Potential of technology and a familiar context to enhance students\u27 concept of rate of change

Students\u27 concept image of rate of change may be incomplete or erroneous. This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students\u27 existing schema of informal understandings of rate of change to more formal mathematical representations. Students developed numerous \u27models of\u27 rate of change in a motion context which then transferred to serve as a \u27model for\u27 rate of change in other contexts.<br /

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

Potential of technology and a familiar context to enhance students’ concept of rate of change

Students’ concept image of rate of change may be incomplete or erroneous This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students’ existing schema of informal understandings of rate of change to more formal mathematical representations Students developed numerous ‘models of’ rate of change in a motion context which then transferred to serve as a ‘model for’ rate of change in other contextsE

Emerging models for the slope of a curve: a reinvention activity for upper secondary school

Would it be useful to invert the standard teaching trend and start designing lessons and activities able to push students to "reinvent" a mathematical notion? The research described in this thesis partially answers to this question, focusing on the concept of "slope of a curve". The whole research investigates if an already planned and progressively improved task- whose design is based on the Theory of Didactical Situations (TDS) and Realistic Mathematics Education (RME)- has the potential to make relevant informal models of the concept of slope of a curve in one point emerge from the students' solution; the consequent question inquired is whether the teacher could effectively build a rigorous lesson and present the formal knowledge about the topic, starting from these models. The whole research, conducted in secondary schools in the Netherlands, was conceived in the context of the Erasmus+ project “MERIA” (Mathematics Education Relevant, Interesting and Applicable)

Conceptualizaciones de pendiente: contenido que enseñan los profesores del bachillerato

Este artículo describe los resultados de una investigación que exploró las conceptualizaciones de pendiente en el contenido que enseñan los profesores de matemáticas de bachillerato. Para ello, analizamos las notas de clase del cuaderno de matemáticas de sus estudiantes por medio del método de análisis de contenido y empleamos las once conceptualizaciones de pendiente reportadas por otros investigadores como marco de referencia. Los resultados indican que las conceptualizaciones razón algebraica, trigonométrica y coeficiente paramétrico, enfatizadas en lo procedimental, fueron las que más promueven los profesores al definir, explicar, ejemplificar y proponer actividades vinculadas al concepto de pendiente

Explorando las conceptualizaciones de la pendiente en estudiantes universitarios

Este escrito reporta las conceptualizaciones de la pendiente de 21 estudiantes universitarios. Para la recolección de datos se utilizó una entrevista basada en tareas y para su análisis se identificaron las frases y procedimientos claves relacionados con las once conceptualizaciones reportadas por investigadores en educación matemática sobre el concepto de pendiente. Los estudiantes evidenciaron de una a ocho conceptualizaciones, entre las cuales se identificaron: Propiedad Física, Razón Algebraica, Propiedad Determinante, Constante Lineal, Coeficiente Paramétrico, Razón Geométrica, Indicador de Comportamiento y Situación Mundo Real (Situación Física)

Estabilidad y cambio conceptual acerca de las razones de cambio en situación escolar

Este artículo da cuenta de una investigación cuyo objetivo se centró en estudiar la estabilidad y el cambio conceptual acerca de algunas razones de cambio en estudiantes de bachillerato. Para ello se diseñó, aplicó y valoró, una secuencia de aprendizaje que tuvo como escenario el salón de clase de una escuela de bachillerato tecnológico. Los resultados fueron valorados mediante una evaluación pre-post test, a través de la cual fueron contrastadas las ideas previas con las ideas manifestadas al final de la aplicación de la secuencia de aprendizaje. Los cambios conceptuales van, de interpretar a la velocidad en una gráfica distancia-tiempo “como punto” o como “magnitud de la distancia” a la concepción geométrica del “desplazamiento vertical” respecto del “desplazamiento horizontal”; de la fijación por la fórmula v = d/t a la utilización del cociente de diferencias v = s/t. Se notó estabilidad en la concepción que asocia a la ordenadas de mayor magnitud de una gráfica tiempo-estatura, como las que representan la “mayor rapidez de crecimiento”

Using Manipulatives to Investigate ESOL Students\u27 Achievement and Dispositions in Algebra

The purpose of this embedded quasi-experimental mixed methods research was to investigate the effectiveness of concrete and virtual manipulatives on the achievement of English Speakers of Other Languages (ESOL) as they employ them to explore linear and exponential functions in high school Sheltered Common Core Coordinate Algebra. Also of interest were the effects concrete and virtual manipulatives have on their disposition towards mathematics and math class. Another goal was to investigate the benefits and disadvantages of using concrete and virtual manipulatives versus traditional instructional practices. This was a 5-week study. The control group (N=20) was instructed through the use of mathematics textbooks and Power Points (traditional) and compared to the treatment group (N=19), which was instructed using concrete and virtual manipulatives. One ESOL mathematics teacher implemented this study, teaching both groups by utilizing the sheltered instruction observation protocol (SIOP) (2012) model to integrate content and language. Qualitative research methods, teacher interviews, recorded field notes, students’ work samples and artifacts were utilized. Quantitative data analysis techniques were used to analyze departmentalized Linear and Exponential Functions Summative Assessments (pretest and posttest) to measure mathematics achievement. The one-way ANOVA uncovered no statistically significant difference between the control group and treatment group as they explored linear and exponential functions. The Quantitative Understanding: Amplifying Student Achievement and Reasoning Students Disposition instrument (pre-questionnaire and post- questionnaire) measured dispositions about mathematics and math class. The one-way ANOVA indicated no statistically significant difference between the control and the treatment group’s dispositions about mathematics and math class

Business Calculus Students’ Reasoning about Optimization Problems: A Study of Quantitative Reasoning in an Economic Context

While the opportunity to learn mathematics via textbooks is well documented at the secondary and elementary levels, research on the opportunity to learn mathematics via textbooks at the undergraduate level has received little attention. Furthermore, research that examines the role of mathematics textbooks in students’ learning of important concepts such as marginal change in applied calculus is scarce. Research on students’ quantitative reasoning at the post-secondary level is lacking. This qualitative study investigated the opportunity to learn about optimization problems, marginal change, and quantitative reasoning in an economic context via a business calculus textbook and from lectures in a business calculus course. The study also investigated students’ quantitative reasoning, using task based interviews conducted with 12 pairs of business calculus students, about optimization problems and marginal change in an economic context. This study found that the textbook’s presentation of optimization problems and marginal change was largely procedural with limited attention to the underlying concepts and that opportunities for students to reason about relationships between or among economic quantities such as the relationship between marginal cost and marginal revenue at a profit maximizing quantity received little attention. The presentation of optimization problems and marginal change in course lectures closely followed the presentation of these topics in the textbook. Students’ interpretations of marginal change varied in different contexts and representations depending on the tasks they were given. This study provided insights into students’ quantitative reasoning when analyzing multivariable situations in an economic context: students created new quantities that helped them to solve the problems in the tasks and helped them to reason about relationships among several quantities. Implications for different stakeholders including business calculus instructors and suggestions for further research are included