Algebra

Algebra Project Mathematics Content and Pedagogy Initiative

This project will scale up, implement, and assess the efficacy of interventions in K-12 mathematics education based on the well-established Algebra Project (AP) pedagogical framework, which seeks to improve performance and participation in mathematics of students in distressed school districts, particularly low-income students from underserved populations.

Lead Organization(s): 
Award Number: 
1621416
Funding Period: 
Thu, 09/15/2016 - Mon, 08/31/2020
Full Description: 

Algebra continues to serve as a gatekeeper and potential barrier for high school students. The Algebra Project Mathematics Content and Pedagogy Initiative (APMCPI) will scale up, implement and assess the efficacy of interventions in K-12 mathematics education based on the well-established Algebra Project (AP) pedagogical framework. The APMCPI project team is comprised of four HBCUs (Virginia State University, Dillard University, Xavier University, Lincoln University), the Southern Initiative Algebra Project (SIAP), and four school districts that are closely aligned with partner universities. The purpose of the Algebra Project is to improve performance and participation in mathematics by members of students in distressed school districts, particularly those with a large population of low-income students from underserved populations including African American and Hispanics. The project will provide professional development and implement the Algebra Project in four districts and study the impact on student learning. The research results will inform the nation's learning how to improve mathematics achievement for all children, particularly those in distressed inner-city school districts.

The study builds on a prior pilot project with a 74% increase in students who passed the state exam. In the early stages of this project, teachers in four districts closely associated with the four universities will receive Algebra Project professional development in Summer Teacher Institutes with ongoing support during the academic year, including a community development plan. The professional development is designed to help teachers combine mathematical problem solving with context-rich lessons, which both strengthen and integrate teachers' understanding of key concepts in mathematics so that they better engage their students. The project also will focus on helping teachers establish a framework for mathematically substantive, conceptually-rich and experientially-grounded conversations with students. The first year of the study will begin a longitudinal quasi-experimental, explanatory, mixed-method design. Over the course of the project, researchers will follow cohorts who are in grade-levels 5 through 12 in Year 1 to allow analyses across crucial transition periods - grades 5 to 6; grades 8 to 9; and grades 12 to college/workforce. Student and teacher data will be collected in September of Project Year 1, and in May of each project year, providing five data points for each student and teacher participant. Student data will include student attitude, belief, anxiety, and relationship to mathematics and science, in addition to student learning outcome measures. Teacher data will include content knowledge, attitudes and beliefs, and practices. Qualitative data will provide information on the implementation in both the experimental and control conditions. Analysis will include hierarchical linear modeling and multivariate analysis of covariance.

Algebra Project Mathematics Content and Pedagogy Initiative

Development and Empirical Recovery for a Learning Progression-Based Assessment of the Function Concept

The project will design an assessment based on learning progressions for the concept of function - a critical concept for algebra learning and understanding. The goal of the assessment and learning progression design is to specifically incorporate findings about the learning of students traditionally under-served and under-performing in algebra courses.

Lead Organization(s): 
Award Number: 
1621117
Funding Period: 
Thu, 09/15/2016 - Mon, 08/31/2020
Full Description: 

The project will design an assessment based on learning progressions for the concept of function. A learning progression describes how students develop understanding of a topic over time. Function is a critical concept for algebra learning and understanding. The goal of the assessment and learning progression design in this project is to specifically incorporate findings about the learning of students traditionally under-served and under-performing in algebra courses. The project will include accounting for the social and cultural experiences of the middle and high school students when creating assessment tasks. The resources developed should impact mathematics instruction (especially for algebra courses) by creating a learning progression which captures the range of student performance and appropriately places them at distinct levels of performance. The important contribution of the work is the development of a learning progression and related assessment tasks that account for the experiences of students often under-served in mathematics. The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools (RMTs). 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 proposed projects.

The learning progression development will begin by comparing and integrating existing learning progressions and current research on function learning. This project will develop an assessment of student knowledge of function based on learning progressions via empirical recovery (looking for the reconstruction of theoretical levels of the learning theory). Empirical recovery is the process through which data will be collected that reconstruct the various levels, stages, or sequences of said learning progression. The development of tasks and task models will include testing computer-delivered, interactive tasks and rubrics that can be used for human and automated scoring (depending on the task). Item response theory methods will be used to evaluate the assessment tasks' incorporation of the learning progression.

Development and Empirical Recovery for a Learning Progression-Based Assessment of the Function Concept

Supporting Success in Algebra: A Study of the Implementation of Transition to Algebra

The project will research the implementation of Transition to Algebra, a year-long mathematics course for underprepared ninth grade students taken concurrently with Algebra 1 to provide additional support, and its impact on students' attitudes and achievement in mathematics in combination with teachers' instruction and the types of supports teachers need to successfully implement the intervention.

Partner Organization(s): 
Award Number: 
1621011
Funding Period: 
Sat, 10/01/2016 - Wed, 09/30/2020
Full Description: 

The project will research the impact and implementation of Transition to Algebra, a year-long mathematics course for underprepared ninth grade students taken concurrently with Algebra 1 to provide additional support. Nationally, there is a need for programs that support students' learning of algebra and that provide research-based resources and models particularly for students in need of additional support. The design of the Transition to Algebra curriculum reflects the idea that students underprepared for Algebra 1 can benefit from very specific help in building the logic of algebra by connecting arithmetic pattern and algebraic structure. The materials feature the use of mental mathematics, puzzles, explorations, and student dialogues to connect arithmetic pattern to algebraic structure. These features should encourage students to expect mathematics ideas to make sense, and to build algebraic habits of mind and problem solving stamina. The research will investigate the effects of the curriculum on students' algebra achievement and their attitudes towards mathematics. The Discovery Research PreK-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools. 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 proposed projects.

The research questions examine the impact of the Transition to Algebra course on students' attitudes and achievement in mathematics in combination with teachers' instruction and the types of supports teachers need to successfully implement the intervention. The project will use a pre-post quasi-experimental design, along with propensity score methods to reduce selection bias threats, to examine the implementation in approximately 35 treatment schools and 35 comparison schools. Qualitative and quantitative data will be collected and analyzed to address research questions. The study will also investigate how teachers use and adapt Transition to Algebra materials, and the supports critical to successful implementation. For example, the study will examine whether and how school and district activities such as common planning time, coaching, and other professional development experiences influence the implementation fidelity of the curriculum. Qualitative data will be collected through interviews and classroom observations. Quantitative data will be collected using student and teacher surveys, an algebra readiness assessment, a standardized end-of-course assessment, and students' scores on state tests.

Supporting Success in Algebra: A Study of the Implementation of Transition to Algebra

CAREER: Designing Learning Environments to Foster Productive and Powerful Discussions Among Linguistically Diverse Students in Secondary Mathematics

This project will design and develop specialized instructional materials and guidelines for teaching secondary algebra in linguistically diverse classrooms. These materials will incorporate current research on student learning in mathematics and research on the role of language in students' mathematical thinking and learning. The work will connect research on mathematics learning generally with research on the mathematics learning of ELLs, and will contribute practical resources and guidance for mathematics teachers who teach ELLs.

Award Number: 
1553708
Funding Period: 
Mon, 02/01/2016 - Sun, 01/31/2021
Full Description: 

The project will design and investigate learning environments in secondary mathematics classrooms focused on meeting the needs of English language learners. An ongoing challenge for mathematics teachers is promoting deep mathematics learning among linguistically diverse groups of students while taking into consideration how students' language background influences their classroom experiences and the mathematical understandings they develop. In response to this challenge, this project will design and develop specialized instructional materials and guidelines for teaching fundamental topics in secondary algebra in linguistically diverse classrooms. The materials will incorporate insights from current research on student learning in mathematics as well as insights from research on the role of language in students' mathematical thinking and learning. A significant contribution of the work will be connecting research on mathematics learning generally with research on the mathematics learning of English language learners. In addition to advancing theoretical understandings, the research will also contribute practical resources and guidance for mathematics teachers who teach English language learners. The Faculty Early Career Development (CAREER) program is a National Science Foundation (NSF)-wide activity that offers awards in support of junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education, and the integration of education and research within the context of the mission of their organizations.

The project is focused on the design of specialized hypothetical learning trajectories that incorporate considerations for linguistically diverse students. One goal for the specialized trajectories is to foster productive and powerful mathematics discussions about linear and exponential rates in linguistically diverse classrooms. The specialized learning trajectories will include both mathematical and language development learning goals. While this project focuses on concepts related to reasoning with linear and exponential functions, the resulting framework should inform the design of specialized hypothetical learning trajectories in other topic areas. Additionally, the project will add to the field's understanding of how linguistically diverse students develop mathematical understandings of a key conceptual domain. The project uses a design-based research framework gathering classroom-based data, assessment data, and interviews with teachers and students to design and refine the learning trajectories. Consistent with a design-based approach, the project results will include development of theory about linguistically diverse students' mathematics learning and development of guidance and resources for secondary mathematics teachers. This research involves sustained collaboration with secondary mathematics teachers and the impacts will include developing capacity of teachers locally, and propagating the results of this work in professional development activities.

CAREER: Designing Learning Environments to Foster Productive and Powerful Discussions Among Linguistically Diverse Students in Secondary Mathematics

Retention of Early Algebraic Understanding

The project will use a quasi-experimental design to explore students' knowledge of core algebraic concepts in middle grades (grade 6), one year after their completion of 3-year, grades 3-5 early algebra intervention. The research questions are: (1) how well students who received a specific intervention retain their understanding of algebraic concepts in future years; and (2) whether and how the intervening year of regular classroom instruction in grade 6 influences the algebra understanding of both intervention and comparison students.

Lead Organization(s): 
Award Number: 
1550897
Funding Period: 
Tue, 09/01/2015 - Wed, 08/31/2016
Full Description: 

The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by prek-12 students and teachers through research and development of innovative resources, models and tools. While national and state standards provide important benchmarks for algebra learning beginning in kindergarten, they do not provide rigorously tested models by which these algebra standards might be attained in elementary grades classrooms in ways that will ensure further mathematics achievement. This work will addresses this need by closely documenting the effectiveness of models and tools, developed in our previous work, for early algebra education

The proposed project will use a quasi-experimental design to explore students' knowledge of core algebraic concepts in middle grades (grade 6), one year after their completion of 3-year, grades 3-5 early algebra intervention. The project will also study the algebraic knowledge of a comparison group of students. The research questions are: (1) how well students who received a specific intervention retain their understanding of algebraic concepts in future years; and (2) whether and how the intervening year of regular classroom instruction in grade 6 influences the algebra understanding of both intervention and comparison students.

Retention of Early Algebraic Understanding

Teaching and Learning Algebraic Thinking Across the Middle Grades: A Research-based Approach Using PhET Interactive Simulations

This project addresses three central challenges: 1) the tendency for students to not engage in real mathematical thinking as they use technologies; 2) the tendency for teachers to not enact pedagogically-effective approaches; and 3) the lack of adoption of effective technologies by teachers due to a variety of barriers. This project will use rich, exploratory, interactive simulations and associated instructional materials as a pathway for making rapid progress and focusing on advancing algebraic thinking in Grades 6-9.

Lead Organization(s): 
Award Number: 
1503510
Funding Period: 
Tue, 09/01/2015 - Fri, 08/31/2018
Full Description: 

The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers through research and development of innovative resources, models and tools. 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 proposed projects.

Widespread, high-quality use of technology has great potential to transform today's mathematics classrooms and enable all students to develop a robust conceptual understanding of mathematics. Critical challenges are currently limiting the realization of this potential, and 69% of US Grade 8 students are scoring below proficient in national studies. In this 3-year Discovery Research K-12 Full Research and Development project, Teaching and Learning Algebraic Thinking Across the Middle Grades: A Research-based Approach Using PhET Interactive Simulations, the PhET Interactive Simulations group at the University of Colorado Boulder is partnering with mathematics education researchers at the University of South Florida St. Petersberg and Florida State University to address three central challenges, as follows: 1) the tendency for students to not engage in real mathematical thinking as they use technologies; 2) the tendency for teachers to not enact pedagogically-effective approaches; and 3) the lack of adoption of effective technologies by teachers due to a variety of barriers. This collaborative effort uses rich, exploratory, interactive simulations and associated instructional materials as a pathway for making rapid progress and focuses on advancing algebraic thinking in Grades 6-9.

This project seeks to enable teachers to fully-leverage the benefits of interactive simulations to advance student engagement and learning of mathematics, moving technology from the margins to a core part of instruction. The project will answer critical research questions, such as: how the design of an interactive simulation can generate pedagogically-productive use; how instruction with simulations can be best structured to support learning of mathematical concepts and engagement in mathematical practices; how sim-based instruction can be made attractive, feasible and effective for teachers; and finally, how student learning is impacted by sim-based instruction. At the same time, this project will produce a collection of open educational resources for teachers and students. These resources will include 15 research-based, student-tested simulations for teaching and learning of algebraic thinking, associated instructional support materials, and teacher professional development resources for effective implementation. Based on the 75 million uses per year of PhET?s science simulations, we expect these resources to transform mathematics instruction for millions of students and thousands of teachers.

This project will employ a variety of research methods to approach these questions. Researchers will use individual interviews from a diverse group of Grades 6-9 students as they use the 15 new simulations to examine usability, engagement, and achievement and to identify design approaches that stimulate productive use. In parallel, classroom-based studies in Colorado and Florida will investigate ways in which simulations can be combined with instructional materials and teacher facilitation to engage groups of students in inquiry, promote rich discussions of important mathematical ideas, and advance achievement in the Common Core State Standards for Mathematics. The project will employ an iterative design and development process involving qualitative and quantitative analysis of diverse measures including the quality of mathematical instruction. Finally, a pilot study and an evaluation of teacher PD supports will examine the feasibility and fidelity with which teachers implement the innovation, and the impact on student learning.

Teaching and Learning Algebraic Thinking Across the Middle Grades: A Research-based Approach Using PhET Interactive Simulations

Math Snacks Early Algebra Using Games and Inquiry to Help Students Transition from Number to Variable

This project will develop games to build conceptual understanding of key early algebra topics. The materials will be freely accessible on the web in both English and Spanish. The project will develop 4-5 games. Each game will include supporting materials for use by students in inquiry-based classroom lessons, and web-based professional development tools for teachers.

Lead Organization(s): 
Award Number: 
1503507
Funding Period: 
Tue, 09/01/2015 - Sat, 08/31/2019
Full Description: 

The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers through research and development of innovative resources, models and tools. 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 proposed projects.

Many U.S. students enter college without the necessary background in algebra to be successful in advanced mathematics and science courses, and are thereby blocked from many rewarding careers. Oftentimes, the problem goes back to early algebra in grades 4-6, where students are introduced to abstract formulations before they understand the underlying ideas and the reasons for the questions being asked. As a result of inadequate preparation many students turn away from mathematics when faced with abstract algebra. Without mathematics, students are not able to enter the STEM field which results in a weakened workforce in these fields in the United States. In this 4-year Full Research and Development project, Math Snacks Early Algebra: Using Games and Inquiry to Help Students Transition from Number to Variable, the interdisciplinary research group from New Mexico State University will build on their success in using games to increase students' understanding of proportional reasoning and fractions. They will develop games to build conceptual understanding of key early algebra topics. The materials will be freely accessible on the web in both English and Spanish. The project will develop 4-5 games. Each game will include supporting materials for use by students in inquiry-based classroom lessons, and web-based professional development tools for teachers.

Most students do not understand the variety of distinct ways that variables are used in mathematics: unknowns to be solved for, related quantities, general properties of numbers, and other uses. Algebra courses often emphasize the rules of manipulation, with less time spent on the underlying ideas. Students see variables as confusing new material, rather than as shortcuts for making sense of numbers, or as powerful tools for analyzing interesting problems. This hinders students' later interest and progress in STEM courses and careers.The intellectual merit for this R & D project includes the development of a new way to learn key underlying concepts in algebra, further investigation of the affordances of games and technology in learning abstract mathematical concepts, and a better understanding of learning assessment in early algebra. The broader impact for this R & D project includes making these tools widely available to students, and the potential shift of teachers towards effective mathematical pedagogy that is engaging and inquiry-based. Development will begin with existing research on early algebraic thinking and learning, and proceed through an iterative process involving design, testing in the NMSU Learning Games Lab, testing in classrooms, and back to design. The project will then study the effect of the developed materials on student understanding and on classroom learning environments. Qualitative and quantitative measures will be used. Researchers will use a custom measure aligned with NAEP (National Assessment of Educational Progress) and other standard tests, interviews and observations with teachers and students, and embedded data collection and self-reports on frequency and extent of game usage. After two earlier pilot studies, in the final year a delayed intervention study will be conducted with 50 teachers and their students. The Math Snacks team has existing partnerships for distribution of games and materials with PBS, GlassLabs, BrainPOP, and others. Academic findings of the project will be shared through conferences and research publications.

Math Snacks Early Algebra Using Games and Inquiry to Help Students Transition from Number to Variable

CAREER: Algebraic Knowledge for Teaching: A Cross-Cultural Perspective

The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. This study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance.

Lead Organization(s): 
Award Number: 
1350068
Funding Period: 
Fri, 08/15/2014 - Wed, 07/31/2019
Full Description: 

What content knowledge is needed for the teaching of mathematics? What practices are more effective for realizing student success? These questions have received considerable attention in the mathematics education community. The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. Focusing on two fundamental mathematical ideas recently emphasized by the Common Core State Standards - inverse relations and properties of operations - this study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance. It will be focused on three objectives: (1) identify AKT that facilitates algebraic thinking and develop preliminary findings into teaching materials; (2) refine research-based teaching materials based on the evaluative data; and (3) integrate research with education through course development at Temple University and teacher outreach in Philadelphia.

The model underlying this research program is that improved pedagogy will improve student learning, both directly and indirectly. A design-based research method will be used to accomplish objectives #1 and #2. Cross-cultural videotaped lessons will be first analyzed to identify AKT, focusing on teachers' use of worked examples, representations, and deep questions. This initial set of findings will then be developed into teaching materials. The U.S. and Chinese expert teachers will re-teach the lessons as part of the refinement process. Data sources will include: baseline and updated survey data (control, context, and process variables), observation, documents, videos, and interviews. The statistical techniques will include descriptive and inferential statistics and HLM will to address the hierarchical nature of the data.

This project involves students and teachers at various levels (elementary, undergraduate, and graduate) at Temple University and the School District of Philadelphia (SDP) in the U.S. and Nanjing Normal University and Nantong School District in China. A total of 600 current and future elementary teachers and many of their students will benefit directly or indirectly from this project. Project findings will be disseminated through various venues. Activities of the project will promote school district-university collaboration, a novice-expert teacher network, and cross-disciplinary and international collaboration. It is anticipated that the videos of expert teaching will also be useful future research by cognitive researchers studying ways to improve mathematics learning.

CAREER: Algebraic Knowledge for Teaching: A Cross-Cultural Perspective

Preparing Urban Middle Grades Mathematics Teachers to Teach Argumentation Throughout the School Year

The objective of this project is to develop a toolkit of resources and practices that will help inservice middle grades mathematics teachers support mathematical argumentation throughout the school year. A coherent, portable, two-year-long professional development program on mathematical argumentation has the potential to increase access to mathematical argumentation for students nationwide and, in particular, to address the needs of teachers and students in urban areas.

Lead Organization(s): 
Award Number: 
1417895
Funding Period: 
Sun, 06/15/2014 - Thu, 05/31/2018
Full Description: 

The project is an important study that builds on prior research to bring a comprehensive professional development program to another urban school district, The District of Columbia Public Schools. The objective of this full research and development project is to develop a toolkit  that provides resources and practices for inservice middle grades mathematics teachers to support mathematical argumentation throughout the school year. Mathematical argumentation, the construction and critique of mathematical conjectures and justifications, is a fundamental disciplinary practice in mathematics that students often never master. Building on a proof of concept of the project's approach ifrom two prior NSF-funded studies, this project expands the model to help teachers support mathematical argumentation all year. A coherent, portable, two-year-long professional development program on mathematical argumentation has the potential to increase access to mathematical argumentation for students nationwide and, in particular, to address the needs of teachers and students in urban areas. Demonstrating this program in the nation's capital will likely attract broad interest and produces important knowledge about how to implement mathematical practices in urban settings. Increasing mathematical argumentation in schools has the potential for dramatic contributions to students' achievement and participation in 21st century workplaces.

Mathematical argumentation is rich discussion in which students take on mathematical authority and co-construct conjectures and justifications. For many teachers, supporting such discourse is challenging; many are most comfortable with Initiate-Respond-Evaluate types of practices and/or have insufficient content understanding. The professional development trains teachers to be disciplined improvisers -- professionals with a toolkit of tools, knowledge, and practices to be deployed creatively and responsively as mathematical argumentation unfolds. This discipline includes establishing classroom norms and planning lessons for argumentation. The model's theory of action has four design principles: provide the toolkit, use simulations of the classroom to practice improvising, support learning of key content, and provide job-embedded, technology-enabled supports for using new practices all year. Three yearlong studies will address design, feasibility, and promise. In Study 1 the team co-designs tools with District of Columbia Public Schools staff. Study 2 is a feasibility study to examine program implementation, identify barriers and facilitators, and inform improvements. Study 3 is a quasi-experimental pilot to test the promise for achieving intended outcomes: expanding teachers' content knowledge and support of mathematical argumentation, and increasing students' mathematical argumentation in the classroom and spoken argumentation proficiency. The studies will result in a yearlong professional development program with documentation of the theory of action, design decisions, pilot data, and instrument technical qualities.

Preparing Urban Middle Grades Mathematics Teachers to Teach Argumentation Throughout the School Year

Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships

This project will use classroom-based research to teach children about important algebraic concepts and to carefully explore how children come to understand these concepts. The primary goal is to identify levels of sophistication in children's thinking as it develops through instruction. Understanding how children's thinking develops will provide a critical foundation for designing curricula, developing content standards, and informing educational policies.

Lead Organization(s): 
Award Number: 
1415509
Funding Period: 
Tue, 07/15/2014 - Thu, 06/30/2016
Full Description: 

Algebra is a central concern in school mathematics education. Its historical gatekeeper role in limiting students' career and life choices is well documented. In recent years, the response has been to reframe algebra as a K-12 endeavor. To this end, research on children's algebraic thinking in grades 3-5 shows that students can begin to understand algebraic concepts in elementary grades that they will later explore more formally. However, there is much that is unknown about how children in grades K-2 make sense of algebraic concepts appropriate for their age. This project aims to understand specific ways in which grades K-2 children begin to think algebraically. It will identify how children understand mathematical relationships, how they represent the relationships they notice, and how they use these relationships as building blocks for more sophisticated thinking. The project will use classroom-based research to teach children about important algebraic concepts and to carefully explore how children come to understand these concepts. The primary goal is to identify levels of sophistication in children's thinking as it develops through instruction. Understanding how children's thinking develops will provide a critical foundation for designing curricula, developing content standards, and informing educational policies, all in ways that can help children become successful in algebra and have wider access to STEM-related careers.

While college and career readiness standards point to the role of algebra beginning in kindergarten, the limited research base in grades K-2 restricts algebra's potential in K-2 classrooms. This project will develop cognitive foundations regarding how children learn to generalize, represent, and reason with algebraic relationships. Such findings will inform both the design of new interventions and resources to strengthen algebra learning in grades K-2 and the improvement of educational policies, practices, and resources. The project will use design research to identify: (1) learning trajectories as cognitive models of how grades K-2 children learn to generalize, represent, and reason with algebraic relationships within content dimensions where these practices can occur (e.g., generalized arithmetic); (2) critical junctures in the development of these trajectories; and (3) characteristics of tasks and instruction that facilitate movement along the trajectories. The project's design will include the use of classroom teaching experiments that incorporate: (1) instructional design and planning; (2) ongoing analysis of classroom events; and (3) retrospective analysis of all data sources generated in the course of the experiment. This will allow for the development and empirical validation of hypothesized trajectories in students' understanding of algebraic relationships. This exploratory research will contribute critical early-grade cognitive foundations of K-12 teaching and learning algebra that can help democratize access to student populations historically marginalized by a traditional approach to teaching algebra. Moreover, the project will occur in demographically diverse school districts, thereby increasing the generalizability of findings across settings.

Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships
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