Characterizing Levels of Reasoning in Graph Theory

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.

[1]  M. Trigueros,et al.  Teaching eigenvalues and eigenvectors using models and APOS Theory , 2015 .

[2]  Evans Kofi Hokor.  Pre-service Teachers’ Probabilistic Reasoning in Constructivist Classroom , 2020 .

[3]  Xiangquan Yao,et al.  Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities , 2021 .

[4]  Maike Schindler,et al.  Designing tasks for mathematically talented students , 2015 .

[5]  Phyllis Z. Chinn,et al.  Discovery-Method Teaching in Graph Theory , 1993 .

[6]  Sybil Derrible,et al.  Applications of Graph Theory and Network Science to Transit Network Design , 2011 .

[7]  J. Mayberry,et al.  The Van Hiele Levels of Geometric Thought in Undergraduate Preservice Teachers. , 1983 .

[8]  P. M. Hiele.  Structure and Insight: A Theory of Mathematics Education , 1985 .

[9]  Silvano Martello,et al.  Graphsj 3: a Modern Didactic Application for Graph Algorithms , 2014, J. Comput. Sci..

[10]  J. Shaughnessy,et al.  CHARACTERIZING THE VAN HIELE LEVELS OF DEVELOPMENT IN GEOMETRY , 1986 .

[11]  R. Chaphalkar,et al.  Students’ Reasoning about Variability in Graphs during an Introductory Statistics Course , 2020 .

[12]  Farzad Radmehr,et al.  Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative , 2018 .

[13]  Nhon V. Do,et al.  Intelligent Educational Software in Discrete Mathematics and Graph Theory , 2018, SoMeT.

[14]  Victor N. Kasyanov.  Support tools for graphs in computer science education , 2001, Proceedings IEEE International Conference on Advanced Learning Technologies.

[15]  John Niman.  Graph theory in the elementary school , 1975 .

[16]  L. Almeida,et al.  Reasoning Abilities and Learning Math: A Möbius Strip? , 2019, International Electronic Journal of Mathematics Education.

[17]  J. Paul Gibson,et al.  Teaching graph algorithms to children of all ages , 2012, ITiCSE '12.

[18]  F. J. Langdon,et al.  The Child's Conception of Space , 1967 .

[19]  G. Stols,et al.  The relationship between teachers' instructional practices and their learners' level of geometrical thinking , 2013 .

[20]  Nicholas H. Wasserman.  Math madness: Coloring, reasoning, and celebrating , 2017 .

[21]  Graphs in Primary School: Playing with Technology , 2017 .

[22]  Richard Johnsonbaugh,et al.  Discrete mathematics (2nd ed.) , 1990 .

[23]  Eva Milková.  Puzzles as Excellent Tool Supporting Graph Problems Understanding , 2014 .

[24]  Robert Benjamin Armah,et al.  Use of the van Hiele Theory in Investigating Teaching Strategies used by College of Education Geometry Tutors , 2019, EURASIA Journal of Mathematics, Science and Technology Education.

[25]  J. Rosenstein.  The Absence of Discrete Mathematics in Primary and Secondary Education in the United States… and Why that Is Counterproductive , 2018 .

[26]  R. Marrades,et al.  Proofs produced by secondary school students learning geometry in a dynamic computer environment , 2000 .

[27]  G. Pólya,et al.  Mathematics and Plausible Reasoning: Vol. I: Induction and Analogy in Mathematics , 1979 .

[28]  Ulrich Kortenkamp,et al.  Visage—Visualization of algorithms in discrete mathematics , 2005 .

[29]  Bernard Parzysz.  “Knowing” vs “seeing”. problems of the plane representation of space geometry figures , 1988 .

[30]  Jordi Deulofeu Piquet,et al.  Perspectivas teóricas en el proceso de elaboración de conjeturas e implicaciones para la práctica : tipos y pasos , 2008 .

[31]  Asuman Oktaç,et al.  Construction of the vector space concept from the viewpoint of APOS theory , 2010 .

[32]  P. Kolman,et al.  The development of e-learning applications solving problems from graph theory , 2013 .

[33]  Ángel Gutiérrez,et al.  On the Assessment of the Van Hiele Levels of Reasoning. , 1998 .

[34]  E. Hart.  Vertex-Edge Graphs: An Essential Topic in High School Geometry. , 2008 .

[35]  Daniela Ferrarello,et al.  Graph Theory in Primary, Middle, and High School , 2018 .

[36]  Ángel Gutiérrez,et al.  Exploring the links between Van Hiele Levels and 3-dimensional geometry , 1992 .

[37]  Lowell W. Beineke,et al.  Graph Connections: Relationships between Graph Theory and Other Areas of Mathematics , 1997 .

[38]  Eko Budi Santoso,et al.  MATHEMATICS CLASSROOM ACTIVITIES BASED ON SOME TOPICS IN GRAPH THEORY TO DEVELOP CRITICAL THINKING OF PRIMARY AND SECONDARY SCHOOL STUDENTS , 2018, IJIET (International Journal of Indonesian Education and Teaching).

[39]  Eric W. Hart,et al.  Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research , 2018 .

[40]  Eva Milková,et al.  Constructing knowledge in graph theory and combinatorial optimization , 2009 .

[41]  Licia Toscano,et al.  Using graph theory for automated electric circuit solving , 2015 .

[42]  W. Martin,et al.  Discrete Mathematics Is Essential Mathematics in a 21st Century School Curriculum , 2018 .

[43]  Zuzana Naštická,et al.  Undergraduate Students’ Solutions of Modeling Problems in Algorithmic Graph Theory , 2019, Mathematics.

[44]  Irit Hadar,et al.  Reducing Abstraction When Learning Graph Theory , 2005 .

[45]  J. M. M. Escolano,et al.  Euler jugando al dominó , 2006 .

[46]  José‐Luis Llorens Fuster,et al.  An extension of van Hide's model to the study of local approximation , 1997 .