Mathematics in
Virtual Knowledge
Spaces
Ontological Structures
of Mathematical Content
MMISS-Meeting Bremen
21-22. April 2004
Dr. Sabina Jeschke
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
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:
Changes in
mathematical
education
- RESEARCH -
- EDUCATION -
Development of new
fields of research
Development of new
methods of research


Sabina Jeschke
Expansion of
necessary
mathematical
competences
Development of new
teaching and learning
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
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
for
Mathematicians
AND
Sabina Jeschke
Oriented towards
understanding
Independence in the
learning process
Interdisziplinarity and
Soft Skills
for Users of
Mathematics!
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
Potential of Digital Media
(within the context eLearning, eTeaching & eResearch)
Modelling (Simulation, Numerics, Visualisation)
Pedagogical
Interactivity (Experiment, Exploration, Instruction)
Cooperation (Communication, Collaboration, Coordination)
& Educational
Aspects
Adaptability (Learning styles & individual requirements)
Organisational
Reusability & Recomposition
& Logistical
Continous Availability (platform independence)
Sabina Jeschke
Aspects
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
Development of eLearning Technology:
First Generation:
Next Generation:
Information distribution
Document management
Passive, statical objects
„Simple“ training scenarios
Electronic presentation
(Isolated) communication scenarios
Used in many national
and international universities
Sabina Jeschke
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
Object of current research
and development
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
We have to face
a huge divergence
between potential
and reality!
So far:
The Potential of
Electronic Media in Education
is Dramatically Wasted
Sabina Jeschke
TU Berlin
!
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
(Technical) Causes for the Divergence:
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
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:
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
Sabina Jeschke
Content
Guidelines:
•Modularity - Granularity
•Mathematical rigidness and precision
•Division between teacher and author
•Division between content and application
•Strict division between content and context
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
•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
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 with content
Course without 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:
Network of the
Internal Structure
of Statements
General Relations
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 of
Mathematical 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“
•
•
•
•
•
•
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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:
Taxonomy of the Field
(content structure and content relations)
Level 2:
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
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)
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
Structural Level 1: Taxonomy
„Network“ Model
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“:
Geometrical Structure:

Spaces and structural
invariants – abstract
and concrete:


Linear algebra without geometry – with norm
(length) added – with scalar product (angles) added
Principal of Duality:

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
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)
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
Structural level 3:
Internal structure of Entities (II)
theorem
Sabina Jeschke
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
Structural level 3:
Internal structure of Entities (III)
proof
history
(biogr.)
Sabina Jeschke
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
Structural level 3:
Internal structure of Entities (IV)
No internal structure provided for the following elements:
•motivation
•application
•remark
•history (field, result)
•demonstration
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 Integration & Research
Fields of Innovation & Research
Sabina Jeschke
TU Berlin
Mathematik in Virtuellen Wissensräumen –
Ontological Structures of Mathematical Content
The End!
Sabina Jeschke
TU Berlin