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Descriptors:
Mathematics Education; Mathematical Concepts; Mathematics; Mathematics Instruction; Mathematical Logic; Mathematics Skills; Curriculum

Abstract:
In this paper we study the mathematical body as an assemblage of human and non-human mathematical concepts. We argue that learners' bodies are always in the process of becoming assemblages of diverse and dynamic materialities. Following the work of the historian of science Karen Barad, we argue that mathematical concepts must be considered dynamic material, and we suggest a "pedagogy of the concept" that animates concepts as both logical and ontological. We draw on the philosopher of mathematics Gilles Chatelet in order to pursue this argument, elaborating on the way that mathematical concepts partake of the mobility of the virtual, while learners, in engaging with this mobility, enter a material process of becoming. We show how the concept of virtuality allows us to look at mathematical concepts in school curriculum in new ways. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Stakeholders; Mathematics; Case Method (Teaching Technique); Elementary School Teachers; Mathematics Curriculum; Elementary School Mathematics; Interviews; Mathematics Teachers

Abstract:
This qualitative study examined perspectives of two key stakeholder groups, instructors and students, on mathematics content courses for prospective elementary teachers (Mathematics for Teachers [MFT] courses). A collective case study approach, which drew from the data of two cases in different but comparable settings, contributed to the robustness of the findings. Cross-case analysis of the interview data revealed several convergent themes: the role of affect in student learning, pedagogy and instructor disposition, connections to the elementary classroom, and mathematics content. The findings included both conflicting and complementary perspectives between the two key stakeholder groups. When juxtaposed, the multiple viewpoints offer insights into some of the central issues related to teaching and learning in MFT courses and suggest potential avenues for improving experiences in these courses. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Foreign Countries; Mathematics Instruction; Grade 1; Mathematics Teachers; Thinking Skills; Mathematical Logic; Grade 2; Academic Achievement; Mathematics; Interviews; Problem Solving

Abstract:
This paper examines the perceptions and understandings of ten grades 1 and 2 Singapore mathematics teachers as they learned to use clinical interviews (Ginsburg, "Human Development" 52:109-128, 2009) to understand students' mathematical thinking. This study challenged teachers' pedagogical assumptions about what it means to teach for student understanding. Clinical task-based interviews opened a window into students' knowledge, problem-solving and reasoning, and helped teachers reflect on their teaching and assessment of student learning. Teachers also learnt about what it means to establish a culture of thoughtful questioning in the classroom and developed an emerging awareness that this requires a readiness to hear students' ideas and connect informal or invented strategies with classroom mathematics. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Equations (Mathematics); Novices; Mathematics Instruction; Expertise; Algebra; Mathematics Education; Secondary School Teachers; Secondary School Students; Mathematical Formulas; Mathematics; Interviews

Abstract:
While an algebraic expression is typically assigned a regular structure, this article introduces the concept of personal structure ("Strukturierung"); here, the structuring of an algebraic expression is understood as the act of forming relationships between its parts. This concept is used for the analysis of interviews in which experts and novices talk about the personal structures they produced when looking at fractional expressions and equations. The results show, firstly, that experts will structure a given fractional expression in various different ways. Secondly, four categories have been identified: structuring can mean that an expression is made more visually straightforward, changed, reinterpreted or classified. This is meaningful for negotiating the appropriateness of the personal structures in teaching algebra. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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College Students; Mathematics Education; Mathematics Instruction; Higher Education; Mathematical Models; Mathematics

Abstract:
This paper describes university students' grasp of inflection points. The participants were asked what inflection points are, to mark inflection points on graphs, to judge the validity of related statements, and to find inflection points by investigating (1) a function, (2) the derivative, and (3) the graph of the derivative. We found four erroneous images of inflection points: (1) f ' (x) = 0 as a necessary condition, (2) f ' (x) is not equal to 0 as a necessary condition, (3) f " (x) = 0 as a sufficient condition, and (4) the location of "a peak point, where the graph bends" as an inflection point. We use the lenses of Fischbein, Tall, and Vinner and Duval's frameworks to analyze students' errors that were rooted in mathematical and in real-life contexts. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Educational Policy; Jews; Foreign Countries; Hidden Curriculum; Textbooks; Educational History; Mathematics; Nationalism; Judaism; Ideology; Teaching Methods; Geography

Abstract:
This paper discusses the attempts of Israeli education, in a similar fashion to other national educational systems, to shape a territorial identity for the pupils of the new State. The Israeli school used a variety of educational means to shape a person who would be modelled on his new birthplace's landscape, including the use of textbooks, illustrations, and maps, to aid in the process of creating a desired image of the homeland's landscape. The hidden curriculum used textbooks employing mathematics questions to learn details about the geographical expanse. Alongside the use of a written curriculum, Israeli education made use of the extra curriculum by becoming physically familiar with a place and creating a local time based on the seasons of the year. Local nature was studied during "moledet" (homeland) lessons, similar to the Weimar Republic of Germany's Heimatkunde studies, as well as during other subjects, such as nature studies and Bible. These studies integrated national goals and progressive humanistic educational schools of thought which viewed a child's encounter with nature as a vital part of his or her education. The readers, which were built on a timeline of the seasons and the school celebrating nature festivals, created a natural time frame for the pupils in which they acted and studied. The discussion about the ways territorial identity was structured by the Israeli education system is another chapter in the wider debate about national education and illustrates the schools' function as one of the State's national social agents, particularly in its early years. (Contains 1 figure and 79 footnotes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Elementary Schools; Foreign Countries; Teaching Assistants; Mathematics; Mathematics Education; Mathematics Instruction; Interviews; Elementary School Mathematics

Abstract:
Teaching Assistants (TAs) in primary schools in England have a growing pedagogic role. For some, this sometimes includes responsibility for the whole class instead of the teacher. This article draws on 24 interview transcripts to examine the practice in the context of primary mathematics lessons and from TAs' viewpoints. Emergency cover is often seen as reasonable where good working relationships exist. The practice of being regularly responsible for mathematics lessons evokes more diverse reactions. Some TAs initially appear to support the "official" view that it is unproblematic to run a lesson from pre-prepared plans, though close inspection reveals a different picture. Others acknowledge that the interactions involved in such lessons are not necessarily susceptible to planning. The findings raise considerable doubt about current policy and question its presentation as a way to raise standards. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Preservice Teacher Education; Teacher Education Programs; Humanities; Mathematics; Social Change; Fundamental Concepts; Racial Discrimination; Resistance (Psychology); Teacher Educators; Equal Education; Culturally Relevant Education

Abstract:
As a response to the attacks on ethnic studies in Arizona and the move to ban certain books, this essay presents theoretical and pedagogical reflections from two professors and addresses the ways teacher preparation programs can offer a resistance. Based on the authors' experience in teacher preparation programs, one in the humanities and the other in mathematics, they discuss fundamental concepts that undergird social change methodology from Gloria Anzaldua (la facultad and conocimiento) and from Isabel Gunning's work (World Traveling). Ultimately, our premise is that teachers of teachers can impact the curricula in significant ways that result in dismantling racism and in teaching that is focused on positive social change. We posit that the university classroom where future teachers are trained must address (1) Equity issues, (2) Cultural identity or cultural framing, and (3) Culturally relevant strategies and teaching, modeled by the university professor. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Elementary Secondary Education; STEM Education; Computation; Thinking Skills; Abstract Reasoning; Problem Solving; Mathematics; Programming; Educational Research

Abstract:
Jeannette Wing's influential article on computational thinking 6 years ago argued for adding this new competency to every child's analytical ability as a vital ingredient of science, technology, engineering, and mathematics (STEM) learning. What is computational thinking? Why did this article resonate with so many and serve as a rallying cry for educators, education researchers, and policy makers? How have they interpreted Wing's definition, and what advances have been made since Wing's article was published? This article frames the current state of discourse on computational thinking in K-12 education by examining mostly recently published academic literature that uses Wing's article as a springboard, identifies gaps in research, and articulates priorities for future inquiries. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Arithmetic; Numeracy; Mathematics Skills; Training Methods; Preschool Children; Factor Structure; Early Childhood Education; Numbers; Number Concepts; Computation; Mathematics; Factor Analysis; Scores; Comparative Analysis; Correlation; Intervention

Abstract:
Validating the structure of informal numeracy skills is critical to understanding the developmental trajectories of mathematics skills at early ages; however, little research has been devoted to construct evaluation of the Numbering, Relations, and Arithmetic Operations domains. This study was designed to address this knowledge gap by examining the structure of these three numeracy skill domains and examining the relations among these domains. Three hundred ninety-three children participated in the study (51.7% girls, 55.7% White, 33.8% African American, and 10.5% other). Results indicated that the relations among the informal numeracy skills were best explained by a three-factor model that included Numbering, Relations, and Arithmetic Operations factors, and this factor structure was the same in both younger and older preschool children. (Contains 9 tables, 1 figure, and 5 notes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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