María Trigueros

Integrating Technology in the Primary School Mathematics Classroom: The Role of the Teacher

In this chapter, we analyse the role of the teacher when using digital resources in the primary school mathematics classroom in Mexico and its relation to students’ mathematical learning. We carry out this analysis through the use of an instrument that we developed in which we relate five different aspects of the role of the teacher we consider important with the three different uses of technology classified by Hughes (Journal of Technology and Teacher Education, 13(2), 277–302, 2005) namely replacement, amplification and transformation. We use an enactivist perspective that considers learning as effective action in a given context (Maturana, H., & Varela, F. The tree of knowledge: The biological roots of human understanding, Revised Edition, Boston: Shambhala, 1992) in order to describe the way in which differences both in the uses of technology and in the role the teacher assumes in the classroom contribute to creating classroom contexts in which mathematical learning is promoted to different degrees.

Research on the Learning and Teaching of Algebra

For many years, the mathematics education community has investigated the difficulties students have with algebra. Different aspects of algebraic thinking, considered to be fundamental to overcome those difficulties, have been analyzed by researchers using a variety of theoretical frameworks.

Teachers Teaching Mathematics with Enciclomedia: A Study of Documentational Genesis

In this chapter, following the documentational approach (Gueudet & Trouche Educational Studies in Mathematics, 71, 199–218, 2009), we examine teachers’ development through the analysis of their appropriation and transformation of resources from “Enciclomedia”, a Mexican national project. To do this, we analyse information about three teachers’ interactions with “Enciclomedia” obtained from different sources: lesson observation, analysis of written materials, and interviews. The three teachers changed their practice in different ways as a result of using “Enciclomedia”. Each produced different kinds of documents in the process of incorporating the digital resources into their activities and in this process they transformed these material resources by using them in particular ways. It is possible to see that the introduction of the program “Enciclomedia” can influence teaching practices and documentational genesis in powerful and different ways, especially when it is accompanied by reflection and discussion with fellow teachers and researchers.

Learning Linear Algebra Using Models and Conceptual Activities

In this chapter, an innovative approach, including challenging modeling situations and tasks sequences to introduce linear algebra concepts is presented. The teaching approach is based on Action, Process, Object, Schema (APOS) Theory. The experience includes the use of several modeling situations designed to introduce some of the main linear algebra concepts. Results obtained in several experiences involving different concepts are presented focusing on crucial moments where students develop new strategies, and on success in terms of student’s understanding of linear algebra concepts. Conclusions related to the success of the use of the approach in promoting student’s understanding are discussed.

Thematization of derivative schema in university students: nuances in constructing relations between a function's successive derivatives

ABSTRACT This study is part of a more extensive research project that addresses the understanding of the derivative concept in university students with prior instruction in differential calculus. In particular, we focus on the analysis of students’ responses to a sequence of tasks that require a high level of understanding of the concept, and complement this information with clinical interviews. APOS (Action-Process-Object-Schema) theory and the configuration of the derivative concept that is characterized by: mathematical elements, logical relations and the representation modes that students use to solve a task were used in the analysis of students’ responses. The results obtained suggest that thematizing the derivative schema is difficult to achieve. In addition, nuances were observed in responses given by those students who succeeded, indicating differences in the construction of relations between the successive derivatives of a function.

La separación ciega de fuentes: un puente entre el álgebra lineal y el análisis de señales

El objetivo de este articulo es presentar un analisis praxeologico enmarcado en la Teoria Antropologica de lo Didactico (TAD) de un metodo proveniente de la ingenieria conocido como Separacion Ciega de Fuentes (BSS). En el metodo estan presentes praxeologias que pueden trasponerse a los cursos iniciales de matematicas dentro de una formacion de ingenieros, concretamen- te dentro del curso de Algebra Lineal. El analisis muestra que la BSS tiene potencial para generar actividades de modelacion que conecten la teoria mate- matica con la practica ingenieril. Se presenta, ademas, una propuesta inicial para una actividad de estudio e investigacion basada en la BSS.

Use of APOS Theory to Teach Mathematics at Elementary School

Throughout the first half of the 1990s, the mathematics team of the Center for Educational Technology, Tel-Aviv, Israel(CET), set out to revise the team’s existing materials for teaching mathematics in Israeli elementary schools (Grades 1–6, ages 6–12). One important aspect of the revision was to introduce the ideas of Piaget and APOS Theory into the teaching sequences. An area of particular interest was the teaching of fractions in grades 4 and 5.

The Teaching of Mathematics Using APOS Theory

This chapter is a discussion of the design and implementation of instruction using APOS Theory. For a particular mathematical concept, this typically begins with a genetic decomposition, a description of the mental constructions an individual might make in coming to understand the concept (see Chap. 4 for more details). Implementation is usually carried out using the ACE Teaching Cycle, an instructional approach that supports development of the mental constructions called for by the genetic decomposition.

Totality as a Possible New Stage and Levels in APOS Theory

The focus of this chapter is a discussion of the emergence of a possible new stage or structure and the use of levels in APOS Theory. The potential new stage, Totality, would lie between Process and Object. At this point, the status of Totality and the use of levels described in this chapter are no more than tentative because evidence for a separate stage and/or the need for levels arose out of just two studies: fractions (Arnon 1998) and an extended study of the infinite repeating decimal \( 0.\bar{9} \) and its relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. in press). It remains for future research to determine if Totality exists as a separate stage, if levels are really needed in these contexts, and to explore what the mental mechanism(s) for constructing them might be. Research is also needed to determine the role of Totality and levels for other contexts, both those involving infinite processes and those involving finite processes. It seems clear that explicit pedagogical strategies are needed to help most students construct each of the stages in APOS Theory and that levels which describe the progressions from one stage to another may point to such strategies. Moreover, observation of levels may serve to help evaluate students’ progress in making those constructions.

Mental Structures and Mechanisms: APOS Theory and the Construction of Mathematical Knowledge

The focus of this chapter is a discussion of the characteristics of the mental structures that constitute APOS Theory, Action, Process, Object, and Schema, and the mechanisms, such as interiorization, encapsulation, coordination, reversal, de-encapsulation, thematization, and generalization, by which those mental structures are constructed.