The technological mediation of mathematics and its learning

This paper examines the extent to which mathematical knowledge, and its related pedagogy, is inextricably linked to the tools – physical, virtual, cultural – in which it is expressed. Our goal is to focus on a few exemplars of computational tools, and to describe with some illustrative examples, how mathematical meanings are shaped by their use. We begin with an appraisal of the role of digital technologies, and our rationale for focusing on them. We present four categories of digital tool-use that distinguish their differing potential to shape mathematical cognition. The four categories are: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. new representational infrastructures, and iv. the implications of highbandwidth connectivity on the nature of mathematics activity. In conclusion, we draw out the implications of this analysis for mathematical epistemology and the mathematical meanings students develop. We also underline the central importance of design, both of the tools themselves and the activities in which they are embedded

Next steps in implementing Kaput's research programme

We explore some key constructs and research themes initiated by Jim Kaput, and attempt to illuminate them further with reference to our own research. These 'design principles' focus on the evolution of digital representations since the early nineties, and we attempt to take forward our collective understanding of the cognitive and cultural affordances they offer. There are two main organising ideas for the paper. The first centres around Kaput's notion of outsourcing of processing power, and explores the implications of this for mathematical learning. We argue that a key component for design is to create visible, transparent views of outsourcing, a transparency without which there may be as many pitfalls as opportunities for mathematical learning. The second organising idea is that of communication, a key notion for Kaput, and the importance of designing for communication in ways that recognise the mutual influence of tools for communication and for mathematical expression

Infusing technology and algebra grant proposal

Includes bibliographical references

The importance of being accessible: The graphics calculator in mathematics education

The first decade of the availability of graphics calculators in secondary schools has just concluded, although evidence for this is easier to find in some countries and schools than in others, since there are gross socio-economic differences in both cases. It is now almost the end of the second decade since the invention of microcomputers and their appearance in mathematics educational settings. Most of the interest in technology for mathematics education has been concerned with microcomputers. But there has been a steady increase in interest in graphics calculators by students, teachers, curriculum developers and examination authorities, in growing recognition that accessibility of technology at the level of the individual student is the key factor in responding appropriately to technological change; the experience of the last decade suggests very strongly that mathematics teachers are well advised to pay more attention to graphics calculators than to microcomputers. There are clear signs that the commercial marketplace, especially in the United States, is acutely aware of this trend. It was recently reported that current US sales of graphics calculators are around six million units per year, and rising. There are now four major corporations developing products aimed directly at the high school market, with all four producing graphics calculators of high quality and beginning to understand the educational needs of students and their teachers. To get some evidence of this interest, I scanned a recent issue (April 1995) of The Mathematics Teacher, the NCTM journal focussed on high school mathematics. The evidence was very strong: of almost 20 full pages devoted to paid advertising, nine featured graphics calculators, while only two featured computer products, with two more featuring both computers and graphics calculators. The main purposes of this paper are to explain and justify this heightened level of interest in graphics calculators at the secondary school level, and to identify some of the resulting implications for mathematics education, both generally, and in the South-East Asian region

On the integration of digital technologies into mathematics classrooms

Trouche‘s (2003) presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing knowledge with technology; the latter concerning the problem of integrating technology into classroom practice. At the Symposium, there was considerable discussion of the idea of situated abstraction, which the current authors have been developing over the last decade. In this paper, we summarise the theory of instrumental genesis and attempt to link it with situated abstraction. We then seek to broaden Trouche‘s discussion of orchestration to elaborate the role of artefacts in the process, and describe how the notion of situated abstraction could be used to make sense of the evolving mathematical knowledge of a community as well as an individual. We conclude by elaborating the ways in which technological artefacts can provide shared means of mathematical expression, and discuss the need to recognise the diversity of student‘s emergent meanings for mathematics, and the legitimacy of mathematical expression that may be initially divergent from institutionalised mathematics

Bridging symbolic computation and economics: a dynamic and interactive tool to analyze the price elasticity of supply

It is not possible to achieve the objectives and skills of a program in economics, at the secondary and undergraduate levels, without resorting to graphic illustrations. In this way, the use of educational software has been increasingly recognized as a useful tool to promote students' motivation to deal with, and understand, new economic concepts. Current digital technology allows students to work with a large number and variety of graphics in an interactive way, complementing the theoretical results and the so often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. Thanks to the symbolic and numerical capabilities of Mathematica, these applications allow the user to interact with the graphical and analytical information in real time. However, Mathematica is a commercially distributed application which makes it difficult for teachers and students to access. The main goal of this paper is to present a new dynamic and interactive tool, created with Mathematica and available in the Computable Document Format. This format allows anyone with a computer to use, at no cost, the PES(Linear)-Tool, even without an active Wolfram Mathematica license. The PES(Linear)-Tool can be used as an active learning tool to promote better student activity and engagement in the learning process, among students enrolled in socio-economic programs. This tool is very intuitive to use which makes it suitable for less experienced users.Funding Agency Portuguese Foundation for Science and Technology UID/ECO/04007/2019info:eu-repo/semantics/publishedVersio

ICT use in the teaching of mathematics: implications for professional development of pre-service teachers in Ghana

Included in the contemporary mathematics curricula in Ghana is the expectation that mathematics teachers will integrate technology in their teaching. However, importance has not been placed on preparing teachers to use ICT in their instruction. This paper reports on a study conducted to explore the feasibility of ICT use in mathematics teaching at senior high school levels in Ghana. Interviews and survey data were used for data collection. Preliminary results showed that mathematics teachers in Ghana do not integrate ICT in their mathematics instruction. Among the major perceived barriers identified were: Lack of knowledge about ways to integrate ICT in lesson and Lack of training opportunities for ICT integration knowledge acquisition. To overcome some of these barriers, opportunities of a professional development arrangement for pre-service mathematics teachers were explored. Findings from the study revealed specific features of a professional development scenario that matters for ICT integration in mathematics teaching in the context of Ghana

Symbolic Manipulators Affect Mathematical Mindsets

Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely offer a convenient way to perform the computations students would have otherwise wanted to do by hand. This paper presents examples from the work of upper level physics majors where Mathematica plays an active role in focusing and sustaining their thought around calculation. These students still engage in powerful mathematical reasoning while they calculate but struggle because of the narrowed breadth of their thinking. Their reasoning is drawn into local attractors where they look to calculation schemes to resolve questions instead of, for example, mapping the mathematics to the physical system at hand. We model the influence of Mathematica as an integral part of the constant feedback that occurs in how students frame, and hence focus, their work

James J. Kaput (1942–2005) imagineer and futurologist of mathematics education

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work