Geometry

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.


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.

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

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


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


Mathematical and Computational Methods for Planning a Sustainable Future II

The project will develop modules for grades 9-12 that integrate mathematics, computing and science in sustainability contexts. The project materials also include information about STEM careers in sustainability to increase the relevancy of the content for students and broaden their understanding of STEM workforce opportunities. It uses summer workshops to pilot test materials and online support and field testing in four states. 

Lead Organization(s): 
Award Number: 
1503414
Funding Period: 
Wed, 07/15/2015 to Sun, 06/30/2019
Full Description: 

The project will develop modules for grades 9-12 that integrate mathematics, computing and science in sustainability contexts. The project materials also include information about STEM careers in sustainability to increase the relevancy of the content for students and broaden their understanding of STEM workforce opportunities. It uses summer workshops to pilot test materials and online support and field testing in four states. Outcomes include the modules, tested and revised; strategies for transfer of learning embedded in the modules; and a compendium of green jobs, explicitly related to the modules. 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 STEM+Computing Partnerships (STEM+C) Program is a joint effort between the Directorate for Education & Human Resources (EHR) and Directorate Computer & Information Science & Engineering (CISE). Reflecting the increasing role of computational approaches in learning across the STEM disciplines, STEM+C supports research and development efforts that integrate computing within one or more STEM disciplines and/or integrate STEM learning in computer science; 2) advance multidisciplinary, collaborative approaches for integrating computing in STEM in and out of school, and 3) build capacity in K-12 computing education through foundational research and focused teacher preparation

The project is a full design and development project in the learning strand of DRK-12. The goal is to enhance transfer of knowledge in mathematics and science via sustainability tasks with an emphasis on mathematical and scientific practices. The research questions focus on how conceptual representations and the modules support students' learning and especially transfer to novel problems. The project design integrates the research with the curriculum development. It includes a mixed methods data collection and analysis from teachers and students (e.g., interviews, content exams, focus groups, implementation logs). Assessment of student work includes both short, focused problems in the content area and longer project-based tasks providing a range of assessments of student learning. The investigators will develop a rubric for scoring student work on the tasks. The curriculum design process includes iterations of the modules over time with feedback from teachers and using data collected from the implementation.

Collaborative Math: Creating Sustainable Excellence in Mathematics for Head Start Programs

This project will adapt and study a promising and replicable teacher professional development (PD) intervention, called Collaborative Math (CM), for use in early childhood programs. Prepared as generalists, preschool teachers typically acquire less math knowledge in pre-service training than their colleagues in upper grades, which reduces their effectiveness in teaching math. To address teacher PD needs, the project will simultaneously develop teacher content knowledge, confidence, and classroom practice by using a whole teacher approach.

Lead Organization(s): 
Award Number: 
1503486
Funding Period: 
Tue, 09/01/2015 to Sat, 08/31/2019
Full Description: 

This project was submitted to the Discovery Research K-12 (DRK-12) program that 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. 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 adapt and study a promising and replicable teacher professional development (PD) intervention, called Collaborative Math (CM), for use in early childhood programs. CM content will focus on nine topics emphasized in preschool mathematics, including sets, number sense, counting, number operations, pattern, measurement, data analysis, spatial relationships, and shape. These concepts are organized around Big Ideas familiar in early math, are developmentally appropriate and foundational to a young child's understanding of mathematics. The project addresses the urgent need for improving early math instruction for low-income children. Prepared as generalists, preschool teachers typically acquire less math knowledge in pre-service training than their colleagues in upper grades, which reduces their effectiveness in teaching math. To address teacher PD needs, the project will simultaneously develop teacher content knowledge, confidence, and classroom practice by using a whole teacher approach. Likewise, the project will involve teachers, teacher aides, and administrators through a whole school approach in PD, which research has shown is more effective than involving only lead teachers. Through several phases of development and research, the project will investigate the contributions of project components on increases in teacher knowledge and classroom practices, student math knowledge, and overall implementation. The project will impact approximately 200 Head Start (HS) teaching staff, better preparing them to provide quality early math experiences to more than 3,000 HS children during the project period. Upon the completion of the project, a range of well-tested CM materials such as resource books and teaching videos will be widely available for early math PD use. Assessment tools that look at math knowledge, attitudes, and teacher practice will also be available. 

The project builds on Erikson Institute research and development work in fields of early math PD and curriculum. Over a 4-year span, project development and research will be implemented in 4 phases: (1) adapting the existing CM and research measures for HS context; (2) conducting a limited field study of revised CM in terms of fidelity and director, teacher/aide, and student outcomes, and study of business as usual (BAU) comparison groups; (3) a study of the promise of the intervention promise with the phase 3 BAU group (who offered baseline in phase 2) and (4) a test of the 2nd year sustainability intervention with phase 3 treatment group. The teacher and student measures are all published, frequently used measures in early childhood education and will be piloted and refined prior to full implementation. The project is a partnership between Erikson, SRI, and Chicago Head Start programs. Project research and resources will be widely disseminated to policy makers, researchers, and practitioners.

TRUmath and Lesson Study: Supporting Fundamental and Sustainable Improvement in High School Mathematics Teaching (Collaborative Research: Schoenfeld)

Given the changes in instructional practices needed to support high quality mathematics teaching and learning based on college and career readiness standards, school districts need to provide professional learning opportunities for teachers that support those changes. The project is based on the TRUmath framework and will build a coherent and scalable plan for providing these opportunities in high school mathematics departments, a traditionally difficult unit of organizational change.

Award Number: 
1503454
Funding Period: 
Wed, 07/01/2015 to Sun, 06/30/2019
Full Description: 

Given the changes in instructional practices needed to support high quality mathematics teaching and learning based on college and career readiness standards, school districts need to provide professional learning opportunities for teachers that support those changes. The project will build a coherent and scalable plan for providing these opportunities in high school mathematics departments, a traditionally difficult unit of organizational change. Based on the TRUmath framework, characterizing the five essential dimensions of powerful mathematics classrooms, the project brings together a focus on curricular materials that support teaching, Lesson Study protocols and materials, and a professional learning community-based professional development model. The project will design and revise professional development and coaching guides and lesson study mathematical resources built around the curricular materials. The project will study changes in instructional practice and impact on student learning. By documenting the supports used in the Oakland Unified School District where the research and development will be conducted, the resources can be used by other districts and in similar work by other research-practice partnerships.

This project hypothesizes that the quality of classroom instruction can be defined by five dimensions - quality of the mathematics; cognitive demand of the tasks; access to mathematics content in the classroom; student agency, authority, and identity; and uses of assessment. The project will use an iterative design process to develop and refine a suite of tool, including a conversation guide to support productive dialogue between teachers and coaches, support materials for building site-based professional learning materials, and formative assessment lessons using Lesson Study as a mechanism to enact reforms of these dimensions. The study will use a pre-post design and natural variation to student the relationships between these dimensions, changes in teachers' instructional practice, and student learning using hierarchical linear modeling with random intercept models with covariates. Qualitative of the changes in teachers' instructional practices will be based on coding of observations based on the TRUmath framework. The study will also use qualitative analysis techniques to identify themes from surveys and interviews on factors that promote or hinder the effectiveness of the intervention.

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