A framework for research and curriculum development in undergraduate mathematics education

Asiala, Mark; Brown, A. H. F.; DeVries, David J.; Dubinsky, Ed; Mathews, David; Thomas, Karen F.

doi:10.1090/cbmath/006/01

Cited by 214 publications

(230 citation statements)

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“…They acknowledged the difficulty in visualizing a horizontal translation in comparison to a vertical one, suggesting that "there is much more involved in visually processing the transformation of f(x) to f(x + k) than in visually processing the transformation of f to f(x) + k" (p. 58). Baker, Hemenway, and Trigueros (2000) have investigated the understanding of transformations of various functions from a perspective of Action-Process-Object-Schema (APOS) theory (for a description of APOS, see for example, Asiala et al, 1996or Dubinsky & MacDonald, 2001. They confirmed the observation that vertical transformations appear easier for the students than horizontal transformations.…”

Section: Introductionmentioning

confidence: 84%

Conceptions of function translation: obstacles, intuitions, and rerouting

Zazkis

Liljedahl

Gadowsky³

2003

The Journal of Mathematical Behavior

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A horizontal translation of a function is the focus of this study. We examine the explanations provided by secondary school students and secondary school teachers to a translation of a function, focusing on the example of the parabola y = (x−3) 2 and its relationship to y = x 2 . The participants' explanations focused on attending to patterns, locating the zero of the function, and the point-wise calculation of function values. The results confirm that the horizontal shift of the parabola is, at least initially, inconsistent with expectations and counterintuitive to most participants. We articulate possible sources of this perceived inconsistency and describe a pedagogical approach aimed at resolving it. © 2003 Elsevier Inc. All rights reserved. Prologue (motivation and questions)Imagine the graph of (a) y = x 2 . Now imagine the graph of (b) y = (x − 3) 2 . Check your imagination with the graphing device. If you are not surprised, it is probably because you have already explored the relationship between the two parabolas in the past. Most people conjecture that graph (b) will appear three units to the left of graph (a). The surprising result is that (b) is actually three units to the right of (a).As students, we memorized this result as one of the counterintuitive facts of mathematics. As teachers, we attempted to explain this phenomenon to our students, often creating a conflict between their intuition and our explanations.

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Section: Introductionmentioning

confidence: 84%

Conceptions of function translation: obstacles, intuitions, and rerouting

Zazkis

Liljedahl

Gadowsky³

2003

The Journal of Mathematical Behavior

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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

“…En la figura 2, se representa en general cómo la comprensión de un concepto y/o noción matemática inicia cuando un estudiante aplica Acciones sobre Objetos previamente construidos (Arnon et al, 2014;Asiala et al, 1996). Estas relaciones son definidas por Dubinsky (1991) como un "circular feedback system".…”

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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“…Ed Dubinsky and others (Asiala et al, 1996) took an apparently different path, following Piaget's idea of reflective abstraction to focus on operations that are seen first as actions, routinized as processes, then encapsulated as mental objects within knowledge schemas.…”

Section: The Evolution Of Theories Of Mathematical Thinking and Proofmentioning

confidence: 99%

Making Sense of Mathematical Reasoning and Proof

Tall

2014

Mathematics &Amp; Mathematics Education: Searching for Common Ground

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The sensori-motor language of mathematicsThe cognitive development of mathematical thinking and proof is based on fundamental human aspects that we all share: human perception, action and the use of language and symbolism that enables us to develop increasingly sophisticated thinkable concepts within increasingly sophisticated knowledge structures. It is based on what I term the sensori-motor language of mathematics, (Tall, forthcoming).Mathematical thinking develops in the child as perceptions are recognised and described using language and as actions become coherent operations to achieve a specific mathematical purpose. According to Bruner (1966), these may be communicated first through enactive gestures, then iconic images, then the use of symbolism, including not only written and spoken language but also the operational symbolism of arithmetic and the axiomatic formal symbolism of logical deduction.The theoretical framework proposed here follows a similar path enriched by the experience over time, building from conceptual embodiment that combines the enactive and iconic modes of human perception and action, developing into the mental world of perceptual and mental thought experiment. Embodied operations, such as counting, adding, sharing, are symbolised as manipulable concepts in arithmetic and algebra in a second mental world of operational symbolism. As the child matures, there is a further shift into a focus on the properties of mental objects as in Euclidean geometry, or the properties of arithmetic operations that are recast as 'rules' that underlie the generalized operations and expressions in algebra. Each of these leads to different forms of mathematical proof: Euclidean proof in geometry and symbolic proof, based on the 'rules of arithmetic' in arithmetic and algebra.

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A framework for research and curriculum development in undergraduate mathematics education

Cited by 214 publications

References 0 publications

Conceptions of function translation: obstacles, intuitions, and rerouting

Conceptions of function translation: obstacles, intuitions, and rerouting

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

Making Sense of Mathematical Reasoning and Proof

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