Teaching eigenvalues and eigenvectors using models and APOS Theory

Salgado, Hilda; Trigueros, María

doi:10.1016/j.jmathb.2015.06.005

Cited by 54 publications

(27 citation statements)

References 11 publications

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“…Numerous frameworks explicitly include a theoretical analysis as an initial step in their research cycle, as is the case of the APOS theory (Arnon et al, 2014), whose theoretical analysis culminates in the socalled preliminary genetic decomposition. This is a hypothetical model of the mental structures and mechanisms that a student may need to build a specific mathematical concept: numerical sequence (Bajo, Gavilán-Izquierdo, & Sánchez-Matamoros, 2019), linear transformation (Roa-Fuentes & Oktaç, 2010), derivative (Borji, Alamolhodaei, & Radmehr, 2018), vector space (Parraguez & Oktaç, 2010), eigenvectors and eigenvalues (Salgado & Trigueros, 2015); among others. In these works, the authors carry out their theoretical analyses based on one or more data sources: researchers' mathematical understanding of the concept, their experiences as teachers, prior research on students' thinking about the concept, historical perspectives on the development of the concept, and/or an analysis of text or instructional materials related to the concept (Arnon et al, 2014).…”

Section: Methodsmentioning

confidence: 99%

Characterizing Levels of Reasoning in Graph Theory

González

Gallego

Izquierdo

et al. 2021

EURASIA J Math Sci Tech Ed

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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.

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Section: Methodsmentioning

confidence: 99%

Characterizing Levels of Reasoning in Graph Theory

González

Gallego

Izquierdo

et al. 2021

EURASIA J Math Sci Tech Ed

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“…[33] studied the geometrical representations of two-variable functions in the three dimensional space using APOS theory while [16] also extended the application of APOS theory to Multivariate Calculus. Salgado and Trigueros [34] investigated classroom experiences of a modelling problem involving eigenvalues, eigenvectors and eigen spaces with successful object conception of difficult concepts; [35] studied the limit concept using APOS theory, [36] managed to study teachers' responses to questions on normal distribution using the APOS theory. [37] used APOS theory to study the students' understanding of the derivative concept.…”

Section: Literature Reviewmentioning

confidence: 99%

Digital Instructional and Pedagogical Gaps in Mathematics Education during COVID-19: APOS Theory Based Laboratory Methodology in Higher Education

Mukavhi¹,

Brijlall²

2021

ujer

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The COVID-19 pandemic globally transformed the face to face teaching and learning into digital online teaching and learning. The uptake and integration of digital technology in mathematics education have been gradual for many decades. However, the COVID-19 pandemic has reshaped the digital teaching and learning and made it an imperative and safe remote instruction and pedagogy of the day in mathematics education and education in general. We explored the uptake and use of virtual mathematics laboratory methodology for undergraduate Calculus teaching and learning as a remote instruction and pedagogy gap revealed during the COVID-19 infections prevention and containment period in Zimbabwe. The main objective of the study was to identify digital instructional and pedagogical gaps that existed in addressing students' mathematical misconceptions in Calculus learning in using the APOS theory and a virtual mathematics laboratory. Data was collected from 160 undergraduate students taking Calculus courses via APOS theory mediated digital mathematics laboratory for a semester. Results revealed that the undergraduate students who took the Calculus courses via the digital mathematics laboratory methodology performed better than those who used to take the same courses using didactical approaches. This COVID-19 crisis ushered in digital teaching and learning experiences that clearly spelt out the existence of digital instructional and pedagogical gaps in mathematics education.

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“…Sus fundamentos se basan en la Abstracción Reflexiva formulada por Piaget. Esta teoría ha sido usada y validada para explicar diferentes nociones y/o conceptos matemáticos avanzados, asociados con el nivel universitario; en áreas como cálculo (Valdivia y Parraguez, 2015;Trigueros y Martínez-Planell, 2010), álgebra lineal Salgado y Trigueros, 2015;Parraguez y Oktaç, 2010), álgebra abstracta (Asiala et al, 1996) ecuaciones diferenciales (Trigueros, 2014), fractales (Villabona y Roa-Fuentes, 2016), probabilidad (Vásquez y Parraguez, 2014), teorema del isomorfismo (Mena et al, 2016), entre otros.…”

Section: La Teoría Apoeunclassified

Estructuras Mentales que Modelan el Aprendizaje de un Teorema del Álgebra Lineal: Un Estudio de Casos en el Contexto Universitario

Fuentes

Parraguez

2017

Form. Univ.

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ResumenSe presenta una investigación cualitativa que indaga sobre cómo estudiantes universitarios chilenos y colombianos aprenden un teorema del álgebra lineal, denominado teorema Transformación Lineal Matricial, TLMA. Para esto se define un modelo cognitivo en términos de las estructuras y mecanismos mentales que propone la teoría APOE (Acción, Proceso, Objeto, Esquema) para interpretar el pensamiento matemático avanzado. Además se describen los niveles de evolución intra, inter y trans del esquema del teorema TLMA y sus indicadores de desarrollo en términos de relaciones, transformaciones e invariantes. Los resultados de la investigación muestran por un lado que la construcción de la representación matricial de una transformación lineal como objeto, es fundamental para que estudiantes universitarios de primer año comprendan el teorema TLMA; y por otro, que dicho objeto debe ser asimilado por el esquema de transformación lineal. Palabras clave: teorema TLMA; teoría APOE; álgebra lineal; descomposición genética; aprendizaje Mental Structures that Model the Learning of a Linear Algebra Theorem: A Case Study in the University Context AbstractIn this paper, a qualitative research that explores how Chilean and Colombian university students learn a linear algebra theorem, called Matrix Linear Transformation theorem, MALT, is presented. For this, a cognitive model is defined in terms of the structures and mechanisms proposed by the APOS theory (Action, Process, Object, and Schema) to interpret the advanced mathematical thinking. Furthermore, the levels of Intra, Inter and trans evolution of the MALT theorem and its development indicators in terms of relationships, transformations and invariants are described. The results of this research show on the one hand, that the construction of the matrix representation of a linear transformation as object is essential for first year students to understand the MALT theorem; and on the other hand that such an object must be assimilated by the linear transformation schema.

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Teaching eigenvalues and eigenvectors using models and APOS Theory

Cited by 54 publications

References 11 publications

Characterizing Levels of Reasoning in Graph Theory

Characterizing Levels of Reasoning in Graph Theory

Digital Instructional and Pedagogical Gaps in Mathematics Education during COVID-19: APOS Theory Based Laboratory Methodology in Higher Education

Estructuras Mentales que Modelan el Aprendizaje de un Teorema del Álgebra Lineal: Un Estudio de Casos en el Contexto Universitario

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