DeltaTick: Applying Calculus to the Real World through Behavioral Modeling

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\''s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus

Mathematics teachers’ ideas about mathematical models: A diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diversoEste artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.Handle:  http://hdl.handle.net/10481/3323

Mathematics teachers’ ideas about mathematical models: a diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds

Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diverso

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Este artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.This study was funded by the National Science Foundation (NSF), Grant # DUE- 0962863, “The Poincaré Institute: A Partnership for Mathematics Education.” The ideas expressed herein are those of the authors and do not necessarily reflect the ideas of the funding agency

Mathematics teachers’ ideas about mathematical models: A diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diversoEste artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.Handle:  http://hdl.handle.net/10481/3323

DeltaTick: Applying Calculus to the Real World through Behavioral Modeling

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\''s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus

DeltaTick: Applying Calculus to the Real World through Behavioral Modeling

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\''s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus

PNA

Resumen basado en el de la publicaciónSe describen las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y se explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Se analizan respuestas de 56 profesores estadounidenses en ejercicio a tres preguntas abiertas, mediante un análisis de contenido. Un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, si los datos pertenecen o no al modelo, y si el modelo es más o menos informativo que los datos. Se encuentran diferencias interesantes entre profesores con y sin formación matemática.ES