• Aim : Use the knowledge on prerequisite relationships between items and cognitive demands also for teaching,
  
• Components (i.e. documents) of the Dexter model
  1. A base component (the unit of information in the component, e.g. a text or a picture),
  2. Two sets of source anchors and destination anchors located on this base component,
  3. Some additional information.
  
• In the relational hypertext model, this has been formalized by defining a component c as a triple c=(b,S_c,D_c) where b denotes a base component and S_c and D_c denote subsets of source and destination anchors respectively.
  
• A link l is defined as a pair of pairs l=((c,s_c),(c',d_c')) where c and c' denote components and s_c and d_c' denote a source and a destination anchor located on c and c' respectively.
  
• Such a construct formalizes a link from the source anchor s_c on component c to the destination anchor d_c' on the component c'.
  
• By defining subsets P of anchors characterized by a specific property also links where these anchors are involved are distinguished: Given a set L of links on a set C of components and a subset of source anchors, a subset of P-links can be defined.
  
• Examples for such properties: `is a prerequisite of' or `is an example of'.
  
• Based on these definitions, link relations between components can be defined directly: For two components , we say if and only if there exists a link for some source anchor and some destination anchor .
  
• The distinction of links with specific properties can also be transferred to this link relation between components defining a link relation by setting if and only if there exists a link for some source anchor and some destination anchor .
  
• Advantage of formalization: (a) Relational information can easily be stored, processed and retrieved in a relational database system. (b) Relational formalization can almost directly be transferred into relation schema definitions of a relational database.
  

The domain of elementary probability

• Two reasons for choosing elementary probability theory as a domain of knowledge
  1. A field which causes many problems to pupils additional support for teaching needed.
  2. Structure of the knowledge in this field was the object of prior investigations (Held, 1992); results were reused in the course development. This research dealt with the sub domain of urn experiments.
  
• Three different problem components:
  1. numerical ratio of differently coloured balls,
  2. way of drawing,
  3. specification of the asked event
  
• For each of the components : two or three possible values (attributes ) with orderings.
  
• 6 different problems
  
• 10 cognitive demands necessary for the solution of the problems.
  
• For the course one introducing demand (labelled ``0'') on random experiments and elementary events was added.

Table: Cognitive demands used in the course on elementary probability theory Random experiments and elementary eve

	Random experiments and elementary events
1	Laplace-probabilities
2	Number of possible events
3	Number of favourable events
4	Drawing multiple balls with a common property
5	Events containing elementary events with different properties
6	Addition of events and probabilities
7	Multiplication of events and probabilities
8	Different proportions of properties
9	Drawing without replacement
10	Generalized (``at least n balls) descriptions of events

• Example problem of the course:
  'An urn contains five yellow and five black balls. Two balls are drawn from the urn successively. Drawing is performed with replacement. The drawn balls are yellow. Compute the probability of this event.'
  For solving this problem demands 0 to 4 and 7 are required.
  
• Partial order on the set of problems and demands identified based on the attribute specifications of problems and demands
  
• NOTE: Edges of the Hasse diagram of this order are the prerequisite links between the documents (lessons and exercises) in the RATH course
  
• Besides lessons and exercises, the RATH course also contains examples for the lessons. Examples in the RATH course contain a prerequisite link to the corresponding lesson.
  

Structure of Demands and Exercises

The RATH system (The current status of RATH: First prototype )

Screenshots

Example for Adaptivity in RATH

Lesson from the RATH Course

Exercise from the RATH Course

Example from the RATH course

RATH Homepage

Further Research

• RATH is a prototype system. Things to do:
  
  • Introduction of an initial assessment of a pupil's knowledge.
    
  • Improvementof the analysis of pupils' answers to the exercises.
    
  • Evaluation of the system.
    
    • Evaluation of the course: Evaluation of course structure exists (Held, 1992); no evaluation of the lessons' quality.
      
    • Evaluation of the underlying (technical) system
• Summary: RATH is an important step in bringing psychological theories into a working tutorial system.

RATH-URL

http://wundt.uni-graz.at/rath/

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