Search

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]

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]

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]

There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the literature on the teaching of proof-based undergraduate mathematics, drawing on 104 published reports from a range of countries and research traditions. We primarily differentiate the papers into those which explore lecture-based pedagogy and student-centered pedagogy. For each type of instruction, we focus on three categories of findings from the literature: description of instruction, instructor beliefs and rationales, and the relationship between instruction and students (cognitive, participatory, affective, and equity oriented). Much is known about the enactment of lecture-based teaching, including instructors' cognitive and affective goals. The student-centered literature focuses on tensions and challenges implementing curricula with a greater focus on participatory goals. Overall, there are few studies that attempt to link instructors' classroom activity and students' learning. Similarly, attention to equity is relatively lacking in the extant research. • In the lecture setting, we have a rich understanding of what mathematicians do and why, but only limited work on its influence on student activity and outcomes. • In the student-centered setting, we have evidence of affective and participation benefits to students, but more work is needed on the teachers' role. • We propose a framework by which to conduct research on teaching in advanced mathematics in both lecture-based and student-centered formats. [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]

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]

This paper investigates data activities in an afterschool setting, offering a deeper understanding of the social nature of students' informal inferences by investigating how informal inferences are negotiated in group interactions, influenced by social norms, and how statistical concepts come into play in learners' informal inferential reasoning (IIR). Analyses take up a multi-sited orientation to investigate how youth used quantitative and contextual resources during a research activity to make meaning of data and negotiate emergent social tensions. Findings show how data activities that are part of informal inferential reasoning, such as collection, interpretation, generalization, inference, and representation unfolded as social, political, and personal. Implications call for designs for learning that better support working with data and understanding real-world phenomena and sociopolitical issues in ways that leverage youths' experiences, enabling them to take part in social action as critical community actors. • We illustrate the consequential nature of youths' negotiations of 'truths' about particular social phenomena (here, gender identity and bathroom restriction laws). • A multi-sited orientation illuminates how statistical concepts such as frequency counts and quantitative reasoning emerged as co-constructed aspects of meaning-making. • Analysis traces how youth used data activities to bridge their truths-under-negotiation with broader sociopolitical understandings of transgender people's experiences. • We show data activities, as part of informal inferential reasoning, such as collection, interpretation, and representation unfolded as social, political, and personal. • Implications focus on informal inferential reasoning as a crucial aspect of citizenship to be leveraged across and outside of disciplinary boundaries in designs for learning. [ABSTRACT FROM AUTHOR]

'Procedural' fluency in mathematics is often judged solely on numerical representations. 'Mathematical' fluency incorporates explaining and justifying as well as producing correct numerical solutions. To observe mathematical fluency, representations additional to a student's numerical work should be considered. This paper presents analysis of students' oral responses. Findings suggested oral responses are important vantage points from which to view fluency – particularly characteristics harder to notice through numerical work such as reasoning. Students' oral responses were particularly important when students' written (language) responses were absent/inconsistent. Findings also revealed the importance of everyday language alongside technical terms for observing reasoning as a fluency characteristic. Students used high modality verbs and language features, such as connectives, to explain concepts and justify their thinking. The results of this study purport that to gain a fuller picture of students' fluency, specifically their explanations or reasoning, students' oral responses should be analyzed, not simply numerical work. • Oral representations are useful for observing mathematical fluency. • Students' oral responses explain actions, concepts and reasons to show mathematical fluency. • Students use high modality words as evidence of well-developed levels of fluency. • Everyday verbs and connectives are important for observing reasoning as a fluency characteristic. • When students' oral and written representations are not aligned, observing mathematical fluency can be difficult. [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]

This paper analyses six years of tutor feedback produced after inquiries made by students in a second linear algebra course at a university mathematics support center (MSC). We utilized Mason's (2002) pedagogical tactics to build a model to analyze MSC tutors' feedback responding to these students' queries. The aim of this research was to investigate the nature of students' difficulties with concepts in a second linear algebra course that emphasizes theories and proof, in addition to examining the tactics employed by tutors to resolve student difficulties. We analyzed 227 feedback comments from 44 tutors based on their interactions with 82 students over six years. Our findings indicated that the most common areas of difficulty were basis, vector space, subspace, span, and proof. Tutor tactics deployed included 'being mathematical', 'simplifying and complexifying', and 'worked examples'. We also discuss some implications for linear algebra tutor training. [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]

When a classroom environment is designed to foster student sense-making and reasoning, resulting in collective argumentation, the teacher's support for collective argumentation requires adaptive teaching. This paper examines the teacher's adaptive support for argumentation during the teaching of mathematics in a digital environment that simulated a laboratory experiment. The teacher and twenty 15-year-old students engaged in multiple episodes of argumentation during the class. Adaptiveness of instruction is evidenced by the teacher's actions in support of collective argumentation both within and across episodes of argumentation. Within episodes, the kinds of questions and other supportive actions reveal the teacher's focus on adapting instruction with respect to the utterances, gestures, and observations of the students. Across episodes, the teacher's support for collective argumentation differed from the beginning of class to the end as evidenced by the number of questions asked in early episodes of argumentation and the preponderance of repeating actions later in the class session. • A teacher's micro-adaptations revealed by differences in support for argumentation. • Adaptive instruction apparent in responses to students' unexpected answers. • Purpose of arguments important to teacher's different supports for argumentation. [ABSTRACT FROM AUTHOR]

Although the notion of proof is important for all learners' mathematical experiences, there has been limited attention to what it might involve and look like to introduce students and prospective teachers to proof. In this paper we argue for the importance of having a coherent approach to introducing students and prospective teachers to proof, and we discuss the theoretical basis of a learning trajectory relevant to both groups. We also discuss an instructional sequence that aimed to promote the learning trajectory among English secondary students and U.S. prospective elementary teachers, drawing on data from two multi-year design experiments. The learning trajectory comprises two milestones: (1) seeing a need to learn about proof and (2) developing an operationally functional conceptualization of proof. The "need" in milestone 1 entails an aspect of epistemological justification applicable to both students and prospective teachers, and a further aspect of pedagogical justification applicable to prospective teachers. • It is unclear what it might involve and look like to introduce learners to proof. • A learning trajectory is proposed and its theoretical basis discussed. • The learning trajectory is applicable to both students and prospective teachers. • An instructional sequence was designed to promote the learning trajectory. • Data from two design experiments support the promise of the instructional sequence. [ABSTRACT FROM AUTHOR]

• Novice facilitators' changes in practices in two different countries are examined. • Despite the different contexts, changes in practices are found to be similar. • Practices are linked to facilitators' resources, orientations, goals and identity. • Managing mathematics teachers' PD discussions can be learned and developed. • Recommendations for supporting novice facilitators of are presented. Novice facilitators of professional development (PD) programs for mathematics teachers often face challenges in leading productive discussions and achieving the goals of these programs. Although research in this area is gradually accumulating, not much is known about how novice facilitators address these challenges and change their practices accordingly. This paper presents case studies of two novice facilitators of PD programs in two different countries. The analyses look at their work over one year, to illustrate the changes in their practices while managing discussions. The results show that although the facilitators operated in different contexts, their practices and their processes of change resembled, suggesting that these processes are not idiosyncratic. We argue that novice facilitators' changes in practices correspond to changes in their resources, orientations, goals, and identities and that PD program teams can support these changes. [ABSTRACT FROM AUTHOR]