Development and application of statistical notions from professional practice: the case of social workers

Daniel Eudave Muñoz

Abstract: The purpose of this research is to analyze the social workers’ activities involving
the generation and processing of numerical and statistical data and how they deal with
these tasks and how they solve them. The work focuses on the analysis of activities,
situations and conceptual structure, following the approach of the French educational
training (didactique professionnelle). We report the use of statistics for the social workers
in Aguascalientes City, Mexico, and the pragmatical concepts that give meaning and
order to their professional knowledge.

Keywords: pragmatical concept, situated learning, education and work, statistical
education, social work.


Research on fractions: experiences inspired in the teaching experiment methodology

José Luis Cortina

Abstract: We discuss the contributions to the field of fractions of four research projects,
conducted by the author and his colleagues. Several aspects of the design research
methodology were central to the implementation of these projects. The contributions
include specifying the big ideas of learning fractions, identifying viable starting points,
and developing instructional innovations to support students’ understanding of this

Keywords: fractions, design experiments, proportional reasoning.



Affection and gender differences in low achieving mathematics students in middle school

Sonia Ursini

Abstract: Attitudes, beliefs and self-confidence in mathematics regarding 192 ninthgrade
low achievers (96 girls and 96 boys) were studied in Mexico City. The sociocultural
and economic environment where these affects appear is stressed. In spite of economic
hardships, cultural gaps, lack of familiar and institutional supports, violence inside and
outside mathematics classroom, and traditional mathematics teaching, these students,
without gender differences, showed a good potential for mathematics learning and the
majority considered that there are no gender differences in the capability to learn this
subject. Gender differences were found in attitudes towards mathematics in general
(more polarized, or positive or negative, among women) and towards specific mathematics
areas. In general, their self-confidence was low, although more polarized for women;
however, men were more self-confident than women.

Keywords: attitudes, beliefs, self-confidence, mathematics achievement, low mathematics
performance, gender and mathematics.


Division of fractions as a multiplicative comparison based on methods invented by students

Alfinio Flores Peñafiel

Abstract: We present several methods invented by students in grades 5th through 8th
to solve division of fractions problems. For each method we discuss how the teacher
can help students develop their understanding of multiplicative comparison of fractions,
by emphasizing fundamental mathematical principles that allow students generalize,
extend, and relate their methods with other methods. The methods presented are repeated
subtraction and interpretation of the remainder; use of the multiplicative identity and
inverses; division as missing factor; inverse and direct proportional thinking; division of
fractions as composition of operations; and division of fractions as a ratio between two

Keywords: fractions, division, multiplicative comparison, proportional reasoning, students



Understanding of the concepts of first order differential equation and solution

María Trigueros

Abstract: The goal of this paper is to describe how the use of modeling in a differential
equations course contributes to foster a change in students’ understanding of first order
differential equation and solution set. The design and results of a research experience
based on APOS theory are presented. The focus of the paper is the description of those
results related with the use of different representations and others that seem to play an
important role in the changes observed in students’ understanding.

Keywords: modeling, differential equations, APOS theory, representations, understanding.


Intuiting and formalizing: coextensive processes

Luis Moreno Armella

Abstract: There is a force that goes through the teaching of Calculus: The tension between
the intuitive and the formal. Calculus continues to be taught as if it were natural
to introduce the study of change and accumulation by means of formalized concepts
known as the mathematics of e and d. It is frequently considered as a failure that
“students still seem to conceptualize limits via the imagination of motion.” This kind of
assertions shows the tension created by traditional teaching between students’ intuitions
and a misdirected formalization. The internal connections of the intuition of change and
accumulation are not correctly translated into that arithmetical approach of e and d.

Keywords: intuition, formalization, infinitesimal, analog cognition, symbol, metaphor,
embodied knowledge, digital medium, mediation.



The enacting perspective in mathematics education: doing is always knowing

María Dolores Lozano

Abstract: In this article I present the enactivist perspective (Maturana and Varela, 1984)
as a theoretical alternative that can be used to investigate and clarify the teaching and
learning of mathematics. The biological roots of the theory are introduced, including fundamental
ideas such as autopoiesis and structural determinism. In addition, the theory’s
ideas on cognition as embodied action are discussed and some of its uses in mathematics
education are exemplified. Finally, I conclude by inviting the reader to use the ideas
exposed in order to investigate, in a complex manner and considering a multiplicity of
dimensions, the phenomena related to the teaching and learning of mathematics.

Keywords: enactivism, autopoiesis, embodied cognition, mathematics education.



Mathematical language games from different forms of life: Wittgenstein y Foucault’s contributions to think about mathematics education

Gelsa Knijnik

Abstract: The paper aims at discussing aspects of mathematics education, which is
understood as the educational processes that happen inside or outside the school
space. Specifically it problematizes the no recognition of the existence of other ways to
mathematize, different from those usually taught in school. This problematization has
as theoretical bases what is called Ethnomatematics Perspective, a theoretical toolbox
built with the interlocution of Wittgenstein’s late ideas and the theorizations of Michel
Foucault. From the empirical side, the paper presents examples of mathematical language
games of different forms of life. The discussion done shows the productivity of
such Ethnomathematics Perspective to increase the possibilities of school mathematics.

Keywords: ethnomatematics perspective, Wittgenstein, Foucault, mathematics education.




Phenomenology, Praxis, and the Question of Mathematical Objects

Luis Radford

Abstract: In this article, I discuss some aspects of the manner in which phenomenology
has dealt with the question of the nature of objects of knowledge and the knowability of
such objects. I focus on Kant’s phenomenology and consider in particular some ontological
presuppositions that make Kant’s phenomenology both Platonist and anti-Platonist.
Then, I make a brief incursion into Hegel’s phenomenological approach and Marx’s
critique of Hegel, Kant, and German idealism in general. In the last part of the article,
I comment on Marx’s idea of praxis as an entirely new path to tackle the question of
the nature and knowability of objects of knowledge. I discuss some of the implications
of such an idea for mathematics education. I end up sketching a Hegelian dialectic
materialist concept of knowledge that provides room for understanding knowledge as
something ineluctably embedded in cultural praxis.

Keywords: phenomenology, sense, sensation, mathematical objects, praxis, dialectical



From the knowledge of experience to the knowledge in the experience: 25 years of research on mathematical knowledge and late schooling in Mexico.

Alicia Ávila

Abstract: This article presents a review of research conducted and published in Mexico
about the mathematical knowledge of the experience and its links with the school
mathematical knowledge. Information sources are the papers published in research or
adult education journals and doctoral theses on the topic identified. Various aspects of
research defined the object of study is shown, which results in its enlargement and complexity.
This enlargement and complexity are necessarily accompanied by the inclusion
or neglect of data collection tools and analysis data.

Keywords: mathematical knowledge from experience, mathematical knowledge in the
experience, school mathematical knowledge, basic education for adult persons, research