Technology Enhanced Math Rehab
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  • 1. EPFL Seminar, 2007-10-30 First Class Mathematics The Technology Enhanced Mathematics Rehabilitation Clinic Ambjörn Naeve The Knowledge Management Research Group The Royal Institute of Technology, and Uppsala University Sweden http:// kmr.nada.kth.se/wiki/Amb http://kmr.nada.kth.se/wiki/Amb/MathematicsEducationProjects
  • 2. The KMR group - what do we do? • We work with Asynchronous Public Service in the form of infrastructures, architectures, frameworks and tools that contribute towards the creation of a Public Knowledge and Learning Management Environment. • This PKLME should enable a global, (synchronous and) asynchronous public discourse that aims to enhance the learning of all participants. • Our main information architecture for this PKLME is the Knowledge Manifold (navigated in Conzilla). • All our software is Open Source and based on Semantic Web technology.
  • 3. We work to enable a fundamental shift • From: Teacher-centric, curricular-oriented “knowledge push” • To: Learner-centric, interest-oriented “knowledge pull”
  • 4. A traditional educational design pattern (Tenured Preacher / Learner Duty) Security Employee Confinement Prisoner Tenure Preacher Pupil School Duties Minimal Life-long learning efforts teaching: * * Teacher Learner Doing time Agent 007 in return for with a right Course a degree to kill interest
  • 5. An emerging educational design pattern (Requested Preacher / Learner Rights) Knowledge Seeker Interest Resource Pedagogical Student Consultant School Rights Requested Life-long * * Teacher Learner teaching: learning: Course Developing You teach as long your interests as somebody is learning
  • 6. Question-Based Learning The three fundamental questions What am I interested in? What is there to know about it? What do I want to know about it?
  • 7. Structure of todayʼs math education system Closed, layered architecture based on: • curricular-oriented ”knowledge push”. • life long teaching with: • lack of subject understanding within the earlier layers. • minimization of teaching duties within the later layers.
  • 8. Problems with todayʼs 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.
  • 9. Long term trend in mathematics education Question How Why Algorithm Proof Mathematical Acti vity
  • 10. Solving vs Eliminating a Problem Elimination Problem Solution Symptom Cause eliminate dissolve Solution Pr oblem Elimination Problem dissolve crystalize Organization Organization
  • 11. Applying the problem/solution pattern to the judicial system Legal System dissolve Crime Jail crystalize Economic Dysfunctionality Organized
  • 12. Applying the problem/solution pattern to the earlier parts of math education Pedagogical System dissolve Conceptual Difficulty Algorithmic Ability behave or degrade crystalize Understanding Dysfunctionality Anticipated nurturing the difficulties isA because Mathematics I never understood it is difficult when I was at school
  • 13. Applying the problem/solution pattern to the later parts of math education Publicational System dissolve Conceptual Difficulty Academic Status publish or perish crystalize Understanding Dysfunctionality Anticipated nurturing the diffculties i isA because I understand it, and Mathematics I am smarter than you is difficult
  • 14. Applying the problem/solution pattern to the commercial parts of math education Extracting the Computing the mathematical Problem algorithmic skeleton solution Formulation Solution Computational Industry dissolve Conceptual Difficulty Algorithmic Ability solve by computation crystalize Understanding Disfunctionality Anticipated marketing the difficulties isA but Mathematics We can help you is difficult to solve your problems
  • 15. The X Calculating with X is hard. I never understood it when I was at school. anxiety isA pattern Defence Parent Attitude Teacher isA Children, we will now start to calculate with this mysterious thing X that you have all heard about. Expectation Child isA Shit, I’m never going to understand this stuff! Conceptual Difficulty isA Slightest sign of mental resistance Confirmation isA Yeah, just as I figured, I simply can’t understand this stuff. time
  • 16. 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.
  • 17. Mathematics as a de-semantization process that transforms ”meaning” into ”form” Appli- cation Meaning Form Mathe- matics Force Vector Velocity
  • 18. Create / Apply Mathematics Apply Math Form Meaning Create Math Definition Existential Operational What is it? How does it act?
  • 19. Create Mathematics Create Math “Forget” meaning Model Extract Turn into semantics structure definition Domain Math model skeleton
  • 20. Apply Apply Mathematics Math Model Create Collect model data Interpret symbols Experimental Over-determined script script Solvable Solve Modify script script script Minimal modification => pseudo-inverse List of “who did it” “CSI-Mathematics”
  • 21. Project Mathemagic • Ideology: Within the mathemagic project we want to emphasize the speculative and creative aspects of mathematics. • Aim: To stimulate interest in mathematics among young and old by emphasizing “week-end mathematics”. • Basic idea: Problematize and dramatize the major mathematical concepts by anchoring them in the history of ideas. • Metdod: Improving the narrative - showing without necessarily proving. • Form: The news of yesterday: Proust: “In Search of Lost Mathematics.” Knowledge components (featuring Pythagoras, Archimedes, Newton, …) are “tied together” by a ”news anchor in space-timequot; who follows different trails along the evolution of mathematical ideas.
  • 22. Nine Mathemagic Stories 1. The story of the people who thought the world was understandable. From Thales and Pythagoras to Demokritos and Aristarkos. 2. The story of the people that wanted to escape the realm of the senses. From Plato via Augustinus and Aquino to the “scolastic age”. 3. The mathematics of the eye: The development of true perspective. From Pappos via de la Franchesca and da Vinci to Desargues, Pascal, Poncelét, Plücker, Grassmann and Klein. 4. Einstein for Flatlanders: Two-dimensional relativity theory. The story about the flatlanders that lived on a sphere and the flatlanders that lived on a torus (“dough-nut”).
  • 23. Nine Mathemagic Stories 5. The story of the people that disregarded almost everything. The evolution of abstract thinking: From induction and abduction to abstraction and deduction. quot;The power of thinking is knowing what not to think about.quot; 6. About the difficulties in overcoming psychological complexes. The story about the development of the concept of number: From “positive” to “negative”, from “rational” to “irrational”, from “real” to “imaginary” and “complex.” 7. What is there between the atoms? Does the world consist of particles or waves - or maybe something else? The historical debate from Thales versus Pythagoras via Newton versus Huygens to Einstein versus Bohr and Heisenberg and the break-up of the particle concept (super-string theory).
  • 24. Nine Mathemagic Stories 8. The mysterious law about the degradation of work: The principles of energy and entropy. The development of the energy concept from Leibniz via Rumford and Carnot to Maier, Joule and Bolzmann. 9. The story of the long-lived demon that was unable to forget. Maxwellʼs demon and the deep connections between information theory and thermodynamics.
  • 25. Ongoing mathematical ILE work at KMR • Virtual Mathematics Explainatorium with Conzilla • Interactive geometrical constructions with PDB • Interacting with mathematical formulas • using LiveGraphics3D / Graphing Calculator • Mathemagic component archive in Confolio • CyberMath: a shared 3D ILE for exploring math
  • 26. Seven different Knowledge Roles in a KM • Knowledge Cartographer • constructs context-maps. • Knowledge Librarian • fills context-maps with content-components. • Knowledge Composer • combines content-components into learning modules. • Knowledge Coach • cultivates questions. • Knowledge Preacher • provides live answers. • Knowledge Plummer • connects questions to relevant preachers. • Knowledge Mentor • supplies motivation and supports self reflection.
  • 27. The Conzilla “Mantra” Content in Contexts through Concept = Outsides of Concept Contexts = Inside of Concept Content Concept = Border between these
  • 28. Conzilla (www.conzilla.org) Right-clicking on a concept or concept-relation brings up a menu with three choices: Contexts, Content, and Information. • Selecting Contexts opens a sub-menu, which lists all the other contexts where this concept or concept-relation appears. • Selecting Content opens a window (to the right) where the content-components of the concept or concept-relation are listed. • Pointing to a content-component brings up information about it, and double-clicking on a content-component opens another window where the content is shown.
  • 29. Conceptual Browsing: Viewing the content Context Content Geometry Projective geometry is the study of the incidences Algebraic What of points, lines and planes How in space. Differential Surf Where It could be called View When the geometry Projective of the e ye Info Who
  • 30. Conceptual Browsing: Filtering the content Aspect Filter Context Elementary Geometry Secondary Algebraic High Differential School Surf W H W o h h Projective w e a View Level r t e Info ... Aspect
  • 31. Where is mathematics done? Depth Contextualize Clarification Context Content inspire Mathematics Magic invok e What illu Mathematics Religion Surf stra te How ap View ply Philosophy Where Info When Science Who
  • 32. How is mathematics applied to science? Depth Contextualize Clarification Context Content Science inspire inspire Mathematics Magic Mathematics Magic assumption ∗ ∗v logical conclusion in ok invok e e A is true ⇒ B is true illu illu Religion Religion stra stra fact te te ↓ ↓ ap ap ply p ly Surf Philosophy Philosophy If A were true then ↓ View ↓ What B would be true Science Science Info ∗ conditional How statement Where Mathematics ∗ When experiment Falsification of assumptions Who by falsification of their logical conclusions
  • 33. The interplay between mathematics and science Experimental Science Theoretical ∗ ∗ <<is an>> Fact Assumption A is true ∗ ∗ ↓ Mathematics ↓ ∗ ∗ B survives If A were true, the test Conditional statement then B would be true ∗ a Test ↓ Therefore B ∗ ∗ <<is a>> must be true ∗ ∗ Logical ∗ Theory 1 ∗ Experiment conclusion ↓ Falsification of assumptions by falsification of their logical conclusions
  • 34. Virtual Mathematics Exploratorium - Entrance
  • 35. Virtual Mathematics Explainatorium - Filtering
  • 36. Virtual Mathematics Explainatorium - Viewing
  • 37. Dynamic Geometry with PDB
  • 38. Taylor Expansion with the Graphing Calculator
  • 39. Mathematical Component Archive www.nada.kth.se/~osu/math/Geometry/Quadrics/one_sheeted_hyp_tang1.html.
  • 40. CyberMath: A Shared Virtual Environment for the Interactive Exploration of Mathematics Goals: The CyberMath system should allow: • teaching of both elementary, intermediate and advanced mathematics and geometry. • the teacher to teach in a direct manner. • teachers to present material that is hard to visualize using standard teaching tools. • students to work together in groups. • global sharing of resources. Means: • Making use of advanced VR technology (e.g. DIVE).
  • 41. CyberMath: an avatar using a laser pointer
  • 42. CyberMath: finding the kernel of a linear map = = =
  • 43. CyberMath: importing a Mathematica object
  • 44. CyberMath: the cylindrical exhibition hall
  • 45. CyberMath: Avatars visiting the Virtual Museum
  • 46. CyberMath: The Solar Energy Exhibit
  • 47. CyberMath: The Cooperative Learning Mode
  • 48. CyberMath: The WASA Platform
  • 49. Mathematical ILE collaborative projects < 2002 Advanced Media Technology Laboratory (KTH) • Mathemagic: Mathematical storytelling Swedish Learning Lab • Content and context of mathematics in engineering education (with DSV, KTH/Kista) • 3D Communication and Visualization Environments for Learning (with DIS, Uppsala Univ) Learning Lab Lower Saxony & Stanford LL • Personalized Access to Distributed Learning Resorces • MathViz: Personalized Mathematical Courselets
  • 50. The Greek Beginning Pythagoras Thales What is the basic stuff Disinterested knowledge that the universe is made of? is the most effective purification of the soul. Herakleitos • everything changes Ionian school Pythagorean school • all is substance (matter) • all is form (number) • nothing changes Parmenides • atoms build up shapes (bodies) Demokritos that move around in empty space
  • 51. The Greek Beginning: Harmony of the Spheres Thales Ionian School Pythagorean School Pythagoras Substance Form Number Figure Relation Formal What is the basic stuff Proofs Number that the universe is the essence is made of? of all things Geometry Astronomy Music Religion Arithmetic Monad odd <=> unity Philolaus Formal # The one that is The earth is spherical Definitions everything and rotates around the C central fire, protected Pythagoras’ by the counter-earth b 2 Extasis theorem 1 + 3 +5 + 7 = 4 1234 Contemplating The quint circle Herakleides -- quot; -- quot; -- quot; -- ---- the structure 1234 5 3 of numbers The earth also rotates 4 even <=> infi nity around its own axis. 2 2 2 3 + 4 quot;5 The central fire Harmony of Armonia is inside the earth the Spheres The five 2:1 = octave regular solids 3:2 = quint The intervals between Aristarchus 4:3 = quart 2345 the heavenly bodies -- ! -- ! -- ! -- ---- The three The earth rotates daily are determined by the 1234 regular tilings around its own axis, and laws of musical harmony Katharsis annually around the sun The holy tetrakys together with the planets The unspeakables Disinterested point knowledge line is the most destroyed plane 2 effective by Pairwise Rational Digital 1 solid purification Commensurability = Mysticism = Worldview 1 of the soul 1 + 2 + 3 + 4 = 10
  • 52. Energy - an overview
  • 53. Function Function2 arguments and parameters Function1 Rule Set of Set of A Arguments Result R Trans- formation Parameters P Two different Function of A perspectives: parametrized by P Function of A and P Function2(A,P) Function1P(A) = Same result:
  • 54. Later Education understand? yes no make money? make money? no yes no yes Academia Business Business creative? creative? creative? creative? yes no yes no no yes yes no Researcher Teacher Consultant Employee Artist socially conscious? yes no Teacher
  • 55. Academia despise Science Humaniora Relational Computational stigmatize Rational Emotional my heart my heart rocks is a rock Artist Non Human Scientist rock heart Sexy at heart Nerdy of rock hot & cool silicon ! Cool Hot Luke Warm rocks rocks Einstein I am I am Springsteen a rock rock Frankenstein
  • 56. Resource Components / Learning Modules Learning Environment * * Resource Component Learning Module connecting separating What to Teach from with What to Learn through through Multiple Narration Component Composition
  • 57. QBL: the 3 performing knowledge roles Knowledge Preacher Coach Plumber ´ ´ ´ Gardener Master Broker you teach you assist you find as long in developing someone as somebody each indidual to discuss is learning learning strategy the question
  • 58. QBL: the 3 performing knowledge roles (cont) fascination methodology opportunity Strategist Source Opportunist Knowledge quality quality quality measured by measured by measured by raised questions given answers lost questions
  • 59. Web links to some of my math projects http://kmr.nada.kth.se/wiki/Amb/FirstClassMathematics http://kmr.nada.kth.se/wiki/Amb/TheGardenOfKnowledge http://kmr.nada.kth.se/wiki/Amb/VirtualMathematicsExplainatorium http://kmr.nada.kth.se/wiki/Amb/MathPRAO http://kmr.nada.kth.se/wiki/Amb/CyberMath http://kmr.nada.kth.se/wiki/Amb/ProjectiveDrawingBoard http://kmr.nada.kth.se/wiki/Amb/MatriksProjektet http://kmr.nada.kth.se/wiki/Amb/FLIT http://kmr.nada.kth.se/wiki/Amb/Projects-Matemagi