz-Escolano, J. M. (2006). Euler jugando al dominó [Euler playing domino]. Revista SUMA, 53, 39-49.
Parraguez, M., & Oktaç, A. (2010). Construction of the vector space concept from the viewpoint of APOS theory. Linear Algebra and its Applications, 432(8), 2112-2124. https://doi.org/10.1016/j.laa.2009.06.034
Parzysz, B. (1988). “Knowing” vs “seeing”. Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79-92. https://doi.org/10.1007/BF00428386
Pegg, J., Gutiérrez, A., & Huerta, P. (1998). Assessing reasoning abilities in geometry. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 275-295). Kluwer.
Piaget, J. (1972). Psicología y epistemología [Psychology and epistemology]. Emecé Editores.
Piaget, J., & Inhelder, B. (1956). The child’s conception of space. The Humanities Press.
Pólya, G. (1954). Mathematics and plausible reasoning, Volume 1: Induction and analogy in mathematics. Princeton University Press. https://doi.org/10.1515/9780691218304
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
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
Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. The Journal of Mathematical Behavior, 39, 100-120. https://doi.org/10.1016/j.jmathb.2015.06.005
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
Schindler, M., & Joklitschke, J. (2015). Designing tasks for mathematically talented students. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (CERME 9) (pp. 1066-1072). Charles University. https://hal.archives-ouvertes.fr/hal-01287313/document
Smithers, D. B. (2005). Graph theory for the secondary school classroom [Master Thesis, East Tennessee State University]. dc.etsu.edu/etd/1015
Tabchi, T. (2018). University teachers-researchers’ practices: The case of teaching discrete mathematics. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of the second conference of the International Network for Didactic Research in University Mathematics (INDRUM 2018) (pp. 432-441). University of Agder and INDRUM. https://hal.archives-ouvertes.fr/hal-01849936/document
Tabchi, T., Sabra, H., & Ouvrier-Buffet, C. (2019). Resources for teaching graph theory for engineers — issue of connectivity. In S. Rezat, L. Fan, M. Hattermann, J. Schumacher, & H. Wuschke, (Eds.), Proceedings of the Third International Conference on Mathematics Textbook Research and Development (pp. 323–328). Universitätsbibliothek Paderborn.
Toscano, L., Stella, S., & Milotti, E. (2015). Using graph theory for automated electric circuit solving. European Journal of Physics, 36(3), 1-12. https://doi.org/10.1088/0143-0807/36/3/035015
Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. CDASSG Project.
Van Hiele, P. M. (1986). Structure and insight. A theory of mathematics education. Academic Press.
Vergel, C., Molina, B., & Echeverry, A. (2005). Grafos en la educación básica [Graphs in basic education]. Revista EMA, 10(2-3), 440-451. http://funes.uniandes.edu.co/1507/
Vidermanová, K., & Melušová, J. (2011). Teaching graph theory with Cinderella and Visage: An undergraduate case. In D. Vallo, O. Šedivý, & K. Vidermanová (Eds.), New Trends in Mathematics Education: DGS in Education (pp. 79-85). Constantine the Philosopher University.
Wasserman, N. H. (2017). Math madness: Coloring, reasoning, and celebrating. Teaching Children Mathematics, 23(8), 468-475. https://doi.org/10.5951/teacchilmath.23.8.0468
Yao, X., & Elia, J. (2021). Connections between empirical and structural reasoning in technology-aided generalization activities. International Electronic Journal of Mathematics Education, 16(2), em0628. https://doi.org/10.29333/iejme/9770

How to cite this article

APA

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), em1990. https://doi.org/10.29333/ejmste/11020

Vancouver

González A, Gallego-Sánchez I, Gavilán-Izquierdo JM, Puertas ML. Characterizing Levels of Reasoning in Graph Theory. EURASIA J Math Sci Tech Ed. 2021;17(8):em1990. https://doi.org/10.29333/ejmste/11020

AMA

González A, Gallego-Sánchez I, Gavilán-Izquierdo JM, Puertas ML. Characterizing Levels of Reasoning in Graph Theory. EURASIA J Math Sci Tech Ed. 2021;17(8), em1990. https://doi.org/10.29333/ejmste/11020

Chicago

González, Antonio, Inés Gallego-Sánchez, José María Gavilán-Izquierdo, and María Luz Puertas. "Characterizing Levels of Reasoning in Graph Theory". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 8 (2021): em1990. https://doi.org/10.29333/ejmste/11020

Harvard

González, A., Gallego-Sánchez, I., Gavilán-Izquierdo, J. M., and Puertas, M. L. (2021). Characterizing Levels of Reasoning in Graph Theory. Eurasia Journal of Mathematics, Science and Technology Education, 17(8), em1990. https://doi.org/10.29333/ejmste/11020

MLA

González, Antonio et al. "Characterizing Levels of Reasoning in Graph Theory". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 8, 2021, em1990. https://doi.org/10.29333/ejmste/11020