Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

Cárcamo, Andrea; Aymemí, Josep Maria Fortuny; Urgellés, Joan Vicenç Gómez i

doi:10.1080/0020739x.2016.1241436

Cited by 18 publications

(12 citation statements)

References 16 publications

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“…Concerning task 2, the conclusions of cycle 1 (Cárcamo et al, 2016) and cycle 2 (Cárcamo et al, 2017) indicate that it promoted the students' visualization of the concepts of spanning set and span in a real-world as well as a mathematical context. This gave them the opportunity to distinguish the connection with a real-world situation, as exemplified in a written response of a student from the group 12 of cycle 2 ( Figure 5).…”

Section: Tasks That Favored the Constructionmentioning

confidence: 98%

“…In the results of the first cycle (Cárcamo et al, 2016) we observed that most students differentiate between spanning set and span, and also they verified whether a vector is belonging to a specific span (Figure 8). In the second cycle (Cárcamo et al, 2017) we found that many students made conjectures and some properties obtained from the spanning set of a specific space (Figure 9). In the third cycle (Cárcamo et al, 2018) we noticed that the students did the same as those from the previous cycles and could even determine if a set of R 4 generated a subset of R 4 ( Figure 10).…”

Section: / 16mentioning

confidence: 98%

“…This suggestion was incorporated into the HLT of the second cycle, so task 3 was redesigned and focused on the properties of these concepts of Linear Algebra. Furthermore, we add an individual task 4 on the applications of these concepts (Cárcamo et al, 2017).…”

Section: Also Inmentioning

confidence: 99%

“…In cycle 2, we identified that the analytical representation of spanning set and span was the main difficulty (Cárcamo et al, 2017). To help the students overcome this difficulty, we proposed adding a task in cycle 3 in which the students work in groups on questions about the applications of spanning set and span, and then also individually on these questions.…”

Section: Also Inmentioning

confidence: 99%

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Local Instruction Theories at the University Level: An Example in a Linear Algebra Course

Cárcamo

Fuentealba

Garzón

2019

EURASIA J MATH SCI T

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In order to promote the design of innovative instructional activities at the Linear Algebra, we perform a design-based research project to explore how to teach Linear Algebra at the university level. In this article, we present the results of three cycles of a teaching experiment that we carried out to design, try out, and improve a local instruction theory (LIT) on the teaching of the concepts of spanning set and span in Linear Algebra with first-year engineering students. In a retrospective analysis, we looked for patterns in the data set of all the experiments, and we identified key learning moment of the students. Based on these patterns, we formulated a LIT to support the construction of the concepts of spanning set and span.

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Section: Tasks That Favored the Constructionmentioning

confidence: 98%

Section: / 16mentioning

confidence: 98%

Section: Also Inmentioning

confidence: 99%

Section: Also Inmentioning

confidence: 99%

See 2 more Smart Citations

Local Instruction Theories at the University Level: An Example in a Linear Algebra Course

Cárcamo

Fuentealba

Garzón

2019

EURASIA J MATH SCI T

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“…En los últimos quince años, la investigación sobre el aprendizaje y la enseñanza del álgebra lineal a nivel universitario ha sido de interés para la comunidad científica de educación matemática (Trigueros, 2019;González y Roa-Fuentes, 2017). Por esta razón, actualmente, existen estudios sobre cómo los estudiantes construyen conceptos en este curso (Thomas y Stewart, 2011) y sobre innovaciones docentes (Andrews-Larson, Wawro y Zandieh, 2017;Cárcamo, Fortuny y Gómez, 2017). Para estos últimos estudios, varios investigadores declaran usar una trayectoria hipotética de aprendizaje (THA) y muestran su utilidad en la enseñanza del álgebra lineal (Cárcamo et al, 2017;Trigueros y Possani, 2013;Wawro, Larson, Zandieh y Rasmussen, 2012).…”

Section: Introductionunclassified

Las trayectorias hipotéticas de aprendizaje: Un ejemplo en un curso de álgebra lineal

2021

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RESUMEN • Este trabajo propone y evalúa una trayectoria hipotética de aprendizaje (THA) para los conceptos de conjunto generador y espacio generado. La THA se diseñó siguiendo el planteamiento de Simon (1995), la heurística de diseño de los modelos emergentes y el mecanismo de reflexión sobre la relación actividad-efecto. Se realizó un experimento con estudiantes universitarios para evaluar si la THA contribuyó al aprendizaje esperado de los conceptos. Para ello, se contrastó la THA con la trayectoria real de aprendizaje (TRA) de dos estudiantes que mostraron aproximarse a las fases clave de la THA. Los resultados muestran que la noción de conjunto es útil para la construcción del concepto de espacio generado y que, además, se requiere incorporar nuevas interrogantes en la THA que faciliten la reflexión sobre la relación actividad-efecto para dar cuenta mejor de la progresión en el aprendizaje. PALABRAS CLAVE: Trayectoria hipotética de aprendizaje; Modelos emergentes; Actividad-efecto; Álgebra lineal. ABSTRACT • This article proposes and evaluates a hypothetical learning trajectory (HLT) for the concepts of span and spanning set. This HLT was designed following the approach of Simon (1995), the design heuristics of emergent models and the mechanism of reflection on activity-effect relationship. We carried out an experiment with university students to determine if this HLT contributed to the expected learning of these concepts. We contrast this HLT with the actual learning trajectory (ALT) of two students showing approximation to key phases of the designed HLT. The results show that the notion of set is useful for the construction of the concept of span. In addition, we found it is necessary to incorporate new questions in HLT that facilitate the reflection on the activity-effect relationship to better account for the learning progression. https://puv.uv.es

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Conceptions of span in linear algebra: from textbook examples to student responses

Gerami,

Khiu,

Mesa

et al. 2024

Educ Stud Math

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Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

Cited by 18 publications

References 16 publications

Local Instruction Theories at the University Level: An Example in a Linear Algebra Course

Local Instruction Theories at the University Level: An Example in a Linear Algebra Course

Las trayectorias hipotéticas de aprendizaje: Un ejemplo en un curso de álgebra lineal

Conceptions of span in linear algebra: from textbook examples to student responses

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