Enhanced Personalised Learning Support of Computer Algebra Systems

1. Enhanced Personalised Learning Support of Computer Algebra Systems • Christian Gütl • Institute of Information Systems and Computer Media (IICM), Graz University of Technology, Austria • School of Information Systems, Curtin Univerity of Technology, Perth, WA • Alexander Nussbaumer (Presenter) • Department of Psychology, University of Graz, Austria • Knowledge Management Institute, Graz University of Technology, Austria • CADGME 2009 • Hangenberg, Austria • 12 July 2009

2. CADGME 2009 Agenda • Introduction • Isabelle for Calculations in Applied Mathematics (ISCA) • Competence-based Knowledge Space Theory (CbKST) • Combining both Approaches • Implemenation • Conclusion and Outlook

3. CADGME 2009 Introduction Isabell for Calculations in Applied Mathematics Project ISAC System Single stepping CAS Focus on micro level (single problems) + help on solving problems - no adaptation, learning path, profile Competence-based Knowledge Space Theory CbKST Tools Psychological theory and application Focus on macro level (combination of problems) + user knowledge, learning paths - assessment items are "black boxes" Combining both approaches => assessment and guidance on micro and macro level => improvement of ISAC through CbKST => improvement of CbKST through ISAC

4. CADGME 2009 Isabell for Calculations in Applied Mathematics (ISAC) • ISAC can automatically solve algebraic tasks … and ….. • ISAC is useful for educational purposes, because: • solution can also be done step by step (single stepping system) by rewriting terms • interaction with learner • learner gets feedback from ISAC after each step • rules applied in each step are revealed to the learner • learner is supported in each step • ISAC "knows" the rules needed to solve a term • ISAC "knows" if the learner can solve a problem AND "knows" which rules the leaner can apply or not apply • Usually mathematical learning systems only "know" if a learner can solve a problem or not

5. CADGME 2009 Isabell for Calculations in Applied Mathematics (ISAC)

6. CADGME 2009 Isabell for Calculations in Applied Mathematics (ISAC)

7. CADGME 2009 Isabell for Calculations in Applied Mathematics (ISAC)

8. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) • Knowledge Space Theory (KST) - behaviouristic theory • knowledge domain (Q) := set of problems • prerequisite relations between problems due to psychological reasons • knowledge state := problems a person can solve (subset of Q) • learning goal := problems a person should be able to solve Example: Q = {a, b, c, d, e} 378 x 605 = ? 58.7 x 0.94 = ? 1/2 x 5/6 = ? 30% of 34? Gwendolyn is 3/4 as old as Rebecca. Rebecca is 2/5 as old as Edwin. Edwin is 20 years old. How old is Gwendolyn?

9. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) { a , b , c , d , e } { a , b , c , d } { a , b , c , e } { a , b , d } { a , b , c } { a , b } { a , c } { c } { a } Æ • Knowledge Space Theory (KST) - behaviouristic theory • knowledge structure := set of possible knowledge states with respect to prerequisite relations between problems

10. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) { { a a , , b b , , c c , , d d , , e e } } { { a a , , b b , , c c , , d d } } { { a a , , b b , , c c , , e e } } { { a a , , b b , , d d } } { { a a , , b b , , c c } } { { a a , , b b } } { { a a , , c c } } { { c c } } { { a a } } Æ Æ • Adaptive Assessment • problem b sloved • problem d solved • problem c not solved Personal Learning Paths χ χ χ χ χ χ χ χ χ

11. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) • Competence-based extension: KST -> CbKST • introducing competences / skills: cognitive constructs • assigning skills to problems: skills needed to solve a problem • assigning skills to learning objects: skills taught by a learning object • assigning skills to learners: learners have available skills • => relation between problems, learning objects, and learners Problems Skills Learning Objects

12. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) • prerequisite relations between skills (due to psychological reasons) • competence state: set of skills which a learner has available • learning goal: set of skills which a learner should have available applying the Pythagorean Theorem stating the sides of a right triangle understanding calculation of the area of a square knowing right triangle knowing square knowing triangle

13. CADGME 2009 Competence-based Knowledge Space Theory (CbKST) • CbKST is a prominent method to achieve adaptivity in e-learning systems (adaptive assessment and adaptive learning paths) • set-theoretic psychological mathematical framework for • structuring knowledege domains • representing knowledge of learners • representing learning goals • adaptive and efficient assessment • personalised learning paths: tailored to learner‘s current competence state and learning goal • performing learning cycle (example): • structuring domains (domain expert) • define learning goal (by teacher/tutor or learner) • adaptive assessment (learner) • personalised learning path based on kowledge/cometence state (learner) • goto 2. • visual feedback in every step

14. CADGME 2009 Combining ISAC and CbKST • Approach 1 • determine the sequence of problems according to CbKST • determine knowledge and competence state according to result of problems

15. CADGME 2009 Combining ISAC and CbKST • Approach 1 • CbKST: domain model has to be created • CbKST: set a learning goal as set of skills • CbKST: calculate which problem should be posed to learner • ISAC: problem is posed to learner, learner goes step by step through • ISAC: result is sent to CbKST • CbKST: if knowledge state has not been found, then according to result, next problem is posed to learner (goto 3) • CbKST: if kowledge state has been found, then calculate competence state (reason available skills) • CbKST: according to knowledge and competence state appropriate learning objects are selected • ISAC: learning objects are presented to the learner

16. CADGME 2009 Combining ISAC and CbKST • Approach 2: • mapping mathematical rules to skills

17. CADGME 2009 Combining ISAC and CbKST • Approach 2: • skill definition: being able to apply a specific mathematical rule • assigning skills to problem: ISAC "knows" which mathematical rules are needed to solve the probleme • Advantages • assignment of skills to problems can be done by ISAC instead of domain expert • instead of reasoning skills from solved (or not solved) problems, skills can be directly assessed • it can be captured if a problem is solved only partly (without correct result) • direct help if learner has difficulties at a certain step

18. CADGME 2009 Combining ISAC and CbKST: Implementation • Implementation • CbKST module implemented as Web Service • ISAC connects to CbKST module • report results • get information about next objects CbKST Extension Browser HTTP SOAP ISAC

19. CADGME 2009 Authoring • Content author has to • create a domain model (in CbKST Web Service) • skill definitions • skill assignment to learning objects • problems and learning objects are referenced in CbKST domain model

20. CADGME 2009 Conclusion and Outlook • Conclusion • combining ISAC and CbKST • approach 1: sequencing problems and learning objects • approach 2: direct monitoring of skills • adaptation on micro and macro level • feedback on micro and macro level • Outlook • implementation of approach 2 • on level of algorithm • implemenation

21. Thank you for your attention! • Contact information: • Christian Gütl • cguetl@iicm.edu • http://www.iicm.tugraz.at/cguetl • Alexander Nussbaumer • alexander.nussbaumer@uni-graz.at • http://css.uni-graz.at/staff/nussbaumer/