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

2. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content 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. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Part A: Background

4. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Change in mathematical Power (II)...leads to: Changes in the fields of mathematics: - RESEARCH - Changes in mathematical education - EDUCATION - Development of new fields of research Development of new methods of research Expansion of necessary mathematical competences Development of new teaching and learning

5. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content 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 New Focus on Mathematical Competence: for Mathematicians for Users of Mathematics! AND Oriented towards understanding Independence in the learning process Interdisziplinarity and Soft Skills

6. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Potential of Digital Media (within the context eLearning, eTeaching & eResearch) Reusability & Recomposition Continous Availability (platform independence) Pedagogical & Educational Aspects Organisational & Logistical Aspects Modelling (Simulation, Numerics, Visualisation) Interactivity (Experiment, Exploration, Instruction) Cooperation (Communication, Collaboration, Coordination) Adaptability (Learning styles & individual requirements)

7. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Development of eLearning Technology: Object of current research and development Used in many national and international universities First 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 Next Generation:

8. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content between potential and reality! We have to face a huge divergence Electronic Media in Education is Dramatically Wasted ! So far: The Potential of

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

10. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Part B: The Mumie – A Virtual Knowledge Space for Mathematics

11. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie - Philosophy: Support of multiple learning scenarios Support of classroom teaching Open Source General Design Approaches: Pedagogical Concept: Content Guidelines: Technical Concept: 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 Modularity - Granularity Mathematical rigidness and precision Division between teacher and author Division between content and application Strict division between content and context 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

12. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie – Fields of Learning: Courses from granular elements of knowledge Composition with the CourseCreator tool Interactive multimedia elements Nonlinear navigation Exercises Combined into exercise paths Interactive, constructive Embedded in an exercise network Intelligent input mechanisms Intelligent control mechanisms Knowledge networks User defined construction Includes an encyclopaedia

13. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie – Interrelation of Fields of Learning:

14. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie (Content) - CourseCreator: Course without content Course with content

15. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie (Practice) – Exercise Network:

16. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie (Retrieval) – Knowledge Nets II: General Relations Network of the Internal Structure of Statements

17. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Mumie Technology – Core Architecture: Database (Central Content Storage) Java Application Server (processing of queries, delivery of documents) Browser

18. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Part C: Structures of Mathematical Content

19. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content 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. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Structure levels within mathematical texts (1-2): Level 1: Level 2: Taxonomy of the Field (content structure and content relations) Entities and their interrelations (structure of text and relation of its parts)

21. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Level 3: Level 4: Internal structure of the entities (structure of the text within the entities) Syntax and Semantics of mathematical language (analysis of symbols and relations between the symbols) Structure levels within mathematical texts (3-4):

22. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Structural Level 1: Taxonomy Hierarchical Model

23. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Network Model Structural Level 1: Taxonomy

24. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Taxonomy of Linear Algebra – The Cube: hu

25. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content The Dimensions of The Cube: 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) Principal of Duality: Geometrical Structure: Linear algebra without geometry – with norm (length) added – with scalar product (angles) added Spaces and structural invariants – abstract and concrete: 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. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Detailed View of The Cube:... Just to add to the confusion... ;-)

27. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Structural level 2: Entities & the Rules of their Arrangement - Content ClassElement NameAttribute flavourParent Object 1 motivation -Any node or element 1 application -Any node or element 1 remark alert, reflective, associative, generalAny node or element 1 history biography, field, resultAny node or element 2 definition -Element container 2 theorem theorem, lemma, corollar, algorithmElement container 2 axiom -Element container 3 proof pre-sketch/complete, post-sketch/completeElement name=theorem 3 demonstration example, visualisationAny Element

28. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Structural level 3: Internal structure of Entities (I) definition axiom

29. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content theorem Structural level 3: Internal structure of Entities (II)

30. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content history (biogr.) proof Structural level 3: Internal structure of Entities (III)

31. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content No internal structure provided for the following elements: motivation application remark history (field, result) demonstration Structural level 3: Internal structure of Entities (IV)

32. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Part D: Next Steps - Vision

33. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content From Mumie... to Multiverse!!

34. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content 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. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content Multiverse – Fields: Fields of Innovation & Research Fields of Integration & Research

36. Sabina JeschkeTU Berlin Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content The End!