Paul Drijvers

Little is known about strategic flexibility in collaborative mathematical problem-solving. In this study, we investigate how cognitive levels and dialogic moves, together with their collective uptake, relate to strategic flexibility. To do so, we combine the Florida taxonomy of cognitive behavior (FTCB) with the teacher scheme for educational dialogue analysis (T-SEDA) to analyze a 29-minute dialogue among four pre-service mathematics teachers solving a geometry problem. Most dialogue contributions involved lower-order thinking and surface-level dialogic moves. The combined framework distinguished actual strategy shifts, unrealized opportunities for strategic reconsideration, and local checking-and-correction moments. Opportunities to reconsider strategy arose when higher-order contributions coincided with making reasoning explicit or challenging a proposal, but were infrequent, largely steered by one participant, and rarely taken up collectively. From this, we conclude that strategic flexibility may benefit from the collective uptake of cognitively demanding contributions. Methodologically, integrating FTCB and T-SEDA traces how cognitive depth and dialogic function relate to such uptake.

Understanding the values that variables can take is a central challenge in students’ early encounters with algebra. While previous research has documented a range of numerical preconceptions related to variable values, much less is known about how they co-occur and develop. This study examines German Year 7/8 students’ (mis)understandings of variable values before and after four weeks of regular algebra teaching. Data were collected from 2205 students who completed the SMART ‘Values for Letters’ test. The test targets well-documented numerical preconceptions, including difficulties with substituting given values, alphabet-based value assumptions, treating multiple occurrences of the same letter as referring to independent values, and assuming that different letters must represent different numbers. To capture qualitative differences in students’ response patterns and their development over time, we applied multilevel Latent Transition Analysis (LTA), accounting for students nested within teachers. The analysis identified four latent classes representing distinct ways of reasoning about variable values. Although overall response rates changed only modestly, the LTA revealed substantial movement between latent classes. Alphabet-based value assumptions were rare, whereas treating multiple occurrences of the same letter as referring to independent values emerged as a persistent, intermediate bottleneck in students’ learning. Moreover, the assumption that different letters must refer to different values appeared as an overgeneralization rather than a foundational difficulty, occurring primarily among students in the highest-performing latent class. These findings show how LTA reveals learning trajectories invisible in aggregate analyses and supports diagnostic work in algebra instruction. Limitations include attrition and the short period between timepoints.

Understanding how mathematics teachers change after disruption is essential for supporting long-term professional growth. With the COVID-19 pandemic as a case, this longitudinal study examines how secondary mathematics teachers in Flanders, Germany, and the Netherlands responded over time to this disruption in terms of teaching practices, teachers’ beliefs, and assessment practices. To explore which changes persisted, which reverted, and how teachers reflected on their experiences, data were collected through two large-scale surveys, conducted in June 2020 and June 2021. A total of 2,058 teachers completed the follow-up questionnaire. Concerning teaching practices, we found a significant increase in videoconferencing and the use of homemade video clips. Learning management systems became a central part of teaching practice. However, there was little change in the use of online learning environments, exercise platforms, and digital mathematics tools. Regarding teachers’ beliefs, teachers reported improved digital skills, particularly with tools they used frequently, such as videoconferencing. However, they reported only slight changes in their perceptions of effective mathematics teaching and limited confidence in the use of digital formative assessment tools. Assessment practices shifted toward more interactive formats, including live questions and chat, while reliance on email declined. Our findings suggest that the impact of the crisis extended beyond the initial period of school closure and that the longer-term picture is more complex than early responses indicated.

Problem solving is a crucial skill in today’s world that needs to be nurtured in education, particularly in mathematics education. However, teaching problem solving is complex. To get to grips with this complexity, we undertook a systematic literature review to identify the knowledge and skills teachers require to create a learning environment that fosters mathematical problem solving. A literature search on problem solving, mathematics, and teacher knowledge and skills in the databases ERIC, PsycInfo, Scopus, and Web of Science led to a final corpus of 104 articles. These articles were categorized into five categories based on Schoenfeld’s (ZDM Math Educ 52(2):359–376, 2020. https://doi.org/10.1007/s11858-019-01057-5 ) teaching for robust understanding (TRU) framework, which we adapted to the context of problem solving: (1) problem-solving skills and knowledge of teaching problem solving, (2) teaching practices to adjust cognitive demand, (3) problem solving for all, (4) students’ active involvement in problem solving, and (5) teacher noticing for problem solving. The findings highlight that the essential knowledge and skills for teaching mathematical problem solving span all dimensions and pose significant challenges for teachers. These results align with existing research on teacher noticing, indicating that teachers’ ability to perceive and interpret students’ work is pivotal in teaching problem solving. Therefore, we suggest to enhance the quality of teaching problem solving through teacher education or professional development programs on teacher noticing within the context of problem solving and on facilitating mathematical discussions in class.

Computer-aided formative assessment can be used to enhance a learning process, for instance, by providing feedback. There are many design choices for delivering feedback that lead to a feedback strategy. In an informative feedback strategy, students do not immediately receive information about the correct response but are offered the opportunity to retry a task to apply feedback information. In this small-scale qualitative study, we explore an informative feedback strategy designed to offer a balance between room for exploration and mitigation of learning barriers. The research questions concern the ways in which students interact with the feedback strategy and their appreciation of error-specific feedback as opposed to worked-out solutions. To answer these questions, 25 15- to 17-year-old senior general secondary education students worked for approximately 20 minutes on linear and exponential extrapolation tasks in an online environment. Data included screen captures of students working with the environment and post-intervention interviews. Results showed that room for exploration offered opportunities for self-guidance, while mitigation of learning barriers prevented disengagement. Furthermore, students appreciated balanced feedback. We conclude that the balanced feedback strategy yielded fruitful student–environment interactions.

Covariational reasoning is critical for understanding functional relationships in mathematics. Yet, many students struggle to understand varying quantities and to conceptualize covariation. This study explores the research question of how bimanual movements within a digital-embodied learning environment can support students’ covariational reasoning, appropriating Thompson and Carlson’s (2017) covariational reasoning taxonomy. The teaching experiment consisted of three lessons involving 76 Grade 9 students. Data included hand-tracking data, posttask probes, and pretest and posttest results. We examined the relationship between students’ bimanual coordination and their covariational reasoning levels, addressing two hypotheses: (H1) Higher levels of covariational reasoning correlate with a shorter time to reach fluency in the bimanual coordination and (H2) higher levels of learning gains in covariational reasoning correlate with longer time spent on fluently performing the bimanual coordination. As a result, the pretest and posttest scores demonstrated significant improvements in students’ covariational reasoning after the experiment. Quantitative analyses of hand-tracking data indicated that students with higher initial covariational reasoning levels achieved fluent bimanual movement more quickly than others (supporting H1), while those with greater learning gains spent more time consolidating their understanding in fluency phases (supporting H2). Qualitative findings showed how the interplay of perception–action loops, attentional anchors, and real-time feedback facilitated the internalization of covariational relationships. This study highlights the potential of an embodied approach to fostering covariational reasoning and introduces a framework for analyzing embodied learning through the integration of hand tracking, probes, and assessments.

Where embodied cognition perspectives are on the rise in mathematics education, little is known on how mathematics teachers can foster embodied learning. Especially when aiming for abstraction by students, knowledge on embodied teaching practices lags behind. This paper focuses on teaching practices that might assist primary school teachers in fostering embodied learning in geometry, specifically embodied abstraction, a process by which learners verbalize and reflect on performed actions and observed structures. To this end, a case study involving two teachers was carried out, with the aim to set up an inventory of embodied teaching practices. The teachers participated in a training program. The results show seven teaching practices (four inspired from literature and three inspired from data) and illustrate how these are adopted. Teachers use students’ multimodal expressions as a starting point for intervention, interpretation and discussion. They direct students’ perception to emerging structures (attentional anchors) by referring to the action space and previous experiences. Sometimes teachers slide back to teacher-oriented practices when describing and introducing mathematical structures they distinguish themselves. Overall, participation in the training program on embodied learning enhanced their awareness of students’ multimodal behavior and their own teaching practices aimed at fostering abstraction.

In recent years, hybrid education has gained attention for teaching and interacting with two groups simultaneously, where both teachers and students face challenges with interaction. We hypothesise that transactional distance plays a role in this. This study, aims to explore the characteristics of hybrid education in terms of interaction and transactional distance. In this case study, we observed four hybrid lectures of a university course on the history of physics. After each lecture, the students (N = 6) completed a transactional distance survey and students and teachers participated in individual semi-structured interviews. Our findings indicate that online students experienced limited engagement in group discussions, mainly asking for clarification when teachers asked questions. Moreover, online students experienced a greater transactional distance compared to their onsite counterparts. This disparity in engagement and perceived distance may stem from the online students’ inability to partake in the informal interactions occurring before, during, and after the lectures. Both students and teachers stressed the importance of this small talk when discussing the differences between online and onsite students. In conclusion, our research underscores the essential role of informal interactions in hybrid education, highlighting the need to facilitate these exchanges to build community and enhance learning.