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Different teacher programs and university courses in teacher education aim at changing the participants’ beliefs towards a view of mathematics as a process and of mathematics learning as a constructivist act. They often include vivid experiences of problem solving and epistemic reflection. This paper describes a case study based on a pre-post interview design with pre-service teachers during such a course. One participant’s belief system is analysed with respect to the beliefs before and after the course, the way of reasoning expressing these beliefs and the structure of the belief system. The results show amongst others that new beliefs evolve without the old ones being rejected. This leads to a more or less conscious ambiguity and to a conflict within the belief system. General results are briefly presented by describing a typology of belief change taking into account the whole sample of eight students.

Diagnosing student achievement in a formative way is a crucial skill for planning and carrying out effective mathematics lessons. This study takes a subject-specific view and aims at investigating diagnostic competence in the field of mathematical functions at secondary level and how to improve it. Following three evidence-based design principles, a Continuous Professional Development (CPD) unit has been designed for a statewide official in-service teacher program of the Ministry for Education in Baden-Wurttemberg, Germany. The workshops of this CPD unit are presented in this paper and were investigated in a first pilot study (N = 26). To this end, we qualitatively analyzed the participants’ diagnostic competence before and after the training. We can report different diagnostic profiles of teachers that contain increased application of pedagogical content knowledge through the training in the field of functions. A number of teachers carried out increasingly profound diagnostic judgments after the training and shifted from a corrective to a descriptive or analytical view on student achievement. We furthermore report on steps to implement the unit on a statewide scale as part of a facilitators’ training and adjustments of the training’s content and methods based on our results of the pilot study.

Research on personal epistemology is confronted with theoretical issues as there exist conflicting data regarding its coherence, discipline-relation and contextdependence as well as methodological issues regarding the often used questionnaires to measure epistemological beliefs. We claim that it is necessary to distinguish between relatively stable “epistemological beliefs” and situationspecific “epistemological judgments”. In a sequence of interviews with regard to the topic of “certainty of mathematical knowledge”, we show that the usual categories used in questionnaires to measure epistemological beliefs have to be differentiated. We argue that epistemological judgments provide a promising framework to interpret the statements of the interviewees.

Welches fachwissenschaftliche Wissen sollen kunftige Lehrkrafte fur das Lehramt Grundschule erwerben? Und welche Formate konnen geeignete Lerngelegenheiten in der universitaren Phase haben? Die oft nur wenigen Studienanteile fur fachliche Lernprozesse erfordern eine fachwissenschaftlich substantielle und zugleich berufsfeldbezogene elementarmathematische Bildung von hoherem Standpunkt (F. Klein), etwa in der Geometrie oder Arithmetik. Sollen daruber hinaus aber auch Grunderfahrungen zur Mathematik moderner Pragung als Anwendungswissenschaft, als Strukturwissenschaft und als Feld individueller kreativer und heuristischer Betatigung (H. Winter) vermittelt werden, braucht es einen Gegenstand, der ebenso mathematisch reichhaltig wie zuganglich ist. Mit Bezug auf nationale und internationale Erfahrungen wird in diesem Beitrag die Theorie und Anwendung von Graphen vorgeschlagen und bildungstheoretisch und kompetenztheoretisch diskutiert. Konkrete Beispiele aus der Grundschullehrerbildung untermauern, wie weitreichend das Thema epistemologische, wissenschaftstheoretische und auch didaktische Fragestellungen eroffnet.

Sie mogen fragen: Was machen Farbeprobleme in der Mathematik? Wird hier et-wa eine Asthetik des Farbempfindens in Zahlen verpackt und verrechnet? Ganz und gar nicht, es geht weiterhin um kombinatorische Optimierung. Wir treffen hier nur auf einen sympathischen Zug der Mathematik: manchmal bleiben die Bezeichnungen der Situationen, aus denen eine mathematische Theorie entstand, an ihr haften, so auch bei dem folgenden Problem: Wenn man eine Landkarte so farben mochte, dass Lander mit gemeinsamer Grenze immer verschiedene Farben haben, mit wie vielen Farben kommt man dann aus? (siehe Abbildung 1)