Algebra

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.

Partner Organization(s): 
Award Number: 
1418267
Funding Period: 
Fri, 08/01/2014 to 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.

GRIDS: Graphing Research on Inquiry with Data in Science

The Graphing Research on Inquiry with Data in Science (GRIDS) project will investigate strategies to improve middle school students' science learning by focusing on student ability to interpret and use graphs. GRIDS will undertake a comprehensive program to address the need for improved graph comprehension. The project will create, study, and disseminate technology-based assessments, technologies that aid graph interpretation, instructional designs, professional development, and learning materials.

Award Number: 
1418423
Funding Period: 
Mon, 09/01/2014 to Sat, 08/31/2019
Full Description: 

The Graphing Research on Inquiry with Data in Science (GRIDS) project is a four-year full design and development proposal, addressing the learning strand, submitted to the DR K-12 program at the NSF. GRIDS will investigate strategies to improve middle school students' science learning by focusing on student ability to interpret and use graphs. In middle school math, students typically graph only linear functions and rarely encounter features used in science, such as units, scientific notation, non-integer values, noise, cycles, and exponentials. Science teachers rarely teach about the graph features needed in science, so students are left to learn science without recourse to what is inarguably a key tool in learning and doing science. GRIDS will undertake a comprehensive program to address the need for improved graph comprehension. The project will create, study, and disseminate technology-based assessments, technologies that aid graph interpretation, instructional designs, professional development, and learning materials.

GRIDS will start by developing the GRIDS Graphing Inventory (GGI), an online, research-based measure of graphing skills that are relevant to middle school science. The project will address gaps revealed by the GGI by designing instructional activities that feature powerful digital technologies including automated guidance based on analysis of student generated graphs and student writing about graphs. These materials will be tested in classroom comparison studies using the GGI to assess both annual and longitudinal progress. Approximately 30 teachers selected from 10 public middle schools will participate in the project, along with approximately 4,000 students in their classrooms. A series of design studies will be conducted to create and test ten units of study and associated assessments, and a minimum of 30 comparison studies will be conducted to optimize instructional strategies. The comparison studies will include a minimum of 5 experiments per term, each with 6 teachers and their 600-800 students. The project will develop supports for teachers to guide students to use graphs and science knowledge to deepen understanding, and to develop agency and identity as science learners.

Changing Culture in Robotics Classroom (Collaborative Research: Shoop)

Computational and algorithmic thinking are new basic skills for the 21st century. Unfortunately few K-12 schools in the United States offer significant courses that address learning these skills. However many schools do offer robotics courses. These courses can incorporate computational thinking instruction but frequently do not. This research project aims to address this problem by developing a comprehensive set of resources designed to address teacher preparation, course content, and access to resources.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1418199
Funding Period: 
Mon, 09/01/2014 to Thu, 08/31/2017
Full Description: 

Computational and algorithmic thinking are new basic skills for the 21st century. Unfortunately few K-12 schools in the United States offer significant courses that address learning these skills. However many schools do offer robotics courses. These courses can incorporate computational thinking instruction but frequently do not. This research project aims to address this problem by developing a comprehensive set of resources designed to address teacher preparation, course content, and access to resources. This project builds upon a ten year collaboration between Carnegie Mellon's Robotics Academy and the University of Pittsburgh's Learning Research and Development Center that studied how teachers implement robotics education in their classrooms and developed curricula that led to significant learning gains. This project will address the following three questions:

1.What kinds of resources are useful for motivating and preparing teachers to teach computational thinking and for students to learn computational thinking?

2.Where do teachers struggle most in teaching computational thinking principles and what kinds of supports are needed to address these weaknesses?

3.Can virtual environments be used to significantly increase access to computational thinking principles?

The project will augment traditional robotics classrooms and competitions with Robot Virtual World (RVW) that will scaffold student access to higher-order problems. These virtual robots look just like real-world robots and will be programmed using identical tools but have zero mechanical error. Because dealing with sensor, mechanical, and actuator error adds significant noise to the feedback students' receive when programming traditional robots (thus decreasing the learning of computational principles), the use of virtual robots will increase the learning of robot planning tasks which increases learning of computational thinking principles. The use of RVW will allow the development of new Model-Eliciting Activities using new virtual robotics challenges that reward creativity, abstraction, algorithms, and higher level programming concepts to solve them. New curriculum will be developed for the advanced concepts to be incorporated into existing curriculum materials. The curriculum and learning strategies will be implemented in the classroom following teacher professional development focusing on computational thinking principles. The opportunities for incorporating computationally thinking principles in the RVW challenges will be assessed using detailed task analyses. Additionally regression analyses of log-files will be done to determine where students have difficulties. Observations of classrooms, surveys of students and teachers, and think-alouds will be used to assess the effectiveness of the curricula in addition to pre-and post- tests to determine student learning outcomes.

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): 
Partner Organization(s): 
Award Number: 
1338176
Funding Period: 
Mon, 07/01/2013 to 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.

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 to 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.

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): 
Partner Organization(s): 
Award Number: 
1317034
Funding Period: 
Tue, 10/01/2013 to 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.

Developing Rich Media-Based Materials for Practice-Based Teacher Education

This research and development project is premised on the notion that recent technological developments have made it feasible to represent classroom work in new ways. In addition to watching recorded videos of classroom interactions or reading written cases, teacher educators and teachers can now watch animations and image sequences, realized with cartoon characters, and made to depict activities that happened, or could have happened, in a mathematics classroom.

Award Number: 
1316241
Funding Period: 
Thu, 08/15/2013 to Tue, 07/31/2018
Full Description: 

The 4-year research and development project, Developing Rich Media-based Materials for Practice-based Teacher Education, is premised on the notion that recent technological developments have made it feasible to represent classroom work in new ways. In addition to watching recorded videos of classroom interactions or reading written cases, teacher educators and teachers can now watch animations and image sequences, realized with cartoon characters, and made to depict activities that happened, or could have happened, in a mathematics classroom. Furthermore, teacher educators and teachers can react to such animations or image sequences by making their own depictions of alternative moves by students or teachers in classroom interaction. And all of that can take place in an on-line, cloud-based environment that also supports discussion fora, questionnaires, and the kinds of capabilities associated with learning management systems. Such technologies offer important affordances to teacher educators seeking to provide candidates with course-based experiences that emphasize the development of practice-based skills. The focus of the project is on mathematics teacher education. This joint project of the University of Maryland Center for Mathematics Education and the University of Michigan will produce 6 to 8 field-tested modules for use in different courses that are a part of mathematics teacher preparation programs. The following two-pronged research question will be resolved: What are the affordances and constraints of the modules and the environment as supports for: (1) practice based teacher education and (2) a shift toward blended teacher education?

The project involves the following activities: (1) a teacher education materials development component; (2) a related evaluation component; and (3) two research components. The development phase seeks to develop both the LessonSketch.org platform and six to eight mathematics teacher education modules for use in preservice teacher education programs from around the country. The modules will be written with practice-based teacher education goals in mind and will use the capacities of the LessonSketch.org platform as a vehicle for using rich-media artifacts of teaching with preservice teacher candidates. LessonSketch Teacher Education Research and Development Fellows will be chosen through a competitive application process. They will develop their respective modules along with teams of colleagues that will be recruited to form their inquiry group and pilot the module activities. The evaluation activity will focus on the materials development aspect of the project. Data will be collected by the LessonSketch platform, which includes interviews with Fellows and their teams, perspectives of module writers, descriptive statistics of module use, and feedback from both teacher educator and preservice teacher end-users about the quality of their experiences. The first research activity of the project is design research on the kinds of technological infrastructure that are useful for practice-based teacher education. The PIs will identify tools that teacher educators need and want beyond the current capabilities for web-based support for use of rich media and will produce prototype tools inside the LessonSketch environment to meet these needs. The second research activity of the project will supplement the evaluation activity by examining the implementation of two of the modules in detail. This aspect of the research will examine the goals of the intended curriculum, the proposed modes of media use, the fidelity of the implemented curriculum, and learnings produced by preservice teachers. This research activity will help the field understand the degree to which practice-based teacher education that is mediated by an online access to rich media would be a kind of practice that could be easily incorporated into existing teacher education structures.

The project will produce 6 to 8 LessonSketch modules for use in teacher education classes. Each module will be implemented in at least eight teacher education classes across the country, which means that between 720 and 960 preservice teacher candidates will study the materials. The project aims to shift the field toward practice-based teacher education by supporting university programs to implement classroom-driven activities that will produce mathematics teachers with strong capabilities to teach mathematics effectively and meaningfully.

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 to Fri, 07/31/2020
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.


Project Videos

2019 STEM for All Video Showcase

Title: Differentiating Mathematics Instruction for Middle School

Presenter(s): Amy Hackenberg, Rebecca Borowski, Mihyun Jeon, Robin Jones, & Rob Matyska


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.

Award Number: 
1219606
Funding Period: 
Mon, 10/01/2012 to 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)

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.

Award Number: 
1219605
Funding Period: 
Mon, 10/01/2012 to 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.

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