Resumen: El objetivo de esta investigación es identificar características de la
comprensión de las figuras geométricas en estudiantes de 6 a 9 años. Usamos
las nociones de aprehensión perceptual, operativa y discursiva de Duval, así
como la idea de concepto figural de Fischbein, para analizar la transición entre
lo perceptual y lo conceptual en el desarrollo de la comprensión de las figuras.
Entrevistamos a 30 niños (de 6 a 9 años de edad) y analizamos cómo reconocen
y relacionan diferentes atributos en las figuras geométricas para clasificarlas.
Los resultados indican tres características relevantes: (a) La existencia de un
desfase entre el uso de los nombres de las figuras prototípicas y su comprensión
conceptual; (b) Que el reconocimiento de los atributos para clasificar las figuras
no es un proceso uniforme y depende del atributo considerado; y (c) La influencia
que desempeñan las figuras prototípicas —y el dominio semántico restringido
del término usado para nombrarla— en el proceso de reconocer y clasificar las
figuras. Estos resultados proporcionan información sobre cómo los estudiantes
establecen relaciones entre lo perceptual y lo conceptual que es relevante para
caracterizar la progresión de la comprensión de las figuras geométricas.
Palabras clave: Comprensión de figuras, concepto figural, entrevistas clínicas
basadas en tareas, pensamiento geométrico, figuras geométricas.
Rubén Elizondo Ramírez
Abstract: Conceiving of space as the space around us, and the inflexibility of
Euclidean geometry to analyze physical space for over 20 centuries led to a crisis
in the conception of mathematics. Eventually this created a tension between
cognition and logic. We explain how such developments challenge learning
paradigms and mathematical inquiry for learners today. We introduce this epistemological
analysis to help us think about the nature of mathematical knowledge
and hence learning today. The direct correspondence between physical
space and mathematical structure was broken after the discovery of non-Euclidean
geometry. Similarly, this is what is occurring with the implementation of
dynamic mathematical environments that breaks the correspondence with static
With our digital model of non-Euclidean geometry we focus on the affordances
of new technological environments and the knowing of mathematics
learners. This illustrates a representational re-description of mathematics and
how it can modify our natural ontology to accommodate change and
Keywords: epistemology, model, representation, intuition, learning.
Abstract: In a pre-service Euclidean plane geometry course, the teacher tries to
encourage student participation in the collective production of proofs. During
his interaction with them, he delivers different types of messages. In this article,
we present a characterization of teacher messages, which arose while analyzing,
from a semiotic perspective, teacher and student interactions within the class.
Initially, we present the theory that underlies the semiotic analysis of the dialogues;
then we discuss in detail the proposed typology. We then describe the
context and methodology of the study. As an example of the use of the characterization,
we analyze the teacher’s messages during a real class situation.
Keywords: Semiotic activity, proof, teacher’s messages, meaning making.
Difariney González Gómez
Abstract: This article discusses teachers’ images on statistics and its teaching. It
began with their trajectories to explain the formation of images on statistics and
how through participation in a professional development program these images
were transformed. The participants of this research were 10 in-service teachers
who were responsible for teaching statistics either in elementary (6-11 years),
middle (11-15 years) or high school (15-17 years) in public schools in Medellin,
Colombia. The teachers participated in a professional development program
inspired by the social learning theory in which they designed statistical lessons,
put them into practice and reflected on the implementation. The sources of
information were their discourses in the program, autobiographical writings,
reflective writings and semi-structured interviews. The results revealed that teachers
began with static, technical, and un-transformable images of statistics and
they started to construct images of statistics as a tool for empirical inquiry.
Keywords: teacher professional development, school statistics, empirical inquiry,
social theory of learning, images about statistics.
Alfonso J. Bustamante-Santos
Rosa del Carmen Flores-Macías
Abstract: This study describes changes in the meanings of the written representation
of the division algorithm within a partition word problem in Andrea, a
student from sixth grade in a public school. Through a clinical interview, she
solves various word problems. Based on the theory of conceptual fields, we
discuss the changes in different theorems-in-act and concepts-in-act that she
uses. She reflects on the relationships between the components of the division,
and reformulates her ideas on how these relationships must be expressed. The
most significant change observed in Andrea during the interview is related to
her understanding of the division algorithm; she initially writes it based on ideas
like “the bigger number must be inside” but, at the end of the interview, she
showed understanding of the relationship between the word problem and the
written algorithm. Likewise, are observed changes in the way she conceives
the measured-magnitude relationship. Discussion analyzes the relevance of
taken into consideration student’s concepts and theorems-in-act to strengthen
the mathematic learning.
Keywords: problem solving, simple proportionality, division, conceptualization,
Abstract: Results are reported of an exploratory study of how the teaching projects
show the spatial orientation in Childhood Education in the Basque Country; to
this end, 9 educational projects from a total of 11 publishers were selected. After
analyzing the treatment of static orientation and the orientation of the subject
in real spaces in the second cycle of this educational period, we conclude that
despite the evidence of a large dispersion in the approach to it, the importance
given to orientation is limited, where in the early years there is a greater emphasis
on static orientation, and activities related with real spaces are subsequently
introducing. Finally, we show some reflections about the need to do more activities
about orientation in this Educational period.
Keywords: Mathematics education, spatial orientation, didactic projects, mathematics
competency, Childhood Education
Abstract: This paper aims to characterize the training that is offered to Mathematics’
future teachers on the grounds of the mathematical knowledge for
teaching(MKT) elemental analytical geometry. It is analyzed in terms of the
curricular contribution of the subject Geometry I, in a career of mathematics’
teachers’ training in Argentina. It seeks to identify in detail theMKT domains
through the actions of the teacher and the contents of the first classes of analytic
geometry. The research, which is carried out through a case study, has got a
qualitative focus and a mainly descriptive purpose. The results reveal that all the
domains of the MKT were activated in the observed classes. The diversity and
specificity of such activations can serve as a methodological guide for studies
that are based on the theoretical model of reference, in the line of teachers’
training in analytical geometry.
Key words: teachers’ training — mathematical knowledge for teaching — analytical
Rosa Araceli Rotaeche Guerrero
Gisela Montiel Espinosa
Abstract: We present on this paper a didactic experience which main goal was
students learning the concept of angle. To achieve it a design was conducted
by a Didactic Engineering. The preliminary analysis integrates previous research
results that highlight specificity of didactic phenomena associated to the school
concept of angle, and some historic and didactical considerations.
Even when the didactic experience was satisfactory for the teacher and her
students, from an initial evaluation is concluded that the school concept of angle
is not “learned” in a formal and absolute sense, but associated meanings to its
nature are developed; so we say students can use angularity in order to interact
with their environment.
Keywords: Angle, Meanings of angle, Angularity, Didactic Engineering, Didactic
Abstract: In this paper a teacher professional development program composed
of three strategies and whose articulating strand is problem solving is presented.
These strategies, which we call RPAction Workshop, RPContent and RPClassroom
Workshop, are intended to promote the development of mathematics
skills and strengthen mathematical knowledge, both for mathematics teachers
and students from different grade levels, thus meeting the requirements
of the new mathematics curriculum. This article describes each of these workshops,
its foundations and pilot schemes. These workshops form the core of a
Fondef development project.
Key-words: Professional teacher development, mathematics education, problem
solving, mathematical skills.
Resumen: En este artículo se presenta un programa de desarrollo profesional
docente compuesto por tres estrategias y cuyo eje articulador es la resolución
de problemas. Estas estrategias, que denominamos Taller RPAcción, Taller RPContenido
y Taller RPAula, tienen como propósito promover el desarrollo de las
habilidades matemáticas y fortalecer los conocimientos matemáticos, tanto de
los docentes de matemática como de los estudiantes de los diferentes niveles
escolares, atendiendo así a los requerimientos del nuevo currículo de matemática.
En el artículo se describe cada uno de estos talleres, sus fundamentos y experiencias
piloto realizadas. Estos talleres constituyen el núcleo central de un
proyecto Fondef en desarrollo.
Palabras clave: desarrollo profesional docente, educación matemática, resolución
de problemas, habilidades matemáticas.