Algebra

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


CAREER: Designing Learning Environments to Foster Productive and Powerful Discussions Among Linguistically Diverse Students in Secondary Mathematics

Lead Organization(s): 
Award Number: 
1553708
Funding Period: 
Mon, 02/01/2016 to Sun, 01/31/2021
Full Description: 

The project will design and investigate learning environments in secondary mathematics classrooms focused on meeting the needs of English language learners. An ongoing challenge for mathematics teachers is promoting deep mathematics learning among linguistically diverse groups of students while taking into consideration how students' language background influences their classroom experiences and the mathematical understandings they develop. In response to this challenge, this project will design and develop specialized instructional materials and guidelines for teaching fundamental topics in secondary algebra in linguistically diverse classrooms. The materials will incorporate insights from current research on student learning in mathematics as well as insights from research on the role of language in students' mathematical thinking and learning. A significant contribution of the work will be connecting research on mathematics learning generally with research on the mathematics learning of English language learners. In addition to advancing theoretical understandings, the research will also contribute practical resources and guidance for mathematics teachers who teach English language learners. The Faculty Early Career Development (CAREER) program is a National Science Foundation (NSF)-wide activity that offers awards in support of junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education, and the integration of education and research within the context of the mission of their organizations.

The project is focused on the design of specialized hypothetical learning trajectories that incorporate considerations for linguistically diverse students. One goal for the specialized trajectories is to foster productive and powerful mathematics discussions about linear and exponential rates in linguistically diverse classrooms. The specialized learning trajectories will include both mathematical and language development learning goals. While this project focuses on concepts related to reasoning with linear and exponential functions, the resulting framework should inform the design of specialized hypothetical learning trajectories in other topic areas. Additionally, the project will add to the field's understanding of how linguistically diverse students develop mathematical understandings of a key conceptual domain. The project uses a design-based research framework gathering classroom-based data, assessment data, and interviews with teachers and students to design and refine the learning trajectories. Consistent with a design-based approach, the project results will include development of theory about linguistically diverse students' mathematics learning and development of guidance and resources for secondary mathematics teachers. This research involves sustained collaboration with secondary mathematics teachers and the impacts will include developing capacity of teachers locally, and propagating the results of this work in professional development activities.


Project Videos

2020 STEM for All Video Showcase

Title: Engaging Multilingual Secondary Math Learners in Discussions

Presenter(s): William Zahner, Ernesto Calleros, & Kevin Pelaez

2019 STEM for All Video Showcase

Title: Fostering Math Discussions among English Learners

Presenter(s): William Zahner


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.

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.

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

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: 
1503342
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.

Visual Access to Mathematics: Professional Development for Teachers of English Learners

This project addresses a critical need, developing professional development materials to address the teachers of ELLs. The project will create resources to help teachers build ELLs' mathematical proficiency through the design and development of professional development materials building on visual representations (VRs) for mathematical reasoning across a range of mathematical topics.

Award Number: 
1503057
Funding Period: 
Sat, 08/01/2015 to Fri, 07/31/2020
Full Description: 

The demands placed on mathematics teachers of all students have increased with the introduction of college and career readiness standards. At the same time, the mathematics achievement of English Language Learners (ELLs) lags behind that of their peers. This project addresses a critical need, developing professional development materials to address the teachers of ELLs. The project will create resources to help teachers build ELLs' mathematical proficiency through the design and development of professional development materials building on visual representations (VRs) for mathematical reasoning across a range of mathematical topics. The project will study how to enhance teachers' pedagogical content knowledge that is critical to fostering ELLs' mathematical problem solving and communication to help support fluency in using VRs among teachers and students. To broaden the participation of students who have traditionally not demonstrated high levels of achievement in mathematics, a critical underpinning to further success in the sciences and engineering, there will need to be greater support for teachers of these students using techniques that have been demonstrated to improve student learning. 

The project will use an iterative design and development process to develop a blended learning model of professional development on using VRs with a 30-hour face-to-face summer institute and sixteen 2-hour online learning sessions. Teachers and teacher-leaders will help support the development of the professional development materials. A cluster randomized control trial will study the piloting of the materials and their impact on teacher outcomes. Thirty middle schools from Massachusetts and Maine serving high numbers of ELLs, with approximately 120 teachers, will be randomly assigned to receive the treatment or control conditions. Using a two-level random intercepts hierarchical linear model, the study will explore the impact of participation in the professional development on teachers' mathematical knowledge for teaching and instructional practice. The pilot study will also explore the feasibility of delivering the professional development model more broadly. It builds on prior work that has shown efficacy in geometry, but expands the work beyond a single area in mathematics. At the same time, they will test the model for feasibility of broad implementation.


Project Videos

2019 STEM for All Video Showcase

Title: Designing PD for Math Educators of Students Who are ELs

Presenter(s): Peter Tierney-Fife, Pamela Buffington, Josephine Louie, Jill Neumayer Depiper, & Johannah Nikula

2016 STEM for All Video Showcase

Title: Visual Access to Mathematics: Supporting Teachers of ELs

Presenter(s): Johannah Nikula, Pam Buffington, Mark Driscoll & Peter Tierney-Fife


Teaching and Learning Algebraic Thinking Across the Middle Grades: A Research-based Approach Using PhET Interactive Simulations

This project addresses three central challenges: 1) the tendency for students to not engage in real mathematical thinking as they use technologies; 2) the tendency for teachers to not enact pedagogically-effective approaches; and 3) the lack of adoption of effective technologies by teachers due to a variety of barriers. This project will use rich, exploratory, interactive simulations and associated instructional materials as a pathway for making rapid progress and focusing on advancing algebraic thinking in Grades 6-9.

Lead Organization(s): 
Award Number: 
1503510
Funding Period: 
Tue, 09/01/2015 to Mon, 08/31/2020
Full Description: 

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. 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. 

Widespread, high-quality use of technology has great potential to transform today's mathematics classrooms and enable all students to develop a robust conceptual understanding of mathematics. Critical challenges are currently limiting the realization of this potential, and 69% of US Grade 8 students are scoring below proficient in national studies. In this 3-year Discovery Research K-12 Full Research and Development project, Teaching and Learning Algebraic Thinking Across the Middle Grades: A Research-based Approach Using PhET Interactive Simulations, the PhET Interactive Simulations group at the University of Colorado Boulder is partnering with mathematics education researchers at the University of South Florida St. Petersberg and Florida State University to address three central challenges, as follows: 1) the tendency for students to not engage in real mathematical thinking as they use technologies; 2) the tendency for teachers to not enact pedagogically-effective approaches; and 3) the lack of adoption of effective technologies by teachers due to a variety of barriers. This collaborative effort uses rich, exploratory, interactive simulations and associated instructional materials as a pathway for making rapid progress and focuses on advancing algebraic thinking in Grades 6-9.

This project seeks to enable teachers to fully-leverage the benefits of interactive simulations to advance student engagement and learning of mathematics, moving technology from the margins to a core part of instruction. The project will answer critical research questions, such as: how the design of an interactive simulation can generate pedagogically-productive use; how instruction with simulations can be best structured to support learning of mathematical concepts and engagement in mathematical practices; how sim-based instruction can be made attractive, feasible and effective for teachers; and finally, how student learning is impacted by sim-based instruction. At the same time, this project will produce a collection of open educational resources for teachers and students. These resources will include 15 research-based, student-tested simulations for teaching and learning of algebraic thinking, associated instructional support materials, and teacher professional development resources for effective implementation. Based on the 75 million uses per year of PhET?s science simulations, we expect these resources to transform mathematics instruction for millions of students and thousands of teachers.

This project will employ a variety of research methods to approach these questions. Researchers will use individual interviews from a diverse group of Grades 6-9 students as they use the 15 new simulations to examine usability, engagement, and achievement and to identify design approaches that stimulate productive use. In parallel, classroom-based studies in Colorado and Florida will investigate ways in which simulations can be combined with instructional materials and teacher facilitation to engage groups of students in inquiry, promote rich discussions of important mathematical ideas, and advance achievement in the Common Core State Standards for Mathematics. The project will employ an iterative design and development process involving qualitative and quantitative analysis of diverse measures including the quality of mathematical instruction. Finally, a pilot study and an evaluation of teacher PD supports will examine the feasibility and fidelity with which teachers implement the innovation, and the impact on student learning.

Sample Publications

Hensberry, KKR, Whitacre, I., Findley, K., Schellinger, J., & Burr, M. (2018). Engaging students with mathematics through play. Mathematics Teaching in the Middle School, 24(3), 197-183. (https://www.jstor.org/stable/10.5951/mathteacmiddscho.24.3.0179)

Ian Whitacre, Karina Hensberry, Jennifer Schellinger & Kelly Findley (2019) Variations on play with interactive computer simulations: balancing competing priorities, International Journal of Mathematical Education in Science and Technology, 50:5 , 665-681. (https://www.tandfonline.com/doi/full/10.1080/0020739X.2018.1532536

Findley, K., Whitacre, I., Schellinger, J. & Hensberry, K. (2019). Orchestrating Mathematics Lessons with Interactive Simulations: Exploring Roles in the Classroom. Journal of Technology and Teacher Education, 27(1), 37-62. (https://www.learntechlib.org/noaccess/184666/)

Jeffrey B. Bush, David C. Webb, Nancy Emerson Kress, Wanqiu Yang and Katherine K. Perkins, Classroom Activities for Digital Interactive Simulations to Support Realistic Mathematics Education, Paper presented at the 6th International Realistic Mathematics Education Conference Georgetown, Cayman Islands, September 20, 2018. (https://www.icrme.net/uploads/1/0/9/8/109819470/bush_etal_phet_rme6paper_final.pdf)


Project Videos

2019 STEM for All Video Showcase

Title: Transforming Math Classrooms with PhET Simulations

Presenter(s): Kathy Perkins, Sebnem Atabas, Jeff Bush, Karina Hensberry, Amanda McGarry, Corinne Singleton, David Webb, & Ian Whitacre

2017 STEM for All Video Showcase

Title: Teaching and Learning Math with PhET Simulations

Presenter(s): Kathy Perkins, Karina Hensberry, Amanda McGarry, David Webb, & Ian Whitacre


Math Snacks Early Algebra Using Games and Inquiry to Help Students Transition from Number to Variable

This project will develop games to build conceptual understanding of key early algebra topics. The materials will be freely accessible on the web in both English and Spanish. The project will develop 4-5 games. Each game will include supporting materials for use by students in inquiry-based classroom lessons, and web-based professional development tools for teachers.

Lead Organization(s): 
Award Number: 
1503507
Funding Period: 
Tue, 09/01/2015 to Sat, 08/31/2019
Full Description: 

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. 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.

Many U.S. students enter college without the necessary background in algebra to be successful in advanced mathematics and science courses, and are thereby blocked from many rewarding careers. Oftentimes, the problem goes back to early algebra in grades 4-6, where students are introduced to abstract formulations before they understand the underlying ideas and the reasons for the questions being asked. As a result of inadequate preparation many students turn away from mathematics when faced with abstract algebra. Without mathematics, students are not able to enter the STEM field which results in a weakened workforce in these fields in the United States. In this 4-year Full Research and Development project, Math Snacks Early Algebra: Using Games and Inquiry to Help Students Transition from Number to Variable, the interdisciplinary research group from New Mexico State University will build on their success in using games to increase students' understanding of proportional reasoning and fractions. They will develop games to build conceptual understanding of key early algebra topics. The materials will be freely accessible on the web in both English and Spanish. The project will develop 4-5 games. Each game will include supporting materials for use by students in inquiry-based classroom lessons, and web-based professional development tools for teachers.

Most students do not understand the variety of distinct ways that variables are used in mathematics: unknowns to be solved for, related quantities, general properties of numbers, and other uses. Algebra courses often emphasize the rules of manipulation, with less time spent on the underlying ideas. Students see variables as confusing new material, rather than as shortcuts for making sense of numbers, or as powerful tools for analyzing interesting problems. This hinders students' later interest and progress in STEM courses and careers.The intellectual merit for this R & D project includes the development of a new way to learn key underlying concepts in algebra, further investigation of the affordances of games and technology in learning abstract mathematical concepts, and a better understanding of learning assessment in early algebra. The broader impact for this R & D project includes making these tools widely available to students, and the potential shift of teachers towards effective mathematical pedagogy that is engaging and inquiry-based. Development will begin with existing research on early algebraic thinking and learning, and proceed through an iterative process involving design, testing in the NMSU Learning Games Lab, testing in classrooms, and back to design. The project will then study the effect of the developed materials on student understanding and on classroom learning environments. Qualitative and quantitative measures will be used. Researchers will use a custom measure aligned with NAEP (National Assessment of Educational Progress) and other standard tests, interviews and observations with teachers and students, and embedded data collection and self-reports on frequency and extent of game usage. After two earlier pilot studies, in the final year a delayed intervention study will be conducted with 50 teachers and their students. The Math Snacks team has existing partnerships for distribution of games and materials with PBS, GlassLabs, BrainPOP, and others. Academic findings of the project will be shared through conferences and research publications.

CAREER: Advancing Secondary Mathematics Teachers' Quantitative Reasoning

Advancing Reasoning addresses the lack of materials for teacher education by investigating pre-service secondary mathematics teachers' quantitative reasoning in the context of secondary mathematics concepts including function and algebra. The project extends prior research in quantitative reasoning to develop differentiated instructional experiences and curriculum that support prospective teachers' quantitative reasoning and produce shifts in their knowledge.

Award Number: 
1350342
Funding Period: 
Tue, 07/15/2014 to Tue, 06/30/2020
Full Description: 

Science, Technology, Engineering and Mathematics [STEM] and STEM education researchers and policy documents have directed mathematics educators at all levels to increase emphasis on quantitative reasoning so that students are prepared for continued studies in mathematics and other STEM fields. Often, teachers are not sufficiently prepared to support their students' quantitative reasoning. The products generated by this project fill a need for concrete materials at the pre-service level that embody research-based knowledge in the area of quantitative reasoning. The accessible collection of research and educational products provides a model program for changing prospective mathematics teachers' quantitative reasoning that is adoptable at other institutions across the nation. Additionally, the support of early CAREER scholars in mathematics education will add to the capacity of the country to address issues in mathematics education in the future.

Advancing Reasoning addresses the lack of materials for teacher education by investigating pre-service secondary mathematics teachers' quantitative reasoning in the context of secondary mathematics concepts including function and algebra. The project extends prior research in quantitative reasoning to develop differentiated instructional experiences and curriculum that support prospective teachers' quantitative reasoning and produce shifts in their knowledge. Three interrelated research questions guide the project: (i) What aspects of quantitative reasoning provide support for prospective teachers' understanding of major secondary mathematics concepts such as function and algebra? (ii) How can instruction support prospective teachers' quantitative reasoning in the context of the teaching and learning of major secondary mathematics concepts such as function and algebra? (iii) How do the understandings prospective teachers hold upon entering a pre-service program support or inhibit their quantitative reasoning? Advancing Reasoning addresses these questions by enacting an iterative, multi-phase study with 200 prospective teachers enrolled in a secondary mathematics education content course over 5 years. The main phase of the study implements a series of classroom design experiments to produce knowledge on central aspects of prospective teachers' quantitative reasoning and the instructional experiences that support such reasoning. By drawing this knowledge from a classroom setting, Advancing Reasoning contributes research-based and practice-driven deliverables that improve the teaching and learning of mathematics.

CAREER: Algebraic Knowledge for Teaching: A Cross-Cultural Perspective

The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. This study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance.

Lead Organization(s): 
Award Number: 
1350068
Funding Period: 
Fri, 08/15/2014 to Fri, 07/31/2020
Full Description: 

What content knowledge is needed for the teaching of mathematics? What practices are more effective for realizing student success? These questions have received considerable attention in the mathematics education community. The goal of this CAREER program of research is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students' algebraic thinking. Focusing on two fundamental mathematical ideas recently emphasized by the Common Core State Standards - inverse relations and properties of operations - this study explores AKT based on integrated insights of the U.S. and Chinese expert teachers' classroom performance. It will be focused on three objectives: (1) identify AKT that facilitates algebraic thinking and develop preliminary findings into teaching materials; (2) refine research-based teaching materials based on the evaluative data; and (3) integrate research with education through course development at Temple University and teacher outreach in Philadelphia.

The model underlying this research program is that improved pedagogy will improve student learning, both directly and indirectly. A design-based research method will be used to accomplish objectives #1 and #2. Cross-cultural videotaped lessons will be first analyzed to identify AKT, focusing on teachers' use of worked examples, representations, and deep questions. This initial set of findings will then be developed into teaching materials. The U.S. and Chinese expert teachers will re-teach the lessons as part of the refinement process. Data sources will include: baseline and updated survey data (control, context, and process variables), observation, documents, videos, and interviews. The statistical techniques will include descriptive and inferential statistics and HLM will to address the hierarchical nature of the data.

This project involves students and teachers at various levels (elementary, undergraduate, and graduate) at Temple University and the School District of Philadelphia (SDP) in the U.S. and Nanjing Normal University and Nantong School District in China. A total of 600 current and future elementary teachers and many of their students will benefit directly or indirectly from this project. Project findings will be disseminated through various venues. Activities of the project will promote school district-university collaboration, a novice-expert teacher network, and cross-disciplinary and international collaboration. It is anticipated that the videos of expert teaching will also be useful future research by cognitive researchers studying ways to improve mathematics learning.

Publications
G indicates graduate student author; U indicates undergraduate student author

Journal Articles in English

  1. Ding, M., G Chen, W., & G Hassler, R. (2019). Linear quantity models in the US and Chinese elementary mathematics classrooms. Mathematical Thinking and Learning, 21, 105-130 doi: 10.1080/10986065.2019.1570834 . PDF
  2. Barnett, E., & Ding, M. (2019). Teaching of the associative property: A natural classroom investigation. Investigations of Mathematics Learning, 11, 148-166. doi: 10.1080/19477503.2018.1425592  PDF
  3. Ding, M., & G Heffernan, K. (2018). Transferring specialized content knowledge to elementary classrooms: Preservice teachers’ learning to teach the associative property. International Journal of Mathematics Educational in Science and Technology, 49, 899-921.doi: 10.1080/0020739X.2018.1426793 PDF
  4. Ding, M. (2018). Modeling with tape diagrams. Teaching Children Mathematics25, 158-165. doi: 10.5951/teacchilmath.25.3.0158  PDF
  5. G Chen, W., & Ding, M.* (2018). Transitioning from mathematics textbook to classroom instruction: The case of a Chinese expert teacher. Frontiers of Education in China, 13, 601-632. doi: 10.1007/s11516-018-0031-z (*Both authors contributed equally). PDF
  6. Ding, M., & G Auxter, A. (2017). Children’s strategies to solving additive inverse problems: A preliminary analysis. Mathematics Education Research Journal, 29, 73-92. doi:10.1007/s13394-017-0188-4  PDF
  7. Ding, M. (2016).  Developing preservice elementary teachers’ specialized content knowledge for teaching fundamental mathematical ideas: The case of associative property. International Journal of STEM Education, 3(9), 1-19doi: 10.1186/s40594-016-0041-4  PDF
  8. Ding, M. (2016). Opportunities to learn: Inverse operations in U.S. and Chinese elementary mathematics textbooks. Mathematical Thinking and Learning, 18, 45-68. doi: 10.1080/10986065.2016.1107819  PDF

Journal Articles in Chinese
Note: The Chinese journals Educational Research and Evaluation (Elementary Education and Instruction教育研究与评论 (小学教育教学) and Curriculum and Instructional Methods (课程教材教法) are both official, core journals in mathematics education field in China.

  1. Chen, W. (2018). Strategies to deal with mathematical representations – an analysis of expert’s classroom instruction. Curriculum and Instructional Methods. 数学教学的表征处理策略——基于专家教师的课堂教学分析. 课程教材教法. PDF
  2. Ma, F. ( 2018) – Necessary algebraic knowledge for elementary teachers- an ongoing cross-cultural study. Educational Research and Evaluation (Elementary Education and Instruction), 2, 5-7.  小学教师必备的代数学科知识-跨文化研究进行时。教育研究与评论 (小学教育教学), 2, 5-7. PDF
  3. Chen, J. (2018) Infusion and development of children’s early algebraic thinking – a comparative study of the US and Chinese elementary mathematics teaching. Educational Research and Evaluation (Elementary Education and Instruction), 2, 8-13.  儿童早期代数思维的渗透与培养-中美小学数学教学比较研究。教育研究与评论(小学教育教学),28-13.  PDF
  4. Zong, L. (2018). A comparative study on the infusion of inverse relations in the US and Chinese classroom teaching. Educational Research and Evaluation (Elementary Education and Instruction), 2, 14-19.  中美逆运算渗透教学对比研究。教育研究与评论(小学教育教学,2,14-19.  PDF
  5. Wu, X. (2018). Mathematical representations and development of children’s mathematical thinking: A perspective of US-Chinese comparison. Educational Research and Evaluation (Elementary Education and Instruction), 2, 20-24.  数学表征与儿童数学思维发展-基于中美比较视角。教育研究与评论(小学教育教学,2, 20-24.  PDF

Dissertations

  1. Hassler, R. (2016). Mathematical comprehension facilitated by situation models: Learning opportunities for inverse relations in elementary school.Published dissertation, Temple University, Philadelphia, PA. (Chair: Dr. Meixia Ding)  PDF
  2. Chen, W. (2018). Elementary mathematics teachers’ professional growth: A perspectives of TPACK (TPACK 视角下小学数学教师专业发展的研究). Dissertation, Nanjing Normal University. Nanjing, China. PDF

National Presentations
G indicates graduate student author; U indicates undergraduate student author

  • Ding, M (symposium organizer, 2019, April). Enhancing elementary mathematics instruction: A U.S.-China collaboration. Papers presented at NCTM research conference (Discussant: Jinfa Cai). (The following three action research papers were written by my NSF project teachers under my guidance).
      • Milewski Moskal, M., & Varano, A. (2019). The teaching of worked examples: Chinese approaches in U.S. classrooms. Paper 
      • Larese, T., Milewski Moskal, M., Ottinger, M., & Varano, A., (2019). Introducing Investigations math games in China: Successes and surprises. Paper
      • Murray, D., Seidman, J., Blackmon, E., Maimon, G., & Domsky, A. (2019). Mathematic instruction across two cultures: A teacher perspective. Paper
    • Ding, M., & Ying Y. (2018, June). CAREER: Algebraic knowledge for teaching: A cross-cultural perspective. Poster presentation at the National Science Foundation (NSF) PI meeting, Washington, DC.  Poster
    • Ding, M., Brynes, J., G Barnett, E., & Hassler, R. (2018, April). When classroom instruction predicts students’ learning of early algebra: A cross-cultural opportunity-propensity analysis. Paper presented at 2018 AERA conference. New York, NY.  Paper
    • Ding, M., Li, X., Manfredonia, M., & Luo, W. (2018, April). Video as a tool to support teacher learning: A Cross-cultural analysis. Paper presented at 2018 NCTM conference. Washington, DC.  PPT
    • GBarnett, E., & Ding, M. (2018, April). Teaching the basic properties of arithmetic: A natural classroom investigation of associativity. Poster presentation at 2018AERA conference, New York, NY.  Poster
    • Hassler, R., & Ding, M. (2018, April). The role of deep questions in promoting elementary students’ mathematical comprehension. Poster presentation at 2018AERA conference, New York, NY.
    • Ding, M., G Chen, W., G Hassler, R., Li, X., & G Barnett, E. (April, 2017). Comparisons in the US and Chinese elementary mathematics classrooms. Poster presentation at AERA 2017 conference (In the session of “Advancing Mathematics Education Through NSF’s DRK-12 Program”). San Antonio, TX. Poster
    • Ding, M., Li, X., G Hassler, R., & G Barnett, E. (April, 2017). Understanding the basic properties of operations in US and Chinese elementary School. Paper presented at AERA 2017 conference. San Antonio, TX.  Paper
    • Ding, M., G Chen, W., & G Hassler, R. (April, 2017). Tape diagrams in the US and Chinese elementary mathematics classrooms. Paper presented at NCTM 2017 conference. San Antonio, TX.  Paper
    • Ding, M., & G Hassler, R. (2016, June). CAREER: Algebraic knowledge for teaching in elementary school: A cross-cultural perspective. Poster presentation at the NSF PI meeting, Washington, DC. Poster
    • Ding, M. (symposium organizer, 2016, April). Early algebraic in elementary school: A cross-cultural perspective. Proposals presented at 2016 AERA conference, Washington, DC.
        • Ding, M. (2016, April). A comparative analysis of inverse operations in U.S. and Chinese elementary mathematics textbooks. Paper 
        • G Hassler, R. (2016, April). Elementary Textbooks to Classroom Teaching: A Situation Model Perspective. Paper
        • G Chen, W., & Ding, M. (2016, April). Transitioning textbooks into classroom teaching: An action research on Chinese elementary mathematics lessons. Paper
        • Li, X., G Hassler, R., & Ding, M. (2016, April). Elementary students’ understanding of inverse relations in the U.S. and China.  Paper
        • Stull, J., Ding, M., G Hassler, R., Li, X., & U George, C. (2016, April). The impact of algebraic knowledge for teaching on student learning: A Preliminary analysis. Paper
      • Ding, M., G Hassler, R., Li., X., & G Chen, W. (2016, April). Algebraic knowledge for teaching: An analysis of US experts' lessons on inverse relations. Paper presented at 2016 NCTM conference, San Francisco, CA. Paper
      • G Hassler. R., & Ding, M. (2016, April). Situation model perspective on mathematics classroom teaching: A case study on inverse relations. Paper presented at 2016 NCTM conference, San Francisco, CA.  Paper
      • Ding, M., & G Copeland, K. (2015, April). Transforming specialized content knowledge: Preservice elementary teachers’ learning to teach the associative property of multiplication. Paper presented at AERA 2015 conference, Chicago, IL. Paper PPT
      • Ding, M., & G Auxter, A. (2015, April). Children’s strategies to solving additive inverse problems: A preliminary analysis. Paper presented at AERA 2015 conference, Chicago, IL.  Paper

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