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

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


Content/Form

Evaluation

User Role:Author/Recipient

Creator/AggregatorBeginner/Expert

User Status:Profile

Inner/Outer Motivation

Social Context in MKM


Content/Form

User Status:Profile

Inner/Outer Motivation

Evaluation

User Role:Author/Recipient

Creator/AggregatorBeginner/Expert

The Relevance of Practices


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

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?

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


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


Communityreference network

Discourse Level

(inter-document)

Document Collection

Paper #4

Paper #222



Paper #132

Paper #26 Paper #224

Where is my place in a CoP?


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>


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


Conclusion: Added-Value Services by CoP Model

Document Collection




Boundarymembership

LearningExploring a

shared repertoire

Thank you!


Document Collection