Susanne Prediger

The Lattice of Concept Graphs of a Relationally Scaled Context

The aim of this paper is to contribute to Data Analysis by clarifying how concept graphs may be derived from data tables. First it is shown how, by the method of relational scaling, a many-valued data context can be treinsformed into a power context family. Then it is proved that the concept graphs of a power context family form a lattice which can be described as a subdirect product of specific intervals of the concept lattices of the power context fajnily (each extended by a new top-element). How this may become practical is demonstrated using a data table about the domestic flights in Austria. Finally, the lattice of syntactic concept graphs over an alphabet of object, concept, and relation names is determined and related to the lattices of concept graphs of the power context families which are semantic models of the given contextual syntax.

Logical Scaling in Formal Concept Analysis

Logical scaling is a new method to transform data matrices which are based on object-attribute-value-relationships into data matrices from which conceptual hierarchies can be explored. The derivation of concept lattices is determined by terminologies expressed in a formallogical language.

Diversity of theories in mathematics education—How can we deal with it?

This article discusses the central question of how to deal with the diversity and the richness of existing theories in mathematics education research. To do this, we propose ways to structure building and discussing theories and we contrast the demand for integrating theories with the idea of networking theories.

Networking of Theories—An Approach for Exploiting the Diversity of Theoretical Approaches

Internationally, mathematics education research is shaped by a diversity of theories. This contribution suggests an approach for exploiting this diversity as a resource for richness by the so-called networking of theories. For being able to include different traditions, this approach is based on a tolerant and dynamic understanding of theories that conceptualizes theories in their dual character as frame and as result of research practices. Networking strategies are presented in a landscape, linearly ordered according to their degree of integration. These networking strategies can contribute to the development of theories and their connectivity and, hence, offer an interesting research strategy for the didactics of mathematics as scientific discipline.

Simple Concept Graphs: A Logic Approach

Conceptual Graphs and Formal Concept Analysis are combined by developing a logical theory for concept graphs of relational contexts. Therefore, concept graphs are introduced as syntactical constructs, and their semantics is defined based on relational contexts. For this contextual logic, a sound and complete system of inference rules is presented and a standard graph is introduced that entails all concept graphs being valid in a given relational context. A possible use for conceptual knowledge representation and processing is suggested.

Terminologische Merkmalslogik in Der Formalen Begriffsanalyse

In den letzten Jahren gab es immer wieder Anstose, logische Sprachelemente in dieFormale Begriffsanalyseeinzufuhren. Vor allem bei den Aktivitaten im Bereich der begrifflichen Wissensverarbeitung entstand zunehmend das Bedurfnis nach formal-logischen Ausdrucksmitteln. Den Anlas, eine logische Sprache fur die Formale Begriffsanalyse zu entwerfen, lieferte das Vorhaben, die von Edwin Diday entwickelteSymbolische Datenanalysein engere Verbindung zur Formalen Begriffsanalyse zu bringen (vgl. [Pre96]). Dieser Anlas legte nahe, eine Merkmalslogik mit extensionaler Semantik bereitzustellen.

Networking of Theories as a Research Practice in Mathematics Education

PART A: Introduction.- Chapter 0: Susanne Prediger & Angelika Bikner-Ahsbahs: Preface.- Chapter 1: Angelika Bikner-Ahsbahs, Susanne Prediger, Michele Artigue, Ferdinando Arzarello, Marianna Bosch, Tommy Dreyfus, Stefan Halverscheid, Mariam Haspekian, Ivy Kidron, Alexander Meyer, Cristina Sabena, & Ingolf Schafer: Starting points for dealing with the diversity of theory.- Chapter 2: Cristina Sabena: Description of the data: Introducing the session of Ciro, Gabriele and the exponential function.- PART B: Diversity of theories.- Chapter 3: Ferdinando Arzarello & Cristina Sabena: Introduction to the approach of Action, Production and Communication (APC).- Chapter 4: Michele Artigue & Mariam Haspekian & Agnes Lenfant: Introduction to the Theory of Didactical Situations (TDS).- Chapter 5: Marianna Bosch & Josep Gascon: Introduction to the Anthropological Theory of the Didactic (ATD).- Chapter 6: Tommy Dreyfus & Ivy Kidron: Introduction to Abstraction in Context (AiC).- Chapter 7: Angelika Bikner-Ahsbahs & Stefan Halverscheid: Introduction to the Theory of Interest-Dense Situations (IDS).- PART C: Case studies of Networking.- Chapter 8: Susanne Prediger & Angelika Bikner-Ahsbahs: Introduction to networking: Networking strategies and their background.- Chapter 9: Tommy Dreyfus, Cristina Sabena, Ivy Kidron, Ferdinando Arzarello: The Epistemic Role of Gestures - A case study on networking of APC and AiC.- Chapter 10: Ivy Kidron, Michele Artigue, Marianna Bosch, Tommy Dreyfus, Mariam Haspekian: Context, milieu and media-milieu dialectic - A case study on networking of AiC, TDS, and ATD.- Chapter 11: Cristina Sabena, Ferdinando Arzarello, Angelika Bikner-Ahsbahs, Ingolf Schafer: The epistemological gap - A case study on networking of APC and IDS.- CHAPTER 12 Angelika Bikner-Ahsbahs, Michele Artigue & Mariam Haspekian: Topaze Effect - A case study on networking of IDS and TDS.- PART D: Reflections.- Chapter 13: Stefan Halverscheid: Beyond the official academic stage - Dialogical intermezzo.- Chapter 14: Angelika Bikner-Ahsbahs & Susanne Prediger: Networking as research practices : methodological lessons learnt from the case studies.- Chapter 15: Michele Artigue & Marianna Bosch: Reflection on Networking through the praxeological lens.- Chapter 16: Kenneth Ruthven: From networked theories to modular tools?.- Chapter 17: Luis Radford: Theories and their networking - A Heideggerian commentary.- Appendix.- Index.

Formal concept analysis for general objects

General objects are classes of individual objects that are considered to be extents of concepts of a formal context. In this paper, different contexts with general objects are defined and their conceptual structure and relation to other contexts is analyzed with methods of Formal Concept Analysis.

Purposefully Relating Multilingual Registers: Building Theory and Teaching Strategies for Bilingual Learners Based on an Integration of Three Traditions

Starting from revisiting three traditions of reflecting on linguistic transitions between registers and representations, we suggest the integrated approach of purposefully relating registers. The result is likely to enhance language-sensitive teaching strategies in multilingual classrooms that aim at conceptual understanding. Two empirical snapshots from design experiments illustrate this potential for teaching and learning mathematics.

Mathematiklernen als interkulturelles Lernen -Entwurf fur einen didaktischen Ansatz

ZusammenfassungAusgehend von der Transfer-Problematik wird ein Ansatz entworfen, Mathematiklernen als interkulturelles Lernen aufzufassen. Dabei wird Mathematik als Kultur formalen Denkens betrachtet, die dem individuellen Alltagsdenken entgegentritt und dann integriert werden muss. Pädagogische Konzepte zum interkulturellen Lernen sind hilf-reich, diesen Prozess zu analysieren und Ideen für seine Unterstützung zu entwickeln. Grundlage für eine solche Perspektive auf das Mathematiklernen ist die in der neueren Philosophie der Mathematik vertretene kulturalistische Sicht auf Mathematik.AbstractStarting with the problem of transfer, an approach is developed to consider mathematics learning as intercultural learning. For this, mathematics is understood as a culture of formal thinking, that shall be integrated into the general thinking of each individual learner. In order to analyse this process and to give ideas how to support it, pedagogical conceptions about intercultural learning are activated. The basis of such an approach to mathematics learning is given by newer tendencies in the philosophy of mathematics where a culturalistic viewpoint becomes more and more common.