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Anna Sfard - One of the best experts on this subject based on the ideXlab platform.
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introduction developing Mathematical Discourse some insights from communicational research
International Journal of Educational Research, 2012Co-Authors: Anna SfardAbstract:Abstract Quite diverse in their foci and specific themes, the seven articles collected in this special issue are unified by their common conceptual framework. Grounded in the premise that thinking can be usefully defined as self-communicating and that mathematics can thus be viewed as a Discourse, the communicational framework provides a unified set of conceptual tools with which to investigate cognitive, affective and social aspects of mathematics learning. The communicational tools are employed by the authors as they investigate diverse aspects of Mathematical Discourse and explore its development in the classroom and beyond. The seven studies combine together to produce a set of insights, some of which go against widespread beliefs about teaching and learning mathematics.
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how does language impact the learning of mathematics comparison of english and korean speaking university students Discourses on infinity
International Journal of Educational Research, 2012Co-Authors: Dongjoong Kim, Joan Ferrinimundy, Anna SfardAbstract:This study investigates the impact of language on students’ learning of mathematics. A comparison has been made between English and Korean speaking university students’ Discourses on infinity. In Korean, unlike in English, there is a disconnection between colloquial and Mathematical Discourses on infinity, in that the Mathematical word for infinity is not a formalized version of a colloquial word but a novel sound, inspired by a Chinese term for infinity. This difference was expected to be paralleled by certain dissimilarity between the ways the Discourses of the two groups developed toward the Mathematical Discourse on infinity. Data with the help of which we intended to test this hypothesis were collected through surveys and interviews. A total of 132 English speakers and 126 Korean speakers participated in the survey and then twenty paired representatives were selected from each group for follow-up interviews. It was found that in spite of the comparable levels of Mathematical performance, there was, indeed, a visible dissimilarity between Mathematical Discourses on infinity of Korean- and English-speaking students. In general, whereas no group could pride itself on a well-developed Mathematical Discourse on infinity, the Mathematical Discourse of the English speakers, just like their colloquial Discourse, was predominantly processual, whereas the Korean-speaking students’ talk on infinity was more structural and, in an admittedly superficial way, closer to the formal Mathematical Discourse.
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doing the same mathematics exploring changes over time in students participation in Mathematical Discourse through responses to gcse questions
2012Co-Authors: Candia Morgan, Sarah Tang, Anna SfardAbstract:The project “The Evolution of the Discourse of School Mathematics” uses the lens of GCSE examinations to investigate changes over the last three decades in what is expected of students in England. We have identified differences in the discursive features of examination questions through this period and now seek to investigate how these differences may have affected the nature of student participation in mathematics Discourse. Students have been tested using questions varying in characteristics typical of different points in time. We discuss the design of the test, and present some preliminary results.
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Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing
2008Co-Authors: Anna SfardAbstract:Introduction Part I. Discourse on Thinking: 1. Puzzling about (Mathematical) thinking 2. Objectification 3. Commognition: thinking as communicating 4. Thinking in language Part II. Mathematics as Discourse: 5. Mathematics as a form of communication 6. Objects of Mathematical Discourse: what mathematizing is all about 7. Routines: how we mathematize 8. Explorations, deeds, and rituals: what we mathematize for 9. Looking back and ahead: solving old quandaries and facing new ones.
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when the rules of Discourse change but nobody tells you making sense of mathematics learning from a commognitive standpoint
The Journal of the Learning Sciences, 2007Co-Authors: Anna SfardAbstract:In this article we introduce a research framework grounded in the assumption that thinking is a form of communication and that learning a school subject such as mathematics is modifying and extending one’s Discourse. This framework is then applied in the study devoted to the learning of negative numbers. The analysis of data is guided by questions about (a) the Discourse on negative numbers as such, and the features that set it apart from the Mathematical Discourse with which the students have been familiar when the learning began; (b) students’ and teacher’s efforts toward the necessary transition to the new meta-discursive rules, and (c) effects of the learning teaching process, that is, the extent of discursive change resulting from these efforts. Our findings lead to the conclusion that discursive change, rather than being necessitated by an extradiscursive reality, is spurred by communicational conflict, that is, by the situation that arises whenever different interlocutors seem to be acting according to differing discursive rules. Another conclusion is that school learning requires an active lead of an experienced interlocutor and is fuelled by a realistic communicational agreement between her and the learners.
Judit N Moschkovich - One of the best experts on this subject based on the ideXlab platform.
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using the academic literacy in mathematics framework to uncover multiple aspects of activity during peer Mathematical discussions
Zdm, 2018Co-Authors: Judit N Moschkovich, William ZahnerAbstract:This paper illustrates how the academic literacy in mathematics framework (Moschkovich, J Math Behav 40:43–62, 2015) can be used to uncover the multiple layers of work bilingual learners accomplish during Mathematical discussions. Using this framework allows researchers to examine students’ joint Mathematical activity in terms of Mathematical proficiency, Mathematical practices, and Mathematical Discourse. The use of the framework is illustrated through analysis of two Mathematical discussions among middle school students. We conclude with reflections on the utility of the framework and consider possible pedagogical implications of this work.
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academic literacy in mathematics for english learners
The Journal of Mathematical Behavior, 2015Co-Authors: Judit N MoschkovichAbstract:Abstract 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.
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how equity concerns lead to attention to Mathematical Discourse
2012Co-Authors: Judit N MoschkovichAbstract:This chapter examines the connections between equity and Mathematical Discourse and explores how Discourse is relevant to equity. Through commentary on the preceding three chapters, I discuss four issues raised by different approaches to equity and to Discourse: multiple approaches to equity, definitions of ‘Discourse’, aspects of school Discourse practices, and challenges with ethno-Mathematical approaches. Next, I summarize what research tells us about equitable Discourse practices for students from non-dominant communities in mathematics classrooms. In closing, I use the four chapters and my own work (Moschkovich, Language(s) and learning mathematics: Resources, challenges, and issues for research. In Moschkovich, J. (Ed.), Language and mathematics education: Multiple perspectives and directions for research (pp. 1–28). Charlotte: Information Age Publishing, 2010) to make recommendations for future research.
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examining Mathematical Discourse practices
2007Co-Authors: Judit N MoschkovichAbstract:What are the features of Discourse practices? Are there characteristic Mathematical Discourse practices? Can we distinguish everyday and academic Mathematical Discourse practices? This article [1] considers these questions from a socio-cultural and situated perspective of Mathematical Discourse practices (Moschkovich, 2002a, 2004) [2]. To ground that discussion, I first present an excerpt of a classroom discussion about quadrilaterals. The excerpt comes from a lesson in a third grade (students are 8-9 years old) bilingual classroom in an urban California school. The students have been working on a unit on two-dimensional geometric figures. During the past weeks, instruction had included technical vocabulary such as the names and definitions for different quadrilaterals. Students had been talking about shapes and the teacher had asked them to point to, touch, and identify different quadrilaterals. In this lesson, students were describing quadrilaterals as they folded and cut paper to form Tangram pieces (see Figure 1).
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what counts as Mathematical Discourse
International Group for the Psychology of Mathematics Education, 2003Co-Authors: Judit N MoschkovichAbstract:The distinction between everyday and Mathematical Discourses can be useful for describing mathematics learning as moving from everyday to more Mathematical ways of talking. However, this distinction has limited uses in the classroom. First, it is difficult to use this distinction to categorize student talk since it is not always possible to tell whether a student’s competence in communicating Mathematically originates in their everyday or school experience. And, while learning mathematics certainly involves learning to use more Mathematical language, everyday Discourse practices should not be seen only as obstacles to learning mathematics. During Mathematical discussions students use multiple resources from student experiences both outside and inside school. Before we label student talk as everyday or Mathematical, we need to seriously consider what we include or exclude in our definition of Mathematical Discourse practices. If we assume that Mathematical Discourse consists only of textbook definitions or those practices that mathematicians use in formal settings, we may miss the Mathematical competence in student talk.
Estrella Johnson - One of the best experts on this subject based on the ideXlab platform.
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teachers Mathematical activity in inquiry oriented instruction
The Journal of Mathematical Behavior, 2013Co-Authors: Estrella JohnsonAbstract:Abstract This work investigates the relationship between teachers’ Mathematical activity and the Mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in Mathematical activity in response to the Mathematical activity of their students. Further, my analysis considered the interactions between teachers’ Mathematical activity and the Mathematical activity of their students. This analysis suggests that teachers’ Mathematical activity can play a significant role in supporting students’ Mathematical development, in that it has the potential to both support students’ Mathematical activity and influence the Mathematical Discourse of the classroom community.
Erin E Turner - One of the best experts on this subject based on the ideXlab platform.
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explicame tu respuesta supporting the development of Mathematical Discourse in emergent bilingual kindergarten students
Bilingual Research Journal, 2012Co-Authors: Sylvia Celedonpattichis, Erin E TurnerAbstract:This study investigated Spanish-speaking kindergarten students' participation in Mathematical Discourse as they solved and discussed a range of word problems. Specifically, we draw upon sociocultural perspectives on mathematics learning to frame Mathematical Discourse and to examine specific teacher and student actions that seemed to support the development of Mathematical Discourse over the course of the kindergarten year. Data sources included pre- and post-task-based clinical interview assessments and weekly (videotaped) observations of problem-solving lessons. Findings demonstrated ways that teachers supported and students appropriated discursive habits such as using more precise Mathematical language, explaining solutions in ways that referenced actions on quantities in the problem, and using multiple visual representations to mediate communication. In addition, the findings point to the critical role the teacher plays in supporting the development of students' Mathematical Discourse.
Magdalena Wolska - One of the best experts on this subject based on the ideXlab platform.
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a language engineering architecture for processing informal Mathematical Discourse
Towards Digital Mathematics Library. Birmingham United Kingdom July 27th 2008, 2008Co-Authors: Magdalena WolskaAbstract:We present a modular architecture for processing informal Mathematical language as found in textbooks and Mathematical publica- tions. We point at its properties relevant in addressing three aspects of informal Mathematical Discourse: (i) the interleaved symbolic and natu- ral language, (ii) the linguistic, domain, and notational context, and (iii) the imprecision of the informal language. The objective in the modular approach is to enable parameterisation of the system with respect to the natural language of the text and the Mathematical domain of Discourse. Informal Mathematical Discourse in textbooks and Mathematical publications is partly written in natural language and partly in a symbolic notation—even within a single utterance. Be it information retrieval or text mining Mathematical documents, flexible human-oriented Mathematical user interfaces, or automated verification of informal proofs crucially rely on automated analysis of the informal language. In (8,9,2) we presented methods of, respectively: parsing, lexical analysis, and domain-specific interpretation of informal Mathematical proofs, and introduced linguistic resources necessary for processing. In this paper, we present a modular architecture of a system for processing Mathematical language based on those resources and emphasise three core aspects of the informal Mathematical Discourse it addresses: the interleaved symbolic and natural language, the linguistic, domain, and notational context, and the imprecision of the informal language. The objective in the modular approach is to enable parameterisation of the system with respect to the natural language of the text in question, the Mathematical domain of the Discourse, and the Mathematical notation. We first briefly discuss the above-mentioned aspects of the Mathematical language based on example utterances, then we present our processing architecture, and finally outline the related work on Mathematical Discourse and our further work.
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building a dependency based grammar for parsing informal Mathematical Discourse
Lecture Notes in Computer Science, 2004Co-Authors: Magdalena Wolska, Ivana KruijffkorbayovaAbstract:Discourse in formal domains, such as mathematics, is characterized by a mixture of natural language and embedded formal expressions. Based on an investigation of a collected corpus of informal dialogues on naive set theory proofs, we are developing a dependency-based lexicalist grammar for parsing input with different degrees of verbalization of the Mathematical content: ranging from symbolic alone to fully worded Mathematical expressions. In this paper, we describe our approach to analysis, focusing on the underlying semantic representations.
