Diese Arbeit untersucht das Verhalten und die Einstellungen von Mathematiklehrkräften im Implementationsprozess der nationalen Bildungsstandards. Als top-down eingeführte Bildungsinnovation wird eine Umstellung des Unterrichts intendiert, der auf die in den Standards formulierten Kompetenzen fokussiert. Es wird herausgearbeitet, was unter einem kompetenzorientierten Mathematikunterricht zu verstehen ist und anschließend aufgezeigt, wie eine solche Intention durch ein vom Institut zur Qualitätsentwicklung im Bildungswesen (IQB) entwickeltes Implementationskonzept in die Praxis überführt werden kann. Anhand von zwei Studien wurden jeweils zehn Merkmale eines kompetenzorientierten Unterrichts an Lehrkräften und Schülern im Längsschnitt analysiert und einzeln ausgewertet. Zusätzlich erhielten die Schülerinnen und Schüler einen kompetenzorientierten Leistungstest. Angelegt in einem Experimentaldesign erhielten die Projektschulen über die Projektlaufzeit eine externe Unterstützung durch eine Setkoordinatorin, die regelmäßig einen Input und beratende Hilfestellung bei der Entwicklung und Erprobung von kompetenzorientierten Lehr-Lernarrangements anbot. Angelehnt an das SINUS-Programm wurden weitere zentrale Fortbildungsveranstaltungen angeboten. Die Drop-Out-Analysen zeigen, dass Lehrkräfte, die den Bildungsstandards gegenüber positiv eingestellt sind und in einem höheren Grad miteinander kooperieren, Engagement im Projekt zeigen. Der Unterricht in den Projektschulen erfuhr eine stärkere Orientierung hin zum mathematischen Argumentieren, Modellieren und Probleme lösen. In den Projektschulen der Studie 1 nehmen die Schülerinnen und Schüler eine verstärkte Kompetenzorientierung im Bereich der Verwendung von mathematischen Darstellungen wahr. In einer zweiten Studie konnte das IQB-Implementationskonzept mit dem SINUS-Folgeprojekt „Kompetenzorientiert unterrichten in Mathematik und Naturwissenschaften (KOU)“ erfolgreich evaluiert werden. / This doctoral thesis focuses on teachers’ development during the implementation process of the national educational standards in Germany. Consider the standards, which were introduced by resolution, classroom activities are focused to improve student learning. To reach the standards, a change in teaching culture is necessary or at least an addition of teachers’ classroom activities. Therefore, the concept of competence-orientation especially for mathematics is described and discussed. Further implementation theories and implementation strategies are suggested and discussed. When educational standards became active, implementation concepts were missing so far. So, the Institute for Educational Quality Improvement (IQB) developed an implementation concept and got support by nine schools to test it. The professional learning communities in mathematics of the schools got external support by a supervisor and input by researchers during their intended change process. Study 1 focuses on analyzing teachers’ engagements and concerns as well as their classroom activities in a longitudinal design. Furthermore, students of ninth grade were tested in mathematics and asked about their classroom activities. Eight schools worked as a control group. Study 2 added another ten schools with a slightly different implementation concept. So, the IQB-concept could be evaluated. Even in study 2, the ten factors are analysed and differentiated by concept. Drop-out analyses show, teachers with a positive attitude towards the standards and a higher degree in cooperation, support the project and continued as participants in the longitudinal study. Furthermore, their classes changed in aspects of arguing, modeling and problems solving mathematically. Students of study 1 observe more activities in working with mathematical figures and diagrams. In study 2, the IQB implementation concept could be evaluated successfully by comparing teachers’ changes with the development in other schools.

Mit der vorliegenden Arbeit wurde ein Konzept zur differenzierten Beurteilung und Rückmeldung mathematischer Kompetenzen entwickelt und exemplarisch erprobt. Damit liegt ein Beurteilungskonzept vor, dass fachdidaktischen Anforderungen an einen kompetenzorientierten Mathematikunterricht genügt aber auch alltagstauglich ist. Es erlaubt dem Lehrer seinen Schülern ihre Stärken und Schwächen bei der Bearbeitung mathematischer Aufgaben differenziert und konkret aufzuzeigen und Hinweise zum selbständigen Lernen zu geben. Die Differenzierung in die vier Kompetenzbereiche darstellend-interpretativ, heuristisch-experimentell, formal-operativ und kritisch-argumentativ spiegelt fachdidaktische Zielsetzungen an den Mathematikunterricht wider. Das mathematische Grundwissen bildet als Grundlage der Ausprägung von Kompetenzen die fünfte Beurteilungskomponente. Für den spezifischen Zweck der differenzierten Beurteilung wurde ein Aufgabenanalyseinstrument entwickelt. Dieses vereint Kriterien an Aufgaben zum Leisten, allgemeine Kriterien der Aufgabenqualität und Kriterien verbunden mit der Anregung vielseitiger mathematischer Kompetenzen. Darüber hinaus werden kompetenzspezifische Anforderungen beschrieben. Die Beurteilung der bei der Aufgabenbearbeitung dargelegten Kompetenzen erfolgt durch kompetenz- und aufgabenspezifische Beurteilungskriterien. Individuelle Rückmeldungsbögen enthalten für die erfassten Kompetenzen kriterienorientierte Beurteilungen und lernfördernde Hinweise. Bei der exemplarischen Erprobung des Konzeptes in drei mathematisch profilierten Klassen zeigten sich unterschiedliche Leistungsverteilungen für die einzelnen Kompetenzbereiche. Es wurden zum einen Stärken und Schwächen der Schüler deutlich. Zum anderen zeigten aufgabenabhängige Unterschiede innerhalb der Kompetenzbereiche, dass die Kompetenzdarlegungen von der Art der Aufgabe oder vom speziellen angesprochenen Inhalt abhängen. In einer weiteren Untersuchung wurden die mit einer Aufgabe intendierten Aktivitäten den tatsächlich ausgeführten gegenübergestellt. Aus der Umsetzung verschiedener Aufgaben wurden Hinweise zur Aufgabengestaltung für die differenzierte Beurteilung im Mathematikunterricht abgeleitet. / With the presented paper a concept of differentiated assessment and feedback of mathematical competencies was developed and exemplary tested. It allows the teacher to show his students in a differentiated and task-bounded way their strong and weak points connected with recommendations for independent learning. The differentiation in descriptive, heuristic, formal and argumentative competencies reflects didactically goals for mathematical lessons. Any development and demonstration of competencies is only possible if the necessary knowledge is available. For this reason the mathematical knowledge is another important element of competence-orientated assessment. To choose and construct tasks for the differentiated assessment an analysis instrument was developed. It includes criterions on tasks for testing, general criterions on task quality and the claim to stimulate various mathematical activities. In addition the analysis instrument describes properties of competence specific claims. For a transparent and objective assessment of ambitious competencies like arguing or problem solving competence and task specific criterions of assessment were prepared. Individual feedback-sheets include for the competencies related to the task a quantitative criterion-orientated assessment, verbal interpretations and learning-supporting recommendations. The concept was exemplary tested in three mathematical profiled classes. Differentiated performance distributions were observed for the different areas of competencies. Strong and weak points of the student could be shown in nearly constant levels, but also task-depending variations. The clearly task-depending variations within an area of competencies indicates that the shown competencies are depending on the kind of the task or on there special mathematical topic. This speaks for a concrete task-related feedback. In a further investigation the potential of the task to induce activities was confronted with the activities really carried out by the students. From the realization of different tasks recommendations to design tasks for the differentiated assessment of mathematical competencies were deduced.

In this paper the notion of “mathematics profiles” and “instructional behaviour profiles is introduced. A brief explanation of what these profiles are and how they were constructed and represented for preservice mathematics teachers is provided. An example of one of the participants’ profiles is included as an example. The influence of the pre-service teachers’ mathematics profiles on their instructional behaviour is then discussed. This is done with regard to using the mathematics profiles as a potential tool to optimise the development of pre-service mathematics teachers’ instructional behaviour towards a more reform-oriented approach.

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.

We report a class of the mathematics in which an animation technology (calculating and plotting capabilities) of the software Mathematica is utilized. This class is taught for university students in a computer laboratory during a second semester. It is our purpose to make a student realize the usefulness and the importance of mathematics easily through visualization. In addition, we hope that students will acquire a new power of mathematics needed in the 21st century. For several years, we have continued this kind of class, and have continued to investigate the effectiveness that our teaching method (especially visualization) brings in the understanding of the mathematics. In this paper, we present some of this teaching method, which is performed in our class. From the questionnaire survey, it is found that our teaching method not only convinces students that the mathematics is useful or important but also deepens the mathematic understanding of students more.

In this paper the notion of “mathematics profiles” and “instructional behaviour profiles is introduced. A brief explanation of what these profiles are and how they were constructed and represented for preservice mathematics teachers is provided. An example of one of the participants’ profiles is included as an example. The influence of the pre-service teachers’ mathematics profiles on their instructional behaviour is then discussed. This is done with regard to using the mathematics profiles as a potential tool to optimise the development of pre-service mathematics teachers’ instructional behaviour towards a more reform-oriented approach.

Mathematics learning seems to be a demanding and time-consuming task for many learners. Information and communication technology (ICT) is an attractive tool of learning for students at any level and it can provide an effective atmosphere for understanding mathematics. The question is how to combine mathematics teaching contents, approaches, curricula, and syllabus with new media. The key issue in European educational policy (and other countries as well) is exchange and sharing digital learning resources (learning objects) among countries. In order to accumulate the practice of various countries and use the best digital resources created by different countries, it is necessary to localize learning objects (LO). The paper deals with some problems connected with localization of LO, developed for mathematics education, and presents some solution. Software localization is mainly referred to as language translation (e.g., translation of user interface texts and help documents). However, there are many other important elements depending on the country and people who will use the localized software. In this paper, the main attention is paid to localization of learning objects used for teaching and learning mathematics.

The purpose of this study is to determine the impact on student achievement of elementary school teachers who participated in professional development in the content area of mathematics. Teachers participated in professional development courses and have accumulated a range from three to eighteen total credits from the summers of 1998 through 2007. The impact is measured by student achievement data collected on standardized tests.

Computer Algebra Systems (CAS) have been changing the mathematics instruction requirements for many years. Since the tendency of using CAS in mathematics instruction has been rising for decades and reports have often been positive, the implementation of notebook classes seems to be the consequent next step of mathematics instruction supported by computers. Experiences that have been made with the use of CAS in PC-rooms can be transformed directly into the classroom. Hence the use of CAS is no longer limited to certain rooms. The permanent availability of the notebook with installed CAS offers the chance to realize these concepts that have already been approved with the use of CAS so far. The following speech shall show what these concepts could look like and that the use of notebooks is not only the further development of teaching in PC-classes. Examples from personal experience in teaching will especially show meanders and thought-provoking impulses in order to support teachers finding their way into teaching mathematics instruction in notebook classes successfully. Please allow me to point out two things in the beginning: (1) Yes, I am a vehement supporter of the use of notebooks (and the use of CAS in particular) in mathematics instruction. (2) No, I do not believe that teachers who have chosen another path (or at least partly) are teaching badly.

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.