Number Sense

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.

Development of the Electronic Test of Early Numeracy

The project will develop and refine an electronic Test of Early Numeracy (e-TEN) in English and Spanish that will assess informal and formal knowledge of number and operations in domains including verbal counting, numbering, numerical relationships, and mental addition/subtraction. The overarching goal of the assessment design is to create a measure that is more accurate, more accessible to a wider range of children, and easier to administer than existing measures.

Partner Organization(s): 
Award Number: 
1621470
Funding Period: 
Thu, 09/15/2016 to Tue, 08/31/2021
Full Description: 

The project will develop and refine an electronic Test of Early Numeracy (e-TEN) in English and Spanish, focused on number and operations. The assessment will incorporate a learning trajectory that describes students' development of the understanding of number. The electronic assessment will allow for the test to adapt to students' responses and incorporate games to increase children's engagement with the tasks. These features take advantage of the electronic format. The achievement test will be designed to be efficient, user-friendly, affordable, and accessible for a variety of learning environments and a broad age range (3 to 8 years old). The overarching goal of the assessment design is to create a measure that is more accurate, more accessible to a wider range of children, and easier to administer than existing measures. This project is funded by the Discovery Research Pre-K-12 Program, which funds research and development of STEM innovations and approaches in assessment, teaching and learning.

The e-TEN will assess informal and formal knowledge of number and operations in domains including verbal counting, numbering, numerical relationships, and mental addition/subtraction. The items will be designed using domain-based learning trajectories that describe students' development of understanding of the topics. The test will be designed with some key characteristics. First, it will be semi-adaptive over six-month age spans. Second, it will have an electronic format that allows for uniform implementation and an efficient, user-friendly administration. The test will also be accessible to Spanish speakers using an inclusive assessment model. Finally, the game-based aspect should increase children's engagement and present more meaningful questions. The user-friendly aspect includes simplifying the assessment process compared to other tests of numeracy in early-childhood. The first phase of the development will test a preliminary version of the e-TEN to test its functionality and feasibility. The second phase will focus on norming of the items, reliability and validity. Reliability will be assessed using Item Response Theory methods and test-retest reliability measures. Validity will be examined using criterion-prediction validity and construct validity. The final phase of the work will include creating a Spanish version of the test including collecting data from bilingual children using both versions of the e-TEN.

Strengthening Mathematics Intervention: Identifying and Addressing Challenges to Improve Instruction for Struggling Learners

This project's first goal is to study the national landscape of mathematics intervention classes, which are additional classes provided to struggling students, including learners with and without identified disabilities. We administered a survey to a nationally representative sample of 2,024 urban and suburban public schools with grades 6-8 to find out how these classes are being implemented and the types of challenges faced.

Award Number: 
1621294
Funding Period: 
Thu, 09/15/2016 to Mon, 08/31/2020
Full Description: 

Across the nation, schools face a pressing need to improve instruction for middle grades students who are not reaching proficiency on standardized assessments. One approach is to schedule additional mathematics classes to provide struggling learners with more time for instruction and support. For our study, we defined mathematics interventional classes as classes taken by struggling students during the regular school day in addition to their general education mathematics classes. These classes are for students who have difficulties learning mathematics, including learners who do not have identified disabilities and those with Individualized Education Programs (IEPs). 

While recommendations for intervention practices are present in the research literature, little is known about how schools are actually implementing intervention classes, including how often the classes meet, the number of students enrolled, who teaches them and the content focus. To address this gap in the knowledge base, we conducted an observational study and a national survey of current practices and challenges in mathematics intervention classes. The survey was administered to a nationally representative sample of 2,024 urban or suburban public schools with grades 6-8.  Approximately, 43% of schools (876 schools) responded; the findings revealed widespread implementation of mathematics intervention classes and variations in class structures and practices. 

The final aspect of the project involves the design of professional development for mathematics intervention teachers based on the needs identified in the earlier phases of the project. We are developing and testing a blended professional development course to help teachers build the knowledge and practices needed to provide high-quality, targeted instruction to struggling learners in mathematics intervention classes.

Related Resource:

Proportions Playground: A Dynamic World to Support Teachers' Proportional Reasoning

This project focuses on the creation of the initial functionality for a dynamic microworld, Proportions Playground, designed to support teachers in developing a coherent understanding of proportional reasoning. The Proportions Playground project seeks to both develop a unique pilot software application for the iPad and explore how it supports teachers in developing a coherent, robust definition of proportions.

Award Number: 
1621290
Funding Period: 
Thu, 09/01/2016 to Thu, 02/28/2019
Full Description: 

Proportions are a critical topic in mathematics that is simultaneously complicated and over-simplified in typical instruction. Current research undertaken by the research team suggests that the over-simplification is related to limitations in teachers' understandings of proportional relationships. Presenting proportions in a dynamic environment offers teachers the opportunity to create key developmental understandings related to this area of mathematics. This project focuses on the creation of the initial functionality for a dynamic microworld, Proportions Playground, designed to support teachers in developing a coherent understanding of proportional reasoning. Proportions Playground is conceptualized as a tool for supporting the development of coherent understandings by allowing teachers to interact in concrete ways with otherwise abstract ideas and by allowing teachers easy access to dynamic objects and other representations. It is meant to address the significant limitations for reasoning about the relationships between measurable aspects of two objects as well as in manipulating those relationships. Building from work currently underway, Proportions Playground will explore key areas in which there are opportunities for engaging teachers in the development of a coherent and robust understanding of proportional reasoning that extends beyond the typical "3 given, 1 unknown" proportion problem. This approach attempts to engage teachers in an array of dynamic, visually-rich sets of tasks designed to challenge teachers' preconceptions of proportions and to strengthen their connections between proportions and related areas of mathematics. This project is funded by the Discovery Research PreK-12 (DRK-12) and EHR Core Research (ECR) Programs. the DRK-12 program supports research and development on STEM education innovations and approaches to teaching, learning, and assessment. The ECR program emphasizes fundamental STEM education research that generates foundational knowledge in the field.

The Proportions Playground project seeks to both develop a unique pilot software application for the iPad and explore how it supports teachers in developing a coherent, robust definition of proportions. The software will be designed to support either numeric manipulation (e.g., graphing software) or geometric constructions (e.g., dynamic geometry software). Specifically, for this project the mathematics of interest will include the relationships between similarity and proportion and the nature of covariation. The research will focus on how teachers are developing a robust and coherent understanding of proportions and how the dynamic environment promotes such understandings. Working with six teacher advisors, the project will develop three task sets. Using teaching experiments and individual interviews, results will be used to refine the task sets. The revised task sets will be piloted with 40 teachers. Data will be collected on participants' thinking and any changes seen in the knowledge resources they are using. The researchers will be looking for factors that seem to impact teachers' thinking as well as evidence to support or deny the assertion that the Proportions Playground activities engage teachers in (a) different ways of reasoning about proportions and (b) support them in drawing from a wide array of resources so that coherence may be developed were the teachers to have a prolonged engagement with the tools. The project will rely on Epistemic Network Analysis to identify the connections between knowledge resources.

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.

Doing the Math with Paraeducators: A Research and Development Project

This project will design and pilot professional development that focuses on developing the confidence, mathematical knowledge, and teaching strategies of paraeducators using classroom activities that they are expected to implement. The planned professional development will enable them to make a greater difference in the classroom, but it will also increase their access to continuing education and workplace opportunities.

Lead Organization(s): 
Award Number: 
1621151
Funding Period: 
Thu, 09/15/2016 to Sat, 08/31/2019
Full Description: 

Over one million paraeducators (teaching assistants and volunteers) currently assist in classrooms, and another 100,000 are likely to be added in the next ten years. Paraeducators (paras) are often required to teach content, such as mathematics, but there are few efforts to provide them with the knowledge or supervision they need to be effective when working with a range of students, including those with disabilities and for whom English is a second language. The project will focus on developing the confidence, mathematical knowledge, and teaching strategies of paras using classroom activities that they are expected to implement. The planned professional development will enable them to make a greater difference in the classroom, but it will also increase their access to continuing education and workplace opportunities. The work will be conducted in the Boston Public Schools (BPS) and will focus on grades K-3, where the largest numbers of paras are employed. Given the importance of early math learning in predicting mathematical achievement, supporting paras who work in the early grades is particularly important.

The project will design and pilot professional development that supports paraeducator knowledge development and addresses instructional challenges in teaching mathematics. The project will address the following goals: research the current roles of paras in mathematics instruction, the preparation of their collaborating teachers, and the opportunities for collaboration and planning between supervising teachers and paras in BPS; pilot, develop, implement, and research a model for professional development program for paras that targets specific activities they can implement that are key to student learning in number and operation in K-3; document how paras assume new roles that increase student engagement and empower them as mathematical learners; pilot, develop, implement, and research a supervisory component to help teachers set expectations, and structures for debriefing and reflecting along with their paras; and identify next steps for an early stage development study based on our findings. A needs assessment survey will investigate the context in which paras work. The iterative process of design-based research will develop, test, and implement the targeted professional development with paras, measuring how prepared they feel to implement new ideas and how they translate their learning into new pedagogical practices. Crosscase analyses, descriptive statistics, tallies and coded behaviors from observations, and themes from paras, and teacher and administrator interviews will be collected, coded, and analyzed. Furthermore, an efficacy survey will be administered periodically to document longitudinal changes in paras, which will be integrated in the crosscase analyses.

Developing Formative Assessment Tools and Routines for Additive Reasoning

This design and development project is an expansion of the Ongoing Assessment Project (OGAP), an established model for research-based formative assessment in grades 3-8, to the early elementary grades. The project will translate findings from research on student learning of early number, addition, and subtraction into tools and routines that teachers can use to formatively assess their students' understanding on a regular basis and develop targeted instructional responses.

Lead Organization(s): 
Award Number: 
1620888
Funding Period: 
Thu, 09/01/2016 to Thu, 02/28/2019
Full Description: 

This design and development project is an expansion of the Ongoing Assessment Project (OGAP), an established model for research-based formative assessment in grades 3-8, to the early elementary grades. OGAP brings together two powerful ideas in mathematics education - formative assessment and research based learning trajectories - to enhance teacher knowledge, instructional practices, and student learning. Building on a proven track record of success with this model, the current project will translate findings from research on student learning of early number, addition, and subtraction into tools and routines that teachers can use to formatively assess their students' understanding on a regular basis and develop targeted instructional responses. The project involves a development component focused on producing and field testing new resources (including frameworks, item banks, pre-assessments and professional development materials) and a research component designed to improve the implementation of these resources in school settings. The materials that are developed from this project will help teachers be able to more precisely assess student understanding in the major mathematical work of grades K-2 in order to better meet the needs of diverse learners. With the addition of these new early elementary materials, OGAP formative assessment resources will be available for use from kindergarten through grade 8.

Although much attention has been paid to the improvement of early literacy, building strong mathematical foundations and early computational fluency is equally critical for later success in school and preparation for STEM careers. This project will develop and field test tools, resources, and routines that teachers can employ to help young students develop deeper conceptual understandings and more powerful and efficient strategies in the early grades. The project emerged from the needs of school-based practitioners looking for instructional support in the primary grades and uses design-based research methodology. The new materials will be developed, tested, and revised through multiple iterations of implementation in schools. Research-based learning trajectories will be consolidated into simplified frameworks that illustrate the overall progression of major levels of student thinking in the domains of counting, addition, and subtraction. A bank of formative assessment items will be developed, field tested, and refined through a three-phase validation process. Professional development modules will be designed and field tested to support teacher knowledge and effective use of the formative assessment tools and routines. Data collected on key activities in the formative assessment process (including teacher selection of items, analysis of student work, instructional implications, and enacted instructional response) will be used to continually inform development as well as illuminate the conditions under which formative assessment leads to productive changes in instruction and student learning in the classroom. The project will yield a set of field tested tools and resources ready for both broader dissemination and further research on the promise of the intervention, as well as an understanding of how to support effective 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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