Search

In this paper, we discuss the relevance of 'Grundvorstellungen' (GVs), a didactical category to analyze students' mental models in comparison to the intended mathematical meanings in the context of Linear Algebra and Analytic Geometry. Diagnostic tasks were used to reveal students' conceptual understanding in this field of expertise. In particular, an open item format was chosen to elicit students' individual GVs and to explore how they use them while working on mathematical tasks. 30 students from upper secondary school participated in our study; data was collected by a paper-and-pencil test. The results show that elaborated representations of GVs foster students' understanding of mathematics and facilitate the process of finding problem solving strategies. [For the complete proceedings, see ED597799.]

Recent studies in early mathematics education highlight the importance of patterning abilities and their influence on mathematics learning and the development of mathematical reasoning in young children. This paper focuses on young children's repeating patterning abilities and reports results from an ongoing four-year longitudinal study that investigates the development of early numeracy understanding of 408 children from one year prior to school until the end of grade 2. The analyses in this paper reveal a significant influence of young children's repeating patterning abilities one year prior to school on their mathematical competencies at the end of grade one. [For the complete proceedings, see ED597799.]

In this paper we report our findings from a study, in which 177 undergraduate pre-service teachers had to verify a statement of elementary number theory at the onset of a lecture serving as a bridging course. The answers were categorized to investigate the qualities of the given argumentations. We also separated the results of three different subsets: (1) the of students in their first semester, to get to know their level of arguing after having past their A-Levels, (2) the students, that take part in this course the first time and are in a higher semester and (3) the subset of repeaters to investigate their problems in detail. [For the complete proceedings, see ED597799.]

Argumentation and proof have been in the focus of attention in mathematics education research for several decades. While it has often been pointed out that it is important to give argumentation a prominent position in the mathematics classroom, it is far from clear how to teach argumentation, particularly to students from non-privileged backgrounds. In this paper, I show how Habermas' theory of communicative action gives a valuable perspective on what makes argumentation likely to occur. The context of a logical game situation in which argumentation happened is analysed to support the following result: the exclusion of force and a cooperative mode of communication are helpful elements in understanding the fostering of argumentation. [For the complete proceedings, see ED597799.]

Results of the last thirty years in mathematics education have shown the importance of an operational concept development. One of the geometrical concepts that has been researched for years already, however, not with the particular focus on its systematic teaching in school mathematics, is the concept of angle. In this paper we focus on children's understanding of the angular size of 1 [degree] and its development obtained through a test followed by a task-based interview. The interview results with 9 pupils showed that they have a fragmented understanding of the angle concept, enabling them to fully grasp what 1 [degree] angle is. Moreover, many of the children's misconceptions were directly connected to the measuring tool, namely set square, and angle notation. Implications for systematic teaching of the angle concept are given at the end. [For the complete proceedings, see ED597799.]

The paper deals with the question of the practicability and the effectiveness of different approaches to foster students' mathematical modelling competencies. Within the modelling project ERMO (Acquirement of modelling competencies) a holistic and an atomistic approach of mathematical modelling were compared in order to find out which approach is more effective in fostering the students' modelling competencies. The results of modelling tests with three measurement points show that both approaches foster students' modelling competencies, but both approaches have strengths and weaknesses. The data indicates that the holistic approach is more effectively for students with weaker performance in mathematics. [For the complete proceedings, see ED597799.]

This paper refers to the concept of semiotic and theoretic control describing resources to conduct decisions in epistemic processes. We consider an argumentation process from a complex problem-solving activity involving different conceptual frames related to parabolas. Using a micro-analytical interpretative lens, we will show that, in order to carry out the argumentation activity, semiotic and theoretic control within conceptual frames (local control) needs to be co-ordinated with control across different conceptual frames (global control). [For the complete proceedings, see ED597799.]

This paper contributes to the discourse in stochastic education of how young students deal with learning settings that allow a data-based approach to probability. By using the micro-structure of arguments by Toulmin (1958), it explores which arguments students use and which role they play in the learning process. The data stems from design experiments with students at the beginning of their stochastic career (aged 11 to 13) and is analysed with an interpretative approach. [For the complete proceedings, see ED597799.]

Currently, language competences in mathematics lessons gain more attention in Germany. The paper reports an interdisciplinary study of linguistics and mathematics education on reasoning. A model to rate the competences in arithmetic reasoning at primary level will be presented for discussion: mathematical reasoning is coded separately from its linguistic realization. In a pilot study, 243 students of 3rd, 4th, and 6th grade solved different arithmetic reasoning tasks. The results show a one-dimensional scale for the model of reasoning. Its specific components provide differentiated requirements, which are formulated concretely in the coding guidelines. They may unfold didactical potential for language support in mathematical reasoning as well as in mathematics lessons itself at primary level. [For the complete proceedings, see ED597799.]

When asked to draw different kinds of triangles, children reveal many creative ways to express variety. In this paper, the drawings of 81 children in the age between 4 and 6 will be examined and illustrated what kind of understanding of the concept "triangle" precedes the drawings. Therefore, different categories of the children's drawings were generated and also compared to their explanations of a triangle, which sometimes might not be in agreement with their drawings. [For the complete proceedings, see ED597799.]

This paper focuses on an inferential view on introducing new concepts in mathematics classrooms. A theoretical framework is presented which helps to analyse and reflect on the processes of teaching and learning mathematical concepts. The framework is based on the philosophies by Ludwig Wittgenstein and Robert Brandom. Wittgenstein's language-game metaphor and especially its core, the primacy of the use of words, provide insight into the processes of giving meaning to words. Concerning the inferentialism by Brandom, the use of words in inferences can be regarded as an indicator of the understanding of a concept. The theoretical considerations are exemplified by the interpretation of a scene of real classroom communication. [For the complete proceedings, see ED597799.]

Place value is a key concept for numbers that is introduced in early mathematics. It is necessary to have a flexible understanding of place value for efficient arithmetic strategies and success in written algorithmic arithmetic. In our research we explore typical mistakes and misconceptions that occur with 3rd graders in German elementary school and investigate the various underlying actions when manipulating numbers in a place value chart. We also report on a follow-up quantitative study that compares real and virtual manipulatives for place value and their effect on learning of place value. [For the complete proceedings, see ED597799.]

Recent psychological studies as well as research findings in mathematics education highlight the significance of early number skills for the child's achievement in mathematics at the end of primary school. In this context, first results from an ongoing four-year longitudinal study are reported. The study investigates the development of early numeracy understanding of 408 children from one year to school entry until the end of grade 2. The study seeks to identify children that struggle with respect to their mathematics learning after the first year of school mathematics and compare their performance with their number concept development one year prior to school as well as immediately prior to school entry. [For the complete proceedings, see ED597799.]

Teacher cognition is seen as an important factor for the quality of instruction and, accordingly, student learning of a subject. However, in-depth research on these relations can only be done if a sound theoretical model for subject-specific teacher cognition (knowledge and competence/practical skills) and corresponding measures are available. This work aims at the development of reliable and valid measures for primary school teachers in mathematics. The subject-specific cognition is modeled as basic professional knowledge (BK) and further two components of reflective competence (RC) and action-related competence (AC). Using video-based items, we developed a computer-based standardized test and collected data of N = 85 primary mathematics teachers. We present results on the quality of the measures and have a closer look on the structure of teacher cognition. [For the complete proceedings, see ED597799.]

Educational research assumes that error reflections are efficient if they include the rationale behind the own error instead of just correcting the error. However, thus far there is a lack of empirical evidence regarding this aspect. Thus, we conducted a field experiment with pre-post-follow-up design and with 7th and 8th grade students (N = 174). The study was conducted during standard mathematics lessons. We compared two different error-handling strategies. Our findings indicate that students who reflected the rationales behind their errors enhanced their procedural knowledge more than students who reflected on the corresponding correct solution only. Regarding conceptual knowledge we found this effect only at the follow-up-test. The implications for theory and school instructions are discussed. [For the complete proceedings, see ED597799.]

A teacher's instructional planning that is enacted in his classroom practice and that potentially impact on his students' knowledge and beliefs could be understood as an individual belief system dependent from his actual teaching and learning experience. Individual belief systems might be contradictory when we regard different teachers or one teacher concerning different mathematical disciplines. For this reason, this report focuses on thirty teachers' beliefs about their teaching of a specific mathematical domain, i.e. calculus that is a central part of the (German) curriculum at upper secondary level. After a brief outline of the theoretical framework and methodology of this research project, results of the qualitative reconstruction of different aspects of teachers' belief systems on calculus will be explained. [For the complete proceedings, see ED597799.]

Given the key role of conversions of representations for mathematical understanding, it is highly relevant to investigate in detail competencies regarding conversions of representations. In particular, a corresponding competence model should not only be developed theoretically, but also examined empirically. However, such empirical studies are rather scarce, especially concerning content domains other than functions. Consequently, this study focuses on the design and empirical validation of a competence structure model regarding conversions of representations in the domain of fractions using multidimensional item response modelling. The results suggest that the data support the theoretically developed structure of the model and moreover, they indicate a hierarchical relationship which may give rise to a competence level model. [For the complete proceedings, see ED597799.]

In the project MultiMa (Multiple solutions for mathematics teaching oriented toward students' self-regulation), we investigated the effects of prompting students to use multiple solution methods while solving real-world problems on their learning. In this quasi-experimental study, we compared three treatment conditions. In one condition, students solved real-world problems by using multiple solution methods. These solution methods consisted of a solution using a table and a solution using differences. In the other two conditions, the same real-world problems were solved using only one of the methods. About 307 ninth graders from twelve middle track classes took part in this study during four lessons. Before and after a teaching unit, students' self-regulation was tested. [For the complete proceedings, see ED597799.]

We report on a long-term study which was executed in a German secondary school with 128 eights graders (ages 14 to 15) in four different classes. Two of these classes served as control groups. The mathematics lessons of the other two classes (treatment groups) were frequently enriched by distinguished phases in which structured argumentation and the use of heuristics was trained. The study aimed at investigating the development of the argumentation competence of the students over that period. For this report, the products of four different geometry tasks of 15 students from one of the treatment groups and 15 from one of the control groups respectively were evaluated. [For the complete proceedings, see ED597799.]

To support university students' understanding of mathematical proofs, pictures accompanying text are frequently used in textbooks as well as in lectures. However, it is unclear if such pictures influence the individual's reading behaviour. By recording the eye movements of eight mathematicians, we investigated whether and how adults with high expertise in mathematics pay attention to additional pictures when reading a written mathematical proof. We found that all participants paid attention to the pictures. As expected, in two out of three items, the text was fixated upon significantly longer than the picture. The data suggest that the participants tried to integrate information from text and picture by alternating between these representations. [For the complete proceedings, see ED597799.]

This exemplary case study describes the learning process of two sixth-graders that learn from an animated worked-out example and an accompanying self-explanation prompt in the domain of fractions. It is based on a corresponding field study. The analysis focuses on the interaction with the computer, the communication between the students, the metacognitive aspects of the learning process and self-explanations. Supported with quantitative data, the qualitative results show that worked-out examples are proper materials for learning in groups of two. Furthermore, it is shown that self-explanation prompts have positive effects on the learning process and the analysed aspects. With detailed scenes it is elucidated, how the interactive capabilities and the animations are used during the learning process. [For the complete proceedings, see ED597799.]

We conducted an experimental study with 192 ninth graders in which we investigated a connection between performance and students' interest and enjoyment using task-unspecific and task-specific questionnaires. Students were randomly assigned to experimental group 1 or to experimental group 2. In group 1, they were asked about their affective measures after task processing, and in group 2, they were asked before task processing. In both groups, students who achieved higher scores on the performance test reported stronger interest and enjoyment. The connection of performance to the task-unspecific and task-specific affective scales did not differ significantly and ranged between 0.15 and 0.47 for problems with and without a connection to the real world. [For the complete proceedings, see ED597799.]

This study uses data from TEDS-M to explore and compare possible individual-based and institutional-based predictors of future secondary mathematics teachers' readiness to teach in Taiwan, Germany, and the United States. Across the three countries, future teachers' intrinsic motivation to become teachers and the consistency of courses arrangement in the institutions where they studied were significant predictors of teaching readiness. Future teachers' highest grade level of mathematics studied at secondary school was a predictor of teaching readiness only in Taiwan, whereas the motivation derived from the empathy of prior learning experience was a predictor of teaching readiness in Germany and the United States, but not in Taiwan. [For the complete proceedings, see ED597799.]

There has been a controversial debate if individuals solve fraction comparison tasks componentially by comparing the numerators and denominators, or holistically by considering the numerical magnitudes of both fractions. Recent research suggested that expert mathematicians predominantly use componential strategies for fraction pairs with common components and holistic strategies for pairs without common components. This study for the first time used eye movements to test if this method allows distinguishing strategy use on specific problem types in expert mathematicians. We found the expected fixation differences between numerators and denominators in problems with common components but not in problems without common components. [For the complete proceedings, see ED597799.]

Symmetric objects are known to be perceived easier than asymmetric objects, because less information has to be processed. Therefore, symmetric objects are often used for spatial tasks. However, in perspective-taking the use of symmetric objects can also cause difficulties, as two side-views of these objects are mirror-images of each other. To examine this influence, 95 children at the beginning of first grade were asked to solve a systematically varied set of tasks in interview sessions. It was assumed that they have more difficulties to solve the tasks with symmetric objects than with asymmetric ones. Against expectation, this effect could not be confirmed based on the number of correct answers. However, the types of errors and the children's explanations show the difficulties of perspective tasks with symmetric objects. [For the complete proceedings, see ED597799.]