Learning by Understanding: The Role of Multiple Representations in                Learning Algebra

Brenner, Mary E.; Mayer, Richard E.; Moseley, Bryan; Brar, Theresa; Durán, Richard P.; Reed, Barbara Smith; Webb, David J.

doi:10.3102/00028312034004663

Cited by 153 publications

(99 citation statements)

References 23 publications

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“…Adding a visual representation of an arithmetic procedure to the more traditional symbolic representation can be particularly appropriate for less skilled students who lack formal academic training in the subject domain by building on their existing intuitive knowledge (English, 1997;Fuson, 1992aFuson, , 1992bHalford, 1993;Hiebert & Carpenter, 1992;Kintsch & Greeno, 1985). We chose the number line for a visual representation because it has been implicated as a central conceptual structure underlying number sense (Case & Okamoto, 1996;Griffin & Case, 1996), as a concrete manipulative for understanding arithmetic (Brenner et al, 1997;Hiebert & Carpenter, 1992;Lewis, 1989;Moreno & Mayer, 1999b), and as a grounding metaphor for arithmetic (Lakoff & Nunez, 1997). According to the walkingalong-a-path metaphor, "numbers are locations on a path," "the mathematical agent is a traveler along that path," "arithmetic operations are acts of moving along the path," and "the result of an arithmetic operation is a location on the path" (Lakoff & Nunez, 1997, p. 37).…”

Section: The Case For Multiple Representationsmentioning

confidence: 99%

“…Representing concepts or procedures in more than one format, which is called providing multiple knowledge representations (Brenner et al, 1997;Spiro, Feltovich, Jacobson, & Coulson, 1992), allows learners to construct understandings that prepare them better for transfer, with each example and representation adding connections and perspectives that others miss (Sternberg & Frensch, 1993).…”

Section: The Case For Multiple Representationsmentioning

confidence: 99%

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Do Multiple Representations Need Explanations? The Role of Verbal Guidance and Individual Differences in Multimedia Mathematics Learning.

Moreno

Durán

2004

Journal of Educational Psychology

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Elementary school children, some of whom were nonnative speakers of English, learned to add and subtract integers in a discovery-based multimedia game either with or without verbal guidance in English or optionally in Spanish (Groups G-verbal guidance and No-G-no verbal guidance, respectively). Group G members chose to listen to verbal explanations in their first language and showed larger posttest scores than Group No-G. High-computer-experience students in Group G outperformed the rest of the students on training session scores and a transfer test. Longer latencies to respond to practice problems affected all learning measures positively. Results support the use of verbal guidance for discovery-based multimedia games and show that multimedia games may not be equally effective for all learners.What is the role of verbal guidance in promoting mathematics learning from a discovery-based multimedia game? Are there important individual differences for which a multimedia game helps some kinds of learners more than others? These are important questions both for research and for the application of research, to improve instruction and learning outcomes in learning mathematics with multimedia programs. Our first question is concerned with the issue of how one should design multimedia games to enhance student learning and focuses on the role of verbal guidance as an aid to meaning making. To answer this question, we examined the cognitive consequences of learning how to add and subtract positive and negative numbers within a discovery-based multimedia game either with or without verbal guidance. Our second question is concerned with the issue of how individual differences influence learning from interactive multimedia games. More specifically, we were interested in examining the role that students' computer experience and latency to solve problems had when students were asked to learn from a discovery-based computer game that presented multiple representations (symbolic and visual) of the arithmetic procedure.The main goal of this study was to investigate if the benefits of guidance that had been documented in past research on discoverybased strategies in the classroom extended into the realm of multimedia learning. In our study, elementary students studied 16 addition and subtraction problems involving positive and negative numbers in each of four training sessions with two representations of the arithmetic procedure: a symbolic representation consisting of the number sentence (i.e., 2 Ϫ Ϫ5 ϭ ___) and a visual representation using a number line and a computer animation to show how the symbolic number sentence relates to a bunny's movements along the number line. One group of students learned without verbal guidance and was presented with the symbolic and visual representations alone (Group No-G). Another group of students learned with identical symbolic and visual representations but was presented, additionally, with a verbal description (in English or optionally in Spanish) explaining how the symbols related to the bun...

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Section: The Case For Multiple Representationsmentioning

confidence: 99%

Section: The Case For Multiple Representationsmentioning

confidence: 99%

Do Multiple Representations Need Explanations? The Role of Verbal Guidance and Individual Differences in Multimedia Mathematics Learning.

Moreno

Durán

2004

Journal of Educational Psychology

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“…Esta característica nos diferencia de los animales y de la inteligencia artificial y es quizá una de las razones que justifica el hecho de que la investigación sobre el lugar de las representaciones en el aprendizaje de las matemáticas y en la resolución de problemas haya experimentado un crecimiento importante en los últimos años. Como resultado de estas investigaciones se considera indiscutible la importancia de las múltiples representaciones en el desarrollo del pensamiento matemático (Brenner et al 1997;Cuoco y Curzio, 2001), de cuya evidencia principal dan cuenta las agendas de prioridades establecidas en comités y reuniones cientí-ficas de rango internacional (Goldin, 1998b;Hitt, 2002).…”

Section: Introductionunclassified

Representaciones en Resolución de Problemas: Un estudio de caso con problemas de optimización

Villegas

Castro²,

Gutiérrez

2017

EJREP

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Introducción. Las representaciones juegan un papel fundamental en el pensamiento matemático, favorecen la comprensión de los conceptos matemáticos y estimulan el desarrollo de un pensamiento flexible y versátil en la resolución de problemas. En este trabajo nos ceñimos a problemas de optimización, de gran importancia y predicamento en la enseñanza y aprendiza-je de la matemática a nivel superiorMétodo. Mediante metodología observacional presentamos los patrones de representación empleados por tres estudiantes de quinto curso de la licenciatura de matemáticas de la Universidad de Granada (España), obtenidos mediante el análisis de protocolos, en el que los registros escritos van acompañados de protocolos de pensar en voz alta. El instrumento empleado fue construido ad hoc y consta de tres problemas de optimización. Las sesiones de resolución de problemas fueron individuales, en un ambiente aislado y fueron grabadas en vídeo.Resultados. Para el estudio de los datos se diseñó un marco para el análisis de protocolos con el que se investigan las transcripciones de las producciones de los sujetos. Los resultados se exponen en forma de análisis microscópico de caso en los que se pormenorizan los registros de representación empleados, las traducciones entre registros y el tiempo empleado en estas acciones, a partir de los cuales se realiza un perfil diferenciador de los resolutores. Según este perfil los tres participantes tienen una tipología distinta.Discusión y Conclusiones. El marco para el análisis microscópico de los protocolos de resolución de problemas se ha mostrado adecuado para describir las representaciones y traducción entre representaciones en resolución de problemas. El proceso de segmentación y codificación nos ha llevado a considerar necesario incluir episodios calificados en principio como eventos no catalogables como representaciones. La caracterización de los resolutotes muestra la fuerte relación entre el éxito en la resolución de problemas de optimización y la habilidad en el manejo de las representaciones.

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“…These four features of multiple-strategy instruction, along with the two types of instructional goals, emerged from research around the learning and teaching of mathematics in elementary school, particularly the work of the Cognitively Guided Instruction (CGI) project (e.g. Carpenter, Franke, Jacobs, Fennema, & Empson, 1998) combined with an emphasis on the use of multiple representations in algebra (e.g., Brenner et al, 1997;Star & Rittle-Johnson, 2009b).…”

Section: Multiple Strategiesmentioning

confidence: 99%

Views of Struggling Students on Instruction Incorporating Multiple Strategies in Algebra I: An Exploratory Study

Lynch¹,

Star²

2014

JRME

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Although policy documents promote teaching students multiple strategies for solving mathematics problems, some practitioners and researchers argue that struggling learners will be confused and overwhelmed by this instructional practice. In the current exploratory study, we explore how 6 struggling students viewed the practice of learning multiple strategies at the end of a yearlong algebra course that emphasized this practice. Interviews with these students indicated that they preferred instruction with multiple strategies to their regular instruction, often noting that it reduced their confusion. We discuss directions for future research that emerged from this work.

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Learning by Understanding: The Role of Multiple Representations in Learning Algebra

Cited by 153 publications

References 23 publications

Do Multiple Representations Need Explanations? The Role of Verbal Guidance and Individual Differences in Multimedia Mathematics Learning.

Do Multiple Representations Need Explanations? The Role of Verbal Guidance and Individual Differences in Multimedia Mathematics Learning.

Representaciones en Resolución de Problemas: Un estudio de caso con problemas de optimización

Views of Struggling Students on Instruction Incorporating Multiple Strategies in Algebra I: An Exploratory Study

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