Algebra

Algebraic Learning and Cognition in Learning Disabled Students

The project is a longitudinal assessment of the prerequisite (e.g. fractions), cognitive (e.g. working memory), and non-cognitive (e.g. math anxiety) factors that dynamically influence 7-9th grade students' algebraic learning and cognition, with a focus on students with learning disabilities in mathematics.

Lead Organization(s): 
Award Number: 
1659133
Funding Period: 
Tue, 08/15/2017 to Sat, 07/31/2021
Full Description: 

High school algebra is the gateway to a career in science, technology, engineering, and mathematics (STEM), and can influence employability and wages in many non-STEM occupations. Students who struggle with or fail high school algebra have compromised occupational prospects, and nations that do not produce mathematically competent citizens may compromise their economic growth. Much is known about the factors that contribute to students' difficulties with arithmetic learning and interventions are being developed to address these difficulties. Little is known, however, about why some students struggle with algebra. Accordingly, the project will follow at risk students (including for example, those with dyslexia) from 7th grade through high school algebra and assess their prerequisite knowledge (e.g. fractions skills), cognitive systems (e.g., memory), attitudes and reactions to mathematics (e.g. math anxiety) and their attentiveness in math classrooms. The comprehensive evaluation of these students will allow us to identify the factors that influence difficulties in learning different aspects of algebra and risk of failing algebra more generally. The results will provide unique scientific insights into the cognitive and motivational influences on students' understanding and learning of algebra and identify areas for intervention with at-risk students. The results will also be used to develop a screening measure for the early identification of at-risk students and to identify specific areas for targeted intervention. The measure will be made freely available to interested school districts throughout the United States.

The project is a 7th to 9th grade longitudinal assessment of the prerequisite (e.g. fractions), cognitive (e.g. working memory), and non-cognitive (e.g. math anxiety) factors that dynamically influence students' algebraic learning and cognition, with a focus on students with learning disabilities in mathematics. The study will provide the most comprehensive assessment of the development of algebra competence ever conducted and is organized by an integrative model of cognitive and non-cognitive influences on students' engagement in math classrooms and on the learning of procedural and spatial-related aspects of algebra. The focus on students at risk for failing high school algebra is informed by research on the number and arithmetic deficits of these students, providing continuity with previous work, and a strong a priori framework for assessing their most likely difficulties in learning algebra; specifically, we developed novel measures that assess different aspects of procedural algebra (e.g. memory for the structure of algebra equations) and spatial-related algebra (e.g. recognizing how common functions map to coordinate space) that will allow for the study of different types of learning deficits and a determination of how more basic cognitive abilities, such as visuospatial working memory, may underlie these deficits. Prior cognitive studies of at-risk students have largely ignored the contributions of non-cognitive factors, such as math anxiety, on their learning or how their learning difficulties change attitudes about and reactions to mathematics (e.g. increasing math anxiety). The proposed research will address this important oversight and integrate these non-cognitive factors with assessments of teacher-rated student engagement in pre-algebra and algebra classrooms (and language arts classrooms as a contrast) and how engagement in the classroom influences the learning of procedural and spatial-related algebra. The research will also provide a thorough analysis of cognitive and non-cognitive influences on algebraic learning and cognition more generally, and thus inform general educational practices. In all, the proposed research will provide a comprehensive model for the study algebraic learning and cognition generally, and will provide a comprehensive assessment of associated deficits of learning disabled students and students at risk for failing high school algebra. The research will also make available revised or newly developed cognitive measures of procedural and spatial-related algebra skills that should facilitate future cognitive science and educational studies of algebra learning.

Developing Preservice Teachers' Capacity to Teach Students with Learning Disabilities in Algebra I

Project researchers are training pre-service teachers to tutor students with learning disabilities in Algebra 1, combining principles from special education, mathematics education, and cognitive psychology. The trainings emphasize the use of gestures and strategic questioning to support students with learning disabilities and to build students’ understanding in Algebra 1.

Project Email: 
Lead Organization(s): 
Award Number: 
1813903
Funding Period: 
Wed, 08/01/2018 to Sat, 07/31/2021
Full Description: 

This project is implementing a program to train pre-service teachers to tutor students with learning disabilities in Algebra 1, combining principles from special education, mathematics education, and cognitive psychology. The project trains tutors to utilize gestures and strategic questioning to support students with LD to build connections between procedural knowledge and conceptual understanding in Algebra 1, while supporting students’ dispositions towards doing mathematics. The training will prepare tutors to address the challenges that students with LD often face—especially challenges of working memory and processing—and to build on their strengths as they engage with Algebra 1. The project will measure changes in tutors’ ability to use gestures and questioning to support the learning of students with LD during and after the completion of our training. It will also collect and analyze data on the knowledge and dispositions of students with LD in Algebra 1 for use in the ongoing refinement of the training and in documenting the impact of the training program.

Measuring Early Mathematical Reasoning Skills: Developing Tests of Numeric Relational Reasoning and Spatial Reasoning

The primary aim of this study is to develop mathematics screening assessment tools for Grades K-2 over the course of four years that measure students' abilities in numeric relational reasoning and spatial reasoning. The team of researchers will develop Measures of Mathematical Reasoning Skills system, which will contain Tests of Numeric Relational Reasoning (T-NRR) and Tests of Spatial Reasoning (T-SR).

Award Number: 
1721100
Funding Period: 
Fri, 09/15/2017 to Tue, 08/31/2021
Full Description: 

Numeric relational reasoning and spatial reasoning are critical to success in later mathematics coursework, including Algebra 1, a gatekeeper to success at the post-secondary level, and success in additional STEM domains, such as chemistry, geology, biology, and engineering. Given the importance of these skills for later success, it is imperative that there are high-quality screening tools available to identify students at-risk for difficulty in these areas. The primary aim of this study is to develop mathematics screening assessment tools for Grades K-2 over the course of four years that measure students' abilities in numeric relational reasoning and spatial reasoning. The team of researchers will develop Measures of Mathematical Reasoning Skills system, which will contain Tests of Numeric Relational Reasoning (T-NRR) and Tests of Spatial Reasoning (T-SR). The measures will be intended for use by teachers and school systems to screen students to determine who is at-risk for difficulty in early mathematics, including students with disabilities. The measures will help provide important information about the intensity of support that may be needed for a given student. Three forms per grade level will be developed for both the T-NRR and T-SR with accompanying validity and reliability evidence collected. The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools (RMTs). Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for proposed projects.

The development of the T-NRR and T-SR measures will follow an iterative process across five phases. The phases include (1) refining the construct; (2) developing test specifications and item models; (3) developing items; (4) field testing the items; and (5) conducting validity studies. The evidence collected and evaluated during each phase will contribute to the overall evaluation of the reliability of the measures and the validity of the interpretations made using the measures. Item models, test specifications, and item development will be continuously evaluated and refined based on data from cognitive interviews, field tests, and reviews by mathematics educators, teachers of struggling students, teachers of culturally and linguistically diverse populations, and a Technical Advisory Board. In the final phase of development of the T-NRR and T-SR, reliability of the results will be estimated and multiple sources of validity evidence will be collected to examine the concurrent and predictive relation with other criterion measures, classification accuracy, and sensitivity to growth. Approximately 4,500 students in Grades K-2 will be involved in all phases of the research including field tests and cognitive interviews. Data will be analyzed using a two-parameter IRT model to ensure item and test form comparability.


Project Videos

2020 STEM for All Video Showcase

Title: Measuring Early Mathematical Reasoning Skills

Presenter(s): Leanne Ketterlin Geller


The Mathematical Knowledge for Teaching Measures: Refreshing the Item Pool

This project proposes an assessment study that focuses on improving existing measures of teachers' Mathematical Knowledge for Teaching (MKT). The research team will update existing measures, adding new items and aligning the instrument to new standards in school mathematics.

Lead Organization(s): 
Award Number: 
1620914
Funding Period: 
Thu, 12/01/2016 to Sat, 11/30/2019
Full Description: 

This project proposes an assessment study that focuses on improving existing measures of teachers' Mathematical Knowledge for Teaching (MKT). The research team will update existing measures, adding new items and aligning the instrument to new standards in school mathematics. In addition, the team will update the delivery system for the assessment to Qualtrics, a more flexible online system.

The research team will build an updated measure of teachers' Mathematical Knowledge for Teaching (MKT). Project researchers will conduct item writing camps, develop new items, cognitively pilot and revise items, and factor analyze items. The researchers will also determine item constructs and calibrate items (and constructs) through an innovative application of Item Response Theory (IRT) employing a variant of the standard 2-parameter IRT model. Finally, the team will oversee the transition of the Teacher Knowledge Assessment System to the Qualtrics data collection environment to allow for more flexible item specification.

Identifying Effective Instructional Practices that Foster the Development of Algebraic Thinking in Elementary School

This project seeks to identify teaching practices that can be linked to students' early algebra learning in grades three, four and five. The goal of the project is to use assessment data and videos of classroom teaching in order to create a tool that can be used to document effective instructional practices. This observation tool can then be used to support teacher professional development in early algebra and research about how teachers' actions can be linked to students' learning.

Lead Organization(s): 
Award Number: 
1721192
Funding Period: 
Thu, 06/01/2017 to Mon, 05/31/2021
Full Description: 

There is a critical need to better prepare all students for learning algebra. Part of this preparation involves developing a strong foundation for algebra in the elementary grades by building on students' informal intuitions about patterns, relationships and structure into more formalized ways of mathematical thinking. This project seeks to identify teaching practices that can be linked to students' early algebra learning in grades three, four and five. The goal of the project is to use assessment data and videos of classroom teaching in order to create a tool that can be used to document effective instructional practices. This observation tool can then be used to support teacher professional development in early algebra and research about how teachers' actions can be linked to students' learning. The project is unique in its work to link an early algebra curriculum with understanding of teachers' practices in implementing that curriculum and students' learning of mathematics.

The project aims to address two research questions. First, what profiles of instructional practice are associated with greater student performance in early algebra? Second, to what extent do these profiles of effective instructional practices vary by grade level? The primary product of the work is an early algebra observation protocol that will capture non-domain and non-grade level specific practices of effective teaching in combination with practices specific to early algebra. Videos of early algebra classrooms will be used to design the observation protocol, which in turn, will then be used along with student assessment data to identify profiles of instructional practices associated with students' learning. Multiple phases of testing and revision will be used to create the observation protocol. The observation protocol will also generate profiles of teacher practices that can be used to describe different models for effectively teaching early algebra. The project will also examine implications of their work for teacher preparation and professional development.

Examining Relationships Between Flipped Instruction and Students' Learning of Mathematics

This study can provide a basis for design research focused on developing effective materials and programs for flipped instruction in secondary mathematics, which is already occurring at an increasing rate, but it is not yet informed by empirical evidence. This project will result in a framework for flipped instruction robust enough to be useful at a variety of grade levels and contexts. The framework will provide a better understanding of the relationships among various implementations of flipped instruction and student learning.

Lead Organization(s): 
Award Number: 
1721025
Funding Period: 
Tue, 08/01/2017 to Fri, 07/31/2020
Full Description: 

Instead of presenting new material in class and then assigning problems to be completed outside of class, flipped instruction involves students watching videos or reading new material outside of class and then completing their "homework" in class. Teachers' implementation of flipped instruction has increased dramatically in recent years, with more than two-thirds of teachers now reporting flipping a lesson, if not an entire course. Although popular media and philanthropic organizations have given a great deal of attention and financial support to flipped instruction, little is known about how teachers implement it and what benefits and drawbacks flipped instruction has in contrast with non-flipped instruction. This study can provide a basis for design research focused on developing effective materials and programs for flipped instruction in secondary mathematics. This design and development is already occurring at an increasing rate, but it is not yet informed by empirical evidence. This project will result in a framework for flipped instruction robust enough to be useful at a variety of grade levels and contexts. The framework will provide a better understanding of the relationships among various implementations of flipped instruction and student learning. These findings can inform teacher educators in better aligning their instruction to instructional formats that correlate with increased student learning outcomes.

Using mixed-methods techniques, the study will look at the nature of the activities and interactions occurring in mathematics classrooms and assess their quality so that the researchers may distinguish high-quality from low-quality univocal discourse, high-quality from low-quality dialogic discourse, and high cognitive demand from low cognitive demand tasks. Working in 40 algebra classrooms -- 20 implementing some form of flipped instruction and 20 serving as a non-flipped basis for comparison -- the project will address the following research questions using a correlational design and multilevel modeling techniques: RQ1. What are salient factors entailed in flipped instruction in secondary algebra? RQ2. What associations, if any, exist among factors entailed in teachers' implementation of flipped algebra instruction and students' learning of algebra as measured on a state-mandated end-of-course assessment and on a concept-of-variable inventory?

CAREER: Designing and Enacting Mathematically Captivating Learning Experiences for High School Mathematics

This project explores how secondary mathematics teachers can plan and enact learning experiences that spur student curiosity, captivate students with complex mathematical content, and compel students to engage and persevere (referred to as "mathematically captivating learning experiences" or "MCLEs"). The study will examine how high school teachers can design lessons so that mathematical content itself is the source of student intrigue, pursuit, and passion.

Lead Organization(s): 
Award Number: 
1652513
Funding Period: 
Wed, 02/15/2017 to Mon, 01/31/2022
Full Description: 

This design and development project explores how secondary mathematics teachers can plan and enact learning experiences that spur student curiosity, captivate students with complex mathematical content, and compel students to engage and persevere (referred to as "mathematically captivating learning experiences" or "MCLEs"). This study is important because of persistent disinterest by secondary students in mathematics in the United States. This study will examine how high school teachers can design lessons so that mathematical content itself is the source of student intrigue, pursuit, and passion. To do this, the content within mathematical lessons (both planned and enacted) is framed as mathematical stories and the felt tension between how information is revealed and withheld from students as the mathematical story unfolds is framed as its mathematical plot. The Mathematical Story Framework (Dietiker, 2013, 2015) foregrounds both the coherence (does the story make sense?) and aesthetic (does it stimulate anticipation for what is to come, and if so, how?) dimensions of mathematics lessons. The project will generate principles for lesson design usable by teachers in other settings and exemplar lessons that can be shared.

Specifically, this project draws from prior curriculum research and design to (a) develop a theory of teacher MCLE design and enactment with the Mathematical Story Framework, (b) increase the understanding(s) of the aesthetic nature of mathematics curriculum by both researchers and teachers, and (c) generate detailed MCLE exemplars that demonstrate curricular coherence, cognitive demand, and aesthetic dimensions of mathematical lessons. The project is grounded in a design-based research framework for education research. A team of experienced high school teachers will design and test MCLEs (four per teacher) with researchers through three year-long cycles. Prior to the first cycle, data will be collected (interview, observations) to record initial teacher curricular strategies regarding student dispositions toward mathematics. Then, a professional development experience will introduce the Mathematical Story Framework, along with other curricular frameworks to support the planning and enacting of lessons (i.e., cognitive demand and coherence). During the design cycles, videotaped observations and student aesthetic measures (surveys and interviews) for both MCLEs and a non-MCLEs (randomly selected to be the lesson before or after the MCLE) will be collected to enable comparison. Also, student dispositional measures, collected at the beginning and end of each cycle, will be used to learn whether and how student attitudes in mathematics change over time. Of the MCLEs designed and tested, a sample will be selected (based on aesthetic and mathematical differences) and developed into models, complete with the rationale for and description of aesthetic dimensions.

CAREER: Investigating Changes in Students' Prior Mathematical Reasoning: An Exploration of Backward Transfer Effects in School Algebra

This project explores "backward transfer", or the ways in which new learning impacts previously-established ways of reasoning. The PI will observe and evaluate algebra I students as they learn quadratic functions and examine how different kinds of instruction about the new concept of quadratic functions helps or hinders students' prior mathematical knowledge of the previous concept of linear functions.

Lead Organization(s): 
Award Number: 
1651571
Funding Period: 
Sat, 07/01/2017 to Thu, 06/30/2022
Full Description: 

As students learn new mathematical concepts, teachers need to ensure that prior knowledge and prior ways understanding are not negatively affected. This award explores "backward transfer", or the ways in which new learning impacts previously-established ways of reasoning. The PI will observe and evaluate students in four Algebra I classrooms as they learn quadratic functions. The PI will examine how different kinds of instruction about the new concept of quadratic functions helps or hinders students' prior mathematical knowledge of the previous concept of linear functions. More generally, this award will contribute to the field of mathematics education by expanding the application of knowledge transfer, moving it from only a forward focused direction to include, also, a backward focused direction. An advisory board of scholars with expertise in mathematics education, assessment, social interactions, quantitative reasoning and measurement will support the project. The research will occur in diverse classrooms and result in presentations at the annual conferences of national organizations, peer-reviewed publications, as well as a website for teachers which will explain both the theoretical model and the findings from the project. An undergraduate university course and professional development workshops using video data from the project are also being developed for pre-service and in-service teachers. Ultimately, the research findings will generate new knowledge and offer guidance to elementary school teachers as they prepare their students for algebra.

The research involves three phases. The first phase includes observations and recordings of four Algebra I classrooms and will test students' understanding of linear functions before and after the lessons on quadratic functions. This phase will also include interviews with students to better understand their reasoning about linear function problems. The class sessions will be coded for the kind of reasoning that they promote. The second phase of the project will involve four cycles of design research to create quadratic and linear function activities that can be used as instructional interventions. In conjunction with this phase, pre-service teachers will observe teaching sessions through a course that will be offered concurrently with the design research. The final phase of the project will involve pilot-applied research which will test the effects of the instructional activities on students' linear function reasoning in classroom settings. This phase will include treatment and control groups and further test the hypotheses and instructional products developed in the first two phases.

Building a Next Generation Diagnostic Assessment and Reporting System within a Learning Trajectory-Based Mathematics Learning Map for Grades 6-8

This project will build on prior funding to design a next generation diagnostic assessment using learning progressions and other learning sciences research to support middle grades mathematics teaching and learning. The project will contribute to the nationally supported move to create, use, and apply research based open educational resources at scale.

Award Number: 
1621254
Funding Period: 
Thu, 09/15/2016 to Sat, 08/31/2019
Full Description: 

This project seeks to design a next generation diagnostic assessment using learning progressions and other research (in the learning sciences) to support middle grades mathematics teaching and learning. It will focus on nine large content ideas, and associated Common Core State Standards for Mathematics. The PIs will track students over time, and work within school districts to ensure feasibility and use of the assessment system.

The research will build on prior funding by multiple funding agencies and address four major goals. The partnership seeks to address these goals: 1) revising and strengthening the diagnostic assessments in mathematics by adding new item types and dynamic tools for data gathering 2) studying alternative ways to use measurement models to assess student mathematical progress over time using the concept of learning trajectories, 3) investigating how to assist students and teachers to effectively interpret reports on math progress, both at the individual and the class level, and 4) engineering and studying instructional strategies based on student results and interpretations, as they are implemented within competency-based and personalized learning classrooms. The learning map, assessment system, and analytics are open source and can be used by other research and implementation teams. The project will exhibit broad impact due to the number of states, school districts and varied kinds of schools seeking this kind of resource as a means to improve instruction. Finally, the research project contributes to the nationally supported move to create, use, and apply research based open educational resources at scale.

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

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

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

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

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


Project Videos

2020 STEM for All Video Showcase

Title: Practitioners’ Use of the Five-Step Curricular Process

Presenter(s): Edith Graf, Frank Davis, Cheryl Eames, Chad Milner, & Maisha Moses

2019 STEM for All Video Showcase

Title: Concept of Function Learning Progression

Presenter(s): Edith Graf, Frank Davis, Chad Milner, Maisha Moses, & Sarah Ohls


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