The Garden of Knowledge as a Knowledge Manifold

Speaker: Ambjörn Naeve

Affiliation: Centre for user-oriented IT-Design (CID) Dept. of Numerical Analysis and Computing ScienceRoyal Institute of Technology (KTH) Stockholm, Sweden

email-address: amb@nada.kth.se

web-sites: cid.nada.kth.se kmr.nada.kth.se

Structure of today’s math education system

Closed layered architecture based on:

• lack of subject understanding in the early layers.

• minimization of teaching duties in the final layers.

• life long teaching with:

• curricular-oriented ”knowledge pushing”.

Problems with today’s math education

It does not:

• promote understanding.

• support personalization.

• integrate mathematics with human culture.

• stimulate interest.

• integrate abstractions with applications.

• support transition between the different layers.

Possibilities for improving math education

• visualizing the concepts.

• interacting with the formulas.

• using ICT to increase the ”cognitive contact” by:

• personalizing the presentation.

Promoting life-long learning based on interest by:

• improving the narrative by:

• showing before proving.

• focusing on the evolutional history.

• routing the questions to live resources.

• proving only when the need is evident.

A traditional educational design pattern

(Tenured Preacher / Learner Duty)

School Duties

Course

Preacher

PrisonerEmployee

Teacher Learner

Life-longteaching:

Minimallearning efforts

Security

Doing time in return fora degree

Pupil

Confinement

Tenure

Agent 007with a rightto kill interest

* *

An emerging educational design pattern

(Requested Preacher / Learner Rights)

School Rights

Resource Seeker

Learner

Student

Knowledge

Life-longTeacher

Consultant

learning:

Pedagogical

Interest

Developingyour interests

Requestedteaching:

You teach as longas somebody is learning

Course

* *

QBL: the 3 performing knowledge roles

Preacher

Knowledge

Coach Plumber

Master Gardener Broker

you teachas longas somebodyis learning

you assistin developingeach indiduallearning strategy

you findsomeoneto discussthe question

´ ´ ´

QBL: the 3 performing knowledge roles (cont)

Source Strategist Opportunist

Knowledge

fascination methodology opportunity

qualitymeasured bygiven answers

qualitymeasured bylost questions

qualitymeasured byraised questions

The Garden of Knowledge

• is an interdisciplinary project at CID that is developing new forms of interactive learning environments.

• can be regarded as a form of Knowledge Patchwork, with a number of linked Knowledge Patches, each with its own Knowledge Gardener.

• has a first prototype that focuses on symmetry in order to illuminate some of the structural connections between mathematics and music.

Taking a walk in the Garden of Knowledge

What follows are some snapshots from a walk in the first prototype of the Garden of Knowledge.

The prototype is available on CD-rom and part of it is accessible on the web at the address: http://cid.nada.kth.se/il/kt/ktproto

This prototype was developed at CID during 1996/97 in collaboration with the Royal College of Music and the Shift New Media Group.

A Knowledge Manifold

• is designed in a way that reflects a strong effort to comply with emerging international IT standards.

• can be regarded as a Knowledge Patchwork, with a number of linked Knowledge Patches, each with its own Knowledge Gardener.

• gives the users the opportunity to ask questions and search for live certified Knowledge Sources.

• is a learner-centric educational architecture that supports question-based learning.

A Knowledge Manifold (cont.)

• allows teachers to compose components and construct customized learning environments.

• makes use of conceptual modeling to support the separation of content from context.

• contains a conceptual exploration tool (concept browser) that supports these principles.

• has access to distributed archives of resource components.

Resource Components / Learning Modules

Learning Environment

Learning Module

What to Teach

Resource Component* *

What to Learn

separating connecting

from with

through

Multiple Narration Component Composition

through

The seven different Knowledge Roles of a KM

• The knowledge cartographer

• The knowledge composer

• The knowledge librarian

• The knowledge coach

• The knowledge preacher

• The knowledge plumber

• The knowledge mentor

• constructs context-maps.

• fills the context-maps with content.

• composes content components into learning modules.

• cultivated questions.

• provides live answers.

• connects questions with appropriate preachers.

• provides motivation and supports self-reflection.

The Garden of Mathematical Knowledge

• Mathematical component archives

• Mathematical Knowledge Manifolds

• Interacting with mathematical formulas, using

• Shared 3D interactive learning environments

• Virtual Mathematics Exploratorium (Conzilla)

• Graphing Calculator (Ron Avitzur)

• LiveGraphics3D (Martin Kraus).

• framework for archiving and accessing components

• CyberMath.

created with these (and other) techniques.

Design principles for Concept Browsers

• separate context (= relationships) from content.

• describe each context in terms of a context-map.

• assign an appropriate set of components as the content of a concept and/or a conceptual relationship.

• filter the components through different aspects.

• label the components with a standardized data description (meta-data) scheme (IMS-LOM).

• transform a content component which is a context-map into a context by contextualizing it.

Surfing

Browsing

Viewing Checking

Context Content Description

Conceptual

Operating modes for Concept Browsers

Conzilla - a prototype of a Concept Browser

• object-oriented implementation in Java.

• dynamic creation of XML files that represent the context maps.

• clean separation of logic and graphics.

• uses URI:s to abstract the location of the content.

• prepared for LDAP based component location lookup.

• masters of science thesis work (“exjobb”) by Mikael Nilsson and Matthias Palmér.

Conzilla - ongoing work

• improvement of the interface.

• implementation of aspect filtering capacity.

• implementation of a help-service system for the localization of live knowledge resourses.

Conzilla - first official release

• will be avaliable as freeware in September 2000.

• check the URL http://www.conzilla.org

Geometry

Mathematics

Algebra

Combinatorics

Analysis

Surf

View

Info

Context Content

Surfing the context: Opening the Geometry field

Context Content

Viewing the content of Projective Geometry

Projective

Geometry

Algebraic

Differential Surf

View

Info

What

How

Where

When

Who

Projective geometry is the studyof the incidencesof points, lines

in space.

It could be calledthe geometryof the eye

and planes

Surf

View

Info

Geometry

Projective

Algebraic

Differential

Context

Aspect

Level

School

Elementary

Secondary

High

W H W

...

hat

ow

here

Aspect Filter

Filtering the content of Projective Geometry

Mathematics

Universe Brain

ContentContext

Mathematics = Hom(U, Brain)

Mathematics is the study of structuresthat the human brain is able to perceive

= Hom(O, Sapiens)

Surf

View

InfoWhat

How

Where

When

Who

Viewing the content: What is mathematics?

Clarification

Depth

Structure-

preserving

Content

In mathematics we study changescalled functions, which we try to restore = revert = invertas well as possible.

To determine the inverseof the minimally disturbed changethat is restorable, is calledto determine the pseudo-inverse of the change.

The perfect restorer = the inverseof a mathematical changein most cases does not exist.

Context

Mathematics Surf

View

InfoWhat

How

Where

When

Who

Viewing the content: How is mathematics done?

Clarification

Depth

Surf

View

Info

What

How

Where

When

Who

Mathematics

Viewing the content: Where is mathematics done?

Content

Clarification

Depth

Context

Science

Magic

Religion

Philosophy

Mathematicsinvoke

illustrateapply

inspire

Surf

View

InfoWhat

How

Where

When

Who

How is mathematics applied to science?

Magic

Philosophy

Religion

Science

Mathematicsinvoke

illustrateapply

inspire

Content

Clarification

DepthContextualize

Context

A is true

Science

assumption

conditional statement

logical conclusion

B is true

If A were truethen

B would be true

Mathematics

Magic

Philosophy

Religion

Science

Mathematicsinvoke

illustrateapply

inspire

How is mathematics applied to science? (cont)

Content

Surf

View

InfoWhat

How

Where

When

Who

Magic

Philosophy

Religion

Science

Mathematicsinvoke

illustrateapply

inspire

Clarification

DepthContextualize

Context

A is true

Science

assumption

conditional statement

logical conclusion

B is true

If A were truethen

B would be true

Mathematics

Falsification of assumptionsby falsification of their logical conclusions

experiment

fact

Conzilla - context of mathematical subtypes

Viewing the content-list of the Geometry node

Checking the metadata on Geometric Calculus

Viewing the content of Geometric Calculus

Viewing the content-list of Euclidean Geometry

Checking the metadata on Point Focus

Checking the metadata on Point Focus (cont.)

Viewing the content of Point Focus

CyberMath: an avatar using a laser pointer

CyberMath: finding the kernel of a linear map

CyberMath: importing a Mathematica object

CyberMath: the gen. cylinder exhibition hall

CyberMath: avatars visiting the virtual museum

CyberMath: the virtual solar energy exhibit

Pointfocusing experiment (Gunnarskog 1995)

Solar horse-power (Gunnarskog 1995)

References

• Taxén G. & Naeve, A., CyberMath - A Shared 3D Virtual Environment for Exploring Mathematics, CID, KTH, 2000, http://www.nada.kth.se/~gustavt/cybermath

Reports are available in pdf at http://kmr.nada.kth.se

• Naeve, A., The Garden of Knowledge as a Knowledge Manifold - a conceptual framework for computer supported subjective education, CID-17, KTH, 1997.

• Naeve, A., Conceptual Navigation and Multiple Scale Narration in a Knowledge Manifold, CID-52, KTH, 1999.

• Nilsson, M. & Palmér M., Conzilla - Towards a Concept Browser, (CID-53), KTH, 1999.