Jonas Bergman Ärlebäck

First Results from a Study Investigating Swedish Upper Secondary Students’ Mathematical Modelling Competencies

This paper reports on the first results from a study investigating Swedish upper secondary students’ (11th – 12th grade) mathematical modelling competency. Using non-parametric statistical methods ...

On the Use of Realistic Fermi Problems in Introducing Mathematical Modelling in Upper Secondary Mathematics

This paper reports a study investigating potential uses of Fermi problems to introduce mathematical modelling to Swedish upper secondary school students. The work of three groups of students’ is analysed using an analytic tool referred to as the “modelling activity diagram”, adapted and developed by the author. It was observed that the processes involved in a mathematical modelling cycle were richly represented in groups’ solution. The students also frequently used their personal extra-mathematical knowledge in the solving process in three different ways: a creative, a verifying and a social way.

A Modeling Perspective on Interpreting Rates of Change in Context

Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students’ abilities to describe and interpret rates of change in the context of exponential decay. In this article, we elaborate the characteristics of the model development sequence and we examine how students interpreted and described non-constant rates of change in context. We provide evidence for how a focus on the context made visible students’ reasoning about rates of change, including difficulties related to the use of language when describing changes in the negative direction. We argue that context and the use of language, forefronted in a modeling approach, should play an important role in supporting the development of students’ reasoning about changing phenomena.

Reflections on Progress in Mathematical Modelling Research

The terms, models and modelling, have been used variously in the literature, including in reference to solving word problems, conducting mathematical simulations, generating representations of problem situations (including constructing explanations of natural phenomena), creating cognitive representations while solving a particular problem, and engaging in a bidirectional process of translating between a real-world situation and mathematics (e.g., Cai, Cirillo, Pelesko, Borromeo Ferri, Borba, Geiger, Stillman, English, Wake, Kaiser, & Kwon, 2014; Doerr & Tripp, 1999; English & Halford, 1995; Gravemeijer, 1999; Greer, 1997; Lesh & Doerr, 2003; Romberg, Carpenter, & Dremock, 2005)...

Modelling from the Perspective of Commognition – An Emerging Framework

This chapter explores an emerging framework on mathematical models and modelling using the theoretical perspective of commognition to analyse and discern discursive objects in a dialogue between two students engaged in a modelling activity. The results, partly presented as realization trees, show a variety of signifiers from different discourses coming into play during the modelling, and examples are given of the activity of negotiation, which plays an important role in any modelling activity. In addition, it is argued that the framework has potential to bridge different research perspectives on mathematical models and modelling.

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.

Moving Beyond a Single Modelling Activity

In this theoretical chapter, we draw on a models and modelling perspective on teaching and learning to elaborate on the components of model development sequences using the lens of variation theory. We give empirical examples of how model exploration activities and model application activities can be described and understood from a variation theory perspective. The chapter concludes by presenting tentative principles for the design of such activities within a model development sequence.

Teaching Practices and Exploratory Computer Simulations

While computing technologies are widely available in secondary schools, these technologies have had only limited impact on changing classroom practices. In part, this can be attributed to an underdeveloped understanding of the role of the teacher in engaging in classroom practices that can support student learning with technology. In this qualitative study, we analyzed the teaching practices that supported pre-college students’ learning of a conceptually rich and deep topic (the average rate of change) when using an exploratory computer simulation environment. The results illustrate the demands placed on teachers when faced with the generation of student ideas from their interactions with the simulation and three broad categories of teaching practices in response to these demands: (a) pressing students for representations, (b) harvesting student ideas, and (c) sorting out and refining student ideas. These findings contribute to evolving frameworks for understanding meaningful and productive technology use in teaching secondary mathematics.

The Bach tensor and other divergence-free tensors

In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

Initial Results of an Intervention Using a Mobile Game App to Simulate a Pandemic Outbreak

This chapter presents the design and results from the first iteration of a classroom activity using the context and simulation provided by a commercial game app for smartphones and tablets. The aim was to study students’ experiences of an intervention entailing the game app in order to inform a more mature design with the long-term goal of being able to develop principles to design and implement modelling activities using game apps. An analysis focusing on the interplay between the designed intervention environment and students’ work from two upper secondary classes resulted in information in terms of affordances to inform the redesign. In addition, students experienced the activity as interesting and engaging, but with significant gender differences. The results inform a discussion of the role of new technology and simulation within classroom teaching and learning of mathematical modelling.