3. Educational Design Patterns in Mathematics Ambjörn Naeve The Knowledge Management Research group Centre for user-oriented IT Design (CID) Numerical Analysis and Computer Science (NADA) Royal Institute of Technology (KTH) Stockholm / Sweden amb@nada.kth.se http://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. 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.

10. 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.

11. 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.

12. 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 ´ ´ ´

13. 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

14. Long term trend in mathematics education

15. SolutionElimination Problem eliminate ProblemElimination Cause Organization e dissolvProblemSolutioncrystalizeSymptom Organization e dissolv Problem / Solution versus Problem / Elimination

16. dissolveCrime Jail crystalize OrganizedEconomic Dysfunctionality Legal System Problem / Solution applied to Crime

17. dissolve Conceptual DifficultyAlgorithmic Ability crystalize AnticipatedUnderstanding Dysfunctionality Pedagogical System isA Mathematics is difficult nurturing the difficulties I never understood it when I was at school because behave or degrade Problem/Solution applied to early math education

18. dissolve Conceptual DifficultyAcademic Statuscrystalize AnticipatedUnderstanding Dysfunctionality Publicational System isA Mathematics is difficult nurturing the difficulties I understand it, and I am smarter than you because publish or perish Problem/Solution applied to late math education

19. Problem/Solution applied to calculator industry

20. Researcher Artist ConsultantTeacher AcademiaBusiness Later Education Business Employee no yes no understand? make money? socially conscious? creative?creative?creative?creative? make money? Teacher yes

21. The digital view - from Pythagoras to Heisenberg

25. Einstein Frankenstein Springsteen

26. Attityd Förväntan FörsvarFörälder Lärare Bekräftelse Barn tid Att räkna med x är svårt. Jag förstod det aldrig när jag gick i skolan. Barn, vi ska nu börja räkna med det där mystiska talet x som ni alla har hört talas om. Jag kommer aldrig att begripa det här! Just det, precis som jag trodde, jag kan helt enkelt inte förstå det här! Begreppsmässig svårighet Minsta tecken på mentalt motstånd

28. References Naeve, A., IT-baserade matematikverktyg på KTH, CID-49, NADA/KTH, 2001. These 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, NADA/KTH, 1997. Naeve, A., Conceptual Navigation and Multiple Scale Narration in a Knowledge Manifold, CID-52, NADA/KTH,1999. Naeve, A., The Work of Ambjörn Naeve in the Field of Math. Educational Reform,CID-110, NADA/KTH, 2001. Naeve, A., Begreppsmodellering och matematik, CID-109, NADA/KTH, 2001.