Morten Blomhøj

What’s all the Fuss about Competencies?

This paper deals with applying a description of a set of mathematical competencies with the aim of developing mathematics education in general and in particular the work with mathematical modelling. Hence it offers a presentation of the general idea of working with mathematical competencies as well as an analysis of some potentials of putting this idea into educational practice. Three challenges form the basis of the analysis: The fight against syllabusitis, the dilemma of teaching directed autonomy and the description of progress in mathematical modelling competency.

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.

Towards a didactical theory for mathematical modelling

Mathematics education research has been somewhat short of its own paradigmatic theories (see ZDM, 2005, issue 6 and ZDM 2006, issue 1 for an ongoing discussion). Theories are often borrowed from the background sciences and applied to the field of mathematics education e.g. general learning theories from pedagogy, sociology, psychology etc. This has led to our field being a subject to a host of criticisms (e,g., Lerman, 2006; Lester, 2005; Sriraman & English, 2005, 2006; Steen, 1999). Therefore it is relevant to look for areas in mathematics education where theories may emerge from studying the processes of teaching and learning mathematics. In this issue of ZDM we present and discuss what we consider to be an example of a developing theory within the field of mathematics education research, namely a theory for the teaching and learning of mathematical modelling. During the last decades a coherent theoretical understanding of mathematical modelling processes and connected teaching and learning processes has been emerging. This has happened through a close interplay between the development of curricula, teaching practices, experimental teaching, theoretical reflections and research. Together with the use of information technology, the introduction of mathematical modelling and applications is a prominent general feature of the recent developments in the practice of mathematics teaching, especially with regard to secondary level teaching. In this development, didactical research has already played an important role and we find it of general interest for mathematics education research to examine the development of a theory for mathematical modelling as a paradigmatic case for developing theory in close interplay with teaching practice.

Modelling Competency: Teaching, Learning and Assessing Competencies – Overview

This chapter frames eight papers which are all addressing questions and issues related to the teaching, learning and assessing of mathematical modelling competency. As a main commonality in the papers, the authors take point of departure in the existence of (such a thing as) mathematical modelling competency (as a concept). A competency in general is understood as a person’s mental capacity to cope with a certain type of challenge in a knowledgeable and reflective way. According to this, a person possesses mathematical modelling competency if he or she is capable of carrying through a mathematical modelling process in order to solve a problem or to understand a situation within a certain domain.

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.

A brief survey of the state of mathematical modeling aroung the world

We briefly survey and discuss the issues and perspectives presented in the two special issues of the Zentralblatt fur Didaktik der Mathematik on mathematical modeling and pose open questions to the community of researchers interested in this domain of inquiry.

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.

Interdiciplinary Problem Oriented Project Work – a Learning Environment for Mathematical Modelling

In this chapter, we analyze the problem oriented project work practiced at the Bachelor Study Program in Natural Science (Nat Bach) at Roskilde University (RU) as a learning environment for developing students’ mathematical modelling competence. The projects are conducted in a rather sophisticated and radical learning environment grounded on the four pedagogical key principles of problem orientation, participant directed group work, interdisciplinarity and exemplarity. We illustrate and discuss the interplay between the aim of developing the students’ modelling competence on the one hand and on the other hand the key principles of the project work and the learning environment in which this pedagogy is brought into life at Nat Bach. In particular, we focus on the role and function of the feedback in the supervision process, and the peer‐review seminars. Through the analyses we aim at contributing to the research on modelling in mathematics education especially regarding the importance of the students’ autonomy in the modelling process, the exemplarity of their work and the interdisciplinary nature of modelling.

Interdisziplinäre problemorientierte Projektarbeit – Eine Lernumgebung für mathematisches Modellieren

Hier analysieren wir die problemorientierte Projektarbeit, die im Rahmen eines naturwissenschaftlichen Bachelor‐Studienprogramms (Nat Bach) an der Roskilde Universitat (RU) durchgefuhrt wurde und als Lernumgebung zur Entwicklung der mathematischen Modellierungskompetenz von Studierenden diente. Die Projekte wurden in einer anspruchsvollen Lernumgebung durchgefuhrt, die auf den vier padagogischen Grundprinzipien der Problemorientierung, der partizipativen Gruppenarbeit, der Interdisziplinaritat und der Exemplaritat basiert. Wir illustrieren und diskutieren das Zusammenspiel zwischen dem Ziel auf der einen Seite, die Modellierungskompetenz der Studierenden zu fordern, und den Grundprinzipien der Projektarbeit und der Lernumgebung auf der anderen Seite. Insbesondere werden die Rolle und die Funktion von Feedback im Betreuungsprozess und die Peer‐Review‐Seminare im Vordergrund stehen.

The Bachelor Programmes and the Roskilde Model

In this chapter we take a closer look at the structure and organization of the four bachelor programmes offered at Roskilde University. These bachelor programmes, and in particular the possibility they offer for students to gradually specialize in two subjects – possibly even from two different faculties – are quite unique in university education. We explain the historical background for the programmes and how they have developed in the 2012 reform. We analyse how the four pedagogical principles of student-directed project work, problem orientation, exemplarity and interdisciplinarity are implemented differently in the structures of each of the four bachelor programmes and how they are realized in the pedagogical practices. In particular, we analyse the organization and evaluation of student-directed project work across the four programmes. We conclude with some reflections on the relation of the bachelor programmes to other aspects of the university and to broader society.