The Garden of Knowledge as a Knowledge Manifold

3. 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

4. Structure of today’s math education system Closed layered architecture based on: • curricular-oriented ”knowledge pushing”. • life long teaching with: • lack of subject understanding in the early layers. • minimization of teaching duties in the final layers.

5. Problems with today’s math education It does not: • stimulate interest. • promote understanding. • support personalization. • support transition between the different layers. • integrate abstractions with applications. • integrate mathematics with human culture.

6. Possibilities for improving math education Promoting life-long learning based on interest by: • using ICT to increase the ”cognitive contact” by: • visualizing the concepts. • interacting with the formulas. • personalizing the presentation. • routing the questions to live resources. • improving the narrative by: • showing before proving. • proving only when the need is evident. • focusing on the evolutional history.

7. Security Employee Confinement Prisoner Tenure Pupil Preacher School Duties * * Teacher Learner Course A traditional educational design pattern (Tenured Preacher / Learner Duty) Minimal Life-long learning efforts teaching: Doing time Agent 007 in return for with a right a degree to kill interest

8. Knowledge Consultant Seeker Interest Pedagogical Student Resource School Rights * * Teacher Lear ner Course An emerging educational design pattern (Requested Preacher / Learner Rights) Requested Life-long teaching: learning: Developing You teach as long your interests as somebody is learning

9. Knowledge Preacher Coach Plumber Gardener Broker Master you teach you assist you find as long in developing someone as somebody each indidual to discuss is learning learning strategy the question QBL: the 3 performing knowledge roles ´ ´ ´

10. fascination methodology opportunity Strategist Source Opportunist Knowledge quality quality quality measured by measur ed by measured by raised questions given answers lost questions QBL: the 3 performing knowledge roles (cont)

11. The Garden of Knowledge • is an interdisciplinary project at CID that is developing new forms of interactive learning environments. • has a first prototype that focuses on symmetry in order to illuminate some of the structural connections between mathematics and music. • can be regarded as a form of Knowledge Patchwork, with a number of linked Knowledge Patches, each with its own Knowledge Gardener.

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

13. A Knowledge Manifold • is a learner-centric educational architecture that supports question-based learning. • 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 livecertifiedKnowledge Sources.

14. A Knowledge Manifold (cont.) • has access to distributed archives of resource components. • 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.

15. Lear ning En vir onment * * Resource Component Lear ning Module separating connecting What to Teach from with What to Learn through through Multiple Narration Component Composition Resource Components / Learning Modules

16. The seven different Knowledge Roles of a KM • The knowledge cartographer • constructs context-maps. • The knowledge librarian • fills the context-maps with content. • The knowledge composer • composes content components into learning modules. • The knowledge coach • cultivated questions. • The knowledge preacher • provides live answers. • The knowledge plumber • connects questions with appropriate preachers. • The knowledge mentor • provides motivation and supports self-reflection.

17. The Garden of Mathematical Knowledge • Mathematical Knowledge Manifolds • Virtual Mathematics Exploratorium (Conzilla) • Interacting with mathematical formulas, using • LiveGraphics3D (Martin Kraus). • Graphing Calculator (Ron Avitzur) • Mathematical component archives • framework for archiving and accessing components created with these (and other) techniques. • Shared 3D interactive learning environments • CyberMath.

18. 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. • label the components with a standardized data description (meta-data) scheme (IMS-LOM). • filter the components through different aspects. • transform a content component which is a context-map into a context by contextualizing it.

19. Browsing Viewing Checking Surfing Context Content Description Conceptual Operating modes for Concept Browsers

20. Conzilla - a prototype of a Concept Browser • masters of science thesis work (“exjobb”) by Mikael Nilsson and Matthias Palmér. • object-oriented implementation in Java. • clean separation of logic and graphics. • dynamic creation of XML files that represent the context maps. • uses URI:s to abstract the location of the content. • prepared for LDAP based component location lookup.

21. 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

22. Context Content Mathematics Surf Geometry View Info Algebra Analysis Combinatorics Surfing the context: Opening the Geometry field

23. Context Content Geometry Algebraic What How Differential Surf Where V iew When Projective Info Who Viewing the content of Projective Geometry

24. Aspect Filter Context Elementary Geometry Secondary Algebraic High Differential School Surf W H W o h h Projective w e a V iew Le v el r t e Info ... Aspect Filtering the content of Projective Geometry

25. Depth Clarification Context Content Mathematics Surf V iew What Structure- Info Universe Brain How preserving Where When Who Viewing the content: What is mathematics? Mathematics is the study of structures that the human brain is able to perceive Mathematics = Hom(U, Brain) = Hom(O, Sapiens)

26. Depth Clarification Context Content Mathematics Surf View What Info Ho w Where When Who Viewing the content: How is mathematics done? In mathematics we study changes called functions, which we try to restore = revert = invert as well as possible. The perfect restorer = the inverse of a mathematical change in most cases does not exist. To determine the inverse of the minimally disturbed change that is restorable, is called to determine the pseudo-inverse of the change.

27. Depth Clarification Context Content Mathematics Magic What Mathematics Surf Religion Ho w V iew Where Philosophy Inf o When Science Who Viewing the content: Where is mathematics done? inspire invoke illustrate apply

28. Depth Contextualize Clarification Context Content inspire inspire Science Mathematics Mathematics Magic Magic invoke invoke * * logical conclusion A is true B is true illustrate illustrate Religion Religion ¯ ¯ apply apply Surf If A w ere true Philosophy Philosophy then V iew What B w ould be true Inf o Science Science * Ho w conditional statement Where Mathematics When Who How is mathematics applied to science? assumption Þ

29. Depth Contextualize Clarification Context Content inspire Science Mathematics Magic assumption invoke * * logical conclusion Þ A is true B is true illustrate Religion ¯ ¯ apply Surf If A w ere true Philosophy ¯ then ¯ V iew What B w ould be true Inf o Science * Ho w conditional statement Where Mathematics * experiment When Who How is mathematics applied to science? (cont) fact Falsification of assumptions by falsification of their logical conclusions

30. Conzilla - context of mathematical subtypes

31. Viewing the content-list of the Geometry node

32. Checking the metadata on Geometric Calculus

33. Viewing the content of Geometric Calculus

34. Viewing the content-list of Euclidean Geometry

35. Checking the metadata on Point Focus

36. Checking the metadata on Point Focus (cont.)

37. Viewing the content of Point Focus

38. CyberMath: an avatar using a laser pointer

39. CyberMath: finding the kernel of a linear map

40. CyberMath: importing a Mathematica object

41. CyberMath: the gen. cylinder exhibition hall

42. CyberMath: avatars visiting the virtual museum

43. CyberMath: the virtual solar energy exhibit

44. Pointfocusing experiment (Gunnarskog 1995)

45. Solar horse-power (Gunnarskog 1995)

46. References • 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. • Taxén G. & Naeve, A., CyberMath - A Shared 3D Virtual Environment for Exploring Mathematics, CID, KTH, 2000, http://www.nada.kth.se/~gustavt/cybermath • Nilsson, M. & Palmér M.,Conzilla - Towards a Concept Browser, (CID-53), KTH, 1999. Reports are available in pdf at http://kmr.nada.kth.se