Reconfiguring mathematical settings and activity through multi-party,whole-body collaboration

This study examines the consequences of whole-body, multi-party activity for mathematics learning, both in and out of the classroom. We develop a theoretical framework that brings together contemporary theories related to social space, embodied cognition, and mathematical activity. Then, drawing on micro-ethnographic and case-comparative techniques, we examine and juxtapose two cases of implementing whole-body, collaborative movement to engage learners in the mathematics of number sense and ratio and proportion. Analytically foregrounding the interdependence among setting, embodied activity, and mathematical tools and practices, we illustrate how whole-body collaboration can transform how learners experience learning environments and make sense of important mathematical ideas. The analysis enriches our understanding of the changing spatial landscapes for learning and doing mathematics as well as how re-instating bodies in mathematics education can open up new forms of collective mathematical sense-making and activity.     

The role of gestures in the mathematical practices of those who do not see with their eyes

In this paper, we aim to contribute to the discussion of the role of the human body and of the concrete artefacts and signs created by humankind in the constitution of meanings for mathematical practices. We argue that cognition is both embodied and situated in the activities through which it occurs and that mathematics learning involves the appropriation of practices associated with the sets of artefacts that have historically come to represent the body of knowledge we call mathematics. This process of appropriation involves a coordination of a variety of the semiotic resources??spoken and written languages, mathematical representation systems, drawings, gestures and the like??through which mathematical objects and relationships might be experienced and expressed. To highlight the connections between perceptual activities and cultural concepts in the meanings associated with this process, we concentrate on learners who do not have access to the visual field. More specifically, we present three examples of gesture use in the practices of blind mathematics students??all involving the exploration of geometrical objects and relationships. On the basis of our analysis of these examples, we argue that gestures are illustrative of imagined reenactions of previously experienced activities and that they emerge in instructional situations as embodied abstractions, serving a central role in the sense-making practices associated with the appropriation of mathematical meanings.     

Proofs and refutations in the undergraduate mathematics classroom

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention.     

Social representations as mediators of mathematics learning in multiethnic classrooms

Drawing on socio-cultural theory, we understand the norms regulating the practices within the mathematics classroom as resulting from the social representations of the socially dominant groups and of the school culture related to what constitutes learning mathematics. Immigrant studients, having their own personal histories as members of particular social groups, and having been in school traditions other than the one predominant in the host society, have their own images of what mathematics in school is about. Individuals interacting in the classroom are all re-interpreting the different episodes from the perspective of the social representations of the larger groups with which they identify themselves. In multiethnic classrooms different re-interpretations of the same norms clash. The lack of negotiation gives rise to obstacles to immigrant students’ participation in the mathematical conversations and, therefore, interferes with the students’ learning process.     

Multiple representations as sites for teacher reflection about mathematics learning

This documentary account situates teacher educator, prospective teacher, and elementary students’ mathematical thinking in relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning skill—creating and analyzing representations. The author examines responses of prospective teachers to a visual representation task and, in turn, their examination of school children’s responses to mathematical tasks. The analysis revealed the initial tendency of prospective teachers to create pictorial representations and highlights the importance of looking beyond the pictures created to how prospective teachers use mathematical models. In addition, the challenges prospective teachers face in moving beyond a ruled-based conception of mathematics and a right/wrong framework for assessing student work are documented. Findings suggest that analyzing representations helps prospective teachers (and teacher educators) rethink their teaching practices by engaging with a culture of teaching focused on reading for multiple meanings and posing questions about student thinking and curriculum materials.     

Affective productions of mathematical experience

In underscoring the affective elements of mathematics experience, we work with contemporary readings of the work of Spinoza on the politics of affect, to understand what is included in the cognitive repertoire of the Subject. We draw on those resources to tell a pedagogical tale about the relation between cognition and affect in settings of mathematical learning. Our interest is first captured in the way in which one teacher’s priority of establishing an inclusive learning community occasionally harboured what appeared to be pedagogically restrictive conceptions of mathematics. Yet, the classroom practices that produced these conceptions promoted the students’ motivation and provided meaningful access to mathematical learning within the classroom collectivity. In a second example, the postponement of scientific encapsulation in bodily imitations of planetary movement kept alive a shared dynamic sense of an elliptical orbit. In both of these cases, we draw on Spinoza’s work to show how the affectivity of classroom practice constituted conceptions of cognition and of mathematical activity crucially linked to the imperatives of participation.     

Participating in Classroom Mathematical Practices

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.     

More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models

Modeling is a major topic of interest in mathematics education. However, the field’s definition of models is diverse. Less is known about what teachers identify as mathematical models, even though it is teachers who ultimately enact modeling activities in the classroom. In this study, we asked nine middle school teachers with a variety of academic backgrounds and teaching experience to collect data related to one familiar physical phenomenon, cooling liquid. We then asked each participant to construct a model of that phenomenon, describe why it was a model, and identify whether a variety of artifacts representing the phenomenon also counted as models during a semi-structured interview. We sought to identify: what do mathematics teachers attend to when describing what constitutes a model? And, how do their attentions shift as they engage in different activities related to models? Using content analysis, we documented what features and purposes teachers attended to when describing a mathematical model. When constructing their own model, they focused on the visual form of the model and what quantitative information it should include. When deciding whether particular representational artifacts constituted models, they focused on how the representations reflected the system under study, and what purposes those representations could serve in further understanding that system. These findings suggest teachers may have multiple understandings of models, which are active at different times and reflect different perspectives. This has implications for research, teacher education, and professional development.     

Social representations as mediators of practice in mathematics classrooms with immigrant students

This article suggests that a critical perspective of the notion of social representations can offer useful insights into understanding practices of teaching and learning in mathematics classrooms with immigrant students. Drawing on literature using social representations, previous empirical studies are revisited to examine three specific questions: what are the dominant social representations that permeate the mathematics classroom with immigrant students? What impact do these social representations have on classroom practices? What are the spaces for changing these practices through becoming reflective and critically aware of these representations? These questions are addressed mostly in relation to teachers’ representations, though the article also draws on data from research with students and parents to illustrate the diversity of representations and to argue for a critical and reflective perspective.     

Teachers’ beliefs and practices related to mathematics instruction

Beliefs and practices related to mathematics were assessed for 21 fourth- through sixth-grade teachers. At the beginning and the end of the school year teachers’ beliefs about (1) the nature of mathematics (i.e., procedures to solve problems versus a tool for thought), (2) mathematics learning (i.e., focusing on getting correct solutions versus understanding mathematical concepts), (3) who should control students’ mathematical activity, (4) the nature of mathematical ability (i.e., fixed versus malleable), and (5) the value of extrinsic rewards for getting students to engage in mathematics activities were assessed. (6) Teachers self-confidence and enjoyment of mathematics and mathematics teaching were also assessed. Analyses were conducted to assess the coherence among these beliefs and associations between teachers’ beliefs and their observed classroom practices and self-reported evaluation criteria. Findings showed substantial coherence among teachers’ beliefs and consistent associations between their beliefs and their practices. Teachers’ self-confidence as mathematics teachers was also significantly associated with their students’ self-confidence as mathematical learners.     

The preparation experiences of elementary mathematics specialists: examining influences on beliefs,content knowledge,and teaching practices

Many in the field of mathematics education call for elementary schools to have elementary mathematics specialists (EMSs) who provide needed mathematical expertise and support for children and teachers. EMSs serve as a reasonable, immediate alternative to the challenges generated by elementary teachers needing improved mathematical knowledge for teaching in the classroom. However, limited inquiry has explored how to best prepare EMSs and how program features and learning activities influence their development. This mixed-method study identifies some of the interrelated benefits from a K-5 Mathematics Endorsement Program designed to prepare EMSs through examining changes in mathematical beliefs, specialized content knowledge (SCK), and classroom teaching practices during the program. Data (n = 32) were collected over the 2-semester program via belief surveys, a content knowledge assessment, observations of teaching practices, and individual interviews from elementary teachers participating in the program. The findings show some changes in beliefs can be made relatively quickly, other shifts in beliefs take more time and continued support, and changes in SCK and adoption of various aspects of standard-based pedagogy require considerably greater opportunities to learn. The described program features and learning experiences provided a context for these changes and offer considerations for EMS preparation programs.     

Teachers’ assessments of students’ learning of mathematics

This paper reports a study of the classroom assessment practices of 12 sixth form college mathematics teachers in Malta. It explores the extent to which these teachers are knowledgeable about their students’ learning of mathematics and the implications that this has for their classroom practices. It reveals that these teachers’ knowledge of their students’ understanding of certain mathematical concepts is fairly limited. It then goes on to discuss this phenomenon in terms of a process that can hinder rather than promote learning. The research illuminates the position of teachers who appear to lack certain detailed information about their students which, it can be argued, could inform more effective teaching strategies. The paper concludes by exploring possible implications for similar situations where learning could be enhanced by more effective classroom assessment strategies and their use to inform future teaching and learning activities.     

UNDERSTANDING STUDENTS�� OUT-OF-SCHOOL MATHEMATICS AND SCIENCE PRACTICE

This study examined standard 6 and 8 (Standards 6 and 8 are the sixth and eighth years, respectively, of primary level schooling in Kenya.) students’ perceptions of how they use mathematics and science outside the classroom in an attempt to learn more about students’ everyday mathematics and science practice. The knowledge of students’ everyday mathematics and science practice may assist teachers in helping students be more powerful mathematically and scientifically both in doing mathematics and science in school and out of school. Thirty-six students at an urban school and a rural school in Kenya were interviewed before and after keeping a log for a week where they recorded their everyday mathematics and science usage. Through the interviews and log sheets, we found that the mathematics that these students perceived they used outside the classroom could be classified as 1 of the 6 activities that Bishop (Educ Stud Math 19:179–191, 1988) has called the 6 fundamental mathematical activities and was also connected to their perception of whether they learned mathematics outside school. Five categories of students’ perceptions of their out-of-school science usage emerged from the data, and we found that 4 of our codes coincided with 2 activities identified by Lederman & Lederman (Sci Child 43(2):53, 2005) as part of the nature of science and 2 of Bishop’s categories. We found that the science these students perceived that they used was connected to their views of what science is.     

A Case Study of Systemic Aspects of Assessment Technologies

Current designs and classroom uses of assessment technology do not adequately reflect or support human practices that produce meaningful information about student learning. We base this argument on findings from a study of the diverse practices making up an assessment system in a fifth-grade mathematics classroom where students have extensive access to personal computing. Videotaped records of classroom activities, reflecting both teachers' and students' perspectives on the classroom, are analyzed with teachers to develop a grounded theoretical framework for describing assessment practices. We present the framework, use it to compare different examples of assessment activities, and propose new criteria for designing systemically valid assessments as kinds of technology that support and extend existing classroom practices.     

Making mathematical meaning through dialogue: ``Once you think of it, the Z minus three seems pretty weird'

In our ongoing qualitative classroom research, we adopt a sociocultural perspective to investigate discourse, and its role in how children and teachers make meaning of mathematics in a fifth grade inquiry classroom. Our theoretical perspective draws primarily on Vygotsky (1978, 1986) and Bakhtin (1981, 1986) each of whom examines how social forms of meaning influence individual cognition. The episode described in this paper examines the process whereby individual and group developmental trajectories are constructed, and allows us to explore the relationship between discourse and knowing. We combine a longitudinal design with a case study approach to focus on the collaborative mathematical problem solving. We use video capture to help us listen to children's discussions in classroom activities and small group interactions. Our analysis of the verbal data recorded on video identifies patterns of interaction, development and change in participants' use of mathematical language and concepts, and their evolving understanding, through discussion and argument, of an algebraic expression constructed by one of the children. The findings lead us to argue for i) a more generative view of the zone of proximal development as a site of learning and of identity formation, ii) an expanded view of the role of the teacher in inquiry classrooms, and iii) an appreciation for the valuable roles difference and resistance play in knowledge building.     

核心素养背景下的数学课堂教学策略探究

核心素养背景下,数学课堂教学研究重点的一方面是探索“如何教”,另一方面是思考学生“如何学”。文章从营造积极学习氛围、设置趣味学习活动、应用现代信息技术、巧妙利用数学小课题四方面,探讨核心素养背景下的数学课堂教学策略,以改变学生的数学学习方式,提高学生的解题能力。     

The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students?? knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment involved five experimental and six control classes (N?=?106 and 138, respectively) of third-grade students. The experiment comprised 20 lessons with 73 word problems, providing a systematic overview of the basic word problem types. Teachers of the experimental classes received a booklet containing lesson plans and overhead transparencies with different types of visual representations attached to the word problems. Students themselves were invited to make drawings for each task, and group work and teacher-led discussion shaped their beliefs about the role of visual representations in word problem solving. The effect sizes of the experiment were calculated from the results of two tests: an arithmetic skill and a word problem test, and the unbiased estimates for Cohen??s d proved to be 0.20 and 0.62. There were significant changes also in experimental group students?? beliefs about mathematics. The experiment pointed to the possibility, feasibility, and importance of learning about visual representations in mathematical word problem solving as early as in grade?3 (around age 9?C10).     

Practices more likely to occur in kindergarten mathematics teachers’ instruction when English learners are present

Considering the rising proportion of English learners (ELs) in general education, little is known about how this change in classroom composition relates to the frequency of instructional activities and skills. This study uses a multimodal perspective on learning mathematics in a multilingual context, which recognizes the diverse modes of communication that students draw on in mathematics classrooms. We examined national data using quantitative analyses and investigated the mathematics practices that were more likely to occur in kindergarten classrooms that did and did not have ELs. We found teachers reported greater frequency of manipulatives and count/calendar activities in mathematics instruction when there were EL students in the classroom. Teachers in classrooms with ELs also reported a greater frequency of mathematics content that involved teaching shapes and practical skills. The findings indicate that teachers tended towards a supportive multimodal approach to instruction when ELs were present.