Rita Borromeo Ferri

Trends in Teaching and Learning of Mathematical Modelling: ICTMA14

This book contains suggestions for and reflections on the teaching, learning and assessing of mathematical modelling and applications in a rapidly changing world, including teaching and learning environments. It addresses all levels of education from universities and technical colleges to secondary and primary schools. Sponsored by the International Community of Teachers of Mathematical Modelling and Applications (ICTMA), it reflects recent ideas and methods contributed by specialists from 30 countries in Africa, the Americas, Asia, Australia and Europe. Inspired by contributions to the Fourteenth Conference on the Teaching of Mathematical Modelling and Applications (ICTMA14) in Hamburg, 2009, the book describes the latest trends in the teaching and learning of mathematical modelling at school and university including teacher education. The broad and versatile range of topics will stress the international state-of-the-art on the following issues:Theoretical reflections on the teaching and learning of modellingModelling competenciesCognitive perspectives on modellingModelling examples for all educational levelsPractice of modelling in school and at university levelPractices in Engineering and Applications

Methodological reflections on a three-step-design combining observation, stimulated recall and interview

In this paper a tripartite qualitative design combining abservation, stimulated recall and interview is presented and discussed. This three-step-design makes it possible to get insight into the interaction of internal and external processes when solving mathematical tasks. The data analysis depends on the research question and the methodological approach. In the light of two research projects in mathematics education two different methods of data analysis are presented and methodologically reflected.

On the Influence of Mathematical Thinking Styles on Learners’ Modeling Behavior

Analyzing modeling processes with a cognitive psychological perspective is much neglected in the national and international discussion on mathematical modeling. In this paper, I will report about the COM2-project (Cognitive psychological analysis of modelling processes in mathematics lessons) which had the goal to investigate pupils’ and teachers’ cognitive processes during modeling activities in mathematics lessons. In this article, the focus lies on the pupils’ behavior, whose individual modeling routes were reconstructed. These routes are described in detail and the influence of the mathematical thinking styles of pupils on these modeling routes is shown.ZusammenfassungDie Analyse von Modellierungsprozessen unter einer kognitionspsychologischen Perspektive wurde sowohl in der nationalen als auch internationalen Modellierungsdiskussion sehr vernachlässigt. In diesem Artikel soll über das KOM2-Projekt (Kognitionspsychologische Analysen von Modellierungsprozessen im Mathematikunterricht) berichtet werden, bei dem das Ziel war, Modellierungsprozesse von Lehrenden und Lernenden im Mathematikunterricht unter einer kognitiven Perspektive zu untersuchen. In diesem Artikel liegt der Fokus auf dem Verhalten der Schülerinnen und Schüler, deren individuelle Modellierungsverläufe rekonstruiert wurden. Diese werden im Detail beschrieben und es wird verdeutlicht, inwiefern mathematische Denkstile der Lernenden Einfluss auf diese Verläufe haben.

Modelling in Mathematics Classroom Instruction: An Innovative Approach for Transforming Mathematics Education

The attitude of many students all over the world is shaped by the experience of learning impractical algorithms without any relevance for their actual or future life. Many students only learn algorithms and concepts in order to pass examinations and forget them afterwards. The inclusion of mathematical modelling in schools is one current innovative approach, which has the potential to offer students insight into the usefulness of mathematics in their life. In this chapter, the development of the current discussion on teaching and learning mathematical modelling is described by detailing the goals of implementing mathematical modelling in schools and ways of integrating modelling into classrooms. Innovative projects for the integration of modelling into classrooms are described, displaying the innovative power of the teaching and learning of mathematical modelling in school. Based on the results of empirical studies, scaffolding as an approach to support students’ independent modelling processes is discussed in detail distinguishing approaches at a macro- and a micro-level.

Insights into Teachers' Unconscious Behaviour in Modeling Contexts

In this paper, we will describe from a cognitive perspective how teachers deal with mathematical modeling problems in the classroom (grades 8–10, 14–16-year-olds) and how their behaviour is obviously dependent on certain features that are widely unknown to them. Our examples are taken from two projects: the COM² project where, among other things, the influence of the teachers’ mathematical thinking styles on their way of dealing with modeling problems is investigated, and the DISUM project where, among other things, teachers’ interventions during their coaching of students’ problem solving processes are analysed.

Mathematisches Modellieren – Eine Einführung in theoretische und didaktische Hintergründe

Mit mathematischem Modellieren wird ein bestimmter Aspekt der angewandten Mathematik bezeichnet. Die starkere Betonung des Modellierungsaspekts im Zusammenhang mit angewandter Mathematik hat vor allem Henry Pollak in den 70er Jahren des letzten Jahrhunderts angestosen.

Should Interpretation Systems Be Considered to Be Models if They Only Function Implicitly

The term “mathematical model” or just “model” is interpreted differently by different people in current international discussions about mathematical modelling. For many, the term “model” is restricted to interpretation systems which are explicit objects of thought. In this paper we ask the question, if interpretation systems should be considered to be models if they only function implicitly. Furthermore we describe characteristics of what we mean by “implicit models” – as well as possible transitions from implicit to explicit models, and what these transitions look like from a cognitive-psychological perspective.

Anwendungen und Modellieren

In diesem Kapitel wird die Relevanz von Anwendungen und Modellieren im Mathematikunterricht begrundet und werden verschiedene Ziele und damit verbundene Begrundungen fur die Integration von Anwendungen und Modellieren in den Mathematikunterricht wie auch verschiedene Schematisierungen des Modelbildungsprozesses dargestellt. Diese theoriebezogene Diskussion wird durch zwei Unterrichtsbeispiele konkretisiert. Des Weiteren wird die Rolle von technologischen Hilfsmitteln diskutiert und es werden Moglichkeiten von deren Integration in den Unterricht aufgezeigt. Der Beitrag schliest mit einem Uberblick uber empirische Untersuchungen zum Lehren und Lernen des Modellierens.

Effective Mathematical Modelling without Blockages – A Commentary

In this chapter, I comment on the paper by Stillman on “Applying metacognitive knowledge and strategies in applications and modelling tasks in secondary school.” In her paper, Stillman highlighted very important aspects concerning learning, teaching, and understanding modelling in the classroom. It is again impressive to see which important role meta-cognitive-activity plays while modelling and also to note that not all meta-cognitive acts are productive for getting a solution. It is quite common that the field of research in meta-cognition is very wide. A lot of research can be found concerning meta-cognition and problem solving. There also exists research on mathematical modelling and meta-cognition, as it was pointed out in the paper and the described project is a wonderful example for this. In my opinion, it is important that we have to learn a lot more on how several aspects of meta-cognition interplay with the learning and understanding of mathematical modelling. So, in my view, one central result of the study of Stillman et al. is that (productive) meta-cognitive activities can be seen as key for effective modelling behavior without “blockages,” respectively for providing “blockages.” Because I see this also as one important goal for teaching and learning mathematical modelling, it will be the starting point of my commentary as well. In the following lines, I discuss the paper by Stillman on the basis of well-known research areas of meta-cognition.

Interdisciplinary Mathematics Education: A State of the Art

This book provides an introductory reading of the State of the Art on Interdisciplinary Mathematics Education. It begins with an outline of the relevant historical, conceptual and theoretical backgrounds to this topic, what ‘discipline’ means and how inter-, trans-, and meta-disciplinary activity might be understood. Key ideas in theory involve boundaries, discourses, identity, and the division of labour in practice. Second, it reviews empirical research findings of the extant literature. For example, we report that a common theme in studies in middle and high schools is the motivational benefits for the learner of the subsumption of disciplinary motives to extra-academic problem-solving activity; this is counter-balanced by the effort needed to overcome the disciplinary boundaries in academic institutions, and in professional identities. Finally, we offer some case studies of current R&D in practice.