Autor: somidem

Silvia Bernardis
Liliana Nitti
Sara Scaglia

Abstract. In this article, we inquire about the use of inequalities in the history
of mathematics, so as to help improve the quality of teaching this theme. The
historical analysis of a concept provides clues to interpret the productions,
conceptions, and difficulties of students and to design experiences that favor
their understanding.
From a phenomenological study (Freudenthal, 2002) we characterize mathematical
phenomena that are organized by the concept of inequalities. In this
article, we present some evidence of the manifestation of these phenomena in
the history of mathematics.
We reflect on the type of experiences that are necessary for students, for
them to create “mental objects” of inequality in an elementary stage, that will
enable them to study this advanced mathematics topic, in optimal conditions.

Keywords: historical inquiry, mathematical inequalities, organized phenomena,
concept, frames.

DOI: 10.24844/EM2903.06

Diego Garzón Castro1

Abstract. This paper analyzes the decisions made by two teachers during “teaching
moments” in which pedagogical opportunities emerge. These correspond
to examples of classroom discourse in which the mathematical thinking of the
student and the construction of mathematical meanings are made manifest.
For this purpose, the MOST-Noticing instrument was designed and evaluated,
allowing for such an analysis. An exploratory case study was carried out that
included the observation of videotaped classes of high school teachers. For the
analysis, we took into account: professional observation of the teaching of
mathematics, emphasizing the teacher’s ability to respond to the mathematical
understanding of the student, and the study of teaching moments that relate
the student’s mathematical thinking, the significant from the mathematical point
of view and the pedagogical opportunities. In the analysis, two teaching moments
were recognized that allowed to characterize the decisions in relation to
the actions from the applied instrument and the constant comparison.

Keywords: Teacher decisions, teachable moments, student’s mathematical thinking,
mathematical discourse in the classroom, teaching practice .

DOI: 10.24844/EM2903.05

Alfonso Jiménez Espinosa
Alba Soraida Gutiérrez Sierra

Abstract. The article presents results of an investigation that had as objective
to analyze realities of classes of mathematics teachers in an institution of basic
and average education. In the theoretical reference are considered aspects like
beliefs, conceptions, interactions in the classroom, pedagogical practices and
didactic models. The research was developed under a qualitative approach,
where it is interesting to establish the realities that are lived inside the classrooms
in the mathematics classes, it tries to emphasize complex understandings
and relationships that are given in the classrooms. There is still a traditional
didactic approach, with constructivism, deriving from conceptions of mathematics.
Teaching is synonymous of exposition of contents, with few actions that
favor the development of mathematical thinking. The results were analyzed
jointly with teachers, especially on possible reasons for these practices, and
how they could be improved.

Key Words: School realities, mathematics, didactic models, (re)signification.

DOI: 10.24844/EM2903.04

Claudia Vásquez Ortiz
Ángel Alsina

Abstract. This paper analyzes the didactic-mathematical knowledge, to teach
probability, of Elementary Scool teachers, focusing specifically on the subcategory
of common knowledge of content. For this, the mathematical practices of
93 active Chilean teachers were analyzed from a questionnaire composed of 7
items that evaluate partial and initial aspects of this knowledge. The results
show an insufficient level of knowledge, with 4.75 average points of correct
answers over 14. It is concluded that it is necessary to design a training program
that allow improving the level of knowledge to teach probability in classrooms.

Key words: Mathematical and Didactic Knowledge of Teacher, Probability, Teachers,
Elementary school.

DOI: 10.24844/EM2903.03

Adriana Berenice Valencia Álvarez
Jaime Ricardo Valenzuela González

Abstract. Given the importance of solving mathematical problems placed in
real contexts and given the influence of the textbook in the teaching and
learning of mathematics, it is important to study to what kind of mathematical
problems students are exposed. In this article, we present a distinction between
conventional mathematical problems and mathematical modeling problems in
terms of nine aspects related to the problem statement and its solution (Green
and Emerson, 2010). Using these nine criteria, we analyzed and classified a
total of 188 examples and 1,114 exercises found in the unit about functions in
a sample of Calculus textbooks from Mexican and American publishing companies.
As a result, the analysis showed that between 65% and 87% of the
examples and exercises in the books studied were classified as completely
conventional problems. Modeling problems were found in only three of the
books analyzed and these correspond to merely 1% or 2% of the examples and
exercises of the unit.

Keywords: Analysis of textbooks, mathematical modeling problems, conventional
problems, solving problems, Calculus.

DOI: 10.24844/EM2903.02

Verónica Parra
María Rita Otero

Abstract. This paper proposes and describes an analysis methodology based
on the formulation of a set of didactic-mathematical indicators of the “dialectics”
of study and research. These indicators were constructed based on the data
obtained from de design, implementation and evaluation of a study and research
path (SRP). The SRP’s initial questions are connected to market equilibrium in
a microeconomic model of the supply and demand. Two implementations were
developed in the last year of the Argentine secondary level (16-17years) (N = 60).
The teacher proposed to the students the following type of questions: How to
determine the market equilibrium? If the parameters of the model are modified:
How to describe the variation of the point of equilibrium?
How much does the
point of equilibrium change exactly in each case? We conclude that the most
identified dialectics are: the individual and collective, theme and out of theme,
study and research, praxeological analysis-synthesis and didactic of analysis-
synthesis, and the dialectic of black boxes and clear boxes.

Keywords: Market balance; Study and Research Path; Anthropological Theory
of the Didactic; dialectics; didactic-mathematical indicators.

DOI: 10.24844/EM2903.01

Romy Adriana Cortez Godinez

Resumen. El presente texto muestra una experiencia de divulgación de las
matemáticas a través de la exploración de patrones numéricos con material
concreto; se fundamenta en las bondades del juego (De Guzmán, 2007) y en
el empleo del razonamiento inductivo en la construcción de generalizaciones
(Osorio, 2012; Cañadas, Castro y Castro, 2008). Los resultados revelan el empleo
de estrategias en la búsqueda de patrones, trabajo colaborativo y motivación
hacia las matemáticas. Se concluye que la propuesta es un valioso escenario
para las tareas de generalización y la divulgación de la ciencia.

Palabras clave: patrones numéricos, juego, divulgación.

DOI: 10.24844/EM2903.08

Ana Luisa Gómez-Blancarte
Rebeca Guirette
Felipe Morales-Colorado

Resumen. Basados en la vía de interpretación global que Duval (1988) sugiere
para el tratamiento de las representaciones gráficas, en este artículo se
presenta una propuesta de interpretación global para la función cuadrática
f (x) = ax 2 + bx + c , mediante el uso del software GeoGebra. A fin de ilustrar la
pertinencia de la propuesta en la enseñanza, se presenta un estudio de caso
con un grupo de estudiantes de Educación Media Superior del sistema de
Telebachillerato. Los resultados muestran la potencialidad del software para
realizar un análisis de congruencia entre los registros de representación gráfica
y algebraica de la función y reconocer cualitativamente la asociación de
las variables visuales del registro gráfico y las unidades simbólicas significativas
del registro algebraico.

Palabras clave: Función cuadrática, interpretación global, coordinación entre
registros, variables visuales y unidades simbólicas significati vas.

DOI: 10.24844/EM2903.07

Silvia Bernardis
Liliana Nitti
Sara Scaglia

Resumen. En este artículo presentamos una indagación histórica sobre los
usos de las desigualdades en la historia de la matemática, realizada en el
marco de una investigación en torno al tema mencionado, cuyo propósito es
contribuir a mejorar la calidad de su enseñanza. Esto porque el análisis histórico
de un concepto proporciona indicios para interpretar las producciones,
concepciones y dificultades de los estudiantes y para diseñar experiencias que
favorezcan su comprensión.
A partir de un estudio fenomenológico (Freudenthal, 2002) caracterizamos
fenómenos matemáticos organizados por el concepto de desigualdad. En este
artículo presentamos algunas evidencias de la manifestación de esos fenómenos
en la historia de la matemática.
Reflexionamos respecto del tipo de experiencias que es necesario ofrecer
a los estudiantes para la construcción de buenos “objetos mentales” de la

desigualdad en la etapa elemental, para abordar en condiciones óptimas el
estudio de la matemática avanzada.

Palabras clave: indagación histórica, desigualdad matemática, fenómenos organizados,
concepto-marcos.

DOI: 10.24844/EM2903.06

Diego Garzón Castro

Resumen. Esta investigación analiza las decisiones que toman dos profesores
en “momentos de enseñanza” en los que emergen oportunidades pedagógicas.
Éstas, corresponden a ejemplos del discurso en el aula en las que se hace
manifiesto el pensamiento matemático del estudiante y la construcción de
significados matemáticos. Con esta finalidad, se diseñó y se evaluó el instrumento
MOST-Noticing que permite dicho análisis. Se llevó a cabo un estudio
de casos exploratorio que incluyó la observación de clases video-grabadas de
profesores de secundaria. Para el análisis, se tuvieron en cuenta: la observación
profesional de la enseñanza de las matemáticas, enfatizando en la habilidad
del profesor para responder a la comprensión matemática del alumno, y el
estudio de momentos de enseñanza que ponen en relación el pensamiento
matemático del alumno, lo significativo desde el punto de vista matemático y
las oportunidades pedagógicas. En el análisis se reconocieron dos momentos de enseñanza que permitieron caracterizar las decisiones en relación con las
acciones a partir del instrumento aplicado y la comparación constante.

Palabras clave: Decisiones del profesor, momentos de enseñanza, pensamiento
matemático del estudiante, discurso matemático en clase, prácticas de enseñanza.