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Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres' rights of the learner • Topic modeling and sentiment analysis helps to quantify and describe valence that arises from large amounts of text. • 32 TCs identified children's confusion, generalized struggle, and mistakes (Torres' RotL 1 and 2) in their MPTC across 50 instances. • Some TCs described children's confusion and mistakes as rights to exercise, a potential rehumanizing stance. [ABSTRACT FROM AUTHOR]

• Pre-service mathematics teachers' narrated failure, identity and resilience are explored. • Mindset framework fruitfully brings together the constructs of failure, identity and resilience. • Self-expectations and goals are a key component of one's subjective understanding of failure. • Rethinking and redefining self-expectations and goals is a strategy that leads to resilience. • The constructs of failure and failure resilience need a subjective approach to investigation. In educational research, failure has often been touched upon as a by-product, but rarely has failure been investigated without attempting to find ways around it instead of attempting to simply understand it better. This paper provides insight into a pilot study on understanding math failure through narrative. I analyse two pre-service mathematics teachers' narratives about failure and identity in order to illustrate their personal understandings of their own failures. The analysed stories provide insight into the subjects and their failure resilience, and initiate a discussion on subjective understanding of resilience. These stories also illustrate the need to further research regarding the subjective understandings of failure in general and in math specifically. [ABSTRACT FROM AUTHOR]

Anyone involved in mathematics education will at some point encounter a classroom situation in which a student's statement provides an opportunity for the teacher to enhance students' mathematical understanding. This paper aims to characterize such classroom situations, known as critical events, as noticed by pre-service mathematics teachers during classroom observations and teaching. Analysis of 80 critical events reported by 20 pre-service mathematics teachers during a field-based teaching preparation program yielded a model that enables teacher-educators and researchers to characterize critical events while reflecting on the wide variations within these events. The model comprises three axes: the participants in the event, the focus of the event, and the dimensions in learning and teaching mathematics (i.e., cognitive, affective and social). The paper discusses the theoretical and empirical construction of the model and demonstrates its potential contribution using examples taken from our research. • Pre-service teachers identified critical events during classroom observations. • A model for characterizing critical events was constructed theoretically and empirically. • The model affords eliciting the multidimensional nature and complexity of critical events. • The model's potential contribution is presented using examples from our research. [ABSTRACT FROM AUTHOR]

This study aimed to identify the role and nature of spatial visualization in the problem solutions of pre-service teachers solving school-mathematics tasks requiring measurement reasoning. The nuances in the pre-service teachers' strategies were examined for the role of spatial visualization in the solution process. The findings suggest that inadequacies in visualizing the spatial configurations of the tasks led to incorrect numerical solutions despite the presence of conceptual knowledge. Furthermore, the tendency to rely on formula-based approaches appeared to have suppressed the preliminary spatial processing of the configurations. Theoretically, the paper offers insights into the mechanism that may be involved in the solution of spatially-related mathematical tasks. The findings imply that pre-service teachers need to be sufficiently engaged in spatial reasoning activities. • Spatial manipulation on mathematical tasks influences numerical solutions. • Spatial visualization is an enabler in solving spatially-related mathematical tasks. • Formula-based thinking may suppress the preliminary spatial inspection of the tasks. • Spatial reasoning needs to be explicitly taught to pre-service teachers. • Pre-service teachers should be empowered to analyze spatially-related tasks. [ABSTRACT FROM AUTHOR]

• A new tool for assessing mathematics lessons. • The tool visualizes the mathematical objects that arise during classroom discussion. • The tool captures the level of students' authority during the lesson. • The tool captures the opportunities for objectification that are provided to students. • Different lessons have different affordances for explorative participation. In this paper we offer a new tool for assessing explorative participation in mathematics lessons, namely participation where students focus on constructing narratives about mathematical objects based on their own authority and on previously established mathematical narratives. This tool, named the Realization Tree Assessment tool (RTA), aims to record opportunities for objectification and the level of students' authority in whole classroom discussions. Inspired by the commognitive theory of mathematical objectification, the RTA visualizes the realizations of mathematical objects that arise during classroom discussions and the extent to which students authored the different realizations. We exemplify the tool on 14 lessons, all of which were based on an identical task – the Hexagons Task. The exemplification shows how the tool captures narratives about realizations and connections between these realizations offered by students and the teacher. We discuss the affordances of the tool for analysing and evaluating instruction that aims towards explorative participation. [ABSTRACT FROM AUTHOR]