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Algebra

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)

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

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): 
Partner Organization(s): 
Award Number: 
1219605
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: Blanton)

Core Math Tools

This project is developing Core Math Tools, a suite of Java-based software including a computer algebra system (CAS), interactive geometry, statistics, and simulation tools together with custom apps for exploring specific mathematical or statistical topics. Core Math Tools is freely available to all learners, teachers, and teacher educators through a dedicated portal at the National Council of Teachers of Mathematics (NCTM) web site.

Lead Organization(s): 
Award Number: 
1201917
Funding Period: 
Sun, 01/15/2012 - Mon, 12/31/2012
Project Evaluator: 
David Barnes, NCTM
Full Description: 

Core Math Tools is a project from Western Michigan University that meets the urgent need of providing mathematical tools that students can use as they explore and learn mathematical concepts that are aligned with the Common Core State Standards in Mathematics (CCSSM). The developers have repurposed and modified tools originally designed for an NSF-funded curriculum project (e.g., Core-Plus Mathematics), creating a suite of tools that supports student learning of mathematics regardless of the curricula choice. Core math Tools is Java-based software that includes a computer algebra system(CAS, interactive geometry, statistics, and simulation tools together with custom apps for exploring specific mathematical and statistical topics. The designers provide exemplary lessons illustrating how the software can be used in the spirit of the new CCSSM. The goal of the project is to provide equitable and easy access to mathematical software both in school and outside of school. The tools are available to all learners and teachers through the web site of the National Council of Teachers of Mathematics (NCTM). The web site includes feedback loops for teachers to provide information about the tools. By using the NCTM website, the tools can be downloaded for use by teachers and students. The dedicated portal on the NCTM website allows supervisors to use the tools in professional development, teachers to use the tools as an integral part of instruction, and students to use the tools for exploring, conjecturing, and problem solving.

Core Math Tools

Gateways to Algebraic Motivation, Engagement and Success (GAMES): Supporting and Assessing Fraction Proficiency with Game-Based, Mobile Applications and Devices

This project is designing digital games for middle school students that will help them prepare for success in Algebra. The games are intended to help students gain a deep understanding of measurement and fraction concepts that are critical as they begin to learn algebra. The project studies students' development of fraction concepts, their engagement in the tasks, and the use of hand-held devices as a useful platform for games.

Award Number: 
1118571
Funding Period: 
Mon, 08/15/2011 - Wed, 07/31/2013
Full Description: 

The Gateways to Algebraic Motivation, Engagement and Success (GAMES) project is designing digital games for middle school students that will help them prepare for success in Algebra. The games are intended to help students gain a deep understanding of measurement and fraction concepts that are critical as they begin to learn algebra. The design of the games is based on research on learning fractions and research on engagement. The researchers at Virginia Polytechnic Institute and State University are studying students' development of fraction concepts, their engagement in the tasks, and the use of hand-held devices as a useful platform for games. They are providing valuable information on how students develop fraction concepts and contributing to the development of a learning trajectory that will guide the teaching of measurement and fraction concepts.

The design of the games is based on engagement states that are known to facilitate learning, with specific attention to cognitive, behavioral, and affective domains. The mathematical framework driving the games is based on how students learn fraction concepts. Most grade 6 students think of fractions from a part-whole conception, but this is not an adequate base for developing algebraic concepts. The games help students develop splitting concepts by moving through activities that focus on sequencing, partitioning, and iterating. The games are designed for iOS platforms that provide ease of engagement and data collection flexibility.

The project offers a variety of products ranging from theories to games. The research is building a conceptual framework that identifies features of engagement that lead to learning, and contributing to the development of a learning trajectory related to fraction concepts. The work will produce a scalable model for developing and using digital games to increase engagement and learning of middle school students. In addition, three games and associated tasks are being developed for use with current curricula to enhance students' understanding of fractions and prepare them for learning algebra.

Gateways to Algebraic Motivation, Engagement and Success (GAMES): Supporting and Assessing Fraction Proficiency with Game-Based, Mobile Applications and Devices

Children's Understanding of Functions in Grades K-2

This project is studying how young children in grades K-2 understand mathematical concepts that are foundational for developing algebraic thinking. Researchers are contributing to an ongoing effort to develop a learning trajectory that describes how algebraic concepts are developed. The project uses teaching experiments, with researchers talking directly to students as they explore algebraic ideas. They explore how students think about and develop concepts related to covariation, representations of functions, relationships among variable, and generalization.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1154355
Funding Period: 
Mon, 08/01/2011 - Thu, 07/31/2014
Full Description: 

The researchers in the Children's Understanding of Functions project are studying how young children in grades K-2 understand mathematical concepts that are foundational for developing algebraic thinking. Researchers at University of Massachusetts at Dartmouth and Tufts University are contributing to an ongoing effort to develop a learning trajectory that describes how algebraic concepts are developed. Most research has focused on student development at the upper elementary and middle school levels, but this project will add information about early elementary learners.

The project's research methodology uses teaching experiments which allow researchers to talk directly to students as they explore algebraic ideas. They explore how students think about and develop concepts related to covariation, representations of functions, relationships among variable, and generalization. Researchers have designed tasks that help students explain their thinking and solve problems where some quantities vary and others are constant. They are analyzing videos and students' written work as they build case studies about the development of algebraic thinking. External evaluation of this exploratory project is one of the responsibilities of its advisory board.

This project is connecting the algebraic thinking of younger children to what has been documented for older children. This process enables them to build an evidence-based learning trajectory about students' development of algebraic thinking. The products of this research can be used to build curricula and lessons that are aligned with what students know and can learn at various points in their development. Project findings, tasks and videos are being disseminated not only to researchers, but also to practitioners through professional publications and the DRK-12 Resource Network.

Children's Understanding of Functions in Grades K-2
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