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Aufgabengeeignete Rechenstrategien flexibel zu nutzen ist ein wichtiges Ziel mathematischer Bildung und Bestandteil der Bildungsstandards der Grundschulmathematik. Kinder sollen spontan entscheiden, ob sie arithmetische Aufgaben in üblicher Weise berechnen oder ob sie Zeit und Aufwand investieren, um nach Vereinfachungsstrategien zu suchen und diese anzuwenden. Der Schwerpunkt der aktuellen Arbeit ist, wie Schüler beim flexiblen Erkennen und Anwenden von Vereinfachungsstrategien unterstützt werden können. Kontextfaktoren werden untersucht, welche die spontane Nutzung von Vereinfachungsstrategien unterstützen und den Transfer zwischen ihnen beeinflussen. Kognitive Theorien über die Entwicklung von mathematischen Konzepten und Strategien wurden mit Erkenntnissen aus der Expertise Forschung verbunden, welche die Unterschiede in der Flexibilität zwischen Experten und Novizen offen legen. Im Rahmen der iterativen Entwicklung von mathematischen Konzepten könnte ein erfolgreiches Erkennen und Anwenden einer Vereinfachungsstrategie von Faktoren, die konzeptionelles und/oder prozedurales Wissen aktivieren, profitieren. Am Beispiel von Vereinfachungsstrategien, die auf dem Kommutativgesetz (a + b = b + a) basieren, werden drei Kontextfaktoren (Instruktion, Assoziation und Schätzen), die spontanen Strategiegebrauch unterstützen oder behindern, untersucht. Insgesamt zeigt die Dissertation, dass spontane Strategienutzung durch bestimmte Kontextfaktoren unterstützt und durch Andere behindert werden kann. Diese Kontextfaktoren können im Prinzip in der Schulumgebung gesteuert werden., Flexible use of task-appropriate solving strategies is an important goal in mathematical education and educational standard of elementary school mathematics. Children need to decide spontaneously whether they calculate arithmetic problems the usual way or whether they invest time and effort to search for shortcut options and apply them. The focus of the current work lies on how students can be supported in spotting and applying shortcut strategies flexibly. Contextual factors are investigated that support the spontaneous usage of shortcuts and influences the transfer between them. Cognitive theories about how mathematical concepts and strategies develop were combined with findings from research on expertise, which disclose differences between the flexibility of experts and novices. In line with iterativ development of mathematical concepts successfully spotting and applying a shortcut might thus benefit from factors activating conceptual and/or procedural knowledge. Shortcuts based on commutativity (a + b = b + a) are used as a test case to investigat three contextual factors (instruction, association and estimation), which support or hinder spontaneous strategy use. Overall, the dissertation shows that spontaneous strategy use can be supported by some contextual factors and impeded by others. These contextual factors can, in principle, be controlled in school environment.

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.

One crucial feature of expertise is the ability to spontaneously recognize where and when knowledge can be applied to simplify task processing. Mental arithmetic is one domain in which people should start to develop such expert knowledge in primary school by integrating conceptual knowledge about mathematical principles and procedural knowledge about shortcuts. If successful, knowledge integration should lead to transfer between procedurally different shortcuts that are based on the same mathematical principle and therefore likely are both associated to the respective conceptual knowledge. Taking commutativity principle as a model case, we tested this conjecture in two experiments with primary school children. In Experiment 1, we obtained eye tracking data suggesting that students indeed engaged in search processes when confronted with mental arithmetic problems to which a formerly feasible shortcut no longer applied. In Experiment 2, children who were first provided material allowing for one commutativity-based shortcut later profited from material allowing for a different shortcut based on the same principle. This was not the case for a control group, who had first worked on material that allowed for a shortcut not based on commutativity. The results suggest that spontaneous shortcut usage triggers knowledge about different shortcuts based on the same principle. This is in line with the notion of adaptive expertise linking conceptual and procedural knowledge.