1. George Goguadze, Eric Andrès Universität des Saarlandes Johan Jeuring, Bastiaan Heeren Open Universiteit Nederland Generation of Interactive Exercises using Domain Reasoners

2. 2 Introduction One of the crucial assets of ITS is interactive problem solving support: constant assessment of student’s progress training the mastery of concepts Systems like Andes, PUMP Algebra Tutor are targeting problem solving Complex learning environment (such as ActiveMath) need to support problem solving in multiple disciplines

3. 3 CTAT Example Tracing Tutors The Solution graph is authored in a “Behavior Recorder” Each possible correct answer has to be authored Feedback and hints are authored Can encode exercises in multiple domains The Solution graph is generated Feedback and hints are generated Sequencing and types of feedback are fixed Domain is fixed, but technology is applicable to other domains such as chemistry ANDES

4. 4 ActiveMath Platform Supports multiple math. disciplines Relies on exercises for the student assessment Needs different types of exercises for training domain skills Adapts to students’ knowledge and other parameters

5. 5 ActiveMath Exercises Well Done! calculate (x^2)’+x’ calculate (x^2)’+x’ calculate (x^2)’+1 calculate (x^2)’+1 calculate 2x+x’ calculate 2x+x’ 2x + 1 Correct ! Incorrect Try again Incorrect Try again Correct! Incorrect Try again Incorrect Try again Incorrect Try again Incorrect Try again calculate (x^2)’+x’ calculate (x^2)’+x’ calculate (x^2)’+1 calculate (x^2)’+1 calculate 2x+x’ calculate 2x+x’ 2x+1 Domain Resoner

6. 6 ActiveMath Exercises Exercise is represented as a state graph the structure is domain-independent system knows how to “play” the exercise External reasoners provide domain intelligence domain related queries are sent to the reasoners diagnosis, hints, solutions, feedback is generated multiple reasoners can be adressed simultaneously

7. 7 Generic Query Format A query to an external reasoning service consists of : action of the query, e.g.’compare’, or ‘getUserSolutionPath’ (list of) input expressions context of the query, e.g. ‘arith’ number of iterations (optional)

8. 8 query ( compare, userInput, (x^2)’+x’, diff_arith, 2) matching all possible correct answers using queries reduces the solution space (x^2)’+x’ (x^2)’+1 linear_rule 2x+x’ 2x+1 (x^2)’+x’ 2x+1 query(compare, userInput, 2x+1, diff_arith, 2) power_rule linear_rule

9. 9 query ( compare, userInput, (x^2)’+x’, diff_arith+buggy, 2) x+x’ (x^2)’+x ’ (x^2)’+1 linear_rule 2x+x’ power_rule linear_rule 2x+1 buggy_power1 x+1 linear_rule query(compare, (x^2)’+x’,x+1, diff_arith) = false query(compare, (x^2)’+x’, x+1, diff_arith+buggy) = true query(compare, x+x’,x+1, diff_arith) = true query(compare, (x^2)’+x’,x+1, diff_arith) = false query(compare, (x^2)’+x’, x+1, diff_arith+buggy) = true query(compare, x+x’,x+1, diff_arith) = true

10. 10 Generation using Domain Reasoners An exercise generator produces general-purpose exercise can use different tutorial strategies keeps track of the student’s state Diagnosis is generated by domain reasoner queries: compare, getBuggyRules, getUserSolutionPaths, getNOfStepsLeft Hint and Feedback generation: basic correct/incorrect feedback ( compare ) task related conceptual and procedural feedback ( getConcepts ) product feedback ( getNextStep, getExpertSolutionPath(s) ) error feedback ( getBuggyRules, getUserSolutionPath(s) )

11. 11 Generation using Domain Reasoners

12. 12 Diagnosis and Feedback Generation

13. 13 Combining Domain Reasoners Equation Solver Reasoner Derivatives Reasoner

14. 14 Conclusion: Current State Domain Reasoner TechnologyDomain(s) IDEAS (1500 exercises) Haskell Algebra Arithmetic Calculus Linear Algebra Logic FractionDRYACASFraction Arithmetics IntegralDRYACASIntegration MathCoachPROLOGDifferentiation SLOPERTPROLOGDifferentiation

15. 15 Conclusion/Future Work Seemless Integration of different domain reasoners most major mathematical subjects are covered simultaneous access to multiple reasoners Exercises with different tutorial behaviour possible immediate or delayed feedback strategies possible different types of feedback generated on request Need to connect other domains (physics, computer science) challenge: reuse domain reasoners of existing ITS Need more advanced interaction interfaces allow for more complex reasoning alternative graphical interaction

16. 16 Thank you!