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This paper presents an intervention study whose aim was to promote teacher beliefs about mathematics and learning mathematics and student competences in mathematical modeling. In the intervention, teachers received written curriculum materials about mathematical modeling. The concept underlying the materials was based on constructivist ideas and findings from mathematics education. Teacher beliefs about mathematics, learning and self-efficacy were expected to have a major impact on their classroom practices. We therefore assessed teacher beliefs about the usefulness of mathematics, learning (constructivist and socio-constructivist beliefs) and teacher self-efficacy when teaching modeling (teacher or class level variable). The student level variables assessed were modeling competence and other individual factors, such as basic mathematical skills and cognitive abilities. The effectiveness of the intervention was measured in a pre-post control group design using multilevel structural equation modeling. The results showed no direct effect of the intervention on student modeling competence. However, they did reveal that the intervention had a significant effect on teacher beliefs about learning (constructivist and socio-constructivist view) and an effect of these teacher beliefs about learning on student modeling competence. Further, the results showed that students' gains in modeling competence is not only mediated by teacher beliefs, but also influenced by individual factors. Implications for teaching as well as limitations of the study are discussed.

Feedback is an important component of interactive learning software. A conclusion from cognitive learning theory is that good software must give the learner more information about what he did. Following the ideas of constructivist learning theory the user should be in control of both the time and the level of feedback he receives. At the same time the feedback system must identify and review different possible solution strategies in an open learning environment. The interactive geometry software Cinderella offers an easy-to-use programming interface. It can be used to implement application specific feedback by the author of learning units. In this paper we present two exemplary learning units implementing two kinds of interactive feedback: feedback on demand and immediate feedback. The presented units come from discrete mathematics and from the theory of line reflections and congruencies in geometry. The units are implemented in a process-oriented design. Various directly given or hidden hints help the students to understand the mathematical principles behind the given problems. Our tools analyses the student's solution processes automatically and generates additional feedback on demand. The second learning environment can also be used in conjunction with recording of user actions. This allows additional feedback given later by the teacher whenever the automatic feedback system fails in analyzing the users' learning processes. First experiences using the units in teaching are presented.

Data about processes and outcomes of preschool teacher education is scarce. This paper examines the opportunities to learn (OTL) of prospective preschool teachers (N = 1,851) at different types and stages of preschool teacher education and their relation to general pedagogical knowledge (GPK), mathematics pedagogical content knowledge (MPCK), and mathematical content knowledge (MCK) with standardized tests. Process indicators in terms of OTL and outcome indicators in terms of knowledge varied substantially across teacher education types and stages. Controlling for preschool teachers' background, multilevel models revealed that OTL in general pedagogy and mathematics pedagogy provided during teacher education were significantly related to GPK and MPCK. Effect sizes reached up to 2 thirds of a standard deviation. OTL in mathematics pedagogy were in turn significantly related to the type of institution that offered a program in favor of pedagogical colleges compared with vocational schools. OTL were also significantly related to program stage in favor of the last year of preschool teacher education compared with the beginning. Process characteristics in terms of OTL mediated fully or partly structural characteristics of teacher education such as type of institution or program stage. These results suggest that the OTL provided are more important than whether prospective preschool teachers were at the beginning or the end of their program or whether they were prepared at vocational schools or pedagogical colleges (although entrance differences have still be taken into account). It may be an important responsibility of policymakers then to ensure that all prospective preschool teachers receive sufficient OTL.

The Programme for the International Assessment of Adult Competencies (PIAAC) has recently drawn additional attention to "mathematical literacy" as an important influential factor for individuals' life chances. High levels of mathematical literacy have thereby been linked to using mathematics in daily and working life frequently. In this paper, based on the data from Germany, we focus on the construct "use of mathematics" in two ways: First, we analyze in depth how it can be utilized to describe different groups of adults. Second, we investigate its role as predictor of mathematical competence and mediator of other relevant background variables. Results show that three groups of adults can be distinguished that use mathematics differently in daily and working life. However, the construct can sensibly be described as unidimensional. In a path model, "use of mathematics" turns out to be the strongest predictor of mathematical competence. In addition, it mediates effects of the mathematical requirements of the job, duration of education, and gender.

The paper presents results from a computer-based assessment in which 171 early career mathematics teachers from Germany were asked to anticipate typical student errors on a given mathematical topic and identify them under time constraints. Fast and accurate perception and knowledge-based judgments are widely accepted characteristics of teacher competence. The item-wise length of anticipation time, the complexity of mathematical topics and the frequency of right or wrong given answers were used as indicators for teacher competence. The data revealed that anticipation time and the complexity of mathematical topics were related with each other. The groups of test persons with correct and incorrect answers behaved contrarily to the length of the anticipation time. Whereas test persons with correct answers needed more time to anticipate typical errors with an increasing complexity of the errors, the test persons giving false answers responded very quickly, even with an increasing error complexity. This finding confirms results of the expertise research, which emphasize that expert teachers focus more intensively if this is required by a complex task.

This paper reports upon particular aspects of a study carried out by Delice in 2003, the main aim of which was to compare the performance of students in the 16-19 age group from Turkey and England on trigonometry of "A-level standard" and then to compare the curriculum and assessment provision in each country to seek possible explanations for differences in performance. Overall, the Turkish students did much better with the algebraic, manipulative aspects of trigonometry, the English students did better with the application of trigonometry to practical situations. The comparison of curriculum and assessment provision in the light of these results raises several issues about the current A-level mathematics syllabus. It may be that there is a tendency to think that mathematics is very much the same in any country, however this research supports previous work [Kawanaka, T., Stigler, J.W. and Hiebart, J. (1999) Studying Mathematics Classrooms in Germany, Japan and the United States: Lessons from the TIMSS Video Study] where mathematics was indeed found to be taught and learnt differently and the roles of teachers and students were different across three countries (Germany, Japan and the USA). The paper begins by briefly describing the research and presenting some of the results. There then follows a discussion of these results within the context of curriculum and assessment provision within each country. Finally, issues for the current A-level syllabus arising out the results are presented.

This paper investigates the impact of early noncognitive skills on social outcomes in adolescence. The child's attention span, approach, prevailing mood and distractibility in early childhood may be crucial predictors for school achievements, health risk behavior, delinquency and autonomy as adolescent. We investigate this issue using a longitudinal epidemiological cohort study of 384 children at risk from the Rhine-Neckar region in Germany. Our results indicate that noncognitive skills in early childhood are important predictors of educational success, tobacco and alcohol use, delinquency and autonomy in adolescence. In particular, the attention span has emerged as a dominant factor among noncognitive skills regarding educational performance, health behavior and delinquency in our study. Further, we find that boys with low noncognitive skills have significantly lower social outcomes compared with girls.

Argues that the paradox of mathematical knowledge--that mathematics cannot be separated from empirical experience and yet cannot be explained by empiricist epistemology--can only be resolved if the causal interactions between knower and environment are accepted. Contains 26 references. (DDR)

Using Progress in International Reading Literacy Study 2001 and Programme for International Student Assessment 2003 data for Germany, this paper examines whether second-generation immigrants and girls are graded worse in math than comparable natives and boys, respectively. Once all grading-relevant characteristics, namely math skills and oral participation, are accounted for, pupils should obtain same school grades. Results of a matching approach and class fixed effects regressions suggest that second-generation immigrants have grade disadvantages in primary education which could bias their secondary school track choice. Regarding secondary school, most immigrants are not affected by grade discrimination and girls enro lled in upper-secondary school are systematically graded better.

Whereas several studies investigated the effects of implementation of state-wide exit exams on student achievement, there is still little known about the impacts of the exams on students' self-regulated learning. This paper examines the question as to whether the implementation of state-wide high school exit exams is associated with a change in the self-regulated learning of students in mathematics or English. We conducted a standardized questionnaire survey of students in two German states for a period of 3 years. In mathematics no significant effects of the immediate introduction of state-wide exit exams were identified. In English the results show significant positive and negative effects. The results are discussed and implications for further research are given. (Contains 3 tables.)

This essay paper discusses the decline in the numbers of students studying mathematics at German universities. Sections on mathematics in German secondary schools, reasons for the decline in numbers, and issues related to reversing the trend are included. The article also highlights the German "Diplom" and mathematics departments in Baden-Wurttemberg. (MM)