dr. sabina jeschke mmiss-meeting bremen 21-22. april 2004 mathematics in virtual knowledge spaces...
TRANSCRIPT
Dr. Sabina JeschkeMMISS-Meeting Bremen
21-22. April 2004
Mathematics inVirtual
Knowledge Spaces
Ontological Structures
of Mathematical Content
Sabina Jeschke TU 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
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Part A:Background
Sabina Jeschke TU 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
Sabina Jeschke TU 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 onMathematical Competence:
forMathematicia
ns
for Users ofMathematics!
AND
Oriented towards
understanding
Independence in the
learning process
Interdisziplinarity and
Soft Skills
Sabina Jeschke TU 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)
Sabina Jeschke TU 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:
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
between potential
and reality!
We have to face
a huge divergence
Electronic Media in Educationis Dramatically
Wasted !
So far:
The Potential of
Sabina Jeschke TU 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
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Part B:The „Mumie“ – A
Virtual Knowledge Space for Mathematics
Sabina Jeschke TU 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:
PedagogicalConcept:
Content Guidelines:
TechnicalConcept:
•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
Sabina Jeschke TU 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“
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Mumie – Interrelation of Fields of Learning:
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Mumie (Content) - CourseCreator:
Course without content
Course with content
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Mumie (Practice) – Exercise Network:
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Mumie (Retrieval) – Knowledge Nets II:
General Relations
Network of the Internal
Structureof Statements
Sabina Jeschke TU 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
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Part C:Structures ofMathematical Content
Sabina Jeschke TU 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
Sabina Jeschke TU 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)
Sabina Jeschke TU 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):
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Structural Level 1: Taxonomy
Hierarchical Model
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
„Network“ Model
Structural Level 1: Taxonomy
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Taxonomy of Linear Algebra – „The Cube“:
hu
Sabina Jeschke TU 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)
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Detailed View of „The Cube“:
... Just to add to the confusion ... ;-)
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Structural level 2: Entities & the Rules of their Arrangement - Content
Class Element Name Attribute „flavour“ Parent Object
1 motivation - Any node or element
1 application - Any node or element
1 remark alert, reflective, associative, general Any node or element
1 history biography, field, result Any node or element
2 definition - Element container
2 theorem theorem, lemma, corollar, algorithm Element container
2 axiom - Element container
3 proof pre-sketch/complete, post-sketch/complete Element name=theorem
3 demonstration example, visualisation Any Element
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Structural level 3: Internal structure of Entities (I)
definition
axiom
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
theorem
Structural level 3: Internal structure of Entities (II)
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
history (biogr.)
proof
Structural level 3: Internal structure of Entities (III)
Sabina Jeschke TU 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)
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Part D:Next Steps - Vision
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
From Mumie ... to Multiverse!!
Sabina Jeschke TU 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
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
Multiverse – Fields:
Fields of Innovation & Research
Fie
lds o
f In
teg
rati
on
&
Researc
h
Sabina Jeschke TU Berlin
Mathematik in Virtuellen Wissensräumen – Ontological Structures of Mathematical Content
The End!