Young children’s ‘alternative’ notions of science are well documented but their unorthodox ideas about arithmetic are less well known. For example, studies have shown that young children initially treat numbers as position markers rather than size symbols. Also, children often hold a transformational view of operations; that is, they are reluctant to accept the commutativity of addition and multiplication. This ‘alternative’ view of operations is often overlooked by teachers, keen to demonstrate the so called ‘laws’ of arithmetic. However, this paper argues that we should not be in any haste to replace these primitive intuitions; instead, we should show that transformational operations actually reflect how objects behave when acted on in the physical world. The paper draws on earlier research of the writer in which young children used signs for transformational arithmetic in game scenarios. In particular, it examines the feasibility of ‘sums’ in which the operator is distinguished from the operand. In short, this paper presents the theory behind an entirely new way of teaching arithmetic, based on children’s ‘alternative’ intuitions about numbers and operations.

Young children’s ‘alternative’ notions of science are well documented but their unorthodox ideas about arithmetic are less well known. For example, studies have shown that young children initially treat numbers as position markers rather than size symbols. Also, children often hold a transformational view of operations; that is, they are reluctant to accept the commutativity of addition and multiplication. This ‘alternative’ view of operations is often overlooked by teachers, keen to demonstrate the so called ‘laws’ of arithmetic. However, this paper argues that we should not be in any haste to replace these primitive intuitions; instead, we should show that transformational operations actually reflect how objects behave when acted on in the physical world. The paper draws on earlier research of the writer in which young children used signs for transformational arithmetic in game scenarios. In particular, it examines the feasibility of ‘sums’ in which the operator is distinguished from the operand. In short, this paper presents the theory behind an entirely new way of teaching arithmetic, based on children’s ‘alternative’ intuitions about numbers and operations.

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.

Die Theorie disjunkter NP-Paare, die auf natürliche Weise statt einzelner Sprachen Paare von NP-Mengen zum Objekt ihres Studiums macht, ist vor allem wegen ihrer Anwendungen in der Kryptografie und Beweistheorie interessant. Im Zentrum dieser Dissertation steht die Analyse der Beziehung zwischen disjunkten NP-Paaren und aussagenlogischen Beweissystemen. Haben die Anwendungen der NP-Paare in der Beweistheorie maßgeblich das Verständnis aussagenlogischer Beweissysteme gefördert, so beschreiten wir in dieser Arbeit gewissermaßen den umgekehrten Weg, indem wir Methoden der Beweistheorie zur genaueren Untersuchung des Verbands disjunkter NP-Paare heranziehen. Insbesondere ordnen wir jedem Beweissystem P eine Klasse DNPP(P) von NP-Paaren zu, deren Disjunktheit in dem Beweissystem P mit polynomiell langen Beweisen gezeigt werden kann. Zu diesen Klassen DNPP(P) zeigen wir eine Reihe von Resultaten, die illustrieren, dass robust definierten Beweissystemen sinnvolle Komplexitätsklassen DNPP(P) entsprechen. Als wichtiges Hilfsmittel zur Untersuchung aussagenlogischer Beweissysteme und der daraus abgeleiteten Klassen von NP-Paaren benutzen wir die Korrespondenz starker Beweissysteme zu erststufigen arithmetischen Theorien, die gemeinhin unter dem Schlagwort beschränkte Arithmetik zusammengefasst werden. In der Praxis trifft man statt auf zwei häufig auf eine größere Zahl konkurrierender Bedingungen. Daher widmen wir uns der Erweiterung der Theorie disjunkter NP-Paare auf disjunkte Tupel von NP-Mengen. Unser Hauptergebnis in diesem Bereich besteht in der Charakterisierung der Fragen nach der Existenz optimaler Beweissysteme und vollständiger NP-Paare mit Hilfe disjunkter Tupel. / Disjoint NP-pairs are an interesting complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this dissertation we explore the connection between disjoint NP-pairs and propositional proof complexity. This connection is fruitful for both fields. Various disjoint NP-pairs have been associated with propositional proof systems which characterize important properties of these systems, yielding applications to areas such as automated theorem proving. Further, conditional and unconditional lower bounds for the separation of disjoint NP-pairs can be translated to results on lower bounds to the length of propositional proofs. In this way disjoint NP-pairs have substantially contributed to the understanding of propositional proof systems. Conversely, this dissertation aims to transfer proof-theoretic knowledge to the theory of NP-pairs to gain a more detailed understanding of the structure of the class of disjoint NP-pairs and in particular of the NP-pairs defined from propositional proof systems. For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist. As an important tool for our investigation we use the connection of propositional proof systems and disjoint NP-pairs to theories of bounded arithmetic.

Working with problems and making investigations is an activity one has to learn already very early. Therefore in primary school children should not only learn concepts and solve given tasks. They also should find out knowledge and reasons by themselves. Here you will find some problem fields in elementary arithmetic within children of primary school can make different investigations and find as well as give reasons for special statements. The topics concerned are partitions of numbers, sums of consecutive numbers, figured numbers, sequences and chains, table of hundred and numberwalls.

Working with problems and making investigations is an activity one has to learn already very early. Therefore in primary school children should not only learn concepts and solve given tasks. They also should find out knowledge and reasons by themselves. Here you will find some problem fields in elementary arithmetic within children of primary school can make different investigations and find as well as give reasons for special statements. The topics concerned are partitions of numbers, sums of consecutive numbers, figured numbers, sequences and chains, table of hundred and numberwalls.

Kognitive Diagnosemodelle (DCMs) sind konfirmatorische probabilistische Modelle mit kategorialen latenten Variablen, die Mehrfachladungsstrukturen erlauben. Sie ermöglichen die Abbildung der Kompetenzen in mehrdimensionalen Profilen, die zur Erstellung informativer Rückmeldungen dienen können. Diese Dissertation untersucht in zwei Anwendungsstudien und einer Simulationsstudie wichtige methodische Aspekte bei der Schätzung der DCMs. In der Arbeit wurde ein neuer Mathematiktest entwickelt basierend auf theoriegeleiteten vorab definierten Q-Matrizen. In den Anwendungsstudien (a) illustrierten wir die Anwendung der DCMs für empirische Daten für den neu entwickelten Mathematiktest, (b) verglichen die DCMs mit konfirmatorischen Faktorenanalysemodellen (CFAs), (c) untersuchten die inkrementelle Validität der mehrdimensionalen Profile und (d) schlugen eine Methode zum Vergleich konkurrierender DCMs vor. Ergebnisse der Anwendungsstudien zeigten, dass die geschätzten DCMs meist einen nicht akzeptablen Modellfit aufwiesen. Zudem fanden wir nur eine vernachlässigbare inkrementelle Validität der mehrdimensionalen Profile nach der Kontrolle der Personenparameter bei der Vorhersage der Mathematiknote. Zusammengenommen sprechen diese Ergebnisse dafür, dass DCMs per se keine zusätzliche Information über die mehrdimensionalen CFA-Modelle hinaus bereitstellen. DCMs erlauben jedoch eine andere Aufbereitung der Information. In der Simulationsstudie wurde die Präzision der Parameterschätzungen in log-linearen DCMs sowie die Sensitivität ausgewählter Indizes der Modellpassung auf verschiedene Formen der Fehlspezifikation der Interaktionsterme oder der Q-Matrix untersucht. Die Ergebnisse der Simulationsstudie zeigen, dass die Parameterwerte für große Stichproben korrekt geschätzt werden, während die Akkuratheit der Parameterschätzungen bei kleineren Stichproben z. T. beeinträchtigt ist. Ein großer Teil der Personen wird in Modellen mit fehlspezifizierten Q-Matrizen falsch klassifiziert. / Cognitive diagnostic classification models (DCMs) have been developed to assess the cognitive processes underlying assessment responses. Current dissertation aims to provide theoretical and practical considerations for estimation of DCMs for educational applications by investigating several important underexplored issues. To avoid problems related to retrofitting of DCMs to an already existing data, test construction of the newly mathematics assessment for primary school DMA was based on a-priori defined Q-matrices. In this dissertation we compared DCMs with established psychometric models and investigated the incremental validity of DCMs profiles over traditional IRT scores. Furthermore, we addressed the issue of the verification of the Q-matrix definition. Moreover, we examined the impact of invalid Q-matrix specification on item, respondent parameter recovery, and sensitivity of selected fit measures. In order to address these issues one simulation study and two empirical studies illustrating applications of several DCMs were conducted. In the first study we have applied DCMs in general diagnostic modelling framework and compared those models to factor analysis models. In the second study we implemented a complex simulation study and investigated the implications of Q-matrix misspecification on parameter recovery and classification accuracy for DCMs in log-linear framework. In the third study we applied results of the simulation study to a practical application based on the data for 2032 students for the DMA. Presenting arguments for additional gain of DCMs over traditional psychometric models remains challenging. Furthermore, we found only a negligible incremental validity of multivariate proficiency profiles compared to the one-dimensional IRT ability estimate. Findings from the simulation study revealed that invalid Q-matrix specifications led to decreased classification accuracy. Information-based fit indices were sensitive to strong model misspecifications.

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restrictions on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained in this way, using appropriate bisimulation characterizations and 0–1 laws as tools to differentiate between the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite, as required by the standard semantics for QFBAPA. Here we dispense with these restrictions to ease the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes. / The first version of this report was put online on April 6, 2017. The current version, containing more information on related work, was put online on July 13, 2017. This is an extended version of a paper published in the proceedings of FroCoS 2017.

Diese Dissertation untersucht Galois-Erweiterungen von Zahlkörpern und die Unverzweigte Fontaine-Mazur-Vermutung für p-adische Galois-Darstellungen und deren Verallgemeinerungen. Wir beweisen viele grundlegende Fälle der Vermutung und liefern einige nützliche Kriterien zur Überprüfung. Darüber hinaus schlagen wir mehrere verschiedene Strategien vor, um die Vermutung anzugreifen und auf einige spezielle Fälle zu reduzieren. Wir beweisen auch viele neue Ergebnisse der Vermutung im zweidimensionalen Fall. Als Anwendung beweisen wir die Endlichkeit der unverzweigten Galois-Deformationsringe unter der Annahme eines speziellen Falles der Vermutung und geben einige Gegenbeispiele zur sogenannten Dimension-Vermutung für Galois-Deformationsringe unter der Annahme der Vermutung. / This thesis studies Galois extensions of number fields, and the Unramified Fontaine-Mazur Conjecture for p-adic Galois representations and its generalizations. We prove many basic cases of the conjecture, and provide some useful criterions for verifying it. In addition, we propose several different strategies to attack the conjecture and reduce it to some special cases. We also prove many new results of the conjecture in the two-dimensional case. As an application, we prove the finiteness of unramified Galois deformation rings assuming a special case of the conjecture, and we give some counterexamples to the so-called dimension conjecture for Galois deformation rings assuming the conjecture.