Nigel Calder

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.

Forming Conjectures Within a Spreadsheet Environment

This paper is concerned with the use of spreadsheets within mathematical investigational tasks. Considering the learning of both children and pre-service teaching students, it examines how mathematical phenomena can be seen as a function of the pedagogical media through which they are encountered. In particular, it shows how pedagogical apparatus influence patterns of social interaction, and how this interaction shapes the mathematical ideas that are engaged with. Notions of conjecture, along with the particular faculty of the spreadsheet setting, are considered with regard to the facilitation of mathematical thinking. Employing an interpretive perspective, a key focus is on how alternative pedagogical media and associated discursive networks influence the way that students form and test informal conjectures.

Apps: Appropriate, Applicable, and Appealing?

Mobile technologies, and the apps associated with them, are an increasing presence and influence in everyday life. This is also reflected in their burgeoning incidence as a digital pedagogical medium, both in classrooms and for private use. This chapter considers the implications of this growth and how they might best be utilized for the enhancement of mathematical thinking. It traverses present research concerned with digital technologies in mathematics education, their attendant affordances, and digital games, before casting a discerning eye with regards to the appropriateness, applicability and appeal of apps in the teaching and learning of mathematics. While some concerns are raised, the considerable potential of using apps for mathematics learning is clearly evident. How might we optimise this potential?

Transformations of teaching and learning through digital technologies

This chapter is a critical synthesis of research related to the transformations that take place when digital technologies are incorporated into teaching and learning practices. In developing this synthesis, research from all levels of education was reviewed with a focus on the opportunities digital technologies offer for cognitive, pedagogical, affective and professional change. The chapter is structured in alignment with Pierce and Stacey’s (Pierce and Stacey, Int J Comput Math Learn 15(1):1–20 2010) map of pedagogical opportunities in which three dimensions for educational transformation were identified: tasks, classroom, and subject. A discussion of future directions for research into technology enhanced mathematics education concludes the review.

How Might Apps Reshape the Mathematical Learning Experience

This chapter reports on how the use of mathematics apps has the potential to reshape the learning experience , a particular aspect of learning with apps that emerged from a research project examining the ways mobile technologies are used in primary-school mathematics. The chapter will consider a number of key themes related to student learning that have emerged through the research. When using some of the apps in the study, students used different digital tools within the app to solve word problems, while the affordances, including multi-representation, dynamic and haptic, made the learning experience different from when using pencil-and-paper technology. Other themes that were identified included: collaboration, socio-material assemblages, and personalisation. All of these appeared influential in the development of mathematical thinking. While the affordances of the mobile technologies are important, the teacher’s pedagogical approach and the dialogue that the apps evoked were central in the learning.

Mobile Technologies: How Might Using Mobile Technologies Reshape the Learning and Teaching of Mathematics?

As our attention moves to the opportunities and constraints that mobile technologies (MT) might afford, app developers, teachers and researchers have become more adept at identifying and enacting opportunities for enhancing mathematical thinking. These opportunities emerge through the various environments, both hardware (i.e., tablets) and software (i.e., applications), and the mathematical activity that these facilitate. The features of MT, for instance the ability to use in-built video and audio tools, allows users to capture authentic data in their everyday world and use the data for modelling, or statistical inference.

Reshaping the learning experience through apps: Affordances

This paper reports on how the affordances of the app Math Shake reshaped the learning experience—an aspect of a research project examining the ways mobile technologies are used in primary-school mathematics. Students used different digital tools within the app to solve word problems, while the affordances, including simultaneous linking, focussed constraint, creative variation, dynamic and haptic, made the learning experience different from when using pencil-and-paper technology. However, while the affordances of the mobile technologies are important, the teacher’s pedagogical approach was also influential in the learning.

Using spreadsheets with 8 year olds: Easy to visuatise?

Despite the prominence information and communication technology (ICT) receives in resource allocation and teacher professional development, little is known about the extent it actually enhances the learning process. A recent study investigating the link between integrating spreadsheets into a primary mathematics programme, and the development of some specific numerical skills, suggested exploring number patterns and spreadsheet templates did enhance their understanding of additive strategies. This paper discusses the role of visualisation of number patterns in fostering learner progression from relying solely on concrete materials, to abstract thinking.

Ways of Knowing

This chapter describes and illustrates a particular version of the hermeneutic process used to examine the nature of the learning when mathematical tasks are engaged through digital technologies.Hermeneutics lends its title to a spread of theoretical perspectives,from a conservative position that endeavours to decipher literal interpretations of written text,through to a radical one that is inherently suspicious of language and opposes the transformative process of interpretation,seeing it more as a play of possible meanings.This chapter unfolds a moderate perspective,and uses data to illustrate how this view can be used productively to provide informative insights into the learning process through digital media.

What the Future Might Reveal

The previous chapter examined how the learning process in primary-school mathematics might be transformed under the gaze of digital technologies.It considered the pedagogical implications for students engaging with mathematics through various digital pedagogical media and the ways that this might influence the learning process and students’understanding.It considered this through the potential offered by hardware and applications that are already available.Yet we don’t really know how these might evolve in the future.