1. Introduction to Knowledge Space Theory: Part II Christina Steiner, University of Graz, Austria April 4, 2005

2. Surmise Function  for mastering a problem p  there is a minimal set of problems that must have been mastered before -prerequisites for problem p -example: c b a d e

3. Surmise Function  there may be more than one set of prerequisites to a problem -these different sets of prerequisites may represent alternative ways of solving a problem -example: for the mastery of problem d, the problems (a and b) or e must have been mastered before  to capture the fact, that a problem may have more than one set of prerequisites, the notion of a surmise function has been introduced -generalisation of the concept of a surmise relation -allows for assigning multiple sets of prerequisites to a problem

4. Surmise Function  assigns to each problem p a family of subsets of problems, called ‚clauses‘ -denoted by σ(p) -they represent all possible ways of acquiring the mastery of problem p -minimal states containing problem p  can be depicted by an And/Or-Graph  example: σ(a) = {{a}} σ(b) = {{a, b}} σ(c) = {{a, b, c}} σ(d) = {{a, b, d}, {d, e}} σ(e) = {{e}}  if a person is found to have mastered a given problem, then at least one ot the clauses for the problem must be included in the person‘s knowledge state c b a d e v

5. Surmise Function  clauses satisfy the following conditions -for each problem p, there is at least one clause for p -every clause for a problem p contains p -if a problem q is in some clause C for p, then there must be some clause D for q included in C -example: σ(a) = {{a}} σ(b) = {{a, b}} σ(c) = {{a, b, c}} σ(d) = {{a, b, d}, {d, e}} σ(e) = {{e}} -any two clauses C, C‘ for the same problem are incomparable, i.e. neither C C‘ nor C‘ C

6. Surmise Function  a knowledge structure conforming to a surmise function  is closed under union but  not necessarily under intersection  example: c b a d e v K = { Ø, {a}, {e}, {a, b}, {a, e}, {d, e}, {a, b, c}, {a, b, d}, {a, b, e}, {a, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, b, c, d, e}}

7. Exercise  Let us assume the following surmise function for the domain Q = {a, b, c, d, e} a)What are the clauses for the problems? b)Find the collection of possible knowledge states corresponding to the surmise function! K ={ Ø, {d}, {e}, {b,d}, {d,e}, {a,b,d}, {b,d,e}, {c,d,e}, {a,b,d,e}, {a,c,d,e}, {b,c,d,e}, {a,b,c,d,e}} σ(a) = {{a,b,d}, {a,c,d,e}} σ(b) = {{b,d}} σ(c) = {{c,d,e}} σ(d) = {{d}} σ(e) = {{e}}

8. Base of a Knowledge Space  in practical application knowledge spaces can grow very large  the base B of a knowledge space provides a way of describing such a structure economically  exploiting the property of being closed under union  smallest subcollection of a knowledge space from which the complete knowledge space can be reconstructed by closure under union

9. Base of a Knowledge Space  example: K ={ Ø, {a}, {e}, {a, b}, {a, e}, {a, b, e}, {a, b, c}, {a, b, c, e}, {a, b, d, e}, {a, b, c, d, e}} all states of the given knowledge space can be obtained by taking all arbitrary unions of the states included in the subcollection: B = {{a}, {e}, {a, b}, {a, b, c}, {a, b, d, e}}

10. Base of a Knowledge Space  the base of a knowledge space is formed by the family of all knowledge states that are minimal for at least one problem  atoms of a knowledge space -for any problem p, an atom at p is a minimal knowledge state containing p -a knowledge state K is minimal for an item p if for any other knowledge state K‘ the condition K‘ K holds

11. Base of a Knowledge Space  example: K = { Ø, {a}, {e}, {a, b}, {a, e}, {a, b, e}, {a, b, c}, {a, b, c, e}, {a, b, d, e}, {a, b, c, d, e}} atom at a:{a} atom at b:{a, b} atom at c:{a, b, c} atom at d:{a, b, d, e} atom at e: {e} B = {{a}, {e}, {a, b}, {a, b, c}, {a, b, d, e}}  in case of a knowledge space induced by a surmise function -each of the clauses is an element of the base -each element of the base is a clause for some problem c b a d e

12. Exercise  Let us assume the following base of a knowledge space for the domain Q = {a, b, c, d, e} B = {{b}, {c}, {c, d}, {a, b, c}, {c, d, e}} -Find the collection of all possible knowledge states! K ={ Ø, {b}, {c}, {b, c}, {c, d}, {a, b, c}, {b, c, d}, {c, d, e}, {a, b, c, d}, {b, c, d, e}, {a, b, c, d, e}}

13. Exercise  Let us assume the following surmise relation and the corresponding knowledge space for the domain Q = {a, b, c, d, e} -Determine the base! B = {{a}, {b}, {a, c}, {a, b, c, d}, {a, b, c, e}, {a, b, c, d, e, f}} a cb de f K ={ Ø, {a}, {b}, {a, b}, {a, c}, {a, b, c}, {a, b, c, d}, {a, b, c, e}, {a, b, c, d, e}, {a, b, c, d, e, f}}

14. Learning Paths  a knowledge structure allows several learning paths  starting from the naive knowledge state  leading to the knowledge state of full mastery Ø {a}{e} {a,e} {a,b} {a,b,c}{a,b,e} {a,b,c,e} {a,b,d,e} {a,b,c,d,e} Ø  a  e  b  d  c Ø  a  b  c  e  d

15. Exercise  How many learning paths are possible for the given knowledge structure? -Which sequences of problems do they suggest for learning? Ø {a}{e} {a,e} {a,b} {a,b,c}{a,b,e} {a,b,c,e} {a,b,d,e} {a,b,c,d,e} cd edc bced aebcd Ødc eabcd dc

16. Well-Graded Knowledge Structure  a knowledge structure where learning can take place step by step is called well-graded  each knowledge state has at least one immediate successor state  containing all the same problems, plus exactly one  each knowledge state has at least one predecessor state  containing exactly the same problems, except one Ø {a}{e} {a,e} {a,b} {a,b,c}{a,b,e} {a,b,c,e} {a,b,d,e} {a,b,c,d,e}

17. Fringes of a Knowledge State  outer fringe  set of all problems p such that adding p to K forms another knowledge state -learning proceeds by mastering a new problem in the outer fringe  inner fringe  set of all problems p such that removing p from K forms another knowledge state -reviewing previous material should take place in the inner fringe of the current knowledge state Ø {a}{e} {a,e} {a,b} {a,b,c}{a,b,e} {a,b,c,e} {a,b,d,e} {a,b,c,d,e}

18. Fringes of a Knowledge State  for a well-graded knowledge structure the two fringes suffice to completely specify the knowledge state  summarising the results of assessment  the knowledge state of a learner can be characterized by two lists -the inner fringe specifies what the student can do (the most sophisticated problems in the knowledge state) -the outer fringe specifies what the student is ready to learn

19. Exercise  Let us assume the following knowledge structure for the domain Q = {a, b, c, d, e} -Determine the fringes of the encircled knowledge states! Ø {d}{e} {d,e}{b,d} {c,d,e} {b,d,e}{a,b,d} {b,c,d,e}{a,c,d,e}{a,b,d,e} {a,b,c,d,e} knowledge state outer fringe inner fringe {b,d}{a,e}{b} {b,d,e}{a,c}{b,e} {a,b,d,e}{c}{a,e}

20. Thank you for your attention!