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

Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tmes20 International Journal of Mathematical Education in Science and Technology ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tmes20 Proof levels of graph theory students under the lens of the Van Hiele model Antonio González, Víctor Manero, Alberto Arnal-Bailera & María Luz Puertas To cite this article: Antonio González, Víctor Manero, Alberto Arnal-Bailera & María Luz Puertas (2022): Proof levels of graph theory students under the lens of the Van Hiele model, International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2022.2113467 To link to this article: https://doi.org/10.1080/0020739X.2022.2113467 Published online: 12 Oct 2022. Submit your article to this journal Article views: 118 View related articles View Crossmark data Citing articles: 1 View citing articles INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY https://doi.org/10.1080/0020739X.2022.2113467 ARTICLE Proof levels of graph theory students under the lens of the Van Hiele model Antonio González a, Víctor Manero b, Alberto Arnal-Bailera band María Luz Puertas c aDepartamento de Didáctica de las Matemáticas, Universidad de Sevilla, Sevilla, Spain; bDepartamento de Matemáticas, Universidad de Zaragoza, Zaragoza, Spain; cDepartamento de Matemáticas, Universidad de Almería, Almería, Spain ABSTRACT This work is devoted to exploring proof abilities in Graph Theory of undergraduate students of the Degree in Computer Engineering and Technology of the University of Seville. To do this, we have designed a questionnaire consisting of five open-ended items that serve as instrument to collect data concerning their proof skills when dealing with graphs. We have thus analysed them adapting the methodology for computing the degrees of acquisition of the Van Hiele levels. OuranalysisleadstodifferentproofprofilesofGraphTheorystudents whose characteristics provide empirical support to consider proof levels in Graph Theory from the perspective of the Van Hiele model. ARTICLE HISTORY Received 8 February 2022 KEYWORDS Levels of reasoning; proof; graph theory; Van Hiele model SUBJECT CLASSIFICATIONS CODE 97K30 1. Introduction Mathematics Education at the university level has emerged as a focus of research interest in the last decade (Durand-Guerrier et al., 2021). As a sign of this, some of the most relevant international conferences on Mathematics Education have dedicated specific working groups or specific meetings to this topic in recent years. Indeed, CERME1has a Thematic Working Group (TWG)onUniversity Mathematics Education and ICME2on Mathematics Education at Tertiary Level. Likewise, new conferences publishing proceedings have recently emerged such as the congress of the INDRUM3or the RUME4Conference in the United States. Moreover, there are long-established journals that have University Mathematics Education among their priority objects of study, such as The College Mathematics Journal and PRIMUS5, published since 1970 and 1991, respectively. In addition to these publications,since2006 the PolishjournalAnnales Universitates Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia publishes studies more focused on didactic aspects. All this educational research at tertiary level has helped to deepen into the study of the skills characterising the so called Advanced Mathematical Thinking (Tall, 1992), this is, precise mathematical definitions and logical deductions of theorems based upon them. This type of thinking is directly involved in our work since we are explicitly concerned with CONTACT Víctor Manero [email protected] Departamento de Matemáticas, Universidad de Zaragoza, Zaragoza, Spain © 2022 Informa UK Limited, trading as Taylor & Francis Group 2A. GONZÁLEZ ET AL. the ability of proof, whose relevance is reflected in a number of research papers focusing on this mathematical ability (Arnal-Bailera & Oller-Marcén, 2017; Demiray & Işiksal, 2017; Stylianides et al., 2007; Uğurel et al., 2015). A relevant mathematical topic at tertiary level is Graph Theory because of its multipleapplicationsinotherareassuchasEcology,Chemistry,Transportation,Biology,Social Networks, Information and Communication, Circuits, Computer Networks, and Software Design; among others (Rosen, 2019). Furthermore, several studies in Mathematics Education(Cartier,2008;DeBellis&Rosenstein,2004;Grenier&Payan,1999;Heinzeetal.,2004; Leon et al., 2020)showtheimportanceoftherolethatgraphscanplayintheacquisition of mathematical skills, specifically proof. Besides these works, much educational research has centred its attention in different aspects of the teaching and learning of Graph Theory such as task design (Niman, 1975;Santoso,2018), resources for enhancing learning (Costa et al., 2014;Geschkeetal.,2005) or the teaching of graphs in levels other than tertiary (Rosenstein, 2018), among others. However, different authors (Hazzan & Hadar, 2005;Medováetal.,2019) highlight the lack of research on the reasoning of Graph Theory students, which requires the development of adequate theoretical frameworks. This last issue is approached by Gavilán-Izquierdo and González (2016),whofirstpointouttheVan Hiele model to be applied in the teaching and learning of Graph Theory, and subsequently by Ferrarello and Mammana (2018), who present an experimental teaching activity which considers the nature of Van Hiele levels. The Van Hiele model has mainly been applied to the field of Geometry (Burger & Shaughnessy, 1986;Chenetal.,2019;Gutiérrez&Jaime,1995;Hoffer,1983;Lee,2015; Pandiscio & Knight, 2010;Perdikaris,2004;Usiskin,1982;Wang&Kinzel,2014), specifically to undertake studies on the abilities of students in proof practices (Gutiérrez et al., 2004; Manero & Arnal-Bailera, 2021;Senk,1989).Inaddition,thismodelhasprovedtobe useful to describe the learning of other mathematical topics such as local approximation (Llorens & Pérez-Carreras, 1997), convergence of sequences (Navarro & Pérez-Carreras, 2006), convergence of series (Jaramillo, 2000) or functions (Nisawa, 2018). Thus, González et al. (2021) perform a theoretical analysis that produces a characterization of the learning of Graph Theory through four levels of reasoning under the lens of the Van Hiele model, whose validity must be tested in empirical studies as the present paper. This characterization is organised through the development of the processes of reasoning that students may activate when dealing with graphs: recognition, use and formulation of definitions, classification, and proof, thus providing descriptors of levels for each process. The purpose of this work is to provide empirical support for the validity of the levels of proof in Graph Theory through the lens of the Van Hiele model proposed by González et al. (2021).Indeed,theresultsofGonzálezetal.(2021) are based on (1) prior research on students’ mathematical thinking in areas different from Graph Theory, (2) their mathematical understanding of the concept of graph as researchers in Mathematics and Didactics of Mathematics, and (3) their experience as Graph Theory teachers. Thus, it is necessary to develop empirical studies as the present work that analyses students’ answers that allow to evaluate the validity of this theoretical characterization. This is the natural process in the elaboration of an educational model (Jaime & Gutiérrez, 1990;O’Leary-Kelly&Vokurka, 1998)which requires empirical validation after the theoretical analysis. This implies the search for evidence of the indicators provided by these authors for the levels, as well as the study of their adequacy to the particularities of the Van Hiele model. INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 3 2. Theoretical considerations 2.1. The Van Hiele model for geometry In the 1950s, Van Hiele (1957)andVanHiele-Geldof(1957)proposeamodelthatcharacterizes the development of geometric thinking through a sequence of five levels. However, we do not consider here the fifth level since it mainly concerns professional mathematicians’ reasoning which are out of the scope of this study. Thus, students at level 1 (visualization) recognize geometric figures by their appearance and as a whole; level 2 (analysis) is characterised by the students’ ability to handle the parts and mathematical properties of figures; the reasoning of level 3 (informal deduction) uses logical deductions, which enable students to interrelate properties of geometric figures; and students at level 4 (formal deduction) can produce formal proofs and deal with equivalent definitions of a concept. For a more detailed description of the levels, we refer the reader to the work of Van Hiele (1986). Theselevelshaveaseriesofcharacteristics(Jaime&Gutiérrez,1990) that differentiate them from levels proposed in other theoretical frameworks (Arnon et al., 2014;Biggs & Collis, 1982; Pirie & Kieren, 1989). Specifically, Van Hiele levels are (1) hierarchical and sequential,whichmeansthatforstudents to completely acquireacertainlevelitisnecessary that they go across the preceding ones in a specific order (e.g. students cannot completely acquire level 3 before having completely acquired level 2); (2) highly related with language, this is, each level has specific vocabulary and different ways to understand mathematical concepts; and (3) continuous, which means that the acquisition of a certain level is not instantaneous and can start before the complete acquisition of a preceding level (e.g. students can start the acquisition of level 3 before completely having acquired level 2). Gutiérrez and Jaime (1998) propose a way to regard the geometrical reasoning as decomposed into different processes of reasoning: recognition (i.e. identification of types of figures, as well as their components and properties), use of definitions (i.e. handling of geometrical concepts), formulation of definitions (i.e. elaboration of descriptions or characterizations of geometrical notions), classification (i.e. placement of geometrical objects into different families), and proof (i.e. explanation in some convincing way that a statement is true). Thus, these authors characterize the Van Hiele levels according to the degree of development of each of these processes. We next describe the development of the proof process,whichisthefocusofthispaper. Proof at level 1 is not considered by Gutiérrez and Jaime (1998).Atlevel2,proofsare characterized byverificationsinparticularcases. Students atlevel3 canverify statementsby using informal explanations based on mathematical properties. Moreover, they are able to understand formal proofs and even reproduce a few logical steps but they cannot produce themselves formal proofs, which characterizes level 4 students. 2.2. The Van Hiele model for graphs González et al. (2021) provide a theoretical characterization of students’ reasoning in Graph Theory based on Van Hiele levels. To present this characterisation, we first provide some fundamentals of Graph Theory for the sake of completeness. Indeed, a graph G consists of a pair (V,E)whereVis any set, which is called the vertex set,andEis a set of pair of non-ordered pairs of elements from V, which is called edge set.Thepictorial representation, 4A. GONZÁLEZ ET AL. which is the most common way to represent graphs, consists of drawing vertices as points in the plane which are joined by (non-necessarily straight) segments whenever the corresponding pairs of vertices form edges. Delving into the parts and properties of graphs, we say that two vertices are adjacent whenever they form an edge, and the degree of a vertex is the number of vertices adjacent to it. Finally, a graph is said to be Eulerian when it admits a sequence of adjacent vertices containing each of its edges exactly once and starting and finishing at the same vertex. Eulerian graphs are characterized by having all their vertices of even degree. Note that properties of graphs can be divided into local (i.e. associated with parts of the graph) and global (i.e. associated with the whole graph). Thus, the degree is local, while the Eulerian character is global. (We refer the reader to the book of Rosen (2019) for more information on Graph Theory.) WecannowdescribethemainindicatorsoftheVanHielelevelsforGraphTheory (González et al., 2021).ItiseasytoseethattheyhavethesamenatureastheVan Hiele levels for Geometry. Indeed, students at level 1 (visualization) have a visual type of recognition that limits them when identifying graphs and global properties; the reasoning at level 2 (analysis) is mainly supported by students’ ability to identify global and local mathematical properties of graphs, which enables them to distinguish graphs independently from their representations; students at level 3 (informal deduction) can interrelate graph properties and provide logical arguments; and students at level 4 (formal deduction) perceive graphs as formal mathematical objects, and so they can work with equivalent definitions of the same concept and construct formal proofs of mathematical results. More details on the descriptors of each level can be found in the work of González et al. (2021). Furthermore, these authors organize the descriptors of each level according to the processes proposed by Gutiérrez and Jaime (1998)adaptedtothefieldofgraphs.Inparticular, the evolution of the process of proof in Graph Theory and in Geometry are analogous from levels 2–4. However, the proof of level 1, which is not considered by Gutiérrez and Jaime (1998), is taken into account by González et al. (2021) due to the peculiarities of graphs. Indeed,proofsatlevel2aregivenbyverificationsinspecificexamples;studentsatlevel3 are able to produce informal proofs to justify the truth of a statement, and they can understand the steps of a formal proof but they cannot write it themselves; and level 4 students can elaborate formal proofs, thus being able to perform classic techniques in Graph Theory such as induction, proof by contradiction, or proof by contraposition. Level 1 students just provide visual arguments to justify the truth of a statement or verify it in specific examples, as well as level 2 students do, but being very limited by the representations that they know for a graph. 3. Method 3.1. Data collection instrument InordertoevaluatetheproofprocessinGraphTheorywehavedesignedafiveitemsquestionnaire focused on detecting evidence of the indicators of the proof levels proposed by González et al. (2021). This instrument has been developed following the same ideas as the proof tasks proposed by Gutiérrez and Jaime (1998)toevaluatetheproofprocess,which are open ended tasks that bring to light the reasoning of the students more clearly than INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 5 multiple choice questions (Jaime & Gutiérrez, 1994).Formoreexamplesofthistypeof tasks, we refer the reader to the works of Aravena et al. (2016)andBurgerandShaughnessy (1986). The first version of this questionnaire was validated by experts in Mathematics Education and experts in Graph Theory who are familiar with the Van Hiele model, none of them being involved in this work. Subsequently, this questionnaire was first administered to students enrolled in the course Logic and Discrete Mathematics of the Degree in Software Engineering of the Polytechnical University of Madrid, which allowed us to create an improved version of the questionnaire employed in this work. Moreover, we studied the scalability and internal coherence of the questionnaire, which have been measured through the Guttman and Cronbach’s Alpha coefficients, respectively. The Guttman coefficient applied to our context (Mayberry, 1983)isgivenbyG=1−e l·N,whereeis the total number of errors, listhenumberoflevelsandNis the number of participants in the study. This coefficient provides an idea of the extent to which an observed set of responses patterns agrees with that expected from a perfect scale (i.e. without errors), which reveals in our context to what extent the hierarchical nature of the Van Hiele levels appears in our results. We have obtained for the Guttman coefficient a value of 0.931, which is an indicator of the reliability of the questionnaire as it is greater than 0.9 (Torgerson, 1967). The Cronbach’s Alpha coefficient is given by α=k k−1·1−k i=1S2 i S2 t,wherekis the number of items, Si2is the variance of item i, and St2is the variance of the total observed values. Thus, this coefficient shows to which extent the answers given by each student to the different tasks of the test manifest similar levels of reasoning. We have obtained a value of 0.874 for this parameter, which is another indicator of the reliability of the test since it is greater than 0.7 (Fraenkel & Wallen, 1996). The questionnaire, that we next describe, consists of five items to evaluate students’ proof skills in Graph Theory, which were answered independently. The tasks included in each item, all containing concepts familiar to the students, were selected in order to detect the different types of proof that can appear when dealing with graphs: visual (level 1), empirical (level 2), informal (level 3) and formal (level 4). Item 1. Remember that the degree of a vertex is the number of vertices adjacent to it (i.e. they are connected by an edge). What is the sum of the degrees of all the vertices of a graph? Give a proof of your answer. The first item, which demands to prove that the sum of the degrees of the vertices of a graphistwicethenumberofitsedges,assesseslevelsfrom2to4.Indeed,theitemdoesnot serve to evaluate level 1 because the statement to be proved necessarily induces to explore a local property such as the degree of each vertex, which is not manageable at level 1. Thus, expected level 2 answers contain a specific value for the required sum and verifications in concrete examples. In contrast, level 3 answers, which may also contain verifications in specific examples, provide general mathematical reasoning but they do not explicitly show thedoubleedgecountingthatappearsinlevel4answers.Moreover,thislasttypeofanswers might be given using mathematical induction. Item 2. Fill in the gaps of the following sentences: For a complete graph with 4 vertices, the number of its edges is ... and the sum of the degrees ofallitsverticesis... 6A. GONZÁLEZ ET AL. Foracyclewith5vertices,thenumberofitsedgesis...andthesumofthedegreesofallits verticesis... For a graph on m edges, the sum of the degrees of all vertices is ... Justify your answer. The second item provides a scaffolding for the proof of the preceding statement, thus demanding students to first count the sum of degrees and the number of edges of two specific graphs. Hence, this scaffolding helps students to find the general relation for any graph and deduce the corresponding proof, which precisely requires a double counting of itsedges.Notethatstudentscouldnotgobacktopreviousitemsduringthereplyingof the questionnaire. Thus, this item does not assess level 1 for the same reasons provided for item 1 since it requires a proof of the same result. Level 2 expected answers, just like in item 1, are based on verifications in particular graphs. Also, any answers containing reasoning beyond verifications in examples reflect level 3, even if they explicitly contain the double edge counting idea. Indeed, this item does not assess level 4 because students have enough information to give the required proof just by linking a few logical steps, which is feasible by level 3 students. Item3.RememberthatthecompletebipartitegraphK n,m iscomposedoftwosetsofvertices suchthatnoedgehasbothverticesinthesameset,andhaseverypossibleedgeconnecting vertices of both sets. Figure 1shows the graph K3,5. Try to prove that no complete bipartite graph with an odd number of vertices is Eulerian. Thethirditem,whichservestoassessalllevels,asksforprovingthatnocompletebipartite graph with an odd number of vertices is Eulerian. Answers with evidence of indicators oflevel1couldcontainvisualargumentssuchastheimpossibilitytodrawthegraphwith- outliftingthepencilfromthepaper,forinstanceintheexampleofgraphprovidedin the questionnaire. Level 2 answers are limited to verifications in specific complete bipartite graphs, whilst level 3 answers contain reasoning made on mathematical properties and relations between these properties. This type of answers may contain gaps and provide conclusions obtained via a non-rigorous process, for instance, they could avoid a justification of the fact that a complete bipartite graph with an odd number of vertices must contain vertices of odd degree. Finally, level 4 answers are characterised by its degree of formality, showing proofs with a series of justified steps that logically lead us from the hypothesis to the thesis of the statement. Thus, these answers necessarily include the following ideas: (1) the degree of a vertex of a set of the complete bipartite graph equals the cardinality of the other set, and (2) in a complete bipartite graph with an odd number of vertices, exactly one of the two sets has odd cardinality. Figure 1. The complete bipartite graph K3,5. INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 7 Item 4. Claim 1. If a graph has a vertex of odd degree, then that graph is not Eulerian. Claim 2. If the complete bipartite graph Kn,m has an odd number of vertices, then either n is oddandmisevenorviceversa. Considering the preceding claims, prove that no complete bipartite graph with an odd number of vertices is Eulerian. The fourth item provides some hints to proof the statement of the preceding item. Thus, if this scaffolding does not work as a deterrent for students that use visual arguments, then they are assigned level 1. Level 2 expected answers, just like in item 3, lie in specific verifications in examples, in contrast with level 3 answers, which provide mathematical reasoning beyond examples. This item does not allow to assess level 4 of proof since students have enough information to construct the required proof just by properly relating the claims provided in the statement of the item. Item 5. Here you have a proof of the fact that no complete bipartite graph with an odd number of vertices is Eulerian. Read it and try to understand it: •The vertex set of the complete bipartite graph is partitioned into two sets, one having n vertices and the other having m vertices. Each vertex of a set is adjacent to all the vertices of the other set, and there are no adjacencies among vertices of the same set. (See Figure 2). •Thus, the vertices ofone set have degree m and the vertices of the other set have degree n. •As the total number of vertices is n +m, which is an odd number, then either m is odd, and n is even or vice versa. •Therefore, the graph has vertices of odd degree and so it is not Eulerian. You have just seen a proof of the fact that no complete bipartite graph with an odd number of vertices is Eulerian. Give a similar proof for the following statement: AcompletebipartitegraphK n,m is Eulerian if and only if n and m are both even. The last item exhibits a formal proof of the result of item 3, and then it asks students to understand it and provide a similar proof for a different statement. Thus, this item does notassessthefeaturesoflevel1sinceitisrequiredtobeatlevel2toatleasthandlethe mathematical properties appearing in the given proof. Again, level 2 answers should be made of specific verifications. Level 3 expected answers include general reasonings to prove the statement but only concerning one of the implications. Although the task demands the Figure 2. The complete bipartite graph Kn,m. 8A. GONZÁLEZ ET AL. Table 1. Levels assessed by each item. Item Level 1 234 223 3 1234 4 123 5 234 replication of a proof, which could be done in principle by level 3 students, it is remarkable that the proof provided to students requires arguments for only one implication, while the proof demanded to students requires to consider two implications. Thus, it is necessary to possess sufficient formality to be aware of the necessity of proving both implications, which is characteristic of level 4 students. We summarize the information provided in this section in Table 1for the sake of readability. 3.2. Sample The questionnaire was administered to 59 students (labelled from S1 to S59) enrolled in the course Discrete Mathematics, which corresponds to the second year of the Degree in Computer Engineering and Technology of the University of Seville. We point out that we have discarded the productions of 5 students (namely S1, S3, S7, S27, and S55) because they provided non-assessable answers to at least three items of the questionnaire (i.e. more than a half of the items). 3.3. Assessment of the proof process in graph theory Thedataobtainedinthisstudyhavebeenanalysedbymeansofthemethodintroduced by Gutiérrez et al. (1991) to compute the degrees of acquisition of the Van Hiele levels, which has been applied in several studies in the field of Geometry (Abdullah & Zakaria, 2013; Aravena et al., 2016;Gutiérrez&Jaime,1995;Gutiérrez&Jaime,1998;Huerta,1999; Manero & Arnal-Bailera, 2021). This method, which provides a description of the development of the students’ skills associated with each of the Van Hiele levels, is adequate in our study since the descriptors of the Van Hiele levels of graphs (González et al., 2021) have been mainly obtained by analogy with the descriptors of the Van Hiele levels for the geometrical case. We next summarize the steps of the procedure for applying such method. Giventheanswersofastudenttothefiveitems,thismethodfirstassignsalevel to each answer, according to the criteria developed in the subsection devoted to the data collection instrument, and also a type in accordance with the indicators described in the second column of Table 2. Subsequently, each answer is marked with a percentage of acquisition for each of the levelsevaluatedbythecorrespondingitem.Indeed,thepercentageofacquisitionofthe level assigned to the answer corresponds with those given in the third column of Table 2; higher levels evaluated by the item are assigned 0%; and lower levels evaluated by the item INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 15 levels.(Obviously,thisdoesnothavesenseforstudentsofprofileP1sincetheyhavereached the maximum degree of reasoning in the proof process). Indeed, we have only found level 2 students who have not started the acquisition of higher levels, which are precisely those of P4profile.Thispointsoutthefactthatthetransitionbetweenlevels2and3couldoccur less gradually than in other levels, at least for the proof process. This is not surprising since achieving level 3 implies to understand what a proof really is, as remarked by Senk (1989) for the geometrical context, who points out that this implies students to move from verifications in concrete examples to proofs based on general mathematical arguments. We have detected certain anomalies in the results whose possible causes could be explored in future works. Indeed, we have obtained a remarkable number of students with more acquisition of a certain level than the previous one, mostly students with more acquisition of level 2 than 1, in contrast to the hierarchical nature expected for the levels of proof under study. This could be due to two main reasons, both of methodological nature. The first reason could be related to our questionnaire because it only contains two items evaluating level 1, which might generate unreliable values. This could be fixed by increasing the number of tasks in the corresponding questionnaire that serve to assess level 1. The second, according to the methodology of Gutiérrez et al. (1991), refers to non-assessable responses (type 1), which are assigned no acquisition for all levels evaluated by the corresponding task. Thus, these responses are weighted with 0 at level 1 (whenever it is assessed bythetask)evenforstudentswhocouldhaveattainedthislevel.Wecouldexplorepossible modifications of this methodology in future studies in order to solve this issue. In addition, we could increase the number of respondents considering several University degrees from different countries. Our work has shown empirical evidence of the suitability of employing the Van Hiele model to analyse students’ development in the proof process for Graph Theory. Thus, we contribute to the literature on the reasoning in Graph Theory, which is scarce as noticed byauthorslikeHazzanandHadar(2005)andMedováetal.(2019). In future works, we could search for empirical support for the rest of the processes undertaken in this theoretical model: recognition, use and formulation of definitions, and classification. Regarding teaching issues, even though our results already serve to identify difficulties of Graph Theory students, it would be also interesting to explore whether the instructional aspects of the Van Hiele model, such as the sequencing according to its phases, produce a better acquisition of the levels in Graph Theory. Notes 1. Congress of the European Society for Research in Mathematics Education. 2. International Congress on Mathematical Education. 3. International Network for Didactic Research in University Mathematics. 4. Research in Undergraduate Mathematics Education. 5. Problems, Resources, and Issues in Mathematics Undergraduate Studies. Acknowledgements The first author would like to thank to Universidad de Zaragoza for its support during his stay in Zaragoza (January-February 2022) and the second author would like to thank to Universidad de Sevilla for its support during his stay in Sevilla (November-December 2021). 16 A. GONZÁLEZ ET AL. Disclosure statement No potential conflict of interest was reported by the authors. Funding The first author is partially supported by Universidad de Sevilla (Spain) [‘VI Plan Propio de Investigación y Transferencia’], the second author is partially supported by Spanish Agencia Estatal de Investigación (Spain) [grant number PID2020-115652GB-I00], the third author is partially supported by Spanish MICINN (Spanish Ministry of Science and Innovation) [grant number PID2019- 104964GB-I00]. The first author belongs to the research group FQM-226 (Junta de Andalucía, Spain), the second and third authors belong to the research group S60_20R (Gobierno de Aragón, Spain), and the fourth author belongs to the research group TIC-146 (Junta de Andalucía, Spain). ORCID Antonio González http://orcid.org/0000-0002-2137-5835 Víctor Manero http://orcid.org/0000-0001-9884-7995 Alberto Arnal-Bailera http://orcid.org/0000-0002-0516-0463 María Luz Puertas http://orcid.org/0000-0002-9093-5461 References Abdullah, A. H., & Zakaria, E. (2013).TheeffectsofVanHiele’sphasesoflearninggeometryon 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).EstudiodelosnivelesderazonamientodeVanHiele 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/10.1590/1980-44 15v31n57a07 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.AcademicPress. 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).Legraphecommeoutilpourenseignerlapreuveetlamodélisation[Thegraphas 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).Discretemathematicsinprimaryandsecondaryschoolsin the United States. ZDM: The International Journal on Mathematics Education,36(2), 46–55. https://doi.org/10.1007/BF02655758 INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 17 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).Towardsthedesignofastandardtestfortheassessmentofthe 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).Characterizationofstudents’ reasoningandproofabilities 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.), Teoríayprácticaeneducación 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. 18 A. GONZÁLEZ ET AL. 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).Theabsenceofdiscretemathematicsinprimaryandsecondaryeducationin 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 INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 19 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. VanHiele,P.M.(1986). Structure and insight. A theory of mathematics education.AcademicPress. 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