PhD thesis,
Modelling early mathematical competencies and misconceptions: An empirical approach based on knowledge space theory and information systems
(2003)
Abstract
The present study was concerned with laying the foundations for an adaptive instrument accounting
for the assessment of correct knowledge elements as well as misconceptions, in the domain of early
mathematics. It was intended not only to serve for the diagnosis of correct knowledge and
misconceptions, but also for their mediation and correction. The construction of the instrument was
done in three steps.
In investigation 1, a number of 26 correct knowledge elements was modelled. As the methodological
framework for modelling, the competence-performance theory - an extension of the knowledge space
theory - was chosen. By means of this theory a number of 49 problems was constructed and structured.
The solution dependencies among the problems were obtained by the way of ordering the subsets of
elementary competencies, accounting for their solution, according to the inclusion principle. A number
of 94 kindergarten children aged 62.84 months on average (SD=9.89) were separately presented six
subsets of thirteen problems each. A number of validation measures showed their solution patterns,
that is, the subsets of solved problems, not to deviate much from the hypothesised patterns, so that the
model was proven valid.
In investigation 2, a number of five misconceptions and seven correct knowledge elements was
modelled. As the methodological framework for modelling an approach based on information systems
was chosen. This approach does not only account for dependencies among knowledge elements and
hence problems, but also for incompatibilities among them. Thirteen problem types, consisting of
three parallel problems each, were constructed. Each of the problem types could either be solved
correctly or incorrect responses caused by misconceptions could be given. For validating the response
patterns, that is, combinations of correct and buggy answers, which arose from the structure on the set
of knowledge elements, 64 preschoolers, aged 61.84 months on average (SD=9.40), were separately
tested. Each proband was presented one of five subsets of problems types, twelve problems each. The
validation measures suggested the model not to be valid.
Finally, in a survey a glossary of training tasks, accounting for the mediation and correction of
knowledge elements, specified in investigations 1 and 2, was made out. The author constructed one
training task for each knowledge element. The construction was guided by four predefined teaching
principles: concrete materials, natural context, child-adequate language, possibility to act. Twentyseven
experts from respective professions such as nursery school teachers or elementary school
teachers were given the training tasks for judging them concerning suitability for the mediation and
correction of knowledge. In addition, the experts had to make suggestions for training tasks
themselves. From the proposals of the experts the best one per knowledge entity was selected
according to the instructional principles.
The results of the three parts of the work on hand were integrated and by means of a practical example
the applicability of the results to the assessment and mediation of knowledge was demonstrated.
