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 rates 0,95 Character Set ContextNetworkingSyntax GlyphsDataInformationKnowledge Form Content Social Life of Information Mathematical Knowledge Space (Kohlhase, Kohlhase, 2005)

“0“,“9“, “5“,“,“ Exchange rate 1 $ = 0,95 € Market mechanisms concerning exchange rates 0,95 Character Set ContextNetworkingSyntax GlyphsDataInformationKnowledge Presentation Dis- ambiguation Social Context Relation Representation 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/Aggregator Beginner/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/Aggregator Beginner/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 ParticipationReification 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 CoP as a set of CoP artifacts Set of Documents A Living CoP Object

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? Meaning Social negotiation Community Discourse as connection between individuals and knowledge Boundary membership Learning Exploring a shared repertoire Document Collection

Added-Value Support for Presentation of Mathematics Social Level Where is my CoP? Boundary membership

Boundary membership 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 Community reference network Discourse Level (inter-document) Document Collection Paper #4 Paper #222 Community Discourse 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 SemigroupsGroups Example: Strings with concatenation Example: Functions with composition Example: Functions with composition Learning Exploring a shared repertoire

OMDoc <presentation for=“binomial” role=“applied” xml:lang=“en”> … Added-Value Support for Presentation of Mathematics OMDoc <presentation for=“binomial” role=“applied” xml:lang=“en”> … Binomial Coefficient: nknk () n! k! * (n-k)! := Document Collection We use the binomial coefficient A Generated Document nknk () Formula Level Meaning Social negotiation

Conclusion: Added-Value Services by CoP Model Document Collection Meaning Social negotiation Community Discourse as connection between individuals and knowledge Boundary membership Learning Exploring 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

Added-Value Support for Presentation of Mathematics Boundary value judgments Social Level Where is my CoP? Monoid Strings Functions Concatenation Group Theory Social Bookmarking

Added-Value Support for Presentation of Mathematics Community reference network Discourse Level (inter-document) Beginner Expert ??? Paper #4 Paper #6Paper #139 Document Collection Paper #4 Paper #6 Paper #139 Paper #8 Paper # Community Discourse as connection between individuals and knowledge

Characteristics of Communities of Practice  Added-Value Services? Meaning notation flexibility Community reference network Boundary value judgments Learning discourse optimization Document Collection