A link between students’ discernment of variation in unidirectional change and their use of quantitative variational reasoning

Abstract

In this design experiment study, we investigated the problem: What conditions might foster students' discernment of a critical aspect-variation in unidirectional change (e.g., discerning a Bdecreasing^increase)? At a public middle school in a large US city, we led a sequence of three days of whole class lessons, followed by task-based, clinical interviews with 14 seventh grade students (~13 years old). Students interacted with researcher-developed dynamic computer environments, which linked filling polygon animations with dynamic graphs. We report results to demonstrate a range of students' work along with detailed analyses of student interview episodes. We found connections between students' predictions of graph type (linear, Bmostly^linear, nonlinear), students' conceptions of attributes represented by a graph, and students' discernment of variation in unidirectional change. Students who discerned variation in unidirectional change also engaged in quantitative variational reasoning. To foster students' discernment of variation in unidirectional change, task designers should provide opportunities for students to conceive of attributes as capable of varying and possible to measure.

Opportunities for Reasoning: Digital Task Design to Promote Students’ Conceptions of Graphs as Representing Relationships between Quantities

“…Finally, students can interpret graphs as representing a single attribute varying with elapsing time (Janvier 1998;Johnson and McClintock 2018). For example, consider the co-ordinate plane shown in Fig.…”

Eleventh Graders’ Increasingly Elaborate Language Use for Disentangling Amount and Change: A Case Study on the Epistemic Role of Syntactic Language Complexity

“…Phase 1 Recognize variation and co-variation in graphs (Clement 1989;Johnson and McClintock 2018) Phase 2 Distinction of amount and change (Nemirovsky and Rubin 1992;Hahn and Prediger 2008) and the interplay of their symbolic and graphical representations (Kinley 2016) Phase 3 Concept of rate of change as a quotient concept (Thompson and Thompson 1994) Phase 4 Dealing with limits and infinitesimal aspects of the derivative (Marx 2006;Mundy and Graham 1994) In order to promote conceptual understanding of concepts in calculus, approaches in qualitative calculus (Thompson and Thompson 1994;Stroup 2002) suggest strengthening Phase 1 and 2 long before change is mathematized as average and instantaneous rate of change (Phase 3) and the derivatives and their procedural rules (Phase 4). Instructional approaches of qualitative calculus engage students in constructing meanings for the core concepts of amount, change, and change of change and in investigating their mutual relationship in different context situations and representations.…”

Reconfiguração De Figuras Para O Cálculo De Áreas Com a Ferramenta Polígono Rígido Do Geogebra

“…El uso de recursos tecnológicos como los sistemas de geometría dinámica como GeoGebra llegan a cumplir un papel fundamental al trabajar con nociones de geometría, dado que aportan dinamismo y permiten realizar interacciones que generan diversas representaciones simultáneas de los objetos de estudio, transformándose en una fuente de exploración de múltiples soluciones de una tarea (POVEDA, 2020). Asimismo, incorporar atributos que permiten a los estudiantes realizar manipulaciones y mediciones al trabajar en tareas diseñadas para fomentar su razonamiento, en temas de cambio y variación (JOHNSON, MCCLINTOCK, 2018;MARTÍNEZ-MIRAVAL et al, 2023), así como aportar a la mejora en la habilidad de representar visualmente un objeto geométrico (AZIZAH et al, 2021), acciones que guardan una relación profunda con la actividad de reconfiguración como procedimiento de resolución.…”