Construction and mental mechanisms for learning matrix theorem associated with a linear transformation

María Trigueros Gaisman, Isabel Maturana Peña,
Marcela Parraguez González y Miguel Rodríguez Jara

Abstract:Based on apos theory (Actions, Processes, Objects and Schemes) as a theoretical and methodological framework, we investigate the mental constructions and mechanisms required to construct the associated matrix of a Linear Transformation (matl). We design a genetic decomposition
(dg) of the theorem matl to discuss how college students learn it. We report three cases of study. The results show how students construct the concept of coordinates of a vector as an object, but have difficulty using it in the construction of the coordinate matrix associated to the image of vectors. Results also show students’ difficulties to construct matl as an object and the crucial role of considering tl as a function in order to attain a deep comprehension of tmatl.

Key words: Matrix associate, Linear Transformation, apos Theory, Linear Algebra.

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Rule of signs of the multiplication: a proposal didactic

José Benjamín Chan Domínguez y Genny Rocío Uicab Ballote

Abstract:This project describes the design and implementation of a methodological approach to introduce the rules of multiplications signs through the geometric representation of the
product like the area of a rectangle. The proposal considers two sequences of activities organized, both on a worksheet as well as the software Cabri II Plus. This proposal is designed in the focus of Brousseau’s theory of didactic situations, and it has the student as the principal actor, which is in charge of an experimentation and explanation process during the resolution of learning activities, for building of the signs rules. This proposal is designed for middle school students.

Key words: Rules of the signs, learning environment, school treatment, theory of the didactic
situations, didactic engineering.

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Analysis of the coupling process between the epistemic and cognitive facets of mathematical knowledge in the context of an exploratory-investigative task about patterns

Luis R. Pino-Fan, Adriana Assis y Juan D. Godino

Abstract: In this paper, we explore the coupling between the intended and implemented meanings by the teachers and the meanings achieved by the students in the context of an exploratory- investigative task on patterns, implemented in a class of the 7th grade of the basic education in Brazil. For this, we use the theoretical model known as Onto-Semiotic Approach (osa) of mathematical knowledge and instruction, which allowed us to describe and interpret the interaction processes in the classroom and establish relations between the facets that, according to this model, compose the process of teaching and learning. As a result, we emphasize the complexity of the teacher’s work, which should be able to recognize and negotiate the
meanings produced by the students –not always contemplated in her initial planning– making decisions about the type of performance that should be adopted, moment-to-moment, throughout the process of study to ensure the optimization of learning.

Key words: Instructional process analysis. Patterns. Classroom interaction. Onto-semiotic
approach.

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An overview of the mathematical beliefs system of pre-service teachers

Santiago Hidalgo Alonso, Ana Maroto Sáez y Andrés Palacios Picos

Abstract:The study of affective factors has responded to uncertainties present in some processes of teaching and learning mathematics. In this paper we propose to study one of the components of the affective domain: mathematical beliefs; in particular, we are interested in making a categorization system of mathematical beliefs of future teachers and knowing the verbal associations engaged on mathematics. In a sample of more than a thousand students of the Degree of Master of Primary Education of ten public universities in Spain, we applied a Likert scale of mathematical beliefs. Statistical analysis indicates that the associations that future teachers do about maths are at the cognitive and metacognitive (knowledge strategies). However, regarding their experiences with mathematics, the students show a strong, mostly negative, with respect to the self-concept, the tasting to mathematics, the affective component and the causal attributions and the positive
assessment of their social connection: usefulness and necessity.

Keywords: pre-service teachers, affective domain, mathematical beliefs system, academic performance, statistic analysis.

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A mathematical modelling as a means to detection student’s obstacles and difficulties related to the function concept: extension of a spring under a weight

Jose Benito Búa Ares, Teresa Fernández Blanco y Mª Jesús Salinas Portugal

Abstract:In this article we present a research about a mathematical modelling for the secondary school. Students start with a data table, it obtained experimentally by themselves, on the behavior of a spring subjected to a weight. From the data table and using the GeoGebra program, students obtain an analytical expression that relates the two variables (mass and length of the spring). After they obtained the model, students have to answer questions related to the mathematical model and the original extramathematical situation. The main objectives of the research are to show how students use their unserstanding about functions during the process to obtain the model and if students interpret the mathematical result in the context of the original extramathematical situation. The investigation results show it is possible obtain a mathematical model without the use of implicit mathematical knowledge and without the interpretation of the mathematical result in the original extramathematical situation.

Keywords: Modelling, functions, GeoGebra, lower secondary education, upper secondary education

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A path to reasoning: Using problems from Mathematical Olympiad

Claudia Gómez Wulschner y Esteban Landerreche Cardillo

Abstract:From the students’ answers in the Olimpiada de Mayo, which is applied to elementary and secondary school students all over Mexico, it is intended to pose a different approach on the instruction of mathematics in elementary education. The “Olympic Type” problems transcend the study plan and the students’ academic baggage. It is why we postulate that presenting problems of this sort, that push the student to consider different ways of reaching an answer, have a positive impact on the development of the youth’s mathematical logic, which translates on a better understanding of mathematics.

Keywords: Mathematics Competitions, Olympiad problems, open-ended questions.

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The notion of relationships among calculations and the production of explanations in maths class as teaching objects. Their configuration within a collaborative work between researchers and teachers

Patricia Sadovsky, María Emilia Quaranta, Horacio Itzcovich,
María Mónica Becerril, Patricia García

Abstract: This study accounts for the process through which, in the context of a collaborative work between researchers and teachers, two primary school teaching objects arise: the practice of appealing to relations among calculations to get results and the production of explanations in mathematics class. We try to show links between the configuration of these objects —the analysis of its meaning and possible conditions for them to arise at a math class— and the collective construction of a didactic intent. This research belongs to a larger project aimed at understanding the type of mathematical and didactic production which could be obtained when a group of researchers attends regularly a school to work with teachers and heads in order to think jointly about mathematics education issues.

Keywords: collaborative work, analysis of the mathematics teaching practices, explanations
in math class, relationships between calculations.

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Abriendo las puertas del razonamiento: los “problemas de Olimpiada” como herramienta

Claudia Gómez Wulschner y Esteban Landerreche Cardillo

Resumen: A partir de las respuestas de los alumnos en la Olimpiada de Mayo, evento donde son aplicadas pruebas a estudiantes de primaria y secundaria en todo México, se intenta  plantear un enfoque diferente para la enseñanza de las matemáticas en la educación básica. Los problemas “Tipo Olimpiada” trascienden el plan de estudios y el bagaje académico de los estudiantes. Es por ello que postulamos que presentar problemas de este tipo, los cuales llevan al alumno a considerar diferentes formas de llegar a una respuesta, tiene un impacto positivo en el desarrollo lógico matemático de los jóvenes y se traduce en un mejor entendimiento de las matemáticas.

Palabras clave: concursos de matemáticas, problemas de olimpiada, problemas de respuesta abierta.

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Una modelización matemática como medio de detección de obstáculos y dificultades de los alumnos sobre el concepto de función: alargamiento de un muelle sometido a un peso

Jose Benito Búa Ares, Teresa Fernández Blanco y Mª Jesús Salinas Portugal

Resumen:En este trabajo se presenta una investigación sobre una actividad de modelización matemática en la Enseñanza Secundaria. Los alumnos generan experimentalmente una tabla  de datos sobre el comportamiento de un muelle sometido a un peso. Partiendo de esos datos y utilizando GeoGebra, llegan a una expresión analítica que relaciona las dos variables presentes(masa y longitud del muelle). Una vez obtenido el modelo, los estudiantes responden a una serie de cuestiones relacionadas con el modelo matemático obtenido y con la situación extra-matemática origen de la modelización. Los objetivos principales de la investigación son, por un lado, mostrar hasta qué punto los alumnos integran los conocimientos adquiridos sobre funciones en el proceso de modelización. Y, por otro, mostrar hasta qué punto los alumnos interpretan el resultado matemático en la situación extra-matemática original. Los resultados de investigación ilustran que es posible que los alumnos obtengan un modelo matemático sin integrar los conocimientos matemáticos implícitos y sin interpretar el resultado matemático en la situación extra-matemática, origen de la modelización.

Palabras clave: Modelización, Funciones, GeoGebra, Enseñanza Secundaria, Bachillerato.

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Una aproximación al sistema de creencias matemáticas en futuros maestros

Santiago Hidalgo Alonso, Ana Maroto Sáez y Andrés Palacios Picos

Resumen:El estudio de los factores afectivos ha dado respuesta a incógnitas presentes en algunos procesos de enseñanza-aprendizaje de las matemáticas. En este trabajo nos proponemos
estudiar uno de los componentes del dominio afectivo: las creencias matemáticas; en particular, nos interesa realizar una categorización del sistema de creencias matemáticas de los futuros maestros y conocer las asociaciones verbales que realizan sobre las matemáticas. A una muestra de más de mil estudiantes del Grado de Maestro en Educación Primaria de diez centros universitarios públicos de España, se les aplicó una escala tipo Likert sobre
creencias matemáticas. El análisis estadístico nos indica que las asociaciones que los futuros docentes realizan sobre las matemáticas se sitúan en lo cognitivo y metacognitivo(estrategias de conocimiento). Sin embargo, en lo relativo a sus vivencias con las matemáticas, manifiestan un fuerte componente afectivo, mayoritariamente negativo, respecto al autoconcepto, al gusto y a las atribuciones de causalidad y una valoración positiva de su conexión
social: utilidad y necesidad.

Palabras clave: maestros, dominio afectivo, sistema de creencias matemáticas, rendimiento académico, análisis estadístico.


An overview of the mathematical beliefs system of pre-service teachers

Abstract:The study of affective factors has responded to uncertainties present in some processes of teaching and learning mathematics. In this paper we propose to study one of the components of the affective domain: mathematical beliefs; in particular, we are interested in making a categorization system of mathematical beliefs of future teachers and knowing the verbal associations engaged on mathematics. In a sample of more than a thousand students of the Degree of Master of Primary Education of ten public universities in Spain, we applied a Likert scale of mathematical beliefs. Statistical analysis indicates that the associations that future teachers do about maths are at the cognitive and metacognitive (knowledge strategies). However, regarding their experiences with mathematics, the students show a strong, mostly negative, with respect to the self-concept, the tasting to mathematics, the affective component and the causal attributions and the positive
assessment of their social connection: usefulness and necessity.

Keywords: pre-service teachers, affective domain, mathematical beliefs system, academic performance, statistic analysis.

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