Artículos (Didáctica de la Matemáticas)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/44316

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A Case Study of the Practices of Conjecturing and Proving of Research Mathematicians
(Taylor & Francis, 2019) Fernández León, Aurora; Gavilán Izquierdo, José María; Toscano Barragán, Rocío; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigación en Educación Matemática
In this paper we investigate how a research mathematician conjectures and proves when conducting her research. To be precise, the aim of this study is to achieve a comprehensive understanding of the way this research mathematician develops these mathematical practices and thus gain insight to improve the teaching and learning of these two practices in an educational context. For this purpose, we consider Rasmussen, Zandieh, King and Teppo’s [1] theoretical constructs of horizontal and vertical mathematising. In particular, we have adopted a case study methodological approach with a single research mathematician. Analysis of the data lead to the identification of eleven categories of activities. Each category is linked either to the practice of conjecturing or proving and also to the horizontal or vertical dimension of such practice. Finally, we compare and contrast our results with other related studies and give some educational suggestions that may derive from our work
Combining theoretical approaches: socio-didactic-mathematical norms and perspectives in pre-service secondary mathematics teachers’ discourse
(Modestum Limited; Modestum Publishing Ltd, 2019) Toscano Barragán, Rocío; Sanchez Garcia, Maria Victoria; Garcia Blanco, María Mercedes; Universidad de Sevilla. Departamento de Teoría e Historia de la Educación y Pedagogía Social; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
This study is part of wider research that seeks to investigate the existence (or not) of socio- didactic-mathematical norms in the discourse arises within a group of pre-service secondary mathematics teachers when they are solving a didactic-mathematical task. In addition, we try to analyze whether any of these norms could have relationships with teachers’ perspectives that they can adopt in their future practice. The data comes from the transcriptions of the audio recordings of the dialogues among the pre-service teachers when they are solving a didactic-mathematical task in the classroom. Based on our analysis, we have been able to identify five socio-didactic- mathematical norms. Three of them were in some way related to the mathematical content and its learning. The other two are related to teachers’ role, providing information about characteristics that the future teachers associate with said role. Furthermore, we have identified features related to teachers’ perspectives through the above-mentioned norms
Developing TPACK through Task Design: Exploring the Use of Multiple Modes of Representation and the Promotion of Mathematical Processes
(Taylor & Francis, 2024-10-29) Gavilán Izquierdo, José María; Gallego Sánchez, Inés Magdalena; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; FQM-226
This research considers the need to train teachers who can effectively integrate technology for teaching and learning mathematics. The primary goal of this research is to showcase a successful training experience by identifying and describing some indicators of the technological pedagogical content knowledge acquired by pre-service early childhood education teachers during their participation in this initial training programme. Specifically, we focus on the different ways of representing mathematical concepts and the mathematical processes that can be activated by including such modes of representation. The participants in this research were 28 PECTs, who were organised into groups that planned a set of tasks using technology to teach geometric concepts. The results show that the groups employed a variety of modes of representation, translations and transformations (connections), fostering the emergence of different mathematical processes. Thus, our proposal to integrate technology was satisfactory in this sense, suggesting its potential applicability across diverse subjects, particularly those in which the ways of representing concepts play a relevant role. Also, the analysis conducted may be extrapolated to subjects others than mathematics.
Algebraic Approach to the Minimum-Cost Multi-Impulse Orbit-Transfer Problem
(American Institute of Aeronautics and Astronautics [Society Publisher], 2016) Avendaño, M.; Martín Molina, Verónica; Martín-Morales, J.; Ortigas-Galindo, J.; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Junta de Andalucía; Gobierno de Aragón; Centro Universitario de la Defensa de Zaragoza; Ministerio de Economía y Competitividad (MINECO). España; Universidad de Sevilla. FQM327: Geometria (Semi)Riemanniana y Aplicaciones
A purely algebraic formulation (i.e., polynomial equations only) of the minimum-cost multi-impulse orbit-transfer problem without time constraints is presented, while keeping all the variables with a precise physical meaning. General algebraic techniques are applied to solve these equations (resultants, Gröbner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, a proof of the optimality of the Hohmann transfer for the minimum-fuel two-impulse circular-to-circular orbit-transfer problem is provided. Finally, a general formula is also provided for the optimal two-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which, it is conjectured, can be removed).
A Banzhaf value for games with fuzzy communication structure: Computing the power of the political groups in the European Parliament
(Elsevier, 2014-11) Gallego Sánchez, Inés Magdalena; Fernández García, Julio R.; Jiménez Losada, Andrés; Ordóñez Sánchez, Manuel; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI); Universidad de Sevilla. FQM237: Juegos con Estructuras Combinatorias y de Orden
In 2013, Jiménez-Losada et al. introduced several extensions of the Myerson value for games with fuzzy communication structure. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Now we study a Banzhaf value for these situations. The Myerson model is followed to define the fuzzy graph Banzhaf value taking as base point the Choquet integral. We propose an axiomatization for this value introducing leveled amalgamation of players. An algorithm to calculate this value is provided and its complexity is studied. Finally we show an applied example computing by this fuzzy value the power of the groups in the European Parliament.
Cooperation among agents with a proximity relation
(Elsevier, 2016-04-16) Fernández García, Julio R.; Gallego Sánchez, Inés Magdalena; Jiménez Losada, Andrés; Ordóñez Sánchez, Manuel; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI); Andrés Jiménez-Losada; Universidad de Sevilla. FQM237: Juegos con Estructuras Combinatorias y de Orden
A cooperative game consists of a set of players and a characteristic function determining the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate, regardless of the actions that the other players take. The relationships of closeness among the players should modify the bargaining among them and therefore their payoffs. The first models that have studied this closeness used a priori unions or undirected graphs. In the a priori union model a partition of the big coalition is supposed. Each element of the partition represents a group of players with the same interests. The groups negotiate among them to form the grand coalition and later, inside each one, players bargain among them. Now we propose to use proximity relations to represent leveled closeness of the interests among the players and extending the a priori unions model.
Characterizing the role of technology in mathematics teachers’ practices when teaching about the derivative
(Taylor & Francis, 2021) Gavilán Izquierdo, José María; García, Mercedes; Martín Molina, Verónica; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Junta de Andalucía; Universidad de Sevilla; Universidad de Sevilla. FQM226: Grupo de investigación en Educación Matemática
A current research problem in mathematics education is the characterization of the role of teachers in the processes of technology integration in mathematics classrooms. This paper shows how two secondary mathematics teachers teach the concept of derivative of a function at a point and the concept of derivative function, one of them using digital technology and the other one without using it. Their teaching was characterized by describing their hypothetical learning trajectories (learning goals, learning activities and the hypothetical learning processes). APOS Theory (which stands for Action, Process, Object and Schema) was used to describe the hypothetical learning processes. The results show that the use of digital technology in class may promote reflection among students without excessive computations, thus helping them to construct the concept of derivative.
Identifying routines in the discourse of undergraduate students when defining
(Springer, 2021) Fernández León, Aurora; Gavilán Izquierdo, José María; González Regaña, Alfonso José; Martín Molina, Verónica; Toscano Barragán, Rocío; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Junta de Andalucía; Universidad de Sevilla; Universidad de Sevilla. FQM226: Grupo de investigación en Educación Matemática
In this paper, we study how undergraduate students define 3D geometrical solids. With this aim, we have identified the routines that are present in the discourse of the students when describing and defining these solids. These routines are one of the properties that characterise the mathematical discourse in the theory of commognition (Sfard 2008). Our results show three different types of routines. The first type is related to the process of describing the solids, the second one to the process of defining the solids and the rest of the routines have a transversal nature. All of them together give us a global vision of the mathematical practice of defining of these undergraduate students. For instance, it seems that some of these students do not have a clear idea of what a definition is. Moreover, there are also differences between the discourse of students when defining 2D figures and the discourse of students when defining 3D solids.
Pre-service mathematics teachers’ discourse: Differences between defining in task situations involving prototypical and non-prototypical solids
(Elsevier, 2024) Toscano Barragán, Rocío; Fernández Leon, Aurora; Gavilán Izquierdo, José María; González Regaña, Alfonso José; Martín Molina, Verónica; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
The literature has highlighted the significant role of definitions and defining in mathematics learning and teaching. Furthermore, non-prototypical figures are particularly important when teaching geometry, but teachers and pre-service teachers still have problems defining them. For these reasons, we investigated whether there were differences in the way that pre-service mathematics teachers constructed and selected definitions for prototypical and non- prototypical solids. In particular, the commognitive framework was employed to investigate the differences in the discourse of 33 pre-service secondary-school teachers when constructing and selecting definitions in task situations that involved prototypical and non-prototypical solids. Moreover, we studied if some commognitive conflicts appeared in task situations involving non- prototypical solids but not in similar task situations involving prototypical solids. The findings show some differences between the pre-service teachers’ discourses in both types of task situa tions. Additionally, some commognitive conflicts appeared only in task situations with non- prototypical solids. Lastly, we classified those commognitive conflicts.
Mathematical naming and explaining in teaching talk: Noticing work with two groups of mathematics teachers
(Springer, 2024) Planas, Núria; Alfonso, José M.; Arnal-Bailera, Alberto; Martín Molina, Verónica; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
Research shows the salient place of mathematical teaching talk, including the mathematical-linguistic practices of naming and explaining, in the enactment of students’ mathematical talk and learning with understanding in the classroom. Our study was developed to examine the noticing of two groups of secondary-school mathematics teachers in one-day workshops with tasks about these practices. The two workshops were mathematically content-specific, with teaching and learning accounts and prompts aimed at guiding focused attention to naming and explaining in the teaching of linear equations and probability. Thematic text analyses led to identify three foci of the two groups’ noticing: (i) missing practices of mathematical naming in own teaching talk; (ii) relative impact of mathematical explaining in teaching talk; and (iii) tensions around mathemati cal naming and explaining in teaching talk. Our results show that the social construction of teacher noticing is a feature of noticing development that can be documented in the context of one-day workshops. Whereas time for individual thinking and responses to the tasks created a context of support for noticing development, participation in the group discussions allowed the teachers to notice nuances of mathematical naming and explaining in teaching talk unaddressed in the task prompts. The group discussions thus amplified and opened up the opportunities to develop some focused noticing on the content of the workshops, specifically in connection with the teachers’ own teaching practice.
Researching how professional mathematicians construct new mathematical definitions: A case study
(Taylor & Francis, 2018) Martín Molina, Verónica; González Regaña, Alfonso José; Gavilán Izquierdo, José María; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Ministerio de Economía y Competitividad (MINECO). España; Junta de Andalucía; Universidad de Sevilla. FQM226: Grupo de investigación en Educación Matemática
In this work, we study the mathematical practice of defining by mathematics researchers. Since research is an important part of many professional mathematicians, understanding how they do research is a necessary step before thinking about future researchers’ undergraduate and postgraduate education. We focus on the defining process associated with the generalization of existing definitions as a way of constructing new ones. Data of this qualitative study come from a case study whose subject is a mathematics researcher in the area of differential geometry. We have interviewed this researcher and collected her research documents. From our analysis of the data, we have identified four phases in the defining process (Finding an opportunity to generalize an existing concept, Proposing a new definition, Justifying that the new definition is valid and Continuing the chain of definitions), which we will describe in detail in the Results section.
Characterising pre-service primary school teachers’ discursive activity when defining
(Modestum Publishing, 2023) Martín Molina, Verónica; Toscano Barragán, Rocío; Fernández León, Aurora; Gavilán Izquierdo, José María; González Regaña, Alfonso José; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM-226: GRUPO DE INVESTIGACION EN EDUCACION MATEMATICA
This paper studies how pre-service primary school teachers construct and select mathematical definitions through the analysis of their discursive activity. Specifically, the theory of commognition (Sfard, 2008) is employed to determine whether the existence of different meta-rules always leads to the existence of a commognitive conflict. Moreover, we study the reasons that give rise to the commognitive conflicts found and whether they are resolved. To this end, we studied the discourse of 45 pre-service primary school teachers while they answered several questions on defining geometric solids. The data in this study consisted of audio recordings of their discussions and their written answers. In this paper, three vignettes showing different meta-rules are presented. In the first, discussions regarding the characteristics of a definition promoted the appearance of different meta-rules that existed in incommensurable discourses, which meant the existence of a commognitive conflict. This conflict highlights the fact that certain pre-service teachers confuse the processes of describing and defining. Both the second and third vignettes featured the appearance of two different meta-rules. However, in both cases, those meta-rules could coexist in the same discourse, and therefore a commognitive conflict could not be inferred
Resolución de problemas de geometría con Cabri II
(Sociedad Canaria Isaac Newton de Profesores de Matemáticas, 2003) Barroso Campos, Ricardo; Gavilán Izquierdo, José María; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
Caracterizando cómo conjeturan los investigadores en matemáticas: un estudio de caso
(Seiem.es, 2023) Gavilán Izquierdo, José María; Fernández León, Aurora; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
Este trabajo se enmarca en la rama de investigación en educación matemática que estudia las actividades matemáticas que los investigadores en matemáticas desarrollan al construir conocimiento matemático. En concreto, tiene por objeto avanzar en la caracterización de la práctica matemática de con-jeturar de esta comunidad de profesionales. Para ello, se analiza qué usa y qué crea (según Rasmussen et al., 2005) una investigadora concreta del área de análisis matemático cuando construye conjeturas en su investigación. La metodología cualitativa que se sigue es el estudio de caso. Los resultados de este estudio muestran el relevante papel que juegan los ejemplos en la dimensión horizontal de la práctica matemática de conjeturar, destacando cómo se crean esos ejemplos y en qué momentos de la actividad investigadora se usan y se crean los ejemplos.
A study of pre-service primary teachers' discourse when solving didactic-mathematical tasks
(Modestum, 2019) Toscano Barragán, Rocío; Gavilán Izquierdo, José María; Sánchez García, María Victoria; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM-226: Grupo de investigación en Educación Matemáticas
From a commognitive approach, this article focuses on the discourse generated by pre service primary teachers who are solving didactic-mathematical tasks. Our aims are to study the characteristics of the aforementioned discourse and, through these characteristics, identify whether a discourse close to the one of primary teachers is beginning to emerge. The sources of data were audio-recordings of group discussions and group reports. Two different discourses were identified in our results. One is the discourse generated by pre-service teachers when adopt the role of students of any level who have to solve a task proposed in the classroom. The other discourse is linked to the adoption of a role close to their future professional work. If we consider that the acquisition of a specific discourse enables future teachers to integrate into the community of practice of primary teachers, the role of the different discourses becomes a relevant element in teacher education
Spanish primary school students’ engagement with a non-traditional method when adding and subtracting: ritualised versus exploratory participation
(Taylor & Francis, 2023) Gallego Sánchez, Inés Magdalena; Martín Molina, Verónica; Caro Torró, Isabel; Gavilán Izquierdo, José María; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM-226: Grupo de Investigación en Educación Matemática
Our work investigated how six primary school students used a non- traditional method for adding and subtracting: the ABN method, a Spanish acronym for Open (method) Based on Numbers. Commognitive theory [Sfard, A. 2008. Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. New York: Cambridge University Press] was employed to study the students’ mathematical routines. In particular, we studied whether their routines were exploratory or ritualised and found that four students had a ritualised use of the ABN method, while two others showed some signs of incipient exploratory use (manifested as an increase in flexibility, applicability, performer’s agentivity, and substantiability). The results show some of the problems that students have when applying this method. Nuestro trabajo investigó cómo seis alumnos de Educación Primaria utilizaban un método no tradicional para sumar y restar: el método ABN, acrónimo español de (método) Abierto Basado en Números. La teoría comognitiva [Sfard, A. 2008. Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. Nueva York: Cambridge University Press] se empleó para estudiar las rutinas matemáticas de los alumnos. En concreto, estudiamos si sus rutinas eran exploratorias o ritualizadas y descubrimos que cuatro estudiantes hacían un uso ritualizado del método ABN, mientras que los otros dos mostraban algunos indicios de un uso exploratorio incipiente (manifestado como un aumento de la flexibilidad, la aplicabilidad, la autonomía y la justificación en las rutinas utilizadas). Los resultados muestran algunos de los problemas que tienen los estudiantes a la hora de aplicar este método.
An Example of Connections between the Mathematics Teacher’s Conceptions and Specialised Knowledge
(Modestum, 2018) Aguilar González, Álvaro; Muñoz Catalán, María Cinta; Carrillo, José; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla.FQM226: Grupo de Investigacion en Educacion Matematica
The motivation for this study is to understand the professional knowledge that a teacher displays in her classroom when she teaches mathematics classes. To this end, our goal is to describe the possible relationships of the subdomains of Mathematics Teacher’s Specialised Knowledge (MTSK) model and the Conceptions about Mathematics Teaching and Learning (CMTL) that are integrated in it. This article presents a position on professional knowledge, the methodological design used has been an interpretative approach with a case study design of a 5th grade teacher in Primary Education, and some results which exemplify how these relationships have been identified and analyzed, and how they have helped to explain and understand the knowledge that the teacher mobilizes in her classroom. Finally, we express how this study can be used for teacher training in mathematics.
Early childhood teachers' specialised knowledge to promote algebraic thinking as from a task of additive decomposition
(Taylor & Francis, 2020) Muñoz Catalán, María Cinta; Ramírez García, Mónica; Joglar Prieto, Nuria; Carrillo Yáñez, José; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; Universidad de Sevilla. FQM226: Grupo de Investigacion en Educacion Matematica
In this article we aim to deepen our understanding of the content and nature of the early childhood teacher’s knowledge, focusing on those aspects which might promote students’ algebraic thinking. Approaching arithmetic from the viewpoint of algebra as an advanced perspective and considering the analytical model Mathematics Teachers’ Specialized Knowledge, we analyze the specialized knowledge in a classroom of 5-year-olds handled by an experienced teacher in a lesson on the decomposition of the number 6. Moreover, alternative management of the session is proposed in order to promote early algebraic thinking. In the domain of mathematical knowledge, this analysis has revealed the specificity of the knowledge that this professional must have of the natural number. In the domain of pedagogical content knowledge, it has highlighted the many elements of the knowledge of mathematics teaching that should be possessed to promote algebraic thinking at this educational stage. These elements appear to be more closely related to a profound knowledge of the mathematics taught than to pedagogical knowledge of a more general nature.
A New Tool for the Teaching of Graph Theory: Identification of Commognitive Conflicts
(Teaching Research NYC MODEL, 2022) Gavilán Izquierdo, José María; Gallego Sánchez, Inés Magdalena; González Herrera, Antonio; Puertas, María Luz; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; GRUPO DE INVESTIGACION EN EDUCACION MATEMATICA.(FQM-226) (Universidad de Sevilla)
MATHEMATICS TEACHING RESEARCH JOURNAL 186 SUMMER 2022 Vol 14 no 2 This content is covered by a Creative Commons license, Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0). This license allows re-users to distribute, remix, adapt, and build upon the material in any medium or format for noncommercial purposes only, and only so long as attribution is given to the creator. If you remix, adapt, or build upon the material, you must license the modified material under identical terms. A New Tool for the Teaching of Graph Theory: Identification of Commognitive Conflicts José María Gavilán-Izquierdo1, Inés Gallego-Sánchez1*, Antonio González1, María Luz Puertas2 1Universidad de Sevilla, Spain, 2Universidad de Almería, Spain gavilan@us.es, inesgal@us.es*, gonzalezh@us.es, mpuertas@ual.es Abstract: In this exploratory work, the discourse of first-year computer engineering undergraduate students of graph theory was analyzed with the aim of improving the teaching of this branch of mathematics. The theoretical framework used is the theory of commognition, specifically, we focus on commognitive conflicts because they are learning opportunities since they foster the learning process when resolved by students, and so teachers should consider them in their practice. A qualitative analysis of the written responses to a questionnaire dealing with definitions and the concepts of path and cycle graphs was performed. Thus, several commognitive conflicts were found, coming from the confluence of discourses governed by different discursive rules. Furthermore, the conflicts encountered were classified according to their origin, into object-level and metalevel conflicts. Concretely, the object-level conflicts had to do with the school discourse of geometry or sequences, and with the discourse of directed graphs; the metalevel commognitive conflicts were associated with the school discourse of the mathematical practice of defining. Finally, our findings are contrasted with related works in the literature, and also a series of implications for the teaching of graph theory are presented
Conocimiento especializado para enseñar la operación de resta en educación infantil
(Cariniana, 2017) Muñoz Catalán, María Cinta; Liñán García, María del Mar; Ribeiro, Miguel; Universidad de Sevilla. Departamento de Didáctica de las Matemáticas; GRUPO DE INVESTIGACION EN EDUCACION MATEMATICA - FQM-226 (Universidad de Sevilla)
En este trabajo reflexionamos sobre el conocimiento profesional del profesor de Educación Infantil en relación con la resta. La atención al conocimiento deseable de este profesional es relativamente reciente, existiendo escasos estudios que aborden este tema. Partimos de la consideración de que este conocimiento es especializado; diferente al del profesor de Primaria o Secundaria; y su caracterización debe realizarse desde un enfoque centrado en la propia matemática. Esto nos lleva a considerar el modelo del Conocimiento Especializado del Profesor de Matemáticas (Mathematics Teachers’ Specialised Knowledge, MTSK) como herramienta teórica y analítica idónea para ofrecer una propuesta de elementos de conocimiento especializado que, a la luz de nuestra experiencia docente e investigadora, consideramos que deseables para este profesional en relación con la resta, incluyendo aspectos de conocimiento del contenido (Mathematical Knowledge, MK) y de conocimiento didáctico del contenido (Pedagogical Content Knowledge, PCK). La reflexión sobre esta propuesta nos lleva a destacar tres elementos principales que parecen caracterizadores de la naturaleza de este conocimiento: su densidad y cohesión, la profundidad de los conocimientos matemáticos implicados y su repercusión para construir un adecuado PCK; y la relevancia del conocimiento de las fases que los niños siguen en su proceso de comprensión del número. Se aportan sugerencias para la formación inicial y continua de estos profesionales.

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