Werner Blum

Applied mathematical problem solving, modelling, applications, and links to other subjects — State, trends and issues in mathematics instruction

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Modelling and Applications in Mathematics Education

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Pedagogical content knowledge and content knowledge of secondary mathematics teachers

Drawing on the work of L. S. Shulman (1986), the authors present a conceptualization of the pedagogical content knowledge and content knowledge of secondary-level mathematics teachers. They describe the theory-based construction of tests to assess these knowledge categories and the implementation of these tests in a sample of German mathematics teachers (N=198). Analyses investigate whether pedagogical content knowledge and content knowledge can be distinguished empirically, and whether the mean level of knowledge and the degree of connectedness between the two knowledge categories depends on mathematical expertise. Findings show that mathematics teachers with an in-depth mathematical training (i.e., teachers qualified to teach at the academic-track Gymnasium) outscore teachers from other school types on both knowledge categories and exhibit a higher degree of cognitive connectedness between the two knowledge categories.

How do Students and Teachers Deal with Modelling Problems

In this paper, we shall report on some of the work that has been, and is being, done in the DISUM project. In §, we shall describe the starting point of DISUM, the SINUS project aimed at developing high-quality teaching. In §, we shall briefly describe the DISUM project itself, and in §3 we shall present and analyse a modelling task from DISUM, the “Sugarloaf” problem. How students dealt with this task will be the topic of §4, the core part of this paper. How experienced SINUS teachers dealt with this task in the classroom will be reported in §5. Finally, in §6, we shall briefly describe future plans for the DISUM project.

Can Modelling Be Taught and Learnt? Some Answers from Empirical Research

This chapter deals with empirical findings on the teaching and learning of mathematical modelling, with a focus on grades 8–10, that is, 14–16-year-old students. The emphasis lies on the actual behaviour of students and teachers in learning environments with modelling tasks. Most examples in this chapter are taken from our own empirical investigations in the context of the project DISUM. In the first section, the terms used in this chapter are recollected from a cognitive point of view by means of examples, and reasons are summarised why modelling is an important and also demanding activity for students and teachers. In the second section, examples are given of students’ difficulties when solving modelling tasks, and some important findings concerning students dealing with modelling tasks are presented. The third section concentrates on teachers; examples of successful interventions are given, as well as some findings concerning teachers treating modelling examples in the classroom. In the fourth section, some implications for teaching modelling are summarised, and some encouraging (though not yet fully satisfying) results on the advancement of modelling competency are presented.

Preformal proving: examples and reflections

The starting point of our reflections is a classroom situation in grade 12 in which it was to be proved intuitively that non-trivial solutions of the differential equationf′=f have no zeros. We give a working definition of the concept of preformal proving, as well as three examples of preformal proofs. Then we furnish several such proofs of the aforesaid fact, and we analyse these proofs in detail. Finally, we draw some conclusions for mathematics in school and in teacher training.

Die Untersuchung des professionellen Wissens deutscher Mathematik-Lehrerinnen und -Lehrer im Rahmen der COACTIV-Studie

ZusammenfassungIn der COACTIV-Studie wurden die Mathematiklehrkräfte der PISA-Klassen 2003/04 befragt und getestet. Zentraler Bestandteil von COACTIV sind die Tests zum fachdidaktischen Wissen und zum Fachwissen von Mathematiklehrkräften der Sekundarstufe. Die vorliegende Publikation stellt Konzeptualisierung und Operationalisierung der beiden Wissensbereiche erstmals umfassend vor und beschreibt die Testkonstruktion ausführlich, wobei zur Illustration auch auf bislang noch unveröffentlichtes Itemmaterial zurückgegriffen wird. Unter anderem die folgenden wichtigen Fragen werden mit den Tests untersucht: Welche Unterschiede gibt es hinsichtlich der Schulformen? Wie hängen fachdidaktisches Wissen und Fachwissen mit der Berufserfahrung zusammen? Welche Zusammenhänge bestehen zwischen den beiden Wissensbereichen und subjektiven Überzeugungen der Lehrkräfte sowie Aspekten des Unterrichts? Inwieweit trägt das professionelle Wissen einer Mathematiklehrkraft zum Lernfortschritt der Schülerinnen und Schüler bei?AbstractThe COACTIV study surveyed and tested the mathematics teachers of the classes sampled for PISA 2003/04 in Germany. The study’s key components were newly developed tests of teachers’ pedagogical content knowledge and content knowledge. This article gives a comprehensive report of the conceptualization and operationalization of both domains of knowledge and describes the construction of the COACTIV tests in detail, presenting previously unpublished items as illustrative examples. Findings from the tests are used to address questions including the following: What differences are there across school types? How are pedagogical content knowledge and content knowledge related to teaching experience? How are the two domains of knowledge related to teachers’ subjective beliefs, on the one hand, and to aspects of their instruction, on the other? To what extent does teacher’s professional knowledge contribute to students’ learning gains?

Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do?

The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.

Grundlagen der Ergänzung des internationalen PISA-Mathematik-Tests in der deutschen Zusatzerhebung

Im Mai 2000 wurden in 33 Landern im Auftrag der OECD die Tests der PISA-Studie (PISA=Programme for International Student Assessment) durchgefuhrt; im Herbst 2001 ist ein erster Bericht zu erwarten. Die Studien im Rahmen von PISA finden in Deutschland aufgeteilt in den internationalen Test und nationale zusatzerhebungen statt. Beide Testteile erganzen sich. In diesem Framework wird die Notwendigkeit einer deutschen Erganzung dargelegt, deren Schwerpunkte im Vergleich zum internationalen Test beschrieben, sowie die Einordnung des Gesamt-Tests in deutsche curriculare Gegebenheiten durch eine geeignete Klassifikation der Items vorgenommen. Die Entwicklung des deutschen Frameworks ist am Aufbau des internationalen PISA-Frameworks fur den Untersuchungsteil „mathematical literacy” orientiert. Es erweitert und differenziert dieses jedoch aufgrund in Deutschland vorliegender mathematikdidaktischer Sichtweisen und spezifischer Ausrichtungen des deutschen Mathematikunterrichts.

ICMI Study 14: Application and Modelling in Mathematics Education — Discussion Document

ZusammenfassungICMI hat in der Reihe der ICMI Studies eine neue Studie über Anwendungen und Modellbildung im Mathematikunterricht initiiert. Der vorliegende Beitrag ist das Discussion Document zu dieser Studie.AbstractIn the series of ICMI Studies, ICMI has mounted a new study on applications and modelling in mathematics education. This paper is the Discussion Document for this study.