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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...
We present new results on nonlocal Dirichlet problems established by means of suitable spectral theoretic and variational methods, taking care of the nonlocal feature of the operators. We mainly address: First, we estimate the Morse index of radially symmetric sign changing bounded weak solutions to a semilinear Dirichlet problem involving the fractional Laplacian. In particular, we derive a conjecture due to Bañuelos and Kulczycki on the geometric structure of the second Dirichlet eigenfunctions. Secondly, we study a small order asymptotics with respect to the parameter s of the Dirichlet eigenvalues problem for the fractional Laplacian. Thirdly, we deal with the logarithmic Schrödinger operator. In particular, we provide an alternative to derive the singular integral representation corresponding to the associated Fourier symbol and introduce tools and functional analytic framework for variational studies. Finaly, we study nonlocal operators of order strictly below one. In particular, we investigate interior regularity properties of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side.
Das Ziel des Beitrags besteht darin, jene Formelideale in der Mathematik und in der Ökonomik zu identifizieren, um anschließend Überlegungen darüber anzustellen, auf welche Art und Weise diese Ideale den logischen Spielraum der Problemlösung erweitern oder einengen. Ganz entscheidend hierfür ist das Verhältnis von Internalitäten und Externalitäten; also ob bzw. welche Umweltbezüge in der Formelsprache eingeschlossen oder ausgeschlossen werden. Die Analyse erfolgt aus wissenssoziologischer Perspektive, die die mathematische, mit Blick auf die Ökonomik vorsichtiger formuliert: die pseudo-mathematische Formelsprache als eine spezifische Semantik versteht. Hierbei handelt es sich nicht nur um eine Semantik, die sich der durch die gesellschaftliche Differenzierung ausgelösten Komplexitätssteigerung anpasst. Die Evolution der mathematischen und wirtschaftswissenschaftlichen Semantik befolgt zudem, so meine These, paradigmatische Ideale, die in den spezifischen Disziplinen angelegt sind. Dies spiegelt sich in der Art und Weise wider, wie in den jeweiligen Disziplinen auf mathematisch-kalkulatorischer Modellebene einerseits eine Geschlossenheit und Binnenreferenzialität des Formelhorizonts und andererseits eine Projektionsfläche für die Entdeckung von Zusammenhängen und das Auffinden formaler Muster generiert wird. Die Ausgangsannahme im Hinblick auf die Ökonomik ist, dass auch die Operationsweise ökonomischer Modelle, insofern sie auf mathematischer Grundlage gebildet werden, auf einer Binnenreferenzialität beruht. Die Stringenz solcher Modellierungen wird gerade deswegen zum vorherrschenden Gütekriterium, weil sie nicht nur jegliche Externalitäten im Sinne einer Irritation des formalen Modells ausschließt, sondern zugleich Beobachtungseffekte durch Sinnüberschüsse ermöglicht, die durch eine Entdeckung formaler Muster aktualisiert werden. Im ersten Schritt wird hierzu das Formelideal der reinen Mathematik auf der Grundlage wissenssoziologischer Überlegungen und mathematischer Selbstreflexionen herausgearbeitet. Daran anschließend wird das Formelideal pseudo-mathematischer Modelle in der Ökonomik untersucht. Die Basis bilden vor allem William Stanley Jevons' und Alfred Marshalls wegweisende Überlegungen zur Mathematisierung der Ökonomik. Die Beobachtungen werden abschließend auf den Aspekt des Erweiterns oder Einengens des logischen Spielraums zur Problemlösung in der Mathematik und der Ökonomik hin zugespitzt.