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Literacy acquisition is one of the primary goals of school education, and usually it takes place in the national language of the respective country. At the same time, schools accommodate pupils with different home languages who might or might not be fluent in the national language and who start from other linguistic backgrounds in their acquisition of literacy. While it is safe to say that schools with a monolingual policy are not prepared to deal with the factual multilingualism in their classrooms in a systematic way, bilingual pupils have to deal with it nonetheless.
The interdisciplinary and comparative research project “Literacy Acquisition in Schools in the Context of Migration and Multilingualism” (LAS) investigated the practical processes of literacy acquisition in two countries, Germany and Turkey, where the monolingual orientation of schools is as much a reality as are the multilingual backgrounds of many of their pupils. The basic assumption was that pupils cope with the ways they are engaged by the school – both socially and academically – based on their cultural and linguistic repertoires acquired biographically, providing them with more or less productive options regarding the acquisition of literary skills. By comparing the literary development of bilingual children with that of their monolingual classmates throughout one school year in the first and the seventh grade in Germany and Turkey, respectively, we found out that the restricting potential of multilingualism is located rather on the part of the schools than on the part of the pupils. While the individual bilingual pupil almost naturally uses his/her home language as a resource for literacy acquisition in the school language, schools still tend to regard the multilingual backgrounds of their pupils as irrelevant or even as an impediment to adequate schooling. We argue that by ignoring or even suppressing the specific linguistic potentials of bilingualism, bilingual pupils are put at a structural disadvantage.
This research report is the slightly revised but full version of the final study project report from 2011 that was until now not available as a quotable publication. While several years have passed since the primary research was finalized, the addressed issues have lost none of their relevance. The report is accompanied by numerous publications in the frame of the LAS project, as well as by a web page (https://www.uni-potsdam.de/de/daf/projekte/las), which also contains the presentations from the final LAS-Conference, including valuable discussions of the report from renowed experts in the field.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all (odd) natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number 1, as conjectured by Lothar Collatz.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number one, as conjectured by Lothar Collatz.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto unavailable. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. Furthermore, we derive and prove several formulas that can be used to traverse the graph. The analysis covers the Collatz problem both in it’s original form 3x + 1 as well as in the generalized variant kx + 1. Finally, we transform the Collatz graph into a binary tree, following the approach of Kleinnijenhuis, which could form the basis for a comprehensive proof of the conjecture.