Timo Leuders

Outcome standards and core curricula: a new orientation for mathematics teachers in Germany

One of the pivotal innovations in Germany after publication and discussion of the results of international large scale assessment studies is the introduction of national educational standards for major subjects. The 16 German federal states have committed themselves to implement these standards by developing core curricula for the use in schools and by regularly testing the achievement of students. This development is expected to have considerable impact on the outcome of education. In this article we give, a picture of the current development in the German education system by describing the design of the core curriculum in one of the federal states (NRW) and discuss its function in bridging the gap between national educational standards and teaching practice

Diagnostic Competence of Mathematics Teachers: Unpacking a Complex Construct

Although diagnostic competence of teachers is regarded as a key component of successful teaching, there are many open questions regarding the structure, the development and the impact of diagnostic competence. This chapter presents an overview of different approaches to pinpoint diagnostic competence theoretically and to investigate it empirically: measuring judgment accuracy, assessing competences in diagnostic situations or analyzing judgment processes. These approaches are discussed with respect to their advantages, restrictions as well as some of their main findings and they are allocated within an overarching model of diagnostic competence as a continuum, comprising diagnostic dispositions, diagnostic thinking and diagnostic performance.

Development, Validation, and Application of a Competence Model for Mathematical Problem Solving by Using and Translating Representations of Functions

In mathematics education, the student’s ability to translate between different representations of functions is regarded as a key competence for mastering situations that can be described by mathematical functions. Students are supposed to interpret common representations like numerical tables (N), function graphs (G), verbally or pictorially represented situations (S), and algebraic expressions (A). In a multi-step project (1) a theoretical competence model was constructed by identifying key processes and key dimensions and corresponding item pools, (2) different psychometric models assuming theory-based concurrent competence structures were tested empirically, and (3) finally, a computerized adaptive assessment tool was developed and applied in school practice.

Assessing the Development of Educational Research Literacy: The Effect of Courses on Research Methods in Studies of Educational Science

The ability to purposefully access, reflect, and use evidence from educational research (Educational Research Literacy) is expected of future professionals in educational practice. Based on the presented conceptual framework, a test instrument was developed to assess the different competency aspects: Information Literacy, Statistical Literacy, and Evidence-based Reasoning. This paper addresses the question of whether this test allows monitoring students’ development during courses on research methods in educational science. In winter semester 2013/14, two courses in Early Education and Teacher Training degree programs were accompanied. Both cross-sectional (multidimensional latent regression models) and longitudinal (paired t tests of plausible values) comparisons are reported. The results indicate no significant differences in competencies between the degree programs at start of the semester. For the end of semester, only data of the Early Education course was available. These course participants showed higher proficiency in all aspects of educational research literacy. In conclusion, the results are discussed with regard to the goals of the assessment of competencies in higher education, particularly the use of test feedback for course development, and desiderata for future research (e.g., ability to transfer research into practice) are stated.

Quality development in mathematics education by focussing on the outcome: new answers or new questions?

1. The paradigm shift: from input to outcome At present the German education system undergoes a drastic change, introducing new steering instruments which are expected to influence teaching practice considerably. As a consequence of the mediocre results of German students in international large scale assessment studies the governments of the sixteen Federal States unanimously decreed that quality improvement of schools should henceforth be steered by the “outcome” instead of the “input” (cf. article by Blum et al. in this issue). The education system has for a long time nurtured the false hope that quality could be achieved and secured by making detailed prescriptions with regard to teachers’ education, by equipping schools and by defining the obligatory content in great detail. After seeing that this approach has not proven sufficiently successful, the new hope grows that quality improvement is better reached by defining expected competencies and by assessing them centrally. In central guidelines that appear radically narrow teachers and schools receive increased responsibility. Since this steering system combines central and decentral strategies it is often referred to as “integrated quality management” (Rolff 2004). The change we just outlined is the most fundamental reform in Germany since the education reform at the end of the 1960s or even since the restoration of the diversified school system after the Second World War (cf. Baumert, Cortina & Leschinsky 2003). The enormous speed and the wide extent of the recently initiated change can be better understood when seen in the historical context:

Vernetzte Kompetenzen statt trägen Wissens – Ein Studienmodell zur konsequenten Vernetzung von Fachwissenschaft, Fachdidaktik und Schulpraxis

Der mathematikdidaktische Teil eines Lehramtsstudiums zielt auf den Erwerb fachwissenschaftlicher, fachdidaktischer und unterrichtspraktischer Kompetenzen und muss dabei die Anforderungen von Wissenschafts- und Berufsorientierung zugleich beachten. Ein in die einzelnen Bereiche fragmentierter Kompetenzerwerb birgt die Gefahr, trages Wissen zu produzieren, welches in der spateren Praxis nicht genutzt werden kann. Bereits im Studium sollte daher die Integration der verschiedenen Kompetenzbereiche angelegt und systematisch gefordert werden. Im vorliegenden Beitrag wird das Studienmodell des IMBF (Institut fur Mathematische Bildung Freiburg) vorgestellt, das einen solchen integrierten Kompetenzerwerb realisiert.

Inductive and Deductive Justification of Knowledge: Flexible Judgments Underneath Stable Beliefs in Teacher Education

ABSTRACT Personal epistemological beliefs are considered to play an important role for processes of learning and teaching. However, research on personal epistemology is confronted with theoretical issues as there is conflicting evidence regarding the structure, stability, and context-dependence of epistemological beliefs. We give evidence how theoretical and methodological issues can partly be resolved by distinguishing between relatively stable “epistemological beliefs” and situation-specific “epistemological judgments.” A qualitative content analysis of a series of semistructured interviews (study 1) with pre-service teachers, teachers, and teacher educators as well as a statistical analysis of pre-service teachers’ extensive answers in questionnaires (study 2), both on the topic of “mathematical discovery,” reveal not only beliefs of the participants but also different qualities of judgments. Therefore, in further research both aspects of beliefs should be considered in a more differentiated manner when categorizing belief structures.

Diagnostische Prozesse und Ressourcen von Mathematiklehrpersonen

Im Unterricht gibt es zahlreiche Situationen, in denen diagnostische Tatigkeiten von Lehrkraften notwendig sind und verschiedene Kompetenzen erfordern. Schrader (2006) definiert diagnostische Kompetenz als die „Fahigkeit eines Urteilers, Personen zutreffend zu beurteilen“ (ebd., S. 95). Im Mathematikunterricht bezieht sich Diagnose neben der Beurteilung von Personenmerkmalen auch auf die Einschatzung von Lernund Aufgabenanforderungen (z.B. Brunner et al., 2011). Wahrend die Bedeutung diagnostischer Kompetenz fur den Mathematikunterricht als hoch eingestuft wird (Helmke et al., 2004; Anders et al., 2010), wird Lehrkraften eine unzureichende Ausbildung diagnostischer Kompetenz bescheinigt (Krauss & Brunner, 2011). Als Indikator diagnostischer Kompetenz dient dabei haufig die Genauigkeit eines Urteils (Veridikalitat). Allerdings bestehen noch Wissenslucken im Hinblick auf die Struktur und die Genese diagnostischer Urteile, ein empirisch fundiertes und uberpruftes Modell diagnostischer Kompetenz fehlt bislang (Schrader, 2011; Anders et al., 2010).

Ein diagnostisches Kompetenzstrukturmodell für ein heuristisches Arbeiten mit Repräsentationen von Funktionen und seine empirische Überprüfung

Ziel des von der DFG im Schwerpunktprogramm „Kompetenzmodelle zur Erfassung individueller Lernergebnisse und zur Bilanzierung von Bildungsprozessen“ seit 2007 geförderten Projektes ist die Konstruktion und Überprüfung eines Kompetenzmodells für das mathematische Problemlösen und Modellieren von Schülerinnen und Schülern der Sekundarstufe I in solchen Situationen, in denen Prozesse des Wachstums und der Veränderung mathematisch erfasst werden. Zentrale Bedeutung für Problemlösen und Modellieren in diesen Zusammenhängen besitzt die heuristische Verwendung der fundamentalen mathematischen Darstellungsarten „numerisch“, „graphisch“, „symbolisch“ und „verbal“ sowie der Wechsel zwischen ihnen. National und international bewährte theoretische Fähigkeitsmodelle aus der Mathematikdidaktik werden operationalisiert und in Form eines Kompetenzstrukturmodells empirisch überprüft. Langfristiges Ziel ist die Bereitstellung eines empirisch fundierten Instrumentariums für die Schulpraxis zur Diagnostik der Stärken und Schwächen von Lerngruppen und Individuen im Inhaltsbereich „Funktionale Veränderung“, eine Bestimmung von Förderbedarf und Abbildung von Kompetenzentwicklungen.

Investigating the Relationship Between Prospective Elementary Teachers’ Math-Specific Knowledge Domains

Notwithstanding long term efforts to differentiate between domains of mathematics-related teacher knowledge, there is no doubt that different forms and aspects of teacher knowledge are interrelated and mutually influence each other. However, the nature of this relation is still open to scholarly debate. First, we give an overview of empirical studies that investigated the relation between different domains of mathematics teachers’ knowledge, notably, the domains of “content knowledge” and “pedagogical content knowledge”. We demonstrate that the research on the relationship turns out to be multifaceted and we point to the need of cognitive orientated research on the integration of knowledge domains. Second, we present our own ongoing research on the integration of prospective elementary teachers’ math-specific knowledge domains by describing our use and analysis of task-based interviews. Preliminary findings indicate that our approach can help to identify mental processes that illuminate the integration of math-specific knowledge domains.