Luis Moreno-Armella

The articulation of symbol and mediation in mathematics education

In this paper we include topics which we consider are relevant building blocks to design a theory of mathematics education. In doing so, we introduce a pretheory consisting of a set of interdisciplinary ideas which lead to an understanding of what occurs in the “central nervous system”—our metaphor for the classroom and eventually in more global settings. In particular we highlight the crucial role of representations, symbols viewed from an evolutionary perspective and mathematics as symbolic technology in which representations are embedded and executable.

Constructivism and mathematical education

Philosophical postures and epistemological theories relative to mathematical knowledge have a determining influence on mathematical education, consequently, the following questions are pertinent: (a) What sort of didactics does a given conception of math and mathematical knowledge lead to?; and (b) What conception of math and mathematical knowledge does a given type of educational practice follow from?. We will attempt to provide some elements of analysis which may allow us to contribute to finding an answer to these questions by contrasting two different ways of conceiving mathematics; one of them leads us to regard math as an object of teaching and the other implies that math is an object of learning.

Symbols and Mediation in Mathematics Education

In this paper we discuss topics that are relevant for designing a theory of mathematics education. More precisely, they are elements of a pre-theory of mathematics education and consist of a set of interdisciplinary ideas which may lead to understand what occurs in the central nervous system—our metaphor for the classroom, and eventually, in larger educational settings. In particular, we highlight the crucial role of representations, the mediation role of artifacts, symbols viewed from an evolutionary perspective, and mathematics as symbolic technology.

Technological Supports for Mathematical Thinking and Learning: Co-action and Designing to Democratize Access to Powerful Ideas

The enterprise of understanding and supporting processes of mathematical cognition is both epistemologically deep and politically urgent. We cannot ignore that new technology-mediated learning environments have the potential to democratize access to powerful ideas. The importance of technology in this respect is bound up with the essential nature of mathematical objects as symbolic entities that can only be expressed and conjured up through the mediation of representations. A key question for the design of technology-enhanced learning environments is whether the cognitive tools—material and digital-symbolic—that have been developed in recent decades might offer learners access to modes of activity with disciplinary structures that have historically been achievable only by ‘maestros’ of the discipline. In this article we elaborate the construct of “co-action” as a means of describing humans’ mathematical interactions with the support of such tools.

The Use of Digital Technology to Frame and Foster Learners’ Problem-Solving Experiences

The purpose of this chapter is to analyze and discuss the extent to which the use of digital technology offers learners opportunities to understand and appropriate mathematical knowledge. We focus on discussing several examples in which the use of digital technology provides distinct affordances for learners to represent, explore, and solve mathematical tasks. In this context, looking for multiple ways to solve a task becomes a powerful strategy for learners to think of different concepts in problem-solving approaches. Thus, the use of a dynamic geometry system such as GeoGebra becomes important to represent and analyze tasks from visual, dynamic, and graphic approaches.

Impact of Classroom Connectivity on Learning and Participation

Our study reports the development of the SimCalc environment that integrates interactive and connected mathematical representations with the latest affordances in wireless connectivity. We describe the development of several instruments to measure the interaction between learning and participation in terms of changes in attitude over time. In addition to empirical data from a quasi-experimental study, we qualify our findings through analysis of discourse patterns in specific forms of participation and explain how the communication infrastructure of an algebra classroom can be radically enhanced.

Mathematics: a historical and didactic perspective

Two related statements are increasingly accepted in educational circles: (1) the teachers conception of mathematics determines, to a great extent, the way (s)he teaches; and (2) mathematics should be conceived of as an activityand not (only) as a product.However the majority of teachers share in practice a position that can be called an expository pedagogy.That is, they conceive of mathematics as a product to be transmitted to the student. The persistence of this pedagogical approach is explained by using a historical and epistemological perspective which sheds some light on the two theses expressed at the beginning of this abstract, and an alternative conception of teaching and learning more in accord with constructivist perspectives is suggested.

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

ABSTRACT This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.

Preface to Part XI

The school is the institution conceived to communicate, re-produce, and appropriate the knowledge that is socially important. The curriculum is the basic instrument to reach this goal. Of course, socially important depends on a long list of factors that includes political, cultural and historical factors as well. Mathematics education has been an important institutional goal. The centrality of this goal has been captured in the field of research we call math education.