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Algebra

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.

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

Intensified Algebra Efficacy Study

Students who fail algebra in the ninth grade are significantly less likely than their peers to graduate from high school on time. This project intends to test a common support strategy for at-risk students that provides an extra period of algebra, commonly known as a "double dose" condition. The Intensified Algebra (IA) is an intervention that addresses both the academic and non-academic needs of students.

Lead Organization(s): 
Award Number: 
1418267
Funding Period: 
Fri, 08/01/2014 - Tue, 07/31/2018
Full Description: 

Student success in algebra continues to be a problem as many U.S. students are underprepared when they enter high school. Students who fail algebra in the ninth grade are significantly less likely than their peers to graduate from high school on time. This project intends to test a common support strategy for at-risk students that provides an extra period of algebra, commonly known as a "double dose" condition. The Intensified Algebra (IA) is an intervention that addresses both the academic and non-academic needs of students. It is set of cohesive, integrated, and rigorous resources that builds student motivation and confidence. IA uses a blended model of instruction with a strong technology component designed to support the productive use of expanded instructional time that has shown evidence of promise in earlier studies.

This project is intended to rigorously test the impact of IA on student outcomes in a school-level random assignment design involving 6 districts, 55 high schools and over 4,000 9th grade students across two cohorts. Within each district, eligible schools are randomly assigned to either implement IA or to use the school's already established "double dose" algebra course. Analyses will use hierarchical linear models that explicitly take into account the clustering of students within classrooms and classrooms within schools. The study investigates short-term outcomes including end-of-9th grade algebra learning, passing rate for algebra I and attitudes toward mathematics. Longer-term outcomes include subsequent course-taking patterns and performances. The study examines fidelity of implementation and key implementation factors with descriptive and correlational analyses.

Intensified Algebra Efficacy Study

Illuminating Learning by Splitting: A Learning Analytics Approach to Fraction Game Data Analysis

This project uses learning analytics and educational data mining methods to examine how elementary students learn in an online game designed to teach fractions using the splitting model. The project uses data to examine the following questions: 1) Is splitting an effective way to learn fractions?; 2) How do students learn by splitting?; 3) Are there common pathways students follow as they learn by splitting?; and 4) Are there optimal pathways for diverse learners?

Lead Organization(s): 
Award Number: 
1338176
Funding Period: 
Mon, 07/01/2013 - Wed, 12/31/2014
Full Description: 

Mathematical literacy is a critical need in our increasingly technological society. Fractions have been identified as a key area of understanding, both for success in Algebra and for access to higher-level mathematics. The project uses learning analytics and educational data mining methods to examine how elementary students learn in Refraction, an online game designed to teach fractions using the splitting model. The project uses the data from a pre- and posttest of fraction understanding and log data from 3000 third-grade students' gameplay to examine the following questions:

1) Is splitting an effective way to learn fractions?

2) How do students learn by splitting?

3) Are there common pathways students follow as they learn by splitting?

4) Are there optimal pathways for diverse learners?

Splitting is a well-known theory of fraction learning and has significant expert buy in. However, few of the research questions above can be advanced past the field's present level of understanding with either current qualitative or quantitative methods. By using data mining methods such as cluster analysis, association rule mining, and predictive analysis, the project provides numerous insights about student learning through splitting, including: classification of learning profiles exhibited in unstructured learning environments, common mistakes and sense-making patterns, the value or cost of exploration in learning, and the best path through learning for different students (such as those who score low on a pre-test).

The project staff shares the methods and results through traditional and novel outlets for maximum impact on the field and on policy. In addition to conferences and journal publications, the principal investigator is working in several contexts in which this work is an exemplar of new ways the field can develop understanding of learning. In addition, many of these contexts have connections to efforts such as the Chief State School Officers' Shared Learning Collaborative, leading to a high probability that the findings and products can quickly impact large numbers of schools across the country.

Illuminating Learning by Splitting: A Learning Analytics Approach to Fraction Game Data Analysis

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

Improving Formative Assessment Practices: Using Learning Trajectories to Develop Resources That Support Teacher Instructional Practice and Student Learning in CMP2

The overarching goal of this project is to develop innovative instructional resources and professional development to support middle grades teachers in meeting the challenges set by college- and career-ready standards for students' learning of algebra.

Lead Organization(s): 
Award Number: 
1316736
Funding Period: 
Tue, 10/01/2013 - Sat, 09/30/2017
Full Description: 

The overarching goal of this project is to develop innovative instructional resources and professional development to support middle grades teachers in meeting the challenges set by college- and career-ready standards for students' learning of algebra. This 4-year project includes three major components: (1) development and empirical testing of learning trajectories for linear functions and linear equations, (2) collaborations with teachers of Connected Mathematics Project 2 (CMP2) to create and test a set of instructional resources focused on formative assessment processes, and (3) iterative refinement of a professional development model for engaging teachers with the instructional resources in ways that optimize students' learning of algebra. The professional development activities provide opportunities for teachers to develop specialized content knowledge of learning trajectories for linear functions and equations in algebra, processes for interpreting students' performances with respect to those trajectories and providing feedback and additional instructional activities based on "where" the student is with respect to the overall learning trajectory. Such changes in teacher knowledge and practice are anticipated to produce improved student learning outcomes for key concepts and procedures in algebra. One of the major stumbling blocks to teachers' implementation of effective formative assessment practice is the sheer volume and management of the information needed to monitor and interpret student performance. The project addresses this impediment by employing the ASSISTments platform, a web-based online system for delivering mathematics problem sets and capable of adapting problem presentation to student performance in real time.

Research on learning trajectories in mathematics has mostly centered on concepts at the primary school level. While research at this level has been prolific and informative in multiple aspects of mathematics education, there are major knowledge gaps in our understanding of learning trajectories in several domains of mathematics, specifically in algebra. Indeed, there is a growing need for new research and development projects to fill these critical knowledge gaps.

This project focuses on two critical areas in mathematics: students' understanding of linear functions and linear equations, and students' ability to use them to solve problems. Empirically validated learning trajectories will support curriculum development in these areas. In addition, this project contributes to the research base to improve the curriculum standards by providing empirical evidence for hypothesized trajectories for selected content standards for middle school students. Finally, the use of CMP2 augmented by the online management system increases the probability of widespread impact of the professional development model targeted at teachers' formative assessment practices. Although we are using a specific curriculum program, the treatment of linear functions and equations topics in CMP is consistent with other functions-based curricula in the U.S. Thus, the work done in the context of this project will be useful in examining learning trajectories and formative assessment in other instructional programs.

Improving Formative Assessment Practices: Using Learning Trajectories to Develop Resources That Support Teacher Instructional Practice and Student Learning in CMP2

Learning Algebra and Methods for Proving (LAMP)

This project tests and refines a hypothetical learning trajectory and corresponding assessments, based on the collective work of 50 years of research in mathematics education and psychology, for improving students' ability to reason, prove, and argue mathematically in the context of algebra. The study produces an evidence-based learning trajectory and appropriate instruments for assessing it.

Lead Organization(s): 
Award Number: 
1317034
Funding Period: 
Tue, 10/01/2013 - Wed, 09/30/2015
Full Description: 

The Learning Algebra and Methods for Proving (LAMP) project tests and refines a hypothetical learning trajectory and corresponding assessments, based on the collective work of 50 years of research in mathematics education and psychology, for improving students' ability to reason, prove, and argue mathematically in the context of algebra. The goals of LAMP are: 1) to produce a set of evidence-based curriculum materials for improving student learning of reasoning, proving, and argumentation in eighth-grade classrooms where algebra is taught; 2) to produce empirical evidence that forms the basis for scaling the project to a full research and development project; and 3) to refine a set of instruments and data collection methods to support a full research and development project. LAMP combines qualitative and quantitative methods to refine and test a hypothetical learning trajectory for learning methods of reasoning, argumentation, and proof in the context of eighth-grade algebra curricula. Using qualitative methods and quantitative methods, the project conducts a pilot study that can be scaled up in future studies. The study produces an evidence-based learning trajectory and appropriate instruments for assessing it.

Over the past two decades, national organizations have called for more attention to the topics of proof, proving, and argumentation at all grade levels. However, the teaching of reasoning and proving remains sparse in classrooms at all levels. LAMP will address this critical need in STEM education by demonstrating ways to improve students' reasoning and argumentation skills to meet the demands of college and career readiness.

This project promises to have broad impacts on future curricula in the United States by creating a detailed description of how to facilitate reasoning and argumentation learning in actual eighth-grade classrooms. At present, a comprehensive understanding of how reasoning and proving skills develop alongside algebraic thinking does not exist. Traditional, entirely formal approaches such as two-column proof have not demonstrated effectiveness in learning about proof and proving, nor in improving other mathematical practices such as problem-solving skills and sense making. While several studies, including studies in the psychology literature, lay the foundation for developing particular understandings, knowledge, and skills needed for writing viable arguments and critiquing the arguments of others, a coherent and complete set of materials that brings all of these foundations together does not exist. The project will test the hypothetical learning trajectory with classrooms with high proportions of Native American students.

Learning Algebra and Methods for Proving (LAMP)

CAREER: Investigating Differentiated Instruction and Relationships Between Rational Number Knowledge and Algebraic Reasoning in Middle School

The proposed project initiates new research and an integrated education plan to address specific problems in middle school mathematics classrooms by investigating (1) how to effectively differentiate instruction for middle school students at different reasoning levels; and (2) how to foster middle school students' algebraic reasoning and rational number knowledge in mutually supportive ways.

Lead Organization(s): 
Award Number: 
1252575
Funding Period: 
Thu, 08/01/2013 - Tue, 07/31/2018
Full Description: 

Middle school mathematics classrooms are marked by increasing cognitive diversity and students' persistent difficulties in learning algebra. Currently middle school mathematics instruction in a single classroom is often not differentiated for different thinkers, which can bore some students or overly challenge others. One way schools often deal with different thinkers at the same grade level is by tracking, which has also been shown to have deleterious effects on students, both cognitively and affectively. In addition, students continue to struggle to learn algebra, and increasing numbers of middle school students are receiving algebra instruction. The proposed project initiates new research and an integrated education plan to address these problems by investigating (1) how to effectively differentiate instruction for middle school students at different reasoning levels; and (2) how to foster middle school students' algebraic reasoning and rational number knowledge in mutually supportive ways. Educational goals of the project are to enhance the abilities of prospective and practicing teachers to teach cognitively diverse students, to improve doctoral students' understanding of relationships between students' learning and teachers' practice, and to form a community of mathematics teachers committed to on-going professional learning about how to differentiate instruction.

Three research-based products are being developed: two learning trajectories, materials for differentiating instruction developed collaboratively with teachers, and a written assessment to evaluate students' levels of reasoning. The first trajectory, elaborated for students at each of three levels of reasoning, focuses on developing algebraic expressions and solving basic equations that involve rational numbers; the second learning trajectory, also elaborated for students at each of three levels of reasoning, focuses on co-variational reasoning in linear contexts. In addition, the project investigates how students' classroom experience is influenced by differentiated instruction, which will allow for comparisons with research findings on student experiences in tracked classrooms. Above all, the project enhances middle school mathematics teachers' abilities to serve cognitively diverse students. This aspect of the project has the potential to decrease opportunity gaps. Finally, the project generates an understanding of the kinds of support needed to help prospective and practicing teachers learn to differentiate instruction.

The project advances discovery and understanding while promoting teaching, training, and learning by (a) integrating research into the teaching of middle school mathematics, (b) fostering the learning of all students by tailoring instruction to their cognitive needs, (c) partnering with practicing teachers to learn how to implement this kind of instruction, (d) improving the training of prospective mathematics teachers and graduate students in mathematics education, and (e) generating a community of mathematics teachers who engage in on-going learning to differentiate instruction. The project broadens participation by including students from underrepresented groups, particularly those with learning disabilities. Results from the project will be broadly disseminated via conference presentations; articles in diverse media outlets; and a project website that will make project products available, be a location for information about the project for the press and the public, and be a tool to foster teacher-to-teacher communication.

CAREER: Investigating Differentiated Instruction and Relationships Between Rational Number Knowledge and Algebraic Reasoning in Middle School

The Impact of Early Algebra on Students' Algebra-Readiness (Collaborative Research: Knuth)

In this project researchers are implementing and studying a research-based curriculum that was designed to help children in grades 3-5 prepare for learning algebra at the middle school level. Researchers are investigating the impact of a long-term, comprehensive early algebra experience on students as they proceed from third grade to sixth grade. Researchers are working to build a learning progression that describes how algebraic concepts develop and mature from early grades through high school.

Lead Organization(s): 
Award Number: 
1219606
Funding Period: 
Mon, 10/01/2012 - Wed, 09/30/2015
Full Description: 

The Impact of Early Algebra on Students' Algebra-Readiness is a collaborative project at the University of Wisconsin and TERC, Inc. They are implementing and studying a research-based curriculum that was designed to help children in grades 3-5 prepare for learning algebra at the middle school level. Researchers are investigating the impact of a long-term, comprehensive early algebra experience on students as they proceed from third grade to sixth grade. Researchers are working to build a learning progression that describes how algebraic concepts develop and mature from early grades through high school. This study helps to build our knowledge about the piece of the progression that is just prior to entering middle school where many students begin formal instruction in algebra.

Building on previous research about early algebra learning, researchers will teach a curriculum that was carefully designed to reflect what we know about learning algebraic concepts. Previous research has shown that young children from very diverse backgrounds have the ability to construct algebraic ideas such as equality, representation, generalization, and functions. Researchers are collecting data about students' algebraic knowledge as well as arithmetical knowledge.

We know that the majority of students in the United States struggle with learning formal algebra. By studying the implementation of the research-based curriculum for an extended period of time, researcher's are learning about how algebraic ideas are connected and whether or not early instruction on algebraic ideas will help students learn more formal ideas in middle school.

The Impact of Early Algebra on Students' Algebra-Readiness (Collaborative Research: Knuth)
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