Geometry

Young Mathematicians: Expanding an Innovative and Promising Model Across Learning Environments to Promote Preschoolers' Mathematics Knowledge

The goal of this design and development project is to address the critical need for innovative resources that transform the mathematics learning environments of preschool children from under-resourced communities by creating a cross-context school-home intervention.

Award Number: 
1907904
Funding Period: 
Mon, 07/01/2019 to Fri, 06/30/2023
Full Description: 

Far too many children in the U.S. start kindergarten lacking the foundational early numeracy skills needed for academic success. This project contributes to the goal of enhancing the learning and teaching of early mathematics in order to build a STEM-capable workforce and STEM-literate citizenry, which are both crucial to our nation's prosperity and competitiveness. Preparation for the STEM-workforce must start early, as young children's mathematics development undergirds cognitive development, building brain architecture, and supporting problem-solving, puzzling, and persevering, while strongly impacting and predicting future success in school. Preschool children from low socio-economic backgrounds are particularly at risk, as their mathematics knowledge may be up to a full year behind their middle-income peers. Despite agreements about the importance of mathematics-rich interactions for young children's learning and development, most early education teachers and families are not trained in evidence-based methods that can facilitate these experiences, making preschool learning environments (such as school and home) a critical target for intervention. The benefit of this project is that it will develop a robust model for a school-based intervention in early mathematics instruction. The model has the potential to broaden participation by providing instructional materials that support adult-child interaction and engagement in mathematics, explicitly promoting school-home connections in mathematics, and addressing educators' and families' attitudes toward mathematics while promoting children's mathematical knowledge and narrowing opportunity gaps.

The goal of this design and development project is to address the critical need for innovative resources that transform the mathematics learning environments of preschool children from under-resourced communities by creating a cross-context school-home intervention. To achieve this goal, qualitative and quantitative research methodologies will be employed, integrating data from multiple sources and stakeholders. Specifically, the project will: (1) engage in a materials design and development process that includes an iterative cycle of design, development, and implementation, collaborating with practitioners and families in real-world settings; (2) collect and analyze data from at least 40 Head Start classrooms, implementing the mathematics materials to ensure that the classroom and family mathematics materials and resources are engaging, usable, and comprehensible to preschoolers, teachers, and families; and (3) conduct an experimental study that will measure the impact of the intervention on preschool children's mathematics learning. The researchers will analyze collected data using hierarchical linear regression modeling to account for the clustering of children within classrooms. The researchers will also use a series of regression models and multi-level models to determine whether the intervention promotes student outcomes and whether it supports teachers' and families' positive attitudes toward mathematics.

Using Animated Contrasting Cases to Improve Procedural and Conceptual Knowledge in Geometry

This project aims to support stronger student outcomes in the teaching and learning of geometry in the middle grades through engaging students in animated contrasting cases of worked examples. The project will design a series of animated geometry curricular materials on a digital platform that ask students to compare different approaches to solving the same geometry problem. The study will measure changes in students' procedural and conceptual knowledge of geometry after engaging with the materials and will explore the ways in which teachers implement the materials in their classrooms.

Award Number: 
1907745
Funding Period: 
Thu, 08/01/2019 to Sun, 07/31/2022
Full Description: 

This project aims to support stronger student outcomes in the teaching and learning of geometry in the middle grades through engaging students in animated contrasting cases of worked examples. Animated contrasting cases are a set of two worked examples for the same geometry problem, approached in different ways. The animations show the visual moves and annotations students would make in solving the problems. Students are asked to compare and discuss the approaches. This theoretically-grounded approach extends the work of cognitive scientists and mathematics educators who have shown this approach supports strong student learning in algebra. The project will design a series of animated geometry curricular materials on a digital platform that ask students to compare different approaches to solving the same geometry problem. The study will measure changes in students' procedural and conceptual knowledge of geometry after engaging with the materials and will explore the ways in which teachers implement the materials in their classrooms. This work is particularly important as geometry is an understudied area in mathematics education, and national and international assessments at the middle school level consistently identify geometry as a mathematics content area in which students score the lowest.

This project draws on prior work that documents the impact of comparison on students' learning in algebra. Providing students with opportunities to compare multiple strategies is recommended by a range of mathematics policy documents, as research has shown this approach promotes flexibility and enhances conceptual knowledge and procedural fluency. More specifically, the approach allows students to compare the effectiveness and efficiency of mathematical arguments in the context of problem solving. An initial pilot study on non-animated contrasting cases in geometry shows promise for the general approach and suggests that animating the cases has the potential for stronger student learning gains. This study will examine the extent to which the animated cases improve students' conceptual and procedural knowledge of geometry and identify factors that relate to changes in knowledge. The project team will develop 24 worked example contrasting cases based on design principles from the prior work in algebra. The materials will be implemented in four treatment classrooms in the first cycle, revised, and then implemented in eight treatment classrooms. Students' written work will be collected along with data on the nature of the classroom discussions and small-group interviews with students. Teachers' perspectives on lessons will also be collected to support revision and strengthening of the materials. Assessments of students' geometry knowledge will be developed using measures with demonstrated validity and reliability to measure changes in student learning.

Prospective Elementary Teachers Making for Mathematical Learning

This study takes an innovative approach to documenting how teacher knowledge can be enhanced by incorporating a design experience into pre-service mathematics education. Teachers will use digital and fabrication technologies (e.g., 3D printers and laser cutters) to design and use manipulatives for K-6 mathematics learning. The goals of the project include describing how this experience influences the prospective teachers' knowledge and identities while creating curriculum for teacher education.

Lead Organization(s): 
Award Number: 
1812887
Funding Period: 
Sat, 09/01/2018 to Mon, 08/31/2020
Full Description: 

What teachers know and believe is central to what they can do in classrooms. This study takes an innovative approach to documenting how teacher knowledge can be enhanced by incorporating a design experience into pre-service mathematics education. The study's participating prospective teachers will use digital and fabrication technologies (e.g., 3D printers and laser cutters) to design and use manipulatives for K-6 mathematics learning. The goals of the project include describing how this experience influences the prospective teachers' knowledge and identities while creating curriculum for teacher education. Also, because more schools and students have access to 3D fabrication capabilities, teacher education can utilize these capabilities to prepare teachers to take advantage of these resources. Prior research by the team demonstrated how the process of making a manipulative can support prospective teachers in learning about mathematics and how to teach elementary mathematics concepts. The project will generate resources for other elementary teacher education programs and research about how prospective elementary teachers learn mathematics for teaching.

The project includes three research questions. First, what forms of knowledge are brought to bear as prospective elementary teachers make new manipulatives and write corresponding tasks to support the teaching and learning of mathematics? Second, how does prospective elementary teachers' knowledge for teaching mathematics develop as they make new manipulatives and write tasks to support the teaching and learning of mathematics? Third, as prospective elementary teachers make new manipulatives and write tasks to support the teaching and learning of mathematics, how do they see themselves in relation to the making, the mathematics, and the mathematics teaching? The project will employ a design-based research methodology with cycles of design, enactment, analysis and redesign to create curriculum modules for teacher education focused on making mathematics manipulatives. Data collection will include video recording of class sessions, participant observation, field notes, artifacts from the participants' design of manipulatives, and assessments of mathematical knowledge for teaching. A qualitative analysis will use multiple frameworks from prior research on mathematics teacher knowledge and identity development.

CAREER: Mechanisms Underlying the Relation Between Mathematical Language and Mathematical Knowledge

The purpose of this project is to examine the process by which math language instruction improves learning of mathematics skills in order to design and translate the most effective interventions into practical classroom instruction.

Lead Organization(s): 
Award Number: 
1749294
Funding Period: 
Wed, 08/01/2018 to Mon, 07/31/2023
Full Description: 

Successful development of numeracy and geometry skills during preschool provides a strong foundation for later academic and career success. Recent evidence shows that learning math language (e.g., concepts such as more, few, less, near, before) during preschool supports this development. The purpose of this Faculty Early Career Development (CAREER) project is to examine the process by which math language instruction improves learning of mathematics skills in order to design and translate the most effective interventions into practical classroom instruction. The first objective of this project is to examine if quantitative and spatial math language effect the development of different aspects of mathematics performance (e.g., numeracy, geometry). The second objective is to examine how quantitative math language versus numeracy instruction, either alone or in combination, effect numeracy development. The findings from this study will not only be used to improve theoretical understanding of how math language and mathematics skills develop, but the instructional materials developed for this study will also result in practical tools for enhancing young children's math language and mathematics skills.

This project is focused on evaluating the role of early math language skills in the acquisition of early mathematics skills. Two randomized control trials (RCTs) will be conducted. The first RCT will be used to evaluate the effects of different types of math language instruction (quantitative, spatial) on distinct aspects of mathematics (numeracy, geometry). It is expected that quantitative language instruction will improve numeracy skills and spatial language instruction will improve geometry skills. The second RCT will be used to examine the unique and joint effects of quantitative language instruction and numeracy instruction on children's numeracy skills. It is expected that both types of instruction alone will be sufficient to generate improvement on numeracy outcomes compared to an active control group, but that the combination of the two will result in enhanced numeracy performance compared to either alone. Educational goals will be integrated with and supported through engaging diverse groups of undergraduate and graduate students in hands-on research experiences, training pre- and in-service teachers on mathematical language instruction, and building collaborative relationships with early career researchers. Intervention materials including storybooks developed for the project and pre- and in-service teacher training/lesson plan materials will be made available at the completion of the project.

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.

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.

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.

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

2019 STEM for All Video Showcase

Title: Concept of Function Learning Progression

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


Enhancing Middle Grades Students' Capacity to Develop and Communicate Their Mathematical Understanding of Big Ideas Using Digital Inscriptional Resources (Collaborative Research: Phillips)

This project will develop and test a digital platform for middle school mathematics classrooms to help students deepen and communicate their understanding of mathematics. The digital platform will allow students to collaboratively create representations of their mathematics thinking, incorporate ideas from other students, and share their work with the class.

Lead Organization(s): 
Award Number: 
1620934
Funding Period: 
Thu, 09/01/2016 to Mon, 08/31/2020
Full Description: 

The primary goal of this project is to help middle school students deepen and communicate their understanding of mathematics. The project will develop and test a digital platform for middle school mathematics classrooms. The digital platform will allow students to collaboratively create representations of their mathematics thinking, incorporate ideas from other students, and share their work with the class. The digital learning environment makes use of a problem-centered mathematics curriculum that evolved from extensive development, field-testing and evaluation, and is widely used in middle schools. The research will also contribute to understanding about the design and innovative use of digital resources and collaboration in classrooms as an increasing number of schools are drawing on these kinds of tools. Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for proposed projects.

The project will support students to collaboratively construct, manipulate, and interpret shared representations of mathematics using digital inscriptional resources. The research activities will significantly enhance our understanding of student learning in mathematics in three important ways. The project will report on how (1) evidence of student thinking is made visible through the use of digital inscriptional resources, (2) student inscriptions are documented, discussed, and manipulated in collaborative settings, and (3) students' conceptual growth of big mathematical ideas grows over time. An iterative design research process will incorporate four phases of development, testing and revision, and will be conducted to study student use of the digital learning space and related inscriptional resources. Data sources will include: classroom observations and artifacts, student and teacher interviews and surveys, student assessment data, and analytics from the digital platform. The process will include close collaboration with teachers to understand the implementation and create revisions to the resources.


Project Videos

2019 STEM for All Video Showcase

Title: Math Understanding in a Digital Collaborative Environment

Presenter(s): Alden Edson, Kristen Bieda, Chad Dorsey, Nathan Kimball, & Elizabeth Phillips


Pages

Subscribe to Geometry