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Enhanced Personalised Learning Support of Computer Algebra Systems

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- 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

CADGME 2009

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

CADGME 2009

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

CADGME 2009

- 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

CADGME 2009

CADGME 2009

CADGME 2009

CADGME 2009

- 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?

CADGME 2009

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- Knowledge Space Theory (KST) - behaviouristic theory
- knowledge structure := set of possible knowledge states with respect to prerequisite relations between problems

CADGME 2009

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- Adaptive Assessment
- problem b sloved
- problem d solved
- problem c not solved

Personal Learning Paths

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CADGME 2009

- 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

CADGME 2009

- 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

CADGME 2009

- 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

CADGME 2009

- Approach 1
- determine the sequence of problems according to CbKST
- determine knowledge and competence state according to result of problems

CADGME 2009

- 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, thenaccording to result, next problem is posed to learner (goto 3)
- CbKST: if kowledge state has been found, thencalculate competence state (reason available skills)
- CbKST: according to knowledge and competence stateappropriate learning objects are selected
- ISAC: learning objects are presented to the learner

CADGME 2009

- Approach 2:
- mapping mathematical rules to skills

CADGME 2009

- 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

CADGME 2009

- 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

CADGME 2009

- 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

CADGME 2009

- 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

Thank you for your attention!

- Contact information:
- Christian Gütl
- [email protected]
- http://www.iicm.tugraz.at/cguetl
- Alexander Nussbaumer
- [email protected]
- http://css.uni-graz.at/staff/nussbaumer/