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Mathematical arguments are central components of mathematics and play a role in certain types of modelling of potential mathematical giftedness. However, particular characteristics of arguments are interpreted differently in the context of mathematical giftedness. Some models of giftedness see no connection, whereas other models consider the formulation of complete and plausible arguments as a partial aspect of giftedness. Furthermore, longitudinal changes in argumentation characteristics remain open. This leads to the research focus of this article, which is to identify and describe the changes of argumentation products in potentially mathematically gifted children over a longer period. For this purpose, the argumentation products of children from third to sixth grade are collected throughout a longitudinal study and examined with respect to the use of examples and generalizations. The analysis of all products results in six different types of changes in the characteristics of the argumentation products identified over the survey period and case studies are used to illustrate student use of examples and generalizations of these types. This not only reveals the general importance of the use of examples in arguments. For one type, an increase in generalized arguments can be observed over the survey period. The article will conclude with a discussion of the role of argument characteristics in describing potential mathematical giftedness.
Interactional niche in the development of geometrical and spatial thinking in the familial context
(2016)
In the analysis of mathematics education in early childhood it is necessary to consider the familial context, which has a significant influence on development in early childhood. Many reputable international research studies emphasize that the more children experience mathematical situations in their families, the more different emerging forms of participation occur for the children that enable them to learn mathematics in the early years. In this sense mathematical activities in the familial context are cornerstones of children’s mathematical development, which is also affected by the ethnic, cultural, educational and linguistic features of their families. Germany has a population of approximately 82 million, about 7.2 million of whom are immigrants (Statisches Bundesamt 2009, pp.28-32). Children in immigrant families grow up with multiculturalism and multilingualism, therefore these children are categorized as a risk group in Germany. “Early Steps in Mathematics Learning – Family Study” (erStMaL-FaSt) is the one of the first familial studies in Germany to deal with the impact of familial socialization on mathematics learning. The study enables us to observe children from different ethnic groups with their family members in different mathematical play situations. The family study (erStMaL-FaSt) is empirically performed within the framework of the erStMaL (Early Steps in Mathematics Learning) project, which relates to the investigation of longitudinal mathematical cognitive development in preschool and early primary-school ages from a socio-constructivist perspective. This study uses two selected mathematical domains, Geometry and Measurement, and four play situations within these two mathematical domains.
My PhD study is situated in erStMaL-FaSt. Therefore, in the beginning of this first chapter, I briefly touch upon IDeA Centre and the erStMaL project and then elaborate on erStMaL-FaSt. As parts of my research concepts, I specify two themes of erStMaL-FaSt: family and play. Thereafter I elaborate upon my research interest. The aim of my study is the research and development of theoretical insights in the functioning of familial interactions for the formation of geometrical (spatial) thinking and learning of children of Turkish ethnic background. Therefore, still in Chapter 1, I present some background on the Turkish people who live in Germany and the spatial development of the children.
This study is designed as a longitudinal study and constructed from interactionist and socio-constructivist perspectives. From a socio-constructivist perspective the cognitive development of an individual is constitutively bound to the participation of this individual in a variety of social interactions. In this regard the presence of each family member provides the child with some “learning opportunities” that are embedded in the interactive process of negotiation of meaning about mathematical play. During the interaction of such various mathematical learning situations, there occur different emerging forms of participation and support. For the purpose of analysing the spatial development of a child in interaction processes in play situations with family members, various statuses of participation are constructed and theoretically described in terms of the concept of the “interactional niche in the development of mathematical thinking in the familial context” (NMT-Family) (Acar & Krummheuer, 2011), which is adapted to the special needs of familial interaction processes. The concept of the “interactional niche in the development of mathematical thinking” (NMT) consists of the “learning offerings” provided by a group or society, which are specific to their culture and are categorized as aspects of “allocation”, and of the situationally emerging performance occurring in the process of meaning negotiation, both of which are subsumed under the aspect of the “situation”, and of the individual contribution of the particular child, which constitutes the aspect of “child’s contribution” (Krummheuer 2011a, 2011b, 2012, 2014; Krummheuer & Schütte 2014). Thereby NMT-Family is constructed as a subconcept of NMT, which offers the advantage of closer analyses and comparisons between familial mathematical learning occasions in early childhood and primary school ages.
Within the scope of NMT-Family, a “mathematics learning support system” (MLSS) is an interactional system which may emerge between the child and the family members in the course of the interaction process of concrete situations in play (Krummheuer & Acar Bayraktar, 2011). All these topics are addressed in Chapter 2 as theoretical approaches and in Chapter 3 as the research method of this study. In Chapter 4 the data collection and analysis is clarified in respect of these approaches...
In an earlier paper we proposed a recursive model for epidemics; in the present paper we generalize this model to include the asymptomatic or unrecorded symptomatic people, which we call dark people (dark sector). We call this the SEPARd-model. A delay differential equation version of the model is added; it allows a better comparison to other models. We carry this out by a comparison with the classical SIR model and indicate why we believe that the SEPARd model may work better for Covid-19 than other approaches.
In the second part of the paper we explain how to deal with the data provided by the JHU, in particular we explain how to derive central model parameters from the data. Other parameters, like the size of the dark sector, are less accessible and have to be estimated more roughly, at best by results of representative serological studies which are accessible, however, only for a few countries. We start our country studies with Switzerland where such data are available. Then we apply the model to a collection of other countries, three European ones (Germany, France, Sweden), the three most stricken countries from three other continents (USA, Brazil, India). Finally we show that even the aggregated world data can be well represented by our approach.
At the end of the paper we discuss the use of the model. Perhaps the most striking application is that it allows a quantitative analysis of the influence of the time until people are sent to quarantine or hospital. This suggests that imposing means to shorten this time is a powerful tool to flatten the curves.
Die Arbeit befasst sich mit einer Vereinfachung des von Devroye (1999) geprägten Begriffs der random split trees und verallgemeinert diesen im Sinne von Janson (2019) auf unbeschränkten Verzweigungsgrad. Diese Verallgemeinerung deckt auch preferential attachment trees mit linearen Gewichten ab, wofür ein Beweis von Janson (2019) aufbereitet wird. Zusätzlich bleiben die von Devroye (1999) nachgewiesenen Eigenschaften über die Tiefe der hinzugefügten Knoten erhalten.
Aus Sicht der Pädagogischen Psychologie ist Lernen ein Prozess, bei dem es zu überdauernden Änderungen im Verhaltenspotenzial als Folge von Erfahrungen kommt. Aus konstruktivistischer Perspektive lässt sich Lernen am besten als eine individuelle Konstruktion von Wissen infolge des Entdeckens, Transformierens und Interpretierens komplexer Informationen durch den Lernenden selbst beschreiben. Erkennt der Lernende den Sinn und übernimmt, erweitert oder verändert ihn für sich selbst, so ist der Grundstein für nachhaltiges Lernen gelegt.
Lernen ist ein sehr individueller Prozess. Schule muss also individuelles Lernen auch im Klassenverband ermöglichen und der Lehrende muss zum Lerncoach werden, da sonst kein individuelles und eigenaktives Lernen möglich ist. Das Unterrichtskonzept des forschend-entdeckenden Lernens bietet genau diese Möglichkeit. Es erlaubt die Erfüllung der drei Grundbedürfnisse eines Menschen nach Kompetenz, Autonomie und sozialer Eingebundenheit und ermöglicht damit Motivation, Leistung und Wohlbefinden (Ryan & Deci, 2004).
Forschend-entdeckendes Lernen im Mathematikunterricht ist schrittweise geprägt von folgenden Merkmalen:
- eine problemorientierte Organisation
- selbstständiges, eigenaktives und eigenverantwortliches Lernen der Schülerinnen und Schüler
- individuelle Lernwege und Lernprozesse
- Entwicklung eigener Fragestellungen und Vorgehensweisen der Lernenden
- eigenes Aufstellen von Hypothesen und Vermutungen; Überprüfung der Vermutungen; Dokumentation, Interpretation und Präsentation der Ergebnisse
- eine fördernde Atmosphäre, in der die Lernenden nach und nach forschende Arbeitstechniken vermitteln bekommen
- kooperative Lernformen und damit Förderung von Team- und Kommunikationsfähigkeit
- Unterrichtsinhalte mit hohem Realitäts- und Sinnbezug, gesellschaftlicher Relevanz, Möglichkeiten der Interdisziplinarität
- Stetige Angebote der Unterstützung
Das entdeckende Lernen kann als Vorstufe des forschenden Lernens gesehen werden, da hier der wissenschaftliche Fokus noch nicht so stark ausgeprägt ist. Um alle Phasen auf dem Weg zu annähernd wissenschaftlichen forschenden Lernens anzusprechen, verwenden wir den Begriff des forschend-entdeckenden Lernens.
Voraussetzung ist, dass die Lehrkräfte das forschende Lernen als aktiven, produktiven und selbstbestimmten Lernprozess selbst zuvor erlebt haben müssen. Unter anderem können die Lehrkräfte Unterrichtsprozesse danach besser planen und währenddessen unterstützen, da sie selbst forschend-entdeckendem Lernen „ausgesetzt“ waren und vergleichbare Prozesse durchlebt haben.
Hiermit wird deutlich, dass forschendes Lernen nicht bedeuten kann, dass die Schülerinnen und Schüler auf sich gestellt sind. Die gezielte Unterstützung der Lernenden beim Entdecken und Forschen durch die Lehrkraft ist für einen ertragreichen Lernerfolg unverzichtbar und muss Teil der Vorbereitung und des Prozesses sein.
Internationale Studien zeigen, dass forschend-entdeckende Unterrichtsansätze (inquiry-based learning IBL) im Mathematikunterricht bei geeigneter Umsetzung Lernen verbessern, Lernerfolg und Lernleistung steigern und Freude gegenüber Mathematikunterricht erhöhen können. Die Implementierung dieses Unterrichtsansatzes ist trotz der positiven Ergebnisse nicht alltäglich.
Um neue Unterrichtskonzepte in den Schulalltag zu bringen beziehungsweise um bestehende Unterrichtskonzepte neu in den Schulalltag zu bringen bedarf es Fortbildungen zur Professionalisierung von Lehrerinnen und Lehrern.
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
We deal with the reconstruction of inclusions in elastic bodies based on monotonicity methods and construct conditions under which a resolution for a given partition can be achieved. These conditions take into account the background error as well as the measurement noise. As a main result, this shows us that the resolution guarantees depend heavily on the Lamé parameter μ and only marginally on λ.
The Calderón problem with finitely many unknowns is equivalent to convex semidefinite optimization
(2023)
We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori.
We will show that this Calderón problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator.
Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.
Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes
(2019)
For the linearized reconstruction problem in electrical impedance tomography with the complete electrode model, Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.