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Automated Theory Formation for Tutoring Tasks in Pure Mathematics

Automated Theory Formation for Tutoring Tasks in Pure Mathematics. Simon Colton, Roy McCasland, Alan Bundy, Toby Walsh. Tools for Maths Teachers. Setting exercises is important in mathematics education Automated tools available for High school mathematics and below

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Automated Theory Formation for Tutoring Tasks in Pure Mathematics

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  1. Automated Theory Formation for Tutoring Tasks in Pure Mathematics Simon Colton, Roy McCasland, Alan Bundy, Toby Walsh

  2. Tools for Maths Teachers • Setting exercises is important in mathematics education • Automated tools available for • High school mathematics and below • No tool available for University level • Use ATF via the HR program • As an aid to setting exercises • Example in group theory

  3. The HR Program • Automated theory formation • Concepts, conjectures, proofs, counters • Concept formation • 10 production rules • Conjecture making • Find patterns empirically • Settling conjectures • Using Otter and MACE (and others) • See CADE system description

  4. Group theory exercises • In most text books/courses: • Determining subgroups • Negative results useful in learning process • Example: centre of a group • Elements which commute with all others • Student has to show: • Closure, associativity, identity, inverse • Or: non-empty and a & b in Z  a * b-1 in Z

  5. Approach using HR • Use HR’s concept formation • Find different types of element • Flag those forming a subgroup empirically • Use Otter • To show that the subset forms a subgroup • Use MACE • To find a counterexample group • Use (human) teacher • Interpret results as exercises

  6. Improvements to HR #1Embed Algebra PR • Designed to be domain independent • Find embedded algebraic structures • In algebras, graphs, integers • Any arity four concept • Triples of subobjects possibly form algebra • User sets algebras to look for • HR abstracts subobjects into Cayley table • MACE used to check axioms • HR checks isomorphism with previous • MACE not asked to search (efficient)

  7. Improvements to HR #2“Reactive” search • Heuristic search  Best first search • BFS often better after delay • Some PRs should be used sparingly • E.g., disjunction, instantiation • Other PRs should be used when poss • E.g., embed algebra rule • HR’s reactive search: after each step: • Java code fragment read by HR • Different to the paper

  8. Experimental Setup • Groups up to order 8 (14 groups) • Reaction to new element type • Force use of embed-algebra • Flag concepts forming non-trivial subgroups • 10,000 theory formation step • No proving or counterexample finding (fast) • Any promising subgroup types • Further investigate with Otter and MACE • Pentium 4 (2.0 Ghz) under Windows XP

  9. Suggestions for Using Results • If subgroup property is proved • Ask student to prove this • If known counterexamples • Ask student to determine smallest • Ask student to identify classes of group • Ask student to characterise all groups •  Caveat  • These problems may be too difficult

  10. Results • 301 seconds to finish search • 330 concepts • 17 element types found • 10 produced subgroups empirically • 8 were non-trivial • Look now at two element types in detail

  11. Concept g93 • [a,b] : all c (exists d (d*c=b & c*d=b)) • Actually defines centre of group (paper) • HR often comes up with usual definition • Empirically true, can we prove it? • Otter employed for three tests: • Closure of: identity, inverse, multiplication • 0.2 secs, 25 secs, 55 secs • Obvious interpretation for tutorial • Nice to see historically interesting result

  12. Concept g43 • [a,b] : exists c (c*c = b) • Diagonal elements on Cayley table • Groups up to order 8: forms a subgroup • Fairly certain not in general case • Suggested trying to disprove first • Passed MACE: axioms, g43 definition • And multiplicative closure property • 143 seconds later, MACE produced:

  13. Smallest(?) “Bad” group forconcept g43 * | 0 1 2 3 4 5 6 7 8 9 10 11 --+------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 1 | 1 0 3 2 5 4 7 6 9 8 11 10 2 | 2 3 4 5 0 1 8 9 10 11 6 7 3 | 3 2 5 4 1 0 9 8 11 10 7 6 4 | 4 5 0 1 2 3 10 11 6 7 8 9 5 | 5 4 1 0 3 2 11 10 7 6 9 8 6 | 6 7 10 11 8 9 1 0 5 4 3 2 7 | 7 6 11 10 9 8 0 1 4 5 2 3 8 | 8 9 6 7 10 11 3 2 1 0 5 4 9 | 9 8 7 6 11 10 2 3 0 1 4 5 10 | 10 11 8 9 6 7 5 4 3 2 1 0 11 | 11 10 9 8 7 6 4 5 2 3 0 1

  14. Possible Use of Concept g43 • Could ask student to find MACE’s c-ex. • Alternatively: • Ask for a subclass of groups with property • (Example Abelian groups) • Ask for a characterisation (honors) • See Appendix A of paper • For 10 tutorial questions which arose

  15. Other ways to use results • Some subsets of elements are contained • In other subsets of elements • Both sets identified by HR (and the conjecture) • Intuition of students • Mathematicians use minimal hypotheses • HR sometimes produces conjectures: • With non-minimal and/or convoluted hypotheses • Useful for students to prove theorems with non-minimal or convoluted hypotheses b*c=d & d*b=c & d*d=b  inv(c)=d

  16. Conclusions and Future Work • Performed an initial feasibility study • HR, Otter, MACE help with maths tutorials • Example in group theory • Novel questions arose from non-human results • Possible to use HR semi-automatically • Maybe HR used as a tool in future

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