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ABSTRACT. The fluency of basic arithmetical operations is a precondition for mathematical problem solving. However, the training of skills plays a minor role in contemporary mathematics instruction. The authors proposed individualization of practice as a means to improve its efficiency, so that the time spent with the training of skills is minimized. As a tool to relieve teachers from the time-consuming tasks of individual diagnosis, selection of problems, and immediate feedback, they developed adaptive training software. The authors evaluated the application of the software in 2 naturalistic studies with 9 third-grade classes. Results show that even a moderate amount of individualized practice was associated with large improvements of arithmetic skills and problem solving, even after a follow-up period of 3 months.

A common view in the research on dynamic system control is that human subjects use exemplar knowledge of system states-at least for controlling small systems. Dissociations between different tasks or stochastic independence between recognition and control tasks, have led to the assumption that part of the exemplar knowledge is implicit. In this paper, I propose an alternative interpretation of these results by demonstrating that subjects learn more than exemplars when they are introduced to a new system. This was achieved by presenting the same material-states of a simple system-with vs. without causal interpretation. If subjects learned exemplars only, then there should be no differences between the conditions and stochastic dependence between various tasks would be expected. However, in an experiment with N=40 subjects the group with causal interpretation is significantly better at completing fragmentary system states and in judging causal relations between switches and lamps, but not in recognizing stimuli as studied. Only in the group without causal interpretation, the contingency between recognition and completion was close to the maximum memory dependence, estimated with Ostergaard's (1992) method. Thus, the results resemble those of other studies only in the condition with causal interpretation. The results are explained by the assumption that subjects under that condition learn and use a second type of knowledge, which is construed as a rudimentary form of structural knowledge. The interpretation is supported by a computational model that is able to reproduce the set of results.

Although individual differences in complex problem solving (CPS) are well–established, relatively little is known about the process demands that are common to different dynamic control (CDC) tasks. A prominent example is the VOTAT strategy that describes the separate variation of input variables (“Vary One Thing At a Time”) for analyzing the causal structure of a system. To investigate such comprehensive knowledge elements and strategies, we devised the real-time driven CDC environment Dynamis2 and compared it with the widely used CPS test MicroDYN in a transfer experiment. One hundred sixty five subjects participated in the experiment, which completely combined the role of MicroDYN and Dynamis2 as source or target problem. Figural reasoning was assessed using a variant of the Raven Test. We found the expected substantial correlations among figural reasoning and performance in both CDC tasks. Moreover, MicroDYN and Dynamis2 share 15.4% unique variance controlling for figural reasoning. We found positive transfer from MicroDYN to Dynamis2, but no transfer in the opposite direction. Contrary to our expectation, transfer was not mediated by VOTAT but by an approach that is characterized by setting all input variables to zero after an intervention and waiting a certain time. This strategy (called PULSE strategy) enables the problem solver to observe the eigendynamics of the system. We conclude that for the study of complex problem solving it is important to employ a range of different CDC tasks in order to identify components of CPS. We propose that besides VOTAT and PULSE other comprehensive knowledge elements and strategies, which contribute to successful CPS, should be investigated. The positive transfer from MicroDYN to the more complex and dynamic Dynamis2 suggests an application of MicroDYN as training device.

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Zusammenfassung: Der Beitrag gibt einen Überblick über individuelle Unterschiede bei der Lösung komplexer Probleme. Betrachtet werden Unterschiede in Wissen und Fertigkeiten, in Parametern der Informationsverarbeitung und in motivationalen Variablen. Das bereichsspezifische Vorwissen wird dabei als stärkste Einflussvariable identifiziert. Charakteristisch für Problemlöseprozesse sind Veränderungen im Wissen und in der Motivation, was den Einfluss vorab erhobener Variablen schmälert. Für die zukünftige Forschung wird gefordert, die zentrale Rolle von Strategien näher zu untersuchen, sie mit Aufgabenanalysen zu verknüpfen und Simulationsmodelle weiter zu entwickeln.

Zusammenfassung. Wer Experimente in Schulklassen durchführt, hat mit hierarchisch strukturierten Daten zu tun, was Mehrebenenanalysen nahelegt. Meist sind solche Experimente so aufwändig, dass die für hierarchisch lineare Modelle üblichen Stichprobengrößen nicht zu erreichen sind. Wenn man an Vorhersagen auf Klassenebene nicht interessiert ist, bieten sich alternativ Varianzanalysen an, die die Klasse als Faktor einbeziehen. In einer Simulationsstudie wurden die Äquivalenz, die Reaktion auf variierte Rahmenbedingungen und die Genauigkeit der Parameterschätzungen der beiden Verfahren geprüft. Dazu wurden acht mal 1000 Datensätze simuliert, die sich systematisch in der Anzahl der Klassen, der Balance der Klassengrößen und der Intraklassenkorrelation unterschieden. Die Datensätze wurden mit hierarchischen Regressionsanalysen nach dem random-intercept Modell und mit Varianzanalysen ausgewertet und die Ergebnisse verglichen. Es zeigte sich, dass die Teststärke der beiden Methoden praktisch gleich ist, dass die Rahmenbedingungen sich nur schwach auswirken und dass die hierarchische Regressionsanalyse die Modellparameter bei Datensätzen einer Größe von zehn Klassen befriedigend reproduziert.

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