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Over the past few decades, Piaget's forms of abstraction have proved productive for developing explanatory models of student and teacher knowledge, yet the broader applicability of his abstraction forms to mathematics education remains an open question. In this paper, we adopt the Piagetian forms of abstraction to accomplish two interrelated goals. Firstly, we analyze instructional tasks to develop hypothetical accounts of the abstractions that might occur during students' engagement with them. Secondly, we draw on middle- and secondary-grades classroom data to discuss the abstractions that occurred during the implementation of those instructional tasks. Because this paper represents an initial attempt at extending the applicability of Piagetian forms of abstraction, we close with potential implications of such use and possible avenues for future research. Most notably, we highlight the complexities involved in supporting abstraction through curriculum and instruction. • We examine task materials and implementation. • We differentiate between forms of abstraction. • We identify abstraction's viability for such analysis. • We differentiate between student learning outcomes. • We suggest avenues for future research. [ABSTRACT FROM AUTHOR]

Touchscreen devices have become prominent in a variety of educational contexts due to their use of hands-on and interactive learning styles, and particularly for their direct engagement, rapid orientation, and multi-touch affordances, which textbooks and oral instruction may not always provide. As such, over the past decade, both the number of touchscreen applications for mathematics learning and the body of research examining their efficacy have increased. Yet, there are still few existing systematic reviews of how these touchscreen-based mathematics applications promote K-12 student learning. In response, this paper provides an in-depth examination of extant studies which adopt practices that utilize touchscreen technology for K-12 mathematics teaching and learning. The Web of Science database was searched for research published between 2010 and 2023, and 34 papers were chosen to review the empirical bases of touchscreen-based mathematics learning. These articles were analyzed based on the educational contexts found, touchscreen actions prompted, and the mathematical meanings elicited from these actions. Most significantly, we propose a touchscreen action matrix that characterizes the actions performed on the touchscreen platform and presents the mathematical meanings derived from these action categories. We hope that this review and characterization provide guidance for researchers, designers, and educators in eliciting mathematical knowing with action-based design in touchscreen medium. • 34 studies of touchscreen-based mathematics applications are systematically reviewed. • Most apps target number concepts via gamified environments at the early primary level. • We proposed a 2 × 2 touchscreen action matrix to characterize actions on touchscreens. • Number of Input, Temporality, Spatiality & Speed are the four dimensions of interest. • We present findings of how mathematical meanings are elicited from touch taxonomies. [ABSTRACT FROM AUTHOR]

In this conceptual paper, we explore a framework of reasoning practices for decision-making in context. We extend prior work related to socioscientific issues (SSI) (Sadler et al., 2007) to consider the applicability of this framework to sociomathematical issues, specifically using the context of fairness in political (re)districting. We illustrate the usefulness of the SSI framework for sociomathematical issues drawing on student work and reflections from two undergraduate courses. We conclude by suggesting adjustments to reasoning practices of the SSI framework that might uniquely reflect the nature of sociomathematical reasoning. We discuss implications of our findings for conceptualizing reasoning practices that include mathematical perspectives within social and political contexts and for mathematics education generally. • Reasoning about a societal issue consists of intertwining personal, practical, contextual, and mathematical meaning. • A framework based on Sadler, Barab & Scott's (2007) analysis of reasoning practices regarding socioscientific issues is useful for analyzing emergent student reasoning in the context of redistricting, a sociomathematical issue. • Students reason in multiple, sophisticated ways about the fairness of specific maps and redistricting processes. [ABSTRACT FROM AUTHOR]

Despite much research exploring affect within a mathematics context, poor student engagement in mathematics persists internationally. There is limited research examining mathematics classroom practice and students' affect, with the potential to inform teachers' practice. This scoping review analysed this limited literature. Of 250 papers analysed, only 26 papers adequately described classroom practices, considered students' affective state in response to the practice, and considered mathematics learning outcomes. Collectively, these explore very few of the practices used in mathematics classrooms, highlighting the need for more and improved practice-focussed research to better inform efforts to improve student engagement in mathematics. • Despite extensive mathematics affect research the problem of poor student engagement in mathematics persists internationally. • This scoping review demonstrates there is limited research examining classroom practices, affect and mathematics outcomes. • Improved focus on classroom practice in research examining mathematics and affect is required. • The explanatory model presented describes the interaction of students' affect, classroom practices, and mathematical learning. [ABSTRACT FROM AUTHOR]

Drawing on data from the Republic of Cyprus, this paper uncovers elementary teachers' epistemological beliefs about mathematics. Twenty-two experienced teachers were invited to individual semi-structured interviews. Thematic data-driven analyses identified three themes and eight sub-themes, which I discuss, taking their socio-cultural context into consideration. This study suggests that applying predetermined frameworks directly taken from the literature when examining teachers’ epistemological beliefs in mathematics can be problematic, as they might hinder other culturally specific beliefs from emerging. In closing, this paper presents some implications for the results on teacher education and professional development, as well as ideas for future research. [ABSTRACT FROM AUTHOR]

• Closely examining the mathematical details of classroom interaction is essential. • Engaging in details of others' mathematical ideas relates to achievement. • Productive teacher support depends on following up on invitations to students. • Interaction varies across classroom participation structures. This paper traces the development of a program of work that seeks to understand the teacher practices that promote productive dialogue and learning in mathematics classrooms. This work originated in cooperative/collaborative learning research, with a focus on understanding group dynamics effective for student learning. Over time, it layered on increasing attention to the teacher's role, and branched out to consider the multiple participation structures in the classroom (whole-class discussion, small-group collaborative work, private student-student conversations). Each phase of this work brought in a heightened focus on the details of classroom interaction around mathematics—both in the thinking that students share and in the practices teachers use to support student participation—to better understand the development of students' mathematics learning. [ABSTRACT FROM AUTHOR]

Abstract How teacher characteristics are related to student outcomes may indicate ways to improve the effectiveness of schooling. This paper examines how consistent such relations are by utilizing TIMSS data on five teacher characteristics (teacher education major, teaching experience, professional development, preparedness and instructional quality) and two outcomes (achievement, motivation) in five countries (England, Norway, South Korea, Thailand, Tunisia) for two grades (4, 8) and two subjects (mathematics, science). Data revealed little consistency, if at all within countries only and regarding instructional quality as a predictor. Policy makers are advised not to make inferences across grades, subjects or outcomes. Highlights • Consistency of TIMSS results across subjects, grades, outcomes, countries was examined. • Instructional quality was consistently positively related to student outcomes. • Other teacher characteristics were mostly inconsistently related to student outcomes. • Instructional quality predicted student outcomes well and with large effect sizes. • Effect sizes of other teacher characteristics were generally moderate only. [ABSTRACT FROM AUTHOR]

Highlights • Psychological operations proposed for product of measures problems. • Differences in how students structure and relate pairs to composite units in combinatorics problems presented. • Differences in how students produce and use arrays presented. Abstract Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students' reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007). [ABSTRACT FROM AUTHOR]

When students share a computer in a mathematics class, the types of interactions that constitute collaboration can vary from more typical group work settings. The way that students position themselves towards one another through utterances and exchanges has implications for how students collaborate. In this paper I illustrate a method that uses techniques from Systemic Functional Linguistics (SFL) to analyze how collaboration can be traced to strings of individual utterances and acts. Drawing on positioning theory and techniques from SFL, I pose the question, how can episodes of collaboration be operationalized through individual utterances and actions? These methods allow for comparison between different models of computer-based interaction and suggest how to foster collaboration in technology-rich settings. Additionally, this study suggests how a broad phenomenon such as collaboration could be described and measured by considering collaborative episodes on a small scale. [ABSTRACT FROM AUTHOR]

The number of students who leave U.S. schools mathematically underprepared has prompted widespread concern. Low achieving students, many of whom have been turned off mathematics, are often placed in low tracks and given remedial, skills-oriented work. This study examines a different approach wherein heterogeneous groups of students were given responsibility and agency and asked to engage in a range of mathematical practices collaboratively. The teaching intervention, which was introduced in the first paper, took place as part of a summer class on algebra, and it gave students the opportunity to participate with mathematics in changed ways. This paper will report evidence that the vast majority responded with increased engagement, achievement, and enjoyment. The students chose collaboration and agency as critical to their improved relationships with mathematics. [ABSTRACT FROM AUTHOR]

Assessment is a key component of all educational systems, and automatic online assessment is becoming increasingly common for formative work in mathematics. This paper reports an investigation of the extent to which contemporary automatic assessment software can automatically mark answers to questions from existing high-stakes mathematics examinations. The questions are taken from a corpus of publicly available core mathematics questions designed for high-achieving students aged approximately eighteen at the school–university interface. We focus on the extent to which objective properties of each final answer may be automatically established and the extent to which automatic marking reasoning by equivalence supports assessment of students' methodology. Our results show that transcribing existing paper-based mathematics examinations into an electronic format is now feasible for a significant proportion of the questions as currently assessed. The most significant barrier to using contemporary automatic assessment is the requirement from examiners that students provide evidence that they have used an appropriate method. [ABSTRACT FROM AUTHOR]

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics. 1. Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions. 2. Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world. 3. Frame 3: Mathematics as a verb (not a noun), a human activity, part of one's identity. After introducing the frames and examining their distinction and their overlap, I discuss their implication with respect to student-centered classroom, context, and culture. • Explicating the definition of mathematics is essential for discussions among educators. • Different definitions of mathematics lead to different implications on education. • Different definitions of mathematics lead to different versions of student-centered mathematics instruction. [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]

The Virginia Mathematics Specialist Initiative (VMSI) is an ongoing collaborative effort among all components of the mathematics and mathematics education community of Virginia spanning more than two and a half decades. This paper describes the use of extensive collaborations to reach shared beliefs about teaching and learning mathematics with the goal of improving student learning; accomplished by preparing and supporting school-based mathematics specialists to coach K-8 teachers. A foundation built on shared beliefs enabled community members to reach consensus on this goal and continues to fuel the accomplishments of the VMSI. Several accomplishments are discussed including identification of the role of K-8 mathematics specialists, the attainment of a Virginia licensure add-on endorsement; and development of a model for a Virginia Mathematics Specialist Preparation Program (VMSPP). Also discussed are the annotated syllabi for core mathematics content and leadership courses in the VMSPP and tools to sustain and extend the VMSI. [ABSTRACT FROM AUTHOR]

Students with learning disabilities are often excluded from conversations about high-quality mathematics instruction as well as conversations about equitable mathematics instruction. As such, they may not be provided many opportunities to engage in mathematics with teachers who implement practices that the field of mathematics education would consider "high-quality" or "equitable". In this paper, we examine how three students with learning differences participate in instruction that demonstrates aspects of high quality and equitable mathematics instruction. Our findings outline practices that appeared to support increased opportunities for students to demonstrate their own reasoning and sense making in mathematics. Contributions of this work include illuminations of specific pedagogical moves that may support students identified as having learning disabilities to engage in more student-centered, conceptually-oriented mathematics. • We examine how three students with learning disabilities participate when math instruction is designed to be student-centered. • Practices that aimed for equity & were conceptually-oriented appeared to improve sense making opportunities for the students. • This work outlines specific pedagogical moves that supported these students in using their own reasoning while "doing math. [ABSTRACT FROM AUTHOR]

National reports have documented deficiencies in the vertical alignment of mathematical learning in K-12 education. Many students fail to master requisite concepts before advancing to more complex ideas, leaving them ill-prepared to succeed in higher level Science, Technology, Engineering, and Mathematics (STEM) coursework. In this paper, we model elementary and middle school students’ performance in mathematics over time as a stochastic process to forecast their proficiency by the eighth grade. We conduct an extensive examination of tens of thousands of student records and extract useful information. We use this data to present a longitudinal analysis of student performance on the North Carolina End-of-Grade mathematics exam and use Markov chain models to probabilistically characterize the movement of students’ scores from one grade level to the next. This work is the first step in developing a framework to forecast individual students’ development of mathematical knowledge over time. [ABSTRACT FROM AUTHOR]

“Behavior which is effective only through the mediation of other persons has so many distinguishing dynamic and topographical properties that a special treatment is justified and indeed demanded” ( Skinner, 1957 , p. 2). Skinner’s demand for a special treatment of verbal behavior can be extended within that field to domains such as music, poetry, drama, and the topic of this paper: mathematics. For centuries, mathematics has been of special concern to philosophers who have continually argued to the present day about what some deem its “special nature.” Two interrelated principal questions have been: (1) Are the subjects of mathematical interest pre-existing in some transcendental realm and thus are “discovered” as one might discover a new planet; and (2) Why is mathematics so effective in the practices of science and engineering even though originally such mathematics was “pure” with applications neither contemplated or even desired? I argue that considering the actual practice of mathematics in its history and in the context of acquired verbal behavior one can address at least some of its apparent mysteries. To this end, I discuss some of the structural and functional features of mathematics including verbal operants, rule-and contingency-modulated behavior, relational frames, the shaping of abstraction, and the development of intuition. How is it possible to understand Nature by properly talking about it? Essentially, it is because nature taught us how to talk. [ABSTRACT FROM AUTHOR]

Failure in US high school courses is common, but little is known about its effects. This paper investigates the extent to which course repeaters in high school mathematics courses exert negative externalities on their course-mates. Using individual and school-specific course fixed effects to control for ability and course selection, it shows that increasing the share of repeaters in a given course results in a moderate, significant increase in the probability of course failure for first-time course-takers. Results suggest that the negative effect is only evident when the share of repeaters reaches a threshold of five to ten percent of the total number of course-takers. The possibility that grading to a curve generates the effect cannot be ruled out, but is not fully supported in the data. Evidence is also presented that course repetition externalities may be distinct from low-ability peer effects. [ABSTRACT FROM AUTHOR]

• Mathematicians discussed common teaching practices as determined by research in undergraduate math education. • Analysis of participants' responses reveal goals and orientations that account for their use of these common teaching practices. • Some goals, like preparing future mathematicians, have so far received limited attention in the math education research literature. • Recommendations are made for math educators that wish to change the teaching practices of mathematicians. Despite mathematics educators' research into more effective modes of teaching, lecture is still the dominant mode of instruction in undergraduate mathematics courses. Surveys suggest this is because most mathematicians believe this is the best way to teach. This paper answers a call by mathematics education researchers to explore mathematicians' needs and goals concerning teaching. We interviewed eight mathematicians about findings in the mathematics education research literature concerning common pedagogical practices of instructors of advanced mathematics classes: "chalk talk," the presentation of formal and informal content, and teacher questioning. We then analyzed the responses for resources, orientations, and goals that might influence the participants to engage in these practices. We describe how participants believed common lecturing practices allowed them to achieve their goals and aligned with their orientations. We discuss these findings in depth and consider what implications they may have for researchers that aim to change mathematicians' teaching practices. [ABSTRACT FROM AUTHOR]