“A minute for Mathematics”. An Experience of Fun, Learning and Outreach Through the Exploration of Numeric Patterns

Romy Adriana Cortez Godinez

Abstract. This paper shows an experience of popularization of mathematics
through the exploration of numeric patterns with concrete material; it is based
on games benefits (De Guzman, 2007) and on the use of inductive reasoning
to construct generalizations (Osorio, 2012; Cañadas, Castro y Castro, 2008). The
results reveal the use of strategies in the search for patterns, collaborative work and motivation toward math. It is concluded that the proposal is a valuable
scenario for the purposes of generalization and dissemination of science.

Key words: number patterns, games and disclosure.

DOI: 10.24844/EM2903.08

Proposal for Treating Global Interpretation of the Quadratic Function Using the Software GeoGebra

Ana Luisa Gómez-Blancarte
Rebeca Guirette
Felipe Morales-Colorado

Abstract. Based on Duval’s (1988) interpretation about the treatment of graphical
representations, this paper presents a proposal to the global interpretation of
the quadratic function f (x) = ax 2 + bx + c by using the Software GeoGebra. In
order to illustrate the relevance of the proposal in the teaching, a case study
of a group of high school students is presented. The results show the potential
of the software for a congruence analysis among graphic and algebraic registers
of representation and its effects on qualitative recognition of the association
of visual variables and significant symbolic units.

Keywords: Quadratic function, global interpretation, coordination among registers,
visual variables and significant symbolic units

DOI: 10.24844/EM2903.07

Inquiring about the history of mathematical inequalities

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

Analysis of Mathematical Teacher’s Decisions in his Classroom Management

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

School Realities in Mathematics Classes

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

Approaching Common Knowledge of Content for Teaching Probability from the Didactic-mathematical Knowledge Model

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

To what Kind of Mathematical Problems are Calculus Students Exposed? A Textbook Analysis

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

Mathematics Teaching Based on the Study and Research Path: Mathematical Teaching Indicators of the “Dialectics”

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

“Un minuto para matemáticas”. Una experiencia de diversión, aprendizaje y divulgación al explorar patrones numéricos

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

Propuesta para el tratamiento de interpretación global de la función cuadrática mediante el uso del software GeoGebra

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