Tinne Hoff Kjeldsen

Teaching mathematical modelling through project work

The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.

Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity

Two simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn (1862–1939) and another by Hermann Minkowski (1864–1909), have been described as the origin of the theory of convex bodies. This article aims to understand and explain (1) how and why the concept of such bodies emerged in these two trajectories of mathematical research; and (2) why Minkowskis – and not Brunns – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunns and Minkowskis mathematical work in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jorg Rheinbergers work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above.

Students’ Reflections in Mathematical Modelling Projects

Students’ reflections play an important role in mathematical modelling competency. In this chapter, we argue that there are two kinds of reflections which have to be challenged and supported in different ways. They can be characterised as respectively internal and external with respect to the modelling process. Internal reflections add meaning and quality to the sub-processes involved in a mathematical modelling process, while the external reflections address the role and function of the model in actual or potential applications. If mathematical modelling competency is an educational goal, the teaching needs to provide students with experiences with modelling and applications of models in a variety of authentic contexts in ways that support the students’ development of both internal and external reflections. Through analyses of two student projects, we illustrate the two kinds of reflections and discuss how they can be developed in students.

New Mathematical Disciplines and Research in the Wake of World War II

This paper focuses on the significance of the Second World War for the rise and establishment of new disciplines in applied mathematics as well as for the renewed interest and growth in some related subjects in pure mathematics. The mathematical topics involved are mathematical programming, operations research, game theory, the theory of convexity, and the theory of systems of linear inequalities. Connections and interactions between different branches of mathematics on the one hand and between different kinds of driving forces in the development of mathematics on the other hand are discussed. Special emphasis is devoted to the significance of the interplay between practical problem solving and basic research in mathematics proper as a consequence of World War II and the post-war organization of science support in the USA.

The emergence of nonlinear programming: interactions between practical mathematics and mathematics proper

ConclusionIt was the duality theorem for linear programming-that is, a purely theoretical result-that sparked the interest of Kuhn and Tucker. It was the duality theory they wanted to extend to the general (quadratic) nonlinear case. It is in this respect that I find the development of the duality theorem in linear programming so crucial for the emergence of nonlinear programming.

Students’ Mathematical Learning in Modelling Activities

Ten years of experience with analyses of students’ learning in a modelling course for first year university students, led us to see modelling as a didactical activity with the dual goal of developing students’ modelling competency and enhancing their conceptual learning of mathematical concepts involved. We argue that progress in students’ conceptual learning needs to be conceptualised separately from that of progress in their modelling competency. Findings are that modelling activities open a window to the students’ images of the mathematical concepts involved; that modelling activities can create and help overcome hidden cognitive conflicts in students’ understanding; that reflections within modelling can play an important role for the students’ learning of mathematics. These findings are illustrated with a modelling project concerning the world population.

A Historical View of Nonlinear Programming: Traces and Emergence

The historical view we propose in this introductory chapter will point out some of the technical difficulties of mathematical problems related to nonlinear programming and some features of the economic and social context (also military) that favored its rootedness in the years of the Second World War and the years immediately following the war. We recall some of the main definitions and basic results of mathematical programming and shortly address the “prehistory” of nonlinear programming. The main part of the chapter deals with the first ideas and developments of linear programming, first in the USSR and then in the USA and with the fundamental researches ofW. Karush, Fritz John, H.W. Kuhn and A.W. Tucker which are analyzed and discussed with respect to their mathematical and historical features.

The Role of History and Philosophy in University Mathematics Education

University level mathematics is organised differently in different universities. In this paper we consider mathematics programmes leading to a graduate degree in mathematics. We briefly introduce a multiple perspective approach to the history of mathematics from its practices, reflections about uses of history and the research direction in philosophy of mathematics denoted ‘Philosophy of Mathematical Practice’. We link history and philosophy of mathematical practices to recent ideas in mathematics education in order to identify different roles history and philosophy can play in the learning of mathematics at university level. We present, analyse and discuss different examples of inclusions of history and philosophy in university programmes in mathematics. These presentations are divided into courses in history and philosophy, respectively, since this is the main way they are organised at the universities. We shall see, however, that the history courses address philosophical questions and that the philosophy courses employ historical material. The chapter ends with comments on how mathematics educations at university level can benefit from history and philosophy of mathematics.

From “Mixed” to “Applied” Mathematics: Tracing an important dimension of mathematics and its history

The workshop investigated historical variations of the ways in which historically boundaries were drawn between ‘pure’ mathematics on the one hand and ‘mixed’ or ‘applied’ mathematics on the other from about 1500 until today. It brought together historians and philosophers of mathematics as well as several mathematicians working on applications. Emphasis was laid upon the clarification of the relation between the historical use and the historiographical usefulness and philosophical soundness of the various cate-

History of Convexity and Mathematical Programming: Connections and Relationships in Two Episodes of Research in Pure and Applied Mathematics of the 20th Century

In this paper, the gradual introduction of the concept of a general convex body in Minkowski’s work and the development of mathematical programming, are presented. Both episodes are exemplary for mathematics of the 20th century, in the sense that the former represents a trend towards a growing abstraction and autonomy in pure mathematics, whereas the latter is an example of the many new disciplines in applied mathematics that emerged as a consequence of efforts to develop mathematics into a useful tool in a wider range of subjects than previously. It will be discussed, how and why these two new areas emerged and developed through different kinds of connections and relations; and how they at some point became connected, and fed and inspired one another. The examples suggest that pure and applied mathematics are more intertwined than the division in ‘pure’ and ‘applied’ signals.