Dr. Sabina Jeschke

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



Part A:Background



„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



• 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



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)



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:



between potential

and reality!

We have to face

a huge divergence

Electronic Media in Educationis Dramatically

Wasted !

So far:

The Potential of



(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



Part B:The „Mumie“ – A

Virtual Knowledge Space for Mathematics



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



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“



Mumie – Interrelation of Fields of Learning:



Mumie (Content) - CourseCreator:

Course without content

Course with content



Mumie (Practice) – Exercise Network:



Mumie (Retrieval) – Knowledge Nets II:

General Relations

Network of the Internal

Structureof Statements



Mumie Technology – Core Architecture:

Database(Central Content Storage)

Java Application Server(processing of queries,delivery of documents)

Browser



Part C:Structures ofMathematical 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



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)



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):



Structural Level 1: Taxonomy

Hierarchical Model



„Network“ Model

Structural Level 1: Taxonomy



Taxonomy of Linear Algebra – „The Cube“:

hu



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)



Detailed View of „The Cube“:

... Just to add to the confusion ... ;-)



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



Structural level 3: Internal structure of Entities (I)

definition

axiom



theorem

Structural level 3: Internal structure of Entities (II)



history (biogr.)

proof

Structural level 3: Internal structure of Entities (III)



No internal structure provided for the following elements:

•motivation•application•remark•history (field, result)

•demonstration

Structural level 3: Internal structure of Entities (IV)



Part D:Next Steps - Vision



From Mumie ... to Multiverse!!



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



Multiverse – Fields:

Fields of Innovation & Research

Fie

lds o

f In

teg

rati

on

&

Researc

h



The End!

Documents

Dr. Sabina Jeschke