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The emergence of the COVID-19 pandemic has brought increased attention to the critical mathematical literacy of citizens in the United States and around the world. A statistically and mathematically literate society is crucial for ensuring that citizens are able to sift through political rhetoric to maintain life-saving procedures such as social distancing and other infection dampening efforts. Additionally, recent civil unrest due to the disproportionate killings of Black men by police provokes investigation into the public's mathematical literacy. In this paper, we investigate adolescent students' critical mathematics consciousness and mathematics literacy as they reason through two interview tasks on the coronavirus and police shooting data. Drawing on Frankenstein's program of Critical Mathematics Education, we introduce an analytic framework for documenting the critical mathematics consciousness of adolescent students. We interviewed fifteen 14- to 16-year-old students as they solved five tasks designed to elicit their critical and ethical mathematical awareness. Our findings indicate that students exhibit very little critical mathematics consciousness in the context of the police problem but show awareness that data can be presented in ways that manipulate the public's emotions in the coronavirus problem. We conclude the paper with a discussion of implications for designing future instruction to support students' growth in critical mathematics consciousness. [ABSTRACT FROM AUTHOR]

Contemporary scholars describe mathematical modeling as a transformation of a real-world problem to a mathematical problem and back again. This paper treats a critical issue in the modeling process: how modelers determine if the transformation from the real world to mathematics was carried out well. I present an empirically derived typology of validating activities explaining how validating functions to ensure a mathematical model will yield a reasonably accurate prediction. The typology arose from analysis of four engineering undergraduates’ production of 276 instances of validating. The nuances of validating suggest that creating and maintaining relationships between reality and mathematics is more complex than a transformation and that we should afford a more prominent role to validation in the modeling process. [ABSTRACT FROM AUTHOR]

This paper discusses the importance of considering bilingual learners' non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper focuses on the word-use, gestures and dragging actions in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners' competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic, touchscreen environments, particularly for bilingual learners. [ABSTRACT FROM AUTHOR]

In a recent paper (Weber & Mejia-Ramos, Educational Studies in Mathematics, 76, 329-344, ), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians ( N = 118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians. [ABSTRACT FROM AUTHOR]

This paper presents the story of a mathematics teacher's transformation that resulted from her engagement in collaborative action research. The collaborative research, conducted by a university mathematics educator and an eighth-grade mathematics teacher (Marylin), centered on investigating the outcomes of implementing a `Hands-On Equations' approach to teaching algebra. This algebraic study, which is briefly reported herein, serves as a vehicle for examining the questions, reflections, and changes brought forth by Marylin throughout the collaborative process. Thus, the project of investigating the teaching and learning of algebra can be viewed as the `sub-study' within the broader study of the effects of collaborative action research on the mathematics teacher involved in the investigation. Within this paper, issues related to the broader study of collaborative action research in the mathematics classroom, that are connected to Marylin's transformation, are discussed. These issues include the characterization of and goals for collaborative action research as well as the challenges for collaborative action research in mathematics classrooms. [ABSTRACT FROM AUTHOR]

This paper investigates how three widely used calculus textbooks in the U.S. realize the derivative as a point-specific object and as a function using Sfard's communicational approach. For this purpose, the study analyzed word-use and visual mediators for the limit process through which the derivative at a point was objectified, and word-use and visual mediators for the derivative process through which the derivative as a function was objectified. The analysis highlighted inconsistency in realizations of the limit process and object, and derivative process and object found in some instances, and implicit discussion on relations among different visual mediators. Specifically, the results of this study led to four observations about issues suggested by one or more of the textbooks: a) visual mediators of the initial and final objects, and the visual mediators of these processes were often inconsistent; b) transformations of visual mediators from initial object, through process, to final object were often implicit both in one visual mediator and across multiple visual mediators; c) visual mediators for the limit processes were sometimes disconnected from the initial and final objects, or the connections were not explicit; and d) realizations of both the derivative at a point and the derivative of a function were mediated with nearly identical symbols suggesting a possible difficulty with understanding the difference between them. [ABSTRACT FROM AUTHOR]

This paper concerns proof presentation at the university level. We report on a study in which we observed ten mathematicians constructing or revising proofs for pedagogical purposes. We highlight the factors that they claimed to consider when completing these tasks. We found that intended audience and medium (lecture or textbook) influenced proof presentation. We also found that, although mathematicians generally valued pedagogical proofs featuring diagrams and emphasizing main ideas, these mathematicians did not always incorporate these aspects in the proofs they constructed or revised. [ABSTRACT FROM AUTHOR]

In this paper, we explored how engagement developed over time during a proportional reasoning unit for a group of US bilingual Latino/a students, with particular attention to aspects of social and cultural activity that supported students' engagement. Our findings suggest that student mathematical engagement developed primarily as a relational process characterized by students' social relations across time, their understandings about their relationships with mathematics, and the important relations emerging across proportional reasoning ideas. [ABSTRACT FROM AUTHOR]

The importance of students' problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students' mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students' mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students' mathematical content knowledge and their problem-posing abilities in mathematics. [ABSTRACT FROM AUTHOR]

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States. [ABSTRACT FROM AUTHOR]

This paper offers for discussion and critique a conceptual framework that applies a situative perspective on learning to the study of learning to teach mathematics. From this perspective, such learning occurs in many different situations -- mathematics and teacher preparation courses, pre-service field experiences, and schools of employment. By participating over time in these varied contexts, mathematics teachers refine their conceptions about their craft -- the big ideas of mathematics, mathematics-specific pedagogy, and sense of self as a mathematics teacher. This framework guides a research project that traces the learning trajectories of teachers from two reform-based teacher preparation programs into their early teaching careers. We provide two examples from this research to illustrate how this framework has helped us understand the process of learning to teach. [ABSTRACT FROM AUTHOR]

The mathematics community in the U.S. has become concerned about the state of doctoral education, including concerns about high attrition rates and the small numbers of women and students from some racial and ethnic groups. This paper proposes a model of doctoral student persistence and attrition, in which student participation in the life of the department and discipline lead to increased student integration, which is crucial for students' success. Ten faculty members and eighteen graduate students were interviewed about their interests, conceptions, and experiences within mathematics, in a case study of one mathematics department. In this department, students experienced four types of obstacles to their participation: obstacles stemming from the program structure, obstacles to participation in class, obstacles to participating with faculty outside of class, and obstacles stemming from faculty beliefs about teaching and learning. Implications for the retention of mathematics doctoral students are discussed. [ABSTRACT FROM AUTHOR]

Multi-base arithmetic blocks (MAB) are the most frequently used physical materials for teaching about decimal numbers, despite published reservations about their appropriateness. This paper presents an alternative, LAB (linear arithmetic blocks) and compares the two materials on the basis of epistemic fidelity and accessibility for students. Two teaching experiments involving 30 matched students indicated that LAB is considerably more accessible for students, and identify three contributing factors (LAB modeling number with length rather than volume, MAB incorporating an apparent dimensional shift and having prior use). Use of LAB was associated with more active engagement by students and deeper discussion. Epistemic fidelity is critical to facilitate teaching with the models, but we attribute the enhanced classroom environment to the greater accessibility of the LAB material. Further research is warranted, so that teaching of mathematics with physical materials can be improved. [ABSTRACT FROM AUTHOR]