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In mathematics, for professionals, the most compelling way to back a statement is to prove it. This could even be true, as recent research indicates, for 9-year-olds (or for even younger learners), given appropriate conditions in the early grades. In this article, based on data drawn from the three preceding papers, I explore some ways in which the building of an argument can emerge in step with the development of understanding. [Copyright &y& Elsevier]

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]

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]

Students make sense of mathematical ideas using a variety of representations including physical models, pictures, diagrams, spoken words, and mathematical symbols. As students’ understanding of mathematical ideas becomes more general and abstract, there is a need to express these ideas using mathematical notation. This paper describes students’ movement from model building and personal notations to elegant use of mathematical symbols that show their understanding of advanced counting ideas. Specifically, this paper shows how earlier ideas from investigations of specific combinatorics problems (questions about making pizzas with different toppings and using cubes to build towers) are retrieved and built upon using the formal mathematical register to explain the meaning of Pascal's Identity, the addition rule of Pascal's Triangle. This analysis also shows the power of shared communication in mathematical problem solving. [ABSTRACT FROM AUTHOR]

This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously. [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]

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed. [ABSTRACT FROM AUTHOR]

• We examine the relationship between teacher meanings and conventions. • Teachers often do not dissociate conventions from logico-mathematical operations. • We provide implications for teaching in relation to conventions of graphing. In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals' meanings for graphs and related topics. Specifically, we draw on data from pre-service (PST) and in-service (IST) teachers to characterize such relationships. We use PSTs' responses during clinical interviews to illustrate three themes: (a) some PSTs' responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs' meanings; (b) some PSTs' responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs' responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs. [ABSTRACT FROM AUTHOR]

• Mathematicians may value pedagogical content knowledge but lack confidence in teaching for its development. • Mathematicians and mathematics educators may value pedagogical content knowledge for similar reasons. • Expectancy-value theory may explain a lack of focus on developing teachers' pedagogical content knowledge. Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancy-value theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interview-based study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers' PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers' PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians' teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values. [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]

Abstract: This paper presents a case study of a highly successful student whose exploration of an advanced mathematical concept relies predominantly on syntactic reasoning, such as developing formal representations of mathematical ideas and making logical deductions. This student is observed as he learns a new mathematical concept and then completes exercises about it. The paper focuses on how Isaac developed an understanding of this concept, how he evaluated whether a mathematical assertion is true or false, how he generated counterexamples to disprove a statement, and the general role examples play for him in concept development and understanding. [Copyright &y& Elsevier]

Abstract: This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student''s problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers. [Copyright &y& Elsevier]

Abstract: The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field. [Copyright &y& Elsevier]

Abstract: This paper describes insights on how to promote mathematical reasoning in problem solving based on the mathematical experiences of participants in a long-term study in which the students engaged in strands of well-defined, open-ended mathematical investigations, as a context for research on the development of particular concepts and ways of reasoning. Over the years, the students demonstrated ways of working in which sense making became a cultural norm and collective and individual sharing and justifying of ideas was a common practice. The paper examines the environment that enhanced the development of these and other qualities. The insights address aspects of task design and researcher role in the students’ mathematical activity. Both were central in enhancing the students’ engagement in thoughtful and meaningful problem-solving activity. The paper addresses the relationship between problem solving and mathematical reasoning from the perspective of the long-term study and provides an examination of students’ problem solving over time, both from a behavioral and introspective perspective. [Copyright &y& Elsevier]

In mathematics, generalizations are the end result of an inductive zigzag path of trial and error, that begin with the construction of examples, within which plausible patterns are detected and lead to the formulation of theorems. This paper examines whether it is possible for high school students to discover and formulate generalizations similar to ways professional mathematicians do. What are the experiences that allow students to become adept at generalization? In this paper, the mathematical experiences of a ninth grade student, which lead to the discovery and the formulation of a mathematical generalization are described, qualitatively analyzed and interpreted using the notion of uniframes. It is found that reflecting on the solutions of a class of seemingly different problem-situations over a prolonged time period facilitates the abstraction of structural similarities in the problems and results in the formulation of mathematical generalizations. [Copyright &y& Elsevier]

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs. [Copyright &y& Elsevier]

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]

The extensive use of National Completion Examinations in Mathematics (NCEM) as a critical filter for educational and social advancement highlights the tension between society’s ambition for high achievement and its ambition for general mathematical literacy. This paper presents an innovative model for a NCEM that demonstrates how such examinations can serve to promote mathematical literacy for all, without losing its ability to promote high achievement. In the course of the paper, the meaning of mathematical literacy for all is also discussed; it is stressed that at the heart of this notion lies students’ openness to mathematics, so that the success of a program directed towards mathematical literacy for all is reasonably measured by the number of students who persist in their study of mathematics and who develop a confident attitude towards it. Data were given suggesting that the new model for the NCEM is successful in this respect. [Copyright &y& Elsevier]

This paper investigates the understanding of a student who, on paper, has presented a picture-perfect solution to a calculus problem. Using a video recording of her work, an analysis of the process through which she went in determining the solution was undertaken using two different theoretical frameworks: Schoenfeld’s levels of analysis and structure and the Pirie–Kieren dynamical theory for the growth of mathematical understanding. It was determined that algebraic formalizations can lead to correct solutions without the student being aware of the relationships among various components of the question. Full understanding can only be achieved if a student has connected knowledge of the various components. It is suggested that “taken as shared” imagery may exist in both understanding and misunderstanding. If teachers are aware of possible misconceptions of students, this may help them to improve instruction. [Copyright &y& Elsevier]

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]

• Summary of interdisciplinary validation process framed on learning trajectories. • Working at scale in an assessment system allows for better understanding of patterns. • Classroom assessments to diagnose student progress in mathematics learning. • Importance of collaboration among learning sciences and psychometrics. • Using measures of learning trajectories to support learner-centered instruction. The paper reports on the design and validation argument for classroom assessments within a digital diagnostic assessment system built on learning trajectories (LTs). It consists of a learning map of nine big ideas, 25 relational learning clusters, and 62 LTs for grades 6−8 mathematics. Students take cluster assessments, and teachers use the data to adapt instruction. An ongoing validation process is presented with data for an algebra cluster. Validation among practitioners, learning scientists, and psychometricians is conceptualized as examining and adjusting inter-level, intra-level, and construct-irrelevant variation in measures of item difficulty and deploying item response theory modeling followed by sequential regressions. Using data from 37,000 assessments collected over three years at 3 middle schools, 167 potentially non-conforming items of the 676 calibrated items (24 %) were identified and revised. The paper discusses how the trajectories and map were refined through a combination of data analysis and feedback from practitioners. [ABSTRACT FROM AUTHOR]

• Mathematical Knowledge for Teaching for secondary mathematics teachers. • Design of tasks for pre-service mathematics teachers. • Mathematics as a human activity. This paper describes theoretical design principles emerging from the development of tasks for standard undergraduate mathematics courses that address applications to teaching secondary mathematics. While researchers recognize that mathematical knowledge for teaching is a form of applied mathematics, applications to teaching remain largely absent from curriculum resources for courses for mathematics majors. We developed various materials that contain applications to teaching that have been integrated into four standard undergraduate mathematics courses. Three primary principles influenced the design of the tasks that prepare future teachers to learn and apply mathematics in a manner central to their future work. Additionally, this paper provides guidance for instructors desiring to develop or implement similar applications. The process of developing these tasks underscores the importance of key features regarding the roles of human beings in the tasks, the intentional focus on advanced content connected to school mathematics, and the integration of active engagement strategies. [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]

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]

This paper reports on part of an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function. Employing APOS theory as our guiding theoretical perspective, we offer a genetic decomposition for the concept of parametric function, and we explore students’ reasoning about an invariant relationship between two quantities varying simultaneously with respect to a third quantity when described in a real-world problem, as such reasoning is important for the study of parametric functions. In particular, we investigate students’ reasoning about an adaptation of the popular bottle problem in which they were asked to graph relationships between (a) time and volume of the water, (b) time and height of the water, and (c) volume and height of the water. Our results illustrate that several issues make reasoning about relationships between variables a complex task. Furthermore, our findings indicate that conceiving an invariant relationship, as it relates to the concept of parametric function, is nontrivial, and various complimentary ways of reasoning are favorable for developing such a conception. We conclude by making connections between our results and our genetic decomposition. [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]

In this paper five classrooms at grades 5–7 (students aged 11–14 years) are studied for one week each during their mathematics lessons. The aim is to study the students’ comments in order to develop categories describing the comments’ different contributions to the mathematical discourse. The main categories developed are student initiatives, explanations, partial answers, teacher-led responses and unexplained answers. The practices analysed are all dominated by the IRE pattern (Initiation–Response–Evaluation), and the different categories of student comments can be seen as a description of the different types of ‘R’ (student response) from the IRE pattern. This also illustrates that different patterns can be hidden behind the IRE-label. The categories can be used to study student comments on a turn-by-turn basis, describing different types of student contribution to the mathematical discourse. [ABSTRACT FROM AUTHOR]

Highlights: [•] Four interrelated case studies of mathematics faculty members teaching proof courses. [•] Observations of actual teaching are combined with interviews of the instructors. [•] The framework developed in this paper describes example usage in proof presentations. [•] The uses of examples are classified by the instructors’ pedagogical intentions. [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]

Abstract: The purpose of this paper is to illustrate cognitive challenges introduced by Common Core State Standards for Mathematics (2010) with regard to conceptualizing fractions. We focus on a strand of standards that appear across grades three through five, which is best represented in grade four, by standard 4.NF.4a: “[Students should] understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5×(1/4), recording the conclusion by the equation 5/4=5×(1/4).” We argue that supporting such conceptualizations is a critical challenge for teachers and researchers to address in US education, but one fraught with conceptual hurdles for students. We adopt a scheme theoretic perspective and share the case study of a sixth-grade student, to illustrate the ways of operating students need to develop for meaningful attainment of the new standards. [Copyright &y& Elsevier]

Abstract: This paper reports results from a written assessment given to 290 third-, fourth-, and fifth-grade students prior to any instructional intervention. We share and discuss students’ responses to items addressing their understanding of equation structure and the meaning of the equal sign. We found that many students held an operational conception of the equal sign and had difficulty recognizing underlying structure in arithmetic equations. Some students, however, were able to recognize underlying structure on particular tasks. Our findings can inform early algebra efforts by highlighting the prevalence of the operational view and by identifying tasks that have the potential to help students begin to think about equations in a structural way at the very beginning of their early algebra experiences. [Copyright &y& Elsevier]

Abstract: Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the cognitive resources (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to examine students’ understanding of the limiting behavior of a function from to at a point P This understanding depends on which definition is used for a limit. Several definitions are considered; two of these concern the notion of a neighborhood of P, while another two are directed at the consistency of limits obtained by restricting the function to lines or half-lines passing through P. A case study is presented involving four university students studying Mathematics. Comments are made about their abilities in working with each definition, associated images that were evoked, and how they related one definition to another (including the issue of logical equivalence). The influence of the students’ previous experience in handling limits for real functions of one variable is also discussed. [Copyright &y& Elsevier]

Abstract: This paper presents a study of the opportunities presented to students that allow them to learn different types of mathematical reasoning during teachers’ ordinary task solving presentations. The characteristics of algorithmic and creative reasoning that are seen in the presentations are analyzed. We find that most task solutions are based on available algorithms, often without arguments that justify the reasoning, which may lead to rote learning. The students are given some opportunities to see aspects of creative reasoning, such as reflection and arguments that are anchored in the mathematical properties of the task components, but in relatively modest ways. [Copyright &y& Elsevier]

Abstract: This paper focuses on three junior high school mathematics teachers and their knowledge of the nature of definitions. The mathematical context of exponentiation is used as a springboard for discussing two aspects of definitions: their corresponding domains and the distinction and relationships between definitions, proofs, and theorems. Through interviews it was shown that some teachers are not aware that definitions and domains are intrinsically connected and some teachers believe that definitions may be proved. Findings also indicate that knowledge of the nature of definitions may be dependent on the context. [Copyright &y& Elsevier]

Abstract: Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated. [Copyright &y& Elsevier]

Abstract: This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically. [Copyright &y& Elsevier]

Abstract: This paper reports the results of an exploratory study of the perceptions of and approaches to mathematical proof of undergraduates enrolled in lecture-based and problem-based “transition to proof” courses. While the students in the lecture-based course demonstrated conceptions of proof that reflect those reported in the research literature as insufficient and typical of undergraduates, the students in the problem-based course were found to hold conceptions of and approach the construction of proofs in ways that demonstrated efforts to make sense of mathematical ideas. This sense-making manifested itself in the ways in which students employed initial strategies, notation, prior knowledge and experiences, and concrete examples in the proof construction process. These differences were also seen when students were asked to determine the validity of completed proofs. These results suggest that such a problem-based course may provide opportunities for students to develop conceptions of proof that are more meaningful and robust than does a traditional lecture-based course. [Copyright &y& Elsevier]

Abstract: Despite general agreement that students should consider more than one way to solve a complex mathematics problem, this practice is rarely observed in U.S. classrooms. In this paper, we examine data that trace the year-long professional development experience of 12 veteran teachers of middle grades mathematics — experienced users of a standards-based mathematics curriculum — to unearth some issues and concerns that appeared to serve as obstacles to their use of multiple solutions in the classroom. We also identify key transformations that appear to account for the success of some teachers in embedding this practice in their instructional repertoire. [Copyright &y& Elsevier]

Abstract: This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs. [Copyright &y& Elsevier]

Abstract: This paper reports on the mathematical thinking of participants of a long-term study, now in its 17th year, who did mathematics together through their public school and early university years. In particular, it describes how fundamental ideas and images of a cohort group of students are elaborated and presented in symbolic expressions of generalized mathematical ideas while exploring problems in grades 10 and 11. From high school and university interview data, we learn from participants how they viewed their mathematical activity in structuring their investigations and justifying their solutions. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study some of the strategies that are possible to use in order to solve the exercises in undergraduate calculus textbooks. It is described in detail how most exercises may be solved by mathematically superficial strategies, often without actually considering the core mathematics of the book section in question. [Copyright &y& Elsevier]

The purpose of this paper is to illustrate a methodological approach for empirically investigating the function of gesturing in the collective development of knowledge. We extend the earlier work of Stephan and Rasmussen [Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior 21, 459–490] who analyzed classroom discourse and symbolizing to document the emergence of six classroom mathematical practices over the course of 22 days of instruction on first-order differential equations. We complement and extend this previous analysis by re-examining the same data for gesturing and coordinate this analysis with the evolution of the classroom mathematical practices as they developed in this particular community of learners. Our illustration of the methodology we developed suggests that (1) gestures and argumentation can function as a unit that supports the establishment of one or more taken-as-shared ideas, and (2) that a gesture/argumentation pair that develops while establishing one practice can change function to support the establishment of ideas embedded in other classroom mathematical practices. [Copyright &y& Elsevier]