The variable concept: an analysis with high school students

Authors: Heli Herrera López, Abraham Cuesta Borges, Juana Elisa Escalante Vega

Abstract: This article analyze the results of the resolutions mode of 48 students
from three different levels of learning in high school, when they try to solve
problems that involve generalization and modeling process through the relation
between algebraic language with figurative, natural and arithmetic language.
Using the 3uv Model (3 uses of the variable) to describe the comprehension of
the different aspects that characterize the variable as the acting of the students
when they face tasks.

Key words: mathematical education, difficulties, high school teaching, algebraic
thinking, variables.

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Substantial Argumentation. An Experience with High School Level Students in Math Classes

Authors: Alma Alicia Benítez Pérez, Héctor Benítez Pérez,  Martha Leticia García Rodríguez

Abstract: This paper presents an analysis on the substantial argumentation on
application problems evoked from a cognitive approach from Algebra and Differential
Calculus produced by high school students. The cognitive approach is
followed in this paper, based of the functional and structural analysis of reasoning
allowing the discursive mechanisms through which reasoning can change the
certainty and validity of the propositions. As a result, it is identified the characteristic
elements of substantial argument in evoked contextualized problems and
organizational presence of connective elements towards the statement-object,
were identified. A particular model based upon Ethnography of the qualitative
research was used for analyzing class records and transcripts. It´s important that
en this paper only one contextualized problem is analyzed.

Keywords: Substantial argument, functional analysis, structural analysis, representations,
contextualized problems evoked.

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How Abundant are the Sets of Numbers? Students Comparing Infinite Sets of Numbers

Authors: Virginia Montoro, Nora Scheuer, Ma. del Puy Pérez Echeverría

Abstract: We studied how students with different mathematical background
conceive infinite cardinality of number sets. We analyzed a task in which high
school and college students were requested to compare infinite number sets.
Students were classified according to their ideas on infinity. Using this classification
together with the students’ level of math education, we performed a
correspondence factorial analysis. A gradient was found in the depth of students’
ideas. At one end we found what we called horror infiniti, based students
propensity avoid infinity and instead construe it as something undefined. These
views were associated with students with less mathematical education. In an
intermediate zone, the finitist conception was placed. It was the most frequent
way of thinking among the participants of the study, with three versions: tacitly
infinitist, explicitly finitist, or taking the integers as model of inclusion. At the
other end, the infinitist conception was placed. It was present among students
with college mathematical education, according to two types: thinking of the
cardinality of number sets as a unique infinite quantity; or conceiving different
infinite cardinals. The latter was found only in advanced mathematics students.

Keywords: comparison, number sets, cardinal infinity, mathematics education.

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Evaluation of the Knowledge of Future Teachers of Mathematics about the Transformation of the Representations of a Function

Authors: Tulio R. Amaya De Armas, Luis R. Pino-Fan, Antonio Medina Rivilla

Abstract: In this paper the mathematical dimension of the Didactical-Mathematical
Knowledge of the future teachers of mathematics is analyzed when they
transform the representations of a function. A questionnaire was applied to 90
future teachers who were distributed in three groups. The homogeneity of the
answers per group is analyzed, and the primary mathematical objects and
the present process in the mathematics practices they develop trough their
answers are characterized. The results show serious difficulties of the future
teachers with some function topics, such as: indetifying and using their intercepts
without a graphical support; analyzing their extreme values and growth ranges;
making a mathematical model or a graphical draft; indetifying the slope of a
linear function. The primary mathematical objects and present processes in the
practices that the prospective teachers develop are very similar, in spite of the difference
in the number of hours od instruction in each of three gropus.

Keywords: future teachers, function, semiotic representations, mathematical
practices.

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The Role of Experimentation in Mathematical Modeling

Authors: Ruth Rodríguez Gallegos, Samantha Quiroz Rivera

Abstract: This research aims at describing the role of experimentation in the
design of a mathematics course for engineers, which was based on mathematical
modeling. The course is Differential Equations (ED), taught in second year
in a private institution in Northeast Mexico. The study, in a qualitative approach,
shows how experimentation provides significant elements for better understanding
the modeling of electrical phenomena by Differential Equations, specifically,
the study of electrical circuit Resistor-Capacitor (RC) and their respective
study from mathematical and physical point of view through the use of specific
technology. The results reveal that the implementation of experimentation in a
math class allows the construction, interpretation, and validation of mathematical
models by the students themselves, which are proposed in the course in
a collaborative way as opposed to traditional teaching of this subject, which it
is limited to the presentation of methods to solve ED without any connection
to reality.

Keywords: Mathematical models, experimentation, Engineering Education, Differential
Equations.

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Análisis de datos bivariados en un ambiente basado en applets y software dinámico

DOI 10.24844/EM2803.03

Autor: Santiago Inzunza Cazares

Resumen: Se reportan resultados de una investigación sobre análisis de datos
bivariados realizada con 34 estudiantes universitarios mientras tomaban un
curso introductorio de estadística en el cual se utilizó un ambiente computacional
integrado por applets y software dinámico para análisis de datos. En la evaluación
diagnóstica los estudiantes exhibieron bajo nivel de razonamiento intuitivo sobre
la covariación en diagramas de dispersión, pero en la actividad final, después
de haber utilizado el ambiente computacional, lograron identificar con éxito la
dirección de la relación entre dos variables y desarrollaron un buen sentido de
la intensidad de la relación y el coeficiente de correlación; sin embargo, la interpretación
de los coeficientes de la ecuación de regresión y el coeficiente de
determinación resultó ser una tarea complicada para ellos. Las estrategias utilizadas
en el análisis de datos privilegiaron el uso de representaciones gráficas y
visuales sobre representaciones simbólicas de cálculo, como resultado del potencial
de visualización dinámica de las herramientas tecnológicas utilizadas.

Palabras clave: Datos bivariados, razonamiento covariacional, tecnologías
digitales.

 

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Research in Mathematics Education in México: Looking at 40 Years of Work

Author:Alicia Avila

Abstract: In México, research in the domain of mathematics education emerged
in the 1970’s. It focused mainly on the cognitive processes of students, and on
the history of mathematical concepts taught in higher education. Over the years,
with the incorporation of new methodological tools, the focus shifted and diversified.
The current state of research is the result of a path with changes and
expansions in both our conceptions of objects and subjects of study, as well as
in the theories and methodologies we use. Based in a review of the work developed
in México, I discuss the evolution of research in mathematics education:
concerns, goals, methodologies and tasks to be performed. I hope that the
experiences of the Mexican community, presented from the author’s interpretation,
are also useful to researchers from other countries to look at their own doing.

Keywords: mathematics education, research history, research development, state
of the art, research methodologies, research approaches.

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Tensions and Challenges in the construction of a Collaborative Work among Teachers and Mathematics Education Researchers

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

Abstract: Findings are reported from a research project that focused on the
mathematical and didactic production emerging from collaboration between
elementary teachers and school principals, and mathematics education researchers.
The reported reflections are mainly oriented towards two pedagogical foci:
a) different ways to problematize the mathematical knowledge with teachers
and b) the complexity of building a collaborative process. The analysis of students’
mathematical productions shared with their teachers became another possible
way to problematize knowledge as it allowed to identify mathematical relationships
linked to learning objectives that expanded what teachers conceived as
possible in relation to the mathematical content knowledge. The study of the
tensions that arise from a collaboration amongst teachers and researchers (training-
collaboration, naturalization-problematization and adaptation to institutional
conditions-preservation of the collaboration) evidences the complexity of this
process which demands specific conditions and professional expertise.

Keywords: collaborative work between teachers and researchers; teaching practices;
mathematics education; teacher training; focus on teaching objects.

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El concepto de variable: un análisis con estudiantes de bachillerato

DOI 10.24844/EM2803.08

Autores: Heli Herrera López, Abraham Cuesta Borges, Juana Elisa Escalante Vega

Resumen: El artículo analiza los resultados de los modos de resolución de 48
estudiantes, pertenecientes a tres niveles diferentes de enseñanza en bachillerato,
cuando responden a problemas donde se manifiestan procesos de generalización
y de modelación a través de la relación del lenguaje algebraico con los lenguajes
figural, natural y aritmético. Empleando el Modelo 3uv (3 Usos de la variable)
se describen, tanto la comprensión de los diferentes aspectos que caracterizan
la variable como las actuaciones de los estudiantes cuando se enfrentan a tareas.

Palabras clave: educación matemática, dificultades, enseñanza en bachillerato,
pensamiento algebraico, variables.

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La argumentación sustancial. Una experiencia con estudiantes de Nivel Medio Superior en clases de matemáticas

DOI 10.24844/EM2803.07

Autores: Alma Alicia Benítez Pérez, Héctor Benítez Pérez,  Martha Leticia García Rodríguez

Resumen: El presente artículo analiza la argumentación sustancial realizada
por estudiantes de 15 a 17 años de Nivel Medio Superior en problemas contextualizados
evocados desde un enfoque cognitivo, en las unidades de aprendizaje
de Álgebra y Cálculo Diferencial. Este enfoque se adopta, desde el análisis
funcional y estructural del razonamiento, que permite evidenciar mecanismos
discursivos por medio de los cuales un razonamiento puede cambiar la certeza
y validez de las proposiciones. Como resultado, se identificaron elementos que
caracterizan a la argumentación sustancial en problemas contextualizados evocados
y la presencia de conectivos organizativos para mostrar la fuerza de los
argumentos y su orientación hacia el enunciado-objeto. En términos generales
los alumnos desarrollaron procesos intuitivos para establecer conjeturas y la
posibilidad de construir proposiciones para emitir afirmaciones razonadas.
La etnografía, modelo particular de la investigación cualitativa, se utilizó para
el análisis de registros y transcripciones de las clases. Es relevante hacer mención
que en el presente escrito sólo se analiza un problema contextualizado.

Palabras clave: Argumentación sustancial, análisis funcional, análisis estructural,
representaciones, problemas contextualizados evocados.

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