An essential tension in mathematics education

There is a problem that goes through the history 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 the formalized ideas and concepts known as the mathematics of ε and δ. It is frequently considered as a failure that “students still seem to conceptualize limits via the imagination of motion.” These kinds of assertions show the tension, the rift created by traditional education between students’ intuitions and a misdirected formalization. In fact, I believe that the internal connections of the intuition of change and accumulation are not correctly translated into that arithmetical approach of ε and δ. There are other routes to formalization which cohere with these intuitions, and those are the ones discussed in this paper. My departing point is epistemic and once this discussion is put forward, I produce a narrative of classroom work, giving a special place to local conceptual organizations.     

Co-action with digital technologies

Circa1895, James M. Baldwin introduced a powerful view regarding Darwinian Evolution. Baldwin suggested that behavioral flexibility could play a role in amplifying natural selection because this ability enables individuals to modify the environment of natural selection affecting the fate of future generations. In this view, behavior can affect evolution but, and this is crucial, without claiming that responses to environmental demands acquired during one’s lifetime could be passed directly to one’s offspring. In the present paper, we want to use this view as a guiding metaphor to cast light on understanding how students and teachers can utilize the environment of digital technologies to scaffold their activities. We present examples of activities from geometry and algebra in high school settings that illustrate the potential role that certain technologies can have in transforming classroom interaction and work.     

Intersecting representation and communication infrastructures

We analyze the intersection of new forms of representation infrastructures in a particular dynamic mathematics software (SimCalc MathWorlds^®) with the affordances of available communication infrastructures (both hardware and software). We describe the fundamental design principles from a software and curriculum perspective of why these two infrastructures can be overlapped in educational environments for important and meaningful learning outcomes. The products of this intersection result in new modes of expression (in terms of gesture, deixis and informal/formal registers), classroom identity formation in mathematically-meaningful ways, and pedagogy in terms of activity structure and intentionality. We exemplify the results of such intersection on classroom learning, participation and motivation.     

Problem solving and the use of digital technologies within the Mathematical Working Space framework

ZDM – Mathematics Education - The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological...     

Structural stability and dynamic geometry: Some ideas on situated proofs

In this paper we survey the historical and contemporary connections in mathematics between classical “conceptual” tools versus modern computing tools. In this process we highlight the interplay between the inductive and deductive, experimental and theoretical, and propose the notion of Situated proofs as a didactic tool for the teaching of geometry in the 21^st century.     

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

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.     

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.