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Introduction: There is a critical need to develop innovative educational strategies that engage youth in meaningful mathematics learning, particularly students from groups that have been historically marginalized in science, technology, engineering, and mathematics (STEM). In this study, we explore youths' participation in two collaborative projects from the Growing Mathletes curriculum which combines baseball contexts and mathematics. Our goal was to understand the potential of these projects to support youths' engagement with mathematical ideas and practices, and the extent to which youth leveraged a range of resources, including prior experiences and funds of knowledge, to inform their decisions and understanding. Methods: The Design a Stadium and Baseball Team Roster projects were implemented in two afterschool setting sites and two summer program sites with 102 youth of all genders in grades 3 to 8. Data sources included video recordings of youth participation in the project, project artifacts, and youth interviews. Results: We found the projects contained specific features that supported youths' engagement in three specific mathematical practices: (1) make sense of problems and persevere in solving them, (2) reason abstractly and quantitatively, and (3) construct viable arguments and critique the reasoning of others. Additionally, there is evidence that while engaging in these projects youth drew on their own funds of knowledge to inform their decisions and understanding. Conclusion: Our findings point to key implications for researchers, educators, and curriculum developers in informal STEM learning spaces. [ABSTRACT FROM AUTHOR]

The Common Core Standards for Mathematical Practice describe mathematical habits of mind foundational for mathematical thinking and understanding. The Statistical Education of Teachers (SET) report, commissioned by the American Statistical Association (ASA), interprets the Mathematical Practices through a statistical lens. In this article, we present tasks focused on variability that can be taught through the lens of Mathematical Practice 6: Attending to Precision (MP6). We show that teacher tasks that focus on variability and its quantification (by computing multiple different measures on different aspects of an entire dataset and constructing arguments), prompts teachers to attend to statistical precision. Additionally, we make the connection that ideas related to variability are present in all components of the statistical investigative process: posing questions that anticipate variability, anticipating variability in data collection, analyzing the variability present, and interpreting the variability to make quantitative statements about error and tasks that aim toward understanding variability require the employment of Statistical MP6.

IntroductionThere is a critical need to develop innovative educational strategies that engage youth in meaningful mathematics learning, particularly students from groups that have been historically marginalized in science, technology, engineering, and mathematics (STEM). In this study, we explore youths’ participation in two collaborative projects from the Growing Mathletes curriculum which combines baseball contexts and mathematics. Our goal was to understand the potential of these projects to support youths’ engagement with mathematical ideas and practices, and the extent to which youth leveraged a range of resources, including prior experiences and funds of knowledge, to inform their decisions and understanding.MethodsThe Design a Stadium and Baseball Team Roster projects were implemented in two afterschool setting sites and two summer program sites with 102 youth of all genders in grades 3 to 8. Data sources included video recordings of youth participation in the project, project artifacts, and youth interviews.ResultsWe found the projects contained specific features that supported youths’ engagement in three specific mathematical practices: (1) make sense of problems and persevere in solving them, (2) reason abstractly and quantitatively, and (3) construct viable arguments and critique the reasoning of others. Additionally, there is evidence that while engaging in these projects youth drew on their own funds of knowledge to inform their decisions and understanding.ConclusionOur findings point to key implications for researchers, educators, and curriculum developers in informal STEM learning spaces.

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Copyright of Estudios Filosóficos is the property of Estudios Filosoficos and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Offering explanations is a central part of teaching mathematics, and understanding those explanations is a vital activity for learners. Given this, it is natural to ask what makes a good mathematical explanation. This question has received surprisingly little attention in the mathematics education literature, perhaps because the field has no agreed method by which explanation quality can be reliably assessed. In this paper, we explore this issue by asking whether mathematicians and undergraduates agree with each other about explanation quality. A corpus of 10 explanations produced by 10 mathematicians was used. Using a comparative judgement method, we analysed 320 paired comparisons from 16 mathematicians and 320 from 32 undergraduate students. We found that both mathematicians and undergraduates were able to reliably assess the quality of a set of mathematical explanations. Furthermore, the assessments were largely consistent across the two groups. Implications for theories of mathematical explanation are discussed. We conclude by arguing that comparative judgement is a promising technique for exploring explanation quality. [ABSTRACT FROM AUTHOR]

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians' notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and claims in the literature regarding the explanatoriness of certain types of proofs. [ABSTRACT FROM AUTHOR]

Teachers' beliefs, professional development, access to hardware/software, and instructional contexts all influence teachers' use of technology (Ertmer et al., 2012; González-Carriedo & Esprívalo Harrell, 2018; Ottenbreit-Leftwich et al., 2010). The first three factors have been studied extensively, but additional research on the instructional contexts of teachers, especially those working in less common settings, may enhance our knowledge of how technology shapes teachers' instruction and, subsequently, student learning. This case study highlights three teachers' beliefs and practices of incorporating technology to enhance mathematics learning and mathematical discourse in public dual-language elementary schools. Through interviews, video-recorded lessons, and workshop discussions, researchers examined why and how teachers leveraged technology to enhance mathematics learning in a bilingual program. The analysis indicates that teachers' reasoning and technology use sought to increase opportunities for discourse, a significant component of the mathematical practices (National Governors Association Center for Best Practices, 2010), strengthening their bilingual students' understanding and application of mathematical concepts, processes, and proficiencies. [ABSTRACT FROM AUTHOR]

Education reforms in the United States and abroad have increased efforts to improve student interest and capacity in STEM (science, technology, engineering, and mathematics). Despite these attempts, students still have little opportunity to engage in STEM learning in K-12 education. This qualitative case study was designed to investigate how incorporating STEM into teacher education can promote STEM teaching as well as enhance math instruction. The study took place in the fourth year of an undergraduate teacher education program spanning a secondary math and science Methods course and student teaching. Guided by the framework of sensemaking, individual interviews, teaching artifacts, and written reflections for four teachers were analyzed to identify moments of dissonance that pushed participants to reach new understandings about the learning and teaching of math. Findings indicated that learning to teach math through the lens of STEM shifted pre-service teachers' instructional emphasis in two ways: 1) figuring out math vs. learning about math; and 2) teaching math through authentic STEM contexts as opposed to focusing purely on mathematics. However, experiences in student teaching can either enhance or stifle these gains. Findings suggest the role of teacher education in promoting STEM education by shifting prospective teachers' mindsets about mathematics and teaching. [ABSTRACT FROM AUTHOR]

This design research investigated the mathematical practices that eight graders (14-year-old students) developed when learning about prisms, cylinders, and their surface areas. An inquiry-based learning environment was created for students to engage in argumentation. The instruction was enriched by use of a dynamic geometry software, namely GeoGebra, to assist students in visualizing and reasoning about 3D shapes. Rasmussen and Stephan's three-phase method (2008) was used to organize the collected data, which in turn were analyzed using Krummheuer's model of argumentation (2015). The results indicated the emergence of three mathematical practices and several taken-as-shared ideas: (1) defining prisms, (2) computing the surface area of a prism, and (3) computing the surface area of a cylinder. The study's results revealed that students' understanding improved when learning concepts simultaneously with argumentation and dynamic geometry software. [ABSTRACT FROM AUTHOR]

Descriptions of mathematical thinking have an extended lineage. Sometimes accurate and sometimes not, sometimes misinterpreted and sometimes not, characterizations of mathematical thought processes have inspired and at times misled people interested in designing or framing mathematics instruction. Challenges the field faces in conceptualizing mathematics instruction include: What can be warranted as legitimate mathematical practices? Which aspects of mathematical practices are relevant and appropriate for K-16 instruction? What kinds of support are necessary? What is viable at scale? This paper provides a description of relevant history from the Western literature, bringing readers up to the present. It then addresses two key issues related to contemporary curricula: the framing of the mathematical enterprise as being fundamentally inquiry-oriented and the need for curricula and instruction to reflect such mathematical values; and the characteristics of mathematical classrooms that support students' development as powerful mathematical thinkers. An emphasis is on problem solving, a major component of "thinking mathematically." The paper concludes with a description of practices that are currently under-emphasized in instruction and that would profit from greater attention. [ABSTRACT FROM AUTHOR]

In their practice, teachers must not only know how to solve mathematics problems; they must also be able to make sense of students' mathematical thinking, understand the organization and intent of curricular materials, and select contexts to motivate and highlight mathematical ideas. Similarly, mathematics content courses for prospective teachers (PTs) should not only seek to convey mathematical content; they should prepare PTs to use mathematical knowledge in ways that enhance school teaching and learning of the subject. Accordingly, mathematics teacher educators (MTEs) should assess not only the mathematics that PTs know but also whether this mathematical knowledge is organized in ways that are likely to support their teaching. In this article, we present some of the existing research on the assessment of mathematical knowledge for teaching and discuss ways in which MTEs can draw upon the work of elementary school teaching to help assess PTs' content knowledge and habits of mind. These include assessments that focus on using representations that occur in elementary textbooks, building mathematical arguments, selecting problems to bring out important ideas, and making sense of students' thinking. [ABSTRACT FROM AUTHOR]

While research in mathematics education has shown that mathematics assessments are highly consequential, traditional assessments often lag behind advancements in instructional methods. One such advancement is the promotion of mathematical habits of mind such as students' abilities to critique others' reasoning. This study explored the use of student work embedded in seventh‐grade curriculum‐based mathematics assessment tasks as a mechanism for critiquing others' thinking. The researcher investigated the prevalence and nature of student work in assessment tasks as compared to textbook tasks from five seventh‐grade, Common Core State Standards for Mathematics‐aligned curriculum series. The text analyses findings revealed that while there were multiple critique types in student work across both assessment and textbook tasks, there were substantial differences between students' opportunities to make sense of someone else's mathematical thinking in curriculum‐based assessments as compared to the student textbooks. These findings reinforced prior curriculum and assessment research that found assessment often lags behind instructional methods. [ABSTRACT FROM AUTHOR]

This article aims to identify and analyze mathematical practices registered in school notebooks of a mixed state school in the city of Antônio Carlos (SC), in the 1930s and 1940s, focused on fractions. Michel Foucault's theorizations are used to guide theoretical and methodological precepts. The results of the research show mathematical practices developed in these schools obeying the Santa Catarina official programs of the time, with quick and succinct solutions and focused on their daily tasks. It is also observed that they are inserted in a historical context, between the Francisco Campos Reform, of 1931, and the beginning of the Modern Mathematics Movement, in the 1960s, in which the fraction receives a new approach, moving away from the relationship between number and measure and approaching the notion of part-whole. [ABSTRACT FROM AUTHOR]

Problem solving has long been a priority in mathematics education, and the first Common Core mathematical practice (SMP1) focuses on this priority through the language of "Make sense of problems and persevere in solving them." We present findings from a survey about how prospective elementary teachers' (PTs) make sense of potential difficulties with fostering SMP1. Findings suggested that PTs' common anticipated difficulties relate to planning a solution pathway and self monitoring whether the solution makes sense. Moreover, a third of PTs disclosed that their anticipated difficulties are linked to their own personal struggles with aspects of SMP1. An alternative interpretation of SMP1 surfaced in which a small number of PTs described SMP1 as necessitating that a teacher teach multiple solution methods to students, instead of engaging students in productive struggle to develop their own strategies. We present a framework illustrating the connections between SMP 1 and Pólya's problem solving phases, and we discuss how these findings connect to and build on previous research of PTs' experiences with problem solving. We offer implications for the targeted support needed in teacher preparation programs to address these struggles, to prevent them from being replicated in their students. [ABSTRACT FROM AUTHOR]

In this research, we explored the potential of using a research-based teaching and learning sequence to promote pupils’ engagement in practices that are coherent with those of real world mathematical and scientific activity. This STEM (Science, Technology, Engineering and Mathematis) sequence was designed and implemented by pre-service teachers and science and mathematics education researchers with the aim of modeling the growth of a real population of rabbits. Results show explicit evidence of pupils’ engagement in relevant mathematical and scientific practices, as well as detailed descriptions of mathematical connections that emerged from those practices. We discuss how these practices and connections allowed the progressive construction of models, and the implications that this proposal may have for STEM task design and for the analysis of extra-mathematical connections.

The purpose of the current article is to analyze the effects of the hypothetical learning trajectory designed for teaching quadrilaterals. This article reports the classroom mathematical practices that emerged in a social learning environment that was developing preservice middle school mathematics teachers' understanding of quadrilaterals. Ten preservice teachers participated in a five-week instructional sequence guided by the hypothetical learning trajectory designed by problem-based learning strategy about quadrilaterals. This teaching experiment study focused on their learning and understanding of quadrilaterals through argumentations. Using case study design, qualitative data was collected through whole class and peer group discussions, field notes, written worksheets, group meetings, and interviews. Data were analyzed by three-phase methodology of Rasmussen and Stephan (2008) based on Toulmin's model of argumentation (1969). Three mathematical practices emerged through the five-week instructional sequence: reasoning on definitions of quadrilaterals, reasoning on relationship among quadrilaterals and reasoning on properties of quadrilaterals. Moreover, it was observed that preservice teachers developed their learning and understanding of quadrilaterals by the hypothetical learning trajectory. [ABSTRACT FROM AUTHOR]

The discipline of mathematics values precision and teachers are accountable for promoting and supporting their students in attending to precision (ATP), which in the USA is an explicit standard for mathematical practice included in the Common Core State Standards for Mathematics. This study used thematic discourse analysis to examine how eight middle and high school teachers understand and interpret the mathematical practice of ATP through discussion within a teacher learning community. Findings suggest that teachers' talk prioritized the themes of precision with numerical quantities, precision with vocabulary, and precision with symbols. In many cases, these themes were discussed through examples from the teachers' experiences with students and the focus was on being precise (or not) rather than attending to issues of precision. Their discourse also highlighted the teachers' influential role in engaging their students in ATP and the relationship between ATP and student learning, with some teachers articulating a direct relationship of teachers explaining ATP to students and other teachers articulating a complex relationship of experiences via ATP. Overall, teachers' perspectives on ATP provide insight into how they create opportunities for students to engage in ATP in their classrooms and may inform the development of shared meaning about ATP across the field. [ABSTRACT FROM AUTHOR]

In this exploratory study, we examine how between-group movement, as an autonomy-promoting practice, might incentivize or disincentivize sixth-grade students' engagement in two mathematical practices: (1) making sense of problems and persevering in solving them; and (2) constructing viable arguments and critiquing the reasoning of others. Between-group movement refers to a pedagogical strategy wherein teachers allow groups to physically move within the classroom while problem-solving to discuss strategies, ask for help, or check their work with other groups. Exploring both the affordances and limitations of between-group movement, we found that between-group movement supported groups to construct viable justifications, among other sense-making mathematical practices. However, we also found that some groups over-scaffolded during between-group conversations which disincentivized meaningful engagement in mathematical practices. Furthermore, between-group movement revealed some equity concerns in relation to status-based privileges. The findings imply that between-group movement can be a constructive pedagogical practice under specific conditions. • Explores between-group movement as an autonomy-promoting teaching practice. • Between-group movement can support learning in elementary mathematics. • Limitations of between-group movement: potential inequity and over-scaffolding. • Between-group movement is constructive under specific conditions. [ABSTRACT FROM AUTHOR]

In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figure-making and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figure-making and sand drawing practices, from one society to another. Finally, we suggest that a philosophical-anthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.