Morten Misfeldt

An Empirical Approach to the Mathematical Values of Problem Choice and Argumentation

In this paper we describe and discuss how mathematical values influence researchers’ choices when practicing mathematics. Our paper is based on a qualitative investigation of mathematicians’ practices, and its goal is to gain an empirically grounded understanding of mathematical values. More specifically, we will analyze the values connected to mathematicians’ choice of problems and their choice of argumentative style when communicating their results. We suggest that these two situations can be understood as relating to the three mathematical values: recognizability, formalizability and believability. Furthermore, we discuss three meta-issues concerning the general nature of mathematical values, namely (1) the origin of mathematical values, (2) the extent to which different values change over time and (3) the situatedness of mathematical values; that is the extent to which mathematical values depend on the specific context in which you are located. We conclude the chapter by recommending a methodological pluralism in future investigations of mathematical values.

Semiotic Scaffolding in Mathematics

This paper investigates the notion of semiotic scaffolding in relation to mathematics by considering its influence on mathematical activities, and on the evolution of mathematics as a research field. We will do this by analyzing the role different representational forms play in mathematical cognition, and more broadly on mathematical activities. In the main part of the paper, we will present and analyze three different cases. For the first case, we investigate the semiotic scaffolding involved in pencil and paper multiplication. For the second case, we investigate how the development of new representational forms influenced the development of the theory of exponentiation. For the third case, we analyze the connection between the development of commutative diagrams and the development of both algebraic topology and category theory. Our main conclusions are that semiotic scaffolding indeed plays a role in both mathematical cognition and in the development of mathematics itself, but mathematical cognition cannot itself be reduced to the use of semiotic scaffolding.

Diffusing scientific knowledge to innovative experts

Communicating science to scientists works well thanks to well-defined communication structures based on both printed material in peer-reviewed publications and oral presentations, e.g. at conferences and seminars. However, when science is communicated to practitioners, the structures become fuzzy. We are looking at how to implement Web2.0 technologies to Danish seed scientists communicating to seed consultants, agricultural advisors, and seed growers, and we are met with the challenge of securing effective knowledge diffusion to the community. Our investigation’s focal point is on Rogers’ theoretical framework “Diffusion of Innovation” (DOI), as we look at how DOI may affect the Danish seed industry if science communication is redesigned in accordance with the framework. During our project workshop, participants recognized trends and characteristics from DOI in the Danish seed community and argued for more collaboration between scientists and practitioners. This can be done by implementing fast-learning via online website, but it needs to be assisted by slowerpaced face-to-face learning to lessen the risk of a digital knowledge divide within the community.

Representations of Modelling in Mathematics Education

Mathematical models have a substantial impact at all levels of society, and hence mathematical modelling stands as an important topic in mathematics education. Mathematical modelling has a particular pedagogical/didactical discourse as modelling continues to garner attention in educational research. Diagrammatic representations of mathematical modelling processes are increasingly being used in curriculum documents on national and transnational levels. In this chapter, we critically discuss one of the most frequently used representations of modelling processes in the literature, namely, that of the modelling cycle, and offer alternative representations to more fully capture multiple aspects of modelling in mathematics education.

The Learning Potentials of Number Blocks

In this paper it is described how an interactive cubic user-configurable modular robotic system can be used to support learning about numbers and their pronunciation. The development is done in collaboration with a class of 7–8 year old children and their mathematics teacher. The tool is called Number Blocks, and it combines physical interaction, learning, and immediate feedback. Number Blocks support the children’s understanding of place value in the sense that it allows them to experiment with large numbers. We found that the blocks contributed to the learning process in several ways. The blocks combined mathematics and play, and they included and supported children at different academic levels. The auditory representation, especially the enhanced rhythmic effects of using speech synthesis, helped the children to pronounce large numbers. This creates a new context for learning mathematical aspects of number names and the place value system.

Instrumental Genesis and Proof: Understanding the Use of Computer Algebra Systems in Proofs in Textbook

In this chapter we investigate the role of Computer Algebra Systems (CAS) in textbook proofs. We describe two cases of CAS use in textbook proofs and use the instrumental approach, and in particular the distinction between epistemic and pragmatic mediations, to understand the consequences of the so-called CAS-assisted proofs. We end with a discussion of the experienced shortcomings of the instrumental approach in relation to CAS use in justification of mathematical results, and suggest the inclusion of alternative frameworks for filling the gap.

A Typology of Mathematical Diagrams

In this paper, we develop and discuss a classification scheme that allows us to distinguish between the types of diagrams used in mathematical research based on the cognitive support offered by diagrams. By cognitive support, we refer to the gain that research mathematicians get from using diagrams. This support transcends the specific mathematical topic and diagram type involved and arises from the cognitive strategies mathematicians tend to use. The overall goal of this classification scheme is to facilitate a large-scale quantitative investigation of the norms and values governing the publication style of mathematical research, as well as trends in the kinds of cognitive support used in mathematics. This paper, however, focuses only on the development of the classification scheme.

Computers as a Source of A Posteriori Knowledge in Mathematics

ABSTRACT Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions.

Analysis of Uses of Technology in the Teaching of Mathematics

This Topic Study Group aimed at providing a forum to discuss the current state of art of the presence of technology in diverse aspects of teaching mathematics conveying a deep analysis of its implications to the future. Technology was understood in a broad sense, encompassing the computers of all types including the hand-held technology, the software of all types, and the technology of communication that includes the electronic board and the Internet. The discussions served as opportunity for all interested in the use of technology in education environment, to understand its diverse aspects and to share the creative and outstanding contributions, with critical analysis of the different uses.

Conceptions of Mathematics at a University Programme in Economics

This study seeks to shed light on perceptions of the role of mathematics in university economics programmes. Previous studies have clearly shown that mathematical knowledge is very important for students’ success in economics programmes, but no coherent overview of qualitative reasons for including mathematics and of proposed teaching practices has been provided. In this paper we rely on qualitative data to articulate why mathematics is perceived as an important part of a specific economics programme. On the basis of a grounded analysis, we conclude that mathematics is of great value partly for content-oriented and instrumental reasons and partly because it has a dual function: it contributes to creating a particular study culture and student community and it filters out students who cannot cope with the high level of mathematics that characterises economics programmes in general. Furthermore, we document potential problems related to mathematics as an integrated and inter-disciplinary part of economics programmes. These problems typically arise because the subject of mathematics has its own ontology and curricular logic, which may be overruled when multiple subjects are organised in clusters.