The domain of elementary probability theory

 There exist two reasons for choosing elementary probability theory as a domain of knowledge for the first prototype course for RATH. First, this is a field which causes many problems to pupils and, therefore, additional support for teaching is needed. Second, the structure of the knowledge in this field was the object of prior investigations (Held, 1993) whose results were reused in the course development. This research dealt with the sub domain of urn experiments.

Held used four different problem components: the numerical ratio of differently coloured balls, the way of drawing, the specification of the asked event, and the wording of the problem. The last component was neglected in our course construction. For each of the components, there existed two or three possible values (attributes ). Problems and demands were characterized by corresponding tuples of attributes. Based on orderings between the possible values for each of the components partial orders for problems and demands were derived.

Held used eighteen different problems, and identified thirteen cognitive demands necessary for the solution of the problems. Due to the neglection of the wording component, these numbers were reduced to six problems and ten demands (labelled from ``1'' to ``10''). For the course, however, one introducing demand (labelled ``0'') on random experiments and elementary events was added. The list of the resulting eleven demands is shown in Table 1.

 

 

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

One example problem of the course is presetned below:

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, the demands 0 to 4 and 7 are required.

Based on the attribute specifications of problems and demands a partial order on the set of problems and demands could be identified. The edges of the Hasse diagram of this order are the prerequisite links between the documents (lessons and exercises) in the RATH course.