Ana Isabel Sacristán

Technology-Driven Developments and Policy Implications for Mathematics Education

The advent of technology has done more than merely increase the range of resources available for mathematics teaching and learning: it represents the emergence of a new culture--a virtual culture with new paradigms--which differs crucially from preceding cultural forms. In this chapter, the implications of this paradigm shift for policies concerning learning, curriculum design, and teacher education will be discussed. Also, the ubiquitous possibility of emergence of ever-new forms of technology brings about both new opportunities for learning and collaborative work (involving students and teachers), as well as potential dangers. Policy measures may give priority to technological access and developments, over the intellectual growth of learners and the professional development of teachers--which should be more demanding goals of mathematics education. Such policy issues will be discussed.

The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

Some Regional Developments in Access and Implementation of Digital Technologies and ICT

Access to and implementation of digital technologies for mathematics teaching and learning across and within countries and regions display similarities and differences. This chapter is derived from regional presentations made at the ICMI Study 17 Conference held in Vietnam in December 2006. The descriptions of the situations in four countries (Russia, Hong Kong, Vietnam, South Africa) and one region (Latin-America) give a sense of the similarities against the general background of a global goal for schooling in the twenty-first century. The complex issue of universal access to digital technologies for meaningful mathematics learning, it is suggested, requires concerted efforts to address a host of mitigating factors.

Introduction to Section 2

This introduction sets the scene for the volume section 2 on the theme of learning and assessing mathematics with and through digital technologies. It first describes the section’s points of departure. Then each of the chapters of the section is briefly addressed. The introduction ends with a short reflection on the section as a whole, noting that the major content emphases are on algebra and geometry, with only limited attention to calculus, statistical reasoning, and proof. In closing, we call for a closer relationship between mathematics education research and educational science in general.

Pyramids in Logo: A School Project in ‘Search’ of the Fourth Dimension

In this paper we present a school project where students constructed three-dimensional pyramids using the Logo programming language, comple-mented with paper-and-pencil, dynamic geometry (Cabri) and spreadsheet (Excel) investigations. The aim of this project was to give, through a fun and meaningful way, and using a constructionist approach, junior secondary students (12-14 year olds), early access to advanced topics such the applications of the Pythagorean Theorem and of trigonometric functions, as well as three-dimensional work, while at the same time covering one of the themes included in the curriculum for this age-group (the pyramid).

The Mexican National Programs on Teaching Mathematics and Science with Technology: The Legacy of a Decade of Experiences of Transformation of School Practices and Interactions

Here we give an overview of the Mexican experience of a national program, begun in 1997, of gradual implementation of computational tools in the lower secondary-school classrooms (children 12-15 years-old) for mathematics and science. This project illustrates, through the benefit of long-term hindsight, the successes and difficulties of large-scale massive implementation of technologies in schools. The key factors for success and for transforming school practices seem to be: adequate planning, gradual implementation, continuous training and support, and enough time (years) for assimilation and integration.

On Visual and Symbolic Representations

Activities intended to enhance the theoretical framework and the methodology of mathematical education have intensified during the last decade. We have witnessed, for instance, the publication of diverse cognitive and epistemological studies - in both books and journals - converging to constructivist stances. See [6], [9]. However, it is still necessary to try to make clear in our own work, the meaning of the terms we are using. At present, the general discussion involves the mise en scene of Representation Theory in the context of computers.