Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds

Altieri, Mike; Schirmer, Evelyn

doi:10.1007/s11858-019-01074-4

Cited by 15 publications

(6 citation statements)

References 29 publications

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“…Students' conceptual understanding of each of these representations may differ, and the relationship of understanding between the representations is often important in research. In order to capture students' understanding of linear algebra concepts in these different types of representations, Tall's model of three worlds developed by Tall ( 2013) is useful and used in some studies: e.g., (Stewart & Thomas, 2009), (Stewart & Thomas, 2010), (Stewart, Troup, & Plaxco, 2019), and (Altieri & Schirmer, 2019). Tall (2013) described the development of mathematical thinking in terms of three worlds: embodied world, symbolic world, and formal world.…”

Section: A Tall's Model Of Three Worldsmentioning

confidence: 99%

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

Kawazoe

2023

epiDEMES

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For the use of embodied notions in teaching linear algebra, some studies indicate that it is helpful, whereas other studies indicate that it could be problematic or become an obstacle. Hence, additional research is needed. This study is focused on linear (in)dependence and basis, and investigates the relation between their understandings in the embodied and symbolic worlds. We also examine whether students' conceptions in the embodied world can be improved by the instruction emphasizing geometric images, as our previous studies identified some limitations of students' understanding in the embodied world. To address these issues, we designed four tasks aiming to assess students' conceptions of linear (in)dependence, basis, and dimension, and also designed linear algebra lessons emphasizing geometric images of these concepts. These tasks were conducted during the lessons and the data of 38 engineering students was collected. The analysis for the data showed that conceptions in the embodied world was positively associated with conceptions in the symbolic world; however, students' conceptions in the embodied world were not sufficiently improved by the geometric instruction implemented in this study.

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Section: A Tall's Model Of Three Worldsmentioning

confidence: 99%

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

Kawazoe

2023

epiDEMES

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“…This is achieved by instructor-researcher proposing a genetic decomposition (GD) of the concept to guide the activities. A GD describes sequence of tasks that demand mental constructions that students ought to attain as they learn a mathematics concept (Altieri & Schirmer, 2019). In this study, APOS theory was used as to evaluate undergraduate students' mental constructions of EROs as a means to solve SLEs after traditional teaching of the concept, as discussed by Arnon et al (2014).…”

Section: Literature and Theorymentioning

confidence: 99%

Undergraduate students’ conceptualization of elementary row operations in solving systems of linear equations

Tatira

2023

EURASIA J Math Sci Tech Ed

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The concept of systems of linear equations (SLEs) is fundamental and core in linear algebra, a subject, which has many applications in a number of disciplines. Gaussian elimination is a versatile method, which can be used to solve almost all types of SLEs by using row-reductions. This study focused on exploring undergraduate students’ conceptualizations of elementary row operations (EROs) as a means to solve SLEs. The purpose of this study was to explore undergraduate students’ conceptualizations of row reductions and their applications to the solutions of systems of equations. The perspectives of the action-process-object-schema theoretical framework were used in analyzing data and discussing the findings. To explore the students’ conceptualization of EROs, a descriptive research approach was followed. I considered a case study of 131 students registered for a mathematics for educators course, where linear algebra was one of the topics. The findings revealed that students attained the action conception of reducing a system with unique solutions but had challenges reducing and interpreting solutions to a system with non-unique solutions. The latter row-reduction implored process and object conceptions especially when variable elements in the augmented matrix were involved. As students find the learning of linear algebra difficult, this study contributes to the debate in literature on how to improve its teaching and make suggestions on the ways make more effective the learning of linear algebra.

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“…Teori ini dapat dijadikan sebagai alat untuk mendeskripsikan skema seseorang ketika membangun pemahaman konsep ketika mempelajari suatu topik tertentu. Teori ini dapat digunakan untuk mendeskripsikan analisa terkait proses kontruksi seseorang dalam membangun pemahaman konsep pada suatu topik matematika (Altieri & Schirmer, 2019). Selain itu, teori ini dapat digunakan untuk membandingkan keberhasilan atau kegagalan seseorang dalam membangun mental yang terbentuk untuk suatu konsep matematika.…”

Section: Pendahuluanunclassified

Implementasi Tahapan APOS pada Analisis Kemampuan Pemahaman Konsep Mahasiswa dalam Menyelesaikan Soal Matriks

Febrinita

Puspitasari

Zaman³

2022

jtm

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Changes in the learning process during the pandemic affected student learning outcomes. As many as 68% of students score below 75 in computational mathematics courses. These results are not comparable to learning outcomes when learning takes place offline. The results of the interviews show that students have difficulty understanding the material when learning is done online. For this reason, a qualitative descriptive study was carried out which aimed to analyze and describe students' conceptual understanding abilities in solving matrix problems using the APOS stages. The research subjects were 3 students of the informatics engineering study program who had taken computational mathematics courses and received matrix material, were in the category of students with low, medium, and high conceptual understanding, and were able to communicate their thoughts orally and in writing. Data collection techniques were carried out through written tests, interviews, and observations. The results showed that students with low conceptual understanding were able to solve matrix problems up to the object stage and made calculation mistakes. Students with a moderate understanding of concepts can solve matrix problems up to the schematic stage but are not careful in doing calculations. Meanwhile, students with high conceptual understanding can solve matrix questions up to the schematic stage, determine the correct answer, make conclusions, and reflect on the problem-solving obtained.

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Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds

Cited by 15 publications

References 29 publications

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

Undergraduate students’ conceptualization of elementary row operations in solving systems of linear equations

Implementasi Tahapan APOS pada Analisis Kemampuan Pemahaman Konsep Mahasiswa dalam Menyelesaikan Soal Matriks

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