Uffe Thomas Jankvist

Special Issue on History and Philosophy of Mathematics in Mathematics Education

Although Science & Education has published papers specifically concerning mathematics over the years—some quite significant for education—mathematics has never received the spotlight. Indeed, when the journal was established it bore the subtitle, Contributions from History, Philosophy and Sociology of Science and Mathematics, but so few papers on mathematics were received in those years that the word ‘‘mathematics’’ was duly dropped. Yet within the community of science and mathematics educators, there is no lack of interest at least in the contributions of history and philosophy of mathematics to mathematics education, particularly the history of mathematics. Besides publications such as those we have listed at the end of this introduction, one can point to the activities of the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM) which dates back to 1972, and which almost from the start became an affiliated study group of the International Commission on Mathematical Instruction (ICMI), the largest and oldest international organization for mathematics education. International conferences of HPM have been held in Europe, Canada, Australia, Taiwan, Mexico, and Korea in association with the quadrennial International Congress on Mathematical Education (ICME). The European Summer University on History and Epistemology in Mathematics Education (ESU), now held every 4 years as well, is another regular venue for meetings dealing with history in mathematics education, as is the working group on History in Mathematics Education at the biennial Congress of European

A teaching module on the history of public-key cryptography and RSA

This article describes some empirical research into using history in mathematics education. More precisely, it discusses a teaching module on the history of public-key cryptography and RSA, implemented in a Danish upper secondary mathematics class in the autumn of 2007. As part of the module the students were expected to write several essays illuminating different aspects of this history. The article discusses whether the students were able to reflect on meta-perspective issues of the evolution and development of mathematics and if so, then whether these reflections were anchored in the taught (and acquired) mathematics of public-key cryptography and RSA. The question of how teaching modules may be designed to take such matters into account is also discussed.This article describes some empirical research into using history in mathematics education. More precisely, it discusses a teaching module on the history of public-key cryptography and RSA, implemented in a Danish upper secondary mathematics class in the autumn of 2007. As part of the module the students were expected to write several essays illuminating different aspects of this history. The article discusses whether the students were able to reflect on meta-perspective issues of the evolution and development of mathematics and if so, then whether these reflections were anchored in the taught (and acquired) mathematics of public-key cryptography and RSA. The question of how teaching modules may be designed to take such matters into account is also discussed.

On the Use of Primary Sources in the Teaching and Learning of Mathematics

In this chapter, an attempt is made to illustrate why the study of primary original sources is, as often stated, rewarding and worth the effort, despite being extremely demanding for both teachers and students. This is done by discussing various reasons for as well as different approaches to using primary original sources in the teaching and learning of mathematics. A selection of these reasons and approaches will be illustrated through a number of examples from the literature on using original sources in mathematics education. In particular, focus is on empirical research findings when choosing illustrative examples. The chapter also includes a section on the background and academic forums of research on using primary original sources in mathematics education. The discussion on using original sources is briefly related to that of using history in mathematics education, and some theoretical constructs are drawn upon to assist this discussion. In the final section of the chapter, the past, present, and future of using primary original sources is discussed, in order to possibly identify new paths which research might follow. In particular, the section addresses the role of primary original sources in relation to the educational problems of recruitment, transition, retention, and interdisciplinarity.

History, Application, and Philosophy of Mathematics in Mathematics Education: Accessing and Assessing Students’ Overview and Judgment

The Regular Lecture addresses the three dimensions of history, application, and philosophy of mathematics in the teaching and learning of mathematics. It is discussed how students’ overview and judgment—interpreted as ‘sets of views’ and beliefs about mathematics as a discipline—may be developed and/or changed through teaching activities embracing all three dimensions of history, application, and philosophy. More precisely, an example of such a teaching activity for upper secondary school is described along with a method for both accessing and assessing students’ overview and judgment. Examples of data analysis are given based on a concrete implementation of the teaching activity.

Analysing the use of history of mathematics through MKT

In this article, we analyse three ‘classical’ cases of using history of mathematics in teacher training and reinterpret them using the framework of mathematical knowledge for teaching (MKT). We continue with a discussion of the potential benefits of applying MKT as a framework for discussion in research related to the use of history in mathematics education. MKT serves as an example of a framework from general mathematics education research which has something to offer in this respect. We argue that the MKT framework provides a powerful language to communicate results from research on the uses of history of mathematics to researchers in other areas of mathematics education research.

Problem solving in the borderland between mathematics and physics

ABSTRACT The article addresses the problématique of where mathematization is taught in the educational system, and who teaches it. Mathematization is usually not a part of mathematics programs at the upper secondary level, but we argue that physics teaching has something to offer in this respect, if it focuses on solving so-called unformalized problems, where a major challenge is to formalize the problems in mathematics and physics terms. We analyse four concrete examples of unformalized problems for which the formalization involves different order of mathematization and applying physics to the problem, but all require mathematization. The analysis leads to the formulation of a model by which we attempt to capture the important steps of the process of solving unformalized problems by means of mathematization and physicalization.

Theories of Mathematics Education, Edited by Bharath Sriraman and Lyn English: Common Ground for Scholars and Scholars in the Making

Theories of mathematics education: Seeking new frontiers is the first book in a new Springer series, Advances in Mathematics Education. To some degree the book is based on a collection of previously published articles from special issues of ZDM—The International Journal on Mathematics Education (previously known as Zentralblatt für Didaktik der Mathematik). These articles, dealing with the role and use of theories in and about mathematics education, originally stem from various conferences and meetings such as PME and CERME. For this reason some of the articles in the book are already well known, a few may even be considered to be “modern classics” within theories of mathematics education such as Frank Lesters “On the theoretical, conceptual, and philosophical foundations for research in mathematics education.” What is new—and non-traditional—in the book, however, is its form of presentation and format, the main articles being accompanied by preludes and commentaries by established researchers and newcomers.

The hidden mathematics of the mars exploration rover mission

ConclusionsA lot of the mathematics of MER is hidden and not only from the public but even from the applied scientists working on the mission. As briefly sketched above, for the scientists, this could be disastrous in a worst-case scenario. The hiding of mathematics, both in our everyday life and within science itself, is a matter not often discussed in public —which in itself is a disaster, taking into account the consequences the hiding of mathematics might have for the public. We like to think that this article may help let in some light.Another question raised by our work is that of beliefs in mathematics. Only occasionally are the beliefs of mathematicians discussed. We found repeatedly that mathematical elements of MER are not actually considered to be mathematics among the applied scientists themselves, not on first hand anyway. Is this due to the fundamentally different views of what mathematics is between applied scientists (including engineers) and pure scientists of the 20th century? We do not know.Finally, we comment on the nature of the mathematics involved in MER. Because of the extreme nature of a Mars mission, one might expect “extreme” mathematics, mathematics developed for the sole purpose of this mission.

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 historical teaching module on ‘the unreasonable effectiveness of mathematics’: Boolean algebra and Shannon circuits

This article describes the design and implementation of a historical teaching module, where Danish upper secondary students studied primary source material from the mathematicians R W Hamming, G Boole and C E Shannon. The module is an example of a so-called HAPh-module, which focuses on elements of the history, application and philosophy of mathematics. A previous version was presented at the History and Pedagogy of Mathematics (HPM) meeting in Deajeon, Korea in 2012.