Die Untersuchungen, deren Ergebnisse in den Heften 32 und 33 der Potsdamer Studien zur Grundschulforschung vorgelegt wurden, gingen den Fragen nach, was Kinder am Anfang und am Ende des ersten Schuljahres über die Größe Geld wissen, welche elementaren Fähigkeiten sie im Umgang mit Geld haben und wie sich diese Kenntnisse und Fähigkeiten im Verlaufe des ersten Schuljahres weiterentwickeln. Mit den nun vorliegenden Ergebnissen unserer dritten Studie zum Thema Geld knüpfen wir an diese vorangegangenen Studien an. In diesem Heft wird dargestellt, welche Kompetenzen die an unseren Untersuchungen beteiligten Kinder am Ende der Klasse 2 besitzen und wie sich diese Kompetenzen im Vergleich zu den anderen Messzeitpunkten verändert haben. Außerdem sind wir der Frage nachgegangen, ob Zusammenhänge zwischen dem Vorwissen der Kinder bezogen auf unseren Untersuchungsgegenstand und den Rechenfähigkeiten am Ende der Klasse 2 bestehen. Im Ergebnis dieser Studien haben wir ein Modell für das Größenkonzept der Größe Geld entwickelt.

This study aims at the presentation of the standardisation method in benchmarking at the conceptual level. This conceptual level is to serve as the basis for the transfer into numerous practical applications. The field example demonstrated is meant to provide additional support. At the same time, this example reveals that the concept is feasible and necessary. The necessity refers to the identification, discussion and solution - oriented towards the goals of the intended benchmarking - of the aspects in the planning and analysis phases of benchmarking which are behind the standardisations. Under certain circumstances this may lead to the fact that this method makes the participating comparative partners sensitive for the conscious perception of the possible significance of standardisations. A more exact analysis may quite as well show that the options in cost accounting are used uniformly to a large extent so that for reasons of materiality the standardisations are not put into practice. Even if this study exclusively examines monetary quantities such as costs we should state that - particularly when benchmarking is extended to further objectives, e.g. quality, environment and time - the need for standardisation is identified for non-monetary quantities as well. Non-monetary quantities also involve - sometimes very complex - questions of evaluation that imply options for subjective evaluation. The basic question of how to evaluate environmental influences or quality aspects is to illustrate this issue. Since standardisations can not at all eliminate subjectivity, which is connected with the determination of options by standards, it may be quite useful in some cases to examine the influence that alternative standards produce. This is particularly useful for the influencing factor parameter, where the Hoechster-Spinne may serve as an analytical tool to illustrate the sensitivity of the quantity which is calculated taking into account this influencing factor.

This study aims at the presentation of the standardisation method in benchmarking at the conceptual level. This conceptual level is to serve as the basis for the transfer into numerous practical applications. The field example demonstrated is meant to provide additional support. At the same time, this example reveals that the concept is feasible and necessary. The necessity refers to the identification, discussion and solution - oriented towards the goals of the intended benchmarking - of the aspects in the planning and analysis phases of benchmarking which are behind the standardisations. Under certain circumstances this may lead to the fact that this method makes the participating comparative partners sensitive for the conscious perception of the possible significance of standardisations. A more exact analysis may quite as well show that the options in cost accounting are used uniformly to a large extent so that for reasons of materiality the standardisations are not put into practice. Even if this study exclusively examines monetary quantities such as costs we should state that - particularly when benchmarking is extended to further objectives, e.g. quality, environment and time - the need for standardisation is identified for non-monetary quantities as well. Non-monetary quantities also involve - sometimes very complex - questions of evaluation that imply options for subjective evaluation. The basic question of how to evaluate environmental influences or quality aspects is to illustrate this issue. Since standardisations can not at all eliminate subjectivity, which is connected with the determination of options by standards, it may be quite useful in some cases to examine the influence that alternative standards produce. This is particularly useful for the influencing factor parameter, where the Hoechster-Spinne may serve as an analytical tool to illustrate the sensitivity of the quantity which is calculated taking into account this influencing factor.

Ziel der Arbeit ist, die Konzepte von Drittklässlern zur Größe Gewicht zu beschreiben. Es handelt sich um eine hypothesengenerierende Arbeit auf Basis einer qualitativen Studie. Dabei soll geklärt werden, welche Konzepte überhaupt auftreten, welchen Einfluss Unterricht auf diese Konzepte haben kann und welche Anhaltspunkte sich finden, Unterricht an die Vorkenntnisse der Kinder anknüpfen lassen zu können. Auf der Basis der Theorie des Konzeptwechsels und der Theorie der Subjektiven Erfahrungsbereiche sowie der mathematisch-physikalischen Hintergründe zur Größe Gewicht und Ergebnissen bereits vorliegender Untersuchungen wird ein Modell eines Gewichtskonzepts erstellt, das die beteiligten Teilkonzepte identifiziert. Vor diesem Hintergrund wurde eine empirische Studie konzipiert, die das Ziel hat, mit Hilfe von Einzelinterviews die Konzepte von Drittklässlern im Kontext der unterrichtlichen Behandlung der Größe Gewicht zu untersuchen. An der Studie nahmen 15 Kinder teil. Die Konzepte zweier Kinder werden in Form von Einzelfallstudien analysiert und durch ausgewählte Konzepte anderer Kinder ergänzt. Es zeigen sich sehr unterschiedlich differenzierte Konzepte zur Größe Gewicht. Manche Kinder verfügen über ein weit ausgebautes Konzept aufgrund vielseitiger Vorerfahrungen. Bei diesen Kindern ist über den Untersuchungszeitraum fast keine Entwicklung der Konzepte feststellbar. Die weniger ausgebauten Konzepte anderer Kinder zeigen insbesondere bezüglich der Teilkonzepte Einheiten, Stützpunktwissen, Waagen sowie Wiegen Entwicklung auf. Direkt nach der unterrichtlichen Behandlung sind sowohl Konzepterweiterungen als auch -neubildungen feststellbar. Später sind diese Umstrukturierungen bei einigen Kindern stabilisiert oder sogar weitergeführt, bei anderen dagegen wieder in den Hintergrund gerückt. Auf der Grundlage der Ergebnisse wird das Modell eines Gewichtskonzepts reflektiert und mögliche Konsequenzen für den Unterricht diskutiert. / This thesis aims at describing German third grade pupils'' concepts concerning the physical quantity weight. It is a hypothesis-generating thesis based on a qualitative study. We are going to clarify those concepts that can be observed, what kind of influence school instruction can have on these concepts and the implications for an instruction that wants to tie in with children''s previous experiences. Based on the theory of conceptual change and the theory of subjective domains of experiences as well as the mathematical and physical background of the quantity weight but also the results of previous studies, we will develop a model of a weight concept by organizing related aspects into subconcepts. We have designed an empirical study with the purpose of examining third graders'' weight concepts in the context of a teaching unit on weights. For the study, 15 children were interviewed. The interviews of two children are evaluated in form of two single case studies. The analysed concepts of these two children are completed with selected concepts of other children to find similarities and differences, allowing us to draw conclusions on both general concept evolution and instructional issues. The interview analyses show varied differentiated weight concepts. Some of the children showed already at the beginning highly developed subconcepts based on all-round individual experience. In the later interviews hardly any change of these children''s concepts were assessed. Other children with initially less sophisticated subconcepts evolved over time the concepts units, base knowledge, scales, and weighing. Immediately after the teaching unit we were able to find both new and evolved concepts. Later, some children''s concepts had stabilised while other children had returned to pre-instructional concepts. Based on the overall results of the study, we revisit our weight concept model.

In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\'s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\'s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.

In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\''s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\''s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.