Algebra

Co-Designing for Statewide Alignment of a Vision for High-Quality Mathematics Instruction (Collaborative Research: Wilson)

This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

Award Number: 
2100903
Funding Period: 
Thu, 07/15/2021 to Mon, 06/30/2025
Full Description: 

Mathematics teaching and learning is influenced by policy and practice at the state, district, and school levels. To support large-scale change, it is important for high-quality mathematics instruction to be aligned and cohesive across each level of the education system. This can be supported through regional partnerships among state, district, and school-based leaders, mathematics teachers, education researchers, and mathematicians. Such partnerships create instructional tools and resources to document the vision for instruction. For example, teams can work together to create instructional frameworks for each grade band that describe standards, mathematics teaching, and units for teaching. This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

This project will develop a shared vision of high-quality mathematics instruction intended to improve systemic coherence during the implementation of education innovations. The project uses a research-practice partnership with a design-based implementation research design. To examine and support implementation of the vision, partners will continue a process of developing instructional frameworks, research and practice briefs, as well as additional resources as needed by stakeholders in the system. Engaging partners at all levels of the system is a central component of developing the shared vision of instruction. This project includes three major research questions. First, what are visions of high-quality mathematics instruction held by educators at different levels of a state educational system? Second, in what ways do educators' visions of high-quality mathematics instruction mediate their use of implementation resources in practice? Finally, in what ways do educators’ visions of high-quality mathematics instruction mediate their participation in the co-design of implementation resources? An activity theory framework is used to understand the interactions between partners at different levels in the system and the creation of artifacts during the design process. The research methods for the study are situated in design-based research to capture the conjectures, instructional resources, design processes, and outcomes of the process. The project will use case studies of partner districts, data gathering from interactions with partners, artifacts of the design process, and other documentation to understand how the vision is created and enacted in different settings and to develop an empirically supported design framework and methodology for implementing STEM innovations at scale that centralizes a shared instructional vision.

Co-Designing for Statewide Alignment of a Vision for High-Quality Mathematics Instruction (Collaborative Research: Mawhinney)

This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

Lead Organization(s): 
Award Number: 
2100833
Funding Period: 
Thu, 07/15/2021 to Mon, 06/30/2025
Full Description: 

Mathematics teaching and learning is influenced by policy and practice at the state, district, and school levels. To support large-scale change, it is important for high-quality mathematics instruction to be aligned and cohesive across each level of the education system. This can be supported through regional partnerships among state, district, and school-based leaders, mathematics teachers, education researchers, and mathematicians. Such partnerships create instructional tools and resources to document the vision for instruction. For example, teams can work together to create instructional frameworks for each grade band that describe standards, mathematics teaching, and units for teaching. This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

This project will develop a shared vision of high-quality mathematics instruction intended to improve systemic coherence during the implementation of education innovations. The project uses a research-practice partnership with a design-based implementation research design. To examine and support implementation of the vision, partners will continue a process of developing instructional frameworks, research and practice briefs, as well as additional resources as needed by stakeholders in the system. Engaging partners at all levels of the system is a central component of developing the shared vision of instruction. This project includes three major research questions. First, what are visions of high-quality mathematics instruction held by educators at different levels of a state educational system? Second, in what ways do educators' visions of high-quality mathematics instruction mediate their use of implementation resources in practice? Finally, in what ways do educators’ visions of high-quality mathematics instruction mediate their participation in the co-design of implementation resources? An activity theory framework is used to understand the interactions between partners at different levels in the system and the creation of artifacts during the design process. The research methods for the study are situated in design-based research to capture the conjectures, instructional resources, design processes, and outcomes of the process. The project will use case studies of partner districts, data gathering from interactions with partners, artifacts of the design process, and other documentation to understand how the vision is created and enacted in different settings and to develop an empirically supported design framework and methodology for implementing STEM innovations at scale that centralizes a shared instructional vision.

Co-Designing for Statewide Alignment of a Vision for High-Quality Mathematics Instruction (Collaborative Research: Schwartz)

This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

Lead Organization(s): 
Award Number: 
2100895
Funding Period: 
Thu, 07/15/2021 to Mon, 06/30/2025
Full Description: 

Mathematics teaching and learning is influenced by policy and practice at the state, district, and school levels. To support large-scale change, it is important for high-quality mathematics instruction to be aligned and cohesive across each level of the education system. This can be supported through regional partnerships among state, district, and school-based leaders, mathematics teachers, education researchers, and mathematicians. Such partnerships create instructional tools and resources to document the vision for instruction. For example, teams can work together to create instructional frameworks for each grade band that describe standards, mathematics teaching, and units for teaching. This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

This project will develop a shared vision of high-quality mathematics instruction intended to improve systemic coherence during the implementation of education innovations. The project uses a research-practice partnership with a design-based implementation research design. To examine and support implementation of the vision, partners will continue a process of developing instructional frameworks, research and practice briefs, as well as additional resources as needed by stakeholders in the system. Engaging partners at all levels of the system is a central component of developing the shared vision of instruction. This project includes three major research questions. First, what are visions of high-quality mathematics instruction held by educators at different levels of a state educational system? Second, in what ways do educators' visions of high-quality mathematics instruction mediate their use of implementation resources in practice? Finally, in what ways do educators’ visions of high-quality mathematics instruction mediate their participation in the co-design of implementation resources? An activity theory framework is used to understand the interactions between partners at different levels in the system and the creation of artifacts during the design process. The research methods for the study are situated in design-based research to capture the conjectures, instructional resources, design processes, and outcomes of the process. The project will use case studies of partner districts, data gathering from interactions with partners, artifacts of the design process, and other documentation to understand how the vision is created and enacted in different settings and to develop an empirically supported design framework and methodology for implementing STEM innovations at scale that centralizes a shared instructional vision.

Co-Designing for Statewide Alignment of a Vision for High-Quality Mathematics Instruction (Collaborative Research: McCulloch)

This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

Award Number: 
2100947
Funding Period: 
Thu, 07/15/2021 to Mon, 06/30/2025
Full Description: 

Mathematics teaching and learning is influenced by policy and practice at the state, district, and school levels. To support large-scale change, it is important for high-quality mathematics instruction to be aligned and cohesive across each level of the education system. This can be supported through regional partnerships among state, district, and school-based leaders, mathematics teachers, education researchers, and mathematicians. Such partnerships create instructional tools and resources to document the vision for instruction. For example, teams can work together to create instructional frameworks for each grade band that describe standards, mathematics teaching, and units for teaching. This project will develop a process for creating a shared, state-wide vision of high-quality mathematics instruction. It will also develop and study the resources to implement that vision at the state, district, and school levels. In addition, the project will investigate a collaborative process of designing and implementing high-quality mathematics instruction at a state level.

This project will develop a shared vision of high-quality mathematics instruction intended to improve systemic coherence during the implementation of education innovations. The project uses a research-practice partnership with a design-based implementation research design. To examine and support implementation of the vision, partners will continue a process of developing instructional frameworks, research and practice briefs, as well as additional resources as needed by stakeholders in the system. Engaging partners at all levels of the system is a central component of developing the shared vision of instruction. This project includes three major research questions. First, what are visions of high-quality mathematics instruction held by educators at different levels of a state educational system? Second, in what ways do educators' visions of high-quality mathematics instruction mediate their use of implementation resources in practice? Finally, in what ways do educators’ visions of high-quality mathematics instruction mediate their participation in the co-design of implementation resources? An activity theory framework is used to understand the interactions between partners at different levels in the system and the creation of artifacts during the design process. The research methods for the study are situated in design-based research to capture the conjectures, instructional resources, design processes, and outcomes of the process. The project will use case studies of partner districts, data gathering from interactions with partners, artifacts of the design process, and other documentation to understand how the vision is created and enacted in different settings and to develop an empirically supported design framework and methodology for implementing STEM innovations at scale that centralizes a shared instructional vision.

Supporting Teachers to Teach Mathematics through Problem Posing

This project aims to support teachers to engage their students in mathematical problem posing (problem-posing-based learning, or P-PBL). P-PBL is a powerful approach to the teaching and learning of mathematics, and provides students with opportunities to engage in authentic mathematical practices.

Lead Organization(s): 
Award Number: 
2101552
Funding Period: 
Sun, 08/01/2021 to Thu, 07/31/2025
Full Description: 

This project aims to support teachers to engage their students in mathematical problem posing (problem-posing-based learning, or P-PBL). P-PBL is a powerful approach to the teaching and learning of mathematics, and provides students with opportunities to engage in authentic mathematical practices. For example, conjecturing in mathematics, a form of problem posing, often plays an important role in solving complex problems, and problem posing is an important component of mathematical modeling. Yet despite its importance, widely used curriculum materials fail to incorporate P-PBL in substantial and consistent ways, leaving teachers with few resources to enact this process. This project will develop problem-posing lessons and illustrative cases of teachers implementing P-PBL that will not only support teachers to develop a vision of what P-PBL looks like and how to implement it in their own classrooms, but will also serve as rich resources for professional development (PD) providers. This project will generate valuable findings about teaching using problem posing for district administrators, mathematics teachers, educators, and researchers as well as curriculum developers and policy makers. The team will develop and pilot a set of 20−30 research-based P-PBL cases that provide critical details for the implementation of P-PBL and reveal “lessons learned” from the development process.

The project promises broader impact on the field of mathematics education as the first goal is to support teachers to teach mathematics through engaging their students in mathematical problem posing. By guiding students to construct and investigate their own problems, P-PBL both helps to create mathematical learning opportunities and develops students’ mathematical agency and positive mathematical identities. A networked improvement community of teachers and researchers will integrate problem posing into daily mathematics instruction and continuously improve the quality of P-PBL through iterative task and lesson design. The intellectual merit of this project is its contribution of new and important insights about teaching mathematics through problem posing. This will be realized through the second project goal which is to longitudinally investigate the promise of supporting teachers to teach with P-PBL for enhancing teachers’ instructional practice and students’ learning. A quasi-experimental design coupled with design-based research methodology and improvement science will be used to understand how, when, and why P-PBL works in practice. Specifically, we plan to follow a sample of 36 teachers and their approximately 3,600 students from six middle schools for multiple years to longitudinally explore the promise of P-PBL for developing teachers’ beliefs about problem posing, their beliefs about P-PBL, and their actual instructional practice. We will also investigate students’ learning as measured by problem-posing performance, problem-solving performance, and mathematics disposition. The findings of the project will add not only to the field’s understanding of the promise of supporting teachers to integrate P-PBL into their mathematics instruction, but also to its understanding of the challenges that teachers face when engaging in a networked improvement community that is focused on improving tasks and lessons by integrating P-PBL.

Practice-Driven Professional Development for Algebra Teachers

This project seeks to develop a personalized, scalable PD approach that centers on and builds from algebra teachers’ practices and individual strengths. The project will focus its PD efforts on instructional actions that are tailored to teachers' existing practice, can be readily adopted, and are easily accessible.

Lead Organization(s): 
Award Number: 
2101508
Funding Period: 
Thu, 07/01/2021 to Mon, 06/30/2025
Full Description: 

Professional development (PD) is a direct attempt to improve the quality of instruction for teachers already in the classroom. Traditional PD is typically costly in terms of time and money, and efforts tend to be delivered as a one-size-fits-all approach. Furthermore, for teachers who adopt novel techniques such as flipped instruction, there may be few resources to support their efforts. This project seeks to develop a personalized, scalable PD approach that centers on and builds from algebra teachers’ practices and individual strengths. The project will focus its PD efforts on instructional actions that are tailored to teachers' existing practice, can be readily adopted, and are easily accessible. The project team have termed such instructional actions high-uptake practices. The project will develop and field test PD materials to support algebra teachers at scale via these high-uptake practices.

In addition to developing the PD materials, the project team will research the efficacy of this PD model in terms of student learning outcomes and teacher instructional practices in approximately 60 algebra classrooms. The main data sources will include teacher observation data, teacher interviews and surveys, student pre/posttests, student surveys, and PD analytics. The research will characterize the immediate and longer term impacts of the PD on teachers’ instructional practices; and characterize the impact of teachers’ participation in the PD on students’ learning outcomes and engagement. The research questions include: 1) In what ways does teachers’ participation in the PD impact their instructional practices? 2) Do students of teachers who participate in the PD demonstrate differential growth in learning outcomes? 3) Do students of teachers who participate in the PD have increased rates of homework completion?; and 4) Do students of teachers who participate in the PD have increased engagement during individual work time? In meeting both our PD development and research aims, this project will contribute knowledge about the effectiveness of an incremental, practice-driven approach to PD and instructional change.

Developing and Researching K-12 Teacher Leaders Enacting Anti-bias Mathematics Education (Collaborative Research: Yeh)

The goal of this project is to study the design and development of community-centered, job-embedded professional development for classroom teachers that supports bias reduction. The project team will partner with three school districts serving racially, ethnically, linguistically, and socio-economically diverse communities, for a two-year professional development program.

Lead Organization(s): 
Award Number: 
2101666
Funding Period: 
Sun, 08/01/2021 to Thu, 07/31/2025
Full Description: 

There is increased recognition that engaging all students in learning mathematics requires an explicit focus on anti-bias mathematics teaching. Teachers, even with positive intentions, have biases, causing them to treat students differently and impacting how they distribute students’ opportunities to learn in K-12 mathematics classrooms. Research is needed to examine models of mathematics teacher professional development that explicitly addresses bias reduction. The goal of this project is to study the design and development of community-centered, job-embedded professional development for classroom teachers that supports bias reduction. The project team will partner with three school districts serving racially, ethnically, linguistically, and socio-economically diverse communities, for a two-year professional development program. The aim is to reduce bias through: analyzing and designing mathematics teaching with colleagues, students, and families to create classrooms and schools based on community-centered mathematics; engaging in anti-bias teaching routines; and building relationships with parents, caretakers, and community members. The project team will study teacher leader professional development, including the professional development model, framework, and tools, along with what teacher leaders across district contexts and grade-levels take up and use in their instructional practice.  This will potentially have wider implications for supporting more equitable mathematics teaching and leadership. Project activities, resources, and tools will be shared with the broader community of mathematics educators and researchers for use in other contexts.

The goal of this two-phase, design based research project is to iteratively design and research teacher leaders’ (TLs) participation in community-centered, job-embedded professional development and investigate their subsequent impact on classrooms, schools, and districts. The project builds on the existing Math Studio professional development model to create a Community Centered Math Studio, integrating the Anti-bias Mathematics Education Framework into the work. The project seeks to understand how the professional development model supports the development of teacher leaders' knowledge, dispositions, and practices for teaching and leading anti-bias mathematics education, and how teachers' subsequent classroom practice can cultivate students' mathematical engagement, discourse, and interests. The project will measure aspects of teacher knowledge and classroom practice by integrating existing classroom observation rubrics and STEM interest surveys to assess the impact on teacher classroom practice and student outcomes. The project will engage 12 TLs and approximately 60 additional teachers working with those TLs in two years of professional development using the Community Centered Math Studio Model to support anti-bias mathematics teaching. Data will be collected for all teachers related to their participation in the professional learning, with six teachers being followed for additional data collection and in-depth case studies. The project's outcomes will contribute to theories of how TLs build adaptive expertise for teaching and leading to reduce bias in classrooms, departments, schools, and districts. In addition, the project will contribute new and adapted research instruments on anti-bias teaching and leading. The research outcomes will add to the growing research base that describes the nature of equitable mathematics teaching in K-12 classrooms and increases access to meaningful mathematics for students, teachers, and communities.

Pandemic Learning Loss in U.S. High Schools: A National Examination of Student Experiences

As a result of the COVID-19 pandemic, schools across much of the U.S. have been closed since mid-March of 2020 and many students have been attempting to continue their education away from schools. Student experiences across the country are likely to be highly variable depending on a variety of factors at the individual, home, school, district, and state levels. This project will use two, nationally representative, existing databases of high school students to study their experiences in STEM education during the COVID-19 pandemic.

Lead Organization(s): 
Award Number: 
2030436
Funding Period: 
Fri, 05/15/2020 to Sat, 04/30/2022
Full Description: 

As a result of the COVID-19 pandemic, schools across much of the U.S. have been closed since mid-March of 2020 and many students have been attempting to continue their education away from schools. Student experiences across the country are likely to be highly variable depending on a variety of factors at the individual, home, school, district, and state levels. This project will use two, nationally representative, existing databases of high school students to study their experiences in STEM education during the COVID-19 pandemic. The study intends to ascertain whether students are taking STEM courses in high school, the nature of the changes made to the courses, and their plans for the fall. The researchers will identify the electronic learning platforms in use, and other modifications made to STEM experiences in formal and informal settings. The study is particularly interested in finding patterns of inequities for students in various demographic groups underserved in STEM and who may be most likely to be affected by a hiatus in formal education.

This study will collect data using the AmeriSpeak Teen Panel of approximately 2,000 students aged 13 to 17 and the Infinite Campus Student Information System with a sample of approximately 2.5 million high school students. The data sets allow for relevant comparisons of student experiences prior to and during the COVID-19 pandemic and offer unique perspectives with nationally representative samples of U.S. high school students. New data collection will focus on formal and informal STEM learning opportunities, engagement, STEM course taking, the nature and frequency of instruction, interactions with teachers, interest in STEM, and career aspirations. Weighted data will be analyzed using descriptive statistics and within and between district analysis will be conducted to assess group differences. Estimates of between group pandemic learning loss will be provided with attention to demographic factors.

This RAPID award is made by the DRK-12 program in the Division of Research on Learning. The Discovery Research PreK-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics by preK-12 students and teachers, through the research and development of new innovations and approaches. Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for the projects.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

 

 

 

 

CAREER: Implementing Mathematical Modeling for Emergent Bilinguals

This project will support teacher capacity for implementing mathematical modeling lessons by engaging teachers in co-planning and co-teaching with researchers skilled in Emergent Bilingual (EB) mathematics instruction. The outcomes of this project will be a framework for teaching mathematical modeling to EB students, teacher professional development materials that can be used widely to support EB mathematics teachers, and a massive open online course (MOOC) for teachers to support their continued learning about teaching mathematics modeling to EB students.

Lead Organization(s): 
Award Number: 
1941668
Funding Period: 
Tue, 09/01/2020 to Sun, 08/31/2025
Full Description: 

This project supports secondary mathematics teachers in teaching mathematical modeling practices to an Emergent Bilingual (EB) population. EB students in linguistically diverse mathematics classrooms are frequently limited to procedural, rote instruction, despite research-based recommendations that suggest that EBs' mathematical and linguistic proficiency can benefit from engaging in complex mathematical tasks based on real-life situations. The project will support teacher capacity for implementing mathematical modeling lessons by engaging teachers in co-planning and co-teaching with researchers skilled in EB mathematics instruction. The project will collect information about the quality of mathematics instruction in modeling lessons, what students learn, and how teachers changed in how they position EB students as knowers and doers of mathematics. The outcomes of this project will be a framework for teaching mathematical modeling to EB students, teacher professional development materials that can be used widely to support EB mathematics teachers, and a massive open online course (MOOC) for teachers to support their continued learning about teaching mathematics modeling to EB students.

The project draws on three important constructs related to teaching mathematics to emergent bilingual (EB) students: research on the mathematics education of EB students; research on mathematical modeling; and positioning theory. Related to mathematics education of EB students, the project supports teachers in enacting high-quality instruction that incldues high cognitive demand tasks, encourages EBs to engage in and explain their problem solving process, and complements that work with linguistic and contextual supports that support EB students' entry into the tasks. Related to mathematical modeling, the project makes use of the conceptualization of modeling as a vehicle for content (as compared to mathematics content of its own), and envisions the use of modeling practices as particularly supportive of EB students' learning of algebra. In particular, the modeling-as-a-vehicle stance invites teachers to engage students in tasks that contain multiple mathematical representations, which has the potential to both build students' conceptual understandings of algebra and to strengthen EBs' language and communication skills in the context of mathematics. With respect to positioning theory, the project seeks to disrupt the finding that secondary mathematics teachers tend underestimate EB students' mathematical abilities due to their English proficiency standards, causing them to choose lower cognitive demand tasks for these students against established research-based recommendations. The project team will engage EB algebra and pre-algebra teachers in Des Moines Public schools in co-planning and co-teaching lessons using mathematical modeling practice. This co-planning and co-teaching activity constitutes in-situ professional development for teachers. Co-planning sessions, co-taught lessons, and regular teacher interviews will be recorded and analyzed for quality of instruction and changes in teacher positioning of EB students. The research team and teachers will co-analyze student learning data from observations and district-administered standardized assessments to better understand the impact of the modeling lessons on students' algebra learning and achievement. Eight teachers will participate in the work over the life of the project, each supporting EB classes of approximately 20 students per teacher. The outcomes of these analyses will guide the development fo a mathematical modeling framework for teaching EBs, teacher professional development materials made available for similar work in other schools and districts, and a massive open online course designed for teachers to develop their skills for teaching secondary mathematics to EB students.

CAREER: Exploring Teacher Noticing of Students' Multimodal Algebraic Thinking

This project investigates and expands teachers' learning to notice in two important ways. First, the research expands beyond teachers' noticing of written and verbal thinking to attend to gesture and other aspects of embodied and multimodal thinking. Second, the project focuses on algebraic thinking and seeks specifically to understand how teacher noticing relates to the content of algebra. Bringing together multimodal thinking and the mathematical ideas in algebra has the potential to support teachers in providing broader access to algebraic thinking for more students.

Award Number: 
1942580
Funding Period: 
Mon, 06/01/2020 to Sat, 05/31/2025
Full Description: 

Effective teachers of mathematics attend to and respond to the substance of students' thinking in supporting classroom learning. Teacher professional development programs have supported teachers in learning to notice students' mathematical thinking and using that noticing to make instructional decisions in the classroom. This project investigates and expands teachers' learning to notice in two important ways. First, the research expands beyond teachers' noticing of written and verbal thinking to attend to gesture and other aspects of embodied and multimodal thinking. Second, the project focuses on algebraic thinking and seeks specifically to understand how teacher noticing relates to the content of algebra. Bringing together multimodal thinking and the mathematical ideas in algebra has the potential to support teachers in providing broader access to algebraic thinking for more students.

To study teacher noticing of multimodal algebraic thinking, this project will facilitate video club sessions in which teachers examine and annotate classroom video. The video will allow text-based and visual annotation of the videos to obtain rich portraits of the thinking that teachers notice as they examine algebra-related middle school practice. The research team will create a video library focused on three main algebraic thinking areas: equality, functional thinking, and proportional reasoning. Clips will be chosen that feature multimodal student thinking about these content areas, and provide moments that would be fruitful for advancing student thinking. Two cohorts of preservice teachers will engage in year-long video clubs using this video library, annotate videos using an advanced technological tool, and engage in reflective interviews about their noticing practices. Follow-up classroom observations will be conducted to see how teachers then notice multimodal algebraic thinking in their classrooms. Materials to conduct the video clubs in other contexts and the curated video library will be made available, along with analyses of the teacher learning that resulted from their implementation.

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