Dr. Sabina Jeschke

1. Mathematics inVirtual Knowledge Spaces Ontological Structures of Mathematical Content MMISS-Meeting Bremen 21-22. April 2004 Dr. Sabina Jeschke

2. Outline: • Part A: Background • Part B: The „Mumie“ – A Virtual Knowledge Space for Mathematics • Part C: Structures of Mathematical Content • Part D: Next Steps - Vision

3. Part A: • Background

4. ...leads to: Changes in the fields of mathematics: - RESEARCH - Changes in mathematical education - EDUCATION - „Change in mathematical Power“ (II) • Development of new fields of research • Development of new methods of research • Expansion of necessary mathematical competences • Development of new teaching and learning

5. New Focus on Mathematical Competence: • Understanding of the the potential and performance of mathematics • Formulating, modelling and solving problems within a given context • Mathematical thinking and drawing of conclusions • Understanding of the interrelations between mathematical concepts and ideas • Mastery of mathematical symbols and formalisms • Communication through and about mathematics • Reflected application of mathematical tools and software • Oriented towards understanding • Independence in the learning process • Interdisziplinarity and Soft Skills for Mathematicians for Users of Mathematics! AND

6. Potential of Digital Media (within the context eLearning, eTeaching & eResearch) Modelling (Simulation, Numerics, Visualisation) Interactivity (Experiment, Exploration, Instruction) Cooperation (Communication, Collaboration, Coordination) Adaptability (Learning styles & individual requirements) Pedagogical & Educational Aspects Reusability & Recomposition Continous Availability (platform independence) Organisational & Logistical Aspects

7. First Generation: • Next Generation: Information distribution Document management Passive, statical objects „Simple“ training scenarios Electronic presentation (Isolated) communication scenarios • Adaptive content authoring • Dynamical content management • Modular, flexible elements of knowledge • High degree of interactivity • Complex training scenarios • Cooperative environments • Support of active, explorative learning processes • Advanced human machine interfaces Used in many national and international universities Object of current research and development Development of eLearning Technology:

8. We have to face a huge divergence between potential and reality! So far: The Potential of Electronic Media in Education is Dramatically Wasted!

9. Monolithic design of most eLearning software Open heterogeneous platform-independent portal solutions integrating virtual cooperative knowledge spaces Missing granularity and missing ontological structure of contents Analysis of self-immanent structures within fields of knowledge and development of granular elements of knowledge Use of statical typographic objects Use of active, executable objects and processes with semantic description (Technical) Causes for the Divergence:

10. Part B: • The „Mumie“ – A Virtual Knowledge Space for Mathematics

11. Mumie - Philosophy: General Design Approaches: Pedagogical Concept: • Support of multiple learning scenarios • Support of classroom teaching • Open Source • Visualisation of intradisciplinary relations • Nonlinear navigation • Visualisation of mathematical objects and concepts • Support of experimental scenarios • Support of explorative learning • Adaptation to individual learning processes Technical Concept: • Field-specific database structure • XML technology • Dynamic „on-the-fly“ page generation • Strict division between content, context & presentation • Customisable presentation • MathML for mathematical symbols • LaTeX (mmTeX) as authoring tool • Transparency and heterogenerity Content Guidelines: • Modularity - Granularity • Mathematical rigidness and precision • Division between teacher and author • Division between content and application • Strict division between content and context

12. Mumie – Fields of Learning: • Courses from granular elements of knowledge • Composition with the CourseCreator tool • Interactive multimedia elements • Nonlinear navigation • Knowledge networks • User defined construction • Includes an „encyclopaedia“ • Exercises • Combined into exercise paths • Interactive, constructive • Embedded in an exercise network • Intelligent input mechanisms • Intelligent control mechanisms

13. Mumie – Interrelation of Fields of Learning:

14. Mumie (Content) - CourseCreator: Course with content Course without content

15. Mumie (Practice) – Exercise Network:

16. Mumie (Retrieval) – Knowledge Nets II: Network of the Internal Structure of Statements General Relations

17. Mumie Technology – Core Architecture: Database (Central Content Storage) Java Application Server (processing of queries, delivery of documents) Browser

18. Part C: • Structures of • Mathematical Content

19. We need a high degree of contentual structuring: Contentual structuring of fields of knowledge „Ontology“ • Formal (~ machine-readable) description of the logical structure of a field of knowledge • Standardised terminology • Integrates objects AND their interrelation • Based on objectifiable (eg logical) structures • „Explicit“ specification is a basic requirement • Ideally: A model of the „natural“ structure independent of use and user preference • Ideally: A model independent of subjective or individual views

20. Structure levels within mathematical texts (1-2): Level 1: Taxonomy of the Field (content structure and content relations) Level 2: Entities and their interrelations (structure of text and relation of its parts)

21. Structure levels within mathematical texts (3-4): Level 3: Internal structure of the entities (structure of the text within the entities) Level 4: Syntax and Semantics of mathematical language (analysis of symbols and relations between the symbols)

22. Structural Level 1: Taxonomy Hierarchical Model

23. Structural Level 1: Taxonomy „Network“ Model

24. Taxonomy of Linear Algebra – „The Cube“: hu

25. The Dimensions of „The Cube“: Geometrical Structure: • Linear algebra without geometry – with norm (length) added – with scalar product (angles) added Spaces and structural invariants – abstract and concrete: Principal of Duality: • Linear Space - Dual Space - Space of bilinear forms - Space of multilinear forms • Linear mapping induces structure through the principle of duality • Concept of inductive sequences (0, 1, ..., n) • Vector Spaces are spaces with a linear structure – linear mappings preserve the linearity between vector spaces • Vector spaces and linear mappings exist in an abstract and in a concrete sense (including coordinates)

26. Detailed View of „The Cube“: ... Just to add to the confusion ... ;-)

27. Structural level 2: Entities & the Rules of their Arrangement - Content

28. Structural level 3: Internal structure of Entities (I) definition axiom

29. Structural level 3: Internal structure of Entities (II) theorem

30. Structural level 3: Internal structure of Entities (III) proof history (biogr.)

31. Structural level 3: Internal structure of Entities (IV) No internal structure provided for the following elements: • motivation • application • remark • history (field, result) • demonstration

32. Part D: • Next Steps - Vision

33. From Mumie ... to Multiverse!!

34. Multiverse – Idee, Programm, Ziele: • Enhancement of existing projects • Development of next-generation technology • Integration of existing separate applications • Enhancement for research applications • Support of cooperative research • Internationalisization of education • Transparency of education in Europe

35. Multiverse – Fields: Fields of Innovation & Research Fields of Integration & Research

36. The End!