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 Science Royal 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: lack of subject understanding in the early layers. minimization of teaching duties in the final layers. life long teaching with: curricular-oriented ”knowledge pushing”.

5. 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.

6. 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.

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

8. An emerging educational design pattern (Requested Preacher / Learner Rights) School RightsResource SeekerLearner Student Knowledge Life-long TeacherConsultant learning: Pedagogical Interest Developing your interests Requested teaching: You teach as long as somebody is learning Course * *

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

10. QBL: the 3 performing knowledge roles (cont) Source Strategist OpportunistKnowledge fascinationmethodologyopportunity quality measured by given answers quality measured by lost questions quality measured by raised questions

11. 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.

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. 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.

13. 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.

14. 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.

15. Resource Components / Learning Modules Learning Environment Learning Module What to Teach Resource Component * * What to Learn separating connecting from with through Multiple NarrationComponent Composition through

16. 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.

17. 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.

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. 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.

19. Surfing Browsing Viewing CheckingContextContentDescription Conceptual Operating modes for Concept Browsers

20. 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.

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. Geometry Mathematics Algebra CombinatoricsAnalysis Surf View Info ContextContent Surfing the context: Opening the Geometry field

23. ContextContent Viewing the content of Projective Geometry Projective GeometryAlgebraic DifferentialSurf View Info What How Where When Who

24. Surf View Info Geometry Projective Algebraic Differential Context Aspect Level School Elementary Secondary High WHW... h a t o w h e r e Aspect Filter Filtering the content of Projective Geometry

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

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

27. Surf View Info What How Where When Who Mathematics Viewing the content: Where is mathematics done? Content Clarification Depth Context Science MagicReligionPhilosophyMathematics invoke illustrate apply inspire

28. Surf View Info What How Where When Who How is mathematics applied to science? Magic Philosophy Religion Science Mathematics invoke illustrate apply inspire Content Clarification Depth Contextualize Context Ais true Science  assumption  conditional statement logical conclusion    Bis true IfA were true then B would be true Mathematics  Magic Philosophy Religion Science Mathematics invoke illustrate apply inspire

29. How is mathematics applied to science? (cont ) Content Surf View Info What How Where When Who Magic Philosophy Religion Science Mathematics invoke illustrate apply inspire Clarification Depth Contextualize Context Ais true Science  assumption  conditional statement logical conclusion    Bis true IfA were true then B would be true Mathematics  Falsification of assumptions by falsification of their logical conclusions experiment    fact

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 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.