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CAREER: Leveraging Contrasting Cases to Investigate Integer Understanding

Most students learn about negative numbers long after they have learned about positive numbers, and they have little time or opportunity to build on their prior understanding by contrasting the two concepts. The purpose of this CAREER project is to identify language factors and instructional sequences that contribute to improving elementary students' understanding of addition and subtraction problems involving negative integers. 

Lead Organization(s): 
Award Number: 
1350281
Funding Period: 
Thu, 05/15/2014 - Tue, 04/30/2019
Full Description: 

Currently, most students learn about negative numbers long after they have learned about positive numbers, and they have little time or opportunity to build on their prior understanding by contrasting the two concepts. Therefore, they struggle to make sense of negative integer concepts, which appear to conflict with their current understanding. The purpose of this CAREER project is to identify language factors and instructional sequences that contribute to improving elementary students' understanding of addition and subtraction problems involving negative integers. A second objective is to identify how elementary teachers interpret their students' integer understanding and use research findings to support their teaching of these concepts. This project is expected to contribute to theories regarding the development of integer understanding as well as what makes a useful contrasting case when learning new, related concepts. Moreover, the results of this project can contribute to our understanding of how to build on students? prior number knowledge rather than contradict it.

The principal investigator will conduct a series of four experimental studies involving a preparation for learning component with students randomly assigned to treatment or control groups. Study 1 will involve second and fourth graders and will test the language factors that support students' understanding of integers. Studies 2-4 will involve second and fifth graders and will test the optimal order in which integer addition and subtraction problems are presented in contrast with each other versus sequentially without contrasts. Using items that measure students? understanding of integers and integer operations, the PI will compare students' gains from pre-tests to post-tests between groups. Further, the investigator will qualitatively code students? solution strategies based on follow-up interviews and written work for additional information on the differences between groups. Following the experimental studies, the PI will work with elementary teachers over three lesson study cycles, during which teachers will implement instruction based on the prior studies? results. The PI will compare the performance of students who participate in the lesson study unit versus control classrooms to measure impact of the unit.

Videos of the lesson study unit, as well as the negative integer lesson plans will be made available for other teachers and teacher educators to use. Further, the investigator will incorporate the research results into an undergraduate mathematics methods course. To ensure that the results of this research reach a wider audience, the investigator will create an integer game and storybook, illustrating key concepts identified through the research, that parents can explore together with their children during family math nights and at home. On a broader scale, this project has the potential to illuminate ways to develop more coherence in the sequencing of mathematics topics to more effectively build on students? current understanding.

CAREER: Leveraging Contrasting Cases to Investigate Integer Understanding

Using Math Pathways and Pitfalls to Promote Algebra Readiness

This project that creates a set of materials for middle grades students and teacher professional development that would support the learning of early algebra. Building on their prior work with an elementary version, the efficacy study focuses on the implementation of the principals underlying the materials, fidelity of use of the materials, and impact on students' learning.

Lead Organization(s): 
Award Number: 
1314416
Funding Period: 
Tue, 10/01/2013 - Sat, 09/30/2017
Full Description: 

Using Math Pathways & Pitfalls to Promote Algebra Readiness is a 4-year Full Research and Development project that creates a set of materials for middle grades students and teacher professional development that would support the learning of early algebra. Building on their prior work with an elementary version, the efficacy study focuses on the implementation of the principals underlying the materials, fidelity of use of the materials, and impact on students' learning.

The project's goals are to: 1) develop an MPP book and companion materials dedicated to algebra readiness content and skills, 2) investigate how MPP transforms pedagogical practices to improve students' algebra readiness and metacognitive skills, and 3) validate MPP's effectiveness for improving students' algebra readiness with a large-scale randomized controlled trial.

The iterative design and efficacy studies produce research-based materials to increase student learning of core concepts in algebra readiness. Though the focus of the project is algebra readiness, the study also examines the validity of the pedagogical approach of MPP. The MPP lesson structures are designed to help students confront common misconceptions, dubbed "pitfalls," through sense-making, class discussions, and the use of multiple visual representations. If the pedagogical approach of MPP proves to be successful, the lesson structures can be presented as an effective framework for instruction that extends to other content areas in mathematics and other disciplines.

The project addresses a critical need in education, and the potential impact is large. Math achievement in the U.S. is not keeping pace with international performance. The current project focuses on algebra readiness skills, an area that is critical for future success in mathematics. Algebra often serves as a gatekeeper to more advanced mathematics, and performance in algebra has been linked to success in college and long-term earnings potential. Longitudinal studies indicate that students taking rigorous high school mathematics courses are twice as likely to graduate from college as those who do not. Thus, adequately preparing students for algebra can dramatically affect educational outcomes for students. The current project broadens the participation of underrepresented groups of students in math and later science classes that require strong math skills. The intervention builds on materials and pedagogical techniques that have demonstrated positive outcomes for diverse students. The targeted districts have large samples of English language learners and students from groups traditionally underrepresented in STEM so that we may evaluate the impact of the intervention on these populations. At the end of the project, the publication quality materials will be readily available to teachers and districts through our website www.wested.org/mpp.

Using Math Pathways and Pitfalls to Promote Algebra Readiness

Examining Formative Assessment Practices for English Language Learners in Science Classrooms (Collaborative Research: Li)

This is an exploratory study to identify critical aspects of effective science formative assessment (FA) practices for English Language Learners (ELLs), and the contextual factors influencing such practices. FA, in the context of the study, is viewed as a process contributing to the science learning of ELLs, as opposed to the administration of discrete sets of instruments to collect data from students. The study targets Spanish-speaking, elementary and middle school students.

Lead Organization(s): 
Award Number: 
1118951
Funding Period: 
Thu, 09/01/2011 - Sat, 08/31/2013
Project Evaluator: 
Advisory board members
Full Description: 

This is a two-year exploratory study to identify critical aspects of effective science formative assessment (FA) practices for English Language Learners (ELLs), and the contextual factors influencing such practices. Three institutions join efforts for this purpose: University of Colorado at Boulder, University of Colorado at Denver, and University of Washington. FA, in the context of the study, is viewed as a process contributing to the science learning of ELLs, as opposed to the administration of discrete sets of instruments to collect data from students. The study targets Spanish-speaking, elementary and middle school students. Findings from this study contribute to advance knowledge and understanding of FA as an inherent component of the science learning process in linguistically diverse classrooms, and to define a research agenda aimed at enhancing science teachers' ability to enact equitable and effective assessment practices for this student subpopulation.

Three research questions guide the work: (1) What FA practices are occurring in science classrooms that serve predominantly mainstream students and in those serving predominantly ELLs?; (2) How are teachers' FA practices for mainstream students different from or similar to those used with ELLs?; and (3) How do contextual factors and teachers' cultural and linguistic competencies influence FA practices? To address these questions, two conceptual frameworks are used--one for characterizing FA events; the other for examining FA events as a communication process. The study employs a mixed-methods research approach with emphasis on case studies. The sample size consists of three school districts in Colorado and Washington, 16 classrooms (8 elementary, 8 middle school), 16 teachers, and 96 ELLs. Classrooms are selected to represent a particular combination of four factors: (a) teacher ethnicity, (b) teacher formal academic preparation in teaching ELLs, (c) type of linguistic student background, and (d) grade level. Students are selected through a stratified random sample, identified by achievement level (i.e., low, medium, high), and linguistic background (i.e., mainstream, ELL). Data collection strategies to document the implementation of FA at the beginning, during, and at the end of a science unit include: (a) classroom observation protocols, (b) classroom video-recording, (c) video/artifact simulated recall, (d) assessment artifacts, (e) student interviews, (f) teacher questionnaires, (g) teacher interviews, (h) school principal interviews, and (i) school observations. Reliability and validity of most of the data-gathering instruments is determined through pilot studies. Data interpretation strategies include: (a) coding based on the two conceptual frameworks, (b) scoring rubrics to identify levels of effectiveness, and (c) narratives and profiles to describe FA patterns. Publications and the development of a website constitute the main dissemination strategies. A technical advisory board is responsible for formative and summative evaluation. Key evaluation questions are: (1) To what extent does the project enhance research on ELL FA practices through case studies?, and (2) How effectively do the project dissemination activities facilitate understanding of FA practices?

Major project outcomes include: (1) a description of the patterns of formal and informal FA practices for ELLs; (2) a comparison of the FA practices observed in classrooms that vary on the dimensions of teacher characteristics and linguistic diversity; and (3) an empirically and theoretically informed set of findings and strategies for supporting teachers to enact and enhance FA practices sensitive to cultural and linguistic diversity. Three main products are developed: (1) a monograph describing the FA practices observed across the different classrooms with concrete examples; (2) a description of possible professional development strategies to improve in-service FA practices for linguistically diverse students; and (3) a research-informed approach for analyzing FA practices. Besides filling the existing research gap on FA with ELLs, outcomes and products serve as a foundation for a future research agenda and a comprehensive project aimed at ensuring equitable science learning for all students, including ELLs.

Examining Formative Assessment Practices for English Language Learners in Science Classrooms (Collaborative Research: Li)

Taking Foundation Science to Scale—Digitally: Transforming a Print Curriculum into an Innovative Learning Tool for Commercial Distribution

This project provides a model of how existing, tested digital enhancements can increase student learning. Increasing the quality of science education requires careful coupling of effective, research-based curricula with innovative digital features that deepen and enhance science learning and teaching. This RAPID is to ensure that the content and pedagogical expertise is present during the development of the digital version of Foundation science.

Partner Organization(s): 
Award Number: 
1119736
Funding Period: 
Fri, 04/01/2011 - Sun, 03/31/2013
Full Description: 

The print version of Foundation Science, a comprehensive high school science curriculum, has been extensively field tested and shown to be effective in increasing student learning and changing teacher practice. Carolina Biological Supply is scheduled to publish a digital version of Biology and Chemistry portions of Foundation Science that goes well beyond the conversion of print text to digital delivery by September 2012. Many digital enhancements have been developed and tested in the biology unit of Foundation Science, which was used as a model to develop a system to incorporate Universal Design for learning features in materials development and in on-line professional development for cross-over teachers. Some of the digital resources include a digital book reader; a notebook in which notes can take various forms such as text, drawing, voice recording; separate unscored assessments; an interactive glossary; graphing capabilities and an online research tool.

Thus this project provides a model of how existing, tested digital enhancements can increase student learning. Increasing the quality of science education requires careful coupling of effective, research-based curricula with innovative digital features that deepen and enhance science learning and teaching. This RAPID is to ensure that the content and pedagogical expertise is present during the development of the digital version of Foundation science.

Taking Foundation Science to Scale—Digitally: Transforming a Print Curriculum into an Innovative Learning Tool for Commercial Distribution

Development of a Cognition-Guided, Formative-Assessment-Intensive, Individualized Computer-Based Dynamic Geometry Learning System for Grades 3-8

This project is focused on creating, testing, refining, and studying a computer-based, individualized, interactive learning system for intermediate/middle school students or by teachers in classrooms. This learning system is called Individualized Dynamic Geometry Instruction and will contain four instructional modules in geometry and measurement that reflect the recommendations of the Common Core State Standards.

Lead Organization(s): 
Award Number: 
1119034
Funding Period: 
Thu, 09/01/2011 - Mon, 08/31/2015
Project Evaluator: 
Jeff Shih
Full Description: 

Developers and researchers at Ohio State University and KCP Technologies are creating, testing, refining, and studying a computer-based, individualized, interactive learning system for intermediate/middle school students that can be used by them independently (online or offline) or by teachers in classrooms. This learning system is called Individualized Dynamic Geometry Instruction (iDGi) and will contain four instructional modules in geometry and measurement that reflect the recommendations of the Common Core State Standards (CCSS). iDGi courseware fully integrates research-based Learning Progressions (LPs) for guiding students' reasoning; formative-assessment linked to LPs; instructional sequencing that interactively adapts to students' locations in LPs; built-in student monitoring, feedback, and guidance; and research-based principles of educational media into the modules. The software platform for iDGi development is an extended version of the dynamic geometry computer environment, The Geometer's Sketchpad.

The development process follows recommendations in Douglas Clements' Curriculum Research Framework and includes sequences of development, trials with students, data collection, and revision. The research and evaluation are based on random assignment of approximately 350 students to treatment and control groups. Achievement data are collected using developer-constructed instruments with items that reflect the mathematics topics in the CCSS. Researchers explore the variability at the student, teacher, and school levels using the appropriate level of hierarchical linear models.

Commercial publishers have expressed strong interest in publishing online and offline computer versions of iDGi, an iPad version of iDGi, an online management system for iDGi, and support materials for users and teachers.

Development of a Cognition-Guided, Formative-Assessment-Intensive, Individualized Computer-Based Dynamic Geometry Learning System for Grades 3-8

Establishing a Roadmap for Large-Scale Improvement of K-12 Education in the Geographical Sciences

This project will engage in a community-wide effort to synthesize the literature from a broad range of fields and to use the findings to create frameworks that will guide the planning, implementation, and scale-up of efforts to improve geographic education over the next decade. This will result in a set of publicly reviewed, consensus reports that will guide collaborative efforts and broaden awareness of the acute need for geographic literacy and geographic science education.

Lead Organization(s): 
Award Number: 
1049437
Funding Period: 
Wed, 09/15/2010 - Fri, 08/31/2012
Project Evaluator: 
Education and Training Institute
Full Description: 

Having a geographically literate population will be critical to the economic stability, physical security, and environmental sustainability of the United States in the 21st century. Yet the U.S. still lags far behind the other developed nations in education in the geographical sciences. Recognizing the risk that geographic illiteracy poses for our country, the National Geographic Society (NGS), in collaboration with the Association of American Geographers, American Geographical Society, and National Council for Geographic Education, proposes to engage in a set of research synthesis and dissemination activities that will provide road maps for the design of assessment, professional development, instructional materials, public information, and educational research for the next decade. The work will be done by a broad range of experts from K-12 institutions as well as the geographical science and educational research communities

Building on a 25 year collaboration, NGS and its partners propose to engage in a community-wide effort to synthesize the literature from a broad range of fields and to use the findings to create frameworks that will guide the planning, implementation, and scale-up of efforts to improve geographic education over the next decade. The result of this effort will be a set of publicly reviewed, consensus reports that will guide the collaborative efforts of the project partners and the larger geographic education community, as well as broaden awareness of the increasingly significant and acute need for geographic literacy and education in the geographical sciences in our country.

This project will create three in-depth "roadmap" reports targeted at practitioners, takeholders, and policymakers. Developed by expert committees, these three reports will be on:

- Assessment frameworks for systematic monitoring and continuous improvement of geographic education programs.

- Professional development for teachers and instructional materials to support large-scale educational improvement across diverse contexts.

- Educational research agenda to set priorities and identify appropriate methodologies for research that will improve geographic education into the future.

These three reports will be summarized in an executive summary written for a broad audience of educators, policymakers, and concerned citizens.

In addition to these consensus reports, the project will also conduct research on public understanding of the nature and importance of geographic literacy, with particular attention to the key audiences of educators, policymakers, and citizens. In addition to shaping the project's reports, this research will inform the broader communications and dissemination efforts of this project and its partners.

Establishing a Roadmap for Large-Scale Improvement of K-12 Education in the Geographical Sciences

The Value of Computational Thinking Across Grade Levels

This project is developing and testing a set of 12 curriculum modules designed to engage high school students and their teachers in the process of applying computational concepts and methods to problem solving in a variety of scientific contexts. The project perspective is that computational thinking can be usefully thought of as a specialized form of mathematical modeling.

Project Email: 
vctal@dimacs.rutgers.edu
Lead Organization(s): 
Award Number: 
1020201
Funding Period: 
Thu, 07/01/2010 - Mon, 06/30/2014
Project Evaluator: 
Len Albright and Andrea Weinberg at CSU
Full Description: 

The Value of Computational Thinking (VCT) project combines the talents and resources of STEM professionals at the Rutgers University DIMACS Center, the Consortium for Mathematics and Its Applications (COMAP), Colorado State University, Hobart and William Smith College, the Computer Science Teachers Association, and five partner school districts to develop and test a set of 12 curriculum modules designed to engage high school students and their teachers in the process of applying computational concepts and methods to problem solving in a variety of scientific contexts. The project perspective is that computational thinking can be usefully thought of as a specialized form of mathematical modeling. The product of computational thinking in a particular domain is a model of a situation, a structuring and representation of the situation, that enables computations to be performed to answer questions, solve problems, control processes, predict consequences, or enhance understanding.

Since computational thinking is a relatively new construct in STEM and STEM education, there are few available curriculum materials to support instruction intended to develop the understanding, habits of mind, and specific techniques that are involved. The fundamental goal of the VCT project is to answer an engineering research question: "What kinds of instructional materials and learning experiences will develop effective computational thinking skills and attitudes?" The VCT project is applying a design research process involving iterative phases of development, pilot testing, and revision to produce prototype instructional materials that will be useful as stand-alone curriculum modules or when collected into different packages to support full high school courses. Project field test evaluation will provide preliminary evidence about the efficacy of the materials in developing desired student learning.

Proponents of computational thinking in STEM and STEM education have argued that it offers a powerful general approach to problem solving in discipline-specific and inter-disciplinary settings. They also argue that, when properly taught, it can provide an effective introduction and attraction to careers in computer science and other computing-intensive fields. Thus the VCT project has a long-term goal of broadening participation in computer science and related technology fields. Materials are being designed with special features to enhance their effectiveness in reaching this objective.

The Value of Computational Thinking Across Grade Levels

CAREER: Teaching Practices That Support Fraction-Based Algorithmic Thinking

The research and educational activities of this project focus on advancing the field in the area of fraction operation algorithm development. The goal of this research is to identify core mathematical teaching practices that engage and support students in algorithmic thinking associated with fraction operations. The educational product of this work will be written educational materials that can be used to support the general population of teachers in this domain.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
0952661
Funding Period: 
Sun, 08/15/2010 - Fri, 07/31/2015
Full Description: 

The research and educational activities of this project focus on advancing the field in the area of fraction operation algorithm development. The goal of the research is to identify core mathematical teaching practices that engage and support students in algorithmic thinking associated with fraction operations. The project has four objectives: (1) understand and document local instructional theories and routines of practice exemplary teachers use as they engage students in algorithmic thinking for fraction operations; (2) develop a prototypical model of core routines of practice generated from exemplary teachers that support students as they engage in algorithmic thinking for fraction operations; (3) design, pilot and study the usability of the prototypical model as a professional development tool with typical teachers as they engage students in algorithmic thinking for fraction operations; and (4) identify specific core routines of practice that are shown to be productive for use with typical teachers and explore ways of disseminating them at a larger scale.

Phase One involves collaborative work with and intensive observational study of the practice of four exemplary teachers while they teach a common unit on fraction operations. Individual teacher case and cross-case examination will focus on mathematical development, use of representational modes, aspects of purposefulness in the teachers' practice, and discursive considerations and positioning of the learner. The identified local instructional theories and routines of practice that emerge will lead to the development of a prototypical model of teaching practice that engages students in algorithmic thinking for fraction operations. The prototypical model will frame the development of activities and tools to be piloted and studied with 15 typical middle grade teachers from urban and suburban districts in Phase Two. A study group focused professional development setting will be used to implement, study, and refine frameworks, activities and tools.

The product of this work will be written educational materials that can be used by other teacher educators to support the general population of teachers in this domain. These materials will identify core routines of practice associated with algorithmic thinking for fraction operations and offer activities and tools to support their development in practice. The domain of rational number has been identified as the most challenging and the most essential topic in elementary and middle school mathematics needed to succeed in higher mathematics and science. Moving forward in this area is critical in the successful preparation of students for STEM careers.

CAREER: Teaching Practices That Support Fraction-Based Algorithmic Thinking

CAREER: A Study of Strategies and Social Processes That Facilitate the Participation of Latino English Language Learners in Elementary Mathematics Classroom Communities

The project aims to: (1) study resources and strategies for teachers that will facilitate participation of 3rd grade Latino English Language Learners (ELLs) in the mathematics classrooms; (2) develop related teacher professional development (PD) materials; and (3) integrate research and teaching activities. The basic research question is: How can 3rd grade teachers facilitate better mathematics instruction for ELLs?

Lead Organization(s): 
Award Number: 
0844556
Funding Period: 
Wed, 07/15/2009 - Mon, 06/30/2014
Full Description: 

The project aims to: (1) study resources and strategies for teachers that will facilitate participation of 3rd grade Latino English Language Learners (ELLs) in the mathematics classrooms; (2) develop related teacher professional development (PD) materials; and (3) integrate research and teaching activities. The basic research question is: How can 3rd grade teachers facilitate better mathematics instruction for ELLs? The PI will conduct a longitudinal study with teachers over three years in nine third-grade classrooms involving 20 Latino ELLs in each classroom. Data (district-administered assessments, one-on-one formal interviews, classroom artifacts, brief conversations with children following the videotaped lessons) will be collected from multiple sources including the use of head-mounted cameras to videotape classroom social processes. Also, existing research base and data from the nine classrooms will be used to develop, test, and publish PD materials for pre-service and practicing elementary teachers. The integrated education activities will have a direct impact on the design of University of Missouri Teacher Development Program, Masters Programs, TESOL Certification Program, and Mathematics Education Doctoral Program as well as local schools. The proposed project is qualitative. Teachers will also participate in a three-year professional development on ELL strategies and use classroom activities in the fall semester that are designed to assist 3rd grade Latinos acquire mathematical competence on various aspects of the number sense strand of the mathematics curriculum. All the relevant instruments will be collected and analyzed. Findings from the proposed research and integrated research/education activities will expand the knowledge base in the fields of elementary mathematics education and multilingual education.

The project research and education activities will inform our understanding of effective strategies in mathematics classrooms that will influence learning processes and ultimately student outcomes relevant to Latino ELLs and potentially other groups of underrepresented students. Findings from the project will promote effective teaching and learning of mathematics at the elementary level and will be broadly disseminated to students, teachers, teacher educators, and education researchers throughout the U.S. The PI will create a vehicle (i.e., PD materials) for U.S. elementary teachers to discuss critical issues related to engaging a group of students that have typically been non-participants in mathematics classrooms and potentially facilitate more effective participation for this growing population.

CAREER: A Study of Strategies and Social Processes That Facilitate the Participation of Latino English Language Learners in Elementary Mathematics Classroom Communities

CAREER: Examining the Role of Context in the Mathematical Learning of Young Children

This project involves a longitudinal, ethnographic study of children's mathematical performances from preschool to first grade in both formal classroom settings and informal settings at school and home. The study seeks to identify opportunities for mathematical learning, to map varied performances of mathematical competence, to chart changes in mathematical performance over time, and to design and assess the impact of case studies for teacher education.

Partner Organization(s): 
Award Number: 
0844445
Funding Period: 
Mon, 06/15/2009 - Tue, 05/31/2011
Full Description: 

This project involves a longitudinal, ethnographic study of children's mathematical performances from preschool to first grade in both formal classroom settings and informal settings at school and home. The proposed site for the study is a small, predominately African-American pk-12 school. The study seeks to identify opportunities for mathematical learning by young children across multiple contexts, to map varied performances of mathematical competence by young children, to chart changes in young children's mathematical performance over time, and to design and assess the impact of case studies for teacher education that explore young children's mathematical competencies. Research questions focus on mathematical opportunities for learning in various contexts, children's development of knowledge, skills, and dispositions over time, the characteristics of competent mathematical performances, and the role of case studies in helping beginning teachers to understand young minority children's mathematical thinking. Data collected will include video tapes of classroom activities, written fieldnotes of formal and informal settings, student work, parent focus group transcripts, and children's interview performances. Analysis will involve both thematic coding and construction of case studies. The overarching goal of this project is to transform the ways that researchers think about and study the mathematical learning of young minority children as well as the quality of schooling these children experience.

CAREER: Examining the Role of Context in the Mathematical Learning of Young Children
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