This study examines upper secondary school-leaving mathematics examination tasks from Poland (2005–2023) and the federal state of North Rhine-Westphalia (2007–2023) with regard to three central mathematical competencies: problem solving, modelling, and reasoning – and their competency levels. Drawing on the competency framework of Niss and Højgaard, a comparative document analysis was conducted using three competency-specific coding schemes. The analysis covers complete corpora of centralized basic-level mathematics examinations, comprising 570 Polish and 2,731 North Rhine-Westphalian tasks. The study first identifies and describes the distribution of competencies and competency levels within each examination system reflected in the examination tasks over the course of time. It then compares both systems statistically, identifying similarities and differences in the distribution of competency demands. The results reveal distinct competency profiles in the mathematics examination tasks from Poland and North Rhine-Westphalia. In North Rhine-Westphalia, modelling constitutes the dominant competency demand, whereas problem solving and reasoning occur considerably less frequently. In contrast, Polish examination tasks show higher levels of problem solving, while modelling and reasoning play only a minor role. In both systems, higher competency levels occur only rarely across all three competencies. By providing the first comprehensive comparison of complete upper secondary school-leaving mathematics examination corpora from Poland and North Rhine-Westphalia, this study contributes to research on competence-oriented assessment. The findings contribute to discussions on examination design, the implementation of competency-oriented curricula, educational quality, and the comparability of upper secondary school-leaving qualifications within the European educational area.

This study presents the development and validation of a comprehensive 35-item assessment instrument for measuring mathematical proficiency in real numbers. Drawing from Ngu and Phan's proficiency framework, the assessment systematically samples four interconnected strands of mathematical competence: Understanding, Fluency, Problem Solving, and Reasoning. The validation study employed a rigorous multi-method approach following the Standards for Educational and Psychological Testing, examining evidence based on test content, response processes, internal structure, and relations to other variables. The instrument was administered to 117 first-semester pre-service secondary mathematics teachers at the University of Cologne, Germany, at the beginning of their studies. A subset of 74 participants completed a second assessment administered at the end of their foundational mathematics course, using parallel forms designed to minimize practice effects. Results demonstrated robust psychometric properties across all validation categories. Content validity was confirmed through strong expert consensus. Response process validation revealed high alignment between intended and observed cognitive processes. Internal structure analysis showed acceptable to good reliability across strands. The assessment demonstrated exceptional sensitivity to instruction, with very large effects for Reasoning and Fluency, indicating that the instrument effectively captures learning gains from systematic instruction. Inter-rater reliability for reasoning scoring was excellent. The assessment enables precise diagnostic capabilities for mathematics education, distinguishing among understanding deficits, fluency limitations, strategic thinking gaps, and reasoning weaknesses while supporting targeted instructional interventions.

Feedback is widely recognized as one of the most influential factors in student learning success. While it is acknowledged that subject-specificity also plays a significant part, there are still only a few models for analyzing feedback based on the characteristics of individual disciplines. Furthermore, many well-known models have not been developed or tested specifically in a classroom setting. Our goal was therefore to develop a tool for analyzing teacher feedback in mathematics teaching. To this end, we videotaped lessons on problem-oriented mathematics teaching in primary school, as solving mathematical problems is particularly challenging for primary school students and requires more specific feedback from teachers. We analyzed these lessons based on levels of feedback and feedback level-specific categories from established feedback models. In doing so, the analysis tool developed could be inductively supplemented with additional feedback level-specific categories, enabling a detailed analysis of teacher feedback. The analysis tool developed was found to be effective in achieving satisfactory levels of intercoder agreement. This will enable future analyses of patterns and practices in teacher feedback.

Epistemic beliefs and their possible effects on human behavior have been a focal point in recent decades. By measuring epistemic beliefs, researchers aim to better understand how individuals perceive and approach knowledge. However, more recently, the construct of epistemic beliefs has become the subject of further investigation, as it is often described as difficult to conceptualize and assess. Hence, in this paper the psychometric properties of one of those instruments, namely the Connotative Aspects of Epistemological Beliefs (CAEB), are further analyzed. Across four data sets, a total of N = 1,108 educational sciences and psychology students were asked to complete the CAEB with regard to educational science and/or psychology. Data was analyzed via confirmatory factor analyses, exploratory structural equation models and analyses of measurement invariance. Results show that neither the originally proposed theoretical, nor the empirically identified dimensions of the CAEB can be replicated. Furthermore, exploratory analyses reveal a new model with a total of 12 items and two factors, which share some similarities with the empirically identified measurement model. Group comparisons between the samples indicate that the metric invariance hypothesis holds, while scalar invariance cannot be substantiated. The findings suggest that the originally proposed CAEB dimensions are not applicable to educational science and psychology. Instead, this preliminary 12-item version appears promising for assessing connotative epistemological beliefs. Nonetheless, the characteristics of the convenience sample, as well as limitations regarding reliability, and the lack of scalar invariance, need to be addressed in future studies in different domains and across different samples.

In the latest educational reform, project-based learning (PjBL) was officially added to the mathematics curriculum in China. According to the curriculum, mathematical problem solving (MPS) is a key activity within PjBL. Teachers’ beliefs are essential for the acceptance, relative teaching design and practice and, thus, success of long-lasting reforms, but studies related to teacher beliefs on MPS within PjBL are relatively scarce. In this paper, two mathematics teachers without prior experience in PjBL were selected as cases to trace and analyse their first PjBL designs and practices in connection with their beliefs. Initially, the participating teachers’ beliefs about MPS within PjBL aligned with the curriculum documents. The following analyses show that while the teachers’ designs and practices were both consistent with their initial beliefs about MPS within PjBL, there was also a trend toward teacher-centred practice in their daily classrooms. The teachers’ beliefs developed after practice, while the development directions were attributable to what they did and what they perceived to be the corresponding outcomes. Both teachers’ beliefs after practice were more justified, evidenced, and probably longer-lasting, and could influence their future MPS teaching within PjBL.

Algebra education in secondary school is a prerequisite for further education; however, many students experience difficulties with algebra. This article focuses on students facing difficulties in algebra education and provides an overview of the research in this area. A scoping review based on a systematic literature search was conducted, which covered 73 papers published between 2019 and 2023. The reviewed articles were inductively categorized according to three objectives: the characteristics of participants, key topics of the studies, and main results. The findings show a variety of student characteristics and a range of interrelated key topics. The main results provide insights into diverse approaches, often used in combination with each other, and describe the group of low-performing students with their difficulties in detail. Implications and suggestions for further research are discussed.

Teachers’ beliefs about mathematics can shape how they approach teaching, learning, and assessment in mathematics classrooms. This study examined Australian pre-service teachers’ (PSTs’) epistemological beliefs about mathematics. A total of 18 PSTs enrolled in Bachelor of Education and Master of Teaching (secondary mathematics) programs participated in the study by answering prompt questions in an open-ended questionnaire. The participants’ responses were content analyzed by categorizing the views evident in the responses into four pre-defined mathematical orientations: formalism-related, scheme-related, process-related, and application-related. Findings have shown that the dominant beliefs for the participants reflected process- or application- related orientations to mathematics. These findings are of value to PST educators who seek to explore and expand the beliefs that future teachers hold about the nature and structure of mathematics.

In this paper, we draw on the distinction of states and traits to investigate the influence of humor and awareness prompts on solving word problems with a reflective component. Two studies with n = 144 and n = 401 mathematics pre-service teachers were conducted, using the three-item Cognitive Reflection test and a ten-item test on mathematical critical thinking, respectively. In both studies, students seeing jokes or awareness prompts showed significantly better results than students solving routine tasks before working on the test items. Drawing on the mindset theory by Gollwitzer, we explain these results with deliberate and implemental mindsets being induced by humoristic and awareness prompts prompts or routine tasks, respectively.

In this exploratory case study, the teaching practices of two secondary level mathematics teachers are examined within the framework of a sequential multi-method design. To this end, data on the teachers’ recurrent patterns of action were collected using video-based classroom observations in problem-solving lessons, semi-structured interviews and video-stimulated recalls. The qualitative analyses of the respective data allowed a differentiated description of the self-reported, observable, and articulable elements of their teaching practices. This includes the requirement situations to which the teachers respond with recurrent patterns of actions, their cognitive and affective dispositions on which they rely, and their situation-specific skills that enable them to adaptively orchestrate their problem-solving lessons. An empirical reconstruction of a selected practice, including a description of its constitutive elements, followed from a superordinate integration of the data by triangulation. This study contributes to the discourse on mathematics teacher education by providing an innovative study design for the empirical reconstruction of teaching practices. Moreover, its findings validate the theoretical assumption that teaching practices are an expression of mathematics teachers’ professional competence and expertise.

Although problem posing is rarely mentioned in the German primary school curriculum, it is recognized as a valuable mathematical activity that requires students to formulate and solve their own problems. However, little is known about how problem-posing tasks are integrated into mathematics textbooks, potentially limiting their effective use in classrooms. This study addresses this gap by analyzing the presence of problem posing in German educational standards and the integration of problem-posing tasks in two widely used German textbooks, Das Zahlenbuch , and Welt der Zahl , across Grades 1–4. The textbook analysis focuses on four key aspects: (a) the number of problem-posing tasks relative to all tasks to assess their significance in the implemented curriculum, (b) their distribution across mathematical content areas (numbers and operations; geometry; measurement; and data, frequencies, and probabilities), (c) the structure of problem-posing situations (structured vs. unstructured), and (d) the nature of the posed problems (routine vs. non-routine). A qualitative content analysis revealed that problem posing is rarely mentioned in educational standards and is almost exclusively linked to mathematical modeling. In textbooks, problem posing appears infrequently, with Das Zahlenbuch including more tasks than Welt der Zahl . Across all grades, tasks predominantly occur in numbers and operations, are mostly structured, and evoke routine tasks. These results provide insights into how problem posing is integrated into textbooks and highlight areas for further discussion on its role in primary mathematics education.

Understanding geometric concepts is an important factor for coping with everyday life. This is also important for students with learning difficulties (LD)-students with significant and persistent difficulties in several school subjects, yet little is known about the understanding of geometric concepts of students with LD: The few previous studies have focused on their overall mathematical performance (usually looking at their products). This study focuses on the identification of triangles-an essential activity with respect to students' understanding of geometric concepts. The aim of this study is to investigate if fifth grade students with LD (n = 20) differ from students without LD (n = 165) in the identification of triangles. We analyzed students' error rates and use of strategies using eye tracking. We found that for representatives of triangles, students with LD used quick strategies, that is, strategies involving only few gazes, more often than students without LD and tended to make fewer errors. For non-representatives, however, we found that students with LD tended to use analytic strategies through attending to more parts of the shapes more often than students without LD but made more errors. The results indicate that students with LD differed from students without LD in the identification of triangles, and that students with LD not only have difficulties in mathematics, but also show certain strengths in geometry. However, the results also indicate that educators need to support students, especially with LD, in attending to critical attributes of triangles.

Following the PRISMA methodology, this systematic literature review addresses instruments that utilize the TPACK framework in studies conducted from 2017 to 2023. The review focuses on pre-service teachers across all subjects and school types, and the facet of selecting digital resources (dRs). The latter was identified as a research gap in existing TPACK reviews. The novelty of the review lies in several key findings. First, consistent with previous reviews, self-report instruments remain the most commonly used TPACK instruments. Second, reviewing the existing TPACK instruments indicated that selecting dRs is assessed in only about a third of the instruments. If so, it is inconsistently linked to TPACK components. However, when the facet of selecting dRs is included, the rationale for selection is consistent with the TPACK component definitions. Third, the analysis of the dRs studied in TPACK assessments underscores the developmental nature of dRs and highlights the need to cultivate and evaluate the digital competence facet of selecting dRs. Notably, dRs that promote “Self-directed learning” and learner “Collaboration” showed a small but increasing trend. Fourth, evaluating the reasoning behind selecting dRs when assessing pre-service teachers is empirically underrepresented. Additionally, concise TPACK assessments utilizing open-text prompts to evaluate pre-service teachers’ rationale for selecting dRs are methodically and empirically underrepresented. In contrast, two TPACK rubrics that examine the alignment of dRs to learning content and teaching strategy have become the de facto standard when evaluating lesson plans and observations.

In many countries, the number line (NL) is an important tool in mathematics education to develop an understanding of numbers. However, students may have difficulties using the NL. To provide students showing NL difficulties with appropriate support, more research is needed on how these students interact with NLs. In this study, we investigated how three different groups of students located numbers on an NL: 20 fifth-grade students with general learning difficulties (LD) from a special school, and 60 fifth-grade students with mathematical learning difficulties (MD) from a school of general education, compared to 55 fifth-grade students without MD/LD. We analyzed students’ strategies based on qualitative analysis of eye-tracking videos and students’ error rates. We found that all students were generally able to solve the tasks correctly. Analyses of students’ strategies showed that the types of strategies used for locating numbers on an NL did not differ between students with LD, with MD, and without MD/LD. Differences were found in the frequency with which certain strategies were used, particularly for numbers between the midpoint and endpoint of the NL—indicating differences in the mathematical development regarding the flexible use of NL strategies between students with LD, with MD, and without MD/LD.

This study investigated the effects of an innovative professional development program aimed at enhancing teacher noticing skills and related professional knowledge in inclusive mathematics education in secondary algebra instruction. A total of 653 participants from three experience groups with different length of teaching experience, namely master’s students, teachers in preparatory service, and in-service teachers from Germany, participated in a pretest–posttest evaluation design that included a control group. The professional development program comprised 18 hours of coursework designed to facilitate teacher noticing skills by integrating novel teaching materials and video-based learning activities that combined both mathematics pedagogical and general pedagogical perspectives. The results indicated significant improvements in teachers’ noticing skills and related professional knowledge for the intervention group across all investigated facets, particularly for teacher noticing under a pedagogical perspective and mathematics pedagogical knowledge, compared to a control group that exhibited no significant changes. Effect sizes ranged from small to medium, suggesting that the professional development program effectively improved participants’ knowledge of inclusive teaching and their abilities to perceive, interpret, and make decisions in inclusive contexts. Notably, master’s students exhibited the most substantial gains in all competencies, while in-service teachers primarily improved their teacher noticing from a pedagogical perspective. The findings underscore the importance of tailored professional development for fostering teacher noticing in inclusive mathematics education and yield valuable insights into this area of teachers’ competencies.

Given the increasing digitalization of education and the variety of available digital learning materials (dLMs) of differing quality, (pre-service) teachers must develop the ability to select appropriate dLMs. Objective, reliable, and valid assessment instruments are necessary to evaluate the effectiveness of that development. This study conceptualized and designed an economical four-item instrument for assessing “selecting dLMs” based on accepted frameworks and competence models. The scientific quality of the instrument was evaluated in Study 1 (n = 164) with four dLMs and empirically investigated in a subsequent Study 2 (n = 395) with pre-service mathematics teachers from two universities. The empirical results indicate that the instrument could objectively and reliably gauge different levels of “selecting dLMs”. Furthermore, the results are consistent with the widely accepted notion that the competence of “selecting dLMs” depends on (content) knowledge; however, that relation was not strong. In addition, the results for objectively assessing “selecting dLMs” paralleled the results of self-assessed TPACK in terms of the academic progression of participants. The proposed approach allows for variations and integration of diverse dLMs, and it has the potential to be adapted in other subject areas and contexts.

This paper offers an overview of current research on the effectiveness of interventions in teacher education. We synthesize 27 reviews published between 2010 and 2024, comprising 510 research publications of empirical intervention studies targeting pre-service and in-service teachers’ professional learning. We applied a specific analysis category system to address five research questions concerning (1) justifications and theoretical frameworks, (2) input and output of the intervention studies, (3) study design, (4) effect sizes, and (5) research gaps. Few reviews had developed their own theoretical framework, and justifications for the interventions’ necessity were not always elaborated. Most assumed the existence of an effect chain—namely, that the reviewed interventions help promote teacher variables (mainly competence facets), which are then positively reflected in teaching quality and students’ learning gains. However, study variables typically indicate only segments of the effect chain while comprehensive operationalization of the total effect chain is rare. Interventions are typically organized as coursework, not necessarily connected to teaching practice or student learning. The interventions’ targeted learning outcomes are heterogeneously operationalized. Basic design features appear typical for intervention studies in teacher education, whereas more challenging designs are less common. Self-report instruments appear more common than rigorous measurement approaches. Based on a subsample of 14 reviews reporting summarized effect sizes, we conducted a meta-meta-analysis that shows an overall significant moderate effect of interventions is observable. Differentiation shows stronger effects for proximal outcomes (e.g., change in teacher knowledge or instructional practice) than for student learning. The synthesis of moderator effects yielded mixed results, and the quality of interventions appears more crucial than the length. Research gaps concern further theoretical development, the individual intervention studies’ research methodological standards, and improvement of literature reviews’ methodological quality. The results offer a theoretical background for future research on teacher education interventions and research progress in teacher education effectiveness. Recommendations for future research are given.

Numerous reviews have synthesized the empirical research on the effectiveness of teacher education, highlighting teacher education effectiveness research (TEER) as an emerging research paradigm. Our systematic search identified 27 reviews related to TEER, wherein teacher education is broadly understood as comprising all stages of teacher professionalization-namely, initial teacher education, teacher induction, and teacher professional development. In reviewing these reviews, we carry out a synthesis of existing research on TEER. Guided by four research questions (RQ), we focused major frameworks (RQ1), outcome measures (RQ2), processes (RQ3), and central research gaps (RQ4). Highlights: Only few reviews provide a background or macro framing, whereas most reviews apply TEER for examining a specific topic (RQ1); outcome measures often relate to the notion of teacher competence, making increased competence the true outcome of TEER (RQ2); coursework is the most dominant category of characteristics-forming processes (RQ3); the frameworks underlying the outcome measures appear to be an object of criticism on a theoretical but even more on a methodological level. Building on these findings, we suggest a processes-and-criteria classification (PCC) grounded in basic distinctions of the various studies synthesized by the reviews. We discuss perspectives on how this classification may provide an orientation for future TEER studies.