Introduction to Knowledge Space Theory: Part II

1
Introduction to Knowledge Space TheoryPart 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

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
• examplefor 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 s(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 s(a) a s(b) a, b s(c)
  a, b, c s(d) a, b, d, d, e s(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 persons
  knowledge state

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 s(a) a s(b) a, b s(c)
  a, b, c s(d) a, b, d, d, e s(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

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 followingsurmise function for
  thedomain Q a, b, c, d, e
• What are the clauses for the problems?
• 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

s(a) a,b,d, a,c,d,e s(b) b,d s(c)
c,d,e s(d) d s(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, eatom 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

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
  domainQ 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

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 ? 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?

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

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

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 domainQ a, b, c, d, e
• Determine the fringes of the encircled knowledge
  states!

20
Thank you for your attention!