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Descriptors:
Computer Software; Computer Assisted Instruction; Multimedia Instruction; Content Analysis; Discourse Analysis; Cooperative Learning; Geometry; Problem Solving; Interpersonal Communication; Group Activities; Research Methodology; Coordination

Abstract:
In CSCL research, collaborative activity is conceptualized along various yet intertwined dimensions. When functioning within these multiple dimensions, participants make use of several resources, which can be social or content-related (and sometimes temporal) in nature. It is the effective coordination of these resources that appears to characterize successful collaborative activity. This study proposes a methodological approach for studying coordination of resources when solving geometry problems with dynamic geometry software. The aim is to suggest a methodological lens to capture both the content-related and social discourse within the context of geometry problem solving using dynamic geometry software. As an example, the paper also provides an analysis of a dyad's face-to-face interaction using the suggested framework. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Academic Achievement; Mathematics Achievement; Geometry; Experimental Groups; Early Childhood Education; Grade 1; Elementary School Students; Gifted; Measurement; Mathematical Concepts; Statistical Analysis; Comparative Analysis; Measures (Individuals)

Abstract:
The goal of Project M2 was to develop and field-test challenging geometry and measurement units for K-2 students. The units were developed using recommendations from gifted, mathematics, and early childhood education. This article reports on achievement results for students in Grade 1 at 12 diverse sites in four states using the Iowa Tests of Basic Skills (ITBS) mathematics concepts subtest and an open-response assessment. Using HLM, as expected there were no statistical differences between the experimental (n = 186) and comparison (n = 174) groups on the ITBS (91% of the items focused on number). Statistically significant differences (p less than 0.001) favoring the experimental group were found on the open-response assessment with a large effect size (d = 1.88). Thus the experimental group exhibited a deeper understanding of geometry and measurement concepts on the open-response assessment while performing as well on a traditional measure that covered all Grade 1 mathematics. (Contains 7 tables and 4 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Liberal Arts; Educational Trends; Educational Change; Poetry; Geometry; Mathematical Logic; Validity; Elementary Secondary Education; Special Education; Middle Class; Higher Education

Abstract:
America was made by and for big ideas. Insofar as big ideas have shaped it, it is ever on the verge of hyperbole and dream. Today's big ideas come with an air of celebrity and accessibility; they glitter with glamour but demand little of the Americans. While they have many manifestations, people see them epitomized in TEDTalks. TED (which stands for Technology, Entertainment, Design), a nonprofit that offers two annual conferences of short lectures on innovative ideas, mixes extreme elitism with extreme accessibility. Honoring the liberal arts may sound like a big idea in itself, but it requires modesty, as its meaning comes clear only in the details. Today's worship of sweeping innovations is preventing more modest, thoughtful ideas from being heard. As those who wish to build on the past are assumed to be protecting the status quo, people risk forsaking the works of lasting beauty and practical significance that are part of the liberal arts tradition. (Contains 18 endnotes.) 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:
Mathematical Logic; Validity; Majors (Students); Undergraduate Students; Algebra; Geometry; Number Concepts; College Mathematics; Logical Thinking; Preservice Teachers; Secondary School Mathematics

Abstract:
Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument. (Contains 6 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Secondary School Mathematics; Geometry; Secondary School Students; Interaction; Problem Solving; Virtual Classrooms; Teamwork; Protocol Analysis; Sequential Learning; Sequential Approach; Pattern Recognition; Object Manipulation; Construction (Process); Reflection; Visual Aids

Abstract:
We analyze the interaction of 3 students working on mathematics problems over several days in a virtual math team. Our analysis traces out how successful collaboration in a later session is contingent upon the work of prior sessions and shows how the development of representational practices is an important aspect of these participants' problem solving. We trace the formation, transformation, and refinement of 2 problem-solving practices (i.e., problem decomposition and inscribe first solve second) and 2 representational practices (i.e., modulate perspective and visualize decomposition). The analysis shows how inscriptions become representations for the group through a historical trajectory of negotiation: Learners appropriate inscriptional resources over time as they develop their representational competencies. We show how analysis of the contingently relevant situation may reach into temporally prior episodes in which interaction relevant resources and practices are constructed. (Contains 4 tables and 9 figures.) 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:
Mathematics Instruction; Mathematics Skills; Class Activities; Mathematics Activities; Worksheets; Lesson Plans; Instructional Materials; Visual Aids; Teaching Guides; Glossaries; Academic Standards; Relevance (Education); Grade 7; Grade 8; Measurement; Geometry; Grade 6; Geometric Concepts

Abstract:
"Setting the Stage with Geometry" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards that is designed to help students in grades 6-8 build and reinforce basic geometry skills for measuring 2D and 3D shapes. Developed by The Actuarial Foundation, this program seeks to provide skill-building math activities to help students become successful in the classroom and in real-world situations outside of school. 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:
Difficulty Level; Mathematics Achievement; Academic Records; Credits; Program Effectiveness; National Competency Tests; Algebra; Geometry; Mathematics Curriculum; High School Graduates; Academic Achievement; Textbooks; Course Content; High School Students; Grade 12; Scores; Racial Differences; Comparative Analysis; Course Selection (Students)

Abstract:
The 2005 National Assessment of Educational Progress (NAEP) High School Transcript Study (HSTS) found that high school graduates in 2005 earned more mathematics credits, took higher level mathematics courses, and obtained higher grades in mathematics courses than in 1990. The report also noted that these improvements in students' academic records were not reflected in twelfth-grade NAEP mathematics and science scores. Why are improvements in student coursetaking not reflected in academic performance, such as higher NAEP scores? The Mathematics Curriculum Study (MCS) explored the relationship between coursetaking and achievement by examining the content and challenge of two mathematics courses taught in the nation's public high schools--algebra I and geometry. Conducted in conjunction with the 2005 NAEP HSTS, the study used textbooks as an indirect measure of what was taught in classrooms, but not how it was taught. In other words, the textbook information is not used to measure classroom instruction. Textbooks served as an indicator of the intended course curriculum (Schmidt, McKnight, and Raizen 1997). The chapter review questions in each textbook were used to identify the mathematics topics covered (or subject matter content) and the complexity of the exercises (or degree of cognitive challenge). Chapter review questions, and not the entire textbook, were coded because the questions have been found to be representative of the chapter content and complexity level in previous studies (Schmidt 2012). The study uses curriculum topics to describe the content of the mathematics courses and course levels to denote the content and complexity of the courses. The results are based on analyses of the curriculum topics and course levels developed from the textbook information, coursetaking data from the 2005 NAEP HSTS, and performance data from the twelfth-grade 2005 NAEP mathematics assessment. The study addresses three broad research questions: (1) What differences exist within the curricula of algebra I and geometry courses?; (2) How accurately do school course titles and descriptions reflect the rigor of what is taught in algebra I and geometry courses compared to textbook content?; and (3) How do the curricula of algebra I and geometry courses relate to subsequent mathematics coursetaking patterns and NAEP performance? In this report, curriculum topics, course levels, and grade 12 NAEP mathematics scale scores are used to describe the findings of the study. Curriculum topics are based on summaries of the textbook content that a school reported covering in an algebra I or geometry course. The six broad categories of curriculum topics used to describe the mathematics content found in both algebra I and geometry textbooks are: elementary and middle school mathematics, introductory algebra, advanced algebra, two-dimensional geometry, advanced geometry, and other high school mathematics. A glossary is included. (Contains 3 charts, 15 figures and 10 tables.) 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:
Mathematics Achievement; State Standards; Algebra; Geometry; Secondary School Mathematics; Evidence; Outcome Measures; Individualized Instruction; Intelligent Tutoring Systems; Program Evaluation; Instructional Effectiveness

Abstract:
"Carnegie Learning Curricula and Cognitive Tutor"[R], published by Carnegie Learning, is a secondary math curricula that offers textbooks and interactive software to provide individualized, self-paced instruction based on student needs. The program includes pre-Algebra, Algebra I, Algebra II, and Geometry, as well as a three-course series that integrates numeric, algebraic, geometric, and statistical content. The developer indicates that the program is aligned with most state standards and the standards set by the National Council of Teachers of Mathematics. The program can be customized to meet other state-specific standards. The What Works Clearinghouse (WWC) identified 27 studies that investigated the effects of "Carnegie Learning Curricula and Cognitive Tutor"[R] on math performance for high school students. The WWC reviewed 11 of those studies against group design evidence standards. Three studies (Cabalo, Jaciw, & Vu, 2007; Campuzano, Dynarski, Agodini, & Rall, 2009; & Pane, McCaffrey, Slaughter, Steele, & Ikemoto, 2010) are randomized controlled trials that meet WWC evidence standards without reservations, and three studies (Shneyderman, 2001; Smith, 2001; & Wolfson, Koedinger, Ritter, & McGuire, 2008) are randomized controlled trials or quasi-experimental designs that meet WWC evidence standards with reservations. These six studies are summarized in this report. Five studies do not meet WWC evidence standards. The remaining 16 studies do not meet WWC eligibility screens for review in this topic area. Appended are: (1) Research details for Cabalo et al., 2007, Campuzano et al., 2009, Pane et al., 2010, and Shneyderman, 2001; (2) Outcome measures for each domain; (3) Findings included in the rating for the mathematics achievement domain; and (4) Summary of supplemental findings for the mathematics achievement domain. A glossary of terms is included. (Contains 7 tables, 4 additional sources and 7 endnotes.) 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:
Music; Mechanics (Physics); Energy Conservation; Optics; Introductory Courses; Science Instruction; Geometry; Scientific Concepts; Acoustics; Scientific Principles; Teaching Methods; Interdisciplinary Approach; Physics

Abstract:
Much of the mathematical reasoning employed in the typical introductory physics course can be traced to Pythagorean roots planted over two thousand years ago. Besides obvious examples involving the Pythagorean theorem, I draw attention to standard physics problems and derivations which often unknowingly rely upon the Pythagoreans' work on proportion, music, geometry, harmony, the golden ratio, and cosmology. Examples are drawn from mechanics, electricity, sound, optics, energy conservation and relativity. An awareness of the primary sources of the mathematical techniques employed in the physics classroom could especially benefit students and educators at schools which encourage integration of their various courses in history, science, philosophy, and the arts. 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:
Student Attitudes; Geometry; Secondary School Mathematics; Educational Technology; Computer Uses in Education; Teaching Methods; Mathematical Concepts; Concept Formation

Abstract:
This paper describes a task-based dynamic geometry platform that is able to record student responses in a collective fashion to pre-designed dragging tasks. The platform provides a new type of data and opens up a quantitative dimension to interpret students' geometrical perception in dynamic geometry environments. The platform is capable of generating a collective image map of student geometrical perceptions for a pre-designed dragging task. This map is interpreted as students' qualitatively different ways of perceiving a geometrical phenomenon under the drag mode, ways which are quantified and categorized in a collective way. The idea of task perceptual landscape is proposed to facilitate discussion on the pedagogical significance of this platform. Specifically, a task case is presented and analysed in which a methodology is developed that provides a way to classify students' geometrical perceptions with respect to the task. The task perceptual landscape is interpreted as a collective example space of student perception of a task. Furthermore, an idea of personal example space is developed through the findings from a qualitative study for the same task. This brings about discussion on possible pedagogical correlation between the quantitative and qualitative aspects of the platform. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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