To be or not to be … a Mathematician
“People don't learn to become [... mathematicians] by memorizing formulas; rather it's the implicit practices that matter most. Indeed, knowing only the explicit, mouthing the formulas, is exactly what gives an outsider away. Insiders know more. By coming to inhabit the relevant community, they get to know not just the “standard” answers, but the real questions, sensibilities, and aesthetics, and why they matter.“ (John Seely Brown, 2003)
Communities of Practice in MKM: an Extensional Model
Michael Kohlhase(International University Bremen)
Joint work with Andrea Kohlhase
(International University Bremen and University Bremen)
Overview
Handles on knowledge and MKM The social context: Mathematical Practices
Modeling a CoP
Communities of Practice (CoP) Meaning/Learning/Boundary/Community
Added-value support for presentation of mathematics via CoP model
Handles on Knowledge
Knowledge Management (Probst, Raub, Romhardt;1997)
“0“,“9“,“5“,“,“
Exchange rate
1 $ = 0,95 €
Market mechanisms concerning
exchange rates0,95
Character Set
Context NetworkingSyntax
Glyphs Data Information Knowledge
Form Content
Social Life of
Information
Mathematical Knowledge Space (Kohlhase, Kohlhase, 2005)
“0“,“9“,“5“,“,“
Exchange rate
1 $ = 0,95 €
Market mechanisms concerning
exchange rates0,95
Character Set
Context NetworkingSyntax
Glyphs Data Information Knowledge
PresentationDis-
ambiguation
Social Context
RelationRepresentation
What has MKM accomplished?
What has MKM not accomplished?
For whom is what when important about knowledge?
Social Context in MKM
For whom is what when important about knowledge?
Content/Form
Evaluation
User Role:Author/Recipient
Creator/AggregatorBeginner/Expert
User Status:Profile
Inner/Outer Motivation
Social Context in MKM
For whom is what when important about knowledge?
Content/Form
User Status:Profile
Inner/Outer Motivation
Evaluation
User Role:Author/Recipient
Creator/AggregatorBeginner/Expert
The Relevance of Practices
“People don't learn to become [... mathematicians] by memorizing formulas; rather it's the implicit practices that matter most. Indeed, knowing only the explicit, mouthing the formulas, is exactly what gives an outsider away. Insiders know more. By coming to inhabit the relevant community, they get to know not just the “standard” answers, but the real questions, sensibilities, and aesthetics, and why they matter.“ (John Seely Brown, 2003)
The Fascination of Practices
“… we interact with each other and with the world and we tune our relations with each other and the world accordingly, we learn. […] Over time, these collective learning results in practices […]” (Etienne Wenger, 1999)
Practices are Results and starting points for human
learning Constitute and represent social relations
Community of Practice (CoP) as Construct for Social Context in MKM
Participation Reification
Action Connection Objectification Evaluation
Meaning
The Problem of “Modeling” a CoP
Communities are no static objects have no clear-cut boundaries
Practices may change
How to model the dynamics of a CoP without inscribing the status quo?
The Artifacts of Math. Practice
Essential mathematical practice: Writing/reading mathematical documents
Document
Participation
Reification
=Living CoP object
The Artifacts of Math. Practice
Essential mathematical practice: Mathematical proofs
Proof
Explanation
Guarantee
=Living CoP object
Q E D
The Extensional CoP Model
Ideas A Collection of documents stays dynamic We treat the document collection as a fuzzy set
by real-valued membership-function Changing practices are congealed in their
artifacts Document Collection
CoP Objects Generalized
CoP as a fuzzy set of documents
Fuzzy, multi-dimensional value judgment function Relevance Soundness Presentation Originality Soundness
Mining Methods Manual evaluation Referee reports
Fuzzy Set of Documents
CoP Object Relations
CoP as collection of semantically annotated, interrelated documents Objective relations (system ontology) Subjective inter-document relations
(e.g. references) Intersubjective
practices
Document Collection
Characteristics of Communities of Practice Added-Value Services?
MeaningSocial negotiation
CommunityDiscourse as connection
between individuals and knowledge
Boundarymembership
LearningExploring a
shared repertoireDocument Collection
Added-Value Support for Presentation of Mathematics
Social Level
Where is my CoP?
Boundarymembership
Boundarymembership
Added-Value Support for Presentation of Mathematics
Social Level
Where is my CoP?
You know or someone may tell you … Statistical analysis of informal mass data
Comparison of value judgments (recommender systems)
Comparison of references (social bookmarking systems)
Paper #139
Added-Value Support for Presentation of Mathematics
Communityreference network
Discourse Level
(inter-document)
Document Collection
Paper #4
Paper #222
CommunityDiscourse as connection
between individuals and knowledge
Paper #132
Paper #26 Paper #224
Where is my place in a CoP?
Added-Value Support for Presentation of Mathematics
Discourse Level
(intra-document)
A Generated Document
Monoids Definition
for Monoid
Document Collection
Monoids
Definition for Monoid
Example: Strings with concatenation
Semigroups Groups
Example: Strings with concatenation
Example: Functions with composition
Example: Functions with composition
LearningExploring a
shared repertoire
OMDoc<presentation for=“binomial” role=“applied” xml:lang=“en”>
…</presentation>
Added-Value Support for Presentation of Mathematics
OMDoc<presentation for=“binomial” role=“applied” xml:lang=“en”>
…</presentation>
Binomial Coefficient:
nk
( ) n!k! * (n-k)!:=
Document Collection
We use the binomial coefficient
A Generated Document
nk
( )
Formula Level
MeaningSocial negotiation
Conclusion: Added-Value Services by CoP Model
Document Collection
MeaningSocial negotiation
CommunityDiscourse as connection
between individuals and knowledge
Boundarymembership
LearningExploring a
shared repertoire
Thank you!
“People don't learn to become [... mathematicians] by memorizing formulas; rather it's the implicit practices that matter most. Indeed, knowing only the explicit, mouthing the formulas, is exactly what gives an outsider away. Insiders know more. By coming to inhabit the relevant community, they get to know not just the “standard” answers, but the real questions, sensibilities, and aesthetics, and why they matter.“ (John Seely Brown, 2003)
Document Collection