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The objective of this research is to understand how a learning trajectory supports pre-service primary teachers in their noticing of students’ mathematical understanding. A total of 95 pre-service primary school teachers used a learning trajectory related to the part-whole meaning of the fraction concept to interpret students’ understanding and provide instructional decisions. The findings indicate that the learning trajectory provided pre-service teachers with a specific language to describe students’ understanding and helped them to notice students’ mathematical understanding. Furthermore, the pre-service primary teachers who produced a more detailed discourse also proposed more suitable activities based on students’ understanding. These findings suggest that the learning trajectory can act as a scaffold to notice students’ mathematical understanding.

Enabling students to apply their energy knowledge to various everyday phenomena is one of the main goals of physics education. Understanding how and why some students achieve this goal and others not is crucial to adapt instruction in order to better support the majority of students. To achieve support, research suggests that it is not sufficient to solely focus on content knowledge, but also include affective and metacognitive variables. To better understand why some students are able to apply their energy understanding while others are not, we developed a ten-week-long instructional unit to collect fine-grained longitudinal data, not only on the energy understanding of students but also their affective and metacognitive characteristics. Using unsupervised machine learning, specifically a k-means longitudinal analysis, we were able to distinguish, from N  = 165 students, three clusters based on students’ learning trajectories, represented by their energy knowledge network coherence. These three clusters were then analyzed on basis of affective and metacognitive variables. The analysis showed disparities in the accumulation of energy knowledge. These disparities were then be analyzed in greater detail by the trajectories of affective and metacognitive variables, mainly showing disparities in the perception of the instructional unit regarding emotions and cognitive load. These findings indicate that affective and metacognitive variables have an impact on the learning outcome of students, which can be used to design instructional units, that address the needs of all students.

This paper comprises the results of a design study that aims at developing a theoretically and empirically based learning trajectory on statistical inference for 9th-grade students. Based on theories of informal statistical inference, an 8-step learning trajectory was designed. The trajectory consisted of two similar four step sequences: (1) experimenting with a physical black box, (2) visualizing distributions, (3) examining sampling distributions using simulation software, and (4) interpreting sampling distributions to make inferences in real -life contexts. Sequence I included only categorical data and Sequence II regarded numerical data. The learning trajectory was implemented in an intervention among 267 students. To examine the effects of the trajectory on students’ understanding of statistical inference, we analyzed their posttest results after the intervention. To investigate how the stepwise trajectory fostered the learning process, students’ worksheets during each learning step were analyzed. The posttest results showed that students who followed the learning trajectory scored significantly higher on statistical inference and on concepts related to each step than students of a comparison group (n = 217) who followed the regular curriculum. Worksheet analysis demonstrated that the 8-step trajectory was beneficial to students’ learning processes. We conclude that ideas of repeated sampling with a black box and statistical modeling seem fruitful for introducing statistical inference. Both ideas invite more advanced follow-up activities, such as hypothesis testing and comparing groups. This suggests that statistics curricula with a descriptive focus can be transformed to a more inferential focus, to anticipate on subsequent steps in students’ statistics education.

The objective of the study was to characterise how pre-service kindergarten teachers used a Hypothetical Learning Trajectory on length and its measurement to notice children’s mathematical thinking. A total of 64 pre-service kindergarten teachers enrolled in an Early Years Education mathematics teaching course were asked to notice teaching situations focusing on kindergarten children’s learning of length. On the one hand, three profiles of pre-service kindergarten teachers were found according to the use they made of the Hypothetical Learning Trajectory. These profiles were characterised by three ways of learning to use the Hypothetical Learning Trajectory based on the type of mathematical elements they identified: only mathematical elements related to the magnitude; only mathematical elements related to the measurement of length; or both magnitude and measurement elements. On the other hand, when considering the three skills of professional noticing, by identifying the mathematical elements required to solve the proposed task, a group of pre-service kindergarten teachers within each profile were able to notice the thinking of these elements by children and to suggest activities. Our findings provide learning opportunities to pre-service kindergarten teachers who use a Hypothetical Learning Trajectory. It provides them with ‘check-points’ to answer the proposed questions, in order to learn the specialised knowledge for teaching length and its measurement as well as to develop the skill of noticing student’s mathematical thinking.

Dynamic process modeling is essential for simulating time‐evolving biochemical systems, particularly those with multistate interactions and combinatorial complexity. Traditional Ordinary Differential Equation (ODE) models offer mechanistic clarity but struggle with scalability and context‐sensitive encoding. Rule‐Based Modeling (RBM) frameworks address these limitations through modular rule ion, yet require manual specification and lack adaptive learning. This study introduces algorithmic innovations within the Neural Ordinary Differential Equation (Neural ODE) paradigm to bridge the gap between mechanistic interpretability and scalable expressivity. Neural ODEs can be considered as a revolutionary approach in the field of modeling dynamic biochemical interactions. They have made it possible to create models of such interactions that are flexible enough to adapt to different scenarios and do so without requiring any manual intervention in terms of rule encoding or predefined reaction schemes. This is achieved by employing differential solvers within the framework of neural networks, thus enabling a learning process that is in accordance with the behavior of the system. Using the DARPP‐32 signaling network—a benchmark system characterized by multivalent phosphorylation and dynamic perturbations—the proposed Neural ODE framework demonstrates the ability to replicate key dynamic behaviors observed in ODE and RBM models. Comparative simulations under baseline and perturbed conditions reveal that Neural ODEs maintain trajectory fidelity while offering enhanced modularity and computational efficiency. Feature importance analysis and latent space visualizations further validate the model's interpretability and robustness. Unlike ODEs and RBMs, Neural ODEs adapt to structural mutations and binding schemes through latent trajectory learning, enabling flexible simulation of biochemical variability without manual rule encoding. This work establishes Neural ODEs as a viable and scalable alternative for modeling complex biochemical systems, combining the strengths of data‐driven learning with the interpretability of differential equations. This study presents algorithmic innovations in Neural Ordinary Differential Equations (NeuralODEs) for dynamic process modeling of complex biochemical systems. Using the DARPP‐32 signaling network as a benchmark, the proposed framework demonstrates accurate replication of ODE and Rule‐Based Model behaviors while offering enhanced modularity, computational efficiency, and adaptability to structural variations. Feature importance analysis and latent trajectory learning highlight the model's interpretability and robustness, establishing NeuralODEs as a scalable alternative for simulating multistate biochemical dynamics.

Learning science in the context of socio-scientific issues (SSI) can promote scientific literacy that links science to everyday life and society. In this position paper, we argue that developing and using multiple models equip students with the appropriate knowledge and skills needed to deal with complex issues. We draw upon literature from science education and philosophy of science and advance our theoretical argument about why it is critical for students to develop and use multiple models as part of their science learning experiences in general, and how the practice benefits students in the context of SSI in particular. We posit that students should engage in both scientific and socio-scientific models as they explore a complex societal issue because (1) engagement in multiple scientific models promotes students’ understanding about the phenomena relevant to the focal issue, and (2) engagement in socio-scientific models helps students to use that scientific knowledge in the larger social contexts and reason about how interacting science and social factors may impact students’ positions on the complex issue. We take COVID-19 as the learning context and present exemplar models students can develop and use as they learn about the pandemic. We conclude the paper by discussing the teaching aspects of the proposed modeling approach for SSI-based instruction as well as identifying possible areas for future research.

This study aims to facilitate and implement inquiry-based exercises in the domain of hydrostatic pressure within the subject of physics education. The design research method was used to support eighth-grade pupils at Palembang State Middle School by developing inquiry-based activities on hydrostatic pressure. Three stages-experimental preparation, classroom experiments (pilot experiments and teaching experiments, and retrospective analysis-were carried out to formulate pupil learning trajectories. Pupils were expected to form the hypothesis that the dam walls are designed to increase the thickness of deeper wall or dam. They then created and conducted an experiment using appropriate tools and materials. The data collected were graphed and the graph was used to determine whether the hypothesis had been proven true. Finally, pupils applied their understanding of hydrostatic pressure and to given problems. Findings demonstrated significant improvements in pupils' conceptual understanding, experimental skills, and problem-solving abilities. Generally, students were able to accurately describe the relationship between hydrostatic pressure and the depth and density of the liquid, as evidenced by their correct interpretation of experimental data and graphical representations. The results also can be used to implement inquiry-based activities with a broader planned learning trajectory, and as a pioneer of further research across different learning contexts.

Identity changes over time through a series of external (Beauchamp &Thomas, 2009; Flores & Day, 2006; Sachs, 2005), but also of internal factors, such as emotions (Rodgers & Scott, 2008). The school placement is the pre-service teachers’ (PST) first year of confrontation with teaching and is characteristically an emotional experience (Zembylas, 2003). Several authors focus on the development of teachers at the beginning of their professional practice (Meijer, 2011). Moirs (1999) identifies five phases: anticipation, survival, disillusionment, rejuvenation, and reflection. The literature encourages us to continue investigating school placement contexts to better analyse and interpret the PSTs’ learning, thus improving teacher education (e.g., Gomes et al., 2023). This case study aimed to examine how PSTs perceive their learning trajectories by reflecting on the most memorable experiences of their school placement and associated emotions. In 2022/2023, three PSTs, two female and one male, aged between 22 and 24-year-olds, attending the 2nd year of the master's Teacher Education programme in Teaching Physical Education, filled out a Timeline (Adriansen, 2012) from September 2022 to March 2023, and attended a focus group for greater depth of their emotional/ learning trajectories. A content analysis (Patton, 2001) revealed that: In September, PST1 and PST2 began their journey with positive emotions (euphoria, enthusiasm, joy and hope) related to the first contact with the school and their classes –anticipation. PST3 reported negative emotions (anxiety, fear, concern) due to the volume and nature of the work he was expecting to deal with. During October, November and December, the PSTs’ 1 and 2 trajectories were marked by negative emotions (sadness, disappointment, frustration, lack of confidence, desire to give up), as they drew a descending curve - survival and disappointment, associated with a poor pedagogical relationship with students, the observation of classes, a bad relationship with the PST peers or the accumulation of work extra school. PST3 described a rising learning line, marked by positive emotions (confidence, pleasure, motivation), related to his success in teaching, the dynamisation of school activities and the good interaction with other teachers - rejuvenation until reaching a plateau of reflection between October and February. In January, February, and March, the PSTs’ 2 and 3 curves took a pointy downward before reaching a new anticipation phase (PST 002), subscribing to negative emotions (boredom, anxiety, sadness, and feelings of incompetence). The PST’s 1 learning curve assumed regular growth since November and December, marked by positive emotions (hope, confidence, motivation, joy), resulting from the success in implementing student-centred teaching strategies - new anticipation. At this point of their school placement, PST2 never reached an uplifting of her emotional/learning curve.

Using a cluster randomized trial design, we evaluated the persistence of effects of a research-based model for scaling up educational interventions. The model was implemented in 42 schools in two city districts serving low-resource communities, randomly assigned to three conditions. In pre-kindergarten, the two experimental interventions were identical, but one included follow-through in the kindergarten and first-grade years, including knowledge of the pre-K intervention and ways to build upon that knowledge using learning trajectories. Students in the experimental group scored significantly higher than control students (g = .51 for those who received follow-through intervention in kindergarten and first grade; g = .28 for non—follow-through), and follow-through students scored significantly higher than non—follow-through students (g = .24).

The aim of this paper is to examine whether the use of a hypothetical learning trajectory as a guide to notice students’ mathematical thinking could improve pre-service teachers professional discourse and enhance pre-service teachers’ noticing. Twenty-nine pre-service primary school teachers participated in a learning environment in which they had to interpret students’ thinking about the fraction concept using a hypothetical learning trajectory as a guide. Results suggest that using a hypothetical learning trajectory as a guide helped pre-service teachers develop a more detailed discourse when interpreting students’ mathematical thinking, enhancing their noticing skill. The enhancement of the skill of noticing, however, was linked to pre-service teachers’ mathematical content knowledge. This research provides teacher educators with resources to help pre-service teachers produce a more detailed professional discourse to attend to the details of students’ answers and their different mathematical levels of thinking in mathematics teacher education programs.