Learning to interpret data in context is an important educational outcome. To assess students' attainment of this outcome, it is necessary to examine the interplay between their contextual and statistical reasoning. We describe a research method designed to do so. The method draws upon Toulmin's (1958, 2003) model of argumentation for the first stage of qualitative data analysis and the Structure of the Observed Learning Outcome (SOLO) (Biggs & Collis, 1991) model for the second stage. Toulmin analyses help identify the justifications and expressions of uncertainty students provide in their interpretive arguments. Subsequent analyses based on the multi-modal conceptualization of SOLO help characterize the quality of student arguments relative to one another. Existing literature and an empirical example are drawn upon to explain how the Toulmin and SOLO models can be used in tandem to analyze students' interpretations of contextualized data. We also explain how pairing Toulmin and SOLO can address theoretical and practical limitations that arise when using just one of the two models on its own.

The COVID pandemic disrupted the schooling of students worldwide resulting in many having had a period of at-home learning. Many parents found themselves assuming responsibility for supporting their children's at-home learning. Parents often find it difficult to support their children's mathematics learning compared with other curriculum areas. There has been limited research exploring parental engagement in mathematics education generally, and little into parental engagement in mathematics education during the COVID pandemic. This paper examines how parents supported their child's mathematics education during the school closures and identifies the factors that impacted this engagement. The Ecologies of Parental Engagement (EPE) model was used to help describe the engagement of different parents in mathematics education during the school closures and to examine the way the home space and available capital shaped parental engagement. Eight parents were selected from a larger Australian study that explored the impact of the pandemic-induced period of at-home schooling on primary school mathematics and science. One-on-one narrative interviews were conducted online with participants. Analysis identified three categories of parental engagement: monitors, facilitators, and enhancers. Parents in each category responded to their role in at-home learning differently, and accessed and activated different capital to support their child's at-home learning in mathematics during the pandemic. Results highlight the value of emotional capital, as well as knowledge of mathematics and mathematics education, with implications for schools hoping to engage parents in mathematics learning. The study offers a typology to be explored in future research concerning parental engagement in mathematics education.

From the socio-critical perspective of mathematical modeling, reflexive discussions about the nature, criteria, and consequences of mathematical models are not a natural consequence of modeling in school. This report is part of a larger study focused on stimulating reflexive discussions in practice employing constructivist grounded theory as a research method. Twenty-seven college algebra students engaged in a 3-week modeling project at a community college in the USA. Audio-recorded group discussions and written reflections were collected to determine how reflexive discussions were taking place. Analysis of students' actions and reflexive discussions during the modeling project produced four concepts: . These concepts are integrated into an overall process for stimulating reflexive discussions and are conceptualized as . The overarching concept of unboxing mathematics represents one interpretation of how reflexive discussions may be constituted during modeling activities and identifies classroom mathematical practices specific to the socio-critical modeling context of this study.

In this article, we report on a typology of the demands of statistical and mathematical products (StaMPs) embedded in media items related to the COVID-19 (coronavirus) pandemic. The typology emerged from a content analysis of a large purposive sample of diverse media items selected from digital news sources based in four countries. The findings encompass nine categories of StaMPs: (1) descriptive quantitative information, (2) models, predictions, causality and risk, (3) representations and displays, (4) data quality and strength of evidence, (5) demographics and comparative thinking, (6) heterogeneity and contextual factors, (7) literacy and language demands, (8) multiple information sources, and (9) critical demands. We illustrate these categories via selected media items, substantiate them through relevant research literature, and point to categories that encompass new or enhanced types of demands. Our findings offer insights into the rich set of capabilities that citizens (including both young people and adults) must possess in order to engage these mass media demands, critically analyze statistical and mathematical information in the media, evaluate the meaning and credibility of news reports, understand public policies, and make evidenced-informed judgments. Our conclusions point to the need to revise current curricular frameworks and conceptual models (e.g., regarding statistical and probability literacy, adult numeracy), to better incorporate notions such as blended knowledge, vagueness, risk, strength of evidence, and criticality. Furthermore, more attention is needed to the literacy and language demands of media items involving statistical and mathematical information. Implications for further research and educational practice are discussed.

The COVID-19 pandemic brought with it a new way of being in a changed and uncertain world. Aotearoa/New Zealand took a well-being approach and in turn, we share the positive outcomes which resulted for some low socio-economic schools and communities in relation to teacher learning and relationships with families. In this article, we report on how teachers and schools connected with diverse students and their families during the period of remote learning. We draw on the responses from 20 teachers and school leaders who participated in interviews. Following the wider government focus, schools took a well-being first approach which led to increased connections and positive home/school relationships. The results highlight how a disruptive event such as COVID-19 can also be a time to focus on strengths of diverse communities and gain insights. We demonstrate that while focusing on mathematics, teachers and school leaders gained insights related to their students' funds of knowledge and saw opportunities for learning for students, parents, and the teachers themselves.

In this research, our objective is to characterize the problem-solving procedures of primary and lower secondary students when they solve problems in real class conditions. To do so, we rely first on the concept of heuristics. As this term is very polysemic, we exploit the definition proposed by Rott (2014) to develop a coding manual and thus analyze students' procedures. Then, we interpret the results of these analyses in a qualitative way by mobilizing the concept of semantic space (Poitrenaud, 1998). This detailed analysis of students' procedures is made possible by collecting audiovisual data as close as possible to the students' work using an action camera mounted on the students' heads. We thus succeed in highlighting three different investigation profiles that we have named explorer, butterfly, and prospector. Our first results tend to show a correlation with these profiles and the success in problem-solving, yet this would need more investigation.

Concerns about health and disease have long pervaded mathematics education research, yet their implications have been underappreciated. This article focuses on three contemporary relationships amplified by the COVID-19 pandemic: (1) school mathematics and national health, (2) mathematics educators' roles in distinguishing the health needs of students, and (3) mathematics instruction as either enhancing or threatening students' mental health and social adjustment. We argue that these concerns are foundational preoccupations of mathematics education research that have persistently shaped debates over who should learn mathematics, how, and to what ends. Our study examines histories of school mathematics and health discourses to explore how particular notions of health entered US mathematics education during the 19th and early twentieth centuries in ways that resonate with recent research trends and responses to COVID-19. We especially attend to how health/pathology distinctions reconfigured hierarchies of nationality, sex, race, and dis/ability within exclusionary, segregated, colonial, and tracked mathematics instruction. By mapping some of the shifting contours of health and pathology over time, we emphasize the potential dangers of the pandemic reanimating long-circulating dividing practices, such as in emerging trends comparing national metrics of well-being, responding to perceived trauma with differentiated instruction, and seeking to calibrate healthy mathematics identities in marginalized groups.

In the spring of 2020, schools and universities around the world were closed because of the COVID-19 pandemic. The relative lockdown affected more than 1.5 billion learners as teachers and students sheltered at home for several weeks. As schooling moved online, teachers were forced to change how they taught. In the research presented here, we focus on university mathematics professors, and we analyze how their practice, knowledge, and beliefs intertwine and change under these circumstances. More specifically, the context of the pandemic and the relative lockdown provides us with the experimental basis to argue that the new practice affected both knowledge and beliefs of mathematics teachers and that practice, knowledge, and beliefs form a system. Being part of a system, the reactions to change in practice can be of two types, namely, the system as a whole tries to resist change, or the system as a whole changes - and it changes significantly. The research presented here proposes a model for describing and analyzing what we called a teaching system and examines three cases that help to better depict the systemic nature of teaching.

Investigating the transition between educational levels is one of the main themes for the future of mathematics education. In particular, the transition from secondary school to STEM degrees is problematic for the widespread students' difficulties and significant for the implications that it has on students' futures. Knowing and understanding the past is key to imagine the future of a research field. For this reason, this paper reports a systematic review of the literature on the secondary-tertiary transition in Mathematics Education from 2008 to 2021. We constructed two corpuses: one from the proceedings of three international conferences in mathematics education (PME, ICME, and INDRUM) and the other from peer reviewed research papers and book chapters returned by the databases ERIC and Google Scholar. A clear evolution in perspectives since 2008 emerges from the analysis of the two corpuses: the research focus changed from a purely cognitive to a more holistic one, including socio-cultural and - to a lesser extent - affective issues. To this end, a variety of research methods were used, and specific theoretical models were developed in the considered papers. The analysis also highlights a worrisome trend of underrepresentation: very little research comes from large geographical areas such as South America or Africa. We argue that this gap in representation is problematic as research on secondary tertiary transition concerns also consideration of socio-cultural and contextual factors.

Findings on children's proportional reasoning abilities strongly vary across studies. This might be due to the different contexts that can be used in proportional problems: fair-sharing, mixtures, and probability. A review of the scientific literature suggests that the context of proportional problems may not only impact the difficulty of the problem, but that it also plays an important role in how children approach the problems. In other words, different contexts might elicit different (erroneous) thinking strategies. The aim of the present study was to investigate the role of context in third graders' ( = 305) proportional reasoning abilities. Results showed that children performed significantly better in a fair-sharing context compared to a mixture and a probability context. No evidence was found for a difference in performance on the mixture and the probability context. However, the kind of erroneous answers that were given in the mixture and probability context differed slightly, with more additive answers in the mixture context and more one-dimensional answers in the probability context. These findings suggest that the type of answers elicited by proportional problems might depend on the specific context in which the problem is presented.

There is a need for a more robust conceptualization of engagement in mathematics education research. Investigating how teachers describe engagement can provide insight into relationships between purposes of engagement and dimensions of engagement. In this exploratory study, we examined how 28 secondary mathematics teachers in two states in the USA talked about their students' engagement. During interviews, we asked teachers to provide their definitions for engagement, describe their teaching strategies for engaging students, and describe their observations of engagement during a video clip from their own classroom. We interpreted teachers' talk to identify how they described the nature of mathematics engagement (dimensions such as behavioral, cognitive, affective, and/or social engagement) and purposes of engagement (engagement in learning or in schooling [Harris, 2011]). When teachers described the purpose of engagement as engagement in learning, they also tended to describe the nature of engagement with cognitive and social dimensions and with multiple dimensions of engagement.

Mathematics discussions are important for helping students to develop conceptual understanding and to learn disciplinary norms and practices. In recent years, there has been increased attention to teaching prospective teachers to lead discussions with students. This paper examines the possibilities of designing a formative assessment that gathers information about prospective elementary teachers' skills with leading problem-based mathematics discussions and makes sense of such information. A decomposition of the practice of leading discussions was developed and used to design the assessment. Nine first-year teachers who graduated from a range of different teacher education programs participated in the study. The findings reveal that our formative assessment works to gather information about teachers' capabilities with leading discussions and that the associated tools support making sense of the information gathered. This suggests that such tools could be useful to support the formative assessment of the developing capabilities of prospective teachers.

The many studies with coin-tossing tasks in literature show that the concept of randomness is challenging for adults as well as children. Systematic errors observed in coin-tossing tasks are often related to the representativeness heuristic, which refers to a mental shortcut that is used to judge randomness by evaluating how well a set of random events represents the typical example for random events we hold in our mind. Representative thinking is explained by our tendency to seek for patterns in our surroundings. In the present study, predictions of coin-tosses of 302 third-graders were explored. Findings suggest that in third grade of elementary school, children make correct as well as different types of erroneous predictions and individual differences exist. Moreover, erroneous predictions that were in line with representative thinking were positively associated with an early spontaneous focus on regularities, which was assessed when they were in second year of preschool. We concluded that previous studies might have underestimated children's reasoning about randomness in coin-tossing contexts and that representative thinking is indeed associated with pattern-based thinking tendencies.

The article introduces a mathematics education measure in response to the COVID-19 epidemic in China and explores students' degree of approval and perception of digital equity towards the response. After the outbreak of the COVID-19, (NCPM) committee had developed a series of micro classes (abbreviated as NCPM micro classes), and more than 25 million teachers and students in China watched the NCPM micro classes during the 3 months social isolation. Then, students' degree of approval towards the NCPM micro classes and perception of digital equity were examined after social isolation. A total of 132,740 pieces of data were collected from Chinese primary school students. Quantitative analysis of student's degree of approval towards different parts of NCPM micro classes indicated that the introduction, interaction, summary and consolidation, curriculum characteristics, and goal achievement parts of the NCPM micro classes have received high approval from students and students with higher former achievement perceived a higher degree of approval towards the NCPM micro classes. Furthermore, we found that gender, socioeconomic status, school location, and learning location had no significant impact on students' degree of approval, indicating a digital equity exists. This study helps researchers or educators understand the mathematics education response to the COVID-19 outbreak in China and extends our understanding of primary students' degree of approval and perception of digital equity with these online classes.

When language is defined narrowly in mathematics classrooms, racially and linguistically minoritized students in classrooms could be systematically positioned as "learners of deficiency." Recent scholarship calls for expanding the notion of language to emphasize embodied expression of mathematical ideas. Taking a critical perspective to understand racialized experiences of using languages in disciplinary learning spaces, this article proposes the reconceptualization of embodiment as a language for racialized multilingual learners. This study was conducted in a Grade 1 classroom in a linguistically and racially diverse school in Canada. Through a series of professional development sessions, we worked with an experienced teacher to redesign the normalized and institutionalized pedagogy toward greater mobility of racialized multilingual learners' bodies, which was intertwined with their intellectual liberation. Focusing on the spatiality of pedagogy, the previously restrictive areas in the school were transformed into a place that augments embodied expression of mathematical ideas and agentive participation of minoritized learners. The analysis focused on the embodied discourse that participating racialized multilingual students used to actively engage in mathematical discussion. Our findings show that the designed pedagogy, characterized by the spatial and temporal expansion of the learning environment, offered more spaces for uncertainty and spontaneity with the decreased control of the teacher as an explicator. Our article furthers anti-colonial approaches to understand the intersection of racialized bodies and language in mathematics education.

Due to the COVID-19 pandemic in Shanghai, China, all school classes were delivered through an online environment from February 24 to May 22, 2020. To support this transition, the Shanghai Education Commission led expert teachers and specialists to develop a series of online video lessons based on the Shanghai unified curriculum, and suggested students watch the online video lessons individually from home, followed by an online synchronous lesson supported by class teachers. This study investigated what primary mathematics teachers learned from addressing these challenges through a case study. By following two purposefully selected teachers over 2 weeks during the transition, multiple data sets including online video lessons, online synchronous lessons, daily reflections, and post-online teacher interviews were collected. A fine-grained analysis of the data from the lens of the documentational approach to didactics found that teachers adaptively used online video lessons as important resources for their online synchronous lessons and virtual Teaching Research Groups as a teachers' collaboration mechanism supported them to develop online video lessons and address various technological constraints. Finally, implications of this case study for mathematics education globally are discussed.

In 2020, the emergency due to the COVID-19 pandemic brought a drastic and sudden change in teaching practices, from the physical space of the classrooms to the virtual space of an e-environment. In this paper, through a qualitative analysis of 44 collected essays composed by Italian mathematics teachers from primary school to undergraduate level during the spring of 2020, we investigate how the Italian teachers perceived the changes due to the unexpected transition from a face-to-face setting to distance education. The analysis is carried out through a double theoretical lens, one concerning the whole didactic system where the knowledge at stake is mathematics and the other regarding affective aspects. The integration of the two theoretical perspectives allows us to identify key elements and their relations in the teachers' narratives and to analyze how teachers have experienced and perceived the dramatic, drastic, and sudden change. The analysis shows the process going from the disruption of the educational setting to the teachers' discovery of key aspects of the didactic system including the teacher's roles, a reflection on mathematics and its teaching, and the attempt to reconstruct the didactic system in a new way.

This study investigates the characteristics of Japanese primary school mathematics lessons that adopt a problem-solving approach. We argue that these characteristics are reflected in three key aspects: collective teaching and learning, the lesson as a "drama" (i.e., its structured flow), and the focus on mathematical knowledge. In this paper, we refer to these aspects as the collective dimension, chronological dimension, and epistemological dimension of Japanese mathematics lessons, respectively. Using the Anthropological Theory of the Didactic (ATD) as a theoretical framework, we analyze the mathematics lessons of a Japanese primary school, focusing on the evolution of and the three geneses of mathematical knowledge (, , and ). Our aim is to highlight these dimensions and further identify the distinct features of Japanese lessons, as well as to demonstrate how, and to what extent, the evolution of effectively captures the structure and dynamics of the Japanese lessons.