Learning Mathematics with

Transcription

1 Deutsches Forschungszentrum für f r Künstliche K Intelligenz Learning Mathematics with Jörg Siekmann German Research Centre for Artificial Intelligence DFKI Universität des Saarlandes

2 e-learning: Systems and Platforms Systems: - Pact-Tutors, - Geometry Tutor, ActiveMath, - ELM-Art.... etc Platforms: - Clix - SAKAI - BLACKBOARD... etc Portals: Nets,... - for example: SaarLearn Net - regional nets, European e-learning Nets, Commercial Companies: - Thompson (USA) - IMC (Scheer), e-tell (Melis), Auralog (France), Giunti Labs (Italy), - Blackboard + WebCT.... etc

3 What is an Intelligent Tutor System? ITS: Cassical Tutor Systems: - precanned solutions Intelligent Features: Adaptive sequencing Interactive problem solving Error diagnosis User Model Personalization of: Content Presentation Dynamically generated suggestions

4 The System

5 Pedagogical Goals: Personalization Contents, exercises Presentation Interaction for individual content in interactive exercises learning tools, reflection tools Natural language dialogs and input editor Diagnosis and feedback motivational support scaffolding

6 Technological Goals Separation of knowledge and functionality Reuse of contents (via standards) Reuse and interoperability of tools and components Semantics for formulae, searchable mathematics content Various output formats and appearances Open web-architecture, modular design, configurability Standard XML-communication Platform-independence open source

7 Interdisciplinary Work AI ITS Computational Linguistics Learning Psychology Web Technology Education Domain knowledge

8 Artificial Intelligence Techniques user modeling presentation planning adaptive user interfaces problem solving systems deduction systems knowledge representation error diagnosis agent-based feedback natural language processing

9 Login :

10 ActiveMath: Start Page created by learner by by author/teacher author/teacher

11 Choice of Book :: Selection of the Scenario

12 Wahl des Buches (in German)

13 Choice of Book :: Chinese

14 Choice of Book : Русский (Russian)

15 Choice of Book :: Français.) ais (and Spanish and...

16 Adaptivity: The Student s Background Mathematics Biology Physics

17 Adaptivity: The Student s Competency Same Content, different Learning Goals Introduction to Calculus Anton Physics University, Master Prep for Examination Eva Mathematics High School, A-level Learning/Understanding

18 Scenario for Anton: Definitions and Exercises

19 Scenario for Eva: Learning in Depth

20 The different Books Web based: Wicky or Google

21 A Book : How good are you? Good mastery Medium mastery Weak mastery

22 Scenario: Unmastered Concepts

23 Scenario: Unmastered Concepts

24 How can this be implemented? Course Generation: MBase MBase OMDoc OMDoc Course Generator Pedagogical Pedagogical Rules Rules User User Model Model

25 Course Generation Course Generator: assembles learning objects according to learner s Goals Capability Context/scenario Using an Operator like: ProvideAdequateExerciseFor(C): if mastery(c)<0.3 then Exercise(easy) if 0.3 mastery(c)<0.7 then Exercise(medium) if 0.7 mastery(c)<1 then Exercise(hard) With instructional ontology: vocabulary at the adequate level of abstraction for defining complex domain independent pedagogical strategies moderate constructivist strategies traditional didactic strategies

26 Course Generation (1) Goal concept 1. Retrieve content from knowledge base Start with goal concept Collect recursively concepts the goal depends on plus additional information (e.g. examples)

27 Course Generation (2) Goal concept 2. Filter concepts According to pedagogical rules According to the user model Defrule PatternExamPrep Allow (definition, exercise) Order (definition, exercise) Defrule ReqAppEx Allow (definition, exercise) Test (user-kb(definition)<0.3) Add ex-for(definition 0.3) ex-for(definition 0.5)

28 Course Generation (3) Goal concept 3. Linearize graph according to mathematical dependency according to pedagogical dependency

29 Knowledge Representation

30 Knowledge Representation Metadata Mathematical dependencies for, theory Pedagogical dependencies prerequisites, against Pedagogical characteristics difficulty, competencies... Domain Data: OMDoc / OpenMath Structures: theory, definition, axiom, example Mathematical semantics

31 Knowledge Representation OpenMath/OMDoc Representation of the semantics of mathematical knowledge - Domain ontology - Structures - Semantics of mathematical objects Metadata characterization and relationships between instructional objects - Semantic handling of formulas: search, copy&paste - Interoperability of mathematical services - Interoperability & intelligent automatic processing at the pedagogical level

32 Definition ε-neighborhood Theorem Proof Example neighborhood-equality for neighborhood-equality for ε-neighborhood Exercise for ε-neighborhood

33 OMDoc Knowledge Representation <definition id="c6s1p4_th2_def_monoid" for="c6s1p4_monoid <metadata> <depends-on> <ref theory="cp1_th3" name="structure" /> </depends-on> <Title xml:lang="en">definition of a monoid</title> </metadata> <CMP xml:lang="en" format="omtext"> A monoid is a <ref xref="cp1_th3_def_structure"> structure </ref> <OMOBJ> <OMS cd="elementary" name="ordered-triple"/> <OMV name="m"/> <OMS cd="cp4_th2" name="times"/> <OMS cd="cp4_th2" name="unit"/> </OMOBJ> in which <OMOBJ> <OMS cd="elementary" name="ordered-pair"/> <OMV name="m"/> <OMS cd="cp4_th2" name="times"/> </OMOBJ> is a semi-group with <ref xref="c6s1p3_th2_def_unit">e</ref> <OMOBJ xmlns=" <OMS cd="cp4_th2" name="unit"/> </OMOBJ>. </CMP> <FMP><OMOBJ>... </OMOBJ></FMP> </definition>

34 Feedback

35 Feedback in Exercises

36 Feedback in Exercises

37 Feedback in Exercises

38 Another Help in an Exercise

39 Interactive Mathematics Interactive Mathematics

40 Interactive Math

41 Interactive Math

42 Interactive Math

43 Interactive Math

44 My Profile my current Knowledge

45 My Profile my current Knowledge

46 IMPLEMENTATION

47 Information Flow in Session Manager xml XSLT html request WebServer http browser Course generator Pedagogical rules CAS Math systems MBase User model profile evaluator history

48 SE: Event Framework Components can publish and subscribe - Event object: Type, timestamp, source, - Loose coupling between components Used for: - Generic component integration - User logging -User modeling - - better scalability: Clients can poll events

49 Example: Mastery Update Event: UserMastery- Change Event: Exercise- Finished User action (http request)

50 Presentation Process for Mathematics

51 Presentation Process Naive approach: Start with table of contents Bring XML-fragments together Apply XSLT transformation Issues Low performance Adaptivity logic defined in XSLT Layout defined in XSLT

52 ActiveMath Presentation Component Idea: 2-stage approach for presentation First stage deals with individual content fragments Second stage combines fragments to user-specific pages, enriched with dynamic data (via templates)

53 Presentation Pipeline: Fetching Collects content from knowledge base The output of this step are XML fragments <definition id="def1"> Definition 1 with a reference to <ref xref="def2">definition 2.</ref> </definition>

54 Presentation Pipeline: Pre-Processing Inserts server-specific information into the XML content <definition id="kb1://def1"> Definition 1 with a reference to <ref xref="kb1://def2">definition 2.</ref> </definition>

55 Presentation Pipeline: Transformation Conversion into the output format by XSLT Output: content fragments <div class="definition" id="kb1://def1"> Definition 1 with a reference to $link.dict("definition 2", "kb1://def2"). </div>

56 Presentation Pipeline: Assembly Joins the fragments to form the requested page <html> <head/><body> This page is generated for $user.name. <!-- begin item --> <div class="definition" id="kb1://def1"> Definition 1 with a reference to $link.dict("definition 2", "kb1://def2"). </div> <!-- end item --> </body></html>

57 Presentation Pipeline: Personalization Adds personalized data to the document This page is generated for $user.name. <!-- begin item --> <div class="definition" id="kb1://def1"> Definition 1 with a reference to <a onclick="openindictionary( kb1://def2 )"> <img src="green.jpg"/> Definition 2 </a>. </div> <!-- end item -->

58 Advantages of this Approach Multi-formats HTML, XHTML+MathML, PDF, SVG, slides Caching real performance much higher Incremental rendering perceived performance much higher High flexibility Separation of concerns (MVC) Eases layout changes

59 More specifically: Presentation of Mathematics Mathematics on the Web is a problem: Mostly just as an image No semantics ActiveMath: HTML, MathML, (SVG) Cross-browser: Internet Explorer, Mozilla Usage of semantics to add invisible information Authorable appearance

60 Math: HMTL

61 Math: XHTML+MathML

62 Math: PDF SVG

63 Mathematical Semantics: Rendering

64 Applications: BRÜCKENKURSE Courses to bridge between School and Further Education at University, Engineering Colleges etc

65 Student Numbers in Engineering Total Number of Students A possible Remedy: Brückenkurse with e-learning Final No: in 1996 there were graduated engineers successful in 2002 there were only approx finalists! Failure Rate: between 43 und 61% (!)

66 Another Application: E-Chalk Integration of ActiveMath into the Classroom

67 E-Chalk: Raul Rojas in Berlin Cooperation with Freie Universität Berlin

68

69 FU Berlin: Formula Recognition

70 Cooperation between Maths Department Saarbrücken and DFKI Prof. F. Schreyer: Computeralgebra and Rendering

71 Cooperation between Maths Department and DFKI Oliver Labs: 3-D Object Definition and Rendering

72

73 ActiveMath in Action Thanks for Listening!