ethanporcaro/eric-fulltext-meta huggingface dataset:
Visual Representations in Mathematics Problem-Solving: Effects of Diagrams and Illustrations
Author: Cooper, Jennifer L.
Date: 2012-11-00
Type: Reports - Research
Subjects: Visual Aids,Mathematics Instruction,Textbook Content,Problem Solving,Illustrations,Trigonometry,Multimedia Materials,Undergraduate Students,Outcomes of Education,Likert Scales,Accuracy

Many types of visual representations are used in math textbooks but not all of them contain mathematically relevant information. Little research directly addresses the effects of different types of representations on mathematics performance. Theories offer differing perspectives about how visual representations such as illustrations influence student learning. Here, we investigated the effects of diagrams and contextual illustrations on trigonometry problem solving. Diagrams helped all students, but the effect of contextual illustrations depended on students' backgrounds. Additionally, not all subgroups of students accurately assessed the effect of illustrations on their performance. We emphasize the need to consider how different types of visual representations interact with student characteristics and the problem-solving task. [For the complete proceedings, see ED584829.]
A Model for Designing Cognition-and-Instruction-Based Goal Trajectories for Research in K-6 Math Curricula
Author: Canty, Reality S.
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Elementary School Mathematics,Algebra,Common Core State Standards,Problem Solving,Models

This paper describes a model for building cognition-and-instruction-based goal trajectories (GT) in the context of a study that examines the validity of curriculum-embedded assessments. The model consists of six design processes and two constraints. The GT is constructed from curriculum-specified learning goals as well as developmental progressions and learning trajectories derived from empirical research. The GT is designed to inform both the selection of assessment activities for data collection and the interpretation of empirical results. Two primary results of the design process are presented: (a) a goal trajectory for promoting algebraic understanding and (b) the relationships between the trajectory and features of the Common Core State Standards. Implications of the design model for building GTs that can be used to assess student reasoning are discussed. [For the complete proceedings, see ED584829.]
A Framework for Mathematicians&apos; Example-Related Activity When Exploring and Proving Mathematical Conjectures
Author: Lockwood, Elise
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Models,Mathematics Instruction,Mathematics,Professional Personnel,Mathematical Logic,Validity,Thinking Skills,Online Surveys,Mathematics Activities

Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, we present a framework that characterizes ways in which mathematicians utilize examples when investigating conjectures and developing proofs. The data consist of 133 mathematicians' responses to two open-ended survey questions. The framework offers categories for the types of examples, uses of examples, and example strategies that mathematicians discussed in reference to their work with conjectures. In addition to presenting the framework, we also discuss potential educational implications of the results. [For the complete proceedings, see ED584829.]
Relationships between Mathematical Proof and Definition
Author: Plaxco, David
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Undergraduate Students,Mathematical Logic,Validity,Probability,Correlation,Definitions,Semi Structured Interviews,Video Technology,Mathematics Instruction

This article discusses results from interviews with two undergraduate students in an introductory proofs course. The researcher assessed the participants' general proof schemes and built models of the participants' conception of probabilistic independence and mutual exclusivity. The participants were then tasked with asserting a relationship between independence and mutual exclusivity and trying to prove the asserted relationship. The results discuss possible interactions between students' conception of mathematical ideas and their approaches to proof. [For the complete proceedings, see ED584829.]
Pre-Fractional Middle School Students&apos; Algebraic Reasoning
Author: Hackenberg, Amy J.
Date: 2012-11-00
Type: Reports - Research
Subjects: Middle School Students,Mathematics Instruction,Logical Thinking,Fractions,Algebra,High School Students,Multiplication,Grade 7,Grade 8,Grade 10,Observation,Semi Structured Interviews,Equations (Mathematics),Generalization,Problem Solving

To understand relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of 3 different multiplicative concepts participated. This paper reports on the 6 students with the most basic multiplicative concept, who were also pre-fractional in that they had yet to construct the first genuine fraction scheme. These students' emerging iterating operations facilitated their algebraic activity, but the lack of a disembedding operation was a significant constraint in developing algebraic equations and expressions. [For the complete proceedings, see ED584829.]
Contrasting Characterizations of Change among Prominent Theoretical Perspectives in Mathematics Education
Author: Tallman, Michael A.
Date: 2012-00-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Education,Epistemology,Educational Change,Mathematics,Mathematics Instruction,Research Methodology

In this paper, we problematize an ontological characterization of change within a complex system by illustrating how epistemological premises of interactionist, individualist, and collectivist theoretical perspectives reveal only specific aspects of a changing system. Methodological considerations resulting from our recognition that change is characterized subjectively within various theoretical perspectives are made. [For the complete proceedings, see ED584829.]
Conceptual Connections between Student Noticing and Productive Changes in Prior Knowledge
Author: Hohensee, Charles
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Attention,Mathematical Concepts,Prior Learning,Knowledge Level,Concept Formation,Middle School Students,Video Technology,Algebra,Classroom Environment,Secondary School Mathematics

In this report, I examine what students noticed as they participated in an instructional unit on quadratic functions and how a shift in what they noticed was conceptually connected to productive changes in their prior knowledge about linear functions. My results show that, over the course of the instructional unit, students' attention shifted toward noticing changes in the quantities involved in quadratic functions. Furthermore, I identify two conceptual connections between the shift toward noticing changes in quantities in quadratic contexts and the productive changes in reasoning that students exhibited on post-interview linear function tasks. [For the complete proceedings, see ED584829.]
Gestures as Facilitators to Proficient Mental Modelers
Author: Garcia, Nancy
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Nonverbal Communication,Mathematics Instruction,Undergraduate Students,Calculus,Workshops,Teaching Methods,Problem Solving,Role,Thinking Skills,Correlation,Classroom Communication,Student Attitudes

Gestures are profoundly integrated into our communication. This study focuses on the impact that gestures have in a mathematical setting, specifically in an undergraduate calculus workshop. There was strong correlation between diagramming and the two types of gestures identified in this study (i.e., dynamic and static gestures). Dynamic and static gestures were part of the students' constructive thinking, whether it was related to the manner in which they viewed the problem or the construction of their diagrams. Nonetheless, gestures played a strong role in the students' problem solving and the manner in which the gestures were utilized provided insight into their constructive thinking. [For the complete proceedings, see ED584829.]
Connections across Representations in Students&apos; Group Discussions of a Non-Routine Problem
Author: Zahner, William
Date: 2012-11-00
Type: Reports - Research
Subjects: Mathematics Skills,Group Discussion,Cooperative Learning,Problem Solving,Algebra,Generalization,Semiotics,Mathematics Activities,High School Students,Grade 9,Bilingual Students,Rural Schools,Abstract Reasoning,Spanish,English (Second Language)

This research report examines how two groups of bilingual algebra students made connections among representations while solving a non-routine generalization problem. Using a socio-cultural orientation to mathematics learning, together with a semiotic lens on students' joint mathematical activity, this report details the type of connections among representations each group of students made as they solved the problem. Follow-up analysis shows that some connections afforded making more productive conclusions while other connections may have constrained the groups' solution processes. Finally, analysis of change across time reveals that the initial connections made by each group persisted across six weeks, despite intervening instruction that suggested other possible connections to solve the problem. The conclusion contains implications for researchers and practitioners. [For the complete proceedings, see ED584829.]
Mathematics, the Common Core Standards, and Language: Mathematics Instruction for ELS Aligned with the Common Core
Author: Moschkovich, Judit N.
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,English Language Learners,Common Core State Standards,Alignment (Education),Guidelines,Mathematics Skills,Language Usage,Teaching Methods

This paper outlines research-based recommendations for mathematics instruction for English Learners (ELs) aligned with the Common Core Standards. The recommendations focus on improving mathematics learning and teaching through language for all students, and especially for ELs. These recommendations are intended to guide teachers and teacher educators in developing approaches to support mathematical reasoning and sense making for ELs. [For the complete proceedings, see ED584829.]
Transitioning into Contemporary Theory: Critical Postmodern Theory in Mathematics Education Research
Author: Stinson, David W.
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Education,Critical Theory,Educational Research,Postmodernism,Research Methodology,Theories

In this theoretical paper, the authors provide an overview of mathematics education as a research domain, identifying and briefly discussing four transitions or historical moments in mathematics education research. Using the &quot;Instructional Triangle&quot; as a point of reference for the dynamics of mathematics instruction, they illustrate how mathematics education researchers working in different moments explore different questions and use different theoretical perspectives. The authors then provide brief summaries of critical theory and postmodern theory, and suggest critical postmodern theory (CPT) as a hybrid theory that offers new possibilities for conceptualizing and conducting mathematics education research. [For the complete proceedings, see ED584829.]
The Impact of Online Activities on Students&apos; Generalizing Strategies and Justifications for Linear Growing Patterns
Author: Beatty, Ruth
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Algebra,Transitional Programs,Grade 8,Teaching Methods,Grade 9,Interviews,Electronic Learning,Semi Structured Interviews,Learning Activities,Thinking Skills,Elementary School Mathematics,Secondary School Mathematics

This study explores the impact of working with online learning activities of linear growing patterns (CLIPS) on students transitioning into Grade 9. Fifty students were interviewed about their understanding of linear growing patterns. Twenty-five students had participated in a study involving an experimental instructional approach that emphasized exploration of multiple representations of linear relationships when they were in Grade 8. They were then assessed five months later, and their reasoning compared to twenty-five students who did not take part in the study. Results indicate that students who worked with CLIPS were able to find explicit, generalized rules for patterns and offered higher levels of justifications than their counterparts. These students were also more likely to refine their thinking. [For the complete proceedings, see ED584829.]
Transitioning from Whole Numbers to Integers
Author: Bofferding, Laura
Date: 2012-11-00
Type: Reports - Research
Subjects: Mathematics Instruction,Numbers,Pretests Posttests,Elementary School Mathematics,Elementary School Students,Grade 1,Number Concepts,Transitional Programs,Comparative Analysis,Elementary School Teachers,Mathematics Teachers

This paper presents the results of a pre-test, instruction, post-test study that investigated students' integer mental models and how their models changed based on instruction. Sixty-one first graders' responses to questions about the values and order of negative numbers were categorized according to a series of mental models. The models reveal that initially, students over-rely on the values and/or order of whole numbers to varying degrees when working with negative numbers. Focused exploration on the properties of negative numbers helped students transition to more sophisticated mental models compared to only learning about moving in positive and negative directions on number lines. [This research was funded by a Stanford University School of Education Dissertation Grant. For the complete proceedings, see ED584829.]
Factors Influencing Middle School Students&apos; Spatial Mathematics Development While Participating in an Integrated STEM Unit
Author: Wilhelm, Jennifer
Date: 2012-11-00
Type: Speeches/Meeting Papers
Subjects: Middle School Students,Spatial Ability,Mathematics Skills,Integrated Curriculum,STEM Education,Gender Differences,Visualization,Quasiexperimental Design,Earth Science,Teaching Methods,Cognitive Tests,Science Tests,Experimental Groups,Control Groups,Geometric Concepts,Grade 6,Mathematics Tests,Pretests Posttests,Statistical Analysis

This study examined differences between two groups of students' spatial-scientific reasoning from pre to post implementation of an Earth/Space unit. Using a quasi-experimental design, researchers explored how instructional method and gender affected learning. Treatment teachers employed an integrated STEM curriculum while the control teacher implemented her regular Earth/Space unit. The Geometric Spatial Assessment (GSA), the Purdue-Spatial Visualization Rotation Test, and the Lunar Phases Concept Inventory (LPCI) were used to assess learning. Experimental groups made gains on periodicity LPCI domains while the control made gains on geometric spatial visualization LPCI domains. Only females made gains on GSA items. This is the first quasi-experimental study to examine students' spatial reasoning as they participate in Earth/Space units and to discover gender's role in this spatial development. [For the complete proceedings, see ED584829.]
What Sense Do Children Make of Negative Dollars?
Author: Whitacre, Ian
Date: 2012-11-00
Type: Reports - Research
Subjects: Mathematics Instruction,Grade 7,Middle School Students,Numbers,Thinking Skills,Number Concepts,Equations (Mathematics),Textbooks,Grade 5,Grade 6,Arithmetic,Qualitative Research

We interviewed 40 students in Grade 7 to investigate their integer reasoning. In one task, children were asked to write and interpret equations related to a story problem about borrowing money from a friend. Their responses reflect different perspectives concerning the relationship between this real-world situation and various numerical representations. We identify distinct ways in which integers were used and interpreted. All of the students solved the story problem correctly. Few thought about the story as involving negative numbers. When asked to interpret an equation involving negative numbers in relation to the story, about half related it to the story in an unconventional fashion, which contrasts with typical textbook approaches. These findings raise questions about the role of money and other contexts in integer instruction. [For the complete proceedings, see ED584829.]
Exponentiation Is Not Repeated Multiplication: Developing Exponentiation as a Continuous Operation
Author: Avitzur, Arnon
Date: 2012-11-00
Type: Reports - Evaluative
Subjects: Multiplication,Mathematics Instruction,Arithmetic,Models,Algebra,Fractions,Numbers,Mathematical Concepts,Learning Activities

The concept of exponents has been shown to be problematic for students, especially when expanding it from the domain of positive whole numbers to that of exponents that are negative and later rational. This paper presents a theoretical analysis of the concept of exponentiation as a continuous operation and examines the deficiencies of existing approaches to teaching it. Two complementary theoretical frameworks are used to suggest an alternative definition for exponentiation and guiding principles for the development of a teaching trajectory, and then to analyze an example of the hypothetical learning of a student who goes through the first task in the trajectory. The paper concludes with some possible implications on curriculum and task design, as well as on the development of mathematical operations. [For the complete proceedings, see ED584829.]
Middle School Students&apos; Example Use in Conjecture Exploration and Justification
Author: Ellis, Amy B.
Date: 2012-11-00
Type: Reports - Research
Subjects: Middle School Students,Mathematics Skills,Mathematical Logic,Problem Solving,Abstract Reasoning,Semi Structured Interviews,Coding,Data Analysis

Although students' difficulties in developing and understanding proofs in mathematics is well documented, less is known about how students' example use may support their proof practices, particularly at the middle school level. Research on example use suggests that strategic thinking with examples could play an important role in exploring conjectures and developing appropriate justifications. This paper introduces a framework of middle-school students' example exploration, distinguishing between the types of examples students use and the uses examples play in making sense of and proving conjectures. Drawing from clinical interviews with 20 students, we present thirteen categories of example types and seven categories of uses, followed by a discussion of each set of categories and their connections to one another. [For the complete proceedings, see ED584829.]
Two Forms of Reasoning about Amounts of Change in Covarying Quantities
Author: Johnson, Heather L.
Date: 2012-00-00
Type: Reports - Research
Subjects: Secondary School Students,Mathematics Skills,Algebra,Abstract Reasoning,Mathematical Concepts,Change,Computation,Mathematical Applications,Climate,Interviews

This paper addresses how secondary students might reason about amounts of change in covarying quantities. Two empirically based forms of covariational reasoning are distinguished. The first form-- reasoning about quantities as varying simultaneously and independently--supports tandem comparison of amounts of change. The second form--coordination of change in one quantity with change in a related quantity--supports coordinated comparison of amounts of change. By expanding the mental actions of Carlson et al.'s (2002) covariation framework, these forms of reasoning provide finer grained distinctions in the &quot;Quantitative Coordination&quot; level of covariational reasoning. Distinctions made between these forms of reasoning might help to explain how students could begin from informal reasoning to transition to more formal reasoning about average and instantaneous rate of change. [For the complete proceedings, see ED584829.]
Unpacking the Common Core State Standards for Mathematics: The Case of Length, Area and Volume
Author: Lee, KoSze
Date: 2012-11-00
Type: Reports - Research
Subjects: Common Core State Standards,Mathematics Instruction,Curriculum Implementation,Epistemology,Geometric Concepts,Learning Strategies,Educational Principles,Misconceptions,Teaching Methods,Educational Practices,Elementary Secondary Education

Adoption of the Common Core State Standards present challenges to school districts, school administrators, and teachers. To assist in this endeavor, we present our work on unpacking the CCSS-M for the length, area, and volume Learning Trajectory (LT). The overarching theme of &quot;genetic epistemology&quot; and a five-characteristic framework guided our work of unpacking the Standards. As a result, we added &quot;Bridging Standards&quot; to mediate students' progression through the length, area, and volume LT to provide a coherent structure through this trajectory. The implications of our work were discussed. [Additional support for this work was provided by the Oak Foundation. For the complete proceedings, see ED584829.]
Analysis of Two-Variable Function Graphing Activities
Author: Martínez-Planell, Rafael
Date: 2012-11-00
Type: Reports - Research
Subjects: Graphs,Geometric Concepts,Spatial Ability,College Students,Mathematics Skills,Learning Theories,Algebra,Mathematics Activities,Problem Solving,Interviews

This is a study about the didactical organization of a research based group of activities designed using APOS theory to help university students make constructions, needed to understand and graph two-variable functions, but found to be lacking in previous studies. The model of the &quot;moments of study&quot; of the Anthropological Theory of Didactics is applied to analyze the activities in terms of their institutional viability. [This project was partially supported by Asociación Mexicana de Cultura A.C. and the Instituto Tecnológico Autónomo de México. For the complete proceedings, see ED584829.]